FEMA P-751 - 2009 NEHRP Recommended Seismic Provisions Design Examples.pdf

2009 NEHRP Recommended Seismic Provisions: Design Examples FEMA P-751 / September 2012

FEMA

2009 NEHRP Recommended Seismic Provisions: Design Examples FEMA P-751 - September 2012

Prepared by the National Institute of Building Sciences Building Seismic Safety Council For the Federal Emergency Management Agency of the Department of Homeland Security

NOTICE: Any opinions, findings, conclusions, or recommendations expressed in this publication do not necessarily reflect the views of the Federal Emergency Management Agency. Additionally, neither FEMA nor any of its employees make any warranty, expressed or implied, nor assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, product or process included in this publication. The opinions expressed herein regarding the requirements of the International Residential Code do not necessarily reflect the official opinion of the International Code Council. The building official in a jurisdiction has the authority to render interpretation of the code. This report was prepared under Contract HSFEHQ-09-R-0147 between the Federal Emergency Management Agency and the National Institute of Building Sciences. For further information on the Building Seismic Safety Council, see the Council’s website — www.bssconline.org — or contact the Building Seismic Safety Council, 1090 Vermont, Avenue, N.W., Suite 700, Washington, D.C. 20005; phone 202-289-7800; fax 202-289-1092; e-mail [emailprotected].

FOREWORD One of the goals of the Department of Homeland Security’s Federal Emergency Management Agency (FEMA) and the National Earthquake Hazards Reduction Program (NEHRP) is to encourage design and building practices that address the earthquake hazard and minimize the resulting risk of damage and injury. The 2009 edition of the NEHRP Recommended Seismic Provisions for New Buildings and Other Structures (FEMA P-750) affirmed FEMA’s ongoing support to improve the seismic safety of construction in this country. The NEHRP Provisions serves as a key resource for the seismic requirements in the ASCE/SEI 7 Standard Minimum Design Loads for Buildings and Other Structures as well as the national model building codes, the International Building Code (IBC), International Residential Code (IRC) and NFPA 5000 Building Construction Safety Code. FEMA welcomes the opportunity to provide this material and to work with these codes and standards organizations. This product provides a series of design examples that will assist the users of the 2009 NEHRP Provisions and the ASCE/SEI 7 standard the Provisions adopted by reference. FEMA wishes to express its gratitude to the authors listed elsewhere for their significant efforts in preparing this material and to the BSSC Board of Direction and staff who made this possible. Their hard work has resulted in a guidance product that will provide important assistance to a significant number of users of the nation’s seismic building codes and their reference documents. Department of Homeland Security/ Federal Emergency Management Agency

iii

PREFACE This volume of design examples is intended for those experienced structural designers who are relatively new to the field of earthquake-resistant design and to the 2009 NEHRP (National Earthquake Hazards Reduction Program) Recommended Seismic Provisions for New Buildings and Other Structures. By extension, it also applies to use of the current model codes and standards because the Provisions is the key resource for updating seismic design requirements in most of those documents including ASCE 7 Standard, Minimum Design Loads for Buildings and Other Structures; and the International Building Code (IBC). Furthermore, the 2009 NEHRP Provisions (FEMA P-750) adopted ASCE7-05 by reference and the 2012 International Building Code adopted ASCE7-10 by reference; therefore, seismic design requirements are essentially equivalent across the Provisions, ASCE7 and the national model code. The design examples, updated in this edition, reflect the technical changes in the 2009 NEHRP Recommended Provisions. The original design examples were developed from an expanded version of an earlier document (entitled Guide to Application of the NEHRP Recommended Provisions, FEMA 140) which reflected the expansion in coverage of the Provisions and the expanding application of the Provisions concepts in codes and standards. The widespread use of the NEHRP Recommended Provisions in the past and the essential equivalency of ASCE7, the Provisions and the national model codes at present attested to the success of the NEHRP at the Federal Emergency Management Agency and the efforts of the Building Seismic Safety Council to ensure that the nation’s building codes and standards reflect the state of the art of earthquake-resistant design. In developing this set of design examples, the BSSC initially decided on the types of structures; types of construction and materials; and specific structural elements that needed to be included to provide the reader with at least a beginning grasp of the new requirements and critical issues frequently encountered when addressing seismic design problems. Many of the examples are from the previous edition of the design examples but updated by the authors to illustrate issues or design requirements not covered or that have changed from the past edition. Because it obviously is not possible to present, in a volume of this type, complete building designs for all the situations and features that were selected, only portions of designs have been used. All users of the Design Examples are recommended to obtain and familiarize themselves with the 2003 and 2009 NEHRP Recommended Provisions (FEMA 450 and FEMA P-750) or ASCE7, Copies of the Provisions are available free of charge from FEMA by calling 1-800-480-2520 (order by FEMA Publication Number). Currently available are the 2003 and 2009 editions as follows: NEHRP (National Earthquake Hazards Reduction Program) Recommended Seismic Provisions for New Buildings and Other Structures, 2009 Edition, FEMA P-750, 1 volume with maps (issued as paper document with a CD attached) NEHRP (National Earthquake Hazards Reduction Program) Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, 2003 Edition, 2 volumes and maps, FEMA 450 (issued as a paper document).The 2003 and 2009 edition of the Provisions can also be downloaded from the BSSC website at www.nibs.org/bssc. Also see the website for information regarding BSSC projects iv

and publications or write to the BSSC at [emailprotected] or at the National Institute of Building Sciences, 1090 Vermont Avenue, NW, Suite 700, Washington, DC 20005 (telephone 202-289-7800). Updated education/training materials to supplement this set of design examples will be published as a separate FEMA product, 2009 NEHRP Recommended Seismic Provisions: Training Material, FEMA P752. The BSSC is grateful to all those individuals and organizations whose assistance made the 2012 edition of the design examples a reality: I Michael T. Valley, Magnusson Klemencic Associates, Seattle, Washington, who served as project manager and managing technical editor for the update. I Ozgur Atlayan, Robert Bachman, Finley A. Charney, Brian Dean, Susan Dowty, John Gillengerten, James Robert Harris, Charles A. Kircher, Suzanne Dow Nakaki, Clinton O. Rex, Frederic R. Rutz, Rafael A. Sabelli, Peter W. Somers, Greg Soules, Adrian Tola Tola and Michael T. Valley for editing the original chapters to prepare this update of the 2006 Edition. I Robert Pekelnicky for preparing a new Introduction; and Nicolas Luco, Michael Valley and C.B. Crouse for preparing a new chapter on Earthquake Ground Motions for this edition. I Lawrence A. Burkett, Kelly Cobeen, Finley Charney, Ned Cleland, Dan Dolan, Jeffrey J. Dragovich, Jay Harris, Robert D. Hanson, Neil Hawkins, Joe Maffei, Greg Soules, and Mai Tong for their reviews of the edited, updated and expanded material. And finally, the BSSC Board is grateful to FEMA Project Officer Mai Tong for his support and guidance and to Deke Smith, Roger Grant and Pamela Towns of the NIBS staff for their efforts preparing the 2012 volume for publication and issuance as an e-document available for download and on CD-ROM. Jim. W. Sealy, Chairman BSSC Board of Direction

Table of Contents

FOREWORD................................................................................................................................................ iii PREFACE......................................................................................................................................................iv

1 INTRODUCTION 1.1

EVOLUTION OF EARTHQUAKE ENGINEERING............................................................... 1-3

1.2

HISTORY AND ROLE OF THE NEHRP PROVISIONS......................................................... 1-6

1.3

THE NEHRP DESIGN EXAMPLES......................................................................................... 1-8

1.4

GUIDE TO USE OF THE PROVISIONS.................................................................................11-1

1.5 REFERENCES......................................................................................................................... 1-38

2 FUNDAMENTALS 2.1

EARTHQUAKE PHENOMENA............................................................................................... 2-3

2.2

STRUCTURAL RESPONSE TO GROUND SHAKING.......................................................... 2-5 2.2.1

Response Spectra....................................................................................................... 2-5

2.2.2

Inelastic Response.....................................................................................................2-11

2.2.3

Building Materials.................................................................................................... 2-14

2.2.4

Building Systems..................................................................................................... 2-16

2.2.5

Supplementary Elements Added to Improve Structural Performance..................... 2-17

2.3

ENGINEERING PHILOSOPHY............................................................................................. 2-18

2.4

STRUCTURAL ANALYSIS.................................................................................................... 2-19

2.5

NONSTRUCTURAL ELEMENTS OF BUILDINGS............................................................. 2-22

2.6

QUALITY ASSURANCE........................................................................................................ 2-23

EARTHQUAKE GROUND MOTION

3.1

BASIS OF EARTHQUAKE GROUND MOTION MAPS........................................................ 3-2 3.1.1

ASCE 7-05 Seismic Maps.......................................................................................... 3-2

vii

FEMA P-751, NEHRP Recommended Provisions: Design Examples 3.1.2 MCER Ground Motions in the Provisions and in ASCE 7-10.................................... 3-3 3.1.3

PGA Maps in the Provisions and in ASCE 7-10........................................................ 3-7

3.1.4

Basis of Vertical Ground Motions in the Provisions and in ASCE 7-10.................... 3-7

3.1.5 Summary.................................................................................................................... 3-7 3.1.6 References.................................................................................................................. 3-8 3.2

3.3

DETERMINATION OF GROUND MOTION VALUES AND SPECTRA.............................. 3-9 3.2.1

ASCE 7-05 Ground Motion Values............................................................................ 3-9

3.2.2

2009 Provisions Ground Motion Values.................................................................. 3-10

3.2.3

ASCE 7-10 Ground Motion Values...........................................................................3-11

3.2.4

Horizontal Response Spectra................................................................................... 3-12

3.2.5

Vertical Response Spectra........................................................................................ 3-13

3.2.6

Peak Ground Accelerations...................................................................................... 3-14

SELECTION AND SCALING OF GROUND MOTION RECORDS.................................... 3-14 3.3.1

Approach to Ground Motion Selection and Scaling................................................ 3-15

3.3.2

Two-Component Records for Three Dimensional Analysis.................................... 3-24

3.3.3

One-Component Records for Two-Dimensional Analysis....................................... 3-27

3.3.4 References................................................................................................................ 3-28

STRUCTURAL ANALYSIS

4.1

IRREGULAR 12-STORY STEEL FRAME BUILDING, STOCKTON, CALIFORNIA......... 4-3 4.1.1 Introduction................................................................................................................ 4-3

4.2

4.1.2

Description of Building and Structure....................................................................... 4-3

4.1.3

Seismic Ground Motion Parameters.......................................................................... 4-4

4.1.4

Dynamic Properties.................................................................................................... 4-8

4.1.5

Equivalent Lateral Force Analysis............................................................................4-11

4.1.6

Modal Response Spectrum Analysis........................................................................ 4-29

4.1.7

Modal Response History Analysis........................................................................... 4-39

4.1.8

Comparison of Results from Various Methods of Analysis..................................... 4-50

4.1.9

Consideration of Higher Modes in Analysis............................................................ 4-53

4.1.10

Commentary on the ASCE 7 Requirements for Analysis........................................ 4-56

SIX-STORY STEEL FRAME BUILDING, SEATTLE, WASHINGTON.............................. 4-57 4.2.1

Description of Structure........................................................................................... 4-57

4.2.2 Loads........................................................................................................................ 4-60

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4.2.3

Preliminaries to Main Structural Analysis............................................................... 4-64

4.2.4

Description of Model Used for Detailed Structural Analysis.................................. 4-72

4.2.5

Nonlinear Static Analysis......................................................................................... 4-94

Table of Contents 4.2.6

Response History Analysis..................................................................................... 4-109

4.2.7

Summary and Conclusions..................................................................................... 4-134

FOUNDATION ANALYSIS AND DESIGN

5.1

SHALLOW FOUNDATIONS FOR A SEVEN-STORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA................................................................................................ 5-3

5.2

5.1.1

Basic Information....................................................................................................... 5-3

5.1.2

Design for Gravity Loads........................................................................................... 5-8

5.1.3

Design for Moment-Resisting Frame System...........................................................5-11

5.1.4

Design for Concentrically Braced Frame System.................................................... 5-16

5.1.5

Cost Comparison...................................................................................................... 5-24

DEEP FOUNDATIONS FOR A 12-STORY BUILDING, SEISMIC DESIGN CATEGORY D......................................................................................................................... 5-25 5.2.1

Basic Information..................................................................................................... 5-25

5.2.2

Pile Analysis, Design and Detailing......................................................................... 5-33

5.2.3

Other Considerations................................................................................................ 5-47

STRUCTURAL STEEL DESIGN

6.1

INDUSTRIAL HIGH-CLEARANCE BUILDING, ASTORIA, OREGON.............................. 6-3 6.1.1

Building Description.................................................................................................. 6-3

6.1.2

Design Parameters...................................................................................................... 6-6

6.1.3

Structural Design Criteria.......................................................................................... 6-7

6.1.4 Analysis.................................................................................................................... 6-10 6.1.5 6.2

6.3

Proportioning and Details........................................................................................ 6-16

SEVEN-STORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA.............................. 6-40 6.2.1

Building Description................................................................................................ 6-40

6.2.2

Basic Requirements.................................................................................................. 6-42

6.2.3

Structural Design Criteria........................................................................................ 6-44

6.2.4

Analysis and Design of Alternative A: SMF............................................................ 6-46

6.2.5

Analysis and Design of Alternative B: SCBF.......................................................... 6-60

6.2.6

Cost Comparison...................................................................................................... 6-72

TEN-STORY HOSPITAL, SEATTLE, WASHINGTON......................................................... 6-72 6.3.1

Building Description................................................................................................ 6-72

6.3.2

Basic Requirements.................................................................................................. 6-76

6.3.3

Structural Design Criteria........................................................................................ 6-78

6.3.4

Elastic Analysis........................................................................................................ 6-80

FEMA P-751, NEHRP Recommended Provisions: Design Examples 6.3.5

Initial Proportioning and Details.............................................................................. 6-86

6.3.6

Nonlinear Response History Analysis...................................................................... 6-93

REINFORCED CONCRETE

7.1

SEISMIC DESIGN REQUIREMENTS..................................................................................... 7-7

7.2

7.3

7.4

7.5

7.1.1

Seismic Response Parameters.................................................................................... 7-7

7.1.2

Seismic Design Category........................................................................................... 7-8

7.1.3

Structural Systems...................................................................................................... 7-8

7.1.4

Structural Configuration............................................................................................. 7-9

7.1.5

Load Combinations.................................................................................................... 7-9

7.1.6

Material Properties................................................................................................... 7-10

DETERMINATION OF SEISMIC FORCES...........................................................................7-11 7.2.1

Modeling Criteria......................................................................................................7-11

7.2.2

Building Mass.......................................................................................................... 7-12

7.2.3

Analysis Procedures................................................................................................. 7-13

7.2.4

Development of Equivalent Lateral Forces.............................................................. 7-13

7.2.5

Direction of Loading................................................................................................ 7-19

7.2.6

Modal Analysis Procedure....................................................................................... 7-20

DRIFT AND P-DELTA EFFECTS........................................................................................... 7-21 7.3.1

Torsion Irregularity Check for the Berkeley Building............................................. 7-21

7.3.2

Drift Check for the Berkeley Building..................................................................... 7-23

7.3.3

P-delta Check for the Berkeley Building................................................................. 7-27

7.3.4

Torsion Irregularity Check for the Honolulu Building............................................. 7-29

7.3.5

Drift Check for the Honolulu Building.................................................................... 7-29

7.3.6

P-Delta Check for the Honolulu Building................................................................ 7-31

STRUCTURAL DESIGN OF THE BERKELEY BUILDING................................................ 7-32 7.4.1

Analysis of Frame-Only Structure for 25 Percent of Lateral Load.......................... 7-33

7.4.2

Design of Moment Frame Members for the Berkeley Building.............................. 7-37

7.4.3

Design of Frame 3 Shear Wall................................................................................. 7-60

STRUCTURAL DESIGN OF THE HONOLULU BUILDING.............................................. 7-66 7.5.1

Compare Seismic Versus Wind Loading.................................................................. 7-66

7.5.2

Design and Detailing of Members of Frame 1......................................................... 7-69

PRECAST CONCRETE DESIGN

8.1

HORIZONTAL DIAPHRAGMS............................................................................................... 8-4 8.1.1

Untopped Precast Concrete Units for Five-Story Masonry Buildings Located in Birmingham, Alabama and New York, New York..................................................... 8-4

Table of Contents 8.1.2 8.2

8.3

8.4

Topped Precast Concrete Units for Five-Story Masonry Building Located in Los Angeles, California (see Sec. 10.2)................................................................... 8-18

THREE-STORY OFFICE BUILDING WITH INTERMEDIATE PRECAST CONCRETE SHEAR WALLS................................................................................................. 8-26 8.2.1

Building Description................................................................................................ 8-27

8.2.2

Design Requirements............................................................................................... 8-28

8.2.3

Load Combinations.................................................................................................. 8-29

8.2.4

Seismic Force Analysis............................................................................................ 8-30

8.2.5

Proportioning and Detailing..................................................................................... 8-33

ONE-STORY PRECAST SHEAR WALL BUILDING........................................................... 8-45 8.3.1

Building Description................................................................................................ 8-45

8.3.2

Design Requirements............................................................................................... 8-48

8.3.3

Load Combinations.................................................................................................. 8-49

8.3.4

Seismic Force Analysis............................................................................................ 8-50

8.3.5

Proportioning and Detailing..................................................................................... 8-52

SPECIAL MOMENT FRAMES CONSTRUCTED USING PRECAST CONCRETE.......... 8-65 8.4.1

Ductile Connections................................................................................................. 8-65

8.4.2

Strong Connections.................................................................................................. 8-67

COMPOSITE STEEL AND CONCRETE

9.1

BUILDING DESCRIPTION...................................................................................................... 9-3

9.2

PARTIALLY RESTRAINED COMPOSITE CONNECTIONS................................................ 9-7

9.3

9.4

9.2.1

Connection Details..................................................................................................... 9-7

9.2.2

Connection Moment-Rotation Curves.........................................................................10

9.2.3

Connection Design................................................................................................... 9-13

LOADS AND LOAD COMBINATIONS................................................................................ 9-17 9.3.1

Gravity Loads and Seismic Weight.......................................................................... 9-17

9.3.2

Seismic Loads.......................................................................................................... 9-18

9.3.3

Wind Loads.............................................................................................................. 9-19

9.3.4

Notional Loads......................................................................................................... 9-19

9.3.5

Load Combinations......................................................................................................20

DESIGN OF C-PRMF SYSTEM............................................................................................. 9-21 9.4.1

Preliminary Design................................................................................................... 9-21

9.4.2

Application of Loading............................................................................................ 9-22

9.4.3

Beam and Column Moment of Inertia..................................................................... 9-23

9.4.4

Connection Behavior Modeling............................................................................... 9-24

9.4.5

Building Drift and P-delta Checks........................................................................... 9-24

FEMA P-751, NEHRP Recommended Provisions: Design Examples 9.4.6

Beam Design............................................................................................................ 9-26

9.4.7

Column Design......................................................................................................... 9-27

9.4.8

Connection Design................................................................................................... 9-28

9.4.9

Column Splices........................................................................................................ 9-29

9.4.10

Column Base Design................................................................................................ 9-29

10 MASONRY 10.1

10.2

WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES, CALIFORNIA.............................................................................................. 10-3 10.1.1

Building Description................................................................................................ 10-3

10.1.2

Design Requirements............................................................................................... 10-4

10.1.3

Load Combinations.................................................................................................. 10-6

10.1.4

Seismic Forces......................................................................................................... 10-8

10.1.5

Side Walls................................................................................................................. 10-9

10.1.6

End Walls............................................................................................................... 10-25

10.1.7

In-Plane Deflection – End Walls............................................................................ 10-44

10.1.8

Bond Beam – Side Walls (and End Walls)............................................................. 10-45

FIVE-STORY MASONRY RESIDENTIAL BUILDINGS IN BIRMINGHAM, ALABAMA; ALBUQUERQUE, NEW MEXICO; AND SAN RAFAEL, CALIFORNIA.............................................................................................. 10-45 10.2.1

Building Description.............................................................................................. 10-45

10.2.2

Design Requirements............................................................................................. 10-48

10.2.3

Load Combinations................................................................................................ 10-50

10.2.4

Seismic Design for Birmingham 1......................................................................... 10-51

10.2.5

Seismic Design for Albuquerque........................................................................... 10-69

10.2.6

Birmingham 2 Seismic Design............................................................................... 10-81

10.2.7

Seismic Design for San Rafael............................................................................... 10-89

10.2.8

Summary of Wall D Design for All Four Locations............................................ 10-101

WOOD DESIGN

11.1

THREE-STORY WOOD APARTMENT BUILDING, SEATTLE, WASHINGTON...............11-3

11.2

11.1.1

Building Description.................................................................................................11-3

11.1.2

Basic Requirements...................................................................................................11-6

11.1.3

Seismic Force Analysis.............................................................................................11-9

11.1.4

Basic Proportioning.................................................................................................11-11

WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES, CALIFORNIA.............................................................................................11-30 11.2.1

xii

Building Description...............................................................................................11-30

Table of Contents 11.2.2

Basic Requirements.................................................................................................11-31

11.2.3

Seismic Force Analysis...........................................................................................11-33

11.2.4

Basic Proportioning of Diaphragm Elements.........................................................11-34

SEISMICALLY ISOLATED STRUCTURES

12.1

BACKGROUND AND BASIC CONCEPTS.......................................................................... 12-4 12.1.1

Types of Isolation Systems....................................................................................... 12-4

12.1.2

Definition of Elements of an Isolated Structure....................................................... 12-5

12.1.3

Design Approach...................................................................................................... 12-6

12.1.4

Effective Stiffness and Effective Damping.............................................................. 12-7

12.2

CRITERIA SELECTION......................................................................................................... 12-7

12.3

EQUIVALENT LATERAL FORCE PROCEDURE................................................................ 12-9

12.4

12.5

12.3.1

Isolation System Displacement................................................................................ 12-9

12.3.2

Design Forces...............................................................................................................11

DYNAMIC LATERAL RESPONSE PROCEDURE............................................................. 12-15 12.4.1

Minimum Design Criteria...................................................................................... 12-15

12.4.2

Modeling Requirements......................................................................................... 12-16

12.4.3

Response Spectrum Analysis................................................................................. 12-18

12.4.4

Response History Analysis..................................................................................... 12-18

EMERGENCY OPERATIONS CENTER USING DOUBLE-CONCAVE FRICTION PENDULUM BEARINGS, OAKLAND, CALIFORNIA.................................. 12-21 12.5.1

System Description................................................................................................ 12-22

12.5.2

Basic Requirements................................................................................................ 12-25

12.5.3

Seismic Force Analysis.......................................................................................... 12-34

12.5.4

Preliminary Design Based on the ELF Procedure.................................................. 12-36

12.5.5

Design Verification Using Nonlinear Response History Analysis......................... 12-51

12.5.6

Design and Testing Criteria for Isolator Units....................................................... 12-61

NONBUILDING STRUCTURE DESIGN

13.1

NONBUILDING STRUCTURES VERSUS NONSTRUCTURAL COMPONENTS........... 13-4

13.2

13.1.1

Nonbuilding Structure.............................................................................................. 13-5

13.1.2

Nonstructural Component........................................................................................ 13-6

PIPE RACK, OXFORD, MISSISSIPPI................................................................................... 13-6 13.2.1 Description............................................................................................................... 13-7 13.2.2

Provisions Parameters.............................................................................................. 13-7

13.2.3

Design in the Transverse Direction......................................................................... 13- 8

13.2.4

Design in the Longitudinal Direction......................................................................13-11

xiii

FEMA P-751, NEHRP Recommended Provisions: Design Examples 13.3

STEEL STORAGE RACK, OXFORD, MISSISSIPPI.......................................................... 13-13 13.3.1 Description............................................................................................................. 13-13

13.4

13.3.2

Provisions Parameters............................................................................................ 13-14

13.3.3

Design of the System............................................................................................. 13-15

ELECTRIC GENERATING POWER PLANT, MERNA, WYOMING................................ 13-17 13.4.1 Description............................................................................................................. 13-17

13.5

13.4.2

Provisions Parameters............................................................................................ 13-19

13.4.3

Design in the North-South Direction..................................................................... 13-20

13.4.4

Design in the East-West Direction......................................................................... 13-21

PIER/WHARF DESIGN, LONG BEACH, CALIFORNIA................................................... 13-21 13.5.1 Description............................................................................................................. 13-21

13.6

13.7

13.5.2

Provisions Parameters............................................................................................ 13-22

13.5.3

Design of the System............................................................................................. 13-23

TANKS AND VESSELS, EVERETT, WASHINGTON........................................................ 13-24 13.6.1

Flat-Bottom Water Storage Tank............................................................................ 13-25

13.6.2

Flat-Bottom Gasoline Tank.................................................................................... 13-28

VERTICAL VESSEL, ASHPORT, TENNESSEE................................................................. 13-31 13.7.1 Description............................................................................................................. 13-31 13.7.2

Provisions Parameters............................................................................................ 13-32

13.7.3

Design of the System............................................................................................. 13-33

DESIGN FOR NONSTRUCTURAL COMPONENTS

14.1

DEVELOPMENT AND BACKGROUND OF THE REQUIREMENTS FOR NONSTRUCTURAL COMPONENTS.......................................................................... 14-3 14.1.1

Approach to Nonstructural Components.................................................................. 14-3

14.1.2

Force Equations........................................................................................................ 14-4

14.1.3

Load Combinations and Acceptance Criteria........................................................... 14-5

14.1.4

Component Amplification Factor............................................................................. 14-6

14.1.5

Seismic Coefficient at Grade.................................................................................... 14-7

14.1.6

Relative Location Factor.......................................................................................... 14-7

14.1.7

Component Response Modification Factor.............................................................. 14-7

14.1.8

Component Importance Factor................................................................................. 14-7

14.1.9

Accommodation of Seismic Relative Displacements.............................................. 14-8

14.1.10 Component Anchorage Factors and Acceptance Criteria......................................... 14-9 14.1.11 Construction Documents.......................................................................................... 14-9 14.2 xiv

ARCHITECTURAL CONCRETE WALL PANEL................................................................ 14-10

Table of Contents

14.3

14.4

14.5

14.6

14.2.1

Example Description.............................................................................................. 14-10

14.2.2

Design Requirements............................................................................................. 14-12

14.2.3

Spandrel Panel........................................................................................................ 14-12

14.2.4

Column Cover........................................................................................................ 14-19

14.2.5

Additional Design Considerations......................................................................... 14-20

HVAC FAN UNIT SUPPORT................................................................................................ 14-21 14.3.1

Example Description.............................................................................................. 14-21

14.3.2

Design Requirements............................................................................................. 14-22

14.3.3

Direct Attachment to Structure............................................................................... 14-23

14.3.4

Support on Vibration Isolation Springs.................................................................. 14-26

14.3.5

Additional Considerations for Support on Vibration Isolators............................... 14-31

ANALYSIS OF PIPING SYSTEMS...................................................................................... 14-33 14.4.1

ASME Code Allowable Stress Approach............................................................... 14-33

14.4.2

Allowable Stress Load Combinations.................................................................... 14-34

14.4.3

Application of the Standard................................................................................... 14-36

PIPING SYSTEM SEISMIC DESIGN.................................................................................. 14-38 14.5.1

Example Description.............................................................................................. 14-38

14.5.2

Design Requirements............................................................................................. 14-43

14.5.3

Piping System Design............................................................................................ 14-45

14.5.4

Pipe Supports and Bracing..................................................................................... 14-48

14.5.5

Design for Displacements...................................................................................... 14-53

ELEVATED VESSEL SEISMIC DESIGN............................................................................ 14-55 14.6.1

Example Description.............................................................................................. 14-55

14.6.2

Design Requirements............................................................................................. 14-58

14.6.3

Load Combinations................................................................................................ 14-60

14.6.4

Forces in Vessel Supports....................................................................................... 14-60

14.6.5

Vessel Support and Attachment.............................................................................. 14-62

14.6.6

Supporting Frame................................................................................................... 14-65

14.6.7

Design Considerations for the Vertical Load-Carrying System............................. 14-69

THE BUILDING SEISMIC SAFETY COUNCIL

1 Introduction Robert G. Pekelnicky, P.E., S.E. and Michael Valley, S.E. 1.1

EVOLUTION OF EARTHQUAKE ENGINEERING .................................................... 3

1.2

HISTORY AND ROLE OF THE NEHRP PROVISIONS .............................................. 6

1.3

THE NEHRP DESIGN EXAMPLES .............................................................................. 8

1.4

GUIDE TO USE OF THE PROVISIONS ..................................................................... 11

1.5

REFERENCES ............................................................................................................... 38

The NEHRP Recommended Provisions: Design Examples are written to illustrate and explain the applications of the 2009 NEHRP Recommended Seismic Provisions for Buildings and Other Structures, ASCE 7-10 Minimum Design Loads for Buildings and Other Structures and the material design standards referenced therein and to provide explanations to help understand them. Designing structures to be resistant to major earthquake is complex and daunting to someone unfamiliar with the philosophy and history of earthquake engineering. The target audience for the Design Examples is broad. College students learning about earthquake engineering, engineers studying for their licensing exam, or those who find themselves presented with the challenge of designing in regions of moderate and high seismicity for the first time should all find this document’s explanation of earthquake engineering and the Provisions helpful. Fortunately, major earthquakes are a rare occurrence, significantly rarer than the other hazards, such as damaging wind and snow storms that one must typically consider in structural design. However, past experiences have shown that the destructive power of a major earthquake can be so great that its effect on the built environment can be underestimated. This presents a challenge since one cannot typically design a practical and economical structure to withstand a major earthquake elastically in the same manner traditionally done for other hazards. Since elastic design is not an economically feasible option for most structures where major earthquakes can occur, there must be a way to design a structure to be damaged but still safe. Unlike designing for strong winds, where the structural elements that resist lateral forces can be proportioned to elastically resist the pressures generated by the wind, in an earthquake the lateral force resisting elements must be proportioned to deform beyond their elastic range in a

FEMA P-751, NEHRP Recommended Provisions: Design Examples

controlled manner. In addition to deforming beyond their elastic range, the lateral force resisting system must be robust enough to provide sufficient stability so the building is not at risk of collapse. While typical structures are designed to be robust enough to have a minimal risk of collapse in major earthquakes, there are other structures whose function or type of occupants warrants higher performance designs. Structures, like hospitals, fire stations and emergency operation centers need to be designed to maintain their function immediately after or returned to function shortly after the earthquake. Structures like schools and places where large numbers of people assemble have been deemed important enough to require a greater margin of safety against collapse than typical buildings. Additionally, earthquake resistant requirements are needed for the design and anchorage of architectural elements and mechanical, electrical and plumbing systems to prevent falling hazards and in some cases loss of system function. Current building standards, specifically the American Society of Civil Engineers (ASCE) 7 Minimum Design Loads for Buildings and Other Structures and the various material design standards published by the American Concrete Institute (ACI), the American Institute of Steel Construction (AISC), the American Iron and Steel Institute (AISI), the American Forest & Paper Association (AF&PA) and The Masonry Society (TMS) provide a means by which an engineer can achieve these design targets. These standards represent the most recent developments in earthquake resistant design. The majority of the information contained in ASCE 7 comes directly from the NEHRP Recommended Seismic Provisions for New Buildings and Other Structures. The stated intent of the NEHRP Provisions is to provide reasonable assurance of seismic performance that will: 1. Avoid serious injury and life loss, 2. Avoid loss of function in critical facilities, and 3. Minimize structural and nonstructural repair costs where practical to do so. The Provisions have explicit requirements to provide life safety for buildings and other structures though the design forces and detailing requirements. The current provisions have adopted a target risk of collapse of 1% over a 50 year lifespan for a structure designed to the Provisions. The Provisions provide prevention of loss of function in critical facilities and minimized repair costs in a more implicit manner though prescriptive requirements. Having good building codes and design standards is only one action necessary to make a community’s buildings resilient to a major earthquake. A community also needs engineers who can carry out designs in accordance with the requirements of the codes and standards and contractors who can construct the designs in accordance with properly prepared construction documents. The first item is what the NEHRP Recommended Provisions: Design Examples seeks to foster. The second item is discussed briefly later in this document in Chapter 1, Section 1.6 Quality Assurance. The purpose of this introduction is to offer general guidance for users of the design examples and to provide an overview. Before introducing the design examples, a brief history of earthquake engineering is presented. That is followed by a history of the NEHRP Provisions and its role in

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Introduction

setting standards for earthquake resistant design. This is done to give the reader a perspective of the evolution of the Provisions and some background for understanding the design examples. Following that is a brief summary of each chapter.

It is helpful to understand the evolution of the earthquake design standards and the evolution of the field of earthquake engineering in general. Much of what is contained within the standards is based on lessons learned from earthquake damage and the ensuing research. Prior to 1900 there was little consideration of earthquakes in the design of buildings. Major earthquakes were experienced in the United States, notably the 1755 Cap Ann Earthquake around Boston, the 1811 and 1812 New Madrid Earthquakes, the 1868 Hayward California Earthquake and the 1886 Charleston Earthquake. However, none of these earthquakes led to substantial changes in the way buildings were constructed. Many things changed with the Great 1906 San Francisco Earthquake. The earthquake and ensuing fire destroyed much of San Francisco and was responsible for approximately 3,000 deaths. To date it is the most deadly earthquake the United States has ever experienced. While there was significant destruction to the built environment, there were some important lessons learned from those buildings that performed well and did not collapse. Most notable was the exemplary performance of steel framed buildings which consisted of riveted wind frames and brick infill, built in the Chicago style. The recently formed San Francisco Section of the American Society of Civil Engineers (ASCE) studied the effects of the earthquake in great detail. An observation was that “a building designed with a proper system of bracing wind pressure at 30 lbs. per square foot will resist safely the stresses caused by a shock of the intensity of the recent earthquake.” (ASCE, 1907) That one statement became the first U.S. guideline on how to provide an earthquake resistant design. The earthquakes in Tokyo in 1923 and Santa Barbara in 1925 spurred major research efforts. Those efforts led to the development of the first seismic recording instruments, shake tables to investigate earthquake effects on buildings, and committees dedicated to creating code provisions for earthquake resistant design. Shortly after these earthquakes, the 1927 Uniform Building Code (UBC) was published (ICBO, 1927). It was the first model building code to contain provisions for earthquake resistant design, albeit in an appendix. In addition to that, a committee began working on what would become California’s first state-wide seismic code in 1939. Another earthquake struck Southern California in Long Beach in 1933. The most significant aspect, of that earthquake was the damage done to school buildings. Fortunately the earthquake occurred after school hours, but it did cause concern over the vulnerabilities of these buildings. That concern led to the Field Act, which set forth standards and regulations for earthquake

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

resistance of school buildings. This was the first instance of what has become a philosophy engrained in the earthquake design standards of requiring higher levels of safety and performance for certain buildings society deems more important that a typical building. In addition to the Field Act, the Long Beach earthquake led to a ban on unreinforced masonry construction in California, which in later years was extended to all areas of moderate and high seismic risk. Following the 1933 Long Beach Earthquake there was significant activity both in Northern and Southern California, with the local Structural Engineers Associations of each region drafting seismic design provisions for Los Angeles in 1943 and San Francisco in 1948. Development of these codes was facilitated greatly by observations from the 1940 El Centro Earthquake. Additionally, that earthquake was the first major earthquake for which the strong ground motion shaking was recorded with an accelerograph. A joint committee of the San Francisco Section of ASCE and the Structural Engineers Association of Northern California began work on seismic design provisions which were published in 1951 as ASCE Proceedings-Separate No. 66. Separate 66, as it is commonly referred to as, was a landmark document which set forth earthquake design provisions which formed the basis of US building codes for almost 40 years. Many concepts and recommendations put forth in Separate 66, such as the a period dependent design spectrum, different design forces based on the ductility of a structure and design provisions for architectural components are still found in today’s standards. Following Separate 66, the Structural Engineers Association of California (SEAOC) formed a Seismology committee and in 1959 put forth the first edition of the Recommended Lateral Force Requirements, commonly referred to as the “The SEAOC Blue Book.” The Blue Book became the base document for updating and expanding the seismic design provisions of the Uniform Building Code (UBC), the model code adopted by most western states including California. SEAOC regularly updated the Blue Book from 1959 until 1999, with the changes made and new recommendations in each new edition of the Blue Book being incorporated in to the subsequent edition of the UBC. The 1964 Anchorage Earthquake and the 1971 San Fernando Earthquake both were significant events. Both earthquakes exposed significant issues with the way reinforced concrete structures would behave if not detailed for ductility. There were failures of large concrete buildings which had been designed to recent standards and those buildings had to be torn down. To most engineers and the public this was unacceptable performance. Following the 1971 San Fernando Earthquake, the National Science Foundation gave the Applied Technology Council (ATC) a grant to develop more advanced earthquake design provisions. That project engaged over 200 preeminent experts in the field of earthquake engineering. The landmark report they produced in 1978, ATC 3-06, Tentative Provisions for the Development of Seismic Regulations for Buildings (1978), has become the basis for the current earthquake design standards. The NEHRP Provisions trace back to ATC 3-06, as will be discussed in more detail in the following section.

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Introduction

There have been additional earthquakes since the 1971 San Fernando Earthquake which have had significant influence on seismic design. Table 1 provides a summary of major North American earthquakes and changes to the building codes that resulted from them through the 1997 UBC. Of specific note are the 1985 Mexico City, 1989 Loma Prieta and 1994 Northridge Earthquakes. Earthquake

UBC Enhancement Edition 1971 San Fernando 1973 Direct positive anchorage of masonry and concrete walls to diaphragms 1976 Seismic Zone 4, with increased base shear requirements Occupancy Importance Factor I for certain buildings Interconnection of individual column foundations Special Inspection requirements 1979 Imperial Valley 1985 Diaphragm continuity ties 1985 Mexico City 1988 Requirements for column supporting discontinuous walls Separation of buildings to avoid pounding Design of steel columns for maximum axial forces Restrictions for irregular structures Ductile detailing of perimeter frames 1987 Whittier Narrows 1991 Revisions to site coefficients Revisions to spectral shape Increased wall anchorage forces for flexible diaphragm buildings 1989 Loma Prieta 1991 Increased restrictions on chevron-braced frames Limitations on b/t ratios for braced frames 1994 Ductile detailing of piles 1994 Northridge 1997 Restrictions on use of battered piles Requirements to consider liquefaction Near-fault zones and corresponding base shear requirements Revised base shear equations using 1/T spectral shape Redundancy requirements Design of collectors for overstrength Increase in wall anchorage requirements More realistic evaluation of design drift Steel moment connection verification by test Table 1: Recent North American Earthquakes and Subsequent Code Changes (from SEOAC, 2009)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

The 1985 Mexico City Earthquake was extremely devastating. Over 10,000 people were killed and there was three to four billion dollars of damage. The most significant aspect of this earthquake was that while the epicenter was located over 200 miles away from Mexico City. The unique geologic nature, that Mexico City was sited on an old (ancient?) lake bed of silt and clay, generated ground shaking with a much longer period and larger amplitudes than would be expected from typical earthquakes. This long period shaking was much more damaging to midrise and larger structures because these buildings were in resonance with the ground motions. In current design practice site factors based on the underlying soil are used to modify the seismic hazard parameters. The 1989 Loma Prieta Earthquake caused an estimated $6 billion in damage, although it was far less deadly than other major earthquakes throughout history. Only sixty-three people lost their lives, a testament to the over 40 years of awareness and consideration of earthquakes in the design of structures. A majority of those deaths, 42, resulted from the collapse of the Cyprus Street Viaduct, a nonductile concrete elevated freeway. In this earthquake the greatest damage occurred in Oakland, parts of Santa Cruz and the Marina District in San Francisco where the soil was soft or poorly compacted fill. As with the Mexico City experience, this indicates the importance of subsurface conditions on the amplification of earthquake shaking. The earthquake also highlighted the vulnerability of soft and weak story buildings because a significant number of the collapsed buildings in the Marina District were wood framed apartment buildings with weak first stories consisting of garages with door openings that greatly reduced the wall area at the first story. Five years later the 1994 Northridge earthquake struck California near Los Angeles. Fifty seven people lost their lives and the damage was estimated at around $20 billion. The high cost of damage repair emphasized the need for engineers to consider overall building performance, in addition to building collapse, and spurred the movement toward Performance-Based design. As with the 1989 Loma Prieta earthquake, there was a disproportionate number of collapses of soft/weak first story wood framed apartment buildings. The 1994 Northridge Earthquake was most significant for the unanticipated damage to steel moment frames that was discovered. Steel moment frames had generally been thought of as the best seismic force resisting system. However, many moment frames experienced fractures of the welds that connected the beam flange to the column flange. This led to a multi-year, FEMA funded problem-focused study to assess and improve the seismic performance of steel moment frames. It also led to requirements for the number of frames in a structure, and penalties for having a lateral force resisting system that does not have sufficient redundancy.

Following the completion of the ATC 3 project in 1978, there was desire to make the ATC 3-06 approach the basis for new regulatory provisions and to update them periodically. FEMA, as the lead agency of the National Earthquake Hazard Reduction Program (NEHRP) at the time, contracted with the then newly formed Building Seismic Safety Council (BSSC) to perform trial designs based on ATC 3-06 to exercise the proposed new provisions. The BSSC put together a

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Introduction

group of experts consisting of consulting engineers, academics, representatives from various building industries and building officials. The result of that effort was the first (1985) edition of the NEHRP Recommended Provisions for the Development of Seismic Regulations for New Buildings. Since the publication of the first edition through the 2003 edition, the NEHRP Provisions were updated every three years. Each update incorporated recent advances in earthquake engineering research and lessons learned from previous earthquakes. The intended purpose of the Provisions was to serve as a code resource document. While the SEAOC Blue Book continued to serve as the basis for the earthquake design provisions in the Uniform Building Code, the BOCA National Building Code and the Standard Building Code both adopted the 1991 NEHRP Provisions in their 1993 and 1994 editions respectively. The 1993 version of the ASCE 7 standard Minimum Design Loads for Buildings and Other Structures (which had formerly been American National Standards Institute (ANSI) Standard A58.1) also utilized the 1991 NEHRP Provisions. In the late 1990’s the three major code organizations, ICBO (publisher of the UBC), BOCA, and SBC decided to merge their three codes into one national model code. When doing so they chose to incorporate the 1997 NEHRP Provisions as the seismic design requirements for the inaugural 2000 edition of the International Building Code (IBC). Thus, the SEAOC Blue Book was no longer the base document for the UBC/IBC. The 1997 NEHRP Provisions had a number of major changes. Most significant was the switch from the older seismic maps of ATC 3-06 to new, uniform hazard spectral value maps produced by USGS in accordance with BSSC Provisions Update Committee (PUC) Project 97. The 1998 edition of ASCE 7 was also based on the 1997 NEHRP Provisions. ASCE 7 continued to incorporate the 2000 and 2003 editions of the Provisions for its 2002 and 2005 editions, respectively. However, the 2000 IBC adopted the 1997 NEHRP Provisions by directly transferring the text from the provisions into the code. In the 2003 IBC the provisions from the 2000 IBC were retained and there was also language, for the first time, which pointed the user to ASCE 7-02 for seismic provisions instead of adopting the 2000 NEHRP Provisions directly. The 2006 IBC explicitly referenced ASCE 7 for the earthquake design provisions, as did the 2009 and 2012 editions. With the shift in the IBC from directly incorporating the NEHRP Provision for their earthquake design requirements to simply referencing the provisions in ASCE 7, the BSSC Provisions Update Committee decided to move the NEHRP Provisions in a new direction. Instead of providing all the seismic design provisions within the NEHRP Provisions, which would essentially be repeating the provisions in ASCE 7, and then modifying them, the PUC chose to adopt ASCE 7-05 by reference and then provide recommendations to modify it as necessary. Therefore, Part 1 of the 2009 NEHRP Provisions contains major technical modifications to ASCE 7-05 which, along with other recommendations from the ASCE 7 Seismic Subcommittee, were the basis for proposed changes that were incorporated into ASCE 7-10 and included associated commentary on those changes. The PUC also developed a detailed commentary to the seismic provisions of ASCE 7-05, which became Part 2 of the 2009 NEHRP Provisions.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

In addition to Part 1 and Part 2 in the 2009 NEHRP Provisions, a new section was introduced – Part 3. The intent of this new portion was to showcase new research and emerging methods, which the PUC did not feel was ready for adoption into national design standards but was important enough to be disseminated to the profession. This new three part format marks a change in the Provisions from a code-language resource document to the key knowledge-based resource for improving the national seismic design standards and codes. The most significant technical change to Part 1 of the 2009 Provisions was the adoption of a “Risk-Targeted” approach to determine the Maximum Considered Earthquake hazard parameters. This was a switch from the Uniform Hazard approach in the 1997, 2000, and 2003 editions. In the “Risk Targeted” approach, the ground motion parameters are adjusted such that they provide a uniform 1% risk of collapse in a 50 year period for a generic building, as opposed to a uniform return period for the seismic hazard. A detailed discussion of this can be found in the commentary in Part 1 of the 2009 Provisions. Today, someone needing to design a seismically resilient building in the U.S. would first go to the local building code which has generally adopted the IBC with or without modifications by the local jurisdiction. For seismic design requirements, the IBC then points to relevant Chapters of ASCE 7. Those chapters of ASCE 7 set forth the seismic hazard, design forces and system detailing requirements. The seismic forces in ASCE 7 are dependent upon the type of detailing and specific requirements of the lateral force resisting system elements. ASCE 7 then points to material specific requirements found in the material design standards published by ACI, AISC, AISI, AF&PA and TMS for those detailing requirements. Within this structure, the NEHRP Provisions serves as a consensus evaluation of the design standards and a vehicle to transfer new knowledge to ASCE 7 and the material design standards.

1.3 Design examples were first prepared for the 1985 NEHRP Provisions in a publication entitled Guide to Application of the NEHRP Recommended Provisions, FEMA 140. These design examples were based on real buildings. The intent was the same as it is now, to show people who are not familiar with seismic design of how to apply the Provisions, the standards referenced by the Provisions and the concepts behind the Provisions. Because of the expanded role that the Provisions were having as the basis for the seismic design requirements for the model codes and standards, it was felt that there should be an update and expansion of the original design examples. Following the publication of the 2003 NEHRP Provisions, FEMA commissioned a project to update and expand the design examples. This resulted in NEHRP Recommended Provisions: Design Examples, FEMA 451. Many of the design problems drew heavily on the examples presented in FEMA 140, but were completely redesigned based on first the 2000 and then the 2003 NEHRP Provisions and the materials standards referenced therein. Additional examples were created to reflect the myriad of structures now covered under the Provisions.

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Introduction

This volume is an update of the design examples in FEMA 451 to reflect the 2009 NEHRP Provisions and the updated standards referenced therein. Many of the design examples are the same as presented in FEMA 451, with only changes made due to changes in the provisions. The Design Examples not only covers the application of ASCE 7, the material design standards and the NEHRP Provisions, it also illustrates the use of analysis methods and earthquake engineering knowledge and judgment in situations which would be encountered in real designs. The authors of the design examples are subject matter experts in the specific area covered by the chapter they authored. Furthermore, the companion NEHRP Recommend Provisions: Training Materials provides greater background information and knowledge, which augment the design examples. It is hoped that with the Part 2 Expanded Commentary in the 2009 NEHRP Provisions, the Design Examples and the Training Materials, an engineer will be able to understand not just how to use the Provisions, but also the philosophical and technical basis behind the provisions. Through this understanding of the intent of the seismic design requirements found in ASCE 7, the material design standards and the 2009 NEHRP Provisions, it is hoped that more engineers will find the application of those standards less daunting and thereby utilize the standards more effectively in creating innovative and safe designs. Chapter 1 – This preceding introduction and the Guide to Use of the Provisions which follows provides background and presents a series of flow charts to walk an engineer through the use of the provisions. Chapter 2 – Fundamentals presents a brief but thorough introduction to the fundamentals of earthquake engineering. While this section does not present any specific applications of the Provisions, it provides the reader with the essential philosophical background to what is contained within the Provisions. The concepts of idealizing a seismic dynamic load as an equivalent static load and providing ductility instead of pure elastic strength are explained. Chapter 3 - Earthquake Ground Motion is new to this edition of the Design Examples. This chapter explains the basis for determining seismic hazard parameters used for design in the Provisions. It discusses the seismic hazard maps in ASCE 7-05 and the new Risk Targeted maps found in the 2009 NEHRP Provisions and ASCE 7-10. The chapter also discusses probabilistic seismic hazard assessment, the maximum direction response parameters and selection and scaling of ground motion histories for use in linear and nonlinear response history analysis. Chapter 4 – Structural Analysis presents the analysis of two different buildings, a 12-story steel moment frame and a 6-story steel moment frame structure. The 12-story structure is irregular and is analyzed using the three linear procedures referenced in ASCE 7 – Equivalent Lateral Force, Modal Response Spectrum, and Linear Response History. The 6-story structure is analyzed using three methods referenced in ASCE 7 - Equivalent Lateral Force, Modal Response Spectrum and Nonlinear Response History – and two methods which are referenced in other documents – Plastic Strength (Virtual Work) and Nonlinear Static Pushover. The intent of this chapter is to show the variations in predicted response based on the chosen analysis method.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Some of the examples have been updated based on advances in the state of the practice with respect to seismic analysis. Chapter 5 – Foundation Analysis and Design presents design examples for both shallow and deep foundations. First, a spread footing foundation for a 7-story steel framed building is presented. Second the design of a pile foundation for a 12-story concrete moment frame building is presented. Designs of the steel and concrete structures whose foundations are designed in this chapter are presented in Chapters 6 and 7 respectively. Chapter 6 – Structural Steel Design presents the design of three different types of steel buildings. The first building is a high-bay industrial warehouse which uses an ordinary concentric braced frame in one direction and an intermediate steel moment frame in the other direction. The second example is a 7-story office building which is designed using two alternate framing systems, special steel moment frames and special concentric braced frames. The third example is new to this edition of the design examples. It is a 10-story hospital using buckling restrained braced frames (BRBF). This replaces an example using eccentrically braced frames (EBF) in the previous edition of the design examples because the profession has moved toward favoring the BRBF system over the EBF system. Chapter 7 – Reinforced Concrete presents the designs of a 12-story office building located in moderate and high seismicity. The same building configuration is used in both cases, but in the moderate seismicity region “Intermediate” member frames are used while “Special” moment frames are used in the high seismicity region. Also in the high seismicity region, special concrete walls are needed in one direction and their design is presented. Chapter 8 – Precast Concrete Design presents examples of four common cases where precast concrete elements are a component of a seismic force resisting system. The first example presents the design of precast concrete panels being used as horizontal diaphragms both with and without a concrete topping slab. The second example presents the design of 3-story office building using intermediate precast concrete shear walls in a region of low or moderate seismicity The third example presents the design of a one-story tilt-up concrete industrial building in a region of high seismicity. The last example, which is new to this edition of the design examples, presents the design of a precast Special Moment Frame. Chapter 9 – Composite Steel and Concrete presents the design of a 4-story medical office building in a region of moderate seismicity. The building uses composite partially restrained moment frames in both directions as the lateral force resisting system. Chapter 10 – Masonry presents the design of two common types of buildings using reinforced masonry walls as their lateral force resisting system. The first example is a single-story masonry warehouse building with tall, slender walls. The second example is a five-story masonry hotel building with a bearing wall system designed in areas with different seismicity. Chapter 11 – Wood Design presents the design of a variety of wood elements in common seismic force resisting applications. The first example is a three-story, wood-frame apartment

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Introduction

building. The second example illustrates the design of the roof diaphragm and wall-to-roof anchorage for the masonry building featured in the first example of Chapter 10. Chapter 12 – Seismically Isolated Structures presents both the basic concepts of seismic isolation and then the design of an essential facility using a seismic isolation system. The example building has a special concentrically braced frame superstructure and uses doubleconcave friction pendulum isolators, which have become the most common type of isolator used in regions of high seismicity. In the previous edition of the design examples, high-damping rubber isolators were used. Chapter 13 – Nonbuilding Structure Design presents the design of various types of structures other than buildings that are covered by the Provisions. First there is a brief discussion about the difference between a nonbuilding structure and a nonstructural component. The first example is the design of a pipe rack. The second example is of an industrial storage rack. The third example is a power generating plant with significant mass irregularities. The third example is a pier. The fourth examples are flat-bottomed storage tanks, which also illustrates how the Provisions are used in conjunction with industry design standards. The last example is of a tall, slender vertical storage vessel containing hazardous materials, which replaces an example of an elevated transformer. Chapter 14 – Design for Nonstructural Components presents a discussion on the design of nonstructural components and their anchorage plus several design examples. The examples are of an architectural concrete wall panel, the supports for a large rooftop fan unit, the analysis and bracing of a piping system (which is greatly expanded from FEMA 451) and an elevated vessel (which is new).

The flow charts that follow are provided to assist the user of the NEHRP Recommended Provisions and, by extension, the seismic provisions of ASCE 7, Minimum Design Loads for Buildings and Other Structures; and the International Building Code. The flow charts provide an overview of the complete process for satisfying the Provisions, including the content of all technical chapters. Part 1 of the 2009 NEHRP Recommended Seismic Provisions for New Buildings and Other Structures (the Provisions) adopts by reference the national consensus design loads standard, ASCE/SEI 7-05, Minimum Design Loads for Buildings and Other Structures (the Standard), including Supplements 1 and 2, and makes modifications to the seismic requirements in the Standard. Part 2 of the Provisions contains an up-to-date, user friendly commentary on the seismic design requirements of the Standard. Part 3 of the Provisions consists of a series of resource papers that clarify aspects of the Provisions and present new seismic design concepts and procedures. The flow charts in this chapter are expected to be of most use to those who are unfamiliar with the scope of the NEHRP Recommended Provisions, but they cannot substitute for a careful reading of the Provisions. The level of detail shown varies, being greater where questions of applicability of the Provisions are pertinent and less where a standard process of structural analysis or detailing is all that is required. The details contained in the many standards referenced in the Provisions are not included;

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FEMA P-751, NEHRP Recommended Provisions: Design Examples therefore, the actual flow of information when proportioning structural members for the seismic load effects specified in the Provisions will be considerably more complex. Cited section numbers (such as Sec. 11.1.2) refer to sections of the Standard. Where reference is to a Provisions Part 1 modification to the Standard, the citation indicates that (such as Provisions Sec. 11.1.2). In a few rare instances, the Provisions Update Committee deferred to the ASCE 7 committee to make needed technical changes; in those cases reference is made specifically to ASCE 7-10 (such as ASCE 710 Sec. 12.12.3). On each chart the flow generally is from a heavy-weight box at the top-left to a medium-weight box at the bottom-right. User decisions are identified by six-sided cells. Optional items and modified flow are indicated by dashed lines. Chart 1.1 provides an overall summary of the process which begins with consideration of the Scope of Coverage and ends with Quality Assurance Requirements. Additions to, changes of use in and alterations of existing structures are covered by the Provisions (see Chart 1.3), but evaluation and rehabilitation of existing structures is not. Nearly two decades of FEMA-sponsored development of technical information to improve seismic safety in existing buildings has culminated in a comprehensive set of codes, standards and guidelines. The International Existing Building Code references the ASCE 31 Standard, Seismic Evaluation of Existing Buildings; and the ASCE 41 Standard, Seismic Rehabilitation of Existing Buildings.

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Introduction

Chart 1.1 Overall Summary of Flow Chart 1.2 Scope of Coverage

Chart 1.5 SDC A

Chart 1.3 Application to Existing Structures Chart 1.4 Basic Requirements

Chart 1.25 Quality Assurance

Chart 1.24 Nonstructural Components

Chart 1.7 Redundancy Factor

Chart 1.6 Structural Design

Chart 1.8 Simplified Design

Chart 1.9 ELF Analysis

Chart 1.11 Modal Analysis

Chart 1.12 Response History Analysis

Chart 1.13 Seismically Isolated

Chart 1.14 Damped

Chart 1.10 Soil-Structure Interaction

Chart 1.15 Deformation Requirements Chart 1.16 Design and Detailing Requirements

Chart 1.17 Steel

Chart 1.19 Composite

Chart 1.18 Concrete

Chart 1.20 Masonry

Chart 1.21 Wood

Chart 1.22 Nonbuilding Structures

Chart 1.23 Foundations

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FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.2 Scope of Coverage

Determine if structure falls in scope of the Standard (Sec. 11.1.2).

Is structure a vehicular bridge, electrical transmission tower, hydraulic structure, buried utility line, or nuclear reactor? No Is the use agricultural storage with only incidental human occupancy? No

Yes

Standard not applicable.

Yes

No requirements.

Determine S S and S1 (Sec. 11.4.1).

S1 ≤ 0.04 and S S ≤ 0.15? No

Assign to Seismic Design Category A. Go to Chart 1.5.

Yes

Is it a detached 1- or 2-family dwelling? No

Yes

S S ≤ 0.4 or SDC A, B, or C? No

Yes

Wood frame dwelling with not more than 2 stories and compliant with the IRC? No Is it an existing structure? No

Yes

No additional requirements.

Go to Chart 1.3. Go to Chart 1.4.

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Introduction Chart 1.3 Application to Existing Structures

* Addition to existing structure (Sec. 11B.2 and 11B.3).

Is addition structurally independent from existing structure? No

Is any element's seismic force increased by more than 10% or its seismic resistance decreased? Yes

Do the affected elements still Yes comply with the Standard? No *

Change of use (Sec. 11B.5).

Change to higher Occupancy Category? Yes Change from Occupancy Category I or II to III and S DS < 0.33? No

* Alteration of existing structure (Sec. 11B.4).

Only addition or alteration designed as new structure. Go to Chart 1.4.

Yes

Entire structure designed as new structure. Go to Chart 1.4. No requirements.

Yes

Does alteration increase seismic Yes forces to or decrease design strength of existing structural elements by more than 10 percent? Such alteration No not permitted. Is seismic force on existing Yes structural elements increased beyond their design strength? No Does alteration create a structural Yes irregularity or make an existing irregularity more severe? No Is the design strength of Yes existing structural elements required to resist seismic forces reduced? New structural elements and new No or relocated nonstructural elements must be detailed and connected as required by the Standard.

* The Standard applies to existing structures only in the cases of additions to, changes of use in, and alterations of such structures.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.4 Basic Requirements Determine Occupancy Category (Sec. 1.5.1) and Importance Factor (Sec. 11.5.1). Using mapped acceleration parameters and risk coefficients (from Fig. 22-1 through 22-6, or online application), determine the Maximum Considered Earthquake (MCER ) spectral response acceleration at short periods (S S ) and at 1 second (S1).

Assign structure to Seismic Design Category A. Go to Chart 1.5.

Yes

S S ≤ 0.15 and S1 ≤ 0.04? No

Use Site Class D unless authority having jurisdiction determines that Site Class E or F is present at the site.

Soil properties known in sufficient No detail to determine Site Class? Yes Classify the site (Ch. 20).

Site Class E or F? No

Yes

Fulfill site limitation (Sec. 11.8.1).

S1 > 0.6 and seismically isolated or with damping system? No Site Class F? No

Yes

Perform ground motion hazard analysis (Sec. 21.2).

Perform site response analysis (Sec. 21.2).

Determine design response spectrum (Sec. 21.3) and design acceleration parameters (Sec. 21.4).

Adjust MCER acceleration parameters for site class (Sec. 11.4.3). Calculate design earthquake acceleration parameters S DS and S D1 (Sec. 11.4.4). Design response spectrum required Yes for the analysis to be used? No

Go to Chart 1.6 for structural requirements.

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Calculate design response spectrum (Sec. 11.4.5). Determine Seismic Design Category (Sec. 11.6).

Go to Chart 1.24 for nonstructural components.

Go to Chart 1.25 for quality assurance requirements.

Introduction Chart 1.5 Seismic Design Category A

Does the structure have a damping system?

Yes

Reassign to Seismic Design Category B (Sec. 18.2.1). Return to Chart 1.4.

Determine static lateral forces and apply them independently in two orthogonal directions (Sec. 11.7.2).

Provide a continuous lateral load path and connect each smaller portion of the structure to the remainder of the structure with elements having a design strength of at least 5 percent of the portions weight (Sec. 11.7.3).

For each beam, girder, or truss, provide positive connections with a design strength of at least 5 percent of the dead plus live load reaction for a horizontal force acting parallel to the member (Sec. 11.7.4).

Opt to perform a more involved analysis? No

Yes

Go to Chart 1.6.

Classify the effects due to the loads described above as E and combine with the effects of other loads in accordance with Sec. 2.3 or 2.4 (Sec. 11.7.1).

Done.

* The requirement to reclassify Seismic Design Category A structures to Seismic Design Category B has been declared editorially erroneous and will be removed via errata for ASCE 7-13. 1– 17

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Chart 1.6 Structural Design

Satisfy limitations of and choose to use Simplified Design Procedure? (Sec. 12.14.1.1) No

Yes

Go to Chart 1.8.

Comply with the stated design basis (Sec. 12.1). Select the seismic force-resisting system (including requirements for height limits, combinations of framing systems, dual systems, cantilevered column systems, and inverted pendulum systems, as applicable) and note R, Ω0, and Cd for later use (Sec. 12.2 and Table 12.2-1). Classify diaphragm flexibility (Sec. 12.3.1). Examine plan and vertical regularity (Sec. 12.3.2) and meet minimum requirements for irregular structures (Sec. 12.3.3). Assign redundancy factor, ρ, (per Sec. 12.3.4.2) using Chart 1.7. Moment frame assigned to Seismic Design Category D, E, or F? No

Yes

Requirements for special moment frame continuity (Sec. 12.2.5.5). Note modified drift limits for moment frames (Sec. 12.12.1.1).

Determine seismic load effects and load combinations applicable to design (Sec. 12.4). Determine applicable direction of loading criteria (Sec. 12.5). Seismically isolated? No Damping system? Yes

Go to Chart 1.14.

1– 18

Go to Chart 1.13.

Yes No

Select a permitted structural analysis procedure (Sec. 12.6). Determine effective seismic weight (Sec. 12.7.2), and model foundation (Sec. 12.7.1) and structure (Sec. 12.7.3), including interaction effects (Sec. 12.7.4).

Go to Chart 1.9 for ELF analysis.

Go to Chart 1.11 for modal analysis.

Go to Chart 1.12 for response history analysis.

Introduction Chart.7 Redundancy Factor

Perform linear analysis with all elements

Yes

Define story X p above which no more than 35% of base shear is resisted

Below X p is item b of Section 12.3.4.2 satisfied? No* Extreme torsional irregularity?

Yes

Does the seismic force-resisting system comprise only shear walls or wall piers with a height-to-length ratio not greater than 1.0?

Prioritize elements based on highest force or force/story shear

Select an element (below X p ) to remove, and perform linear analysis without that element

Yes

Extreme torsional irregularity? No

Does the demand in any remaining element (below X p ) increase by more than 50%?

Yes

No Does plastic mechanism analysis show that element removal decreases story strength by more than 33%?

Yes*

No No

ρ = 1.0

Have all likely elements been considered? Yes

ρ = 1.3

* or not considered

1– 19

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.8 Simplified Design Procedure Comply with the stated design basis (Sec. 12.14.2), including interconnection (Sec. 12.14.7.1). Select the seismic force-resisting system (including requirements for combinations of framing systems, as applicable) and note R for later use (Sec. 12.14.4 and Table 12.14-1). Classify diaphragm flexibility (Sec. 12.14.5). Determine seismic load effects and load combinations applicable to design (Sec. 12.14.3). Determine applicable direction of loading criteria (Sec. 12.14.6). Perform simplified lateral force analysis (Sec. 12.14.8). Satisfy deformation requirements (Sec. 12.14.8.5). Design diaphragms, including appropriate detailing at openings (Sec. 12.14.7.2 and 12.14.7.4). Provide collector elements to transfer seismic forces (Sec. 12.14.7.3). Determine out-of-plane forces for design of structural walls and their anchorage, interconnect wall elements, and satisfy requirements for diaphragm crossties, subdiaphragm aspect ratio, and detailing of wood or metal deck diaphragms (Sec. 12.14.7.5 and 12.14.7.6). For various materials, go to these charts: Chart 1.17 Steel Chart 1.18 Concrete Chart 1.19 Composite Masonry Chart 1.20 Wood Chart 1.21

1– 20

Introduction Chart 1.9 Equivalent Lateral Force (ELF) Analysis

Determine fundamental period of vibration for the structure, carefully noting the upper limit placed on periods calculated from analytical models of the structure (Sec. 12.8.2).

Determine the seismic response coefficient, C s, and the total base shear (Sec. 12.8.1).

Consider soil-structure-interaction? (Optional) No

Yes

Go to Chart 1.10 to calculate reduced base shear.

Distribute the base shear to the stories of the structure (Sec. 12.8.3).

To determine the internal forces, perform a linear elastic analysis with an appropriate distribution of forces within stories due to the relative lateral stiffnesses of vertical elements and diaphragms (Sec. 12.8.4). Include inherent torsion (Sec. 12.8.4.1) and amplified accidental torsion (Sec. 12.8.4.2 and 12.8.4.3). Calculate the overturning effects caused by seismic forces (Sec. 12.8.5).

Determine the story drifts. A re-analysis based upon a period larger than the upper limit is permitted for calculating deformations (Sec. 12.8.6). Check the first order deformation for stability. If the stability coefficient, θ , exceeds 0.10, redesign the structure or demonstrate its stability using nonlinear static or nonlinear response history analysis (Provisions Sec. 12.8.7). Go to Chart 1.15.

1– 21

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.10 Soil-Structure Interaction (SSI) Modal Analysis: Follow SSI procedure for ELF analysis (Sec. 19.2) with these modifications (Sec. 19.3).

ELF Analysis: Follow this procedure (Sec. 19.2).

This SSI procedure applies only to the fundamental mode of vibration (Sec. 19.3.1). Therefore, substitute W 1 for W , T 1 for T, V 1 for V , etc. Calculate the foundation stiffnesses K y and K θ (possibly using equations in Provisions Part 2 Sec. C19.2.1.1) at the expected strain level (Table 19.2-1). Calculate effective gravity load, W (as a fraction of W ), effective height, h (as a fraction of h), and effective stiffness, k, of the fixed base structure.

Calculate the effective seismic weight of the fundamental period of vibration, W 1. Use Eq. 19.3-2 to calculate h.

Calculate effective period using Eq. 19.2-3. Read foundation damping factor from Figure 19.2-1. Point bearing piles? or Uniform soft soils over a stiff deposit? No

Yes

Use Eq. 19.2-12 to modify foundation damping factor.

Calculate effective damping using Eq. 19.2-9. Effective damping is not taken less than 5 percent or more than 20 percent of critical. Calculate reduced base shear, V , per Sec. 19.2.1, which cannot be less than 0.7V.

Calculate reduced base shear for the first mode, V 1, per Sec. 19.3.1, which cannot be less than 0.7V 1. Use standard modal combination techniques (Sec. 19.3.3).

Revise deflections to include foundation rotation (Sec. 19.2.3). Return to Chart 1.9.

1– 22

Return to Chart 1.10.

Introduction Chart 1.11 Modal Response Spectrum Analysis

Use linear elastic analysis to determine periods and mode shapes, including enough modes to obtain at least 90 percent mass participation (Sec. 12.9.1).

Consider soil-structure-interaction? (Optional) No

Yes

Go to Chart 1.10 to calculate reduced base shear.

Determine story forces, individual member forces, displacements, and drifts in each mode (Sec. 12.9.2) and combine modal quantities using either the SRSS or the CQC technique* (Sec. 12.9.3).

Where the base shear is less than 85 percent of that computed using Sec. 12.8 with T ≤ CuT a, amplify design forces. Where the base shear is less than 85 percent of that computed using Sec. 12.8 with C s determined using Eq. 12.8-6, amplify drifts (ASCE 7-10 Sec. 12.9.4.2).

To determine the internal forces, perform a linear elastic analysis. Include inherent and accidental torsions. Amplify torsions that are not in the dynamic model (Sec. 12.9.5).

Check the first order deformation for stability (Sec. 12.9.6). If the stability coefficient, θ , exceeds 0.10, redesign the structure or demonstrate its stability using nonlinear static or nonlinear response history analysis (Provisions Sec. 12.8.7).

Go to Chart 1.15.

*As indicated in the text, use of the CQC technique is required where closely spaced periods in the translational and torsional modes will result in cross-correlation of the modes.

1– 23

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.12 Response History Analysis Linear Response History Analysis: Follow this procedure (Sec. 16.1).

Nonlinear Response History Analysis: Follow this procedure (Sec. 16.2).

Model structure as for other analyses (Sec. 16.1.2).

Global modeling requirements are similar to those for other analyses. Modeling of hysteretic behavior of elements must be consistent with laboratory test results and expected material properties (Sec. 16.2.2).

Select and scale ground motions based on spectral values in the period range of interest (Sec. 16.1.3), as follows. For 2-D analysis, the average is not less than the design spectrum. For 3-D analysis, the average of the SRSS spectra computed for each pair of ground motions is not less than the design spectrum. A narrower period range of interest is used for seismically isolated structures and for structures with damping systems (Sec. 17.3.2).

Select and scale ground motion as for linear response history analysis (Sec. 16.2.3). Nonlinear analyses must directly include dead loads and not less than 25 percent of required live loads.

Scale analysis results so that the maximum base shear is consistent with that from the ELF procedure (Sec. 16.1.4).

Analysis results need not be scaled.

Determine response parameters for use in design as follows. If at least seven ground motions are analyzed, may use the average value. If fewer than seven are analyzed, must use the maximum value (Sec. 16.1.4).

As for linear response history analysis, use average or maximum values depending on number of ground motions analyzed (Sec. 16.2.4).

Subsequent steps of the design process change. For instance, typical load combinations and the overstrength factor do not apply (Sec. 16.2.4.1), member deformations must be considered explicitly (Sec. 16.2.4.2), and story drift limits are increased (Sec. 16.2.4.3). The design must be subjected to independent review (Sec. 16.2.5). Go to Chart 1.15.

1– 24

Introduction Chart 1.13 Seismically Isolated Structures

Do the structure and isolation system satisfy the criteria of Sec. 17.4.1? Yes

Yes Opt to perform dynamic analysis? No

Perform ELF analysis (see Chart 1.9) and satisfy the provisions of Sec. 17.5.

Site Class A, B, C, or D? and isolation system meets the criteria of Sec. 17.4.1, item 7? Yes

Opt to perform response-history analysis? No

Yes

Perform response-history analysis as described in Sec. 17.6.

Perform modal analysis (see Chart 2.11) and satisfy the appropri1te provisions of Sec. 17.6.

Satisfy detailed requirements for isolation system (Sec. 17.2.4) and structural system (Sec. 17.2.5). Satisfy requirements for elements of structures and nonstructural components (Sec. 17.2.6). Perform design review (Sec. 17.7). Satisfy testing requirements (Sec. 17.8).

Go to Chart 1.15.

1– 25

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.14 Structures with Damping Systems

Is the structure located at a site with S1 < 0.6?

Yes

Do the structure and damping system satisfy the criteria of Sec. 18.2.4.3? Yes

Opt to perform ELF analysis? Yes

Perform ELF analysis (see Chart 1.9) and satisfy the provisions of Sec. 18.5.

Do the structure and damping system satisfy the criteria of Sec. 18.2.4.2? Yes

Opt to perform response-history analysis? Yes

Perform nonlinear response-history analysis as described in Sec. 18.3.1.

Perform modal analysis (see Chart 1.11) and satisfy the appropriate provisions of Sec. 18.4.

Modify the response of the structure for the effects of the damping system (Sec. 18.6). It is permitted to use the nonlinear static procedure to calculate the effective ductility demand (Sec. 18.3.2).

Determine seismic load conditions and acceptance criteria (Sec. 18.7). Satisfy general requirements for damping system (Sec. 18.2.2.2 and 18.2.5) and structural system (Sec. 18.2.2.1). Perform design review (Sec. 18.8). Satisfy testing requirements (Sec. 18.9).

1– 26

Go to Chart 1.15.

Introduction Chart 1.15 Deformation Requirements

Enter with story drifts from the analysis of seismic force effects. These drifts must include the deflection amplification factor, Cd , given in Table 12.2-1 (Sec. 12.2).

Compare with the limits established in Table 12.12-1, including reduction by the redundancy factor for systems with moment frames in Seismic Design Category D, E, or F (Sec. 12.12.1.1).

Confirm that diaphragm deflections are not excessive (Sec. 12.12.2).

Separations between adjacent buildings (including at seismic joints) must be sufficient to avoid damaging contact (Sec. 12.12.3).

Consider deformation compatibility for Seismic Design Category D, E, or F structural components that are not part of the seismic force-resisting system (Sec. 12.12.4).

Go to Chart 1.16.

1– 27

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.16 Design and Detailing Requirements

Determine diaphragm design forces, including application of the redundancy factor to transfer forces (Sec. 12.10.1.1).

Design diaphragms, including appropriate detailing at openings (Sec. 12.10.1). Provide collector elements to transfer seismic forces (Sec. 12.10.2).

Determine out-of-plane forces for design of structural walls and their anchorage, and interconnect wall elements (Sec. 12.11.1 and 12.11.2).

Seismic Design Category B? No

Yes

Apply overstrength factor to loads used in design of collector elements, splices, and their connections (Sec. 12.10.2.1).

Use larger forces for wall anchorage to flexible diaphragms (Sec. 12.11.2.1).

Continuous diaphragm crossties required. Limit on subdiaphragm aspect ratio. Special detailing for wood diaphragms, metal deck diaphragms, and embedded straps. Resolve anhorage eccentricity and consider pilaster effects (Sec. 12.11.2.2), with some exceptions for light-frame construction (Provisions Sec. 12.11.2.2.1 and 12.11.2.2.3).

1– 28

For nonbuilding structures, go to Chart 1.22. For various materials, go to these charts: Chart 1.17 Steel Chart 1.18 Concrete Chart 1.19 Composite Chart 1.20 Masonry Chart 1.21 Wood

Introduction Chart 1.17 Steel Structures

Seismic Design Category B or C? No

Yes

Using a "structural steel system not specifically detailed for seismic resistance?" Yes

Select an R value from Provisions Table 12.2-1 for the appropriate steel system.

From Provisions Table 12.2-1, R = 3.

The system must be designed and detailed in accordance with AISC 341 for structural steel or AISI S110 for cold-formed steel or AISI Lateral for light-framed, cold-formed steel construction.

Any of the reference documents in Provisions Sec. 14.1.1 may be used for design.

Provisions Sec. 14.1.1 modifies AISC 341. Provisions Sec. 14.1.4.1 modifies AISI S110. Sec. 14.1.6 applies to steel deck diaphragms. Sec. 14.1.7 applies to steel cables. Sec. 14.1.8 sets forth additional detailing requirements for steel piles in Seismic Design Categories D, E, and F.

Go to Chart 1.23.

1– 29

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.18 Concrete Structures

Modifications to ACI 318 to add definitions and requirements for detailed plain concrete structural walls, ordinary precast structural walls, and wall piers. Additional requirements for intermediate precast structural walls. Revision of requirements for ties at anchor bolts and for size limits on anchors. (Provisions Sec. 14.2.2)

Requirements for concrete piles in Seismic Design Category C, D, E, or F (Provisions Sec. 14.2.3).

Acceptance criteria for special precast structural walls based on validation testing (Provisions Sec. 14.2.4).

Go to Chart 1.23.

1– 30

Introduction Chart 1.19 Composite Steel and Concrete Structures

Select an R value from Table 12.2-1 for the appropriate composite system.

The system must be designed and detailed in accordance with the AISC 341 Parts I and II, ACI 318 excluding Ch. 22, and AISC 360.

Go to Chart 1.23.

1– 31

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 2.20 Masonry Structures

Must construct in accordance with TMS 402 and use materials in conformance to TMS 602.

Clarifications for classification of shear walls (Provisions Sec. 14.4.3) and anchorage forces (Provisions Sec. 14.4.4).

Modifications to TMS 402 for separation joints, flanged shear walls, stress increase, reinforcement details, walls with high axial stress, coupling beams, deep flexural members, shear keys, anchor bolts, and corrugated sheet metal anchors (Provisions Sec. 14.4.5 through 14.4.8).

Modifications to TMS 602 concerning grout placement and control of shrinkage (Provisions Sec. 14.4.9).

Go to Chart 1.23.

1– 32

Introduction Chart 1.21 Wood Structures

Must satisfy quality, testing, design, and construction requirements of AF&PA NDS and AF&PA SDPWS (Sec. 14.5.1).

Additional requirements for end bearing of columns and posts, continuity of wall top plates, and detailing of walls at offsets (Sec. 14.5.2).

Modification of AF&PA SDPWS for calculation of shear values for walls with multiple shear panels applied to the same or opposite faces (Sec. 14.5.3).

Go to Chart 1.23.

1– 33

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.22 Non-building Structures

Yes

Is weight of nonbuilding structure less than 25 percent of combined weight of nonbuilding structure and supporting structure? Yes

Nonbuilding structure supported by another structure? No

Classify system, determine importance factor, and calculate design forces per Sec. 15.4.

Treat nonbuilding structure as nonstructural component (using Chart 2.24) and design supporting structure as a building or nonbuilding structure (Sec. 15.3.1).

Yes

Dynamic response similar to that of building structures?

Structures Similar to Buildings Specific provisions for: pipe racks; steel storage racks; electrical power generating facilities; structural towers for tanks and vessels; and piers and wharves (Sec. 15.5).

Structures Not Similar to Buildings Specific provisions for: earth-retaining structures; stacks and chimneys; amusem*nt structures; special hydraulic structures; and secondary containment systems (Sec. 15.6); and for tanks and vessels (Sec. 15.7).

Go to Chart 1.23.

1– 34

Introduction Chart 1.23 Foundations

Satisy the design basis (Sec. 12.1.5).

May opt to model foundation load-deformation characteristics if bounding analyses for foundation stiffness are performed (Sec. 12.13.3).

May reduce foundation overturning effects at soil-foundation interface for ELF or modal analysis (Sec. 12.13.4).

Seismic Design Category B? No

Yes

Requirements for: geotechnical investigation report (Sec. 11.8.2); pole type structures; ties between piles or piers; and pile anchorage (Sec. 12.13.5).

Seismic Design Category C? No

Yes

Requirements for: additional geotechnical investigation report items (Sec. 11.8.3); ties between spread footings; and other pile design requirements for deformations due to both free-field soil strains and structure response, batter piles, pile anchorage, splices, pile flexibility, and pile group effects (Sec. 12.13.6).

Go to Chart 1.24.

1– 35

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chart 1.24 Nonstructural Components

Assign component importance factors (Sec. 13.1.3). Note component exemptions in Sec. 13.1.4. Design requirements may be satisfied by project-specific design and documentation or by manufacturer certification (Sec. 13.2.1). Option to satisfy design requirements on the basis of experience data (Sec. 13.2.6). Special certications required for designated seismic systems in Seismic Design Category C, D, E, or F (Sec. 13.2.2). Must consider both flexibility and strength for components and support structures (Sec. 13.2.4). Avoid consequential damage by considering functional and physical interrelationship of components (Sec. 13.2.3). Determine the periods of mechanical and electrical components (Sec. 13.6.2). Select a p and R p values from Tables 13.5-1 and 13.6-1. Calculate seismic design forces per Sec. 13.3.1, including vertical load effects. (Don't forget to consider nonseismic horizontal loads.) Compute seismic relative displacements (Sec. 13.3.2) and accommodate such displacements. Components require positive attachment to the structure without reliance on gravity-induced friction (Sec. 13.4).

Architectural Components Specific provisions for: exterior nonstructural wall elements and connections; out-of-plane bending; suspended ceilings; access floors; partitions; and glass (Sec. 13.5).

Mechanical and Electrical Components Specific provisions for: mechanical components; electrical components; component supports; utility and service lines; storage tanks; HVAC ductwork; piping systems; boilers and pressure vessels; and elevator and escalator design (Sec. 13.6).

Satisfy requirements for construction documents (Sec. 13.2.7). Go to Chart 1.25.

1– 36

Introduction Chart 1.25 Quality Assurance

Seismic-force-resisting system assigned to Seismic Design Category C, D, E, or F? or Designated seismic system in structure assigned to Seismic Design Category D, E, or F? Yes

Satisfy exceptions in Sec. 11A.1.1? Yes

Registered design professional must prepare QA plan and affected contractors must submit statements of responsibility (Sec. 11A.1.2).

QA plan not required.

Special inspection is required for some aspects of the following: deep foundations, reinforcing steel, concrete, masonry, steel connections, wood connections, cold-formed steel connections, selected architectural components, selected mechanical and electrical components, isolator units, and energy dissipation devices (Sec. 11A.1.3).

Special testing is required for some aspects of the following: reinforcing and prestressing steel, welded steel, mechanical and electrical components and mounting systems (Sec. 3.4 [2.4]), and seismic isolation systems (Sec. 11A.2).

Reporting and compliance procedures are given (Sec. 11A.4).

Satisfy testing and inspection requirements in the reference standards (Ch. 13 and 14). Seismic Design Category C? No

Yes

Occupancy Category III or IV? or Height > 75 ft? or Seismic Design Category E or F and more than two stories? Yes

Registered design professional must perform structural observations (Sec. 11A.3).

Done.

1– 37

FEMA P-751, NEHRP Recommended Provisions: Design Examples

American Society of Civil Engineers, 1907, The Effects of the San Francisco Earthquake of April 18, 1906., New York, NY. American Society of Civil Engineers, 1951, Proceedings-Separate No. 66., New York, NY. American Society of Civil Engineers, 2010, ASCE 7-10: Minimum Design Loads for Buildings and Other Structures, Reston, VA. Applied Technology Council, 1978, ATC 3-06: Tentative Provisions for the Development of Seismic Regulations for Buildings, Redwood City, California. Building Seismic Safety Council, 2009, 2009 NEHRP Recommended Seismic Provisions for Buildings and Other Structures, prepared for the Federal Emergency Management Agency, Washington, DC. International Conference of Building Officials, 1927, Uniform Building Code. Whittier, CA. Structural Engineers Association of California, SEAOC Blue Book: Seismic Design Recommendations, Sacramento, CA.

1– 38

2 Fundamentals James Robert Harris, P.E., PhD Contents 2.1

EARTHQUAKE PHENOMENA ........................................................................ 3

2.2

STRUCTURAL RESPONSE TO GROUND SHAKING ................................... 5

2.2.1

Response Spectra .......................................................................................... 5

2.2.2

Inelastic Response....................................................................................... 11

2.2.3

Building Materials ...................................................................................... 14

2.2.4

Building Systems ........................................................................................ 16

2.2.5

Supplementary Elements Added to Improve Structural Performance ........ 17

2.3

ENGINEERING PHILOSOPHY ....................................................................... 18

2.4

STRUCTURAL ANALYSIS ............................................................................. 19

2.5

NONSTRUCTURAL ELEMENTS OF BUILDINGS ...................................... 22

2.6

QUALITY ASSURANCE ................................................................................. 23

FEMA P-751, NEHRP Recommended Provisions: Design Examples

In introducing their classic text, Fundamentals of Earthquake Engineering, Newmark and Rosenblueth (1971) comment: In dealing with earthquakes, we must contend with appreciable probabilities that failure will occur in the near future. Otherwise, all the wealth of the world would prove insufficient to fill our needs: the most modest structures would be fortresses. We must also face uncertainty on a large scale, for it is our task to design engineering systems – about whose pertinent properties we know little – to resist future earthquakes and tidal waves – about whose characteristics we know even less. . . . In a way, earthquake engineering is a cartoon. . . . Earthquake effects on structures systematically bring out the mistakes made in design and construction, even the minutest mistakes. Several points essential to an understanding of the theories and practices of earthquakeresistant design bear restating: 1. Ordinarily, a large earthquake produces the most severe loading that a building is expected to survive. The probability that failure will occur is very real and is greater than for other loading phenomena. Also, in the case of earthquakes, the definition of failure is altered to permit certain types of behavior and damage that are considered unacceptable in relation to the effects of other phenomena. 2. The levels of uncertainty are much greater than those encountered in the design of structures to resist other phenomena. This is in spite of the tremendous strides made since the Federal government began strongly supporting research in earthquake engineering and seismology following the 1964 Prince William Sound and 1971 San Fernando earthquakes. The high uncertainty applies both to knowledge of the loading function and to the resistance properties of the materials, members and systems. 3. The details of construction are very important because flaws of no apparent consequence often will cause systematic and unacceptable damage simply because the earthquake loading is so severe and an extended range of behavior is permitted. The remainder of this chapter is devoted to a very abbreviated discussion of fundamentals that reflect the concepts on which earthquake-resistant design are based. When appropriate, important aspects of the NEHRP Recommended Seismic Provisions for New Buildings and Other Structures are mentioned and reference is made to particularly relevant portions of that document or the standards that are incorporated by reference. The 2009 Provisions is composed of three parts: 1) “Provisions”, 2) “Commentary on ASCE/SEI 7-2005” and 3) “Resource Papers on Special Topics in Seismic Design”. Part 1 states the intent and then cites ASCE/SEI 7-2005 Minimum Design Loads for Buildings and Other Structures as the primary reference. The remainder of Part 1 contains recommended changes to update ASCE/SEI 7-2005; the recommended changes include

2-2

Chapter 2: Fundamentals

commentary on each specific recommendation. All three parts are referred to herein as the Provisions, but where pertinent the specific part is referenced and ASCE/SEI 7-2005 is referred to as the Standard. ASCE/SEI 7-2005 itself refers to several other standards for the seismic design of structures composed of specific materials and those standards are essential elements to achieve the intent of the Provisions.

2.1 EARTHQUAKE PHENOMENA According to the most widely held scientific belief, most earthquakes occur when two segments of the earth’s crust suddenly move in relation to one another. The surface along which movement occurs is known as a fault. The sudden movement releases strain energy and causes seismic waves to propagate through the crust surrounding the fault. These waves cause the surface of the ground to shake violently, and it is this ground shaking that is the principal concern of structural engineering to resist earthquakes. Earthquakes have many effects in addition to ground shaking. For various reasons, the other effects generally are not major considerations in the design of buildings and similar structures. For example, seismic sea waves or tsunamis can cause very forceful flood waves in coastal regions, and seiches (long-period sloshing) in lakes and inland seas can have similar effects along shorelines. These are outside the scope of the Provisions. This is not to say, however, that they should not be considered during site exploration and analysis. Designing structures to resist such hydrodynamic forces is a very specialized topic, and it is common to avoid constructing buildings and similar structures where such phenomena are likely to occur. Long-period sloshing of the liquid contents of tanks is addressed by the Provisions. Abrupt ground displacements occur where a fault intersects the ground surface. (This commonly occurs in California earthquakes but apparently did not occur in the historic Charleston, South Carolina, earthquake or the very large New Madrid, Missouri, earthquakes of the nineteenth century.) Mass soil failures such as landslides, liquefaction and gross settlement are the result of ground shaking on susceptible soil formations. Once again, design for such events is specialized, and it is common to locate structures so that mass soil failures and fault breakage are of no major consequence to their performance. Modification of soil properties to protect against liquefaction is one important exception; large portions of a few metropolitan areas with the potential for significant ground shaking are susceptible to liquefaction. Lifelines that cross faults require special design beyond the scope of the Provisions. The structural loads specified in the Provisions are based solely on ground shaking; they do not provide for ground failure. Resource Paper 12 (“Evaluation of Geologic Hazards and Determination of Seismic Lateral Earth Pressures”) in Part 3 of the Provisions includes a description of current procedures for prediction of seismic-induced slope instability, liquefaction and surface fault rupture. Nearly all large earthquakes are tectonic in origin – that is, they are associated with movements of and strains in large segments of the earth’s crust, called plates, and virtually all such earthquakes occur at or near the boundaries of these plates. This is the 2-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples

case with earthquakes in the far western portion of the United States where two very large plates, the North American continent and the Pacific basin, come together. In the central and eastern United States, however, earthquakes are not associated with such a plate boundary, and their causes are not as completely understood. This factor, combined with the smaller amount of data about central and eastern earthquakes (because of their infrequency), means that the uncertainty associated with earthquake loadings is higher in the central and eastern portions of the nation than in the West. Even in the West, the uncertainty (when considered as a fraction of the predicted level) about the hazard level is probably greater in areas where the mapped hazard is low than in areas where the mapped hazard is high. The amplitude of earthquake ground shaking diminishes with distance from the source, and the rate of attenuation is less for lower frequencies of motion than for higher frequencies. This effect is captured, to an extent, by the fact that the Provisions use three parameters to define the hazard of seismic ground shaking for structures. Two are based on statistical analysis of the database of seismological information: the SS values are pertinent for higher frequency motion, and the S1 values are pertinent for other middle frequencies. The third value, TL, defines an important transition point for long period (low frequency) behavior; it is not based upon as robust an analysis as the other two parameters. Two basic data sources are used in establishing the likelihood of earthquake ground shaking, or seismicity, at a given location. The first is the historical record of earthquake effects and the second is the geological record of earthquake effects. Given the infrequency of major earthquakes, there is no place in the United States where the historical record is long enough to be used as a reliable basis for earthquake prediction – certainly not as reliable as with other phenomena such as wind and snow. Even on the eastern seaboard, the historical record is too short to justify sole reliance on the historical record. Thus, the geological record is essential. Such data require very careful interpretation, but they are used widely to improve knowledge of seismicity. Geological data have been developed for many locations as part of the nuclear power plant design process. On the whole, there is more geological data available for the far western United States than for other regions of the country. Both sets of data have been taken into account in the Provisions seismic ground shaking maps. The Commentary provides a more thorough discussion of the development of the maps, their probabilistic basis, the necessarily crude lumping of parameters and other related issues. Prior to its 1997 edition, the basis of the Provisions was to “provide life safety at the design earthquake motion,” which was defined as having a 10 percent probability of being exceeded in a 50-year reference period. As of the 1997 edition, the basis became to “avoid structural collapse at the maximum considered earthquake (MCE) ground motion,” which is defined as having a 2 percent probability of being exceeded in a 50year reference period. In the 2009 edition of the Provisions the design basis has been refined to target a 1% probability of structural collapse for ordinary buildings in a 50 year period. The MCE ground motion has been adjusted to deliver this level of risk combined with a 10% probability of collapse should the MCE ground motion occur. This new

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approach incorporates a fuller consideration of the nature of the seismic hazard at a location than was possible with the earlier definitions of ground shaking hazard, which were tied to a single level of probability of ground shaking occurrence.

2.2 STRUCTURAL RESPONSE TO GROUND SHAKING The first important difference between structural response to an earthquake and response to most other loadings is that the earthquake response is dynamic, not static. For most structures, even the response to wind is essentially static. Forces within the structure are due almost entirely to the pressure loading rather than the acceleration of the mass of the structure. But with earthquake ground shaking, the aboveground portion of a structure is not subjected to any applied force. The stresses and strains within the superstructure are created entirely by its dynamic response to the movement of its base, the ground. Even though the most used design procedure resorts to the use of a concept called the equivalent static force for actual calculations, some knowledge of the theory of vibrations of structures is essential. 2.2.1 Response Spectra Figure 2.2-1 shows accelerograms, records of the acceleration at one point along one axis, for several representative earthquakes. Note the erratic nature of the ground shaking and the different characteristics of the different accelerograms. Precise analysis of the elastic response of an ideal structure to such a pattern of ground motion is possible; however, it is not commonly done for ordinary structures. The increasing power and declining cost of computational aids are making such analyses more common but, at this time, only a small minority of structures designed across the country, are analyzed for specific response to a specific ground motion.

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Northridge (Sylmar 360°) 1994 San Fernando (Pacoima Dam) 1971

Tabas 1978

Kern Taft 1952

Morgan Hill (Gilroy) 1984

El Centro 1940

Kobe 1995

Imperial 6 (Hudson) 1979

Northridge (Sylmar 90°) 1994 Loma Prieta (Oakland Wharf) 1989

San Fernando (Orion Blvd.) 1971

North Palm Springs 1986

Landers (Joshua Tree) 1992

Mexico City 1985

Figure 2.2-1 Earthquake Ground Acceleration in Epicentral Regions (all accelerograms are plotted to the same scale for time and acceleration – the vertical axis is % gravity). Great earthquakes extend for much longer periods of time.

Figure 2.2-2 shows further detail developed from an accelerogram. Part (a) shows the ground acceleration along with the ground velocity and ground displacement derived from it. Part (b) shows the acceleration, velocity and displacement for the same event at 2-6

Chapter 2: Fundamentals

the roof of the building located where the ground motion was recorded. Note that the peak values are larger in the diagrams of Figure 2.2-2(b) (the vertical scales are essentially the same). This increase in response of the structure at the roof level over the motion of the ground itself is known as dynamic amplification. It depends very much on the vibrational characteristics of the structure and the characteristic frequencies of the ground shaking at the site.

Figure 2.2-2 Holiday Inn Ground and Building Roof Motion During the M6.4 1971 San Fernando Earthquake: (a) north-south ground acceleration, velocity and displacement and (b) north-south roof acceleration, velocity and displacement (Housner and Jennings, 1982). The Holiday Inn, a 7-story, reinforced concrete frame building, was approximately 5 miles from the closest portion of the causative fault. The recorded building motions enabled an analysis to be made of the stresses and strains in the structure during the earthquake.

In design, the response of a specific structure to an earthquake is ordinarily estimated from a design response spectrum such as is specified in the Provisions. The first step in creating a design response spectrum is to determine the maximum response of a given structure to a specific ground motion (see Figure 2.2-2). The underlying theory is based entirely on the response of a single-degree-of-freedom oscillator such as a simple onestory frame with the mass concentrated at the roof. The vibrational characteristics of

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such a simple oscillator may be reduced to two: the natural period1 and the amount of damping. By recalculating the record of response versus time to a specific ground motion for a wide range of natural periods and for each of a set of common amounts of damping, the family of response spectra for one ground motion may be determined. It is simply the plot of the maximum value of response for each combination of period and damping. Figure 2.2-3 shows such a result for the ground motion of Figure 2.2-2(a) and illustrates that the erratic nature of ground shaking leads to a response that is very erratic in that a slight change in the natural period of vibration brings about a very large change in response. The figure also illustrates the significance of damping. Different earthquake ground motions lead to response spectra with peaks and valleys at different points with respect to the natural period. Thus, computing response spectra for several different ground motions and then averaging them, based on some normalization for different amplitudes of shaking, will lead to a smoother set of spectra. Such smoothed spectra are an important step in developing a design spectrum.

Much of the literature on dynamic response is written in terms of frequency rather than period. The cyclic frequency (cycles per second, or Hz) is the inverse of period. Mathematically it is often convenient to use the angular frequency expressed as radians per second rather than Hz. The conventional symbols used in earthquake engineering for these quantities are T for period (seconds per cycle), f for cyclic frequency (Hz) and ω for angular frequency (radians per second). The word frequency is often used with no modifier; be careful with the units.

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Spectral Acceleration, Sa (g)

1.5

1 0%

0.5

5% 20% 10%

3 Period, T (s)

Figure 2.2-3 Response spectrum of north-south ground acceleration (0%, 2%, 5%, 10%, 20% of critical damping) recorded at the Holiday Inn, approximately 5 miles from the causative fault in the 1971 San Fernando earthquake.

Figure 2.2-4 is an example of an averaged spectrum. Note that acceleration, velocity, or displacement may be obtained from Figure 2.2-3 or 1.2-4 for a structure with known period and damping.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Acceleration, Sa (g)

mean plus one standard deviation

mean

smoothed elastic spectrum 0

Period, T (s) Figure 2.2-4 Averaged Spectrum(In this case, the statistics are for seven ground motions representative of the de-aggregated hazard at a particular site.)

Prior to the 1997 edition of the Provisions, the maps that characterized the ground shaking hazard were plotted in terms of peak ground acceleration (at period, T, = 0), and design response spectra were created using expressions that amplified (or de-amplified) the ground acceleration as a function of period and damping. With the introduction of the new maps in the 1997 edition, this procedure changed. Now the maps present spectral response accelerations at two periods of vibration, 0.2 and 1.0 second, and the design response spectrum is computed more directly, as implied by the smooth line in Figure 2.2-4. This has removed a portion of the uncertainty in predicting response accelerations. Few structures are so simple as to actually vibrate as a single-degree-of-freedom system. The principles of dynamic modal analysis, however, allow a reasonable approximation of the maximum response of a multi-degree-of-freedom oscillator, such as a multistory building, if many specific conditions are met. The procedure involves dividing the total response into a number of natural modes, modeling each mode as an equivalent singledegree-of-freedom oscillator, determining the maximum response for each mode from a single-degree-of-freedom response spectrum and then estimating the maximum total response by statistically summing the responses of the individual modes. The Provisions does not require consideration of all possible modes of vibration for most buildings because the contribution of the higher modes (lower periods) to the total response is relatively minor. The soil at a site has a significant effect on the characteristics of the ground motion and, therefore, on the structure’s response. Especially at low amplitudes of motion and at longer periods of vibration, soft soils amplify the motion at the surface with respect to 2-10

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bedrock motions. This amplification is diminished somewhat, especially at shorter periods as the amplitude of basic ground motion increases, due to yielding in the soil. The Provisions accounts for this effect by providing amplifiers that are to be applied to the 0.2 and 1.0 second spectral accelerations for various classes of soils. (The ground motion maps in the Provisions are drawn for sites on rock.) Thus, very different design response spectra are specified depending on the type of soil(s) beneath the structure. The Commentary (Part 2) contains a thorough explanation of this feature. 2.2.2 Inelastic Response The preceding discussion assumes elastic behavior of the structure. The principal extension beyond ordinary behavior referenced at the beginning of this chapter is that structures are permitted to strain beyond the elastic limit in responding to earthquake ground shaking. This is dramatically different from the case of design for other types of loads in which stresses and therefore strains, are not permitted to approach the elastic limit. The reason is economic. Figure 2.2-3 shows a peak acceleration response of about 1.0 g (the acceleration due to gravity) for a structure with moderately low damping – for only a moderately large earthquake! Even structures that resist lateral forces well will have a static lateral strength of only 20 to 40 percent of gravity. The dynamic nature of earthquake ground shaking means that a large portion of the shaking energy can be dissipated by inelastic deformations if the structure is ductile and some damage to the structure is accepted. Figure 2.2-5 will be used to illustrate the significant difference between wind and seismic effects. Figure 2.2-5(1) would represent a cantilever beam if the load W were small and a column if W were large. Wind pressures create a force on the structure, which in turn produces a displacement. The force is the independent variable and the displacement is the dependent result. Earthquake ground motion creates displacement between the base and the mass, which in turn produces an internal force. The displacement is the independent variable, and the force is the dependent result. Two graphs are plotted with the independent variables on the horizontal axis and the dependent response on the vertical axis. Thus, part (b) of the figure is characteristic of the response to forces such as wind pressure (or gravity weight), while part (c) is characteristic of induced displacements such as earthquake ground shaking (or foundation settlement). Note that the ultimate resistance (Hu) in a force-controlled system is marginally larger than the yield resistance (Hy), while the ultimate displacement (Δu) in a displacementcontrolled system is much larger than the yield displacement (Δy). The point being made with the figures is that ductile structures have the ability to resist displacements much larger than those that first cause yield. The degree to which a member or structure may deform beyond the elastic limit is referred to as ductility. Different materials and different arrangements of structural members lead to different ductilities. Response spectra may be calculated for oscillators with different levels of ductility. At the risk of gross oversimplification, the following conclusions may be drawn: 2-11

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1. For structures with very long natural periods, the acceleration response is reduced by a factor equivalent to the ductility ratio (the ratio of maximum usable displacement to effective yield displacement – note that this is displacement and not strain). 2. For structures with very short natural periods, the acceleration response of the ductile structure is essentially the same as that of the elastic structure, but the displacement is increased. 3. For intermediate periods (which applies to nearly all buildings), the acceleration response is reduced, but the displacement response is generally about the same for the ductile structure as for the elastic structure strong enough to respond without yielding. W

Δ H

Force control

Displacement control

HY HU

(a)

ΔY

ΔU

HU /H Y = 1

Δ U /Δ Y >> 1

(b)

(c)

Figure 2.2-5 Force Controlled Resistance Versus Displacement Controlled Resistance (after Housner and Jennings 1982). In part (b) the force H is the independent variable. As H is increased, the displacement increases until the yield point stress is reached. If H is given an additional increment (about 15 percent) a plastic hinge forms, giving large displacements. For this kind of system, the force producing the yield point stress is close to the force producing collapse. The ductility does not produce a large increase in load capacity, although in highly redundant structures the increase is more than illustrated for this very simple structure. In part (c) the displacement is the independent variable. As the displacement is increased, the base moment increases until the yield point is reached. As the displacement increases still more, the resistance (H) increases only a small amount. For a highly ductile element, the displacement can be increased 10 to 20 times the yield point displacement before the system collapses under the weight W. (As W increases, this ductility is decreased dramatically.) During an earthquake, the oscillator is excited into vibrations by the ground motion and it behaves essentially as a displacement-controlled system and can survive displacements much beyond the yield point. This explains why ductile structures can survive ground shaking that produces displacements much greater than yield point displacement.

Inelastic response is quite complex. Earthquake ground motions involve a significant number of reversals and repetitions of the strains. Therefore, observation of the inelastic 2-12

Chapter 2: Fundamentals

properties of a material, member, or system under a monotonically increasing load until failure can be very misleading. Cycling the deformation can cause degradation of strength, stiffness, or both. Systems that have a proven capacity to maintain a stable resistance to a large number of cycles of inelastic deformation are allowed to exercise a greater portion of their ultimate ductility in designing for earthquake resistance. This property is often referred to as toughness, but this is not the same as the classic definition used in mechanics of materials. Most structures are designed for seismic response using a linear elastic analysis with the strength of the structure limited by the strength at its critical location. Most structures possess enough complexity so that the peak strength of a ductile structure is not accurately captured by such an analysis. Figure 2.2-6 shows the load versus displacement relation for a simple frame. Yield must develop at four locations before the peak resistance is achieved. The margin from the first yield to the peak strength is referred to as overstrength, and it plays a significant role in resisting strong ground motion. Note that a few key design standards (for example, American Concrete Institute (ACI) 318 for the design of concrete structures) do allow for some redistribution of internal forces from the critical locations based upon ductility; however, the redistributions allowed therein are minor compared to what occurs in response to strong ground motion.

160

HU 5 H

120

Yield

100 80

Maximum Resistance

140

Overstrength

60 40 20 2

0 0

(a) Structures

0.5

1.5

2.5

3.5

4.5

(b) H - δ curve

Figure 2.2-6 Initial Yield Load and Failure for a Ductile Portal Frame (The margin from initial yield to failure (mechanism in this case) is known as overstrength.)

To summarize, the characteristics important in determining a building’s seismic response are natural period, damping, ductility, stability of resistance under repeated reversals of inelastic deformation and overstrength. The natural frequency is dependent on the mass and stiffness of the building. Using the Provisions the designer calculates, or at least approximates, the natural period of vibration (the inverse of natural frequency). Damping, ductility, toughness and overstrength depend primarily on the type of building system, but not the building’s size or shape. Three coefficients – R, Cd and Ω0 – are

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

provided to encompass damping, ductility, stability of resistance and overstrength. R is intended to be a conservatively low estimate of the reduction of acceleration response in a ductile system from that for an elastic oscillator with a certain level of damping. It is used to compute a required strength. Computations of displacement based upon ground motion reduced by the factor R will underestimate the actual displacements. Cd is intended to be a reasonable mean for the amplification necessary to convert the elastic displacement response computed for the reduced ground motion to actual displacements. Ω0 is intended to deliver a reasonably high estimate of the peak force that would develop in the structure. Sets of R, Cd and Ω0 are specified in the Provisions for the most common structural materials and systems. 2.2.3 Building Materials The following brief comments about building materials and systems are included as general guidelines only, not for specific application. 2.2.3.1 Wood Timber structures nearly always resist earthquakes very well, even though wood is a brittle material as far as tension and flexure are concerned. It has some ductility in compression (generally monotonic), and its strength increases significantly for brief loadings, such as earthquake. Conventional timber structures (plywood, oriented strand board, or board sheathing on wood framing) possess much more ductility than the basic material primarily because the nails, and other steel connection devices yield, and the wood compresses against the connector. These structures also possess a much higher degree of damping than the damping that is assumed in developing the basic design spectrum. Much of this damping is caused by slip at the connections. The increased strength, connection ductility, and high damping combine to give timber structures a large reduction from elastic response to design level. This large reduction should not be used if the strength of the structure is actually controlled by bending or tension of the gross timber cross sections. The large reduction in acceleration combined with the light weight timber structures make them very efficient with regard to earthquake ground shaking when they are properly connected. This is confirmed by their generally good performance in earthquakes. Capacities and design and detailing rules for wood elements of seismic force-resisting systems are now found in the Special Design Provisions for Wind and Seismic supplement to the National Design Specification for Wood Construction. 2.2.3.2 Steel Steel is the most ductile of the common building materials. The moderate-to-large reduction from elastic response to design response allowed for steel structures is primarily a reflection of this ductility and the stability of the resistance of steel. Members subject to buckling (such as bracing) and connections subject to brittle fracture (such as

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Chapter 2: Fundamentals

partial penetration welds under tension) are much less ductile and are addressed in the Provisions in various ways. Defects, such as stress concentrations and flaws in welds, also affect earthquake resistance as demonstrated in the Northridge earthquake. The basic and applied research program that grew out of that experience has greatly increased knowledge of how to avoid low ductility details in steel construction. Capacities and design and detailing rules for seismic design of hot-rolled structural steel are found in the Seismic Provisions for Structural Steel Buildings (American Institute of Steel Construction (AISC) Standard 341) and similar provisions for cold-formed steel are found in the “Lateral Design” supplement to the North American Specification for the Design of Cold-Formed Steel Structures published by AISI (American Iron and Steel Institute). 2.2.3.3 Reinforced Concrete Reinforced concrete achieves ductility through careful limits on steel in tension and concrete in compression. Reinforced concrete beams with common proportions can possess ductility under monotonic loading even greater than common steel beams; in which local buckling is usually a limiting factor. Providing stability of the resistance to reversed inelastic strains, however, requires special detailing. Thus, there is a wide range of reduction factors from elastic response to design response depending on the detailing for stable and assured resistance. The Commentary and the commentary with the ACI 318 standard Building Code Requirements for Structural Concrete explain how to design to control premature shear failures in members and joints, buckling of compression bars, concrete compression failures (through confinement with transverse reinforcement), the sequence of plastification and other factors, which can lead to large reductions from the elastic response. 2.2.3.4 Masonry Masonry is a more complex material than those mentioned above and less is known about its inelastic response characteristics. For certain types of members (such as pure cantilever shear walls), reinforced masonry behaves in a fashion similar to reinforced concrete. The nature of masonry construction, however, makes it difficult, if not impossible, to take some of the steps (e.g., confinement of compression members) used with reinforced concrete to increase ductility, and stability. Further, the discrete differences between mortar, grout and the masonry unit create additional failure phenomena. Thus, the response reduction factors for design of reinforced masonry are not quite as large as those for reinforced concrete. Unreinforced masonry possesses little ductility or stability, except for rocking of masonry piers on a firm base and very little reduction from the elastic response is permitted. Capacities and design and detailing rules for seismic design of masonry elements are contained within The Masonry Society (TMS) 402 standard Building Code Requirements for Masonry Structures.

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2.2.3.5 Precast Concrete Precast concrete obviously can behave quite similarly to reinforced concrete but it also can behave quite differently. The connections between pieces of precast concrete commonly are not as strong as the members being connected. Clever arrangements of connections can create systems in which yielding under earthquake motions occurs away from the connections, in which case the similarity to reinforced concrete is very real. Some carefully detailed connections also can mimic the behavior of reinforced concrete. Many common connection schemes, however, will not do so. Successful performance of such systems requires that the connections perform in a ductile manner. This requires some extra effort in design but it can deliver successful performance. As a point of reference, the most common wood seismic resisting systems perform well yet have connections (nails) that are significantly weaker than the connected elements (structural wood panels). The Provisions includes guidance, some only for trial use and comment (Part 3), for seismic design of precast structures. ACI 318 also includes provisions for precast concrete elements resisting seismic forces, and there are also supplemental ACI standards for specialized seismic force-resisting systems of precast concrete. 2.2.3.6 Composite Steel and Concrete Reinforced concrete is a composite material. In the context of the Provisions, composite is a term reserved for structures with elements consisting of structural steel and reinforced concrete acting in a composite manner. These structures generally are an attempt to combine the most beneficial aspects of each material. Capacities and design and detailing rules are found in the Seismic Provisions for Structural Steel Buildings (AISC Standard 341). 2.2.4 Building Systems Three basic lateral-load-resisting elements – walls, braced frames and unbraced frames (moment resisting frames) – are used to build a classification of structural types in the Provisions. Unbraced frames generally are allowed greater reductions from elastic response than walls and braced frames. In part, this is because frames are more redundant, having several different locations with approximately the same stress levels and common beam-column joints frequently exhibit an ability to maintain a stable response through many cycles of reversed inelastic deformations. Systems using connection details that have not exhibited good ductility and toughness, such as unconfined concrete and the welded steel joint used before the Northridge earthquake, are penalized: the R factors permit less reduction from elastic response. Connection details often make development of ductility difficult in braced frames, and buckling of compression members also limits their inelastic response. The actual failure of steel bracing often occurs because local buckling associated with overall member

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Chapter 2: Fundamentals

buckling frequently leads to locally high strains that then lead to brittle fracture when the member subsequently approaches yield in tension. Eccentrically braced steel frames and new proportioning and detailing rules for concentrically braced frames have been developed to overcome these shortcomings. But the newer and potentially more popular bracing system is the buckling-restrained braced frame. This new system has the advantages of a special steel concentrically braced frame, but with performance that is superior as brace buckling is controlled to preserve ductility. Design provisions appear in the Seismic Provisions for Structural Steel Buildings (AISC Standard 341). Shear walls that do not bear gravity load are allowed a greater reduction than walls that are load bearing. Redundancy is one reason; another is that axial compression generally reduces the flexural ductility of concrete and masonry elements (although small amounts of axial compression usually improve the performance of materials weak in tension, such as masonry and concrete). The 2010 earthquake in Chile is expected to lead to improvements in understanding and design of reinforced concrete shear wall systems because of the large number of significant concrete shear wall buildings subjected to strong shaking in that earthquake. Systems that combine different types of elements are generally allowed greater reductions from elastic response because of redundancy. Redundancy is frequently cited as a desirable attribute for seismic resistance. A quantitative measure of redundancy is included in the Provisions in an attempt to prevent use of large reductions from elastic response in structures that actually possess very little redundancy. Only two values of the redundancy factor, ρ, are defined: 1.0 and 1.3. The penalty factor of 1.3 is placed upon systems that do not possess some elementary measures of redundancy based on explicit consideration of the consequence of failure of a single element of the seismic force-resisting system. A simple, deemed-to-comply exception is provided for certain structures. 2.2.5 Supplementary Elements Added to Improve Structural Performance The Standard includes provisions for the design of two systems to significantly alter the response of the structure to ground shaking. Both have specialized rules for response analysis and design detailing. Seismic isolation involves placement of specialized bearings with low lateral stiffness and large lateral displacement capacity between the foundation and the superstructure. It is used to substantially increase the natural period of vibration and thereby decrease the acceleration response of the structures. (Recall the shape of the response spectrum in Figure 2.2-4; the acceleration response beyond a threshold period is roughly proportional to the inverse of the period). Seismic isolation is becoming increasingly common for structures in which superior performance is necessary, such as major hospitals and emergency response centers. Such structures are frequently designed with a stiff superstructure to control story drift, and isolation makes it feasible to design such structures for lower total lateral force. The design of such systems requires a conservative estimate of the likely deformation of the isolator. The early provisions for

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

that factor were a precursor of the changes in ground motion mapping implemented in the 1997 Provisions. Added damping involves placement of specialized energy dissipation devices within stories of the structure. The devices can be similar to a large shock absorber, but other technologies are also available. Added damping is used to reduce the structural response, and the effectiveness of increased damping can be seen in Figure 2.2-3. It is possible to reach effective damping levels of 20 to 30 percent of critical damping, which can reduce response by factors of 2 or 3. The damping does not have to be added in all stories; in fact, it is common to add damping at the isolator level of seismically isolated buildings. Isolation and damping elements require extra procedures for analysis of seismic response. Both also require considerations beyond common building construction to assure quality and durability.

2.3 ENGINEERING PHILOSOPHY The Commentary, under “Intent,” states: ”The primary intent of the NEHRP Recommended Seismic Provisions for normal buildings and structures is to prevent serious injury and life loss caused by damage from earthquake ground shaking. Most earthquake injuries and deaths are caused by structural collapse. Thus, the main thrust of the Provisions is to prevent collapse for very rare and intense ground motion, termed the maximum considered earthquake (MCE) motion…Falling exterior walls and cladding, and falling ceilings, light fixtures, pipes, equipment and other nonstructural components also cause deaths and injuries.” The Provisions states: “The degree to which these goals can be achieved depends on a number of factors including structural framing type, building configuration, materials, as-built details and overall quality of design. In addition, large uncertainties as to the intensity and duration of shaking and the possibility of unfavorable response of a small subset of buildings or other structures may prevent full realization of the intent.” At this point it is worth recalling the criteria mentioned earlier in describing the risktargeted ground motions used for design. The probability of structural collapse due to ground shaking is not zero. One percent in 50 years is actually a higher failure rate than is currently considered acceptable for buildings subject to other natural loads, such as wind and snow. The reason is as stated in the quote at the beginning of this chapter “…all the wealth of the world would prove insufficient…” Damage is to be expected when an earthquake equivalent to the design earthquake occurs. (The “design earthquake” is currently taken as two-thirds of the MCE ground motion). Some collapse 2-18

Chapter 2: Fundamentals

is to be expected when and where ground motion equivalent to the MCE ground motion occurs. The basic structural criteria are strength, stability and distortion. The yield-level strength provided must be at least that required by the design spectrum (which is reduced from the elastic spectrum as described previously). Structural elements that cannot be expected to perform in a ductile manner are to have greater strength, which is achieved by applying the Ω0 amplifier to the design spectral response. The stability criterion is imposed by amplifying the effects of lateral forces for the destabilizing effect of lateral translation of the gravity weight (the P-delta effect). The distortion criterion is a limit on story drift and is calculated by amplifying the linear response to the (reduced) design spectrum by the factor Cd to account for inelastic behavior. Yield-level strengths for steel and concrete structures are easily obtained from common design standards. The most common design standards for timber and masonry are based on allowable stress concepts that are not consistent with the basis of the reduced design spectrum. Although strength-based standards for both materials have been introduced in recent years, the engineering profession has not yet embraced these new methods. In the past, the Provisions stipulated adjustments to common reference standards for timber and masonry to arrive at a strength level equivalent to yield, and compatible with the basis of the design spectrum. Most of these adjustments were simple factors to be applied to conventional allowable stresses. With the deletion of these methods from the Provisions, other methods have been introduced into model building codes, and the ASCE standard Minimum Design Loads for Buildings and Other Structures to factor downward the seismic load effects based on the Provisions for use with allowable stress design methods. The Provisions recognizes that the risk presented by a particular building is a combination of the seismic hazard at the site and the consequence of failure, due to any cause, of the building. Thus, a classification system is established based on the use and size of the building. This classification is called the Occupancy Category (Risk Category in the Standard). A combined classification called the Seismic Design Category (SDC) incorporates both the seismic hazard and the Occupancy Category. The SDC is used throughout the Provisions for decisions regarding the application of various specific requirements. The flow charts in Chapter 2 illustrate how these classifications are used to control application of various portions of the Provisions.

2.4 STRUCTURAL ANALYSIS The Provisions sets forth several procedures for determining the force effect of ground shaking. Analytical procedures are classified by two facets: linear versus nonlinear and dynamic versus equivalent static. The two most fully constrained and frequently used are both linear methods: an equivalent static force procedure and a dynamic modal response spectrum analysis procedure. A third linear method, a full history of dynamic response (previously referred to as a time-history analysis, now referred to as a response-history analysis), and a nonlinear method are also permitted, subject to certain limitations. These 2-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples

methods use real or synthetic ground motions as input but require them to be scaled to the basic response spectrum at the site for the range of periods of interest for the structure in question. Nonlinear analyses are very sensitive to assumptions about structural behavior made in the analysis and to the ground motions used as input, and a peer review is required. A nonlinear static method, also known as a pushover analysis, is described in Part 3 of the Provisions, but it is not included in the Standard. The Provisions also reference ASCE 41, Seismic Rehabilitation of Existing Buildings, for the pushover method. The method is instructive for understanding the development of mechanisms but there is professional disagreement over its utility for validating a structural design. The two most common linear methods make use of the same design spectrum. The reduction from the elastic spectrum to design spectrum is accomplished by dividing the elastic spectrum by the coefficient R, which ranges from 1-1/4 to 8. Because the design computations are carried out with a design spectrum that is two-thirds the MCE spectrum that means the full reduction from elastic response ranges from 1.9 to 12. The specified elastic spectrum is based on a damping level at 5 percent of critical damping, and a part of the R factor accomplishes adjustments in the damping level. Ductility and overstrength make up the larger part of the reduction. The Provisions define the total effect of earthquake actions as a combination of the response to horizontal motions (or forces for the equivalent static force method) with response to vertical ground acceleration. The response to vertical ground motion is roughly estimated as a factor (positive or negative) on the dead load force effect. The resulting internal forces are combined with the effects of gravity loads and then compared to the full strength of the members, reduced by a resistance factor, but not by a factor of safety. With the equivalent static force procedure, the level of the design spectrum is set by determining the appropriate values of basic seismic acceleration, the appropriate soil profile type and the value for R. The particular acceleration for the building is determined from this spectrum by selecting a value for the natural period of vibration. Equations that require only the height and type of structural system are given to approximate the natural period for various building types. (The area and length of shear walls come into play with an optional set of equations.) Calculation of a period based on an analytical model of the structure is encouraged, but limits are placed on the results of such calculations. These limits prevent the use of a very flexible model in order to obtain a large period and correspondingly low acceleration. Once the overall response acceleration is found, the base shear is obtained by multiplying it by the total effective mass of the building, which is generally the total permanent load. Once the total lateral force is determined, the equivalent static force procedure specifies how this force is to be distributed along the height of the building. This distribution is based on the results of dynamic studies of relatively uniform buildings and is intended to give an envelope of shear force at each level that is consistent with these studies. This set of forces will produce, particularly in tall buildings, an envelope of gross overturning moment that is larger than many dynamic studies indicate is necessary. Dynamic analysis is encouraged, and the modal procedure is required for structures with large periods (essentially this means tall structures) in the higher seismic design categories.

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Chapter 2: Fundamentals

With one exception, the remainder of the equivalent static force analysis is basically a standard structural analysis. That exception accounts for uncertainties in the location of the center of mass, uncertainties in the strength and stiffness of the structural elements and rotational components in the basic ground shaking. This concept is referred to as horizontal torsion. The Provisions requires that the center of force be displaced from the calculated center of mass by an arbitrary amount in either direction (this torsion is referred to as accidental torsion). The twist produced by real and accidental torsion is then compared to a threshold and if the threshold is exceeded, the accidental torsion must be amplified. In many respects, the modal analysis procedure is very similar to the equivalent static force procedure. The primary difference is that the natural period and corresponding deflected shape must be known for several of the natural modes of vibration. These are calculated from a mathematical model of the structure. The procedure requires inclusion of enough modes so that the dynamic response of the analytical model captures at least 90 percent of the mass in the structure that can vibrate. The base shear for each mode is determined from a design spectrum that is essentially the same as that for the static procedure. The distribution of displacements and accelerations (forces) and the resulting story shears, overturning moments and story drifts are determined for each mode directly from the procedure. Total values for subsequent analysis and design are determined by taking the square root of the sum of the squares for each mode. This summation gives a statistical estimate of maximum response when the participation of the various modes is random. If two or more of the modes have very similar periods, more advanced techniques for summing the values are required; these procedures must account for coupling in the response of close modes. The sum of the absolute values for each mode is always conservative. A lower limit to the base shear determined from the modal analysis procedure is specified based on the static procedure, and the approximate periods specified in the static procedure. When this limit is violated, which is common, all results are scaled up in direct proportion. The consideration of horizontal torsion is the same as for the static procedure. Because the equivalent static forces applied at each floor, the story shears and the overturning moments are separately obtained from the summing procedure, the results are not statically compatible (that is, the moment calculated from the summed floor forces will not match the moment from the summation of moments). Early recognition of this will avoid considerable problems in later analysis and checking. For structures that are very uniform in a vertical sense, the two procedures give very similar results. The modal analysis method is better for buildings having unequal story heights, stiffnesses, or masses. The modal procedure is required for such structures in higher seismic design categories. Both methods are based on purely elastic behavior, and, thus, neither will give a particularly accurate picture of behavior in an earthquake approaching the design event. Yielding of one component leads to redistribution of the forces within the structural system; while this may be very significant, none of the linear methods can account for it.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Both of the common methods require consideration of the stability of the building as a whole. The technique is based on elastic amplification of horizontal displacements created by the action of gravity on the displaced masses. A simple factor is calculated and the amplification is provided for in designing member strengths when the amplification exceeds about 10 percent. The technique is referred to as the P-delta analysis and is only an approximation of stability at inelastic response levels.

2.5 NONSTRUCTURAL ELEMENTS OF BUILDINGS Severe ground shaking often results in considerable damage to the nonstructural elements of buildings. Damage to nonstructural elements can pose a hazard to life in and of itself, as in the case of heavy partitions or facades, or it can create a hazard if the nonstructural element ceases to function, as in the case of a fire suppression system. Some buildings, such as hospitals and fire stations, need to be functional immediately following an earthquake; therefore, many of their nonstructural elements must remain undamaged. The Provisions treats damage to and from nonstructural elements in three ways. First, indirect protection is provided by an overall limit on structural distortion; the limits specified, however, may not offer enough protection to brittle elements that are rigidly bound by the structure. More restrictive limits are placed upon those Occupancy Categories (Risk Categories in the Standard) for which better performance is desired given the occurrence of strong ground shaking. Second, many components must be anchored for an equivalent static force. Third, the explicit design of some elements (the elements themselves, not just their anchorage) to accommodate specific structural deformations or seismic forces is required. The dynamic response of the structure provides the dynamic input to the nonstructural component. Some components are rigid with respect to the structure (light weights, and small dimensions often lead to fundamental periods of vibration that are very short). Application of the response spectrum concept would indicate that the response history of motion of a building roof to which mechanical equipment is attached looks like a ground motion to the equipment. The response of the component is often amplified above the response of the supporting structure. Response spectra developed from the history of motion of a point on a structure undergoing ground shaking are called floor spectra, and are useful in understanding the demands upon nonstructural components. The Provisions simplifies the concept greatly. The force for which components are checked depends on: 1. The component mass; 2. An estimate of component acceleration that depends on the structural response acceleration for short period structures, the relative height of the component within the structure and a crude approximation of the flexibility of the component or its anchorage; 2-22

Chapter 2: Fundamentals

3. The available ductility of the component or its anchorage; and 4. The function or importance of the component or the building. Also included in the Provisions is a quantitative measure for the deformation imposed upon nonstructural components. The inertial force demands tend to control the seismic design for isolated or heavy components whereas the imposed deformations are important for the seismic design for elements that are continuous through multiple levels of a structure or across expansion joints between adjacent structures, such as cladding or piping.

2.6 QUALITY ASSURANCE Since strong ground shaking has tended to reveal hidden flaws or weak links in buildings, detailed requirements for assuring quality during construction are contained in the Provisions by reference to the Standard, where they are located in an appendix. The actively implemented provisions for quality control are actually contained in the model building codes, such as the International Building Code, and the material design standards, such as Seismic Provisions for Structural Steel Buildings. Loads experienced during construction provide a significant test of the likely performance of ordinary buildings under gravity loads. Tragically, mistakes occasionally will pass this test only to cause failure later, but it is fairly rare. No comparable proof test exists for horizontal loads, and experience has shown that flaws in construction show up in a disappointingly large number of buildings as distress and failure due to earthquakes. This is coupled with the seismic design approach based on excursions into inelastic straining, which is not the case for response to other loads. The quality assurance provisions require a systematic approach with an emphasis on documentation and communication. The designer who conceives the systems to resist the effects of earthquake forces must identify the elements that are critical for successful performance as well as specify the testing and inspection necessary to confirm that those elements are actually built to perform as intended. Minimum levels of testing and inspection are specified in the Provisions for various types of systems and components. The Provisions also requires that the contractor and building official be aware of the requirements specified by the designer. Furthermore, those individuals who carry out the necessary inspection and testing must be technically qualified, and must communicate the results of their work to all concerned parties. In the final analysis, there is no substitute for a sound design, soundly executed.

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3 Earthquake Ground Motion Nicolas Luco1, Ph.D., P.E., Michael Valley2, S.E., C.B. Crouse3, P.E., Ph.D. Contents 3.1

3.1.1

ASCE 7-05 Seismic Maps ..................................................................................................... 2

3.1.2

MCER Ground Motions in the Provisions and in ASCE 7-10 ............................................... 3

3.1.3

PGA Maps in the Provisions and in ASCE 7-10 ................................................................... 7

3.1.4

Basis of Vertical Ground Motions in the Provisions and in ASCE 7-10 .............................. 7

3.1.5

Summary ................................................................................................................................ 7

3.1.6

References.............................................................................................................................. 8

3.2

DETERMINATION OF GROUND MOTION VALUES AND SPECTRA ................................ 9

3.2.1

ASCE 7-05 Ground Motion Values ....................................................................................... 9

3.2.2

2009 Provisions Ground Motion Values ............................................................................. 10

3.2.3

ASCE 7-10 Ground Motion Values ..................................................................................... 11

3.2.4

Horizontal Response Spectra ............................................................................................... 12

3.2.5

Vertical Response Spectra ................................................................................................... 13

3.2.6

Peak Ground Accelerations ................................................................................................. 14

3.3

BASIS OFEARTHQUAKE GROUND MOTION MAPS ............................................................ 2

SELECTION AND SCALING OF GROUND MOTION RECORDS ....................................... 14

3.3.1

Approach to Ground Motion Selection and Scaling ............................................................ 15

3.3.2

Two-Component Records for Three Dimensional Analysis ............................................... 24

3.3.3

One-Component Records for Two-Dimensional Analysis .................................................. 27

3.3.4

References............................................................................................................................ 28

Author of Section 3.1. Author of Sections 3.2 and 3.3. 3 Reviewing author. 2

FEMA P-752, NEHRP Recommended Provisions: Design Examples

Most of the effort in seismic design of buildings and other structures is focused on structural design. This chapter addresses another key aspect of the design process—characterization of earthquake ground motion. Section 3.1 describes the basis of the earthquake ground motion maps in the Provisions and in ASCE 7. Section 3.2 has examples for the determination of ground motion parameters and spectra for use in design. Section 3.3 discusses and provides an example for the selection and scaling of ground motion records for use in response history analysis.

3.1 BASIS OFEARTHQUAKE GROUND MOTION MAPS This section explains the basis of the new Risk-Targeted Maximum Considered Earthquake (MCER) ground motions specified in the 2009 Provisions and mapped in ASCE 7-10. In doing so, it also explains the basis for the uniform-hazard ground motion (SSUH and S1UH) maps, risk coefficient (CRS and CR1) maps and deterministic ground motion (SSD and S1D) maps in the 2009 Provisions. These three sets of maps are combined to form the Site Class B MCER ground motion (SS and S1) maps in ASCE 7-10. The use of SS and S1 ground motions in the 2009 Provisions and ASCE 7-10 to derive a design response spectrum remains the same as it is in ASCE 7-05. This section also explains the basis for the new Peak Ground Acceleration (PGA) maps in the 2009 Provisions and ASCE 7-10 and the new equations for vertical ground motions. The basis for the longperiod transition (TL) maps in the 2009 Provisions and ASCE 7-10, which are identical to those in ASCE 7-05, is also reviewed. In fact, we start with a review of these maps and the Maximum Considered Earthquake (MCE) ground motion maps in ASCE 7-05. 3.1.1

ASCE 7-‐05 Seismic Maps

The bases for the seismic ground motion (MCE) and long-period transition (TL) maps in Chapter 22 of ASCE 7-05 were established by, respectively, the Building Seismic Safety Council (BSSC) Seismic Design Procedures Group, also referred to as Project ’97, and Technical Subcommittee 1 (TS-1) of the 2003 Provisions Update Committee. They are reviewed briefly in the following two subsections. 3.1.1.1 Maximum Considered Earthquake (MCE) Ground Motion Maps The MCE ground motion maps in ASCE 7-05 can be described as applications of its site-specific ground motion hazard analysis procedure in Chapter 21 (Section 21.2), using ground motion values computed by the USGS National Seismic Hazard Mapping Project (in Golden, CO) for a grid of locations and/or polygons that covers the US. In particular, the 1996 USGS update of the ground motion values was used for ASCE 7-98 and ASCE 7-02; the 2002 USGS update was used for ASCE 7-05. The site-specific procedure in all three editions calculates the MCE ground motion as the lesser of a probabilistic and a deterministic ground motion. Hence, the USGS computed both types of ground motions, whereas otherwise it would have only computed probabilistic ground motions. Brief reviews of how the USGS computed the probabilistic and deterministic ground motions are provided in the next few paragraphs. For additional information, see Leyendecker et al. (2000). The USGS computation of the probabilistic ground motions that are part of the basis of the MCE ground motion maps in ASCE 7-05 is explained in detail in Frankel et al. (2002). In short, the USGS combines research on potential sources of earthquakes (e.g., faults and locations of past earthquakes), the potential magnitudes of earthquakes from these sources and their frequencies of occurrence, and the potential ground motions generated by these earthquakes. Uncertainty and randomness in each of these components is accounted for in the computation via contemporary Probabilistic Seismic Hazard Analysis (PSHA), which was originally conceived by Cornell (1968). The primary output of PSHA computations are so-called hazard curves, for locations on a grid covering the US in the case of the USGS computation. 3-2

Chapter 3: Earthquake Ground Motion Each hazard curve provides mean annual frequencies of exceeding various user-specified ground motions amplitudes. From these hazard curves, the ground motion amplitudes for a user-specified mean annual frequency can be interpolated and then mapped. The results are known as uniform-hazard ground motion maps, since the mean annual frequency (or corresponding probability) is uniform geographically. For ASCE 7-05 (as well as ASCE 7-02 and ASCE 7-98), a mean annual exceedance frequency of 1/2475 per year, corresponding to 2% probability of exceedance in 50 years, was specified by the aforementioned BSSC Project ’97. That project also specified that the ground motion parameters be spectral response accelerations at vibration periods of 0.2 seconds and 1 second, for 5% of critical damping. For the average shear wave velocity at small shear strains in the upper 100 feet (30 m) of subsurface below each location (vS,30), the USGS decided on a reference value of 760 m/s. The BSSC subsequently decided to regard this reference value, which is at the boundary of Site Classes B and C, as corresponding to Site Class B. Justifications for the decisions summarized in this paragraph are provided in the Commentary of FEMA 303, FEMA 369 and FEMA 450. The USGS computation of the deterministic ground motions for ASCE 7-05 is detailed in the FEMA 303 Commentary. As defined by Project ’97 and subsequently specified in the site-specific procedure of ASCE 7-05 (Section 21.2.2), each deterministic ground motion is calculated as 150% of the median spectral response acceleration for a characteristic earthquake on a known active fault within the region. The specific characteristic earthquake is that which generates the largest median spectral response acceleration at the given location. As for the probabilistic ground motions, the spectral response accelerations are at vibration periods of 0.2 seconds and 1 second, for 5% of critical damping. The same reference site class (see preceding paragraph) is used as well. Though not applied to probabilistic ground motions, lower limits of 1.5g and 0.6g are applied to the deterministic ground motions. As mentioned at the beginning of this section, the lesser of the probabilistic and deterministic ground motions described above yields the MCE ground motions mapped in ASCE 7-05. Thus, the MCE spectral response accelerations at 0.2 seconds and 1 second are equal to the corresponding probabilistic ground motions wherever they are less than the lower limits of the deterministic ground motions (1.5g and 0.6g, respectively). Where the probabilistic ground motions are greater than the lower limits, the deterministic ground motions sometimes govern, but only if they are less than their probabilistic counterparts. On the MCE ground motion maps in ASCE 7-05, the deterministic ground motions govern mainly near major faults in California (like the San Andreas), in Reno and in parts of the New Madrid Seismic Zone. The deterministic ground motions that govern are as small as 40% of their probabilistic counterparts. 3.1.1.2 Long-Period Transition Period (TL) Maps The details of the procedure and rationale used in developing the TL maps in ASCE 7-05; and now in ASCE 7-10 and the 2009 Provisions, are found in Crouse et al. (2006). In short, the procedure consisted of two steps. First, a relationship between TL and earthquake magnitude was established. Second, the modal magnitude from deaggregation of the USGS 2% in 50-year ground motion hazard at a 2-second period (1 second for Hawaii) was mapped. The long-period transition period (TL) maps that combined these two steps delimit the transition of the design response spectrum from a constant velocity (1/T) to a constant displacement (1/T2) shape. 3.1.2

MCER Ground Motions in the Provisions and in ASCE 7-‐10

Like the MCE ground motion maps in ASCE 7-05 reviewed in the preceding section, the new RiskTargeted Maximum Considered Earthquake (MCER) ground motions in the 2009 Provisions and ASCE 710 can be described as applications of the site-specific ground motion hazard analysis procedure in Chapter 21 (Section 21.2) of both documents. For the MCER ground motions, however, the USGS values (for a grid of site and/or polygons covering the US) that are used in the procedure are from its 2008 update. Still, the site-specific procedure of the Provisions and ASCE 7-10 calculates the MCER ground

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FEMA P-752, NEHRP Recommended Provisions: Design Examples motion as the lesser of a probabilistic and a deterministic ground motion. The definitions of the probabilistic and deterministic ground motions in ASCE 7-10, however, are different than in ASCE 7-05. The definitions were revised for the 2009 Provisions and ASCE 7-10 by the BSSC Seismic Design Procedures Reassessment Group (SDPRG), also referred to as Project ’07. Three revisions were made: 1) The probabilistic ground motions are redefined as so-called risk-targeted ground motions, in lieu of the uniform-hazard (2% in 50-year) ground motions that underlie the ASCE 7-05 MCE ground motion maps, 2) the deterministic ground motions are redefined as 84th-percentile ground motions, in lieu of median ground motions multiplied by 1.5; and 3) the probabilistic and deterministic ground motions are redefined as maximum-direction ground motions, in lieu of geometric mean ground motions. In addition to these three BSSC redefinitions of probabilistic and deterministic ground motions, there is a fourth difference in the ground motion values computed by the USGS for the 2009 Provisions and ASCE 7-10 versus ASCE 7-05: 4) The probabilistic and deterministic ground motions were recomputed using updated earthquake source and ground motion propagation models, e.g., the Unified California Earthquake Rupture Forecast (UCERF, Version 2; Field et al., 2008) and the Next Generation Attenuation (NGA) ground motion models4. Each of the above four differences between the basis of the MCE ground motions (in ASCE 7-05) and that of the MCER ground motions (in the 2009 Provisions and ASCE 7-10) is explained in more detail below. Also explained are the differences in the presentation of MCER ground motions between the 2009 Provisions and ASCE 7-10; the numerical values of the MCER ground motions in the two documents are otherwise identical. 3.1.2.1 Risk-Targeted Probabilistic Ground Motions For the MCE ground motion maps in ASCE 7-05, recall (from Section 3.1.1) that the underlying probabilistic ground motions are specified to be uniform-hazard ground motions that have a 2% probability of being exceeded in 50 years. It has long been recognized, though, that “it really is the probability of structural failure with resultant casualties that is of concern; and the geographical distribution of that probability is not necessarily the same as the distribution of the probability of exceeding some ground motion” (p. 296 of ATC 3-06, 1978). The primary reason that the distributions of the two probabilities are not the same is that there are geographic differences in the shape of the hazard curves from which uniform-hazard ground motions are read. The Commentary of FEMA 303 (p. 289) reports that “because of these differences, questions were raised concerning whether definition of the ground motion based on a constant probability for the entire United States would result in similar levels of seismic safety for all structures”. The changeover to risk-targeted probabilistic ground motions for the 2009 Provisions and ASCE 7-10 takes into account the differences in the shape of hazard curves across the US. Where used in design, the risk-targeted ground motions are expected to result in buildings with a geographically uniform mean annual frequency of collapse, or uniform risk. The BSSC, via Project ’07, decided on a target risk level corresponding to 1% probability of collapse in 50 years. This target is based on the average of the mean annual frequencies of collapse across the Western US (WUS) expected to result from design for the 4

See the February 2008 Earthquake Spectra “Special Issue on the Next Generation Attenuation Project,” Volume 24, Number 1. 3-4

Chapter 3: Earthquake Ground Motion probabilistic ground motions in ASCE 7-05. Consequently, in the WUS the risk-targeted ground motions in the 2009 Provisions and ASCE 7-10 are generally within 15% of the corresponding uniform-hazard (2% in 50-year) ground motions. In the Central and Eastern US, where the shapes of hazard curves are known to differ from those in the WUS, the risk-targeted ground motions generally are smaller. For instance, in the New Madrid Seismic Zone and near Charleston, South Carolina ratios of risk-targeted to uniform-hazard ground motions are as small as 0.7. The computation of risk-targeted probabilistic ground motions for the MCER ground motions in the 2009 Provisions and ASCE 7-10 is detailed in Provisions Part 1 Sections 21.2.1.2 and C21.2.1 and in Luco et al. (2007). While the computation of the risk-targeted ground motions is different than that of the uniform-hazard ground motions specified for the MCE ground motions in ASCE 7-05, both begin with USGS computations of hazard curves. As explained in Section 3.1.1, the uniform-hazard ground motions simply interpolate the hazard curves for a 2% probability of exceedance in 50 years. In contrast, the risktargeted ground motions make use of entire hazard curves. In either case, the end results are probabilistic spectral response accelerations at 0.2 seconds and 1 second, for 5% of critical damping and the reference site class. 3.1.2.2 84th-Percentile Deterministic Ground Motions For the MCE ground motion maps in ASCE 7-05, recall (from Section 3.1.1) that the underlying deterministic ground motions are defined as 150% of median spectral response accelerations. As explained in the FEMA 303 Commentary (p. 296), Increasing the median ground motion estimates by 50 percent [was] deemed to provide an appropriate margin and is similar to some deterministic estimates for a large magnitude characteristic earthquake using ground motion attenuation functions with one standard deviation. Estimated standard deviations for some active fault sources have been determined to be higher than 50 percent, but this increase in the median ground motions was considered reasonable for defining the maximum considered earthquake ground motions for use in design. For the MCER ground motions in the 2009 Provisions and ASCE 7-10, however, the BSSC decided to define directly the underlying deterministic ground motions as those at the level of one standard deviation. More specifically, they are defined as 84th-percentile ground motions (since it has been widely observed that ground motions follow lognormal probability distributions). The remainder of the definition of the deterministic ground motions remains the same as that used for the MCE ground motions maps in ASCE 7-05. For example, the lower limits of 1.5g and 0.6g described in Section 3.1.1 are retained. The USGS applied a simplification specified by the BSSC in computing the 84th-percentile deterministic ground motions for the 2009 Provisions and ASCE 7-10. The 84th-percentile spectral response accelerations were approximated as 180% of median values. This approximation corresponds to a logarithmic ground motion standard deviation of approximately 0.6, as demonstrated in the Provisions Part 1 Section C21.2.2. The computation of deterministic ground motions is further described in Provisions Part 2 Section C21.2.2. 3.1.2.3 Maximum-Direction Probabilistic and Deterministic Ground Motions Due to the ground motion attenuation models used by the USGS in computing them5, overall the MCE spectral response accelerations in ASCE 7-05 represent the geometric mean of two horizontal components of ground motion. Most users of ASCE 7-05 are unaware of this fact, particularly since the discussion notes on the MCE ground motion maps incorrectly state that they represent “the random horizontal component of ground motion.” For the 2009 Provisions and ASCE 7-10, the BSSC decided that it would 5

See the January/February 1997 Seismological Research Letters “Special Issue on Ground Motion Attenuation Relations,” Volume 68, Number 1. 3-5

FEMA P-752, NEHRP Recommended Provisions: Design Examples be an improvement if the MCER ground motions represented the maximum direction of horizontal spectral response acceleration. Reasons for this decision are explained in Provisions Part 1 Section C21.2. Since the attenuation models used in computing the 2008 update of the USGS ground motions also represent (overall) “geomean” spectral response accelerations, for the 2009 Provisions and ASCE 7-10 the BSSC provided factors to convert approximately to “maximum-direction” ground motions. Based on research by Huang et al. (2008) and others, the factors are 1.1 and 1.3 for the spectral response accelerations at 0.2 seconds and 1.0 second, respectively. The basis for these factors is elaborated upon in the Provisions Part 1 Section C21.2. They are applied to both the USGS probabilistic hazard curves from which the risk-targeted ground motions (described in Section 3.1.2.1) are derived and the USGS deterministic ground motions (described in Section 3.1.2.2). 3.1.2.4 Updated Ground Motions from USGS (2008) For the MCE ground motion maps in ASCE 7-05, recall (from Section 3.1.1) that the underlying probabilistic and deterministic ground motions are from the 2002 USGS update. As mentioned above, the MCER ground motions in the 2009 Provisions and ASCE 7-10 are instead based on the 2008 update of the USGS ground motion values. This update is documented in Petersen et al. (2008) and supersedes the 1996 and 2002 USGS ground motions values. It involved interactions with hundreds of scientists and engineers at regional and topical workshops, including advice from working groups, expert panels, state geological surveys, other federal agencies and hazard experts from industry and academia. Based in large part on new published studies, the 2008 update incorporated changes in both earthquake source models (including magnitudes and occurrence frequencies) and models of ground motion propagation. The UCERF and NGA models mentioned above are just two examples of such changes. The end results are updated ground motions that represent the “best available science” as determined by the USGS from an extensive information-gathering and review process. It is important to note that the 2008 USGS hazard curves and uniform-hazard maps (posted at http://earthquake.usgs.gov/hazards/products/conterminous/2008/), like their 2002 counterparts, represent the “geomean” ground motions discussed in the preceding subsection. Only the MCER ground motions and their underlying probabilistic and deterministic ground motions represent the maximum direction of horizontal spectral response acceleration. 3.1.2.5 Differing Presentation of MCER Ground Motions in the Provisions and in ASCE 7-10 Though their numerical values are identical, the MCER ground motions specified in the Provisions and in ASCE 7-10 are presented differently. As replacements to the MCE ground motion maps in ASCE 7-05, ASCE 7-10 presents (in Chapter 22) contour maps of the MCER ground motions for Site Class B, which are still denoted SS and S1 for the 0.2- and 1.0-second spectral response accelerations, respectively. Like the MCE ground motions in ASCE 7-05, the MCER ground motions mapped in ASCE 7-10 are accessible electronically via a USGS web application (see http://earthquake.usgs.gov/designmaps/). In contrast, Provisions Section 11.4 presents equations to calculate MCER ground motions (SS and S1) for Site Class B using maps (in Chapter 22) of uniform-hazard 2% in 50-year ground motions (denoted SSUH and S1UH), so-called risk coefficients (denoted CRS and CR1); and deterministic ground motions (denoted SSD and S1D, not to be confused with the design ground motions SDS and SD1). The risk coefficient maps show the ratio of the risk-targeted probabilistic ground motions (described in Section 3.1.2.1) to corresponding 2% in 50-year ground motions like those used to derive the MCE ground motion maps in ASCE 7-05. The intent of the equations and three sets of maps presented in the Provisions is transparency in the derivation of the MCER ground motions. The mapped values of the uniform-hazard ground motions, risk coefficients and deterministic ground motions are all accessible electronically via http://earthquake.usgs.gov/designmaps/.

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Chapter 3: Earthquake Ground Motion 3.1.3

PGA Maps in the Provisions and in ASCE 7-‐10

The basis of the Peak Ground Acceleration (PGA) maps in the Provisions and in ASCE 7-10 nearly parallels that of the MCE ground motion maps in ASCE 7-05 described in Section 3.1.1.1. More specifically, the mapped PGA values for Site Class B are calculated as the lesser of uniform-hazard (2% in 50-year) probabilistic and deterministic PGA values that represent the geometric mean of two horizontal components of ground motion. Unlike in ASCE 7-05, though, the deterministic values are defined as 84th-percentile ground motions rather than 150% of median ground motions. This definition of deterministic ground motions parallels that which is described above for the MCER ground motions in the 2009 Provisions and ASCE 7-10. The deterministic PGA values, though, are stipulated to be no lower than 0.5g, as opposed to 1.5g and 0.6g (respectively) for the MCER 0.2- and 1.0-second spectral response accelerations. All of these details of the basis of the PGA maps are provided in ASCE 7-10 Section 21.5; the Provisions do not contain a site-specific procedure for PGA values. The USGS-computed PGA values for vS,30 = 760m/s that are mapped, like their MCER ground motion counterparts in the Provisions and in ASCE 7-10, are from the 2008 USGS update. Also like their MCER ground motion counterparts, the 84th-percentile PGA values have been approximated as median values multiplied by 1.8. While the values on and format of the PGA maps in the Provisions and in ASCE 7-10 are identical, the terminology used to label the maps (and values) is different in the two documents. In the Provisions, they are referred to as “MCE Geometric Mean PGA” maps. In ASCE 7-10, they are labeled “Maximum Considered Earthquake Geometric Mean (MCEG) PGA” maps. The MCEG abbreviation is intended to remind users of the differences between the basis of the PGA maps and the MCE R maps also in ASCE 710, namely that the PGA values represent the geometric mean of two horizontal components of ground motion, not the maximum direction; and that the probabilistic PGA values are not risk-targeted ground motions, but rather uniform-hazard (2% in 50-year) ground motions. 3.1.4

Basis of Vertical Ground Motions in the Provisions and in ASCE 7-‐10

Whereas ASCE 7-05 determines vertical seismic load effects via a single constant fraction of the horizontal short-period spectral response acceleration SDS, the 2009 Provisions and ASCE 7-10 determine a vertical design response spectrum, Sav, that is analogous to the horizontal design response spectrum, Sa. The Sav values are determined via functions (for four different ranges of vertical period of vibration) that each depend on SDS and a coefficient Cv representing the ratio of vertical to horizontal spectral response acceleration. This is in contrast to determination of Sa via mapped horizontal spectral response accelerations. The coefficient Cv, in turn, depends on the amplitude of spectral response acceleration (by way of SS) and site class. These dependencies, as well as the period dependence of the equations for Sav, are based on studies by Bozorgnia and Campbell (2004) and others. Those studies observed that the ratio of vertical to horizontal spectral response acceleration is sensitive to period of vibration, site class, earthquake magnitude (for relatively soft sites) and distance to the earthquake. The sensitivity to the latter two characteristics is captured by the dependence of Cv on SS. The basis of the equations for vertical response spectra in the Provisions and in ASCE 7-10 is explained in more detail in the commentary to Chapter 23 of each document. Note that for vertical periods of vibration greater than 2 seconds, Chapter 23 stipulates that the vertical spectral response accelerations be determined via a site-specific procedure. A site-specific study also may be performed for periods less than 2 seconds, in lieu of using the equations for vertical response spectra. 3.1.5

Summary

While the new Risk-Targeted Maximum Considered Earthquake (MCER) ground motions in the Provisions and in ASCE 7-10 are similar to the MCE ground motions in ASCE 7-05, in that they both represent the lesser of probabilistic and deterministic ground motions, there are many differences in their

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FEMA P-752, NEHRP Recommended Provisions: Design Examples development. The definitions of the probabilistic and deterministic ground motions that underlie the MCER ground motions were revised by the BSSC Seismic Design Procedures Reassessment Group (SDPRG, or Project ’07); and the hazard modeling upon which these ground motions are based was updated by the USGS (in 2008). In particular, the underlying probabilistic ground motions were redefined as so-called risk-targeted ground motions, which led to the new “MCER ground motion” terminology. The basis of the new Peak Ground Acceleration (PGA) maps in the Provisions and in ASCE 7-10 nearly parallels that of the 0.2- and 1.0-second MCE spectral response accelerations in ASCE 7-05 (with one important exception); new equations for vertical ground motion spectra are based on recent studies of the ratio of vertical to horizontal ground motions. The long-period transition (TL) maps in the new documents are the same as those in ASCE 7-05. 3.1.6

References

American Society of Civil Engineers. 1998. Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-98. ASCE, Reston, Virginia. American Society of Civil Engineers. 2002. Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-02. ASCE, Reston, Virginia. American Society of Civil Engineers. 2006. Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-05. ASCE, Reston, Virginia. American Society of Civil Engineers. 2010. Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-10. ASCE, Reston, Virginia. Applied Technology Council. 1978. Tentative Provisions for the Development of Seismic Regulations for Buildings, ATC 3-06. ATC, Palo Alto, California. Bozorgnia, Y. and K.W. Campbell. 2004. “The Vertical-to-Horizontal Response Spectral Ratio and Tentative Procedures for Developing Simplified V/H and Vertical Design Spectra,” Journal of Earthquake Engineering, 8:175-207. Building Seismic Safety Council. 1997. NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, Part 2: Commentary, FEMA 303. FEMA, Washington, D.C. Building Seismic Safety Council. 2000. NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, Part 2: Commentary, FEMA 369. FEMA, Washington, D.C. Building Seismic Safety Council. 2003. NEHRP Recommended Provisions and Commentary for Seismic Regulations for New Buildings and Other Structures, FEMA 450. FEMA, Washington, D.C. Building Seismic Safety Council. 2009. NEHRP Recommended Seismic Provisions for New Buildings and Other Structures, FEMA P-750. FEMA, Washington, D.C. Cornell, C.A. 1968. “Engineering Seismic Risk Analysis,” Bulletin of the Seismological Society of America, 58(5):1583-1606. Crouse C.B., E.V. Leyendecker, P.G. Somerville, M. Power and W.J. Silva. 2006. “Development of Seismic Ground-Motion Criteria for the ASCE 7 Standard,” in Proceedings of the 8th US National Conference on Earthquake Engineering. Earthquake Engineering Research Institute, Oakland, California. Field, E.H., T.E. Dawson, K.R. Felzer, A.D. Frankel, V. Gupta, T.H. Jordan, T. Parsons, M.D. Petersen, R.S. Stein, R.J. Weldon and C.J. Wills. 2008. The Uniform California Earthquake Rupture Forecast, Version 2 (UCERF 2), USGS Open File Report 2007-1437 (http://pubs.usgs.gov/of/2007/1437/). USGS, Golden, Colorado.

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Chapter 3: Earthquake Ground Motion Frankel, A.D., M.D. Petersen, C.S. Mueller, K.M. Haller, R.L. Wheeler, E.V. Leyendecker, R.L. Wesson, S.C. Harmsen, C.H. Cramer, D.M. Perkins and K.S. Rukstales. 2002. Documentation for the 2002 Update of the United States National Seismic Hazard Maps, USGS Open File Report 02-420 (http://pubs.usgs.gov/of/2002/ofr-02-420/). USGS, Golden, Colorado. Huang, Y.-N., A.S. Whittaker and N. Luco, 2008. “Maximum Spectral Demands in the Near-Fault Region,” Earthquake Spectra, 24(1):319-341. Leyendecker, E.V., R.J. Hunt, A.D. Frankel and K.S. Rukstales. 2000. “Development of Maximum Considered Earthquake Ground Motion Maps,” Earthquake Spectra, 16(1):21-40. Luco, N. B.R. Ellingwood, R.O. Hamburger, J.D. Hooper, J.K. Kimball and C.A. Kircher. 2007. “RiskTargeted versus Current Seismic Design Maps for the Conterminous United States,” in Proceedings of the SEAOC 76th Annual Convention. Structural Engineers Association of California, Sacramento, California. Petersen, M.D., A.D. Frankel, S.C. Harmsen, C.S. Mueller, K.M. Haller, R.L. Wheeler, R.L. Wesson, Y. Zeng, O.S. Boyd, D.M. Perkins, N. Luco, E.H. Field, C.J. Wills and K.S. Rukstales. 2008. Documentation for the 2008 Update of the United States National Seismic Hazard Maps, USGS Open File Report 2008-1128 (http://pubs.usgs.gov/of/2008/1128/). USGS, Golden, Colorado.

3.2 DETERMINATION OF GROUND MOTION VALUES AND SPECTRA This example illustrates the determination of seismic design parameters for a site in Seattle, Washington. The site is located at 47.65ºN latitude, 122.3ºW longitude. Using the results of a site-specific geotechnical investigation and the procedure specified in Standard Chapter 20, the site is classified as Site Class C. (This is the same site used in Design Example 6.3.) In the sections that follow design ground motion parameters are determined using ASCE 7-05, the 2009 Provisions and ASCE 7-10. Using the 2009 Provisions, horizontal response spectra, vertical response spectra and peak ground accelerations are computed for both design and maximum considered earthquake ground motions. 3.2.1

ASCE 7-‐05 Ground Motion Values

ASCE 7-05 Section 11.4.1 requires that spectral response acceleration parameters SS and S1 be determined using the maps in Chapter 22. Those maps are too small to permit reading values to a sufficient degree of precision for most sites, so in practice the mapped parameters are determined using a software application available at www.earthquake.usgs.gov/designmaps. That application requires that longitude be entered in degrees east of the prime meridian; negative values are used for degrees west. Given the site location, the following values may be determined using the online application (or read from Figures 22-1 and 22-2). SS = 1.306 S1 = 0.444 Using these mapped spectral response acceleration values and the site class, site coefficients Fa and Fv are determined in accordance with Section 11.4.3 using Tables 11.4-1 and 11.4-2. Using Table 11.4-1, for SS = 1.306 > 1.25, Fa = 1.0 for Site Class C. Using Table 11.4-2, read Fv = 1.4 for S1 = 0.4 and Fv = 1.3 for S1 ≥ 0.5 for Site Class C. Using linear interpolation for S1 = 0.444,

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FEMA P-752, NEHRP Recommended Provisions: Design Examples

Fv = 1.4 +

0.444 − 0.4 (1.3 − 1.4) = 1.356 0.5 − 0.4

Using Equations 11.4-1 and 11.4-2 to determine the adjusted maximum considered earthquake spectral response acceleration parameters,

SMS = Fa SS = 1.0(1.306) = 1.306 SM 1 = Fv S1 = 1.356(0.444) = 0.602 Using Equations 11.4-3 and 11.4-4 to determine the design earthquake spectral response acceleration parameters,

2 2 S DS = S MS = (1.306) = 0.870 3 3 2 2 S D1 = S M 1 = (0.602) = 0.401 3 3 Given the site location read Figure 22-15 for the long-period transition period, TL = 6 seconds. 3.2.2

2009 Provisions Ground Motion Values

Part 1 of the 2009 Provisions modifies Chapter 11 of ASCE 7-05 to update the seismic design ground motion parameters and procedures as described in Section 3.1.2 above. Given the site location, the following values may be determined using the online application (or read from Provisions Figures 22-1 through 22-6). SSUH = 1.305 S1UH = 0.522 CRS = 0.988 CR1 = 0.955 SSD = 1.5 S1D = 0.6 “UH” and “D” appear, respectively, in the subscripts to indicate uniform hazard and deterministic values of the spectral response acceleration parameters at short periods and at a period of 1 second, SS and S1. CRS and CR1 are the mapped risk coefficients at short periods and at a period of 1 second. S1D should not be confused with SD1, which is computed below. The spectral response acceleration parameter at short periods, SS, is taken as the lesser of the values computed using Provisions Equations 11.4-1 and 11.4-2. SS = CRS SSUH = 0.988(1.305) = 1.289 SS = SSD = 1.5 Therefore, SS = 1.289. The spectral response acceleration parameter at a period of 1 second, S1, is taken as the lesser of the values computed using Provisions Equations 11.4-3 and 11.4-4.

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Chapter 3: Earthquake Ground Motion S1 = CR1 S1UH = 0.955(0.522) = 0.498 S1 = S1D = 0.6 Therefore, S1 = 0.498. Using these spectral response acceleration values and the site class, site coefficients Fa and Fv are determined in accordance with Section 11.4.3 using Tables 11.4-1 and 11.4-2 (which are identical to the Tables in ASCE 7-05). Using Table 11.4-1, for SS = 1.289 > 1.25, Fa = 1.0 for Site Class C. Using Table 11.4-2, read Fv = 1.4 for S1 = 0.4 and Fv = 1.3 for S1 ≥ 0.5 for Site Class C. Using linear interpolation for S1 = 0.498,

Fv = 1.4 +

0.498 − 0.4 (1.3 − 1.4) = 1.302 0.5 − 0.4

Using Provisions Equations 11.4-5 and 11.4-6 to determine the MCER spectral response acceleration parameters,

SMS = Fa SS = 1.0(1.289) = 1.289 SM 1 = Fv S1 = 1.302(0.498) = 0.649 Using Provisions Equations 11.4-7 and 11.4-8 to determine the design earthquake spectral response acceleration parameters,

2 2 S DS = S MS = (1.289) = 0.859 3 3 2 2 S D1 = S M 1 = (0.649) = 0.433 3 3 Given the site location read Provisions Figure 22-7 for the long-period transition period, TL = 6 seconds. 3.2.3

ASCE 7-‐10 Ground Motion Values

The seismic design ground motion parameters and procedures in Chapter 11 of ASCE 7-10 are consistent with those in the 2009 Provisions. Given the site location, the following values may be determined using the online application (or read from ASCE 7-10 Figures 22-1 and 22-2). SS = 1.289 S1 = 0.498 Using these spectral response acceleration values and the site class, site coefficients Fa and Fv are determined in accordance with Section 11.4.3 using Tables 11.4-1 and 11.4-2 (which are identical to the Tables in ASCE 7-05 and in the 2009 Provisions). Using Table 11.4-1, for SS = 1.289 > 1.25, Fa = 1.0 for Site Class C. Using Table 11.4-2, read Fv = 1.4 for S1 = 0.4 and Fv = 1.3 for S1 ≥ 0.5 for Site Class C. Using linear interpolation for S1 = 0.498,

Fv = 1.4 +

0.498 − 0.4 (1.3 − 1.4) = 1.302 0.5 − 0.4

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FEMA P-752, NEHRP Recommended Provisions: Design Examples Using Equations 11.4-1 and 11.4-2 to determine the MCER spectral response acceleration parameters,

SMS = Fa SS = 1.0(1.289) = 1.289 SM 1 = Fv S1 = 1.302(0.498) = 0.649 Using Equations 11.4-3 and 11.4-4 to determine the design earthquake spectral response acceleration parameters,

2 2 S DS = S MS = (1.289) = 0.859 3 3 2 2 S D1 = S M 1 = (0.649) = 0.433 3 3 Given the site location read ASCE 7-10 Figure 22-12 for the long-period transition period, TL = 6 seconds. The procedure specified in ASCE 7-10 produces seismic design ground motion parameters that are identical to those produced using the 2009 Provisions—but in fewer steps. 3.2.4

Horizontal Response Spectra

The design spectrum is constructed in accordance with Provisions Section 11.4.5 using Provisions Figure 11.4-1 and Provisions Equations 11.4-9, 11.4-10 and 11.4-11. The design spectral response acceleration ordinates, Sa, may be divided into four regions based on period, T, as described below.

S D1 ⎛ 0.433 ⎞ = 0.2 ⎜ ⎟ = 0.101 seconds, Sa varies linearly from 0.4SDS to SDS. S DS ⎝ 0.859 ⎠ S 0.433 From T0 to TS = D1 = = 0.504 seconds, Sa is constant at SDS. S DS 0.859 From TS to TL, Sa is inversely proportional to T, being anchored to SD1 at T = 1 second. S At periods greater than TL, Sa is inversely proportional to the square of T, being anchored to D1 at TL. TL

From T = 0 to T0 = 0.2

As prescribed in Provisions Section 11.4.6, the MCER response spectrum is determined by multiplying the design response spectrum ordinates by 1.5. Figure 3-1 shows the design and MCER response spectra determined using the ground motion parameters computed in Section 3.2.3.

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Chapter 3: Earthquake Ground Motion

1.2 ) g( a S , n iot ar el ec c A la rt ce p S

1.0

MCER spectrum

0.8 0.6 0.4 design spectrum

0.2 T0

0.0 0

TS 1

Period, T (s)

Figure 3-1 Horizontal Response Spectra for Design and MCER Ground Motions 3.2.5

Vertical Response Spectra

Part 1 of the 2009 Provisions adds a new chapter (Chapter 23) to ASCE 7-05 to define vertical ground motions for seismic design. The design vertical response spectrum is constructed in accordance with Provisions Section 23.1 using Provisions Equations 23.1-1, 23.1-2, 23.1-3 and 23.1-4. Vertical ground motion values are related to horizontal ground motion values by a vertical coefficient, Cv, which is determined as a function of site class and the MCER spectral response parameter at short periods, SS. The design vertical spectral response acceleration ordinates, Sav, may be divided into four regions based on vertical period, Tv, as described below. Using Provisions Table 23.1-1, read Cv = 1.3 for SS ≥ 2.0 and Cv = 1.1 for SS = 1.0 for Site Class C. Using linear interpolation for SS = 1.289,

Cv = 1.1 +

1.289 − 1 (1.3 − 1.1) = 1.158 2 −1

From Tv = 0 to 0.025 seconds, Sav is constant at 0.3CvSDS = 0.3(1.158)(0.859) = 0.298. From Tv = 0.025 to 0.05 seconds, Sav varies linearly from 0.3CvSDS = 0.298 to 0.8CvSDS = 0.8(1.158)(0.859) = 0.796. From Tv = 0.05 to 0.15 seconds, Sav is constant at 0.8CvSDS = 0.796. From Tv = 0.15 to 2.0 seconds, Sav is inversely proportional to Tv0.75, being anchored to 0.8CvSDS = 0.796 at Tv = 0.15 seconds. For vertical periods greater than 2.0 seconds, the vertical response spectral acceleration must be determined using site-specific procedures. As prescribed in Provisions Section 23.2, the MCER vertical response spectrum is determined by multiplying the design vertical response spectrum ordinates by 1.5. Figure 3-2 shows the design and MCER vertical response spectra determined using the ground motion parameters computed in Section 3.2.3.

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FEMA P-752, NEHRP Recommended Provisions: Design Examples

1.2

) g(

S , n oi t rae le cc A la rt ce p S la ci tr e V

MCER spectrum

1.0 0.8 0.6

design spectrum

0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Vertical Period, Tv (s)

Figure 3-2 Vertical Response Spectra for Design and MCER Ground Motions 3.2.6

Peak Ground Accelerations

Part 1 of the 2009 Provisions modifies Section 11.8.3 of the ASCE 7-05 to update the calculation of peak ground accelerations used for assessment of the potential for liquefaction and soil strength loss and for determination of lateral earth pressures for design of basem*nt and retaining walls. Given the site location, the following value of maximum considered earthquake geometric mean peak ground acceleration may be determined using the online application (or read from Provisions Figure 22-8). PGA = 0.521 g Using this mapped peak ground acceleration value and the site class, site coefficient FPGA is determined in accordance with Section 11.8.3 using Table 11.8-1. Using Table 11.8-1, for PGA = 0.521 > 0.5, FPGA = 1.0 for Site Class C. Using Provisions Equation 11.8-1 to determine the maximum considered earthquake geometric mean peak ground acceleration adjusted for site class effects, PGAM = FPGA PGA = 1.0(0.521) = 0.521 g This value is used directly to assess the potential for liquefaction or for soil strength loss. The design peak ground acceleration used to determine dynamic seismic lateral earth pressures for design of basem*nt and retaining walls is computed as 2 3 PGAM = 2 3 (0.521) = 0.347 g.

3.3 SELECTION AND SCALING OF GROUND MOTION RECORDS Response history analysis (whether linear or nonlinear) consists of the step-wise application of timevarying ground accelerations to a mathematical model of the subject structure. The selection and scaling of appropriate horizontal ground motion acceleration time histories is essential to produce meaningful

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Chapter 3: Earthquake Ground Motion results. For two-dimensional or three-dimensional structural analysis, single-component or twocomponent records are used, respectively. The sections that follow discuss the approach to selection and scaling of ground motion records as prescribed in the Provisions (and ASCE 7), illustrate the selection and scaling of two-component ground motions for the structure analyzed in Design Example 6.3 located at the site considered in Section 3.2 and discuss differences in the process for single-component ground motions. 3.3.1

Approach to Ground Motion Selection and Scaling

In the simplest terms the goal of ground motion selection and scaling is to produce acceleration histories that are consistent with the ground shaking hazard anticipated for the subject structure at the site in question. As difficult as it is to forecast the occurrence of an earthquake, it is even more difficult to predict the precise waveform and phasing of the resulting accelerations at a given site. Instead it is necessary to approximate (somewhat crudely) what ground motions can be expected based on past observations (and, possibly, geologic modeling). Provisions Section 16.1.3 prescribes the most commonly applied approach to this process. While some aspects of the process are quite prescriptive, others permit considerable latitude in application. The Pacific Earthquake Engineering Research Center makes available a database of ground motions (at http://peer.berkeley.edu/peer_ground_motion_database/) and a web application for the selection and scaling of ground motions (PEER, 2010). As useful as that data and application are, they do not provide a comprehensive solution to the challenge of ground motion selection and scaling in accordance with the Provisions for all U.S. sites. Pertinent limitations include the following. §

The database is limited to shallow crustal earthquakes recorded in “active tectonic regimes,” like parts of the western U.S. It does not include records from subduction zone earthquakes, deep intraplate events, or events in less active tectonic regimes (such as the central and eastern U.S.).

§

The web application allows use of a code design spectrum (from the Provisions or ASCE 7) as a target and includes powerful selection and scaling methods. However, the set of selected and scaled records produced would still require minor adjustment (scaling up) to satisfy the requirements in Provisions Section 16.1.3 over the period range of interest.

3.3.1.1 Number of ground motions. In recognition of the impossibility of predicting the actual ground motion history that should be expected, Section 16.1.3 requires the use of at least three ground motions in any response history analysis. Where at least seven ground motions are used, Sections 16.1.4 and 16.2.4 permit the use of average response quantities for design. The difference is not one of statistical significance; in either case mean response is approximated, but an incentive is given for the use of more records, which could identify a potential sensitivity in the response. The objective of the response history analyses is not to evaluate the response of the building for each record (since none of the records used will actually occur), but to determine the expected (average) response quantities for use in design calculations. If the analysis predicts collapse for one or more ground motions, the average cannot be computed; the structure is deemed inadequate and must be redesigned. 3.3.1.2 Recorded or synthetic ground motions. Horizontal ground motion acceleration records should be selected as single components (for two-dimensional analysis) or as orthogonal pairs (for threedimensional analysis) from actual recorded events. Where the number of appropriate recorded ground motions is insufficient, use of “simulated” records is permitted. While generation of completely artificial records is not directly prohibited, the intent (as expressed in Provisions Section C16.1.3) is that such simulation is limited to modification for site distance and soil conditions.

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FEMA P-752, NEHRP Recommended Provisions: Design Examples

3.3.1.3 “Appropriate” ground motions. The measure of “appropriate” applied to ground motions by the Provisions is consistency with the magnitude, fault distance and source mechanism that control the maximum considered earthquake. (Other characteristics of ground motion, such as duration, may influence response, but are not addressed by the Provisions.) While it is good practice to select ground motions with these characteristics in mind, the available data are quite limited. And even where the available records are very carefully binned and match the target characteristics quite closely, they are far from hom*ogeneous. As discussed in Section 3.1 the mapped ground motion parameters reflect the likelihood that a certain level of spectral acceleration will be exceeded in a selected period, considering numerous sources of earthquake ground shaking. While the mapping process does not sum accelerations from various sources it does aggregate the probabilities of occurrence from those sources. As a result, it is impossible to determine the controlling source characteristics using only the mapped acceleration parameters. In order to identify the magnitude, fault distance and source mechanism that control the maximum considered earthquake at a specific spectral period, it is necessary to “deaggregate” the hazard, which requires reviewing the underlying calculations to note the relative contribution of each source. The USGS provides tools to deaggregate hazard, providing results in three formats: a text tabulation, a graphic presentation binned by distance and magnitude and a graphic presentation projected on a map. Figure 3-3 shows the two graphic formats for the 2-second period spectral acceleration with a 2% probability of exceedance in 50 years (the maximum considered earthquake) at the site considered in Section 3.2.

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Chapter 3: Earthquake Ground Motion

Figure 3-3 Graphic results of deaggregation At most sites deaggregation of hazard reveals that a single source controls the maximum considered earthquake ground motions for all spectral periods. However, at some sites different sources control the maximum considered earthquake ground motions at different spectral periods. Figure 3-4 shows, for one such site, the maximum considered earthquake response spectrum generated from mapped ground motion

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FEMA P-752, NEHRP Recommended Provisions: Design Examples parameters as well as median acceleration response spectra for two of the contributing sources. Since the shape of the uniform hazard spectrum, upon which the design spectrum is based, is artificial (arising from the probabilistic seismic hazard analysis rather than the characteristics of recorded ground motions), there may be conservatisms involved in providing an aggregate match for design purposes (PEER, 2010). However, that aggregate match is exactly what the Provisions requires, so it is prudent to consider how that conservatism may best be balanced.

mapped a

S , n oi t ar el ec c A l ar tc e p S

source 1

source 2

period range of interest

Period, T

Figure 3-4 Response spectra for a site with multiple controlling sources In this example, source 1 can generate moderate magnitude events close to the site; Source 2 can generate very large magnitude events far from the site. Due to differing source and attenuation characteristics, each source can control a portion of the MCER response spectrum. The response of short period structures or very long period structures will be governed by source 1 or source 2, respectively. However, the “controlling” source is less clear for a structure with a fundamental period shown as T1 in the figure. Source 2 appears to control at period T1, but as discussed in Section 3.3.1.5, the Provisions defines a wider period range of interest over which the selected ground motions must be “appropriate.” As outlined below, three approaches are readily apparent. §

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First (and arguably most technically correct), select two full sets of (seven or more) ground motion records conditionally—one set for each source, enveloping the MCER spectrum for the portion of the period range of interest controlled by that source. Since the corresponding portions of the actual and target spectral shapes would be similar, scale factors would be modest. In this case, an independent series of analyses would be performed for each set of ground motion records. Mean response parameters of interest could be computed for each set of analyses and the more conservative of the two mean values for each response parameter could be used for design verification. Although this approach has technical appeal, the Provisions do not outline such a

Chapter 3: Earthquake Ground Motion procedure that makes use of two sets of ground motions, instead requiring use of a single set that on average envelops the entire period range of interest of the target spectrum. §

Second, select a full set of ground motions consistent with Source 2 and then scale the set to envelop the much differently shaped MCER spectrum over the specified period range of interest. While permitted by the Provisions, this approach would require large scale factors that unrealistically exaggerate the long period response. It may seem that this set of ground motions has a desired degree of hom*ogeneity, but that comes at the expense of a very poor fit for the average.

§

Third, select a set with some ground motions for each controlling source type. Select individual scale factors so that the average of their linear elastic spectra envelops the target spectrum (as required by the Provisions) and is shaped similarly to the target. As a result of this process, records consistent with Source 1 will control short periods and those consistent with Source 2 will control long periods. The scale factors will be somewhat larger than those required by the first (conditional) approach, but not excessively large like those in the second approach. Although the record set is less hom*ogeneous than that used in the second approach, the average is much closer to the target. Where used for linear response history analysis, this approach will produce average response quantities consistent with the average linear response spectrum used in the scaling process. Where used for nonlinear response history analysis, this approach (which uses scale factors that are larger than those for the conditional approach) will bias the average response quantities to be slightly more conservative and may increase the prediction of response extremes (collapse). This third method is commonly employed by seismological consultants where multiple source types may govern.

3.3.1.4 Scale factors. The most commonly employed ground motion scaling method involves multiplying all of the acceleration values of the time-acceleration pairs by a scalar value. This timedomain scaling modifies the amplitude of the accelerations (to approximate changes in source magnitude and/or distance) without affecting frequency content or phasing. Although not limited by the Provisions, the scale factors applied to recorded ground motions should be modest (usually falling between 1/3 and 3); if very small or very large scale factors are needed, some aspect of the event that produced the source motion likely is inconsistent with the maximum considered earthquake being modeled. An identical scale factor is applied to both components of a given ground motion to avoid unrealistically biasing one direction of response. Since the response spectra for time-domain scaled ground motions retain their natural jaggedness, the acceptance criterion compares their average to the target spectrum, without imposing limits on the scaling of individual ground motions. That means that there is no single set of scale factors that may be applied to the selected ground motions (as discussed further in Provisions Part 2 Section C16.1.3.2) Another ground motion scaling method involves transforming the time-acceleration data into the frequency domain (such as by means of the fast Fourier transform), making adjustments (to match exactly the target spectrum at multiple, specific frequencies) and transforming back into the time domain. This method affects amplitude, frequency content and phasing (and tends to increase the total input energy). This method makes it possible to estimate mean response with fewer ground motions, but may obscure somewhat the potential variability of response. Use of this method is permitted by the Provisions, but the same number of records is required as for time-domain scaling. Given the jaggedness of individual response spectra, the process of spectral matching (which produces smoother spectra) requires scale factors that can be considerably smaller or larger than those used in time-domain scaling. Since this method applies numerous scale factors to differing frequencies of each ground motion component in order to match spectral ordinates, there is no requirement that the two components be scaled identically. As the spectral ordinates of frequency-domain scaled records may fall below the target spectrum at frequencies 3-19

FEMA P-752, NEHRP Recommended Provisions: Design Examples other than those used for matching, a second round of (minor) scaling is needed to satisfy the Provisions requirements. Where single-component records are being selected for two-dimensional analysis the design response spectrum is used as a target; and Provisions Section 16.1.3.1 requires that the average of the response spectra not fall below the target over the period range of interest. A different approach is needed where two-component records are being selected for three-dimensional analysis. The code writers selected the square root of the sum of the squares (SRSS) of the response spectra for the two components as a measure of the ground motion amplitude for each record. However, the SRSS of two spectra is always larger than the average (and larger than the maximum). In practical terms for ground motion, it is reasonable to expect that the SRSS is larger than the average by a factor of 1.4 to 1.5 and is larger than the maximum (resultant) by a factor of about 1.2. The code writers decided that it is sufficiently conservative to scale two-component records such that the average of the SRSS spectra does not fall below the target over the period range of interest. Given the relationship between SRSS and average, that means that scale factors for ground motions used in threedimensional analysis are only 2/3 of those for ground motions used in two-dimensional analysis. The rationale is that a three-dimensional analysis (using two-component ground motions) subjects the structure to the maximum (resultant) acceleration in some direction due to the interaction of ground motion components, while that is not possible in two-dimensional analysis. Considering other conservative criteria, such as fitting over the entire period range of interest, code writers accepted that the resultant acceleration could be about 20 percent less than the design acceleration at some periods. Note that Provisions Section 16.1.3.2 erroneously requires that the average of the SRSS spectra not fall below the MCER spectrum over the period range of interest. ASCE 7-10 corrects this error by requiring that the average of the SRSS spectra be compared to “the response spectrum used in the design” (rather than to the MCER response spectrum). For the special case described in Section 3.3.1.7 below, both ASCE 7-10 and the Provisions require scaling so that the maximum acceleration exceeds the MCER response spectrum. Apparently, this is an error carried forward from the Provisions to ASCE 7-10. Like the rest of Section 16.1.3.2, the target spectrum used for scaling should be “the response spectrum used in the design” rather than the MCE R response spectrum (which is 1.5 times the design response spectrum). 3.3.1.5 Period range of interest. The smooth spectral acceleration response spectrum constructed using mapped acceleration parameters (and site response coefficients) is a location-specific estimate of the ground shaking hazard. No matter how carefully recorded ground motions are selected and scaled, it is unrealistic to expect a close match to the smooth target spectrum over all periods. On the other hand, selecting and scaling ground motions to match the target spectrum at the natural period for the fundamental mode of vibration of a structure is not enough to produce reasonable estimates of response; important aspects of structural response (including collapse) are affected by both higher modes of response and period elongation due to yielding. To balance these realities, code writers have established a period range of interest (with respect to the fundamental period, T) that extends from 0.2T (to capture higher mode effects) to 1.5T (to include period elongation). Although yielding and period elongation cannot occur in linear response history analysis, for simplicity of application ground motions are selected and scaled considering the same period range of interest as for nonlinear response history analysis. 3.3.1.6 Orientation of ground motion components. Accelerometers record earthquake ground shaking along the vertical axis and two horizontal (orthogonal) instrument axes. Acceleration records can be used in the as-recorded orientation, but orientation in the directions normal to and parallel to the strike of the causative fault (termed the fault-normal and fault-parallel directions, respectively) by means of a simple trigonometric transformation permits greater seismological insights, since some ground motions recorded 3-20

Chapter 3: Earthquake Ground Motion very close to the causative fault contain rupture directivity effects. The differences may be meaningful for selection and scaling, application in analysis, or both. Since the orientation of instrument axes is arbitrary and reorientation along the fault-normal and fault-parallel directions can provide additional insight, it has become common (but not universal) to reorient all horizontal ground motion records in that manner. In the very common condition where a site is not within several miles of the controlling source, the orientation of ground shaking is inconsequential, so the Provisions contain no general requirement to consider orientation. As discussed in Section 3.3.1.7 below, there is a selection and scaling orientation requirement (but no application orientation requirement) for sites close to active controlling faults. Figure 3-5 shows the time series of two components of ground acceleration. Component 1 is faultnormal; component 2 is fault-parallel. What is not apparent in such traces is the interaction of the components. Figure 3-6 shows an orbit plot of ground acceleration pairs (effectively zero-period response) for the same recording. The maximum resultant acceleration occurs along a diagonal direction. 0.8 )g ( 1 t 0.4 en n o p o m 0 oc 0 n oi ta re -0.4 le cc A -0.8

Time (s)

0.8 )g ( 2t 0.4 n e n o p m 0 oc n 0 oi ta re -0.4 le cc A -0.8

Figure 3-5 Horizontal acceleration components for the 1989 Loma Prieta earthquake (Saratoga – Aloha Avenue recording station)

3-21

FEMA P-752, NEHRP Recommended Provisions: Design Examples 0.8

) g( 2 p m o C

0.4

Comp 1 (g)

0 -0.8

-0.4

0.4

0.8

-0.4

-0.8

Figure 3-6 Horizontal acceleration orbit plot for the 1989 Loma Prieta earthquake (Saratoga – Aloha Avenue recording station) Unfortunately, the direction of maximum ground acceleration may or may not correspond to the direction of maximum acceleration response at any other period and the direction of maximum response generally differs at various periods. If bilinear oscillators with various fundamental periods are subjected to the two-component acceleration record, response spectra like those in Figure 3-7 result. The uniaxial response spectra in that figure are identified by component. The “resultant” response spectrum indicates the maximum acceleration along any direction. The SRSS response spectrum is obtained by taking the square root of the sum of the squares of the corresponding component response spectrum ordinates. The case illustrated reflects a possible near-source condition: for long periods, the fault-normal component (component 1) is much larger than the fault-parallel component and is very close to the maximum (resultant) response. The Provisions do not require application of ground motions in multiple possible orientations. Whether using three, seven, or more pairs, it is acceptable to consider a single, arbitrary orientation of a given twocomponent pair. For example, analysis can be performed with “Component 1” applied in the +X direction without considering the implications of applying that component in the -X, +Y, -Y, or other directions. Since the objective of the analyses is to estimate “average” response quantities, it may be advisable (but is not required) to consider whether there is an unwanted directional bias in the selected and scaled ground motions. For instance, in the common case where the controlling source should not produce strongly directional response, records could be oriented when applied so that the average of the component 1 spectra is similar to the average of the component 2 spectra. The much-less-common case, where strongly directional response is expected, is discussed in Section 3.3.1.7. Section 12.4.4 of these Design Examples outlines a more involved approach that is recommended for seismically isolated structures.

3-22

Chapter 3: Earthquake Ground Motion

2.0

1.5 ) g( a S , n 1.0 oi t ar el ec c a la 0.5 rt ce p S

SRSS Resultant Component 1 Component 2

0.0 0.0

0.5

1.0

1.5

2.0 Period, T (s)

2.5

3.0

3.5

4.0

Figure 3-7 Horizontal acceleration response spectra for the 1989 Loma Prieta earthquake (Saratoga – Aloha Avenue recording station) 3.3.1.7 Sites close to controlling active faults. Ground motions at sites close to a causative fault can be strongly directional. At such sites, the maximum long period ground motion often occurs in the faultnormal direction. The last paragraph of Provisions Section 16.1.3.2 addresses this case, where code writers have judged that scaling should be more conservative than that achieved using the SRSS-based method. Although this requirement is well intentioned, the specific language provides a degree of additional conservatism that can vary greatly. The intent is that the maximum spectral acceleration for the scaled motions exceeds the target response spectrum. While it is often true that the fault-normal component is dominant at long periods, some near-field ground motions show no directional bias and some are dominant in the fault-parallel direction. For instance, of the 3182 records in the PEER Ground Motion Database (for shallow crustal earthquakes), only 109 have pulse-like directional effects. Of those, 60 have pulses only in the fault-normal direction, 19 have pulses only in the fault-parallel direction, and 30 have pulses in both directions. As discussed above, it is acceptable to reorient all horizontal ground motion records to the fault-normal and fault-parallel directions. However, that does not assure that the fault-normal component will coincide with the maximum. Figure 3-8 shows response spectra for a ground motion where the fault-parallel direction (component 2) dominates for long periods. Scaling such that the fault-normal component exceeds the target response spectrum, as required in the last paragraph of Section 16.1.3.2, would force the maximum well above the target response spectrum. To obtain the intended result, ground motions should “be scaled so that the average of the fault-normal dominant components is not less than the MCER response spectrum used in the design for the period range from 0.2T to 1.5T.” (Section 3.3.1.4 above explains why all of Section 16.1.3.2 should refer to the response spectrum used in the design rather than to the MCER response spectrum.) While the Provisions set forth orientation requirements for the selection and scaling of ground motions at sites close to controlling active faults, the orientation of ground motion components as applied in analysis is not prescribed. After going to the effort of orienting records in the fault-normal and fault-parallel directions and applying special rules for scaling in recognition of near-source effects, it would be prudent

3-23

FEMA P-752, NEHRP Recommended Provisions: Design Examples (but not required) to apply the records in the analyses consistent with the fault-normal and fault-parallel directions at the actual site. 2.0

) (g

1.5

S , n 1.0 oi t ar el ec c a la 0.5 rt ce p S

SRSS Resultant Component 1 Component 2

0.0 0.0

0.5

1.0

1.5

2.0 Period, T (s)

2.5

3.0

3.5

4.0

Figure 3-8 Horizontal acceleration response spectra for 1999 Duzce, Turkey earthquake (Duzce recording station)

3.3.2

Two-‐Component Records for Three Dimensional Analysis

Design Example 6.3 is a buckling restrained braced frame structure located at the Seattle, Washington site considered in Section 3.2. Some aspects of the design are based on results from three-dimensional nonlinear response history analysis performed in accordance with Provisions Section 16.2. This section illustrates application of the procedures described in Section 3.3.1 for the selection and scaling of twocomponent ground motion records. Pertinent information from Sections 3.2 and 6.3.6.1 is summarized as follows. § § § § § §

Location: 47.65ºN, 122.3ºW Site Class C SMS = 1.289 SM1 = 0.649 TL = 6 seconds Tx = Ty = 2.3 seconds

The period range of interest is from 0.2 × 2.3 = 0.46 seconds to 1.5 × 2.3 = 3.45 seconds. If the two fundamental translational periods differed, the period range of interest would extend from 0.2 times the shorter period to 1.5 times the longer period. The next step is to deaggregate the hazard, as discussed in Section 3.3.1.3, over the period range of interest. Figure 3-9 shows the MCER (target) response spectrum and the relative contributions of three important sources to spectral acceleration at periods between 0 and 4 seconds. For periods greater than about 1.5 seconds, ground shaking hazard is controlled by very large, but distant, subduction zone events. At shorter periods, hazard is controlled by deep intraplate events, with substantial contributions from shallow crustal events. It is necessary to identify not only the magnitude of the controlling event, but also

3-24

Chapter 3: Earthquake Ground Motion the distance and source type. Short and intermediate period response may be more important than long period response (depending on the period of the structure).

1.5 ) g( a S , n iot 1 ar el ec c a la rt ce 0.5 p S

0 0

d r az a h ot n oi t u ib rt n oc ev ti al e R

2 Period, T (s)

Cascadia subduction zone M8.5 - 9.0 d = 110 - 134 km

Deep intraplate M6.6 - 7.0 d = 65 - 74 km

Shallow crustal M6.8 - 7.2 d = 8 - 13 km

Figure 3-9 MCER response spectrum and corresponding hazard contributions Since the MCER response spectrum over the period range of interest is controlled by multiple sources with substantially different spectral shapes, the procedure recommended in Section 3.3.1.3 is used. Table 3-1 provides key information for the selected ground motion records. Few large magnitude subduction zone records are available. Records 1, 2 and 3 are for slightly smaller events than those that control the long period hazard, but at closer distances. These differences are partially offsetting so the required scale factors are acceptable. Record 4 is nearly a perfect match for the hazard that controls short period response; and it is from a past occurrence of a similar event in the same region. Records 5, 6 and 7 are from shallow crustal events with magnitude and distance appropriate for this site. Two of those records include near-source velocity pulses. In a manner similar to that illustrated in Figure 3-4, the actual spectra for the selected ground motion records control different periods of response. Figure 3-10 shows the SRSS spectra for Records 1 and 4, along with the target (MCER) spectrum. The subduction zone event (Record No. 1) dominates long period response; the deep intraplate event (Record No. 4) dominates short period response.

3-25

FEMA P-752, NEHRP Recommended Provisions: Design Examples Table 3-1 Selected and Scaled Ground Motions for Example Site Record No. 1 2 3 4 5 6 7

Year 2003 2003 1968 1949 1989 1999 1995

Earthquake name Tokachi-oki, Japan Tokachi-oki, Japan Tokachi-oki, Japan Western Washington Loma Prieta Duzce, Turkey Kobe, Japan

M 8.3 8.3 8.2 7.1 6.9 7.1 6.9

Source type Subduction zone Subduction zone Subduction zone Deep intraplate Shallow crustal Shallow crustal Shallow crustal

Recording station HKA 094 HKD 092 Hachinohe (S-252) Olympia Saratoga -- Aloha Ave Duzce Nishi-Akashi

Distance (km) 67 46 71 75 9 7 7

Scale factor 2.99 0.96 1.28 1.92 1.28 0.85 1.18

Record no. 4

) g( 1.5 a S , n oi t ar el 1 ec c a la rt ce p0.5 S

Record no. 1

0 0

2 Period, T (s)

Figure 3-10 SRSS response spectra from different source types Figure 3-11 compares the average of the SRSS spectra for the selected ground motions with the target (MCER) response spectrum. It also shows the period range of interest for ground motion selection for this structure. In an average sense the suite of ground motions provides a very good fit to the target. Since seven records are used, average response quantities may be used in design. This suite so well matches the target spectrum that it could be used with no modification for periods from 0.18 to 4.95 seconds, a range much wider than the period range of interest defined in Provisions Section 16.1.3.2. Since this suite of ground motions has been selected and scaled to match the MCER response spectrum, an additional scale factor of 2/3 must be applied when the records are used in an analysis to represent design-level conditions.

3-26

Chapter 3: Earthquake Ground Motion

1.5

Average of SRSS spectra

)g ( a S , n1.0 iot ar el ec ca l ar tc e0.5 p S

MCER spectrum

For fundamental structural period of 2.30 s, the period range of interest is 0.46 to 3.45 s. 0.0 0.0

1.0

2.0

3.0

4.0

Period, T (s)

Figure 3-11 Fit of the selected suite of ground motion records to the target spectrum (for three-dimensional analysis) 3.3.3

One-‐Component Records for Two-‐Dimensional Analysis

As discussed in Section 3.3.1.4, one-component records (for use in two-dimensional analysis) are selected and scaled such that their average fits the design response spectrum, which is two-thirds of the MCER response spectrum. Figure 3-12 compares the average of the 14 component spectra (for the records selected and scaled in Section 3.3.2) to the design response spectrum. These records provide an excellent fit to the target spectrum. The suite of 14 records could be used without modification. If a subset of seven records were selected, some minor adjustment to scale factors might be required. Figure 3-12 also shows the average of the SRSS spectra for those 14 scaled records. As observed in Section 3.3.1.4, the average of the SRSS spectra is about 1.5 times the average of the component spectra. Therefore, if the same suite of records was used for three-dimensional analysis, the scale factors required would be about 2/3 of those required for two-dimensional analysis, due to the difference between average and SRSS spectra (and not due to the purely coincidental 2/3 relationship between design and MCE R response spectra).

3-27

FEMA P-752, NEHRP Recommended Provisions: Design Examples

Average of SRSS spectra

Spectral acceleration, Sa (g)

1.5

1.0

Average of component spectra

0.5

Design spectrum

0.0

For fundamental structural period of 2.30 s, the period range of interest is 0.46 to 3.45 s.

0.0

1.0

2.0

3.0

Period, T (s)

Figure 3-12 Fit of the selected suite of ground motion records to the target spectrum (for two-dimensional analysis)

3.3.4

References

PEER. 2010. Technical Report for the PEER Ground Motion Database Web Application, Pacific Earthquake Engineering Research Center, Berkeley, California.

3-28

4.0

4 Structural Analysis Finley Charney, Adrian Tola Tola and Ozgur Atlayan Contents 4.1

IRREGULAR 12-STORY STEEL FRAME BUILDING, STOCKTON, CALIFORNIA ........... 3

4.1.1

Introduction............................................................................................................................ 3

4.1.2

Description of Building and Structure ................................................................................... 3

4.1.3

Seismic Ground Motion Parameters ...................................................................................... 4

4.1.4

Dynamic Properties ............................................................................................................... 8

4.1.5

Equivalent Lateral Force Analysis....................................................................................... 11

4.1.6

Modal Response Spectrum Analysis ................................................................................... 29

4.1.7

Modal Response History Analysis....................................................................................... 39

4.1.8

Comparison of Results from Various Methods of Analysis ................................................ 50

4.1.9

Consideration of Higher Modes in Analysis ....................................................................... 53

4.1.10

Commentary on the ASCE 7 Requirements for Analysis ................................................... 56

4.2

SIX-STORY STEEL FRAME BUILDING, SEATTLE, WASHINGTON ................................ 57

4.2.1

Description of Structure ....................................................................................................... 57

4.2.2

Loads.................................................................................................................................... 60

4.2.3

Preliminaries to Main Structural Analysis ........................................................................... 64

4.2.4

Description of Model Used for Detailed Structural Analysis .............................................. 72

4.2.5

Nonlinear Static Analysis .................................................................................................... 94

4.2.6

Response History Analysis ................................................................................................ 109

4.2.7

Summary and Conclusions ................................................................................................ 134

FEMA P-751, NEHRP Recommended Provisions: Design Examples This chapter presents two examples that focus on the dynamic analysis of steel frame structures: 1. A 12-story steel frame building in Stockton, California. The highly irregular structure is analyzed using three techniques: equivalent lateral force analysis, modal response spectrum analysis and modal response history analysis. In each case, the structure is modeled in three dimensions and only linear elastic response is considered. The results from each of the analyses are compared and the accuracy and relative merits of the different analytical approaches are discussed. 2. A six-story steel frame building in Seattle, Washington. This regular structure is analyzed using both linear and nonlinear techniques. Due to the regular configuration of the structural system, the analyses are performed for only two dimensions. For the nonlinear analysis, two approaches are used: static pushover analysis and nonlinear response history analysis. The relative merits of pushover analysis versus response history analysis are discussed. Although the Seattle building, as originally designed, responds reasonably well under the design ground motions, a second set of response history analyses is presented for the structure augmented with added viscous fluid damping devices. As shown, the devices have the desired effect of reducing the deformation demands in the critical regions of the structure. In addition to the Standard, the following documents are referenced: AISC 341

American Institute of Steel Construction. 2005. Seismic Provisions for Structural Steel Buildings.

AISC 358

American Institute of Steel Construction. 2005. Prequalified Connections for Special and Intermediate Steel Moment Frames for Seismic Applications.

AISC 360

American Institute of Steel Construction. 2005. Specification for Structural Steel Buildings.

AISC Manual

American Institute of Steel Construction. 2005. Manual of Steel Construction, 13th Edition.

AISC SDM

American Institute of Steel Construction. 2006. Seismic Design Manual.

ASCE 41

American Society of Civil Engineers. 2006. Seismic Rehabilitation of Existing Buildings.

ASCE 7-10

American Society of Civil Engineers. 2010. Minimum Design Loads for Buildings and Other Structures

Charney & Marshall

Charney, F. A. and Marshall, J. D., 2006, “A comparison of the Krawinkler and Scissors models for including beam-column joint deformations in the analysis of steel frames,” Engineering Journal, 43(1), 31-48.

Charney (2008)

Charney, F. A., 2008, “Unintended consequences of modeling damping in structures,” Journal of Structural Engineering, 134(4), 581-592.

Clough & Penzien

Ray W. Clough and Joseph Penzien, Dynamics of Structures, 2nd Edition.

4-2

Chapter 4: Structural Analysis FEMA 440

Federal Emergency Management Agency. 2005. Improvement of Nonlinear Static Seismic Analysis Procedures

FEMA P-750

Federal Emergency Management Agency. 2010. 2009 NEHRP Recommended Seismic Provisions for Buildings and Other Structures

Prakash et al. (1993)

Prakash, V., Powell, G.H. and Campbell, S., 1993, Drain 2DX Base Program Description and User’s Guide, University of California, Berkeley, CA.

Uang & Bertero

Uang C.M. and Bertero V.V., 1990, “Evaluation of Seismic Energy in Structures”, Earthquake Engineering and Structural Dynamics, 19, 77-90.

4.1 4.1.1

IRREGULAR 12-‐STORY STEEL FRAME BUILDING, STOCKTON, CALIFORNIA Introduction

This example presents the analysis of a 12-story steel frame building under seismic effects acting alone. Gravity forces due to dead and live load are not computed. For this reason, member stress checks, member design and detailing are not discussed. Load combinations that include gravity effects are considered, however. For detailed examples of the seismic-resistant design of structural steel buildings, see Chapter 6 of this volume of design examples. The analysis of the structure, shown in Figures 4.1-1 through 4.1-3, is performed using three methods: 1. The Equivalent Lateral Force (ELF) procedure based on the requirements of Standard Section 12.8, 2. The modal response spectrum procedure based on the requirements of Standard Section 12.9 and 3. The modal response history procedure based on the requirements of Chapter 16 of ASCE 7-10. (The 2010 version of the Standard is used for this part of the example because it eliminates several omissions and inconsistencies that were present in Chapter 16 of ASCE 7-05.) In each case, special attention is given to applying the Standard’ rules for direction of loading and for accidental torsion. All analyses were performed in three dimensions using the finite element analysis program SAP2000 (developed by Computers and Structures, Inc., Berkeley, California). 4.1.2

Description of Building and Structure

The building has 12 stories above grade and a one-story basem*nt below grade and is laid out on a rectangular grid with a maximum of seven 30-foot-wide bays in the X direction and seven 25-foot bays in the Y direction. Both the plan and elevation of the structure are irregular with setbacks occurring at Levels 5 and 9. All stories have a height of 12.5 feet except for the first story which is 18 feet high and the basem*nt which extends 18 feet below grade. Reinforced 1-foot-thick concrete walls form the perimeter of the basem*nt. The total height of the building above grade is 155.5 feet. Gravity loads are resisted by composite beams and girders that support a normal-weight concrete slab on metal deck. The slab has an average thickness of 4.0 inches at all levels except Levels G, 5 and 9. The slabs on Levels 5 and 9 have an average thickness of 6.0 inches for more effective shear transfer through the diaphragm. The slab at Level G is 6.0 inches thick to minimize pedestrian-induced vibrations and to support heavy floor loads. The low roofs at Levels 5 and 9 are used as outdoor patios and support heavier live loads than do the upper roofs or typical floors.

4-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples

At the perimeter of the base of the building, the columns are embedded into pilasters cast into the basem*nt walls, with the walls supported on reinforced concrete tie beams over drilled piers. Interior columns are supported by concrete caps over drilled piers. A grid of reinforced concrete grade beams connects all tie beams and pier caps. The lateral load-resisting system consists of special steel moment frames at the perimeter of the building and along Grids C and F. For the frames on Grids C and F, the columns extend down to the foundation, but the lateral load-resisting girders terminate at Level 5 for Grid C and Level 9 for Grid F. Girders below these levels are simply connected. Since the moment-resisting girders terminate in Frames C and F, much of the Y direction seismic shears below Level 9 are transferred through the diaphragms to the frames on Grids A and H. Overturning moments developed in the upper levels of these frames are transferred down to the foundation by axial forces in the columns. Columns in the moment-resisting frame range in size from W24x146 at the roof to W24x229 at Level G. Girders in the moment frames vary from W30x108 at the roof to W30x132 at Level G. Members of the moment-resisting frames have a nominal yield strength of 36 ksi and floor members and interior columns that are sized strictly for gravity forces have a nominal yield strength of 50 ksi. 4.1.3

Seismic Ground Motion Parameters

For this example the relevant seismic ground motion parameters are as follows: §

SS = 1.25

§

S1 = 0.40

§

Site Class C

From Standard Tables 11.4-1 and 11.4-2: §

Fa = 1.0

§

Fv = 1.4

Using Standard Equations 11.4-1 through 11.4-4: §

SMS = FaSs = 1.0(1.25) = 1.25

§

SM1 = FvS1 = 1.4(0.4) = 0.56

§

S DS =

2 2 S MS = (1.25) = 0.833 3 3

§

S D1 =

2 2 S M 1 = (0.56) = 0.373 3 3

As the primary occupancy of the building is business offices, the Occupancy Category is II (Standard Table 1-1) and the Importance Factor (I) is 1.0 (Standard Table 11.5-1). According to Standard Tables 11.6-1 and 11.6-2, the Seismic Design Category (SDC) for this building is D.

4-4

Chapter 4: Structural Analysis

62'-6"

45'-0"

Y X

(a) Level 10

Y X

(b) Level 6

1 2

3 X

4 5 6

Origin for center of mass

7 8 7 at 30'-0"

Figure 4.1-1 Various floor plans of 12-story Stockton building

4-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples

R 12 11 10 11 at 12'-6"

9 All moment connections

8 7 6 5 4 3

2 at 18'-0"

2 G B 7 at 30'-0"

Section A-A 1

R 12 11

Pinned connections

Moment connections

10 11 at 12'-6"

9 8 7 6 5 4 3

2 at 18'-0"

2 G B 7 at 25'-0"

Section B-B

Figure 4.1-2 Sections through Stockton building

4-6

Chapter 4: Structural Analysis

Figure 4.1-3 Three-dimensional wire-frame model of Stockton building The lateral load-resisting system of the building is a special moment-resisting frame of structural steel. For this type of system, Standard Table 12.2-1 has a response modification coefficient (R) of 8 and a deflection amplification coefficient (Cd) of 5.5. There is no height limit for special moment frames. Section 12.2.5.5 of the standard requires that special moment frames in SDC D, where required by Table 12.1-1, be continuous to the foundation. While the girders of the interior moment frames are not present at the lower levels of the interior frames, the frames are continuous to the foundation and the columns are detailed as required for special moment frames. Additionally, there are no other structural system types below the moment frames. Therefore, in the opinion of the author, the requirement is met. Standard Table 12.6-1 is used to determine the minimum level of analysis. Because of the setbacks, the structure clearly has a weight irregularity (Irregularity Type 2 in Standard Table 12.3-2). Thus, the minimum level of analysis required for the SDC D building is modal response spectrum analysis. However, the determination of torsional irregularities, the application of accidental torsion effects and the assessment of P-delta effects are based on ELF analysis procedures. For this reason and for comparison purposes, a complete ELF analysis is carried out and described herein.

4-7

FEMA P-751, NEHRP Recommended Provisions: Design Examples 4.1.4

Dynamic Properties

Before any analysis can be carried out, it is necessary to determine the dynamic properties of the structure. These properties include stiffness, mass and damping. The stiffness of the structure is numerically represented by the system stiffness matrix, which is computed automatically by SAP2000. The terms in this matrix are a function of several modeling choices that are made. These aspects of the analysis are described later in the example. The computer can also determine the mass properties automatically, but for this analysis they are developed by hand and are explicitly included in the computer model. Damping is represented in different ways for the different methods of analysis, as described in Section 4.1.4.2. 4.1.4.1 Seismic Weight. In the past it was often advantageous to model floor plates as rigid diaphragms because this allowed for a reduction in the total number of degrees of freedom used in the analysis and a significant reduction in analysis time. Given the speed and capacity of most personal computers, the use of rigid diaphragms is no longer necessary and the floor plates may be modeled using 4-node shell elements. The use of such elements provides an added benefit of improved accuracy because the true “semi-rigid” behavior of the diaphragms is modeled directly. Where it is not necessary to recover diaphragm stresses, a very coarse element mesh may be used for modeling the diaphragm. Where the diaphragm is modeled using finite elements, the diaphragm mass, including contributions from structural dead weight and superimposed dead weight, is automatically represented by entering the proper density and thickness of the diaphragm elements. The density may be adjusted to represent superimposed dead loads (but the thickness and modulus are “true” values). Line mass, such as window walls and exterior cladding, are modeled with frame element line masses. While complete building masses are easily represented in this manner, the SAP2000 program does not automatically compute the locations of the centers of mass, so these must be computed separately. Center of mass locations are required for the purpose of applying lateral forces in the ELF method and for determining story drift. Due to the various sizes and shapes of the floor plates and to the different dead weights associated with areas within the same floor plate, the computation of mass properties is not easily carried out by hand. For this reason, a special purpose computer program was used. The basic input for the program consists of the shape of the floor plate, its mass density and definitions of auxiliary masses such as line, rectangular and concentrated mass. The uniform area and line masses (in weight units) associated with the various floor plates are given in Tables 4.1-1 and 4.1-2. The line masses are based on a cladding weight of 15.0 psf, story heights of 12.5 or 18.0 feet and parapets 4.0 feet high bordering each roof region. Figure 4.1-4 shows where each mass type occurs. The total computed floor mass, mass moment of inertia and locations of center of mass are shown in Table 4.1-3. Note that the mass moments of inertia are not required for the analysis but are provided in the table for completeness. The reference point for center of mass location is the intersection of Grids A and 8. Table 4.1-3 includes a mass computed for Level G of the building. This mass is associated with an extremely stiff story (the basem*nt level) and is dynamically excited by the earthquake in very high frequency modes of response. As shown later, this mass is not included in equivalent lateral force computations.

4-8

Chapter 4: Structural Analysis Table 4.1-1 Area Weights Contributing to Masses on Floor Diaphragms Area Weight Designation

Mass Type Slab and Deck (psf) Structure (psf) Ceiling and Mechanical (psf) Partition (psf) Roofing (psf) Special (psf) Total (psf)

50 20 15 10 0 0 95

75 20 15 10 0 0 120

50 20 15 0 15 0 100

75 20 15 0 15 60 185

75 50 15 10 0 25 175

See Figure 4.1-4 for mass location. 1.0 psf = 47.9 N/m2.

Table 4.1-2 Line Weights Contributing to Masses on Floor Diaphragms Mass Type From Story Above (plf) From Story Below (plf) Total (plf)

Line Weight Designation 1

60.0 93.8 153.8

93.8 93.8 187.6

93.8 0.0 93.8

93.8 135.0 228.8

135.0 1,350.0 1,485.0

See Figure 4.1-4 for mass location. 1.0 plf = 14.6 N/m.

4-9

FEMA P-751, NEHRP Recommended Provisions: Design Examples

1 D

3 2

Roof

2 2

Levels 3-4

5 5

2 2

Level 5

2 2

Levels 6-8

Levels 9

Levels 10-12

Area Mass

Line Mass

Level 2

Level G

Figure 4.1-4 Key diagram for computation of floor weights Table 4.1-3 Floor Weight, Floor Mass, Mass Moment of Inertia and Center of Mass Locations Level

Weight (kips)

R 12 11 10 9 8 7 6 5 4 3 2 G Σ

1,657 1,596 1,596 1,596 3,403 2,331 2,331 2,331 4,320 3,066 3,066 3,097 6,525 36,912

1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.

4-10

Mass (kip-sec2/in.) 4.287 4.130 4.130 4.130 8.807 6.032 6.032 6.032 11.19 7.935 7.935 8.015 16.89

Mass Moment of Inertia (in.-kipsec2//radian) 2.072x106 2.017x106 2.017x106 2.017x106 5.309x106 3.703x106 3.703x106 3.703x106 9.091x106 6.356x106 6.356x106 6.437x106 1.503x107

X Distance to C.M. (in.) 1,260 1,260 1,260 1,260 1,638 1,553 1,553 1,553 1,160 1,261 1,261 1,262 1,265

Y Distance to C.M. (in.) 1,050 1,050 1,050 1,050 1,175 1,145 1,145 1,145 1,206 1,184 1,184 1,181 1,149

Chapter 4: Structural Analysis 4.1.4.2 Damping. Where an equivalent lateral force analysis or a modal response spectrum analysis is performed, the structure’s damping, assumed to be 5 percent of critical, is included in the development of the spectral accelerations SS and S1. An equivalent viscous damping ratio of 0.05 is appropriate for linear analysis of lightly damaged steel structures. Where recombining the individual modal responses in modal response spectrum analysis, the square root of the sum of the squares (SRSS) technique has generally been replaced in practice by the complete quadratic combination (CQC) approach. Indeed, Standard Section 12.9.3 requires that the CQC approach be used where the modes are closely spaced. Where using the CQC approach, the analyst must correctly specify a damping factor. This factor, which is entered into the SAP2000 program, must match that used in developing the response spectrum. It should be noted that if zero damping is used in CQC, the results are the same as those for SRSS. For modal response history analysis, SAP2000 allows an explicit damping ratio to be used in each mode. For this structure, a damping of 5 percent of critical was specified in each mode. 4.1.5

Equivalent Lateral Force Analysis

Prior to performing modal response spectrum or response history analysis, it is necessary to perform an ELF analysis of the structure. This analysis is used for preliminary design, for evaluating torsional regularity, for computing torsional amplification factors (where needed), for application of accidental torsion, for evaluation of P-delta effects and for development of redundancy factors. The first step in the ELF analysis is to determine the period of vibration of the building. This period can be “accurately” computed from a three-dimensional computer model of the structure. However, it is first necessary to estimate the period using empirical relationships provided by the Standard. Standard Equation 12.8-7 is used to estimate the building period:

Ta = Ct hnx where, from Standard Table 12.8-2, Ct = 0.028 and x = 0.8 for a steel moment frame. Using hn = the total building height (above grade) = 155.5 ft, Ta = 0.028(155.5)0.8 = 1.59 seconds1. Even where the period is accurately computed from a properly substantiated structural analysis (such as an eigenvalue or Rayleigh analysis), the Standard requires that the period used for ELF base shear calculations not exceed CuTa where Cu = 1.4 (from Standard Table 12.8-1 using SD1 = 0.373). For the structure under consideration, CuTa = 1.4(1.59) = 2.23 seconds. Note that where the accurately computed period is less than CuTa, the computed period should be used. In no case, however, is it necessary to use a period less than T = Ta = 1.59 seconds. The use of the Rayleigh method and the eigenvalue method of determining accurate periods of vibration are illustrated in a later part of this example. In anticipation of the accurately computed period of the building being greater than 2.23 seconds, the ELF analysis is based on a period of vibration equal to CuTa = 2.23 seconds2. For the ELF analysis, it is 1

The proper computational units for period of vibration are theoretically “seconds/cycle”. However, it is traditional to use units of “seconds,” and this is done in the remainder of this example. 2 As shown later in this example, the computed period is indeed greater than CuTa.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples assumed that the structure is “fixed” at grade level. Hence, the total effective weight of the structure (see Table 4.1-3) is the total weight minus the grade level weight, or 36,920 – 6,526 = 30,394 kips. 4.1.5.1 Base shear and vertical distribution of force. Using Standard Equation 12.8-1, the total seismic base shear is:

V = CsW where W is the total weight of the structure. From Standard Equation 12.8-2, the maximum (constant acceleration region) spectral acceleration is:

CSmax =

S DS 0.833 = 0.104 = (R / I ) 8 / 1

Standard Equation 12.8-3 controls in the constant velocity region:

CS =

S D1 0.373 = 0.021 = T ( R / I ) 2.23(8 / 1)

However, the acceleration must not be less than that given by Standard Equation 12.8-5:

CSmin = 0.044ISDS ≥ 0.01 = 0.044(1)(0.833) = 0.037 The Cs value determined from Equation 12.8-5 controls the seismic base shear for this building. Using W = 30,394 kips, V = 0.037(30,394) = 1,124 kips. The acceleration response spectrum given by the above equations is plotted in Figure 4.1-5.

12.8-2 0.10 ) g( R /a S t n ei icf 0.05 efo C

12.8-3

12.8-5

0.00 0

Period, T (s)

Figure 4.1-5 Computed ELF total acceleration response spectrum

4-12

Chapter 4: Structural Analysis

35 30 )h cn i( 25 d S ,t n e 20 m ec a l p is 15 d l ra tc e 10 p S

12.8-5

12.8-3

5 12.8-2 0 0

Period, T (s)

Figure 4.1-6 Computed ELF relative displacement response spectrum While it is reasonable to use Equation 12.8-5 to establish a minimum base shear, the equation should not be used as a basis determining lateral forces used in displacement computations. The effect of using Equation 12.8-5 for displacements is shown in Figure 4.1-6, which represents Equations 12.8-2, 12.8-3 and 12.8-5 in the form of a displacement spectrum. It can be seen from this figure that the dotted line, representing Equation 12.8-5, will predict significantly larger displacements than Equation 12.8-3. The problem with the line represented by Equation 12.8-5 is that it gives an exponentially increasing displacement up to unlimited periods, whereas it is expected that the true spectral displacements will converge towards a constant displacement (the maximum ground displacement) at large periods. In other words, Equation 12.8-5 should not be considered as a branch of the response spectrum—it is simply used to represent the lower bound on design base shear. The Standard does not directly recognize this problem. However, Section 12.8.6.2 allows the deflection analysis of the seismic force-resisting system to be based on the accurately computed fundamental period of vibration, without the CuTa upper limit on period. It is the ’authors’ opinion that this clause may be used to justify drift calculations with forces based on Equation 12.8-3 even when Equation 12.8-5 controls the design base shear. ASCE 7-10 has clarified this issue, by providing an exception that specifically states that Equation 12.8-5 need not be considered for computing drift. It is important to note, however, that where Equation 12.8-6 controls the design base shear, drifts must be based on lateral forces consistent with Equation 12.8-6. This is due to the fact that Equation 12.8-6 is an approximation of the long period acceleration spectrum for “near field” ground motions (where S1 is likely to be greater than 0.6 g.) In this example, all ELF analysis is performed using the forces obtained from Equation 12.8-5, but for the purposes of computing drift, the story deflections are computed using the forces from Equation 12.8-3. When using Equation 12.8-3, the upper bound period CuTa was used in lieu of the computed period. This allows for a simple “conversion” of displacements where displacements computed from forces based on Equation 12.8-5 are multiplied by the factor (0.021/0.037 = 0.568) to obtain displacements that would be

4-13

FEMA P-751, NEHRP Recommended Provisions: Design Examples generated from forces based on Equation 12.8-3 and the CuTa limit. If it is found that the factored computed drifts violate the drift limits (which is not the case in this example), it might be advantageous to re-compute the drifts on the basis of Equation 12.8-3 and the computed period T. The seismic base shear computed according to Standard Equation 12.8-1 is distributed along the height of the building using Standard Equations 12.8-11 and 12.8-12:

Fx = CvxV and

Cvx =

wx h k

∑ wi hik i =1

where k = 0.75 + 0.5T = 0.75 + 0.5(2.23) = 1.865. The story forces, story shears and story overturning moments are summarized in Table 4.1-4. Table 4.1-4 Equivalent Lateral Forces for Building Response in X and Y Directions Level wx hx Fx Vx w xh xk Cvx x (kips) (ft) (kips) (kips) R 1,657 155.5 20,272,144 0.1662 186.9 186.9 12 1,596 143.0 16,700,697 0.1370 154.0 340.9 11 1,596 130.5 14,081,412 0.1155 129.9 470.8 10 1,596 118.0 11,670,590 0.0957 107.6 578.4 9 3,403 105.5 20,194,253 0.1656 186.3 764.7 8 2,331 93.0 10,933,595 0.0897 100.8 865.5 7 2,331 80.5 8,353,175 0.0685 77.0 942.5 6 2,331 68.0 6,097,775 0.0500 56.2 998.8 5 4,324 55.5 7,744,477 0.0635 71.4 1,070.2 4 3,066 43.0 3,411,857 0.0280 31.5 1,101.7 3 3,066 30.5 1,798,007 0.0147 16.6 1,118.2 2 3,097 18.0 679,242 0.0056 6.3 1,124.5 30,394 121,937,234 1.00 1124.5 Σ

Mx (ft-kips) 2,336 6,597 12,482 19,712 29,271 40,090 51,871 64,356 77,733 91,505 103,372 120,694

Values in column 4 based on exponent k=1.865. 1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN.

4.1.5.2 Accidental torsion and orthogonal loading effects. Where using the ELF method as the basis for structural design, two effects must be added to the direct lateral forces shown in Table 4.1-4. The first of these effects accounts for the fact that an earthquake can produce inertial forces that act in any direction. For SDC D, E and F buildings, Standard Section 12.5 requires that the structure be investigated for forces that act in the direction that causes the “critical load effects.” Since this direction is not easily defined, the Standard allows the analyst to load the structure with 100 percent of the seismic

4-14

Chapter 4: Structural Analysis force in one direction (along the X axis, for example) simultaneous with the application of 30 percent of the force acting in the orthogonal direction (along the Y axis). The other requirement is that the structure be modeled with additional forces to account for uncertainties in the location of center of mass and center of rigidity, uneven yielding of vertical systems and the possibility of torsional components of ground motion. For torsionally regular buildings, this requirement, given in Standard Section 12.8.4.2, can be satisfied by applying the equivalent lateral force at an “accidental” eccentricity, where the eccentricity is equal to 5 percent of the overall dimension of the structure in the direction perpendicular to the line of the application of force. For torsionally irregular structures in SDC C, D, E, or F, Standard Section 12.8.4.3 requires that the accidental eccentricity be amplified (although the amplification factor may be 1.0). According to Standard Table 12.3-1, a torsional irregularity exists if:

Δ max ≥ 1.2 Δ avg where δmax is the maximum story drift at the edge of the floor diaphragm and Δavg is the average drift at the center of the diaphragm (see Standard Figure 12.8-1). If the ratio of drifts is greater than 1.4, the torsional irregularity is referred to as “extreme.” In computing the drifts, the structure must be loaded with the basic equivalent lateral forces applied at a 5 percent eccentricity. For main loads acting in the X direction, displacements and drifts were determined on Grid Line D. For the main loads acting in the Y direction, the story displacements on Grid Line 1 were used. Because of the architectural setbacks, the locations for determining displacements associated with Δmin and Δmax are not always vertically aligned. This situation is shown in Figure 4.1-7, where it is seen that three displacement monitoring stations are required at Levels 5 and 9. The numerical values shown in Figure 4.1-7 are discussed later in relation to Table 4.1-5b.

4-15

FEMA P-751, NEHRP Recommended Provisions: Design Examples

3.5 l eve

4.4

4.1

3.6

2.5

1.9

5 1.5 3

ire

ctio

L of Ro l 12 ve Le 11 vel Le 10 vel Le 9 vel Le 8 vel Le 7 vel Le 6 vel Le 5 vel Le 4 vel Le 3 vel Le 2 vel Le vel Le d n ou Gr

3.2

4.1 3.7

2.5 1 2.0 9 1.6 2

ctio

e dir

Figure 4.1-7 Drift monitoring stations for determination of torsional irregularity and torsional amplification (deflections in inches, 1.0 in. = 25.4 mm) The analysis of the structure for accidental torsion was performed using SAP2000. The same model was used for ELF, modal response spectrum and modal response history analysis. The following approach was used for the mathematical model of the structure: 1. The floor diaphragm was modeled with shell elements, providing nearly rigid behavior in-plane. 2. Flexural, shear, axial and torsional deformations were included in all columns and beams. 3. Beam-column joints were modeled using centerline dimensions. This approximately accounts for deformations in the panel zone. 4. Section properties for the girders were based on bare steel, ignoring composite action. This is a reasonable assumption since most of the girders are on the perimeter of the building and are under reverse curvature.

4-16

Chapter 4: Structural Analysis 5. Except for those lateral load-resisting columns that terminate at Levels 5 and 9, all columns were assumed to be fixed at their base. 6. The basem*nt walls and grade level slab were explicitly modeled using 4-node shell elements. This was necessary to allow the interior columns to continue through the basem*nt level. No additional lateral restraint was applied at the grade level; thus, the basem*nt level acts as a very stiff first floor of the structure. This basem*nt level was not relevant for the ELF analysis, but it did influence the modal response spectrum and modal response history analyses as described in later sections of this example 7. P-delta effects were not included in the mathematical model. These effects are evaluated separately using the procedures provided in Standard Section 12.8.7. The results of the accidental torsion analysis are shown in Tables 4.1-5a and 4.1-5b. For loading in the X direction, there is no torsional irregularity because all drift ratios (Δmax/Δavg) are less than 1.2. For loading in the Y direction, the largest ratio of maximum to average story drift is 1.24 at Level 9 of the building. Hence, this structure has a Type 1 torsional irregularity, but only marginally so. See Figure 4.1-7 for the source of the dual displacement values shown for Levels 9 and 5 in Table 4.1-5b. Even though the torsional irregularity is marginal, Section 12.8.4.3 of the Standard requires that torsional amplification factors be determined for this SDC D building. The results for these calculations, which are based on story displacement, not drift, are presented in Tables 4.1-6a and 4.1-6b for the main load applied in the X and Y directions, respectively. As may be observed, the calculated amplification factors are significantly less than 1.0 at all levels for both directions of loading. Table 4.1-5a Computation for Torsional Irregularity with ELF Loads Acting in X Direction and Torsional Moment Applied Counterclockwise δ1 δ2 Δ1 Δ2 Δavg Δmax Irregularity Level Δmax/Δavg (in.) (in.) (in.) (in.) (in.) (in.) R 7.27 6.15 0.34 0.29 0.31 0.34 1.08 None 12 6.93 5.87 0.48 0.42 0.45 0.48 1.07 None 11 6.44 5.45 0.60 0.51 0.55 0.60 1.07 None 10 5.85 4.93 0.66 0.56 0.61 0.66 1.08 None 9 5.19 4.37 0.65 0.54 0.59 0.65 1.10 None 8 4.54 3.84 0.69 0.58 0.64 0.69 1.09 None 7 3.84 3.26 0.70 0.59 0.65 0.70 1.09 None 6 3.14 2.67 0.69 0.58 0.63 0.69 1.09 None 5 2.46 2.09 0.60 0.50 0.55 0.60 1.09 None 4 1.86 1.60 0.59 0.50 0.55 0.59 1.08 None 3 1.27 1.10 0.58 0.49 0.53 0.58 1.08 None 2 0.69 0.61 0.69 0.61 0.65 0.69 1.06 None 1.0 in. = 25.4 mm.

4-17

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 4.1-5b Computation for Torsional Irregularity with ELF Loads Acting in Y Direction and Torsional Moment Applied Clockwise δ2 Δ1 Δ2 Δavg Δmax δ1 Level Irregularity Δmax/Δavg (in.) (in.) (in.) (in.) (in.) (in.) R 5.19 4.77 0.15 0.14 0.15 0.15 1.03 None 12 5.03 4.63 0.25 0.23 0.24 0.25 1.03 None 11 4.79 4.40 0.31 0.29 0.30 0.31 1.04 None 10 4.48 4.11 0.38 0.34 0.36 0.38 1.06 None 3.77, 9 4.10 0.46 0.28 0.37 0.46 1.24 Irregularity 3.55 8 3.64 3.26 0.54 0.36 0.45 0.54 1.20 None 7 3.09 2.90 0.56 0.39 0.47 0.56 1.18 None 6 2.53 2.51 0.60 0.42 0.51 0.60 1.18 None 1.93, 5 2.09 0.41 0.47 0.44 0.47 1.06 None 1.95 4 1.53 1.62 0.47 0.50 0.48 0.50 1.03 None 3 1.07 1.12 0.47 0.50 0.48 0.50 1.03 None 2 0.60 0.63 0.60 0.63 0.61 0.63 1.03 None 1.0 in. = 25.4 mm.

Table 4.1-6a Amplification Factor Ax for Accidental Torsional Moment Loads Acting in the X Direction and Torsional Moment Applied Counterclockwise δ1 δ2 δavg δmax Level Ax calculated Ax used (in.) (in.) (in.) (in.) R 7.27 6.15 6.71 7.27 0.81 1.00 12 6.93 5.87 6.40 6.93 0.81 1.00 11 6.44 5.45 5.95 6.44 0.82 1.00 10 5.85 4.93 5.39 5.85 0.82 1.00 9 5.19 4.37 4.78 5.19 0.82 1.00 8 4.54 3.84 4.19 4.54 0.82 1.00 7 3.84 3.26 3.55 3.84 0.81 1.00 6 3.14 2.67 2.90 3.14 0.81 1.00 5 2.46 2.09 2.27 2.46 0.81 1.00 4 1.86 1.60 1.73 1.86 0.80 1.00 3 1.27 1.10 1.18 1.27 0.80 1.00 2 0.69 0.61 0.65 0.69 0.79 1.00 1.0 in. = 25.4 mm.

4-18

Chapter 4: Structural Analysis Table 4.1-6b Amplification Factor Ax for Accidental Torsional Moment Loads Acting in the Y Direction and Torsional Moment applied Clockwise δ1 δ2 δavg δmax Level Ax calculated Ax used (in.) (in.) (in.) (in.) R 5.19 4.77 4.98 5.19 0.75 1.00 12 5.03 4.63 4.83 5.03 0.75 1.00 11 4.79 4.40 4.59 4.79 0.76 1.00 10 4.48 4.11 4.29 4.48 0.76 1.00 9 4.10 3.55 3.82 4.10 0.80 1.00 8 3.64 3.26 3.45 3.64 0.77 1.00 7 3.09 2.90 3.00 3.09 0.74 1.00 6 2.53 2.51 2.52 2.53 0.70 1.00 5 1.95 2.09 2.02 2.09 0.74 1.00 4 1.53 1.62 1.58 1.62 0.73 1.00 3 1.07 1.12 1.10 1.12 0.73 1.00 2 0.60 0.63 0.61 0.63 0.73 1.00 1.0 in. = 25.4 mm.

4.1.5.3 Drift and P-delta effects. Using the basic structural configuration shown in Figure 4.1-1 and the equivalent lateral forces shown in Table 4.1-4, the total story deflections were computed as described in the previous section. In this section, story drifts are computed and compared to the allowable drifts specified by the Standard. For structures with “significant torsional effects”, Standard Section 12.12.1 requires that the maximum drifts include torsional effects, meaning that the accidental torsion, amplified as appropriate, must be included in the drift analysis. The same section of the Standard requires that deflections used to compute drift should be taken at the edges of the structure if the structure is torsionally irregular. For torsionally regular buildings, the drifts may be based on deflections at the center of mass of adjacent levels. As the structure under consideration is only marginally irregular in torsion, the lateral loads were placed at the center of mass and total drifts are based on center of mass deflections and not deflections at the edges of the floor plate. Using the centers of mass of the floor plates to compute story drift is awkward where the centers of mass of the upper and lower floor plates are not aligned vertically. For this reason, the story drift is computed as the difference between displacements of the center of mass of the upper level diaphragm and the displacement at a point on the lower diaphragm which is located directly below the center of mass of the upper level diaphragm. Note that computation of drift in this manner has been adopted in ASCE 7-10 Section 12.8-6. The values in Column 1 of Tables 4.1-7 and 4.1-8 are the total story displacements (δ) at the center of mass of the story as reported by SAP2000 and the values in Column 2 are the story drifts (Δ) computed from these numbers in the manner described earlier. The true elastic “amplified” story drift, which by assumption is equal to Cd (= 5.5) times the SAP2000 drift, is shown in Column 3. As discussed above in Section 4.1.5.1, the values in Column 4 are multiplied by 0.568 to scale the results to the base shear computed using Standard Equation 12.8-3. The allowable story drift of 2.0 percent of the story height per Standard Table 12.12-1 is shown in Column 5. (Recall that this building is assigned to Occupancy Category II.) It is clear from Tables 4.1-7 and 4.1-8 that the allowable drift is not exceeded at any level. It is also evident that the allowable drifts 4-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples would not have been exceeded even if accidental torsion effects were included in the drift calculations, with the drift determined at the edge of the building. Table 4.1-7 ELF Drift for Building Responding in X Direction 1 2 3 Total drift from Story drift from Amplified story Level SAP2000 SAP2000 drift (in.) (in.) (in.) R 6.67 0.32 1.74 12 6.35 0.45 2.48 11 5.90 0.56 3.07 10 5.34 0.62 3.39 9 4.73 0.58 3.20 8 4.15 0.63 3.47 7 3.52 0.64 3.54 6 2.87 0.63 3.47 5 2.24 0.54 2.95 4 1.71 0.54 2.97 3 1.17 0.53 2.90 2 0.64 0.64 3.51

4 Amplified drift times 0.568 (in.) 0.99 1.41 1.75 1.92 1.82 1.97 2.01 1.97 1.67 1.69 1.65 2.00

5 Allowable drift (in.) 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 4.32

Column 4 adjusts for Standard Eq. 12.8-3 (for drift) vs 12.8-5 (for strength). 1.0 in. = 25.4 mm.

Table 4.1-8 ELF Drift for Building Responding in Y Direction 1 2 3 Total drift from Story drift from Amplified story Level SAP2000 SAP2000 drift (in.) (in.) (in.) R 4.86 0.15 0.81 12 4.71 0.24 1.30 11 4.47 0.30 1.64 10 4.17 0.36 1.96 9 3.82 0.37 2.05 8 3.44 0.46 2.54 7 2.98 0.48 2.64 6 2.50 0.48 2.62 5 2.03 0.45 2.49 4 1.57 0.48 2.66 3 1.09 0.48 2.64 2 0.61 0.61 3.35

4 Amplified drift times 0.568 (in.) 0.46 0.74 0.93 1.11 1.16 1.44 1.50 1.49 1.42 1.51 1.50 1.90

Column 4 adjusts for Standard Eq. 12.8-3 (for drift) versus Eq. 12.8-5 (for strength). 1.0 in. = 25.4 mm.

4-20

5 Allowable drift (in.) 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 4.32

Chapter 4: Structural Analysis 4.1.5.3.1 Using ELF forces and drift to compute accurate period. Before continuing with the example, it is advisable to use the computed drifts to more accurately estimate the fundamental periods of vibration of the building. This will serve as a check on the “exact” periods computed by eigenvalue extraction in SAP2000. A Rayleigh analysis will be used to estimate the periods. This procedure, which usually is very accurate, is derived as follows: The exact frequency of vibration ω (a scalar), in units of radians/second, is found from the following eigenvalue equation:

Kφ = ω 2 M φ where K is the structure stiffness matrix, M is the (diagonal) mass matrix and φ is a vector containing the components of the mode shape associated with ω. If an approximate mode shape δ is used instead of φ, where δ is the deflected shape under the equivalent lateral forces F, the frequency ω can be closely approximated. Making the substitution of δ for φ, premultiplying both sides of the above equation by δT (the transpose of the displacement vector), noting that F = Kδ and M = W/g, the following is obtained:

δ T F = ω 2δ T M δ =

ω2 g

δ TW δ

where W is a vector containing the story weights and g is the acceleration due to gravity (a scalar). After rearranging terms, this gives:

ω= g

δTF δ TW δ

Using the relationship between period and frequency, T =

2π

Using F from Table 4.1-4 and δ from Column 1 of Tables 4.1-7 and 4.1-8, the periods of vibration are computed as shown in Tables 4.1-9 and 4.1-10 for the structure loaded in the X and Y directions, respectively. As may be seen from the tables, the X direction period of 2.85 seconds and the Y-direction period of 2.56 seconds are significantly greater than the approximate period of Ta = 1.59 seconds and also exceed the upper limit on period of CuTa = 2.23 seconds. Table 4.1-9 Rayleigh Analysis for X Direction Period of Vibration Drift, δ Force, F Weight, W δF Level (kips) (kips) (in.-kips) (in.) R 6.67 186.9 1,657 1,247 12 6.35 154.0 1,596 979 11 5.90 129.9 1,596 767 10 5.34 107.6 1,596 575 9 4.73 186.3 3,403 881 8 4.15 100.8 2,331 418

δ W/g (in.-kips-sec2) 191 167 144 118 197 104 2

4-21

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 4.1-9 Rayleigh Analysis for X Direction Period of Vibration Drift, δ Force, F Weight, W δF Level (kips) (kips) (in.-kips) (in.) 7 3.52 77.0 2,331 271 6 2.87 56.2 2,331 162 5 2.24 71.4 4,324 160 4 1.71 31.5 3,066 54 3 1.17 16.6 3,066 19 2 0.64 6.3 3,097 4 5,536 Σ ω = (5,536/1,138)0.5 = 2.21 rad/sec. T = 2π/ω = 2.85 sec. 1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.

Table 4.1-10 Rayleigh Analysis for Y Direction Period of Vibration Drift, δ Force, F Weight, W Level δF (kips) (kips) (in.) R 4.86 186.9 1,657 908 12 4.71 154.0 1,596 725 11 4.47 129.9 1,596 581 10 4.17 107.6 1,596 449 9 3.82 186.3 3,403 711 8 3.44 100.8 2,331 347 7 2.98 77.0 2,331 230 6 2.50 56.2 2,331 141 5 2.03 71.4 4,324 145 4 1.57 31.5 3,066 49 3 1.09 16.6 3,066 18 2 0.61 6.3 3,097 4 4,307 Σ

ω = (4,307/716)0.5 = 2.45 rad/sec. T = 2π/ω = 2.56 sec. 1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.

δ W/g (in.-kips-sec2) 75 50 56 23 11 3 1,138 2

δ W/g 2

101 92 83 72 128 72 54 38 46 20 9 3 716

4.1.5.3.2 P-delta effects. P-delta effects are computed for the X direction response in Table 4.1-11. The last column of the table shows the story stability ratio computed according to Standard Equation 12.8-163:

θ=

Px ΔI Vx hsx Cd

Note that the I in the numerator of Equation 12.8-16 was inadvertently omitted in early printings of the Standard.

4-22

Chapter 4: Structural Analysis

Standard Equation 12.8-17 places an upper limit on θ:

θ max =

0.5 β Cd

where β is the ratio of shear demand to shear capacity for the story. Conservatively taking β = 1.0 and using Cd = 5.5, θmax = 0.091. The Δ terms in Table 4.1-11 are taken from the fourth column of Table 4.1-7 because these are consistent with the ELF story shears of Table 4.1-4 and thereby represent the true lateral stiffness of the system. (If 0.568 times the story drifts were used, then 0.568 times the story shears also would need to be used. Hence, the 0.568 factor would cancel out since it would appear in both the numerator and denominator.) The deflections used in P-delta stability ratio calculations must include the deflection amplifier Cd. The live load PL in Table 4.1-11 is based on a 20 psf uniform live load over 100 percent of the floor and roof area. This live load is somewhat conservative because Section 12.8.7 of the Standard states that the gravity load should be the “total vertical design load”. For a 50 psf live load for office buildings, a live load reduction factor of 0.4 would be applicable for each level (see Standard Sec. 4.8), producing a reduced live load of 20 psf at the floor levels. This could be further reduced by a factor of 0.5 as allowed by Section 2.3.2, bringing the effective live load to 10 psf. This value is close to the mean survey live load of 10.9 psf for office buildings, as listed in Table C4.2 of the Standard. Several publications, including ASCE 41 include 25 percent of the unreduced live load in P-delta calculations. This would result in a 12.5 psf live load for the current example. Table 4.1-11 Computation of P-delta Effects for X Direction Response hsx Δ PD PL PT PX Level (in.) (in.) (kips) (kips) (kips) (kips) R 150 1.74 1,656.5 315.0 1,971.5 1,971.5 12 150 2.48 1,595.8 315.0 1,910.8 3,882.3 11 150 3.07 1,595.8 315.0 1,910.8 5,793.1 10 150 3.39 1,595.8 315.0 1,910.8 7,703.9 9 150 3.20 3,403.0 465.0 3,868.0 11,571.9 8 150 3.47 2,330.8 465.0 2,795.8 14,367.7 7 150 3.54 2,330.8 465.0 2,795.8 17,163.5 6 150 3.47 2,330.8 465.0 2,795.8 19,959.3 5 150 2.95 4,323.8 615.0 4,938.8 24,898.1 4 150 2.97 3,066.1 615.0 3,681.1 28,579.2 3 150 2.90 3,066.1 615.0 3,681.1 32,260.3 2 216 3.51 3,097.0 615.0 3,712.0 35,972.3

VX (kips) 186.9 340.9 470.8 578.4 764.7 865.8 942.5 998.8 1,070.2 1,101.7 1,118.2 1,124.5

θX 0.022 0.034 0.046 0.055 0.059 0.070 0.078 0.084 0.083 0.093 0.101 0.095

1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.

The stability ratio just exceeds 0.091 at Levels 2 through 4. However, the live loads were somewhat conservative and β was very conservatively taken as 1.0. Because a more refined analysis would provide somewhat lower live loads and a lower value of β, we will proceed assuming that P-delta effects are not a

4-23

FEMA P-751, NEHRP Recommended Provisions: Design Examples problem for this structure. Calculations for P-delta effects under Y direction loading gave no story stability ratios greater than 0.091 and for brevity, those results are not included herein. It is important to note that for this structure, P-delta effects are a potential issue even though drift limits were easily satisfied. This is often the case when drift limits are based on lateral loads that have been computed using the computed period of vibration (without the CuTa limit, or without the use of Equation 12.8.5). It is the authors’ experience that this is a typical situation in the analysis of steel special steel moment frame systems. Part 1 of the Provisions recommends that a significant change be made to the current P-delta approach. In the recommended approach, Equation 12.8-16 is still used to determine the magnitude of P-delta effects. However, if the stability ratio at any level is greater than 0.1, the designer must either redesign the building such that the stability ratio is less than 0.1, or must perform a static pushover analysis and demonstrate that the slope of the pushover curve of the structure is continuously positive up to the “target displacement” computed in accordance with the requirements of ASCE 41. 4.1.5.4 Computation of member forces. Before member forces may be computed, the proper load cases and combinations of load must be identified such that all critical seismic effects are captured in the analysis. 4.1.5.4.1 Orthogonal loading effects and accidental torsion. For SDC D structures with a Type 5 horizontal structural irregularity, Section 12.5.3 of the Standard requires that orthogonal load effects be considered. For the purposes of this example, it is assumed that such an irregularity does exist because the layout of the frames is not symmetric. (ASCE 7-10 has eliminated non-symmetry as a trigger for invoking the nonparallel system horizontal irregularity. However, as the structural system under consideration has several intersecting frames, it would be advisable to perform the orthogonal load analysis as required under Section 12.5.4 of both the 2005 and 2010 versions of the Standard.) When orthogonal load effects are included in the analysis, four directions of seismic force (+X, -X, +Y, -Y) must be considered and for each direction of force, there are two possible directions in which the accidental eccentricity can apply (causing positive or negative torsion). This requires a total of eight possible combinations of direct force plus accidental torsion. Where the 30 percent orthogonal loading rule is applied (see Standard Sec. 12.5.3 Item “a”), the number of load combinations increases to 16 because, for each direct application of load, a positive or negative orthogonal loading can exist. Orthogonal loads are applied without accidental eccentricity. Figure 4.1-8 illustrates the basic possibilities of application of load. Although this figure shows 16 different load conditions, it may be observed that eight of these conditions—7, 8, 5, 6, 15, 16, 13 and 14—are negatives (opposite signs throughout) of conditions 1, 2, 3, 4, 9, 10, 11 and 12, respectively.

4-24

Chapter 4: Structural Analysis

Figure 4.1-8 Basic load conditions used in ELF analysis 4.1.5.4.2 Load combinations. The basic load combinations for this structure are designated in Chapter 2 of the Standard. Two sets of combinations are provided: one for strength design and the other for allowable stress design. The strength-based combinations that are related to seismic effects are the following: 1.2D + 1.0E + 1.0L + 0.2S 0.9D + 1.0E+1.6H The factor on live load, L, may be reduced to 0.5 if the nominal live load is less than 100 psf. The load due to lateral earth pressure, H, may need to be considered when designing the basem*nt walls.

4-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples Section 12.4 of the Standard divides the earthquake load, E, into two components, Eh and Ev, where the subscripts h and v represent horizontal and vertical seismic effects, respectively. These components are defined as follows: Eh = ρQE Ev = 0.2SDSD where QE is the earthquake load effect and ρ is a redundancy factor, described later. When the above components are substituted into the basic load combinations, the load combinations for strength design with a factor of 0.5 used for live load and with the H load removed are as follows: (1.2 + 0.2SDS)D+ρQE + 0.5L + 0.2S (0.9-0.2SDS)D + ρQE Using SDS = 0.833 and assuming the snow load is negligible in Stockton, California, the basic load combinations for strength design become: 1.37D + 0.5L + ρQE 0.73D + ρQE The redundancy factor, ρ, is determined in accordance with Standard Section 12.3.4. This factor will take a value of 1.0 or 1.3, with the value depending on a variety of conditional tests. None of the conditions specified in Section 12.3.4.1 are applicable, so ρ may not be automatically taken as 1.0, and the more detailed evaluation specified in Section 12.3.4.2 is required. Subparagraph “b” of Section 12.3.4.2 applies to this building (because of the plan irregularities) and therefore, the evaluation described in the second row of Table 12.3-3 must be performed. It can be seen from inspection that the removal of a single beam from the perimeter moment frames will not cause a reduction in strength of 33 percent, nor will an extreme torsional irregularly result from the removal of the beam. Hence, the redundancy factor may be taken as 1.0 for this structure. Hence, the final load conditions to be used for design are as follows: 1.37D + 0.5L + 1.0QE 0.73D + 1.0QE The first load condition will produce the maximum negative moments (tension on the top) at the face of the supports in the girders and maximum compressive forces in columns. The second load condition will produce the maximum positive moments (or minimum negative moment) at the face of the supports of the girders and maximum tension (or minimum compression) in the columns. In addition to the above load condition, the gravity-only load combinations as specified in the Standard also must be checked. Due to the relatively short spans in the moment frames, however, it is not expected that the non-seismic load combinations will control.

4-26

Chapter 4: Structural Analysis 4.1.5.4.3 Setting up the load combinations in SAP2000. The load combinations required for the analysis are shown in Table 4.1-12. It should be noted that 32 different load combinations are required only if one wants to maintain the signs in the member force output, thereby providing complete design envelopes for all members. As mentioned later, these signs are lost in response spectrum analysis and as a result, it is possible to capture the effects of dead load plus live load plus-or-minus earthquake load in a single SAP2000 run containing only four load combinations. Table 4.1-12 Seismic and Gravity Load Combinations as Run on SAP2000 Lateral* Gravity Run Combination A B 1 (Dead) 2 (Live) One 1 [1] 1.37 0.5 2 [1] 0.73 0.0 3 [7] 1.37 0.5 4 [7] 0.73 0.0 5 [2] 1.37 0.5 6 [2] 0.73 0.0 7 [8] 1.37 0.5 8 [8] 0.73 0.0 Two 1 [3] 1. 37 0.5 2 [3] 0.73 0.0 3 [4] 1. 37 0.5 4 [4] 0.73 0.0 5 [5] 1. 37 0.5 6 [5] 0.73 0.0 7 [6] 1. 37 0.5 8 [6] 0.73 0.0 Three 1 [9] 1. 37 0.5 2 [9] 0.73 0.0 3 [10] 1. 37 0.5 4 [10] 0.73 0.0 5 [15] 1. 37 0.5 6 [15] 0.73 0.0 7 [16] 1. 37 0.5 8 [16] 0.73 0.0 Four 1 [11] 1. 37 0.5 2 [11] 0.73 0.0 3 [12] 1. 37 0.5 4 [12] 0.73 0.0 5 [13] 1. 37 0.5 6 [13] 0.73 0.0 7 [14] 1. 37 0.5 8 [14] 0.73 0.0 *Numbers in brackets [ ] in represent load cases shown in Figure 4.1-8.

4-27

FEMA P-751, NEHRP Recommended Provisions: Design Examples

4.1.5.4.4 Member forces. For this portion of the analysis, the earthquake shears in the girders along Gridline 1 are computed. This analysis considers 100 percent of the X direction forces applied in combination with 30 percent of the (positive or negative) Y direction forces. The accidental torsion is not included and will be considered separately. The results of the member force analysis are shown in Figure 4.1-9a. In a later part of this example, the girder shears are compared to those obtained from modal response spectrum and modal response history analyses. Beam shears in the same frame, due to accidental torsion only, are shown in Figure 4.1-9b. The eccentricity was set to produce clockwise torsions (when viewed from above) on the floor plates. These shears would be added to the shears shown in Figure 4.1-9a to produce the total seismic beam shears in the frame. The same torsional shears (from Table 4.1-9b) will be used in the modal response spectrum and modal response history analyses.

8.99

10.3

17.3

18.9

19.0

27.7

28.1

29.5

33.4

33.1

35.7

34.8

34.7

32.2

30.3

13.2

36.4

35.9

33.9

37.8

23.7

41.2

40.1

38.4

41.3

25.8

43.0

40.6

39.3

41.7

26.4

R-12 12-11 11-10 10-9 9-8 8-7 7-6 6-5 14.1

33.1

33.8

36.5

35.5

37.2

24.9

24.1

37.9

32.0

34.6

33.9

34.9

23.9

24.1

37.0

33.3

35.1

34.6

35.4

24.6

22.9

36.9

34.1

35.3

34.9

35.9

23.3

5-4 4-3 3-2 2-G

Figure 4.1-9a Seismic shears (kips) in girders on Frame Line 1 as computed using ELF analysis (analysis includes orthogonal loading but excludes accidental torsion)

4-28

Chapter 4: Structural Analysis 0.56

0.56

0.58

1.13

1.16

1.87

1.77

1.89

2.26

2.12

2.34

2.07

1.97

1.89

1.54

0.76

1.89

1.81

1.72

1.84

1.36

2.17

2.05

1.99

2.06

1.49

2.29

2.09

2.04

2.09

1.51

R-12 12-11 11-10 10-9 9-8 8-7 7-6 6-5 0.59

1.33

1.65

1.72

1.68

1.72

1.27

1.04

1.45

1.34

1.41

1.39

1.42

1.07

1.51

1.45

1.48

1.45

1.47

1.10

1.04

1.58

1.52

1.54

1.53

1.56

1.06

5-4 4-3 3-2 2-G

Figure 4.1-9b Seismic shears in girders (kips) from clockwise torsion only

4.1.6

Modal Response Spectrum Analysis

The first step in the modal response spectrum analysis is the computation of the natural mode shapes and associated periods of vibration. Using the structural masses from Table 4.1-4 and the same mathematical model as used for the ELF and the Rayleigh analyses, the mode shapes and frequencies are automatically computed by SAP2000. This mathematical model included the basem*nt as a separate level. (See Section 4.1.5.2 of this example for a description of the mathematical model used in the analysis). The basem*nt walls were fixed at the base but were unrestrained at grade level. Thus, the basem*nt level is treated as a separate story in the analysis. However, the lateral stiffness of the basem*nt level is significantly greater than that of the upper levels and this causes complications when interpreting the requirements of Standard Section 12.9.1. As shown later, the explicit modeling of the basem*nt can also lead to some unexpected results in the modal response history analysis of the structure. The periods of vibration for the first 12 modes, computed from an eigenvalue analysis, are summarized in the second column of Table 4.1-13. The first eight mode shapes are shown in Figure 4.1-10. The first mode period, 2.87 seconds, corresponds to vibration primarily in the X direction and the second period, 2.60 seconds, corresponds to vibration in the Y direction. The third mode, with a period of 1.57 seconds, is almost purely torsion. The directionality of the modes may be inferred from the effective mass values shown in Columns 3 through 5 of the tables, as well as from the mode shapes. There is very little lateraltorsional coupling in any of the first 12 modes, which is somewhat surprising because of the shifted centers of mass associated with the plan offsets.

4-29

FEMA P-751, NEHRP Recommended Provisions: Design Examples The X- and Y-translation periods of 2.87 and 2.60 seconds, respectively, are somewhat longer than the upper limit on the approximate period, CuTa, of 2.23 seconds. The first and second mode periods are virtually identical to the periods compute by Rayleigh analysis (2.85 and 2.56 seconds in the X and Y directions, respectively). The closeness of the Rayleigh and eigenvalue periods for this building arises from the fact that the first and second modes of vibration act primarily along the orthogonal axes. Had the first and second modes not acted along the orthogonal axes, the Rayleigh periods (based on loads and displacements in the X and Y directions) would have been somewhat less accurate. Standard Section 12.9.1 specifies that “the analysis shall include a sufficient number of modes to obtain a combined modal mass participation of at least 90 percent of the total mass in each of the orthogonal horizontal directions of response considered by the model”. Usually, this is a straightforward requirement and the first twelve modes would be sufficient for a 12-story building. For this building, however, twelve modes capture only about 82 percent of the X and Y direction mass. (The effective mass as a fraction of total mass is shown in brackets [ ] in Columns 3 through 5 of Tables 4.1-13 and 4.1-14.) Most of the remaining effective mass is in the grade-level slab and in the basem*nt walls. This mass does not show up until Mode 112 in the Y direction and Mode 118 in the X direction. This is shown in Table 12.1-14, which provides the periods and effective modal masses in Modes 108 through 119. The intermediate modes (13 through 107) represent primarily vertical vibration of various portions of the floor diaphragms. Analyzing the system with 120 or more modes might provide useful information on the response of the basem*nt level, including shears through the basem*nt and total system base shears at the base of the basem*nt. However, there would be some difficulty in interpreting the results because the model did not include sub-grade soil that would be in contact with the basem*nt walls and which would absorb part of the base shear. Additionally, the computed response of the upper 12 levels of the building, which is the main focus of this analysis, is virtually identical for the 12 and the 120 mode analyses. For this reason, the modal response spectrum analysis discussed in this example was run with only the first 12 modes listed in Table 4.1-13.

Mode 1: T = 2.87 sec

4-30

Mode 2: T = 2.60 sec

Chapter 4: Structural Analysis

Mode 3: T = 1.57 sec

Mode 5: T = 0.98 sec

Mode 7: T = 0.68 sec

Mode 4: T = 1.15 sec

Mode 6: T = 0.71 sec

Mode 8: T = 0.57 sec

Figure 4.1-10 First eight mode shapes Table 4.1-13 Computed Periods and Effective Mass Factors (Lower Modes) Effective Mass Factor [Accum Mass Factor] Period Mode (sec.) X Translation Y Translation Z Rotation 1 2.87 0.6446 [0.64] 0.0003 [0.00] 0.0028 [0.00] 2 2.60 0.0003 [0.65] 0.6804 [0.68] 0.0162 [0.02] 3 1.57 0.0035 [0.65] 0.0005 [0.68] 0.5806 [0.60] 4 1.15 0.1085 [0.76] 0.0000 [0.68] 0.0000 [0.60] 5 0.975 0.0000 [0.76] 0.0939 [0.78] 0.0180 [0.62] 6 0.705 0.0263 [0.78] 0.0000 [0.78] 0.0271 [0.64] 7 0.682 0.0056 [0.79] 0.0006 [0.79] 0.0687 [0.71] 8 0.573 0.0000 [0.79] 0.0188 [0.79] 0.0123 [0.73]

4-31

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 4.1-13 Computed Periods and Effective Mass Factors (Lower Modes) Effective Mass Factor [Accum Mass Factor] Period Mode (sec.) X Translation Y Translation Z Rotation 9 0.434 0.0129 [0.80] 0.0000 [0.79] 0.0084 [0.73] 10 0.387 0.0048 [0.81] 0.0000 [0.79] 0.0191 [0.75] 11 0.339 0.0000 [0.81] 0.0193 [0.81] 0.0010 [0.75] 12 0.300 0.0089 [0.82] 0.0000 [0.81] 0.0003 [0.75]

Table 4.1-14 Computed Periods and Effective Mass Factors (Higher Modes) Period (sec.) 0.0693 0.0673 0.0671 0.0671 0.0669 0.0663 0.0646 0.0629 0.0621 0.0609 0.0575 0.0566

Mode 108 109 110 111 112 113 114 115 116 117 118 119

Effective Mass Factor [Accum Effective Mass] X Translation Y Translation Z Rotation 0.0000 [0.83] 0.0000 [0.83] 0.0000 [0.79] 0.0000 [0.83] 0.0000 [0.83] 0.0000 [0.79] 0.0000 [0.83] 0.0354 [0.86] 0.0000 [0.79] 0.0000 [0.83] 0.0044 [0.87] 0.0000 [0.79] 0.0000 [0.83] 0.1045 [0.97] 0.0000 [0.79] 0.0000 [0.83] 0.0000 [0.97] 0.0000 [0.79] 0.0000 [0.83] 0.0000 [0.97] 0.0000 [0.79] 0.0000 [0.83] 0.0000 [0.97] 0.0000 [0.79] 0.0008 [0.83] 0.0010 [0.97] 0.0000 [0.79] 0.0014 [0.83] 0.0009 [0.97] 0.0000 [0.79] 0.1474 [0.98] 0.0000 [0.97] 0.0035 [0.80] 0.0000 [0.98] 0.0000 [0.97] 0.0000 [0.80]

4.1.6.1 Response spectrum coordinates and computation of modal forces. The coordinates of the response spectrum are based on Standard Section 11.4.5. This spectrum consists of three parts (for periods less than TL = 8.0 seconds) as follows: §

For periods less than T0:

S a = 0.6 §

S DS T + 0.4 S DS T0

For periods between T0 and TS:

Sa = S DS §

For periods greater than TS:

Sa =

S D1 T

where T0 = 0.2SD1/SDS and TS = SD1/SDS.

4-32

Chapter 4: Structural Analysis Using SDS = 0.833 and SD1 = 0.373, TS = 0.448 seconds and T0 = 0.089 seconds. The computed response spectrum coordinates for several period values are shown in Table 4.1-15 and the response spectrum, shown with and without the I/R = 1/8 modification, is plotted in Figure 4.1-11. The spectrum does not include the high period limit on Cs (0.044ISDS), which controlled the ELF base shear for this structure and which ultimately will control the scaling of the results from the response spectrum analysis. (Recall that if the computed base shear falls below 85 percent of the ELF base shear, the computed response must be scaled up such that the computed base shear equals 85 percent of the ELF base shear.) Table 4.1-15 Response Spectrum Coordinates Tm Sa Sa(I/R) (sec.) 0.000 0.333 0.0416 0.089 (T0) 0.833 0.104 0.448 (TS) 0.833 0.104 1.000 0.373 0.0446 1.500 0.249 0.0311 2.000 0.186 0.0235 2.500 0.149 0.0186 3.000 0.124 0.0155 I = 1, R = 8.0.

0.8 ) g ( 0.6 n o it a erl ec c 0.4 a la rt ce p S 0.2

Sa (I/R) 0 0

Period, T (s)

Figure 4.1-11 Total acceleration response spectrum used in analysis Using the response spectrum coordinates listed in Column 3 of Table 4.1-15, the response spectrum analysis was carried out using SAP2000. As mentioned above, the first 12 modes of response were computed and superimposed using the CQC approach. A modal damping ratio of 5 percent of critical was used in the CQC calculations.

4-33

FEMA P-751, NEHRP Recommended Provisions: Design Examples Two analyses were carried out. The first directed the seismic motion along the X axis of the structure and the second directed the motion along the Y axis. Combinations of these two loadings plus accidental torsion are discussed later. 4.1.6.1.1 Dynamic base shear. After specifying member groups, SAP2000 automatically computes the CQC story shears. Groups were defined such that total shears would be obtained for each story of the structure. The shears at the base of the first story above grade are reported as follows: §

X direction base shear = 438.1 kips

§

Y direction base shear = 492.8 kips

These values are much lower that the ELF base shear of 1,124 kips. Recall that the ELF base shear was controlled by Standard Equation 12.8-5. The modal response spectrum shears are less than the ELF shears because the fundamental periods of the structure used in the response spectrum analysis (2.87 seconds and 2.6 seconds in the X and Y directions, respectively) are greater than the upper limit empirical period, CuTa, of 2.23 seconds and because the response spectrum of Figure 4.1-11 does not include the minimum base shear limit imposed by Standard Equation 12.8-5. According to Standard Section 12.9.4, the base shears from the modal response spectrum analysis must not be less than 85 percent of that computed from the ELF analysis. If the response spectrum shears are lower than the ELF shear, then the computed shears must be scaled up such that the response spectrum base shear is 85 percent of that computed from the ELF analysis. Hence, the required scale factors are as follows: §

X direction scale factor = 0.85(1124)/438.1 = 2.18

§

Y direction scale factor = 0.85(1124)/492.8 = 1.94

The computed and scaled story shears are as shown in Table 4.1-16. Since the base shears for the ELF and the modal analysis are different (due to the 0.85 factor), direct comparisons cannot be made between Table 4.1-16 and Table 4.1-4. However, it is clear that the vertical distribution of forces is somewhat similar where computed by ELF and modal response spectrum. Table 4.1-16 Story Shears from Modal Response Spectrum Analysis X Direction (SF = 2.18) Y Direction (SF = 1.94) Story Unscaled Shear Scaled Shear Unscaled Shear Scaled Shear (kips) (kips) (kips) (kips) R-12 82.7 180 77.2 150 12-11 130.9 286 132.0 256 11-10 163.8 357 170.4 330 10-9 191.4 418 201.9 392 9-8 240.1 524 265.1 514 8-7 268.9 587 301.4 585 7-6 292.9 639 328.9 638 6-5 316.1 690 353.9 686 5-4 359.5 784 405.1 786

4-34

Chapter 4: Structural Analysis Table 4.1-16 Story Shears from Modal Response Spectrum Analysis X Direction (SF = 2.18) Y Direction (SF = 1.94) Story Unscaled Shear Scaled Shear Unscaled Shear Scaled Shear (kips) (kips) (kips) (kips) 4-3 384.8 840 435.5 845 3-2 401.4 895 462.8 898 2-G 438.1 956 492.8 956 1.0 kip = 4.45 kN.

4.1.6.2 Drift and P-delta effects. According to Standard Section 12.9.4, the computed displacements and drift (as based on the response spectrum of Figure 4.1-11) need not be scaled by the base shear factors (SF) of 2.18 and 1.94 for the structure loaded in the X and Y directions, respectively. This provides consistency with Section 12.8.6.2, which allows drift from an ELF analysis to be based on the computed period without the upper limit CuTa. Section 12.9.4.2 of ASCE 7-10 requires that drifts from a response spectrum analysis be scaled only if Equation 12.8-6 controls the value of CS. In this example, Equation 12.8-5 controlled the base shear, so drifts need not be scaled in ASCE 7-10. In Tables 4.1-17 and 4.1-18, the story displacement from the response spectrum analysis, the story drift, the amplified story drift (as multiplied by Cd = 5.5) and the allowable story drift are listed. As before the story drifts represent the differences in the displacement at the center of mass of one level, and the displacement at vertical projection of that point at the level below. These values were determined in each mode and then combined using CQC. As may be observed from the tables, the allowable drift is not exceeded at any level. Table 4.1-17 Response Spectrum Drift for Building Responding in X Direction Total Drift from Story Drift Allowable Story Drift Level R.S. Analysis × Cd Story Drift (in.) (in.) (in.) (in.) R 2.23 0.12 0.66 3.00 12 2.10 0.16 0.89 3.00 11 1.94 0.19 1.03 3.00 10 1.76 0.20 1.08 3.00 9 1.56 0.18 0.98 3.00 8 1.38 0.19 1.06 3.00 7 1.19 0.20 1.08 3.00 6 0.99 0.20 1.08 3.00 5 0.80 0.18 0.97 3.00 4 0.62 0.19 1.02 3.00 3 0.43 0.19 1.05 3.00 2 0.24 0.24 1.34 4.32 1.0 in. = 25.4 mm.

4-35

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 4.1-18 Response Spectrum Drift for Building Responding in Y Direction Total Drift from Story Drift Allowable Story Drift Level R.S. Analysis × Cd Story Drift (in.) (in.) (in.) (in.) R 1.81 0.06 0.32 3.00 12 1.76 0.09 0.49 3.00 11 1.67 0.11 0.58 3.00 10 1.56 0.12 0.67 3.00 9 1.44 0.13 0.70 3.00 8 1.31 0.16 0.87 3.00 7 1.15 0.17 0.91 3.00 6 0.99 0.17 0.92 3.00 5 0.92 0.17 0.93 3.00 4 0.65 0.19 1.04 3.00 3 0.46 0.20 1.08 3.00 2 0.26 0.26 1.44 4.32 1.0 in. = 25.4 mm.

According to Standard Section 12.9.6, P-delta effects should be checked using the ELF method. This implies that such effects should not be determined using the results from the modal response spectrum analysis. Thus, the results already shown and discussed in Table 4.1-11 of this example are applicable. Nevertheless, P-delta effects can be assessed using the results of the modal response spectrum analysis if the displacements, drifts and story shears are used as computed from the response spectrum analysis, without the base shear scale factors. However, when computing the stability ratio, the drifts must include the amplifier Cd (because of the presence of Cd in the denominator of Standard Equation 12.8-16). Using this approach, P-delta effects were computed for the X direction response as shown in Table 4.1-19. Note that the stability factors are very similar to those given in Table 4.1-11. As with Table 4.1-11, the stability factors from Table 4.1-19 exceed the limit (θmax = 0.091) only at the bottom three levels of the structure and are only marginally above the limit. Since the β factor was conservatively set at 1.0 inch for computing the limit, it is likely that a refined analysis for β would indicate that P-delta effects are not of particular concern for this structure. Table 4.1-19 Computation of P-delta Effects for X Direction Response hsx Δ PD PL PT PX Level (in.) (in.) (kips) (kips) (kips) (kips) R 150 0.66 1,656.5 315.0 1,971.5 1,971.5 12 150 0.89 1,595.8 315.0 1,910.8 3,882.3 11 150 1.03 1,595.8 315.0 1,910.8 5,793.1 10 150 1.08 1,595.8 315.0 1,910.8 7,703.9 9 150 0.98 3,403.0 465.0 3,868.0 11,571.9 8 150 1.06 2,330.8 465.0 2,795.8 14,367.7 7 150 1.08 2,330.8 465.0 2,795.8 17,163.5 6 150 1.08 2,330.8 465.0 2,795.8 19,959.3 5 150 0.97 4,323.8 615.0 4,938.8 24,898.1

4-36

VX (kips) 82.7 130.9 163.8 191.4 240.1 268.9 292.9 316.1 359.5

θX 0.019 0.032 0.044 0.053 0.057 0.069 0.077 0.083 0.081

Chapter 4: Structural Analysis Table 4.1-19 Computation of P-delta Effects for X Direction Response hsx Δ PD PL PT PX Level (in.) (in.) (kips) (kips) (kips) (kips) 4 150 1.02 3,066.1 615.0 3,681.1 28,579.2 3 150 1.05 3,066.1 615.0 3,681.1 32,260.3 2 216 1.34 3,097.0 615.0 3,712.0 35,972.3

VX (kips) 384.8 401.9 438.1

θX 0.092 0.102 0.093

1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.

4.1.6.3 Torsion, orthogonal loading and load combinations. To determine member design forces, it is necessary to add the effects of accidental torsion and orthogonal loading into the analysis. When including accidental torsion in modal response spectrum analysis, there are generally two approaches that can be taken: §

Displace the center of mass of the floor plate plus or minus 5 percent of the plate dimension perpendicular to the direction of the applied response spectrum. As there are four possible mass locations, this will require four separate modal analyses for torsion with each analysis using a different set of mode shapes and frequencies.

§

Compute the effects of accidental torsion by creating a load condition with the accidental story torques applied as static forces. Member forces created by the accidental torsion are then added directly to the results of the response spectrum analysis. As with the displaced mass method, there are four possible ways to apply the accidental torsion: plus and minus torsion for primary loads in the X and Y directions. Where scaling of the modal response spectrum design forces is required, the torsional loading used for accidental torsion analysis should be multiplied by 0.85.

Each of the above approaches has advantages and disadvantages. The primary disadvantage of the first approach is a practical one: most computer programs do not allow for the extraction of member force maxima from more than one run where the different runs incorporate a different set of mode shapes and frequencies. An advantage of the approach stipulated in Standard Section 12.9.5 is that accidental torsion need not be amplified (when otherwise required by Standard Section 12.8.4.3) because the accidental torsion effect is amplified within the dynamic analysis. For structures that are torsionally regular and which will not require amplification of torsion, the second approach may be preferred. A disadvantage of the approach is the difficulty of combining member forces from a CQC analysis (all results positive), and a separate static torsion analysis (member forces have positive and negative signs as appropriate). In the analysis that follows, the second approach has been used because the structure has excellent torsional rigidity, and amplification of accidental torsion is not required (all amplification factors = 1.0). There are two possible methods for applying the orthogonal loading rule: §

Run two separate response spectrum analyses, one in the X direction and one in the Y direction, with CQC being used for modal combinations in each analysis. Using a direct sum, combine 100 percent of the scaled X direction results with 30 percent of the scaled Y direction results. Perform a similar analysis using 100 percent of the scaled Y direction forces and 30 percent of the scaled X direction forces. All seismic effects can be considered in only two dynamic load cases (one response spectrum analysis in each direction) and two torsion cases (resulting from loads applied at a 5 percent eccentricity in each direction). These are shown in Figure 4.1-12. 4-37

FEMA P-751, NEHRP Recommended Provisions: Design Examples

§

Run two separate response spectrum analyses, one in the X direction and one in the Y direction, with CQC being used for modal combinations in each analysis. Using SRSS, combine 100 percent of the scaled X direction results with 100 percent of the scaled Y direction results (Wilson, 2004).

RSX

T 0.3RSX

0.3RSY

RSY

Figure 4.1-12 Load combinations for response spectrum analysis 4.1.6.4 Member design forces. Earthquake shear forces in the beams of Frame 1 are given in Figure 4.1-13. These member forces are based on 2.18 times the spectrum applied in the X direction and 1.94 times of the spectrum applied independently in the Y direction. Individual member forces from the X and Y directions are obtained by CQC for that analysis and these forces are combined by SRSS. To account of accidental torsion, the forces in Figure 4.1-13 should be added to 0.85 times the forces shown in Figure 4.1-9b.

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Chapter 4: Structural Analysis 8.41

8.72

8.91

14.9

15.6

21.5

21.6

22.5

24.2

24.0

25.8

23.3

21.8

20.0

8.9

23.7

23.5

22.4

24.5

15.8

26.9

26.1

25.4

26.7

17.2

28.4

26.8

26.2

27.3

17.8

R-12 12-11 11-10 10-9 9-8 8-7 7-6 6-5 10.1

22.4

23.6

25.3

24.8

25.5

17.0

17.4

26.6

23.7

24.9

24.6

25.1

17.0

18.5

27.5

25. 9

26.6

26.4

26.8

18.5

29.1

27.8

28.2

28.1

28.7

18.5

5-4 4-3 3-2 2-G

Figure 4.1-13 Seismic shears in girders (kips) as computed using response spectrum analysis (analysis includes orthogonal loading but excludes accidental torsion)

4.1.7

Modal Response History Analysis

Before beginning this section, it is important to note that the analysis performed here is based on the requirements of Chapter 16 of ASCE 7-10. This version contains several important updates that removed inconstancies and omissions that were present in Chapter 16 of ASCE 7-05. In modal response history analysis, the original set of coupled equations of motion is transformed into a set of uncoupled “modal” equations, an explicit displacement history is computed for each mode, the modal histories are transformed back into the original coordinate system, and these responses are added together to produce the response history of the displacements at each of the original degrees of freedom. These displacement histories may then be used to determine histories of story drift, member forces, or story shears. Requirements for response history analysis are provided in Chapter 16 of ASCE 7-10. The same mathematical model of the structure used for the ELF and response spectrum analysis is used for the response history analysis. Five percent damping was used in each mode and as with the response spectrum method, 12 modes were used in the analysis. These 12 modes captured more than 90 percent of the mass of the structure above grade. Several issues related to a reanalysis of the structure with 120 modes are described later. As allowed by ASCE 7-10 Section 16.1, the structure is analyzed using three different pairs of ground acceleration histories. The development of a proper suite of ground motions is one of the most critical

4-39

FEMA P-751, NEHRP Recommended Provisions: Design Examples and difficult aspects of response history approaches. The motions should be characteristic of the site and should be from real (or simulated) ground motions that have a magnitude, distance and source mechanism consistent with those that control the maximum considered earthquake (MCE). For the purposes of this example, however, the emphasis is on the implementation of the response history approach rather than on selection of realistic ground motions. For this reason, the motion suite developed for Example 4.2 is also used for the present example.5 The structure for Example 4.2 is situated in Seattle, Washington and uses three pairs of motions developed specifically for the site. The use of the Seattle motions for a Stockton building analysis is, of course, not strictly consistent with the requirements of the Standard. However, a realistic comparison may still be made between the ELF, response spectrum and response history approaches. 4.1.7.1 The Seattle ground motion suite. It is beneficial to provide some basic information on the Seattle motion suites in Table 4.1-20a below. Refer to Figures 4.2-40 through 4.2-42 for additional information, including plots of the ground acceleration histories and 5-percent damped response spectra for each component of each motion. The acceleration histories for each source motion were downloaded from the PEER NGA Strong Ground Motion Database: http://peer.berkeley.edu/products/strong_ground_motion_db.html The PEER NGA record number is provided in the first column of the table. Note that the magnitude, epicenter distance and site class were obtained from the NGA Flatfile (a large Excel file that contains information about each NGA record). Table 4.1-20a Suite of Ground Motions Used for Response History Analysis Magnitude Number of NGA [Epicenter Site Points and Component Source PGA Record Distance, Class Digitization Motion (g) Number km] Increment 7.28 Landers/LCN260* 0.727 9625 @ 0879 C 0.005 sec [44] Landers/LCN345* 0.789 6.54 SUPERST/B-POE270 0.446 2230 @ 0725 D 0.01 sec [11.2] SUPERST/B-POE360 0.300 7.35 TABAS/DAY-LN 0.328 1192 @ 0139 C 0.02 sec [21] TABAS/DAY-TR 0.406

Record Name (This Example) A00 A90 B00 B90 C00 C90

*Note that the two components of motion for the Landers earthquake are apparently separated by an 85 degree angle, not 90 degrees as is traditional. It is not known whether these are true orientations or whether there is an error in the descriptions provided in the NGA database.

Before the ground motions may be used in the response history analysis, they must be scaled for compatibility with the design spectrum. The scaling procedures for three-dimensional dynamic analysis are provided in Section 16.1.3.2 of ASCE 7-10. These requirements are provided verbatim as follows: 5

See Sec. 3.2.6.2 of this volume of design examples for a detailed discussion of the selection and scaling of ground motions.

4-40

Chapter 4: Structural Analysis “For each pair of horizontal ground motion components a square root of the sum of the squares (SRSS) spectrum shall be constructed by taking the SRSS of the 5-percent damped response spectra for the scaled components (where an identical scale factor is applied to both components of a pair). Each pair of motions shall be scaled such that for each period in the range from 0.2T to 1.5T, the average of the SRSS spectra from all horizontal component pairs does not fall below the corresponding ordinate of the design response spectrum, determined in accordance with Section 11.4.5 or 11.4.7.” ASCE 7-10 does not provide clear guidance as to which fundamental period, T, should be used for determining 0.2T and 1.5T when the periods of vibration are different in the two orthogonal directions of analysis. This issue is resolved herein by taking T as the average of the computed periods in the two principal directions. For this example, the average period, referred to a TAvg, is 0.5(2.87 + 2.60) = 2.74 seconds. (Another possibility would be to use the shorter of the two fundamental periods for computing 0.2T and the longer of the two fundamental periods for computing 1.5T.) It is also noted that the scaling procedure provided by ASCE 7-10 does not provide a unique set of scale factors for each set of ground motions. This “degree of freedom” in the scaling process may be eliminated4 by providing a six-step procedure, as described below: 1. Compute the 5 percent damped pseudo-acceleration spectrum for each unscaled component of each pair of ground motions in the set and produce the SRSS spectrum for each pair of motions within the set. 2. Using the same period values used to compute the ground motion spectra, compute the design spectrum following the procedures in Standard Section 11.4.5. This spectrum is designated as the “target spectrum”. 3. Scale each SRSS spectrum such that the spectral ordinate of the scaled spectrum at TAvg is equal to the spectral ordinate of the design spectrum at the same period. Each SRSS spectrum will have a unique scale factor, S1i, where i is the number of the pair (i ranges from 1 to 3 for the current example). 4. Create a new spectrum that is the average of the S1 scaled SRSS spectra. This spectrum is designated as the “average S1 scaled SRSS spectrum” and should have the same spectral ordinate as the target spectrum at the period TAvg. 5. For each spectral ordinate in the period range 0.2TAvg to 1.5TAvg, divide the ordinate of the target spectrum by the corresponding ordinate of the average S1 scaled SRSS spectrum, producing a set of spectral ratios over the range 0.2TAvg to 1.5TAvg. The largest value among these ratios is designated as S2. 6. Multiply the factor S1i determined in Step 3 for each pair in the set by the factor S2 determined in Step 5. This product, SSi = S1i × S2 is the scale factor that should be applied to each component of ground motion in pair i of the set. The results of the scaling process are summarized in Table 4.1-20b and in Figures 4.1-14 through 4.1-18.

Elimination of the degree of freedom results in consistent scale factors for all persons using the process. This consistency is not required by ASCE 7 and experienced analysts may wish to use the “degree of freedom” to reduce or increase the influence of a given ground motion.

4-41

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Table 4.1-20b Result of 3D Scaling Process Set Number

Designation

SRSS Ordinate at T = TAvg (g)

1 2 3

A00 & A90 B00 & B90 C00 & C90

0.335 0.191 0.104

Target Ordinate at T = TAvg (g) 0.136 0.136 0.136

0.407 0.712 1.310

1.184 1.184 1.184

0.482 0.843 1.551

Figure 4.1-14 shows the unscaled SRSS spectra for each component pair, together with the target spectrum. Figure 4.1-15 shows the average S1 scaled SRSS spectrum and the target spectrum, where it may be seen that both spectra have a common ordinate at the average period of 2.74 seconds. Figure 4.116 is a plot of the spectral ratios computed in Step 5. Figure 4.1-17 is a plot of the SS scaled average SRSS spectrum, together with the target spectrum. From this plot it may be seen that all ordinates of the SS scaled average SRSS spectrum are greater than or equal to the ordinate of the target spectrum over the period range 0.2TAvg to 1.5TAvg. The “controlling” period at which the two spectra in Figure 4.1-17 have exactly the same ordinate is approximately 1.6 seconds. Figure 4.1-18a shows the SS scaled spectra for the “00” components of each earthquake, together with the target spectrum. Figure 4.-18b is similar, but shows the “90” components of the ground motions. Also shown in these plots are vertical lines that represent the first 12 periods of vibration for the structure under consideration. Two additional vertical lines are shown that represent the periods for Modes 112 and 118, at which the basem*nt walls and grade-level slab become dynamically effective. Three important points are noted from Figures 4.1-18: The match for the lower few modes (T > 1.0 sec) is good for the “00” components, but not as good for the “90” components. In particular, the ground motion coordinates for motions A90 and B90 are considerably less than those for the target spectrum. §

Higher mode responses (T < 1.0 sec) will be significantly greater in Earthquake C than in Earthquake A or B. In Modes 10 through 12, the response for Earthquake A is several times greater than for Earthquake B.

§

In Modes 112 and 118, the response for Earthquake A is approximately three times that for the code spectrum.

The impact of these points on the computed response of the structure will be discussed in some detail later in this example.

4-42

Chapter 4: Structural Analysis

SRSS Earthquake A

3.0

SRSS Earthquake B SRSS Earthquake C

)g ( n 2.0 iot ar el ec c A 1.0

Target Spectrum

s 4 7. 2 = T

0.0 0

Period, T (s)

Figure 4.1-14 Unscaled SRSS spectra and target spectrum

Average S1 Scaled SRSS Spectrum Target Spectrum

1.0 ) (g n oi t ar el ec c 0.5 A

s 4 .72 = T

0.0 0

2 Period, T (s)

Figure 4.1-15 Average S1 scaled SRSS spectrum and target spectrum

4-43

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Ratio of target spectrum to average S1 scaled SRSS

1.2 1.0 iot ar la 0.8 rt ce p S 0.6

s8 45 . 0 = T .2 0

0.4

s 11 . 4 = T .5 1

s 4 .72 = T

0.2 0.0 0

2 Period, T (s)

Figure 4.1-16 Ratio of target spectrum to average S1 scaled SRSS spectrum

SS scaled average SRSS spectrum )g (

n iot ar el ec c A

1.0

Target spectrum

0.5

s 4 7. 2 = T

0.0 0

Period, T (s)

Figure 4.1-17 SS scaled average SRSS spectrum and target spectrum

4-44

Chapter 4: Structural Analysis Target Spectrum

Scaled Record A00

Scaled Record B00

Scaled Record C00

8 1 ,1 21 1 se d o M

)g 2.0 ( n iot ar el ec c A

s 47 .2 = T

1.0

0.0 0

Period, T (s)

(a) 00 Components Target Spectrum

Scaled Record A90

Scaled Record B90

Scaled Record C90

8 1 ,1 21 1 se d o M

)g 2.0 ( n oi t ar el ec c A 1.0

s 47 .2 = T

0.0 0

Period, T (s)

(b) 90 Components Figure 4.1-18 SS scaled individual spectra and target spectrum Another detail not directly specified by Chapter 16 of ASCE 7-10 is how ground motions should be oriented when applied. In the analysis presented herein, 12 dynamic analyses were performed with scaled ground motions applied only in one direction, as follows: § §

A00-X: SS scaled component A00 applied in X direction A00-Y: SS scaled component A00 applied in Y direction 4-45

FEMA P-751, NEHRP Recommended Provisions: Design Examples § §

A90-X: SS scaled component A90 applied in X direction A90-Y: SS scaled component A90 applied in Y direction

§ § § §

B00-X: B00-Y: B90-X: B90-Y:

SS scaled component B00 applied in X direction SS scaled component B00 applied in Y direction SS scaled component B90 applied in X direction SS scaled component B90 applied in Y direction

§ § § §

C00-X: C00-Y: C90-X: C90-Y:

SS scaled component C00 applied in X direction SS scaled component C00 applied in Y direction SS scaled component C90 applied in X direction SS scaled component C90 applied in Y direction

The scaled motions, without the (I/R) factor, were applied at the base of the basem*nt walls. Accidental torsion effects are included in a separate static analysis, as described later. All 12 individual response history analyses were carried out using SAP2000. As with the response spectrum analysis, 12 modes were used in the analysis. Five percent of critical damping was used in each mode. The integration timestep used in all analyses was equal to the digitization interval of the ground motion used (see Table 4.1-20a). The results from the analyses are summarized Tables 4.1-21. A summary of base shear and roof displacement results from the analyses using the SS scaled ground motions is provided in Table 4.1-21. As may be observed, the base shears range from a low of 1,392 kips for analysis A90-Y to a high of 5,075 kips for analysis C90-Y. Roof displacements range from a low of 5.16 inches for analysis A90-Y to a high of 20.28 inches for analysis A00-X. This is a remarkable range of behavior when one considers that the ground motions were scaled for consistency with the design spectrum. Table 4.1-21 Result Maxima from Response History Analysis Using SS Scaled Ground Motions Time of Maximum Time of Maximum maximum roof maximum Analysis base shear shear displacement displacement (kips) (sec.) (in.) (sec.) A00-X 3507 11.29 20.28 11.38 A00-Y 3573 11.27 14.25 11.28 A90-X 1588 12.22 7.32 12.70 A90-Y 1392 13.56 5.16 10.80 B00-X 3009 8.28 12.85 9.39 B00-Y 3130 9.37 11.20 10.49 B90-X 2919 8.85 11.99 7.11 B90-Y 3460 7.06 11.12 8.20 C00-X 3130 13.5 9.77 13.54 C00-Y 2407 4.64 6.76 8.58 C90-X 3229 6.92 15.61 6.98 C90-Y 5075 6.88 14.31 7.80 1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN.

4-46

Chapter 4: Structural Analysis The analysis was performed without the (I/R) factor, so in conformance with Section 16.1.4 of ASCE 7-10, all force quantities produced from the analysis were multiplied by this factor. All displacements from the analysis were multiplied by the factor Cd/R. Additionally, the 2010 version of the Standard requires that forces be scaled by the factor 0.85V/Vi where the base shears from the response history analysis, Vi, are less than 0.85 times the base shears, V, produced by the ELF method when either Equation 12.8-5 or 12.8-6 controls the seismic base shear. The displacements must be scaled by the same factor only if Equation 12.8-6 controls when computing the seismic base shear. (It is noted that these requirements are similar to the scaling requirements provided for modal response spectrum analysis [Sections 12.9.4.1 and 12.9.4.2] except that forces from modal response spectrum analysis would be scaled if the shear from the response spectrum analysis is less than 0.85V, regardless of the Cs equation which controls V.) The base shears from the SS scaled motions with the I/R = 1/8 scaling are provided in the first column of Table 4.1-22. These forces are all significantly less than 0.85 times the ELF base shear, which is 0.85(112.5) = 956 kips. The required scale factors to bring the base shears up to the 85 percent requirement are shown in Column 2 of Table 4.2-22. Before proceeding, it is important to remind the reader that three separate sets of scale factors apply to the response history analysis of this structure when member design forces are being obtained: 1. The ground motion SS scale factors 2. The I/R scale factor 3. The 0.85V/Vi factor because the base shear from modal response history analysis (including scale factors 1 and 2 above) is less than 85 percent of that determined from ELF when ELF is governed by Equation 12.8-5 or 12.8-6. Table 4.1-22 I/R Scaled Shears and Required 85% Rule Scale Factors (I/R) times maximum base Required additional scale factor for Analysis shear from analysis V = 0.85VELF = 956 kips (kips) A00-X 438.4 2.18 A00-Y 446.7 2.14 A90-X 198.5 4.81 A90-Y 173.9 5.49 B00-X 376.1 2.54 B00-Y 391.2 2.44 B90-X 364.8 2.62 B90-Y 432.5 2.21 C00-X 391.2 2.44 C00-Y 300.9 3.18 C90-X 403.6 2.37 C90-Y 634.4 1.51 1.0 kip = 4.45 kN

4-47

FEMA P-751, NEHRP Recommended Provisions: Design Examples 4.1.7.2 Drift and P-delta effects. Only two scale factors are required for displacement and drift because Equation 12.8-6 did not control the base shear for this structure: §

The ground motion scale factors SS

§

The Cd/R scale factor

Drift is checked for each individual component of motion acting in the X direction and the envelope values of drift are taken as the design drift values. The procedure is repeated for motions applied in the Y direction. As with the ELF and Modal Response Spectrum analyses, drifts are taken as the difference between the displacement at the center of mass of one level and the displacement at the projection of this point on the level below. The results of the analysis, shown in Table 4.1-23 for the X direction only, indicate that the allowable drift is not exceeded at any level of the structure. Similar results were obtained for Y direction loading. Table 4.1-23

Level R 12 11 10 9 8 7 6 5 4 3 2

Response History Drift for Building Responding in X Direction for All of the Ground Motions in the X Directions Envelope Envelope of drift (in.) for each ground motion of drift for Envelope Allowable drift all the of drift A00-X A90-X B00-X B90-X C00-X C90-X (in.) ground × Cd/R motions 1.17 0.49 0.95 0.81 0.91 1.23 1.23 0.85 3.00 1.64 0.66 1.22 0.95 1.16 1.27 1.64 1.13 3.00 1.97 0.78 1.32 0.99 1.25 1.52 1.97 1.35 3.00 2.05 0.86 1.42 1.04 1.20 1.68 2.05 1.41 3.00 1.79 0.82 1.26 1.25 0.99 1.41 1.79 1.23 3.00 1.83 0.87 1.22 1.42 1.23 1.50 1.83 1.26 3.00 1.82 0.83 1.27 1.36 1.21 1.67 1.82 1.25 3.00 1.77 0.74 1.36 1.35 1.06 1.94 1.94 1.33 3.00 1.50 0.59 1.19 1.21 1.09 1.81 1.81 1.24 3.00 1.55 0.62 1.22 1.32 1.23 1.76 1.76 1.21 3.00 1.56 0.64 1.24 1.30 1.33 1.60 1.60 1.10 3.00 1.97 0.86 1.64 1.58 1.73 1.85 1.97 1.35 4.32

1.0 in. = 25.4 mm.

ASCE 7-10 does not provide information on how P-delta effects should be addressed in response history analysis. It would appear reasonable to use the same procedure as specified for ASCE 7-05 and ASCE 7-10 for Modal Response Spectrum Analysis (Sec. 12.9.6), where it is stated that the Equivalent Lateral Force method of analysis be used. Such an analysis was performed in Section 4.1.5.3.2 of this example, with results provided in Table 4.1-11. These results indicate that allowable stability ratios are marginally exceeded at Levels 2, 3 and 4, but that rigorous analysis with β less than 1.0 would show that the allowable stability ratios are not exceeded. 4.1.7.3 Torsion, orthogonal loading and member design forces. As with ELF or response spectrum analysis, it is necessary to add the effects of accidental torsion and orthogonal loading into the analysis.

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Chapter 4: Structural Analysis Accidental torsion is applied separately with a static analysis in exactly the same manner as done for the response spectrum approach. Member shears for this torsion-only analysis are shown separately in Figure 4.1-9b. These shears must be multiplied by 0.85 before adding to the scaled shears produced by the dynamic response history analysis. Orthogonal loading is automatically accounted for by applying the 100 percent of the ground motions in the X and Y direction simultaneously. For each ground motion pair, these forces are applied in the orientations shown in Figure 4.1-19. The figure also shows the scale factor that was used in each analysis.

Earthquake

Load Combination for Response History Analysis Loading X Direction Loading Y Direction Load Scale Scale Combination Record Factor Record Factor 1 A00-X 2.18 A00-Y 5.49 2 A90-X -4.81 A90-Y 2.14 3 A00-X -2.18 A00-Y -5.49 4 A90-X 4.81 A90-Y -2.14 5 B00-X 2.54 B00-Y 2.21 6 B90-X -2.62 B90-Y 2.44 7 B00-X -2.54 B00-Y -2.21 8 B90-X 2.62 B90-Y -2.44 9 C00-X 2.44 C00-y 1.50 10 C90-X -2.36 C90-Y 3.18 11 C00-X -2.44 C00-Y -1.50 12 C90-X 2.36 C90-Y -3.18

Component 2

Component 1

Component 2

Component 1

Component 2

Component 1

Figure 4.1-19 Orthogonal Loading in Response History Analysis

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Using the load combinations described above, the individual beam shear maxima developed in Frame 1 were computed for each load combination. Envelope values from all combinations are shown in Figure 4.1-20.

14.15

12.82

14.17

21.5

20.6

21.5

29.5

29.4

30.6

33.7

33.2

35.5

32.9

32.0

29.5

28.2

12.1

33.6

32.3

30.7

34.0

21.0

36.3

34.5

33.2

35. 7

22.0

39.0

35.3

34.5

36.2

22.8

R-12 12-11 11-10 10-9 9-8 8-7 7-6 6-5 15.1

32.9

33.9

35.8

35.6

36.0

24.6

25.0

38.5

33.6

35.6

35.5

35.7

24.7

23.7

35.7

33.1

34.3

34.2

34.3

24.0

21.6

34.3

32.3

33.1

33.0

33.5

21.9

5-4 4-3 3-2 2-G

Figure 4.1-20 Envelope of seismic shears in girders (kips) as computed using response history analysis (analysis includes orthogonal loading but excludes accidental torsion)

4.1.8

Comparison of Results from Various Methods of Analysis

A summary of the results from all of the analyses is provided in Tables 4.1-24 through 4.1-28. 4.1.8.1 Comparison of base shear and story shear. The maximum story shears are shown in Table 4.124. For the response history analysis, the shears are the envelope values of story shears for all twelve individual analyses. Note that the modal response spectrum and modal response history shears for the lowest level are both equal to 956 kips, which is 0.85 times the ELF base shear. The story shear is basically of the same character—lower values in upper stories, larger values in lower stories. It appears, however, that the maximum shears from the modal response history analysis occur at stories 2, 3 and 4. This must be due to the amplified energy in the higher modes in the actual ground motions (when compared to the design spectrum).

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Chapter 4: Structural Analysis 4.1.8.2 Comparison of drift. Table 4.1-25 summarizes the drifts computed from each of the analyses. The modal response history drifts are the envelopes among all analyses. The ELF drifts are significantly greater than those determined using modal response spectrum analysis. The drifts from the modal response history analysis are slightly greater than those from the response spectrum analysis. 4.1.8.3 Comparison member forces. The shears developed in Bay D-E of Frame 1 are compared in Table 4.1-26. The shears from the response history analysis are envelope values among all analyses, including torsion and orthogonal load effects. The response history approach produced beam shears similar to those from ELF analysis and somewhat greater than those produced by response spectrum analysis. Table 4.1-24 Summary of Results of Various Methods of Analysis: Story Shear Modal Level ELF response Modal response history spectrum R 187 180 295 12 341 286 349 11 471 357 462 10 578 418 537 9 765 524 672 8 866 587 741 7 943 639 753 6 999 690 943 5 1,070 784 1,135 4 1,102 840 1,099 3 1,118 895 1,008 2 1,124 956 956 Table 4.1-25 Summary of Results from Various Methods of Analysis: Story Drifts X Direction Drift (in.) Level Modal Modal ELF response response spectrum history R 0.99 0.66 0.85 12 1.41 0.89 1.13 11 1.75 1.03 1.35 10 1.92 1.08 1.41 9 1.82 0.98 1.23 8 1.97 1.06 1.26 7 2.01 1.08 1.25 6 1.97 1.08 1.33 5 1.67 0.97 1.24

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FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 4.1-25 Summary of Results from Various Methods of Analysis: Story Drifts X Direction Drift (in.) Level Modal Modal ELF response response spectrum history 4 1.69 1.02 1.21 3 1.65 1.05 1.10 2 2.00 1.34 1.35 1.0 in. = 25.4 mm.

Table 4.1-26 Summary of Results from Various Methods of Analysis: Beam Shear Beam Shear Force in Bay D-E of Frame 1 (kips) Level Modal response Modal response ELF spectrum history R 10.27 8.72 12.82 12 18.91 15.61 20.61 11 28.12 21.61 29.45 10 33.15 24.02 33.22 9 34.69 23.32 32.02 8 35.92 23.47 32.30 7 40.10 26.15 34.53 6 40.58 26.76 35.29 5 36.52 25.29 35.82 4 34.58 24.93 35.65 3 35.08 26.60 34.27 2 35.28 28.25 33.07 1.0 kip = 4.45 kN.

4.1.8.4 Which analysis method is best? In this example, an analysis of an irregular steel moment frame was performed using three different techniques: equivalent lateral force analysis, modal response spectrum analysis and modal response history analysis. Each analysis was performed using a linear elastic model of the structure even though it is recognized that the structure will repeatedly yield during the earthquake. Hence, each analysis has significant shortcomings with respect to providing a reliable prediction of the actual response of the structure during an earthquake. The purpose of analysis, however, is not to predict response but rather to provide information that an engineer can use to proportion members and to estimate whether or not the structure has sufficient stiffness to limit deformations and avoid overall instability. In short, the analysis only has to be “good enough for design.” If, on the basis of any of the above analyses, the elements are properly designed for strength, the stiffness requirements are met and the elements and connections of the structure are detailed for inelastic response according to the requirements of ASCE 7 and AISC 341, the structure will likely survive an earthquake consistent with the MCE ground motion. The exception would be if a highly

4-52

Chapter 4: Structural Analysis irregular structure were analyzed using the ELF procedure. Fortunately, ASCE 7 safeguards against this by requiring three-dimensional dynamic analysis for highly irregular structures. For the structure analyzed in this example, the irregularities were probably not so extreme such that the ELF procedure would produce a “bad design.” However, where computer programs that can perform modal response spectrum analysis with only marginally increased effort over that required for ELF are available (e.g., SAP2000 and ETABS), the modal analysis should always be used for final design in lieu of ELF (even if ELF is allowed by the Provisions). As mentioned in the example, this does not negate the need for or importance of ELF analysis because such an analysis is useful for preliminary design and several components of the ELF analysis are necessary for application of accidental torsion. Modal response history analysis is of limited practical use where applied to a linear elastic model of the structure. The amount of additional effort required to select and scale the ground motions, perform the modal response history analysis, scale the results and determine envelope values for use in design simply is not warranted where compared to the effort required for modal response spectrum analysis. This might change in the future where “standard” suites of ground motions are developed and are made available to the earthquake engineering community. Also, significant improvement is needed in the software available for the preprocessing and, particularly, for the post-processing of the huge amounts of information that produced by the analysis. Scaling the ground motions used for modal response history analysis is also an issue. The Standard requires that the selected motions be consistent with the magnitude, distance and source mechanism of the MCE expected at the site. If the ground motions satisfy this criterion, then why scale at all? Distant earthquakes may have a lower peak acceleration but contain a frequency content that is more significant. Near-source earthquakes may display single damaging pulses. Scaling these two earthquakes to the Standard design spectrum seems to eliminate some of the most important characteristics of the ground motions. The fact that there is a degree of freedom in the ASCE 7 scaling requirements compensates for this effect, but only for very knowledgeable users. The main benefit of modal response history analysis is in the nonlinear dynamic analysis of structures or in the analysis of non-proportionally damped linear systems. This type of analysis is the subject of Example 4.2. 4.1.9

Consideration of Higher Modes in Analysis

All of the computed results for the modal response spectrum and modal response history methods of analysis were based on the first 12 modes of the model with the basem*nt level explicitly modeled. Recall that the basem*nt walls were modeled with 1.0-foot shell elements, that the grade-level diaphragm was modeled using 6.0-inch-thick shell elements and that the grade level was not laterally restrained. The weight associated with the basem*nt-level walls and grade-level slab is 6,526 kips, which is approximately 15 percent of the total weight of the structure (see Table 4.1-3). The accumulated effective modal mass for the first 12 modes (see Table 4.1-14a) is in the neighborhood of 82 percent of the total mass of the structure, which is less than the 90 percent required by Section 12.9-1 of the Standard. However, the first 12 modes capture more than 90 percent of the mass above grade, so it was deemed sufficient to run the analysis with only 12 modes. If the requirement of Section 12.9-1 were satisfied for the structure as modeled, it would have taken 119 modes to capture more than 90 percent of the effective mass of the entire system (see Table 4.1-14b). In the analysis presented so far, all of the seismic base shears were computed at the base of the first story above grade, not the base of the entire structure (the base of the basem*nt walls). It is of some interest to 4-53

FEMA P-751, NEHRP Recommended Provisions: Design Examples examine how the results of the analysis would change if 120 modes were to be used in the analysis. This would definitely satisfy the requirements of Section 12.9-1 for the full structure. 4.1.9.1 Modal response spectrum analysis with higher modes. Table 4.1-27a provides the seismic shears through the basem*nt level and through the first floor above grade for the analysis run with 12, 18, 120 and 200 modes. In this part of the table, the “modes” are the natural mode shapes from an eigenvalue analysis. As may be seen, the shear through the first story above grade is unchanged as the number of modes increases above 12 modes. However, the shears though the basem*nt level are substantially increased when 120 or more modes are used. In the X direction, for example, the ratio of the basem*ntlevel shear for 120 modes to that for 12 modes is 630/439 = 1.44. Thus, in terms of the shear at the base of the structure, the activation of the higher modes increases the shears 44 percent, while the added weight associated with the basem*nt level is only 15 percent. This increase in shear was rather unexpected, so the analysis was re-run using Ritz vectors in lieu of the natural mode shapes. Ritz vectors automatically include the “static corrections” that are sometimes needed for very high frequency modes. As may be seen from Table 4.1-27b, the results using Ritz Vectors are virtually identical to those obtained using the natural mode shapes. 4.1.9.2 Modal response history analysis with higher modes. The comparison of shears using modal response history analysis with 12, 18, 120 and 200 modes are presented in Table 4.1-28. The results are based on the use of natural mode shapes. For brevity, results are given only for motions A00, B00 and C00 applied in the X and Y directions. The analyses include SS ground motion scaling, I/R scaling, but not the 85 percent scaling. As may be observed from Table 4.1-27, the use of the higher modes produces virtually no change in the shears through the first level above grade. However, very significant increases in shear are developed through the basem*nt. The most extreme increase in shears is for ground motion A00, wherein the shears in the basem*nt increase from 439 kips to 744 kips for loading in the X direction and increase from 440 kips to 862 kips for loading in the Y direction. These increases in shear are not unexpected because of the spectral amplitudes of the ground motions at periods associated with modes 112 and 118 (see Figure 4.1-18). 4.1.9.3 Discussion on use of higher modes. Many structures have stiff lower stories or have one or more levels of basem*nt. If the basem*nt is modeled explicitly and if full lateral restraint is not provided at the top of the basem*nt or at subgrade slab levels, the phenomena described herein will likely result. Based on the results presented above, there is some question as to which results should be used for the 85 percent scaling requirements of Standard Section 12.9.4. If the basem*nt level were included in the ELF analysis, the computed period would not significantly change, and the base shear would increase 15 percent to accommodate the added weight associated with the basem*nt walls and grade-level slab. However, the scale factors required to bring the modal response spectrum or modal response history shears up to 85 percent of the ELF shears (with 15 percent increase) could be significantly less than those obtained when the basem*nt level is not included in the model. The net result would be significantly reduced for design shears in the upper levels of the structure. Given these results, it is recommended that scaling always be based on the shears determined at the first level above grade. The question of how many modes to use in the analysis is not as easy to answer. Certainly, a sufficient number of modes must be used to capture at least 90 percent of the above-grade mass. In cases where it is desired to explicitly determine the shears at the base of unrestrained basem*nts, enough modes should be used to capture 90 percent of the mass of the entire structure.

4-54

Chapter 4: Structural Analysis Table 4.1-27 Comparison of Modal Response Spectrum Shears Using 12, 18, 120 and 200 Modes (a) Using Natural Mode Shapes (values from SAP2000 without 85% scaling) Shear Location Base of 1st story Base of structure Base of 1st story Base of structure

Load Case Code spectrum X direction Code spectrum X direction Code spectrum Y direction Code spectrum Y direction

Shear (kips) for number of modes = 12 18 120 200 438 438 439 439 439 439 630 631 492 493 493 493 493 493 686 686

(b) Using Ritz Vectors (values from SAP2000 without 85% scaling) Shear Location Base of 1st story Base of structure Base of 1st story Base of structure

Load Case Code spectrum X direction Code spectrum X direction Code spectrum Y direction Code spectrum Y direction

Shear (kips) for number of modes = 12 18 120 200 435 439 439 439 435 439 467 630 485 494 494 494 485 494 497 686

Table 4.1-28 Comparison of Modal Response History Shears Using 12, 18, 120 and 200 Modes Using Natural Mode Shapes (values from SAP2000 without 85% scaling) Shear Location Base of 1st story Base of structure Base of 1st story Base of structure Base of 1st story Base of structure Base of 1st Story Base of Structure Base of 1st Story Base of Structure Base of 1st Story Base of Structure

Load Case A00 in X direction A00 in X direction B00 in X direction B00 in X direction C00 in X direction C00 in X direction A00 in Y direction A00 in Y direction B00 in Y direction B00 in Y direction C00 in Y direction C00 in Y direction

Shear (kips) for number of modes= 12 18 120 200 438 438 445 445 439 439 744 744 376 376 377 490 377 377 529 530 391 391 392 391 392 392 438 440 447 447 452 452 440 447 861 862 391 391 395 395 397 392 508 508 301 301 301 301 307 302 561 562

4-55

FEMA P-751, NEHRP Recommended Provisions: Design Examples 4.1.10 Commentary on the ASCE 7 Requirements for Analysis As mentioned in this example, ASCE 7-05 contained several inconsistencies in scaling requirements for modal response spectrum analysis and for modal response history analysis. The main source of problems was in Chapter 16 of ASCE 7-05 and fortunately, most of these problems have been eliminated in ASCE 7-10. There are still a few issues that need to be clarified. Some of these are listed below: §

Accidental torsion: The Standard needs to be more specific on how accidental torsion should be applied where used with modal response spectrum and modal response history analyses. The method suggested herein, to apply such torsions as part of a static loading, is easy to implement. However, “automatic” methods based on shifting center of mass need to be explored and, if effective, standardized.

§

Amplification of accidental torsion: Currently, accidental torsion need be amplified only for torsionally irregular structures in SDC C and higher (Sec. 12.8.4.3). However, the torsion need not be amplified if a “dynamic” analysis is performed (Sec. 12.9.5). This implies that the amplification of torsion is a dynamic phenomenon, but the author has found no published technical basis for such amplification. Indeed, most references to amplification are based on problems associated with uneven yielding of lateral load-resisting components. This issue needs to be clarified and resolved.

§

P-Delta effects: It appears that the most efficient method for handling P-delta effects is to perform the analysis without such effects, use a separate ELF analysis to determine if such effects are significant and if so, magnify forces and displacements to include such effects. It would be much more reasonable to include such effects in the analysis directly and establish procedures to determine if such effects are excessive. Comparison of analysis results with and without P-delta effects included is an effective means to assess the significance of the effects.

§

Computing drift: When three-dimensional analysis is performed, drift should be checked at the corners of the building, not the center of mass. Consideration should be given to eliminating the use of story drift in favor of computing shear strain in damageable components. Such calculations can be easily automated.

§

Scaling ground motions for linear response history analysis: The need to scale ground motions over a period range of 0.2T to 1.5T is not appropriate for elastic analysis because the effective period in any mode will never exceed 1.0T. Additionally, placing equal weight on scaling spectral ordinates at higher modes does not seem rational. In some cases a high mode that is only barely contributing to response can dominate the scaling process.

§

Development of standard ground motion histories: The requirement that analysts scan through thousands of ground motion records to find appropriate suites for analysis is unnecessarily burdensome. The Standard should provide tables of ground motion suites that are appropriate to simple parameters such as magnitude, site class and distance.

Finally, it is suggested that requirements for linear response history analysis be removed from Chapter 16 and placed in Chapter 12 (as Section 12.10, for example). Requirements for performing such analysis should be as consistent as possible with those of modal response spectrum analysis.

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Chapter 4: Structural Analysis 4.2

SIX-‐STORY STEEL FRAME BUILDING, SEATTLE, WASHINGTON

In this example, the behavior of a simple, six-story structural steel moment-resisting frame is investigated using a variety of analytical techniques. The structure was initially proportioned using a preliminary analysis and it is this preliminary design that is investigated. The analysis will show that the structure falls short of several performance expectations. In an attempt to improve performance, viscous fluid dampers are considered for use in the structural system. Analysis associated with the added dampers is performed in a very preliminary manner. The following analytical techniques are employed: §

Linear static analysis

§

Plastic strength analysis (using virtual work)

§

Nonlinear static (pushover) analysis

§

Linear dynamic (response history) analysis

§

Nonlinear dynamic (response history) analysis

The primary purpose of this example is to highlight some of the more advanced analytical techniques; hence, more detail is provided on these methods. It is also noted that the linear dynamic analysis was performed only as a precursor and check on the analytical model used for nonlinear dynamic analysis and is not discussed in the example. The 2005 and 2010 versions of the Standard do not provide any guidance on pushover analysis because it is not a permitted method of analysis in Table 12.6-1. Some guidance for pushover analysis is provided in Resource Paper 2 in Part 3 of the Provisions. More detailed information on pushover analysis is provided in FEMA 440 and in ASCE 41. The procedures outlined in ASCE 41 are used in this example. Chapter 16 of the Standard provides some guidance and requirements for linear and nonlinear response history analysis. Certain aspects of these requirements are clarified in ASCE 7-10, but the basic methodology is unchanged. More detailed requirements for response history analysis are provided in Resource Paper 3 of the Provisions. This example follows the recommendations in Resource Paper 3, with certain exceptions, which are noted as the example proceeds. 4.2.1

Description of Structure

The structure analyzed for this example is a six-story office building in Seattle, Washington. According to the descriptions in Standard Table 1-1, the building is assigned to Occupancy Category II. From Standard Table 11.5-1, the importance factor (I) is 1.0. A plan and elevation of the building are shown in Figures 4.2-1 and 4.2-2, respectively. The lateral load-resisting system consists of steel moment-resisting frames on the perimeter of the building. There are five bays at 28 feet on center in the north-south (N-S) direction and six bays at 30 feet on center in the east-west (E-W) direction. The typical story height is 12 feet-6 inches with the exception of the first story, which has a height of 15 feet. There is a 5-foot-tall perimeter parapet at the roof and one basem*nt level that extends 15 feet below grade. For this example, it is assumed that the columns of the moment-resisting frames are embedded into pilasters formed into the reinforced concrete basem*nt wall.

4-57

FEMA P-751, NEHRP Recommended Provisions: Design Examples For the moment-resisting frames in the N-S direction (Frames A and G), all of the columns bend about their strong axes and the girders are attached with fully welded moment-resisting connections. The expected plastic hinge regions of the girders have reduced flange sections, detailed in accordance with AISC 341. For the frames in the E-W direction (Frames 1 and 6), moment-resisting connections are used only at the interior columns. At the exterior bays, the E-W girders are connected to the weak axis of the exterior (corner) columns using non-moment-resisting connections. All interior columns are gravity columns and are not intended to resist lateral loads. A few of these columns, however, would be engaged as part of the added damping system described in the last part of this example. With minor exceptions, all of the analyses in this example are for lateral loads acting in the N-S direction. Analysis for lateral loads acting in the E-W direction would be performed in a similar manner.

B 30'-0"

1'-6" (typical)

C 30'-0"

D 30'-0"

E 30'-0"

F 30'-0"

G 30'-0"

28'-0"

3 Moment connection (typical)

28'-0"

Figure 4.2-1 Plan of structural system

4-58

Chapter 4: Structural Analysis

5 at 12'-6" = 62'-6"

5'-0"

15'-0"

Basem*nt wall

5 at 28'-0" = 140'-0"

Figure 4.2-2 Elevation of structural system Prior to analyzing the structure, a preliminary design was performed in accordance with AISC 341. All members, including miscellaneous plates, were designed using steel with a nominal yield stress of 50 ksi and expected yield strength of 55 ksi. Detailed calculations for the design are beyond the scope of this example. Table 4.2-1 summarizes the members selected for the preliminary design.1 Table 4.2-1 Member Sizes Used in N-S Moment Frames

Member supporting level

Column

Girder

R 6 5 4 3 2

W21x122 W21x122 W21x147 W21x147 W21x201 W21x201

W24x84 W24x84 W27x94 W27x94 W27x94 W27x94

Doubler plate thickness (in.) 1.00 1.00 1.00 1.00 0.875 0.875

The term Level is used in this example to designate a horizontal plane at the same elevation as the centerline of a girder. The top level, Level R, is at the roof elevation; Level 2 is the first level above grade; and Level 1 is at grade. The term Story represents the distance between adjacent levels. The story designation is the same as the designation of the level at the bottom of the story. Hence, Story 1 is the lowest story (between Levels 2 and 1) and Story 6 is the uppermost story (between Levels R and 6).

4-59

FEMA P-751, NEHRP Recommended Provisions: Design Examples

The sections shown in Table 4.2-1 meet the width-to-thickness requirements for special moment frames and the size of the column relative to the girders should ensure that plastic hinges initially will form in the girders. Due to strain hardening, plastic hinges will eventually form in the columns. However, these form under lateral displacements that are in excess of those allowed under the Design Basis Earthquake (DBE). Doubler plates of 0.875 inch thick are used at each of the interior columns at Levels 2 and 3 and 1.00 inch thick plates are used at the interior columns at Levels 4, 5, 6 and R. Doubler plates were not used in the exterior columns. 4.2.2

Loads

4.2.2.1 Gravity loads. It is assumed that the floor system of the building consists of a normal-weight composite concrete slab formed on metal deck. The slab is supported by floor beams that span in the N-S direction. These floor beams have a span of 28 feet and are spaced 10 feet on center. The dead weight of the structural floor system is estimated at 70 psf. Adding 15 psf for ceiling and mechanical units, 10 psf for partitions at Levels 2 through 6 and 10 psf for roofing at Level R, the total dead load at each level is 95 psf. The cladding system is assumed to weigh 15 psf. A basic live load of 50 psf is used at Levels 2 through 6. The roof live load is 20 psf and (based on calculations not shown here) the roof snow load is 25 psf. The reduced floor loads are taken as 0.4(50), or 20 psf. Only half of this load is required in seismic load combinations (see Standard Section 2.3), so the design live loads for the floor is 0.5(20) = 10 psf. The roof live load is not reducible, but never appears in seismic load combinations. The snow load for seismic load combinations is 0.2(25) = 5 psf, which is half of the floor live load. Based on these loads, the total dead load, live or snow load and dead plus live or snow load applied to each level of the entire building are given in Table 4.2-2. The slight difference in dead loads at Levels R and 2 is due to the parapet and the tall first story, respectively. Tributary areas for columns and girders as well as individual element gravity loads used in the analysis are illustrated in Figure 4.2-3. These loads are based on a total dead load of 95 psf, a cladding weight of 15 psf and a live load of 10 psf. Table 4.2-2 Gravity Loads on Seattle Building* Dead load Reduced live or snow load (kips) (kips) Level Story Accumulated Story Accumulated R 2,596 2,596 131 131 6 2,608 5,204 262 393 5 2,608 7,813 262 655 4 2,608 10,421 262 917 3 2,608 13,029 262 1,179 2 2,621 15,650 262 1,441 *Loads are for the entire building.

4-60

Story 2,727 2,739 2,739 2,739 2,739 2,752

Total load (kips) Accumulated 2,727 5,597 8,468 11,338 14,208 17,091

Chapter 4: Structural Analysis 4.2.2.2 Equivalent static earthquake loads. Although the main analysis in this example is nonlinear, equivalent static forces are computed in accordance with Standard Section 12.8. These forces are used in a preliminary static analysis to determine whether the structure, as designed, conforms to the drift requirements limitations imposed by Standard Section 12.2. The structure is situated in Seattle, Washington. The short period and the 1-second mapped spectral acceleration parameters for the site are as follows: §

SS = 1.63

§

S1 = 0.57

The structure is situated on Site Class C materials. From Standard Tables 11.4-1 and 11.4-2: §

Fa = 1.00

§

Fv = 1.30

From Standard Equations 11.4-1 and 11.4-2, the maximum considered spectral acceleration parameters are as follows: SMS = FaSS = 1.00(1.63) = 1.63 SM1 = FvS1 = 1.30(0.57) = 0.741 And from Standard Equations 11.4-3 and 11.4-4, the design acceleration parameters are as follows: SDS = (2/3)SM1 = (2/3)1.63 = 1.09 SD1 = (2/3)SM1 = (2/3)0.741 = 0.494

4-61

FEMA P-751, NEHRP Recommended Provisions: Design Examples

1'-6"

28'-0"

15'-0"

1'-6"

28'-0"

30'-0"

(a) Tributary area for columns 1'-6"

5'-0"

28'-0"

(b) Tributary area for girders P A - RC

PB - 2RC

Figure 4.2-3 Element loads used in analysis

4-62

PB - 2RC

Chapter 4: Structural Analysis

Based on the above coefficients and on Standard Tables 11.6-1 and 11.6-2, the structure is assigned to Seismic Design Category D. For the purpose of analysis, it is assumed that the structure complies with the requirements for a special moment frame, which, according to Standard Table 12.2-1, has the following design values: §

R=8

§

Cd = 5.5

§

Ω0 = 3.0

Note that the overstrength factor, Ω0, is not needed for the analysis presented herein. 4.2.2.2.1 Approximate period of vibration. Standard Equation 12.8-7 is used to estimate the building period:

Ta = Ct hnx where, from Standard Table 12.8-2, Ct = 0.028 and x = 0.8 for a steel moment frame. Using hn (the total building height above grade) = 77.5 feet, Ta = 0.028(77.5)0.8 = 0.91 sec/cycle5. Where the period is determined from a properly substantiated analysis, the Standard requires that the period used for computing base shear not exceed CuTa where, from Standard Table 12.8-1 (using SD1 = 0.494), Cu = 1.4. For the structure under consideration, CuTa = 1.4(0.91) = 1.27 seconds. This period is used for base shear calculation as it is expected that the period computed for the actual structure will be greater than 1.27 seconds. 4.2.2.2.2 Computation of base shear. Using Standard Equation 12.8-1, the total seismic base shear is:

V = CS W where W is the total seismic weight of the structure. From Standard Equation 12.8-2, the maximum (constant acceleration region) seismic response coefficient is:

CSmax =

S DS 1.09 = = 0.136 ( R / I ) (8 / 1)

Equation 12.8-3 controls in the constant velocity region:

CS =

S D1 0.494 = = 0.0485 T ( R / I ) 1.27(8 / 1)

The seismic response coefficient, however, must not be less than that given by Equation 12.8-5:

CSmin = 0.044ISDS = 0.044(1)(1.09) = 0.0480 5

The correct computational units for period of vibration is “seconds per cycle”. However, the traditional units of “seconds” are used in the remainder of this example.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Thus, the value from Equation 12.8-3 controls for this building. Using W = 15,650 kips, V = 0.0485(15,650) = 759 kips. 4.2.2.2.2 Vertical distribution of forces. The seismic base shear is distributed along the height of the building using Standard Equations 12.8-11 and 12.8-12: k Fx = CvxV and C = wx h vx n ∑ wi hik i =1

where k = 0.75 + 0.5T = 0.75 + 0.5(1.27) = 1.385. The lateral forces acting at each level and the story shears acting at the bottom of the story below the indicated level are summarized in Table 4.2-3. These are the forces acting on the whole building. For analysis of a single frame, one-half of the tabulated values are used. Table 4.2-3 Equivalent Lateral Forces for Building Responding in N-S Direction wx hx Fx Level x w xh xk Cvx (kips) (ft) (kips) 77.5 1,080,327 0.321 243.6 R 2,596 65.0 850,539 0.253 191.8 6 2,608 52.5 632,564 0.188 142.6 5 2,608 40.0 433,888 0.129 97.8 4 2,608 27.5 258,095 0.077 58.2 3 2,608 15.0 111,909 0.033 25.2 2 2,621 15,650 3,367,323 1.000 759.3 Σ

4.2.3

Vx (kips) 243.6 435.4 578.0 675.9 734.1 759.3

Preliminaries to Main Structural Analysis

Performing a nonlinear analysis of a structure is an incremental process. The analyst should first perform a linear analysis to obtain some basic information on expected behavior and to serve later as a form of verification for the more advanced analysis. Once the linear behavior is understood (and extrapolated to expected nonlinear behavior), the anticipated nonlinearities are introduced. If more than one type of nonlinear behavior is expected to be of significance, it is advisable to perform a preliminary analysis with each nonlinearity considered separately and then to perform the final analysis with all nonlinearities considered. This is the approach employed in this example. 4.2.3.1 The computer programs NONLIN-Pro and DRAIN 2Dx The computer program NONLIN-Pro was used for all of the analyses described in this example. This program is basically a pre- and post-processor to DRAIN 2Dx (Prakash et al., 1993). While DRAIN is not the most robust program currently available for performing nonlinear response history analysis, it was used because many of the details of the analysis (e.g., panel zone modeling) must be done explicitly. This detail provides insight into the modeling process which is not available when using the automated features of the more robust software. Note that a full version of NONLIN-Pro, as well as input files used for this example, is provided on the CD.

4-64

Chapter 4: Structural Analysis DRAIN has several shortcomings that are related specifically to the example at hand. These shortcomings are listed below. Also provided is a brief explanation of the influence the shortcoming may have on the analysis. §

It is not possible to model strength loss when using the ASCE 41 model for girder plastic hinges. However, as discussed later in the example, this loss of strength generally occurs at plastic hinge rotations well beyond the rotational demands produced under the DBE ground motions. Maximum plastic rotation angles of plastic hinges were checked with the values in Table 5-6 of ASCE 41-06.

§

The DRAIN model for axial-flexural interaction in columns is not particularly accurate. This is of some concern in this example because hinges form at the base of the columns in all of the analyses and in some of the upper columns during analysis with MCE level ground motions.

§

Only two-dimensional analysis may be performed. Such an analysis is reasonable for the structure considered in this example because of its regular shape and because full moment connections are provided only in the N-S direction for the corner columns (see Fig. 4.2-1).

As with any finite element analysis program, DRAIN models the structure as an assembly of nodes and elements. While a variety of element types is available, only three element types were used in the analysis: §

Type 1 inelastic bar (truss) element

§

Type 2 beam-column element

§

Type 4 connection element

Two models of the structure were prepared for DRAIN. The first model, used for preliminary analysis and for verification of the second (more advanced) model, consisted only of Type 2 elements for the main structure and Type 1 elements for modeling P-delta effects. All analyses carried out using this model were linear. For the second, more detailed model, Type 1 elements were used for modeling P-delta effects and the dampers in the damped system. It was assumed that these elements would remain linear elastic throughout the response. Type 2 elements were used to model the beams, the columns and the braces in the damped system, as well as the rigid links associated with the panel zones. Plastic hinges were allowed to form in all columns. The column hinges form through the mechanism provided in DRAIN’s Type 2 element. Plastic behavior in girders and in the panel zone region of the structure was modeled explicitly through the use of Type 4 connection elements. Girder yielding was forced to occur in the Type 4 elements (in lieu of the main span represented by the Type 2 elements) to provide more control in hinge location and modeling. A complete description of the implementation of these elements is provided later. 4.2.3.2 Description of preliminary model and summary of preliminary results The preliminary DRAIN model is shown in Figure 4.2-4. Important characteristics of the model are as follows: §

Only a single frame (Frame A or G) is modeled. Hence one-half of the loads shown in Tables 4.2-2 and 4.2-3 are applied.

4-65

FEMA P-751, NEHRP Recommended Provisions: Design Examples §

Columns are fixed at their base (at grade level; the basem*nt is not modeled).

§

Each beam or column element is modeled using a Type 2 element. For the columns, axial, flexural and shear deformations are included. For the girders, flexural and shear deformations are included but, because of diaphragm slaving, axial deformation is not included. Composite action in the floor slab is ignored for all analysis.

§

All members are modeled using centerline dimensions without rigid end offsets. This approach allows for the effects of panel zone deformation to be included in an approximate but reasonably accurate manner. Note that this model does not provide any increase in beam-column joint stiffness due to the presence of doubler plates. The stiffness of the girders was decreased by 7 percent (in preliminary analyses) to account for the reduced flange sections. Moment rotation properties of the reduced flange sections are used in the detailed analyses.

P-delta effects are modeled using the leaner “ghost” column shown in Figure 4.2-4 at the right of the main frame. This column is modeled with an axially rigid truss element. P-delta effects are activated for this column only (P-delta effects are turned off for the columns of the main frame). The lateral degree of freedom at each level of the P-delta column is slaved to the floor diaphragm at the matching elevation. Where P-delta effects are included in the analysis, a special initial load case was created and executed. This special load case consists of a vertical force equal to one-half of the total story weight (dead load plus 50 percent of the fully reduced live load) applied to the appropriate node of the P-delta column. When P-delta effects are included, modal analysis should be performed after the P-delta load case is applied so that stiffness modification of P-delta effects will increase the period of the structure. P-delta effects are modeled in this manner to provide true column axial forces for assessing strength.

Frame A or G

P-Δ column

R 6 5 4 3 2

Y X

Figure 4.2-4 Simple wire frame model used for preliminary analysis 4.2.3.2.1 Results of preliminary analysis: period of vibration and drift. The computed periods for the first three natural modes of vibration are shown in Table 4.2-4. As expected, the period including P-delta effects is slightly larger than that produced by the analysis without such effects. More significant is the fact that the first mode period is considerably longer than that predicted from Standard 4-66

Chapter 4: Structural Analysis Equation 12.8-7. Recall from previous calculations that this period (Ta) is 0.91 seconds and the upper limit on the computed period CuTa is 1.4(0.91) = 1.27 seconds. Where doubler plate effects are included in the detailed analysis, the period will decrease slightly, but it remains obvious that the structure is quite flexible. Table 4.2-4 Periods of Vibration From Preliminary Analysis (sec/cycle) Mode P-delta excluded P-delta included 1 2.054 2.130 2 0.682 0.698 3 0.373 0.379 The results of the preliminary analysis for drift are shown in Tables 4.2-5 and 4.2-6 for the computations excluding and including P-delta effects, respectively. In each table, the deflection amplification factor (Cd) equals 5.5 and the acceptable story drift (story drift limit) is taken as 2 percent of the story height, which is the limit provided by Standard Table 12.12-1. In the Standard it is permitted to determine the elastic drifts using seismic design forces based on the computed fundamental period of the structure without the upper limit CuTa. Thus a new set of lateral loads based on the computed period of the actual structure is applied to the structure to calculate the elastic drifts. Where P-delta effects are not included, the computed story drift is less than the allowable story drift at each level of the structure. The largest magnified story drift, including Cd = 5.5, is 2.26 inches in Stories 2 and 3. As a preliminary estimate of the importance of P-delta effects, story stability coefficients, θ, were computed in accordance with Standard Section 12.8-7. These are shown in the last column of Table 4.2-5. At Story 2, the stability coefficient is 0.0862. According to the Standard, P-delta effects may be ignored where the stability coefficient is less than 0.10. For this example, however, analyses are performed with and without P-delta effects. When P-delta effects are included (Table 4.2-6), the drifts can also be estimated as the drifts without P-delta times the quantity 1/(1-θ), where θ is the stability coefficient for the story. As can be seen in Table 4.2-6, drifts calculated in this manner are consistent with the results obtained by running the analyses with P-delta effects. The difference is always less than 2 percent. Table 4.2-5 Results of Preliminary Analysis Excluding P-delta Effects Magnified Total drift Story drift Drift limit Story story drift (in.) (in.) (in.) (in.) 6 2.08 0.22 1.21 3.00 5 1.86 0.32 1.76 3.00 4 1.54 0.38 2.09 3.00 3 1.16 0.41 2.26 3.00 2 0.75 0.41 2.26 3.00 1 0.34 0.34 1.87 3.60

Story stability ratio, θ 0.0278 0.0453 0.0608 0.0749 0.0862 0.0691

4-67

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 4.2-6 Results of Preliminary Analysis Including P-delta Effects Magnified Total drift Story drift Drift from θ Story story drift (in.) (in.) (in.) (in.) 6 2.23 0.23 1.27 1.24 5 2.00 0.34 1.87 1.84 4 1.66 0.40 2.20 2.23 3 1.26 0.45 2.48 2.44 2 0.81 0.45 2.48 2.47 1 0.36 0.36 1.98 2.01

Drift limit (in.) 3.00 3.00 3.00 3.00 3.00 3.60

4.2.3.2.2 Results of preliminary analysis: demand-to-capacity ratios. To determine the likelihood and possible order of yielding, demand-to-capacity ratios (DCR) are computed for each element. The results are shown in Figure 4.2-5. For this analysis, the structure is subjected to full dead load plus 0.5 times the fully reduced live load, followed by equivalent lateral forces computed without the R factor. Pdelta effects are included. Figure 4.2-5(a) displays the DCR of columns and girders and Figure 4.2-5(b) displays the DCR of panel zones with and without doubler plates. In Figure 4.2-5b, the values in parentheses represent the DCRs without doubler plates. Since the DCRs in Figure 4.2-5 are found from preliminary analyses, in which the centerline model is used, doubler plates aren’t added into the model. Thus, the demand values of Figure 4.2-5(b) are the same with and without doubler plates. However, since the capacity of the panel zone increases with added doubler plates, the DCRs decrease at the interior beam column joints as the doubler plates are used only at the interior joints. As may be seen in Figure 4.2-5(b), the DCR at the exterior joints are the same with and without doubler plates added. For girders, the DCR is simply the maximum moment in the member divided by the member’s plastic moment capacity where the plastic capacity is ZeFye. Ze is the plastic section modulus at center of reduced beam section and Fye is the expected yield strength. For columns, the ratio is similar except that the plastic flexural capacity is estimated to be Zcol(Fye-Pu/Acol) where Pu is the total axial force in the column. The ratios are computed at the center of the reduced section for beams and at the face of the girder for columns. To find the shear demand at the panel zones, the total moment in the girders (at the left and right sides of the joint) is divided by the effective beam depth to produce the panel shear due to beam flange forces. Then the column shear at above or below the panel zone joint was subtracted from the beam flange shears and the panel zone shear force is obtained. This force is divided by the shear strength capacity, Rv (which is discussed in Section 4.2.4.2) to determine the DCR of the panel zones. Several observations are made regarding the likely inelastic behavior of the frame: §

The structure has considerable overstrength, particularly at the upper levels.

§

The sequence of yielding will progress from the lower-level girders to the upper-level girders. Because of relatively low live load, the DCRs in the girders are almost uniform at each level. Hence, all the hinges in the girders in a level will form almost simultaneously.

§

With the possible exception of the first level, the girders should yield before the columns. While not shown in the table, the DCRs for the lower-story columns are controlled by the moment at the base of the column. It is usually very difficult to prevent yielding of the base of the first-story

4-68

Chapter 4: Structural Analysis columns in moment frames and this frame is no exception. The column on the leeward (right) side of the building will yield first because of the additional axial compressive force arising from the seismic effects. §

The maximum DCR of the columns and girders is 3.475, while the maximum DCR for the panel zones without doubler plates is 4.339. Thus, if doubler plates aren’t used, the first yield in the structure is in the panel zones. However, with doubler plates added, the first yield is at the girders as the maximum DCR of the panel zones reduces to 2.405.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

1.033

Level R

0.973

0.595

1.084 1.837

Level 6

1.060 1.249 1.041

Level 2 3.345

1.575 2.895

2.850

1.483

3.475

1.550 2.903

2.922

1.882 3.189

1.550 2.903

1.203

3.085

1.857 3.198

1.601 3.155

1.712 2.773

1.857 3.198

1.074

2.626

1.692 2.782

1.908 3.406

Level 3

1.482 2.357

1.693 2.782

0.671

1.935

1.482 2.366

1.721 3.025

Level 4

1.082 1.826

1.477 2.366

1.098

1.082 1.815

1.480 2.557

0.971

1.082 1.826

0.971

Level 5

0.968

1.225

3.224

2.850

2.856

4.043

(a) DCRs of columns and girders (0.839)

(1.427)

(1.422)

(1.427)

(1.429)

(0.899)

Level R 0.839

(1.656)

0.574

(3.141)

0.576

(3.149)

0.576

(3.149)

0.577

(3.149)

0.899

(1.757)

(2.021)

1.268

(3.774)

1.272

(3.739)

1.272

(3.732)

1.272

(3.779)

1.757

(2.092)

(2.343)

1.699

(4.334)

1.683

(4.285)

1.680

(4.285)

1.701

(4.339)

2.092

(2.405)

(1.884)

1.951

(3.598)

1.929

(3.567)

1.929

(3.567)

1.953

(3.605)

2.405

(1.932)

(1.686)

2.009

(3.128)

1.991

(3.076)

1.991

(3.076)

2.013

(3.132)

1.932

(1.731)

Level 6 1.656

Level 5 2.021

Level 4 2.343

Level 3 1.884

Level 2 1.686

1.746

1.718

1.749

1.731

(b) DCRs of panel zones with and without doubler plates (DCR values in parentheses are without doubler plates) Figure 4.2-5 DCRs for elements from preliminary analysis with P-delta effects included

4.2.3.2.3 Results of preliminary analysis: overall system strength. The last step in the preliminary analysis is to estimate the total lateral strength (collapse load) of the frame using virtual work. In the analysis, it is assumed that plastic hinges are perfectly plastic. Girders hinge at a value ZeFye and the hinges form at the center of the reduced section (approximately 15 inches from the face of the column). 4-70

Chapter 4: Structural Analysis Columns hinge only at the base and the plastic moment capacity is assumed to be Zcol(Fye-Pu/Acol). The fully plastic mechanism for the system is illustrated in Figure 4.2-6. The inset to the figure shows how the angle modification term, σ, was computed. The strength, V, for the total structure is computed from the following relationships (see Figure 4.2-6 for nomenclature): §

Internal Work = External Work

§

Internal Work = 2[20σθMPA + 40σθMPB + θ(MPC + 4MPD + MPE)]

§

External Work = V θ

nLevels

∑ i =1

Fi H i where

nLevels

∑ i =1

Fi = 1

Three lateral force patterns are used: uniform, upper triangular and Standard (where the Standard pattern is consistent with the vertical force distribution of Table 4.2-3 in this volume of design examples). The results of the analysis are shown in Table 4.2-7. As expected, the strength under uniform load is significantly greater than under triangular or Standard load. The closeness of the Standard and triangular load strengths results from the vertical-load-distributing parameter (k = 1.385) being close to 1.0. The ELF base shear, 759 kips (see Table 4.2-3), when divided by the Standard pattern capacity, 2,616 kips, is 0.29. This is reasonably consistent with the DCRs shown in Figure 4.2-5. Table 4.2-7 Lateral Strength on Basis of Rigid-Plastic Mechanism Lateral strength for Lateral strength Lateral Load Pattern entire structure (kips) single frame (kips) Uniform 3,332 1,666 Upper Triangular 2,747 1,373 Standard 2,616 1,308

4-71

FEMA P-751, NEHRP Recommended Provisions: Design Examples

M PA M PA M PB M PB M PB

(c)

M PB Y

M PC

M PD

M PE

X (a)

e = 0.625bbf + 0.375db + 0.5dc

db θ dc (b)

σθ σθ

θ'

eθ e

(c)

L-2e

2e θ e

(d)

Figure 4.2-6 Plastic mechanism for computing lateral strength Three important points concerning the virtual work analysis are as follows: §

The rigid-plastic analysis does not include strain hardening, which is an additional source of overstrength.

§

The rigid-plastic analysis does not consider the true behavior of the panel zone region of the beam-column joint. Yielding in this area can have a significant effect on system strength.

§

Slightly more than 15 percent of the system strength comes from plastic hinges that form in the columns. If the strength of the column is taken simply as Mp (without the influence of axial force), the difference in total strength is less than 2 percent.

4.2.4

Description of Model Used for Detailed Structural Analysis

Nonlinear static and nonlinear dynamic analyses require a much more detailed model than was used in the linear analysis. The primary reason for the difference is the need to explicitly represent yielding in the girders, columns and panel zone region of the beam-column joints. 4-72

Chapter 4: Structural Analysis

The DRAIN model used for the nonlinear analysis is shown in Figure 4.2-7. A detail of a girder and its connection to two interior columns is shown in Figure 4.2-8. The detail illustrates the two main features of the model: an explicit representation of the panel zone region and the use of concentrated plastic hinges in the girders.

15'-0"

See Figure 4.2-8

5 at 12'-6"

In Figure 4.2-7, the column shown to the right of the structure is used to represent P-delta effects. See Section 4.2.3.2 for details.

28'-0" (typical)

Figure 4.2-7 Detailed analytical model of six-story frame

Panel zone panel spring (typical)

Girder plastic hinge

Panel zone flange spring (typical)

Figure 4.2-8 Model of girder and panel zone region The development of the numerical properties used for panel zone and girder hinge modeling is not straightforward. For this reason, the following theoretical development is provided before proceeding with the example. 4.2.4.1 Plastic hinge modeling and compound nodes. In the analysis described below, much use is made of compound nodes. These nodes are used to model plastic hinges in girders and deformations in the panel zone region of beam-column joints.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

A compound node typically consists of a pair of single nodes with each node sharing the same point in space. The X and Y degrees of freedom of the first node of the pair (the slave node) are constrained to be equal to the X and Y degrees of freedom of the second node of the pair (the master node), respectively. Hence, the compound node has four degrees of freedom: an X displacement, a Y displacement and two independent rotations. In most cases, one or more rotational spring connection elements (DRAIN element Type 4) are placed between the two single nodes of the compound node and these springs develop bending moment in resistance to the relative rotation between the two single nodes. If no spring elements are placed between the two single nodes, the compound node acts as a moment-free hinge. A typical compound node with a single rotational spring is shown in Figure 4.2-9. The figure also shows the assumed bilinear, inelastic moment-rotation behavior for the spring.

4-74

Chapter 4: Structural Analysis

Master

Slave

θMaster

Rotational spring

θSlave

dθ = θMaster - θSlave

Master node Slave node

(b)

(a) My 1

αΚ

Κ 1

dθ

(c)

Figure 4.2-9 A compound node and attached spring 4.2.4.2 Modeling of beam-column joint regions. A very significant portion of the total story drift of a moment-resisting frame is due to deformations that occur in the panel zone region of the beam-column joint. In this example, panel zones are modeled explicitly using an approach developed by Krawinkler (1978) and described in more detail in Charney and Marshall (2006). Only a brief overview is presented here. This model, illustrated in Figure 4.2-10, represents the panel zone stiffness and strength by an assemblage of four rigid links and two rotational springs. The links form the boundary of the panel and the springs are used to provide the desired inelastic behavior. The model has the advantage of being conceptually

4-75

FEMA P-751, NEHRP Recommended Provisions: Design Examples simple, yet robust. The disadvantage of the model is that the number of degrees of freedom required to model a structure is significantly increased.4

A J

Figure 4.2-10 Krawinkler beam-column joint model The Krawinkler model assumes that the panel zone has two resistance mechanisms acting in parallel: §

Shear resistance of the web of the column, including doubler plates

§

Flexural resistance of the flanges of the column

These two resistance mechanisms, apparent in AISC 360 Section J10-11, are used for determining panel zone shear strength:

⎡ 3bcf tcf2 ⎤ Rv = 0.6 Fy d c t p ⎢1 + ⎥ ⎢⎣ db dc t p ⎥⎦ The equation can be rewritten as:

Rv = 0.6Fy dct p +1.8

Fy bcf tcf2 db

= VPanel + 1.8VFlanges

In ASCE 41, the first term of the above equation is taken as 0.55Fye dc t p and the second term is neglected conservatively. In this study, the following equation—in which the first term is taken as the same as in

The numbers of degrees of freedom in the Krawinkler model may be reduced to only four if the rigid links around the perimeter of the model are represented by mathematical constraints instead of stiff elements. Most commercial programs employ this approach for the Krawinkler model.

4-76

Chapter 4: Structural Analysis ASCE 41 and the second term is taken from AISC 360, with the exception of replacing nominal yield stress with expected yield strength (for consistency)—is used to calculate the panel zone shear strength:

Rv = 0.55Fye dct p + 1.8

Fyebcf tcf2 db

= VPanel + 1.8VFlanges

where the first term is the panel shear resistance and the second term is the plastic flexural resistance of the column flange. The terms in the equations are defined as follows: Fye = expected yield strength of the column and the doubler plate dc = total depth of column tp

= thickness of panel zone region = column web thickness plus doubler plate thickness

bcf = width of column flange tcf = thickness of column flange db = total depth of girder Additional terms used in the subsequent discussion are: tbf = girder flange thickness G = shear modulus of steel

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Yielding of column flange

(a)

V Flanges

4M pθ = V Flangesdbθ V Flanges= 4M p db

θ Mp V

(b)

Figure 4.2-11 Column flange component of panel zone resistance The panel zone shear resistance, VPanel, is simply the effective shear area of the panel, dctp, multiplied by the yield stress in shear, assumed as 0.55Fye. (The 0.55 factor is a simplification of the Von Mises yield criterion that gives the yield stress in shear as 1/ 3 = 0.577 times the strength in tension.) The additional plastic flexural resistance provided by yielding in the column flange is neglected in ASCE 41-06 but is included herein. The second term, 1.8VFlanges, is based on experimental observation. Testing of simple beam-column subassemblies show that a “kink” forms in the column flanges as shown in Figure 4.2-11(a). If it can be assumed that the kink is represented by a plastic hinge with a plastic moment capacity of Mp = FyeZ = Fyebcftcf2/4, it follows from virtual work (see Figure 4.2-11b) that the equivalent shear strength of the column flanges is:

VFlanges =

4-78

4M p db

Chapter 4: Structural Analysis and by simple substitution for Mp:

VFlanges =

Fyebcf tcf2 db

This value does not include the 1.8 multiplier that appears in the AISC equation. This multiplier is based on calibration of experimental results. It should be noted that the flange component of strength is small compared to the panel component unless the column has very thick flanges. The shear stiffness of the panel is derived as shown in Figure 4.2-12:

K Panel , γ =

VPanel

VPanel δ db

noting that the displacement δ can be written as follows:

δ=

VPanel db Gt p dc

K Panel ,γ =

VPanel ⎛ VPanel db ⎜⎜ ⎝ Gt p d c

⎞ 1 ⎟⎟ ⎠ db

= Gt p d c

V Panel

Thickness = t p γ=θ

Figure 4.2-12 Column web component of panel zone resistance Krawinkler assumes that the column flange component yields at four times the yield deformation of the panel component, where the panel yield deformation is:

γy =

0.55Fye dct p 0.55Fye VPanel = = K Panel ,γ Gdct p G

At this deformation, the panel zone strength is VPanel + 0.25 Vflanges; at four times this deformation, the strength is VPanel + VFlanges. The inelastic force-deformation behavior of the panel is illustrated in 4-79

FEMA P-751, NEHRP Recommended Provisions: Design Examples Figure 4.2-13. This figure applies also to exterior joints (girder on one side only), roof joints (girders on both sides, column below only) and corner joints (girder on one side only, column below only).

Shear

Total resistance Panel

V Panel Shear

γ, panel

Flanges

V Flanges

γy

Κ γ, flanges Shear strain, γ 4γ y

Figure 4.2-13 Force-deformation behavior of panel zone region The actual Krawinkler model is shown in Figure 4.2-10. This model consists of four rigid links, connected at the corners by compound nodes. The columns and girders frame into the links at right angles at Points I through L. These are moment-resisting connections. Rotational springs are used at the upper left (Point A) and lower right (Point D) compound nodes. These springs are used to represent the panel resistance mechanisms described earlier. The upper right and lower left corners (Points B and C), without rotational springs, act as real hinges. The finite element model of the joint requires 12 individual nodes: one node each at Points I through L and two nodes (compound node pairs) at Points A through D. It is left to the reader to verify that the total number of degrees of freedom in the model is 28 (if the only constraints are associated with the corner compound nodes). The rotational spring properties are related to the panel shear resistance mechanisms by a simple transformation, as shown in Figure 4.2-14. From the figure it may be seen that the moment in the rotational spring is equal to the applied shear times the beam depth. Using this transformation, the properties of the rotational spring representing the panel shear component of resistance are as follows:

M Panel = VPanel db = 0.55Fye dc dbt p K Panel ,θ = K Panel ,γ db = Gdc dbt p

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Chapter 4: Structural Analysis

Shear = V

Moment = Vdb

(a)

(b)

Note θ = γ

Panel spring Web spring

(c)

Figure 4.2-14 Transforming shear deformation to rotational deformation in the Krawinkler model It is interesting to note that the shear strength in terms of the rotation spring is simply 0.55Fye times the volume of the panel and the shear stiffness in terms of the rotational spring is equal to G times the panel volume. The flange component of strength in terms of the rotational spring is determined in a similar manner:

M Flanges = 1.8VFlanges db = 1.8Fyebcf tcf2 Because of the equivalence of rotation and shear deformation, the yield rotation of the panel is the same as the yield strain in shear:

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

θy = γ y =

M Panel 0.55Fye = K Panel ,θ G

To determine the initial stiffness of the flange spring, it is assumed that this spring yields at four times the yield deformation of the panel spring. Hence:

K Flanges,θ =

M Flanges 4θ y

= 0.82Gbcf tcf2

The complete resistance mechanism, in terms of rotational spring properties, is shown in Figure 4.2-13. This trilinear behavior is represented by two elastic-perfectly plastic springs at the opposing corners of the joint assemblage. If desired, strain-hardening may be added to the system. ASCE 41 suggests use of a strain hardening stiffness equal to 6 percent of the initial stiffness of the joint. In this analysis, the strain-hardening component was simply added to both the panel and the flange components:

K SH ,θ = 0.06( K Panel ,θ + K Flanges ,θ ) Before continuing, one minor adjustment is made to the above derivations. Instead of using the nominal total beam and girder depths in the calculations, the distance between the center of the flanges was used as the effective depth. Hence:

dc ≡ dc,nom − tcf where the nom part of the subscript indicates the property listed as the total depth in the AISC Manual. The Krawinkler properties are now computed for a typical interior subassembly of the six-story frame. A summary of the properties used for all connections is shown in Table 4.2-8. Table 4.2-8 Properties for the Krawinkler Beam-Column Joint Model Doubler plate Mpanel,θ Kpanel,θ Connection Girder Column (in.-k) (in.-k/rad) (in.) A W24x84 W21x122 – 8,782 3,251,567 B W24x84 W21x122 1.00 23,419 8,670,846 C W27x94 W21x147 – 11,934 4,418,647 D W27x94 W21x147 1.00 28,510 10,555,656 E W27x94 W21x201 – 15,386 5,696,639 F W27x94 W21x201 0.875 30,180 11,174,176 Example calculations shown for row in bold type.

The sample calculations below are for Connection D in Table 4.2-8. §

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Material Properties:

Mflanges,θ (in.-k) 1,131 1,131 1,637 1,637 3,314 3,314

Kflanges,θ (in.-k/rad) 104,721 104,721 151,486 151,486 306,771 306,771

Chapter 4: Structural Analysis

Fye = 55.0 ksi (girder, column and doubler plate) G = 11,200 ksi §

Girder: W27x94 db,nom = 26.90 in. tbf = 0.745 in. db = 26.16 in.

§

Column: W21x147 dc,nom = 22.10 in. tw = 0.72 in. tcf = 1.150 in. dc = 20.95 in. bcf = 12.50 in.

§

Doubler plate: 1.00 in. Total panel zone thickness = tp = 0.72 + 1.00 = 1.72 in.

VPanel = 0.55Fye dct p = 0.55(55)(20.95)(1.72) = 1,090 kips VFlanges = 1.8

Fyebcf tcf2 db

= 1.8

55(12.50)(1.152 ) = 62.6 kips 26.16

K Panel ,γ = Gt p dc = 11,200(1.72)(20.95) = 403,581 kips/unit shear strain

γ y =θy =

0.55Fye G

0.55(55) = 0.0027 11,200

M Panel = VPanel db = 1,090(26.16) = 28,510 in.-kips

K Panel ,θ = K Panel ,γ db = 403,581(26.16) = 10,555,656 in.-kips/radian

4-83

FEMA P-751, NEHRP Recommended Provisions: Design Examples

M Flanges = VFlanges db = 62.6(26.16) = 1,637 in.-kips K Flanges ,θ =

M Flanges 4γ y

1,637 = 151,486 in.-kips/radian 4(0.0027)

4.2.4.3 Modeling girders. Because this structure is designed in accordance with the strongcolumn/weak-beam principle, it is anticipated that the girders will yield in flexure. Although DRAIN provides special yielding beam elements (Type 2 elements), more control over behavior is obtained through the use of the Type 4 connection element. The modeling of a typical girder is shown in Figure 4.2-8. This figure shows an interior girder, together with the panel zones at the ends. The portion of the girder between the panel zones is modeled as four segments with one simple node at mid-span and one compound node near each end. The mid-span node is used to enhance the deflected shape of the structure.5 The compound nodes are used to represent inelastic behavior in the hinging region. The following information is required to model each plastic hinge: §

The initial stiffness (moment per unit rotation)

§

The effective yield moment

§

The secondary stiffness

§

The location of the hinge with respect to the face of the column

AISC SDM recommends design practices to force the plastic hinge forming in the beam away from the face of the column. There are two methods used to move the plastic hinges of the beam away from the column face. The first one aims to reduce the cross-sectional properties of the beam at a specific location away from the column, and the second one focuses on special detailing of the beam-column connection to provide adequate strength and toughness in the connection so that inelasticity will be forced into the beam adjacent to the column face. In this study the reduced beam section (RBS) was used. A side view of the reduced beam sections is shown in Figure 4.2-15. The distance between the column face and the edge of the reduced beam section was chosen as a = 0.625b f and the reduced section length was assumed as b = 0.75db . Both of these values are just at the middle of the limits stated in AISC 358. Plastic hinges of the beams are modeled at the center of the reduced section length.

A graphic post-processor was used to display the deflected shape of the structure. The program represents each element as a straight line. Although the computational results are unaffected, a better graphical representation is obtained by subdividing the member.

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Chapter 4: Structural Analysis

Reduced Beam Section (RBS)

0.625 bbf

0.75db

Zero Length Inelastic Plastic Hinge

Rigid End Zone (0.5dc)

Figure 4.2-15 Side view of beam element and beam modeling To determine the plastic hinge capacities of the girder cross section, a moment-curvature analysis, which is dependent on the stress-strain curve of the steel, was implemented. The idealized stress-strain curve is shown in Figure 4.2-16. This curve does not display a yield plateau, which is consistent with the assumption that the section has yielded in previous cycles, with the Bauschinger effect eliminating any trace of the yield plateau. The strain hardening ratio is taken as 3 percent of the initial stiffness and the curvature ductility limit used is 20. To compute the moment-curvature relationship, the girder is divided into 50 slices through it’s depth, with 10 slices in each flange and 30 slices at the web. By gradually increasing the rotation, fiber strains, fiber stresses, fiber forces and then the resisting moment are found consecutively. Figure 4.2-17 shows the top view of the assumed reduced beam section in this study. The reduced beam length is divided into seven equal sections and flange widths of each section are calculated using the radius of the cut. The radius of the cut, R, is calculated using the formulas in AISC 358.

(4c 2 + b2 ) 8c

c = 0.175b f b = 0.75db

a = 0.625b f where:

c = depth of cut at center of the RBS, in.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

b f = width of beam flange, in. b = length of RBS cut, in.

db = beam depth, in. a = distance from face of the column to start of RBS cut, in.

R = radius of cut, in.

ESH

i)s 60 k ( ss er 40 t S

20 1 0 0

0.01

0.02

0.03

Strain

Figure 4.2-16 Assumed stress-strain curve for modeling girders

a = 0.625b f

b f1

b f2

b f3

c = 0.65b f

0.75d/14

0.75d/7

b f3

0.75d/7

b f2

b f1

c = 0.175b f b = 0.75d

Figure 4.2-17 Top view of RBS Figure 4.2-18 shows the moment-curvature diagram for the W27x94 girder. As may be seen in the figure, the moment-curvature relationship is different at each segment of the reduced length. The locations of the different reduced beam sections used in Figure 4.2-18, named as “bf1”, “bf2” and “bf3”, can be seen in

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Chapter 4: Structural Analysis Figure 4.2-17. Because of the closely adjacent locations chosen for “0.65bf” and “bf3” (see Figure 4.217), their moment-curvature plots are nearly indistinguishable from each other in Figure 4.2-18. 25,000 bf bf1 bf2

20,000 )s ip khc 15,000 ni ( t n e 10,000 m o M 5,000

bf3, 0.65bf

0 0

0.001

0.002

0.003

Curvature (rad/inch)

Figure 4.2-18 Moment-curvature diagram for W27x94 girder A tip loaded cantilever beam analysis using half of the clear span length is used to generate the momentrotation relationship for the inelastic hinges. For regular beams, where a cantilever beam is tip loaded, the moment diagram is linear and the curvature diagram is also linear as long as the moment along the beam remains in the elastic region (see Figure 4.2-19). If the moment along the beam exceeds the yield moment, the curvature along the beam will be as shown in Figure 4.2-20.

M F

Lspan / 2

Figure 4.2-19 Tip loaded cantilever beam and moment diagram for cantilever beam

4-87

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Lspan / 2

Figure 4.2-20 Curvature diagram for cantilever beam Because a RBS is used in this study, the curvature diagrams are different from those for regular beams. As may be seen in Figure 4.2-21, the curvatures in the reduced flange region of the beam have a distinctive “bump”. Because the moment diagram of the tip loaded cantilever beam is always linear, the moment values can be found easily at the different sections of the reduced flange, and then the corresponding curvature values can be assigned from the moment curvature diagram (Figure 4.2-18) to the curvature diagram along half of the clear span length (Figure 4.2-21).

0.003 )h c n /i d ar ( er 0.002 tu a v r u C 0.001

0 0

48 72 96 120 Position along cantilever beam (inches)

144

Figure 4.2-21 Curvature diagram for cantilever beam with reduced beam section Figure 4.2-21 shows the curvature diagram when the curvature ductility reaches 20. The curvature difference (the bump at the center of RBS in Figure 4.2-21) section is less prominent when the ductility is smaller. Given the curvature distribution along the cantilever beam length, the deflections at the point of load (tip deflections) can be found using the moment area method. Figure 4.2-22 illustrates the forcedisplacement relationship at the end of the half span cantilever for the W27x94 with the reduced flange section.

4-88

Chapter 4: Structural Analysis

120 P2

100 s)p ik ( ec ro F

80 60

Real F-D relationship

Trilinear F-D relationship

20 0 0

Displacement (inches)

Figure 4.2-22 Force displacement diagram for W27x94 with RBS To convert the force-tip displacement diagram into moment-rotation of the plastic hinge, the following procedure is followed: 1. Using the trilinear force displacement relationship shown in Figure 4.2-22, find the moment at the plastic hinge for P1, P2 and P3 load levels and name them M1, M2 and M3. To find the moments, the tip forces (P1, P2 and P3) are multiplied by the distance from the center of the reduced section to the tip of the cantilever. 2. Calculate the change in moment for each added load (for example: dM1 = M2 - M1). 3. Find the flexural rigidity (EI) of the beam given a tip displacement of 1 inch under the first load (P1 in Figure 4.2-22). 4. Calculate the required rotational stiffnesse of the hinge between M1 and M2 and then M2 and M3. 5. Calculate the change in rotation from M1 to M2 and from M2 to M3, by dividing the change in moment found at Step 2 by the required rotational stiffness values calculated at Step 4. 6. Find the specific rotations at M1, M2 and M3 using the change in rotation values found in Step 5. Note that the rotation is zero at M1. 7. Plot a moment-rotation diagram of the plastic hinge using the values calculated at Step 1 and Step 6. Figure 4.2-23 shows the moment-rotation diagrams for the plastic hinges of both of the girders used in the models. Note that two bilinear springs (Components 1 and 2) are needed to represent the trilinear behavior shown in the figure.

4-89

FEMA P-751, NEHRP Recommended Provisions: Design Examples

15,000

W27x94

)s ip 12,000 -k hc ni ( 9,000 t n e m o 6,000 M

W24x84

3,000 0 0

0.01

0.02

0.03

0.04

0.05

Rotation (rad)

Figure 4.2-23 Moment-rotation diagram for girder hinges with RBS The properties for the W24x84 and W27x94 girder are shown in Table 4.2-9. Note that the first yield of the model is the yield moment from Component 1. s Table 4.2-9 Girder Properties as Modeled in DRAIN Property Elastic Properties Inelastic Component 1

Inelastic Component 2 Comparative Property

Moment of Inertia (in. ) Shear Area (in.2) Yield Moment (in.-kip) Initial Stiffness (in.-kip/radian) S.H. Ratio Yield Moment (in.-kip) Initial Stiffness (in.-kip/radian) S.H. Ratio Plastic Moment = ZeFye

Section W24x84 2,370 11.3 8,422 1×1010 0.0 2,075 287,550 0.217 9,200

W27x94 3,270 13.2 10,458 1×1010 0.0 2,615 337,020 0.232 11,539

4.2.4.4 Modeling columns. All columns in the analysis are modeled in DRAIN with Type-2 elements. Preliminary analysis indicated that columns should not yield, except at the base of the first story. Subsequent analysis shows that the columns will yield in the upper portion of the structure as well. For this reason, column yielding must be activated in all of the Type-2 column elements. The columns are modeled using the built-in yielding functionality of the DRAIN program, wherein the yield moment is a function of the axial force in the column. The yield surfaces used by DRAIN for all the columns in the model are shown in Figure 4.2-24.

4-90

Chapter 4: Structural Analysis

4,000 W21x201

3,000 2,000 )s ip 1,000 k ( d a 0 o ll ia x A -1,000

W21x147 W21x122

-2,000 -3,000 -4,000 -30,000

-20,000

-10,000

10,000

20,000

30,000

Moment (inch-kips)

Figure 4.2-24 Yield surface used for modeling columns The rules employed by DRAIN to model column yielding are adequate for event-to-event nonlinear static pushover analysis, but leave much to be desired where dynamic analysis is performed. The greatest difficulty in the dynamic analysis is adequate treatment of the column when unloading and reloading. An assessment of the effect of these potential problems is beyond the scope of this example. 4.2.4.5 Results of detailed analysis. 4.2.4.5.1 Period of vibration. Table 4.2-10 tabulates the first three natural modes of vibration for models with and without doubler plates. While the P-delta effects increase the period, the doubler plates decrease the period because the model becomes stiffer with doubler plates. As may be seen, different period values are obtained from preliminary and detailed analyses (see Table 4.2-4). The detailed model results in a slightly stiffer structure than the preliminary model especially when doubler plates are added. Table 4.2-10 Periods of Vibration From Detailed Analysis (sec/cycle) Model Mode P-delta excluded P-delta included 1 1.912 1.973 Strong Panel with 2 0.627 0.639 Doubler Plates 3 0.334 0.339 1 2.000 2.069 Weak Panel without 2 0.654 0.668 Doubler Plates 3 0.344 0.349 4.2.4.5.2 Demand-to-capacity ratios. DCRs are found for the detailed analyses with the same load combination used for the preliminary analyses. The main reason for repeating the DCR for the detailed

4-91

FEMA P-751, NEHRP Recommended Provisions: Design Examples model is to make a comparison with the DCR of the preliminary model. The detailed analyses include the advanced panel zone, girder and column modeling discussed in Section 4.2.4. Figures 4.2-25(a) and 4.225(b) illustrate the DCR of the beams with columns and panel zones of the detailed model respectively. In both figures, the values in the parentheses represent the DCR with no doubler plates added to the structure. The DCR values of the detailed analyses are similar to those of the preliminary analysis displayed in Figure 4.2-5. The girders of the first and fifth bays at the third level have the maximum DCR in both the preliminary and detailed analyses. Similar trends are also observed for the DCR of the columns in both analyses. Note that the flexural stiffness of the girders is decreased by 7 percent in the preliminary analyses to compensate for the effect of reduced beam sections (with 35 percent flange reduction) which are included in the detailed analyses. Similar to the preliminary DCR, the panel zone DCR increases significantly when doubler plates aren’t used. Since the doubler plates are used only at the interior columns, that is where the difference of the DCRs changes significantly with and without doubler plates. See Figure 4.2-5(b) and Figure 4.2-25 (b).

4-92

Chapter 4: Structural Analysis

0.999 (1.152)

Level R

0.581 (0.665)

1.013 (1.023) 1.113 (1.159)

1.782 (1.945)

Level 6

0.969 (1.064)

Level 5

1.507 (1.546)

1.027 (1.134)

1.731 (1.765)

1.214 (1.292)

1.903 (1.888)

0.935 (1.063)

1.572 (1.654)

3.189 (3.405)

1.729 (1.697)

1.859 (1.772) 1.519 (1.513) 2.375 (2.279) 1.728 (1.697)

2.791 (2.678) 1.885 (1.793)

2.791 (2.643) 1.884 (1.795)

3.181 (3.120) 1.540 (1.558)

2.973 (2.851) 2.777 (2.980)

1.125 (1.143)

2.375 (2.305)

3.181 (3.111)

3.189 (3.337)

Level 2

1.520 (1.514)

2.791 (2.669)

3.302 (3.527)

1.013 (1.016)

1.848 (1.772)

2.375 (2.305)

2.869 (3.120)

Level 3

1.125 (1.143)

1.848 (1.772)

2.427 (2.669)

Level 4

1.009 (1.014)

3.181 (3.094) 1.539 (1.559)

2.981 (2.869) 2.737 (2.839)

2.955 (2.825) 2.737 (2.844)

1.067 (1.217) 1.119 (1.155)

0.659 (0.748)

1.837 (2.032) 1.513 (1.532)

1.072 (1.175)

2.496 (2.747) 1.734 (1.739)

1.166 (1.286)

2.938 (3.181) 1.894 (1.835)

1.439 (1.541)

3.371 (3.596) 1.561 (1.606)

1.106 (1.256)

3.250 (3.397) 2.743 (2.853)

3.845 (4.123)

(a) DCRs of columns and girders with and without doubler plates (DCR values in parentheses are for without doubler plates) (1.535)

(0.948)

(1.528)

(1.526)

(1.540)

(1.008)

Level R 0.827

(1.755)

0.597

(3.091)

0.604

(3.047)

0.601

(3.047)

0.598

(3.101)

0.888

(1.846)

(2.125)

1.271

(3.715)

1.288

(3.638)

1.288

(3.631)

1.274

(3.713)

1.685

(2.174)

(2.401)

1.690

(4.187)

1.688

(4.099)

1.688

(4.099)

1.693

(4.192)

1.985

(2.470)

(1.942)

1.941

(3.548)

1.936

(3.470)

1.936

(3.470)

1.944

(3.548)

2.299

(1.993)

(1.797)

1.994

(3.113)

1.978

(3.026)

1.978

(3.026)

1.994

(3.113)

1.873

(1.841)

Level 6 1.594

Level 5 1.914

Level 4 2.230

Level 3 1.827

Level 2 1.737

1.794

1.768

1.794

1.780

(b) DCRs of panel zones with and without doubler plates (DCR values in parentheses are for without doubler plates) Figure 4.2-25 DCRs for elements from detailed analysis with P-delta effects included

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 4.2.5

Nonlinear Static Analysis

As mentioned in the introduction to this example, nonlinear static (pushover) analysis is not an allowed analysis procedure in the Standard, nor does it appear in ASCE 7-10. The method is allowed in analysis related to rehabilitation of existing buildings and guidance for that use is provided in ASCE 41. The Provisions makes at least two references to pushover analysis. In Section 12.8.7 of Part 1 pushover analysis is used to determine if structures with stability ratios (see Equation 12.8-16) greater than 0.1 are allowed. Such systems have a potential for dynamic instability and the pushover curve is used to determine if the slope of the pushover curve is continuously positive up to the target displacement. If the slope is positive, the system is deemed acceptable. If not, it must be redesigned such that either the stability ratio is less than 0.1, or the slope stays positive. The analysis carried out for this purpose must be performed according to the requirements of ASCE 41. Pushover analysis is also mentioned in Provisions Part 3 Resource Paper 2. The intent of the procedure outlined there is to “determine whether lateral strength is nominally less than that required by the ELF procedure.” The use of nonlinear static analysis for this purpose is limited to structures with a height of less than 40 feet. The building under consideration has a height of 77.5 feet and violates this limit. In this example, pushover analysis is used simply to establish an estimate of the inelastic behavior of the structure under gravity and lateral loads. Of particular interest is the sequence of yielding in the beams, columns and panel zones; the lateral strength of the structure; the expected inelastic displacement; and the basic shape of the pushover curve. In the authors’ opinion, such analysis should always be used as a precursor to nonlinear response history analysis. Without pushover analysis as a precursor, it is difficult to determine if the response history analysis is producing reasonable results. The nonlinear static analysis illustrated in this example follows the recommendations of ASCE 41. The reader is also referred to FEMA 440. The pushover curve obtained from a nonlinear static analysis is a function of both modeling and load application. For this example, the structure is subjected to the full dead load plus 50 percent of the fully reduced live load, followed by the lateral loads. The Provisions states that the lateral load pattern should follow the shape of the first mode. In this example, three different load patterns are initially considered: §

UL = uniform load (equal force at each level)

§

ML = modal load (lateral loads proportional to first mode shape)

§

BL = Provisions load distribution (using the forces indicated in Table 4.2-3)

Relative values of these load patterns are summarized in Table 4.2-11. The loads have been normalized to a value of 15 kips at Level 2. DRAIN analyses are run with P-delta effects included and, for comparison purposes, with such effects excluded. This effect is represented through linearized geometric stiffness, which is the basis of the outrigger column shown in Figure 4.2-4. Consistent geometric stiffness, which may be used to represent the influence of axial forces on the flexural flexibility of individual columns, may not be used directly in

4-94

Chapter 4: Structural Analysis DRAIN. Such effects may be approximated in DRAIN by subdividing columns into several segments and activating the linearized geometric stiffness on a column-by-column basis.6 Table 4.2-11 Lateral Load Patterns Used in Nonlinear Static Analysis Provisions load, Uniform load, UL Modal load, ML Level BL (kips) (kips) (kips) R 15.0 85.1 144.8 6 15.0 77.3 114.0 5 15.0 64.8 84.8 4 15.0 49.5 58.2 3 15.0 32.2 34.6 2 15.0 15.0 15.0 As described later, the pushover analysis indicates most of the yielding in the structure occurs in the clear span of the girders and columns. Panel zone hinging occurs only at the exterior columns where doubler plates weren’t used. To see the effect of doubler plates, the ML analysis is repeated for a structure without doubler plates. These structures are referred to as the strong panel (SP) and weak panel (WP) structures, respectively. The analyses are carried out using the DRAIN-2Dx computer program. Using DRAIN, an analysis may be performed under “load control” or under “displacement control.” Under load control, the structure is subjected to gradually increasing lateral loads. If, at any load step, the tangent stiffness matrix of the structure has a negative on the diagonal, the analysis is terminated. Consequently, loss of strength due to P-delta effects cannot be tracked. Using displacement control, one particular point of the structure (the control point) is forced to undergo a monotonically increasing lateral displacement, and the lateral forces are constrained to follow the desired pattern. In this type of analysis, the structure can display loss of strength because the displacement control algorithm adds artificial stiffness along the diagonal to overcome the stability problem. This approach is meaningful because structures subjected to dynamic loading can display strength loss and remain stable incrementally. It is for this reason that the poststrength-loss realm of the pushover response is of interest. Where performing a displacement controlled pushover analysis in DRAIN with P-delta effects included, one must be careful to recover the base-shear forces properly.7 At any displacement step in the analysis, the true base shear in the system consists of two parts: n

V = ∑VC ,i − i =1

P1Δ1 h1

where the first term represents the sum of all the column shears in the first story, and the second term represents the destabilizing P-delta shear in the first story. The P-delta effects for this structure are 6

DRAIN uses the axial forces at the end of the gravity load analysis to set geometric stiffness for the structure. This is reasonably accurate where consistent geometric stiffness is used, but is questionable where linearized geometric stiffness is used. If P-delta effects have been included, this procedure needs to be used where recovering base shear from column shear forces. This is true for displacement controlled static analysis, force controlled static analysis and dynamic response history analysis.

4-95

FEMA P-751, NEHRP Recommended Provisions: Design Examples included through the use of the outrigger column shown at the right of Figure 4.2-4. Figure 4.2-26 plots two base shear components of the pushover response for the SP structure subjected to the ML loading. Also shown is the total response. The kink in the line representing P-delta forces occurs because these forces are based on first-story displacement, which, for an inelastic system, generally will not be proportional to the roof displacement. For all of the pushover analyses reported in this example, the structure is pushed to a displacement of 37.5 inches at the roof level. This value is approximately 4 percent of the total height.

2,000

Column shear forces

Total base shear 1,000 )s p i (k r a e h S

0 P-Delta forces

-1,000 0

Roof displacement (inches)

Figure 4.2-26 Two base shear components of pushover response 4.2.5.1 Pushover response of strong panel structure. Figure 4.2-27 shows the pushover response of the SP structure to all three lateral load patterns where P-delta effects are excluded. In each case, gravity loads are applied first, and then the lateral loads are applied using the displacement control algorithm. Figure 4.2-28 shows the response curves if P-delta effects are included. In Figure 4.2-29, the response of the structure under ML loading with and without P-delta effects is illustrated. Clearly, P-delta effects are an extremely important aspect of the response of this structure and the influence grows in significance after yielding. This is particularly interesting in the light of the Standard, which ignores P-delta effects in elastic analysis if the maximum stability ratio is less than 0.10 (see Sec. 12.8-7). For this structure, the maximum computed stability ratio is 0.0862 (see Table 4.2-5), which is less than 0.10 and is also less than the upper limit of 0.0909. The upper limit is computed according to Standard Equation 12.8-17 and is based on the very conservative assumption that β = 1.0. While the Standard allows the analyst to exclude P-delta effects in an elastic analysis, this clearly should not be done in the pushover analysis (or in response history analysis). (In the Provisions the upper limit for the stability ratio is eliminated. Where the calculated θ is greater than 0.10, a pushover analysis must be performed in accordance with ASCE 41, and it must be shown that that the slope of the pushover curve is positive up to the target displacement. The pushover analysis must be based on the MCE spectral acceleration and must include P-delta effects [and loss of strength, as appropriate]. If the slope of the pushover curve is negative at displacements less than the target displacement, the structure must be redesigned such that θ is less than 0.10 or the pushover slope is positive up to the target displacement.)

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Chapter 4: Structural Analysis

2,000

UL ML BL

)s pi (k r ae 1,000 h s sea B

0 0

20 Roof displacement (inches)

Figure 4.2-27 Response of strong panel model to three load patterns, excluding P-delta effects

2,000

)s ip k ( r a e h s 1,000 es a B

UL ML BL

0 0

20 Roof displacment (inches)

Figure 4.2-28 Response of strong panel model to three load patterns, including P-delta effects

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

2,000 s)p ik ( r a e h s es a B

Excluding P-Delta Including P-Delta 1,000

0 0

20 Roof displacement (inches)

Figure 4.2-29 Response of strong panel model to ML loads, with and without P-delta effects In Figure 4.2-30, a plot of the tangent stiffness versus roof displacement is shown for the SP structure with ML loading and with P-delta effects excluded or included. This plot, which represents the slope of the pushover curve at each displacement value, is more effective than the pushover plot in determining when yielding occurs. As Figure 4.2-30 illustrates, the first significant yield occurs at a roof displacement of approximately 6.5 inches and that most of the structure’s original stiffness is exhausted by the time the roof displacement reaches 13 inches. 150 120 )h c n i/ sp 90 ik ( ss e 60 fn fi ts t en 30 g n a T

Excluding P-Delta

Including P-Delta

-30 0

Roof displacement (inches)

Figure 4.2-30 Tangent stiffness history for structure under ML loads, with and without P-delta effects

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Chapter 4: Structural Analysis

For the case with P-delta effects excluded, the final tangent stiffness shown in Figure 4.2-30 is approximately 10.2 kips/in., compared to an original value of 139 kips/in. Hence, the strain-hardening stiffness of the structure is 0.073 times the initial stiffness. This is somewhat greater than the 0.03 (3.0 percent) strain hardening ratio used in the development of the model because the entire structure does not yield simultaneously. Where P-delta effects are included, the final tangent stiffness is -1.6 kips per inch. The structure attains this negative tangent stiffness at a displacement of approximately 23 inches. 4.2.5.1.1 Sequence and pattern of plastic hinging. The sequence of yielding in the structure with ML loading and with P-delta effects included is shown in Figure 4.2-31. Part (a) of the figure shows an elevation of the structure with numbers that indicate the sequence of plastic hinge formation. For example, the numeral “1” indicates that this was the first hinge to form. Part (b) of the figure shows a pushover curve with several hinge formation events indicated. These events correspond to numbers shown in Part (a) of the figure. The pushover curve only shows selected events because an illustration showing all events would be difficult to read. Comparing Figure 4.2-31(a) with Figures 4.2-5 and 4.2-25, it can be seen how the DCRs indicate the plastic hinge formation sequence. The highest ratios in Figure 4.2-5 are observed at the girders of the third and the second levels beginning from the bays at the leeward (right) side. As may be seen from Figure 4.2-31(a), first plastic hinges form at the same locations of the building. Similarly, the first panel zone hinge forms at the beam-column joint of the sixth column at the fourth level, and this is where the highest DCR values are obtained for the panel zones in both preliminary and detailed DCR analyses.

27 25

27 11

13 22

27 11

(a)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

2,000

)s pi (k ra e 1,000 h sl at o T

5 1

12 9

21 15 19

22 23

0 0

Drift (inches)

(b) Figure 4.2-31 Patterns of plastic hinge formation: SP model under ML load, including P-delta effects Several important observations are made from Figure 4.2-31: §

There is no hinging in Levels 6 and R.

§

There is panel zone hinging only at the exterior columns at Levels 4 and 5. Panel zone hinges do not form at the interior joints where doubler plates are used.

§

Hinges form at the base of all the Level 1 columns.

§

Plastic hinges form in all columns on Level 3 and all the interior columns on Level 4.

§

Both ends of all the girders at Levels 2 through 5 yield.

It appears the structure is somewhat weak in the middle two stories and is relatively strong at the upper stories. The doubler plates added to the interior columns prevent panel zone yielding. The presence of column hinging at Levels 3 and 4 is a bit troublesome because the structure is designed as a strong-column/weak-beam system. This design philosophy, however, is intended to prevent the formation of complete story mechanisms, not to prevent individual column hinging. While hinges do form at the top of each column in the third story, hinges do not form at the bottom of these columns and a complete story mechanism is avoided. Even though the pattern of hinging is interesting and useful as an evaluation tool, the performance of the structure in the context of various acceptance criteria cannot be assessed until the expected inelastic displacement can be determined. This is done below in Section 4.2.5.3.

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Chapter 4: Structural Analysis 4.2.5.1.2 Comparison with strength from plastic analysis. It is interesting to compare the strength of the structure from pushover analysis with that obtained from the rigid-collapse analysis performed using virtual work. These values are summarized in Table 4.2-12. The strength from the case with P-delta excluded was estimated from the curves shown in Figure 4.2-27 and is taken as the strength at the principal bend in the curve (the estimated yield from a bilinear representation of the pushover curve). Consistent with the upper bound theorem of plastic analysis, the strength from virtual work is greater than that from pushover analysis. The reason for the difference in predicted strengths is related to the pattern of yielding that actually formed in the structure, compared to that assumed in the rigid-plastic analysis. Table 4.2-12 Strength Comparisons: Pushover versus Rigid Plastic Pattern Uniform Modal (Triangular) BSSC

Lateral Strength (kips) P-delta Excluded

P-delta Included

Rigid-Plastic

1,340 1,200 1,170

1,270 1,130 1,105

1,666 1,373 1,308

4.2.5.2 Pushover response of weak panel structure. Before continuing, the structure should be reanalyzed without panel zone reinforcing, and the behavior compared with that determined from the analysis described above. For this exercise, only the modal load pattern is considered, but the analysis is performed with and without P-delta effects. The pushover curves for the structure under modal loading and with weak panels are shown in Figure 4.2-32. Curves for the analyses run with and without P-delta effects are included. Figures 4.2-33 and 4.2-34 are more informative because they compare the response of the structures with and without panel zone reinforcement. Figure 4.2-35 shows the tangent stiffness history comparison for the structures with and without doubler plates. In both cases P-delta effects have been included. Figures 4.2-32 through 4.2-35 show that the doubler plates, which represent approximately 2.0 percent of the volume of the structure, increase the strength and initial stiffness by approximately 10 percent.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

2,000

Excluding P-Delta

s)p ik ( ra e h s 1,000 es a B

Including P-Delta

0 0

Roof Displacement (inches)

Figure 4.2-32 Weak panel zone model under ML load

2,000 Strong panels s) pi k( r ae h s 1,000 es a B

Weak panels

0 0

Roof displacement (inches)

Figure 4.2-33 Comparison of weak panel zone model with strong panel zone model, excluding P-delta effects

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Chapter 4: Structural Analysis

2,000

)s p i k ( ra e h s 1,000 es a B

Strong panels Weak panels

0 0

Roof displacement (inches)

Figure 4.2-34 Comparison of weak panel zone model with strong panel zone model, including P-delta effects

150

Strong panels

) h cn 120 i/ s p i k ( ss 90 e n ffi ts t 60 n eg n a T 30

Weak panels

0 -30 0

20 Roof displacement (inches)

Figure 4.2-35 Tangent stiffness history for structure under ML loads, strong versus weak panels, including P-delta effects The difference between the behavior of the structures with and without doubler plates is attributed to the yielding of the panel zones in the structure without panel zone reinforcement. The sequence of hinging is illustrated in Figure 4.2-36. Part (a) of this figure indicates that panel zone yielding occurs early. (Panel zone yielding is indicated by a numeric sequence label in the corner of the panel zone.) In fact, the first yielding in the structure is due to yielding of a panel zone at the fourth level of the structure, which is consistent with panel zone DCR calculated before where no doubler plates were added to the structure.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples Under very large displacements the flange component of the panel zone yields. Girder and column hinging also occurs, but the column hinging appears relatively late in the response. It is also significant that the upper two levels of the structure display yielding in several of the panel zones. Aside from the relatively marginal loss in stiffness and strength due to removal of the doubler plates, it appears that the structure without panel zone reinforcement behaves adequately. Of course, actual performance cannot be evaluated without predicting the maximum inelastic panel shear strain and assessing the stability of the panel zones under these strains.

23 54

58 53

70 8

2 33 57 5 13 9 14

36 50 26 23 34 29 37 26

23 62

10 69

20 55

62 8

38 64 3 45 42 31 60 6 22

38 65 64 3 46 40 31 59 6 22

63 36 1 41 44 26 61 4 21

11 19

39 27

35 28 39 27

(a)

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10 21

43 51 20 49 12

34 25

37 27

Chapter 4: Structural Analysis

2,000

)s pi k( r ae 1,000 sh l at o T

5 1

30 1925 14 12

43 33 37 40

44 47 51 52 56 64

0 0

Drift (inches)

(b) Figure 4.2-36 Patterns of plastic hinge formation: weak panel zone model under ML load, including P-delta effects 4.2.5.3 Target displacement. In this section, the only loading pattern considered is the modal load pattern discussed earlier. This is consistent with the requirements of ASCE 41 and FEMA 440. The structures with strong and weak panel zones are analyzed including P-delta effects. ASCE 41 uses the coefficient method for calculating target displacement. The target displacement is computed as follows:

δ t = C0C1C2 S a

Te2 g 4π 2

where:

C0 = φ1,r Γ1 = modification factor to relate roof displacement of a multiple degree of freedom building system to the spectral displacement of an equivalent single degree of freedom system

φ1,r = the ordinate of mode shape 1 at the roof (control node) Γ1 = the first mode participation factor C1 = modification factor to relate expected maximum inelastic displacements to displacements calculated for linear elastic response

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

C2 = modification factor to represent the effect of pinched hysteresis shape, cyclic stiffness degradation and strength deterioration on maximum displacement response S a = response spectrum acceleration, at the effective fundamental period and damping ratio of the building in the direction under consideration

Te = Ti

Ki = effective fundamental period of the building in the direction under consideration Ke

Ti = elastic fundamental period in the direction under consideration calculated by elastic dynamic analysis Ki = elastic lateral stiffness of the building in the direction under consideration K e = effective lateral stiffness of the building in the direction under consideration g = acceleration due to gravity To find the coefficient S a , the general horizontal response spectrum defined in ASCE 41 Section 1.6.1.5 is used. The damping of the spectrum is chosen as 2 percent for this study. The same damping ratio is used in the dynamic analysis. The parameters S XS and S X 1 are chosen as the same values as SMS and SM1 which are defined in Section 4.2.2.2 of this study. Note that these are the MCE spectral acceleration parameters. Figure 4.2-37 shows the horizontal response spectrum obtained from ASCE 41. The parameters of this spectrum are discussed further with the dynamic analyses. This spectrum is for the Basic Safety Earthquake 2 (BSE-2) hazard level which has a 2 percent probability of exceedance in 50 years. Coefficient C0 is found using the first mode shape of the model with the mass matrix. For both strong and weak panel models, the C0 coefficient is found a bit higher than 1.3, which is the value provided in Table 3.2 of ASCE41 for the shear buildings with triangular load pattern.

C1 and C 2 are equal to 1.0 for periods greater than 1.0 second and 0.7 second respectively. Since the first mode periods of the strong and weak panel models are both approximately 2 seconds, these coefficients are taken as 1.0.

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Chapter 4: Structural Analysis

)g ( 2 a S , n iot ar el ec c a l 1 ar tc e p S

0 0

Period, T (s)

Figure 4.2-37 Two percent damped horizontal response spectrum from ASCE 41 To find the target displacement, the procedure described in ASCE 41 is followed. The nonlinear forcedisplacement relationship between the base shear and displacement of the control node are replaced with an idealized force-displacement curve. The effective lateral stiffness and the effective period depend on the idealized force-displacement curve. The idealized force-displacement curve is developed using an iterative graphical procedure where the areas below the actual and idealized curves are balanced approximately up to a displacement value of Δ d . Δ d is the displacement at the end of second line segment of the idealized curve and Vd is the base shear at the same displacement. ( Δ d , Vd ) should be a point on the actual force displacement curve either at the calculated target displacement or at the displacement corresponding to the maximum base shear, whichever is the least. The first line segment of the idealized force-displacement curve should begin at the origin and finish at ( Δ y , Vy ), where Vy is the effective yield strength and Δ y is the yield displacement of idealized curve. The slope of the first line segment is equal to the effective lateral stiffness, K e , which should be taken as the secant stiffness calculated at a base shear force equal to 60 percent of the effective yield strength of the structure. See Figures 4.2-38 and 4.2-39 for the actual and idealized force-displacement curves of strong and weak panel models which are under ML loading and both include P-delta effects.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 1,500 ? d,Vd

Actual

? y ,Vy

)s ip k ( 1,000 V r, a e sh es a B 500

Idealized

0 0

20 Roof displacement,

(inches)

Figure 4.2-38 Actual and idealized force displacement curves for strong panel model, under ML load, including P-delta effects 1,500

)s ip (k

V ,r a e h s es a B

? d,Vd ? y ,Vy

1,000

Actual

Idealized

500

0 0

20 Roof displacement,

(inches)

Figure 4.2-39 Actual and idealized force displacement curves for weak panel model, under ML load, including P-delta effects Table 4.2-13 shows the target displacement values of SP and WP models. Story drifts are also shown at the load level of target displacement for both models. Table 4.2-13 Target Displacement for Strong and Weak Panel Models Strong Panel Weak Panel C0 1.303 1.310 C1 1.000 1.000

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Chapter 4: Structural Analysis Table 4.2-13 Target Displacement for Strong and Weak Panel Models Strong Panel Weak Panel C2 1.000 1.000 S a (g) 0.461 0.439 Te (sec) 1.973 2.069 δ t (in.) at Roof Level 22.9 24.1 Drift R-6 (in.) 0.96 1.46 Drift 6-5 (in.) 1.76 2.59 Drift 5-4 (in.) 2.87 3.73 Drift 4-3 (in.) 4.84 4.84 Drift 3-2 (in.) 5.74 5.35 Drift 2-1 (in.) 6.73 6.12 Negative tangent stiffness starts at 22.9 inches and 29.3 inches for strong and weak panel models, respectively. Thus negative tangent stiffness starts after target displacements for both models. Again note that these displacements are computed from the 2 percent-damped MCE horizontal response spectrum of ASCE 41. 4.2.6

Response History Analysis

The response history analysis method, with three ground motions, is used to estimate the inelastic deformation demands for the structure. While an analysis with seven or more ground motions generally is preferable, that was not done here due to time and space limitations. The analysis did consider a number of parameters, as follows: §

Scaling of ground motions to the DBE and MCE level

§

Analysis with and without P-delta effects

§

Two percent and five percent inherent damping

§

Added linear viscous damping

All of the models analyzed have “Strong Panels” (with doubler plates included in the interior beamcolumn joints). 4.2.6.1 Modeling and analysis procedure. The DRAIN-2Dx program is used for each of the response history analyses. With the exception of requirements for including inherent damping, the structural model is identical to that used in the nonlinear static analysis. Second-order effects are included through the use of the leaning column element shown to the right of the actual frame in Figure 4.2-4. Only one-half of the building (a single frame in the N-S direction) is modeled. Inelastic hysteretic behavior is represented through the use of a bilinear model. This model exhibits neither a loss of stiffness nor a loss of strength and for this reason, it will generally have the effect of overestimating the hysteretic energy dissipation in the yielding elements. Fortunately, the error produced by such a model will not be of great concern for this structure because the hysteretic behavior of panel 4-109

FEMA P-751, NEHRP Recommended Provisions: Design Examples zones and flexural plastic hinges should be very robust where inelastic rotations are less than about 0.03 radians. Rayleigh proportional damping was used to represent viscous energy dissipation in the structure. The mass and stiffness proportional damping factors are set initially to produce 2.0 percent damping in the first and third modes. It is generally recognized that this level of damping (in lieu of the 5 percent damping that is traditionally used in elastic analysis) is appropriate for nonlinear response history analysis. Two percent damping is also consistent with that used in the pushover analysis (see Section 4.2.5 of this example). In Rayleigh proportional damping, the damping matrix, C, is a linear combination of the mass matrix, M and the initial stiffness matrix, K:

C = αM + β K where α and β are mass and stiffness proportionality factors, respectively. If the first and third mode frequencies, ω1 and ω3, are known, the proportionality factors may be computed from the following expression (Clough & Penzien):

⎧α ⎫ 2ξ ⎧ w1w3 ⎫ ⎨ ⎬ = ⎨ ⎬ ⎩ β ⎭ w1 + w3 ⎩ 1 ⎭ Note that α and β are directly proportional to ξ. To increase the target damping from 2 percent to 5 percent of critical, all that is required is a multiplying factor of 2.5 on α and β. The targeted structural frequencies and the resulting damping proportionality factors are shown in Table 4.2-14. The frequencies shown in the table are computed from the detailed model shown in Figure 4.2-7. Table 4.2-14 Structural frequencies and damping factors used in response history analysis (damping factors that produce 2 percent damping in modes 1 and 3) Model/Damping Parameters

ω1 (rad/sec)

ω3 (rad/sec)

Strong Panel with P-delta Strong Panel without P-delta

3.184 3.285

18.55 18.81

0.109 0.112

0.00184 0.00181

The stiffness proportional damping factor must not be included in the Type 4 elements used to represent rotational plastic hinges in the structure. These hinges, particularly those in the girders, have a very high initial stiffness. Before the hinge yields, there is virtually no rotational velocity in the hinge. After yielding, the rotational velocity is significant. If a stiffness proportional damping factor is used for the hinge, a viscous moment will develop in the hinge. This artificial viscous moment—the product of the rotational velocity, the initial rotational stiffness of the hinge and the stiffness proportional damping factor—can be quite large. In fact, the viscous moment may even exceed the intended plastic capacity of the hinge. These viscous moments occur in phase with the plastic rotation; hence, the plastic moment and the viscous moments are additive. These large moments transfer to the rest of the structure, affecting the sequence of hinging in the rest of the structure and produce artificially high base shears. The use of

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Chapter 4: Structural Analysis stiffness proportional damping in discrete plastic hinges can produce a totally inaccurate analysis result. See Charney (2008) for details. The structure is subjected to dead load and half of the fully reduced live load, followed by ground acceleration. The incremental differential equations of motion are solved in a step-by-step manner using the Newmark constant average acceleration approach. Time steps and other integration parameters are carefully controlled to minimize errors. The minimum time step used for analysis is as small as 0.0005 second for the first earthquake and 0.001 second for the second and third earthquakes. A smaller integration time step is required for the first earthquake because of its impulsive nature. 4.2.6.2 Development of ground motion records. The ground motion acceleration histories used in the analysis are developed specifically for the site. Basic information for the records is shown in Table 4.120a. Ground acceleration histories and 2- and 5-percent-damped pseudoacceleration spectra for each of the motions are shown in Figures 4.2-40 through 4.2-42. For these two-dimensional analyses performed using DRAIN, single ground motion components areapplied one at a time. For this example, the component that produces the larger spectral acceleration at the structure’s fundamental period (A00, B90 and C90) is used. A complete analysis would require consideration of both components of ground motions and possibly of a rotated set of components. When analyzing structures in two dimensions, Section 16.1.3.1 of the Standard (as well as ASCE 7-10) gives the following instructions for scaling: The ground motions shall be scaled such that the average value of the 5 percent damped response spectra for the suite of motions is not less than the design response spectrum for the site for periods ranging from 0.2T to 1.5T where T is the natural period of the structure in the fundamental mode for the direction of response being analyzed. The scaling requirements in Provisions Part 3 Resource Paper 3are similar, except that the target spectrum for scaling is the MCER spectrum. In this example, the only adjustment is made for scaling when the inherent damping is taken as 2 percent of critical. In this case, the ground motion spectra are based on 2 percent damping and the DBE or MCE spectrum is adjusted from 5 percent damping to 2 percent damping using the modification factors given in ASCE 41. The scaling procedure described above has a “degree of freedom” in that there are an infinite number of scaling factors that can fit the criterion. To avoid this, a two-step scaling process is used wherein each spectrum is initially scaled to match the target spectrum at the structure’s fundamental period and then the average of the scaled spectra are re-scaled such that no ordinate of the scaled average spectrum falls below the target spectrum in the range of periods between 0.2T and 1.5T. The final scale factor for each motion consists of the product of the initial scale factor and the second scale factor. The initial scale factors, referred to as S1i (for each ground motion, i), are different for the three ground motions. The second scale factor, S2, is the same for each ground motion. The scale factors used in the response history analyses are shown in Table 4.2-15. Factors are determined for 2 percent and 5 percent damping and for the DBE and MCE motions. The 2 percent damped target MCE spectrum corresponds to ASCE 41-06 spectrum used in the pushover analysis. If a scale factor of 1.367 is used for the structure with 2 percent damping, Figure 4.2-43 indicates that the scaling criteria specified by the Standard are met for all periods in the range 0.2(1.973) = 0.4 second to 1.5(1.973) = 3.0 seconds. 1.973 seconds is the period of the SP model with P-delta effects included (See Table 4.2-10).

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

A00

Acceleration (g)

0.8 0.4 0.0 -0.4 -0.8 0

A90

0.8

Acceleration (g)

Time (s)

0.4 0.0

-0.4 -0.8 0

Time (s)

A00

5% damped

3 2 1

0 0

2 Period, T (s)

A90

2% damped

Pseudoacceleration (g)

2% damped 5% damped

3 2 1 0 0

2 Period, T (s)

Figure 4.2-40 Ground acceleration histories and response spectra for Record A

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Chapter 4: Structural Analysis

B00

Acceleration (g)

0.8

0.4 0.0 -0.4

-0.8 0

Time (s)

B90

0.8

Acceleration (g)

0.4 0.0 -0.4 -0.8 0

Time (s)

B00

2% damped 5% damped

2 1 0 0

2 Period, T (s)

B90

Pseudoacceleration (g)

2% damped 5% damped

2 1 0 0

2 Period, T (s)

Figure 4.2-41 Ground acceleration histories and response spectra for Record B

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

C00

Acceleration (g)

0.8 0.4 0.0 -0.4 -0.8 0

C90

0.8 Acceleration (g)

Time (s)

0.4

0.0 -0.4 -0.8 0

Time (s)

C00

2% damped 5% damped

3 2 1

C90

Pseudoacceleration (g)

2% damped 5% damped

3 2 1 0

0 0

1 2 Period, T (s)

2 Period, T (s)

Figure 4.2-42 Ground acceleration histories and response spectra for Record C

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Chapter 4: Structural Analysis Table 4.2-15 Ground Motion Scale Factors Used in the Analyses 2% Damped 2% Damped Scale Factor DBE MCE S1 0.919 1.380 S2 1.367 1.367 Motion A00 SS 1.257 1.886 S1 1.495 2.245 S2 1.367 1.367 Motion B90 SS 2.045 3.068 S1 1.332 2.000 S2 1.367 1.367 Motion C90 SS 1.822 2.734

Pseudoacceleration (g)

5%Damped DBE 0.765 1.428 1.092 1.439 1.428 2.056 1.359 1.428 1.941

5% Damped MCE 1.147 1.428 1.638 2.159 1.428 3.084 2.039 1.428 2.911

Average of scaled spectra

3 2 2% damped MCE spectrum

1 0 0

Period, T (s)

(a) Comparison of average of scaled spectra and target spectrum (SF = 1.367)

Spectral ratio

Period, T (s)

(b) Ratio of average of scaled spectra to target spectrum (SF = 1.367) Figure 4.2-43 Ground motion scaling parameters

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 4.2.6.3 Results of response history analysis. The following parameters are varied to determine the sensitivity of the response to that parameter: §

Analyses are run with and without P-delta effects for all three ground motions.

§

Analyses are run with 2 percent and 5 percent inherent damping. The ground motion scale factors are correlated with the corresponding inherent damping of the structure.

§

Added dampers are used for the structure with 2 percent inherent damping. Various added damper configurations are used. These analyses are performed to assess the potential benefit of added viscous fluid damping devices. The SP model with P-delta effects included is used for this analysis and only Ground Motions A00 and B90 are used.

The results from the first series of analyses, all run with 2 or 5 percent of critical damping with and without P-delta effects, are summarized in Tables 4.2-16 through 4.2-23. Selected time history traces are shown in Figures 4.2-44 through 4.2-48. The tabulated shears in the tables are for the single frame analyzed and should be doubled to obtain the total shear in the structure. The tables of story shear also provide two values for each ground motion. The first value is the maximum total elastic column story shear, including P-delta effects if applicable. The second value represents the maximum total inertial force for the structure. The inertial base shear, which is not necessarily concurrent with the column shears, was obtained as a sum of the products of the total horizontal accelerations and nodal mass of each joint. For a system with no damping, the story shears obtained from the two methods should be identical. For a system with damping, the base shear obtained from column forces generally will be less than the shear from inertial forces because the viscous component of column shear is not included. Additionally, the force absorbed by the mass proportional component of damping will be lost (as this is not directly recoverable in DRAIN). The total roof displacement and the story drifts listed in the tables are peak (envelope) values and are not necessarily concurrent. 4.2.6.3.1 Response of structure with 2 and 5 percent of critical damping. Tables 4.2-16 and 4.2-17 summarize the results of the DBE analyses with 2 percent inherent damping, including and excluding P-delta effects. Part (a) of each table provides the maximum base shears, computed either as the sum of column forces (including P-delta effects as applicable), or as the sum of the products of the total acceleration and mass at each level. In each case, the shears computed using the two methods are similar, which serves as a check on the accuracy of the analysis. Had the analysis been run without damping, the shears computed by the two methods should be identical. As expected base shears decrease when P-delta effects are included. The maximum story drifts are shown in the (b) parts of each table. The drift limits in the table, equal to 2 percent of the story height, are the same as provided in Standard Table 12.12-1. Standard Section 16.2.4.3 provides for the allowable drift to be increased by25 percent where nonlinear response history analysis is used; these limits are shown in the tables in parentheses. Provisions Part 2 states that the increase in drift limit is attributed to “the more accurate analysis and the fact that drifts are computed explicitly.” Drifts that exceed the increased limits are shown in bold text in the tables. When a SP frame with 2 percent inherent damping is analyzed under MCE spectrum scaled motions excluding P-delta effects, earthquake A00 results in 62.40-inch displacement at the roof level and approximately between 15- to 20-inch drifts at the first three stories of the structure. These story drifts are well above the limits. When P-delta effects are included with the same level of motion, roof 4-116

Chapter 4: Structural Analysis displacement increases to 101.69 inches with approximately 20- to 40-inch displacement at the first three stories. It is clear from Part (b) of Tables 4.2-16 and 4.2-17 that Ground Motion A00 is much more demanding with respect to drift than are the other two motions. The drifts produced by Ground Motion A00 are particularly large at the lower levels, with the more liberal drift limits being exceeded in the lower four stories of the building. When P-delta effects are included, the drifts produced by Ground Motion A00 increase significantly;drifts produced by Ground Motions B90 and C90 change only slightly. Tables 4.2-18 and 4.2-19 provide result summaries for the structure analyzed with the MCE ground motions. Damping is still set at 2 percent of critical and analysis is run with and without P-delta effects. The drift limits listed in the (b) parts of Tables 4.2-18 and 4.2-19 are based on Provisions Part 3 Resource Paper 3 Section 16.4.5. These limits are 1.5 times those allowed by Standard Section 12.2.1. The 50 percent increase in drift limits is consistent with the increase in ground motion intensity when moving from DBE to MCE ground motions. If all of the increase in drift limit is attributed to the DBE-MCE scaling, there is no apparent adjustment related to “the more accurate analysis and explicit computation of drift”. When P-delta effects are included maximum story shears decrease and the drifts in lower stories increase for all motions. The drifts predicted for Ground Motion A00 (as much as 40 inches) indicate probable collapse of the structure. Loss of strength associated with such large drifts is not included in the analytical model (since DRAIN does not provide a mechanism for decreasing moment capacity under large plastic rotations). It is highly likely, however, that collapse would be predicted by more accurate modeling. Similar trends in response are produced when the inherent damping is increased from 2 percent critical to 5 percent. The results for the 5 percent damped analysis are provided in Tables 4.2-20 through 4.2-23. The first two of these tables, Tables 4.1-20 and 4.1-21, are for the analysis using the DBE ground motions. When compared to the results using 2 percent damping, it is seen that both the base shears and the story drifts decrease significantly. DBE-level drifts at lower stories due to Ground Motion A00 exceed the drift limit but may not indicate collapse. MCE-level drifts produced by the A00 ground motion indicate likely collapse. Table 4.2-16 DBE Results for 2% Damped Strong Panel Model with P- Delta Excluded (a) Maximum Base Shear (kips) Level

Motion A00

Motion B90

Motion C90

Column forces

1,780

1,649

1,543

Inertial forces

1,848

1,650

1,540

(b) Maximum Displacment and Story Drift (in.) Level

Motion A00

Motion B90

Motion C90

Limit*

Roof displacement Drift R-6 Drift 6-5 Drift 5-4

26.80 1.85 2.51 3.75

14.57 1.92 2.60 3.08

13.55 1.71 2.33 3.03

NA 3.00 (3.75) 3.00 (3.75) 3.00 (3.75)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples (b) Maximum Displacment and Story Drift (in.) Level

Motion A00

Motion B90

Motion C90

Limit*

Drift 4-3 Drift 3-2 Drift 2-G

5.62 6.61 8.09

2.98 3.58 4.68

3.03 2.82 3.29

3.00 (3.75) 3.00 (3.75) 3.60 (4.50)

*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.

Table 4.2-17 DBE Results for 2% Damped Strong Panel Model with P-Delta Included (a) Maximum Base Shear (kips) Motion B90 Motion A00 1,467 1,458 1,558 1,481

Level Column forces Inertial forces

Level Roof displacement Drift R-6 Drift 6-5 Drift 5-4 Drift 4-3 Drift 3-2 Drift 2-G

(b) Maximum Displacement and Story Drift (in.) Motion A00 Motion B90 Motion C90

Motion C90 1,417 1,419

Limit*

32.65

14.50

14.75

1.86 2.64 4.08 6.87 8.19 10.40

1.82 2.50 2.81 3.21 3.40 4.69

1.70 2.41 3.19 3.33 2.90 3.44

3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.60 (4.50)

*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.

Table 4.2-18 MCE Results for 2% Damped Strong Panel Model with P-Delta Excluded (a) Maximum Base Shear (kips) Motion B90 Motion A00 2,181 1,851 2,261 1,893

Level Column forces Inertial forces

Level Roof displacement Drift R-6 Drift 6-5 Drift 5-4 Drift 4-3 Drift 3-2 Drift 2-G

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(b) Maximum Displacement and Story Drift (in.) Motion B90 Motion C90 Motion A00 62.40 22.45 20.41 1.98 3.57 7.36 14.61 16.29 19.76

2.30 2.77 3.33 4.61 5.21 6.60

3.05 3.69 4.43 4.45 3.97 5.11

Motion C90 1,723 1,725

Limit NA 4.5 4.5 4.5 4.5 4.5 5.4

Chapter 4: Structural Analysis

Table 4.2-19 MCE Results for 2% Damped Strong Panel Model with P-Delta Included Level Column Forces Inertial Forces

Level Total Roof R-6 6-5 5-4 4-3 3-2 2-G

(a) Maximum Base Shear (kips) Motion A00 Motion B90 1,675 1,584 1,854 1,633

(b) Maximum Story Drifts (in.) Motion A00 Motion B90 Motion C90 101.69 26.10 20.50 1.95 2.32 2.93 2.97 2.60 3.49 6.41 3.62 4.32 20.69 5.61 4.63 31.65 6.32 4.18 40.13 7.03 5.11

Motion C90 1,507 1,515

Limit NA 4.5 4.5 4.5 4.5 4.5 5.4

Table 4.2-20 DBE Results for 5% Damped Strong Panel Model with P-Delta Excluded Level Column Forces Inertial Forces

Level Total Roof R-6 6-5 5-4 4-3 3-2 2-G

(a) Maximum Base Shear (kips) Motion B90 Motion A00 1,622 1,568 1,773 1,576 (b) Maximum Story Drifts (in.) Motion B90 Motion C90 Motion A00 19.17 14.09 13.14 1.33 1.73 1.77 2.18 2.52 2.32 3.06 2.98 2.89 3.97 2.86 2.78 5.02 3.19 2.72 6.13 4.05 3.01

Motion C90 1,483 1,482

*Limit NA 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.60 (4.50)

*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 4.2-21 DBE Results for 5% Damped Strong Panel Model with P-Delta Included (a) Maximum Base Shear (kips Motion B90 Motion A00 1,374 1,419 1,524 1,448

Level Column forces Inertial forces

Level Roof displacement Drift R-6 Drift 6-5 Drift 5-4 Drift 4-3 Drift 3-2 Drift 2-G

(b) Maximum Displacement and Story Drift (in.) Motion B90 Motion C90 Motion A00 21.76 14.07 14.16 1.40 2.25 3.23 4.38 5.60 7.12

1.56 2.42 2.80 3.04 3.28 4.33

1.73 2.33 3.00 3.09 2.77 3.15

Motion C90 1,355 1,361

*Limit NA 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.00 (3.75) 3.60 (4.50)

*Values in ( ) reflect increased drift limits provided by Standard Sec. 16.2.4.3.

Table 4.2-22 MCE Results for 5% Damped Strong Panel Model with P-Delta Excluded Level Column forces Inertial forces

Level oof displacement Drift R-6 Drift 6-5 Drift 5-4 Drift 4-3 Drift 3-2 Drift 2-G

(a) Maximum Base Shear (kips) Motion B90 Motion A00 1,918 1,760 2,139 1,861 (b) Maximum Displacement and Story Drift (in.) Motion A00 Motion B90 Motion C90 40.84 20.17 21.10 1.68 1.94 2.97 2.91 2.61 3.75 4.86 3.12 4.50 9.04 4.18 4.43 10.48 4.77 3.98 13.04 6.09 4.93

Motion C90 1,630 1,633

Limit NA 4.5 4.5 4.5 4.5 4.5 5.4

Table 4.2-23 MCE Results for 5% Damped Strong Panel Model with P-Delta Included (a) Maximum Base Shear (kips) Motion B90 Motion C90 Level Motion A00 Column forces 1,451 1,486 1,413 Inertial forces 1,798 1,607 1,419

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Chapter 4: Structural Analysis

Level Roof displacement Drift R-6 Drift 6-5 Drift 5-4 Drift 4-3 Drift 3-2 Drift 2-G

(b) Maximum Displacement and Story Drift (in.) Motion A00 Motion B90 Motion C90

Limit

54.33

23.12

21.83

1.66 2.65 4.88 12.63 15.27 19.31

2.01 2.38 3.31 5.09 5.66 6.14

2.88 3.64 4.49 4.72 4.28 5.07

4.5 4.5 4.5 4.5 4.5 5.4

4.2.6.3.2 Discussion of response history analyses. The computed structural response to Ground Motion A00 is clearly quite different from that for Ground Motions B90 and C90. This difference in behavior occurs even though the records are all scaled to produce exactly the same spectral acceleration at the structure’s fundamental period. A casual inspection of the ground acceleration histories and response spectra (Figures 4.2-40 through 4.2-42) does not reveal the underlying reason for this difference in behavior. Figure 4.2-44 shows response histories of roof displacement and first story drift for the 2 percent damped SP model subjected to the DBE-scaled A00 ground motion. Two trends are readily apparent. First, the vast majority of the roof displacement results in residual deformation in the first story. Second, the Pdelta effect increases residual deformations by about 50 percent. Such extreme differences in behavior do not appear in plots of base shear, as provided in Figure 4.2-45. The residual deformations shown in Figure 4.2-44 may be real (due to actual system behavior) or may reflect accumulated numerical errors in the analysis. Numerical errors are unlikely because the shears computed from member forces and from inertial forces are similar. The energy response history can provide further validation.. Figure 4.2-46 shows the energy response history for the 2 percent damped DBE analysis with P-delta effects included. If the analysis is accurate, the input energy will coincide with the total energy (sum of kinetic, damping and structural energy). DRAIN 2D produces individual energy values as well as the input energy. See the article by Uang and Bertero for background on computing energy curves. As evident from Figure 4.2-46, the total and input energy curves coincide, so the analysis is numerically accurate. Where this accuracy is in doubt, the analysis should be re-run using a smaller integration time step. A time step of 0.0005 second is required to produce the energy balance shown in Figure 4.2-46. A time step of 0.001 second is sufficient for analyses with Ground Motions B90 and C90. The trends observed for the DBE analysis are even more extreme when the MCE ground motion is used. Figure 4.2-47 shows the displacement histories for the 2 percent damped structure under the MCE scaled A00 ground motion. As may be seen, residual deformations again dominate and in this case the total residual roof displacement with P-delta effects included is five times that without P-delta effects. This behavior indicates dynamic instability and eventual collapse. It is interesting to compare the response computed for Ground Motion B90 with that obtained for ground motion A00. Displacements occurring for the 2 percent damped model under the MCE-scaled B90 ground motions are shown in Figure 4.2-48. While there is some small residual deformation in this

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FEMA P-751, NEHRP Recommended Provisions: Design Examples system, it is not extreme, and it appears that the structure is not in danger of collapse. (The corresponding plastic rotations are less than those that would be associated with significant strength loss.) The characteristic of the ground motion that produces the residual deformations shown in Figures 4.2-44 and 4.2-48 (the DBE and MCE scaled A00 ground motions, respectively) is not evident from the ground acceleration history or from the acceleration response spectrum. The source of the behavior is quite obvious from plots of the ground velocity and ground displacement histories, shown in Figure 4.2-49(a) and (b), respectively. The ground velocity history shows that a very large velocity pulse occurs approximately 10 seconds into the earthquake. This leads to a surge in ground displacement, also occurring approximately 10 seconds into the response. The surge in ground displacement is more than 8 feet, which is somewhat unusual. Recall from Table 4.1-20(a) that the distance between the epicenter and the recording site for this ground motion is 44 kilometers; so, the motion would not be considered as near-field. The unusual characteristics of Ground Motion A00 may be seen in Figure 4.2-49 (c), which is a tripartite spectrum.

Displacement (inches)

Roof, with P-Delta 20

without P-Delta

First story, with P-Delta

without P-Delta

0 -10 0

Time (s)

Figure 4.2-44 Response history of roof and first-story displacement, Ground Motion A00 (DBE)

Base shear (kips)

2000 Total shear with P-Delta Total shear without P-Delta

1000 0 -1000 -2000 0

Time (s)

Figure 4.2-45 Response history of total base shear, Ground Motion A00 (DBE)

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Chapter 4: Structural Analysis

Kinetic

Energy (inch-kips)

40,000

Damping

30,000 Structural

20,000 10,000 0 0

Time (s)

Figure 4.2-46 Energy response history, Ground Motion A00 (DBE), including P-delta effects

Roof, with P-Delta

Displacement (inches)

80 First story, with P-Delta

Roof, without P-Delta First story, without P-Delta

-40 0

Time (s)

Figure 4.2-47 Response history of roof and first-story displacement, Ground Motion A00 (MCE)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Displacement (inches)

Roof with P-Delta Roof without P-Delta First Story with P-delta First Story without P-delta

20 10 0

-10

-20 0

Time (s)

Ground velocity (ft/s)

Figure 4.2-48 Response history of roof and first-story displacement, Ground Motion B90 (MCE)

-2 0

Time (s)

Ground displacement (ft)

(a) Ground velocity history 8 6 4 2

0 -2 0

Time (s)

(b) Ground displacement history 4-124

Chapter 4: Structural Analysis

10 50

2% Damping 5% Damping

Pseudovelocity, ft/sec

1 0.

g 10 .0 1

0.1 0.

ft.

g 05

ft.

g 05

0 0.

g 1

0.01 0.01

0.1

Period, sec

0 .0

(c) Tripartite spectrum Figure 4.2-49 Ground velocity and displacement histories and tripartite spectrum of Ground Motion A00 (unscaled) Figure 4.2-50 shows the pattern of yielding in the structure subjected to a 2 percent damped MCE-scaled Ground Motion B90 including P-delta effects. Recall that the model incorporates panel zone reinforcement at the interior beam-column joints. The circles on the figure represent yielding at any time during the response; consequently, yielding does not necessarily occur at all locations simultaneously. The circles shown at the upper left corner of the beam-column joint region indicate yielding in the rotational spring, which represents the web component of panel zone behavior. There is no yielding in the flange component of the panel zones, as seen in Figure 4.2-50. Yielding patterns for the other ground motions and for analyses run with and without P-delta effects are similar but are not shown here. As expected, there is more yielding in the columns when the structure is subjected to the A00 ground motion. Figure 4.2-50 shows that yielding occurs at both ends of each of the girders at Levels 2, 3, 4 and 5. Yielding occurs at the bottom of all the first-story columns as well as at the top of the interior columns at the third and fourth stories and at bottom of the fifth-story interior columns. The panel zones at the exterior joints of Levels 4 and 5 also yield. The maximum plastic hinge rotations are shown where they occur for the columns, girders and panel zones; values are shown in Table 4.2-24. The maximum plastic shear strain in the web of the panel zone is identical to the computed hinge rotation in the panel zone spring. For the DBE-scaled B90 ground motion, the maximum rotations occurring at the plastic hinges are less than 0.02 radians.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Panel zone,max = 0.00411 rad

Girder,max = 0.03609 rad Column,max = 0.02993 rad

Figure 4.2-50 Yielding locations for structure with strong panels subjected to MCE-scaled B90 motion, including P-delta effects 4.2.6.3.3 Comparison with results from other analyses. Table 4.2-24 compares the results from the response history analysis with those from the ELF and the nonlinear static analyses. Base shears in the table are half of the total shear. The nonlinear static analysis results are for the 2 percent damped MCE target displacement so, for consistency, the tabulated dynamic analysis results are for the 2 percent damped MCE-scaled B90 ground motion. In addition, the lateral forces used to find the ELF drifts in Table 4.2-6 are multiplied by 1.5 forconsistency with MCE-level shaking; the ELF analysis drift values include the deflection amplification factor of 5.5. The results show some similarities and some striking differences, as follows: §

The base shear from nonlinear dynamic analysis is approximately three times the value from ELF analysis. The predicted displacements and story drifts are similar at the top three stories but are significantly different at the bottom three stories. Due to the highly empirical nature of the ELF approach, it is difficult to explain these differences. The ELF method also has no mechanism to include the overstrength that will occur in the structure, although it is represented explicitly in the static and dynamic nonlinear analyses.

§

The nonlinear static analysis predicts base shears and story displacements that are less than those obtained from response history analysis. Excessive drift occurred at the bottom three stories as a result of both pushover and response history analyses.

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Chapter 4: Structural Analysis Table 4.2-24

Summary of All Analyses for Strong Panel Structure, Including P-delta Effects Analysis Method Response Quantity Equivalent Nonlinear Static Nonlinear Lateral Forces Pushover Dynamic Base shear (kips) 569 1,208 1,633 Roof disp. (in.) 18.4 22.9 26.1 Drift R-6 (in.) 1.86 0.96 2.32 Drift 6-5 (in.) 2.78 1.76 2.60 Drift 5-4 (in.) 3.34 2.87 3.62 Drift 4-3 (in.) 3.73 4.84 5.61 Drift 3-2 (in.) 3.67 5.74 6.32 Drift 2-1 (in.) 2.98 6.73 7.03 Girder hinge rot. (rad) NA 0.03304 0.03609 Column hinge rot. (rad) NA 0.02875 0.02993 Panel hinge rot. (rad) NA 0.00335 0.00411 Panel plastic shear strain NA 0.00335 0.00411

Note: Shears are for half of total structure.

Some of the difference between pushover and nonlinear response history results is due to the scale factor (1.367) used to satisfy ground motion scaling requirements for the nonlinear response history analysis, but most of the difference is due to higher mode effects. Figure 4.2-51 shows the inertial forces from the nonlinear response history analyses at the time of peak base shear and the loads applied to the nonlinear static analysis model at the target displacement. The higher mode effects apparent in Figure 4.2-51 likely are the cause of the different hinging patterns and certainly are the reason for the very high base shear developed in the response history analysis. (If the inertial forces were constrained to follow the first mode response, the maximum base shear that could be developed in the system would be in the range of 1200 kips. See, for example, Figure 4.2-28.)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 11k

53k

237k

316k

313k

286k

241k

187k

494k

123k

542k

58k

Figure 4.2-51 Comparison of inertial force patterns 4.2.6.3.4 Effect of increased damping on response. The nonlinear response history analysis of the structure with panel zone reinforcement indicates first story drifts in excess of the allowable limits. The most cost-effective measure to enhance the performance of the structure probably would be to provide additional strength and/or stiffness at this story. However, added damping is also a viable approach. To investigate the viability of added damping, additional analysis that treats individual dampers explicitly is required. Linear viscous damping can be modeled in DRAIN using the stiffness proportional component of Rayleigh damping. Base shear increases with added damping, so in practice added damping systems usually employ viscous fluid devices with a “softening” nonlinear relationship between the deformational velocity in the device and the force in the device, to limit base shears when deformational velocities become large. A linear viscous fluid damping device (Figure 4.2-52) in a selected story can be modeled using a Type 1 (truss bar) element. A damping constant for the device, Cdevice , is obtained as follows: The elastic stiffness of the damper element is simply as follows:

kdevice =

Adevice Edevice Ldevice

where:

Adevice = the cross sectional area Edevice = the modulus of elasticity Ldevice = the length of the Type 1 damper element

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Chapter 4: Structural Analysis

As stiffness proportional damping is used, the damping constant for the element is:

Cdevice = β device kdevice The damper elastic stiffness should be negligible, so consider kdevice = 0.001 kips/in. Thus:

β device =

Cdevice = 1000Cdevice 0.001

Where modeling added dampers in this manner, it is convenient to consider Edevice = 0.001 and Adevice = the damper length Ldevice. This value of βdevice is for the added damper element only. Different dampers may require different values. Also, a different (global) value of β is required to model the stiffness proportional component of damping in the remaining nondamper elements. Modeling the dynamic response using Type 1 elements is exact within the typical limitations of finite element analysis. Using the modal strain energy approach, DRAIN reports a damping value in each mode. These modal damping values are approximate and may be poor estimates of actual modal damping, particularly where there is excessive flexibility in the mechanism that connects the damper to the structure. To determine the effect of added damping on the behavior of the structure, dampers are added to the SP frame with 2 percent inherent damping, and the structure is subjected to the DBE-scaled A00 and B90 ground motions. P-delta effects are included in the analyses. Table 4.2-25 shows the base shear and story drifts of the SP frame with 2 percent inherent damping when it is subjected to DBE-scaled A00 and B90 ground motions. The results summarized in this table can also be found in the tables of Section 4.2.6.3.1.

L j

i Damper

Brace

Figure 4.2-52 Modeling a simple damper

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Table 4.2-25 Maximum Story Drifts (in.) and Base Shear (kips) when SP Model with 2% Inherent Damping is Subjected to DBE Scaled A00 and B90 Ground Motions, including P-delta Effects Level Motion A00 Motion B90 Limit Roof displacement 32.65 14.50 NA Drift R-6 1.86 1.82 3.75 Drift 6-5 2.64 2.50 3.75 Drift 5-4 4.08 2.81 3.75 Drift 4-3 6.87 3.21 3.75 Drift 3-2 8.19 3.40 3.75 Drift 2-G 10.40 4.69 4.50 Column forces 1467 1458 NA Inertial forces 1558 1481 NA As can be seen in Table 4.2-25, drift limits are exceeded at the bottom four stories for the A00 ground motion and only for the bottom story for the B90 ground motion. Four different added damper configurations are used to assess their effect on story drifts and base shear, as summarized in Tables 4.2-26 and 4.2-27. Table 4.2-26 Effect of Different Added Damper Configurations when SP Model is Subjected to DBE-Scaled A00 Ground Motion, including P-delta Effects First Config Second Config Third Config Fourth Config Damper Damper Damper Damper Drift Coeff. Drift Coeff. Drift Coeff. Drift Coeff. Drift Level Limit (kip(in.) (kip(in.) (kip(in.) (kip(in.) (in.) sec/in.) sec/in.) sec/in.) sec/in.) R-6 10.5 1.10 60 1.03 1.82 1.47 3.75 6-5 33.7 1.90 60 1.84 3.56 2.41 3.75 5-4 38.4 2.99 70 2.88 4.86 56.25 3.46 3.75 4-3 32.1 5.46 70 4.42 5.24 56.25 4.47 3.75 3-2 36.5 6.69 80 5.15 160 4.64 112.5 4.76 3.75 2-G 25.6 8.39 80 5.87 160 4.40 112.5 4.96 4.50 Column base 1,629 2,170 2,134 2,267 shear (kips) Inertial base 1,728 2,268 2,215 2,350 shear (kips) Total 10.1 20.4 20.2 20.4 damping (%)

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Chapter 4: Structural Analysis Table 4.2-27 Effect of Different Added Damper Configurations when SP Model is Subjected to DBE-Scaled B90 Ground Motion, including P-delta Effects First Config Second Config Third Config Fourth Config Damper Damper Damper Damper Drift Coeff. Drift Coeff. Drift Coeff. Drift Coeff. Drift Level Limit (kip(in.) (kip(in.) (kip(in.) (kip(in.) (in.) sec/in.) sec/in.) sec/in.) sec/in.) R-6 10.5 1.11 60 0.86 1.53 1.31 3.75 6-5 33.7 1.76 60 1.35 2.11 1.83 3.75 5-4 38.4 2.33 70 1.75 2.51 56.25 2.07 3.75 4-3 32.1 2.67 70 2.11 2.37 56.25 2.16 3.75 3-2 36.5 2.99 80 2.25 160 2.09 112.5 2.13 3.75 2-G 25.6 3.49 80 1.96 160 1.87 112.5 1.82 4.50 Column base 1,481 1,485 1,697 1,637 shear (kips) Inertial base 1,531 1,527 1,739 1,680 shear (kips) Total 10.1 20.4 20.2 20.4 damping (%) These configurations increase total damping of the structure from 2 percent (inherent) to 10 and 20 percent. In the first configuration added dampers are distributed proportionally to approximate story stiffnesses. In the second configuration, dampers are added at all six stories, with larger dampers in lower stories. Since the structure seems to be weak at the bottom stories (where it exceeds drift limits), dampers are concentrated at the bottom stories in the last two configurations. Added dampers are used only at the first and second stories in the third configuration and at the bottom four stories in the fourth configuration. Based on this supplemental damper study, it appears to be impossible to decrease the story drifts for the A00 ground motion below the limits. This is because of the incremental velocity of Ground Motion A00 causes such significant structural damage. The drift limits could be satisfied if the total damping ratio is increased to 33.5 percent, but since that is impractical the results are not reported here. The third configuration of added dampers reduces the first-story drift from 10.40 inches to 4.40 inches All of the configurations easily satisfy drift limits for the B90 ground motion.While the system with 10 percent total damping is sufficient for drift limits, systems with 20 percent damping further improve performance. Although configurations 3 and 4 have the same amount of total damping as configuration 2, story drifts are higher at the top stories since dampers are added only at lower stories. Figures 4.2-53 through 4.2-55 show the effect of added damping of roof displacement, inertial base shear and energy history for the A00 ground motion. As Figure 4.2-53 shows added dampers reduce roof displacement significantly but do not prevent residual displacement. Figure 4.2-54 shows how added damping increases peak base shear. Figure 4.2-55 is an energy response history for the structure with damping configuration 4. It should be compared to Figure 4.2-46, which is the energy history for the structure with 2 percent inherent damping but with no added damping. As should be expected, adding discrete damping reduces the hysteretic energy demand in the structure (designated as structural energy in Figure 4.2-55). A reduction in hysteretic energy demand for the system with added damping corresponds to a reduction in structural damage.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Roof displacement (inches)

Figures 4.2-56 through 4.2-58 display the same response plots for Ground Motion B90. As for Ground Motion A00 roof displacement decreases with added damping, peak base shear increases and hysteretic energy demand (which is related to structural damage) decreases.

2% inherent damping

30 20

Fourth damper configuration (20% damping)

0 -10 0

Time (s) 30

Figure 4.2-53 Roof displacement response histories with added damping (20% total) and inherent damping (2%) for Ground Motion A00

Base shear (kips)

2,000 2% inherent damping

1,000 0 -1,000

-2,000

Fourth damper configuration (20% damping)

-3,000 0

20 Time (s)

Figure 4.2-54 Inertial base shear response histories with added damping (20% total) and inherent damping (2%) for Ground Motion A00

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Chapter 4: Structural Analysis

Energy (inch-kips)

40,000 Kinetic

30,000 20,000

Damping

10,000

Structural

0 0

Time (s)

Figure 4.2-55 Energy response history with added damping of fourth configuration (20% total damping) for Ground Motion A00

Roof displacement (inches)

2% inherent damping Fourth damper configuration (20% damping)

-10

-20 0

Time (s)

Figure 4.2-56 Roof displacement response histories with added damping (20% total) and inherent damping (2%) for Ground Motion B90

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Base shear (kips)

2,000

1,000 0 -1,000

2% inherent damping

-2,000

Fourth damper configuration (20% damping)

-3,000

Time (s)

Figure 4.2-57 Inertial base shear response histories with added damping (20% total) and inherent damping (2%) for Ground Motion B90

Energy (inch-kips)

40,000

30,000

Kinetic

20,000

Damping 10,000 Structural

0 0

Time (s)

Figure 4.2-58 Energy response history with added damping of fourth configuration (20% total damping) for Ground Motion B90

4.2.7

Summary and Conclusions

In this example, five different analytical approaches are used to estimate the deformation demands in a simple structural steel moment-resisting frame structure: 1. Linear static analysis (the equivalent lateral force method) 2. Plastic strength analysis (using virtual work) 3. Nonlinear static (pushover) analysis 4. Linear dynamic (modal response history) analysis

4-134

Chapter 4: Structural Analysis 5. Nonlinear dynamic (response history) analysis The nonlinear structural model includes careful representation of possible inelastic behavior in the panelzone regions of the beam-column joints. The results obtained from the three different analytical approaches 1, 3 and 5 are quite dissimilar. Except for preliminary design, the ELF approach should not be used in explicit performance evaluation since it cannot reflect the location and extent of yielding in the structure. Due to higher mode effects, pushover analysis, where used alone, is inadequate. This leaves nonlinear response history analysis as the most viable approach. Given the speed and memory capacity of personal computers, nonlinear response history analysis is increasingly common in the seismic analysis of buildings. However, significant shortcomings, limitations and uncertainties in response history analysis still exist. Among the most pressing problems is the need for a suitable suite of ground motions. All ground motions must adequately reflect site conditions and where applicable, the suite must include near-field effects. Through future research and the efforts of code writing bodies, it may be possible to develop standard suites of ground motions that could be published together with selection tools and scaling methodologies. The scaling techniques currently recommended in the Standard are a start but need improvement. Systematic methods need to be developed for identifying uncertainties in the modeling of the structure and for quantifying the effect of such uncertainties on the response. While probabilistic methods for dealing with such uncertainties seem like a natural extension of the analytical approach, the authors believe that deterministic methods should not be abandoned entirely. In the context of performance-based design, improved methods for assessing the effect of inelastic response and acceptance criteria based on such measures need to be developed. Methods based on explicit quantification of damage should be considered seriously. The ideas presented above certainly are not original. They have been presented by many academics and practicing engineers. What is still lacking is a comprehensive approach to seismic-resistant design based on these principles.

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5 Foundation Analysis and Design Michael Valley, S.E. Contents 5.1

SHALLOW FOUNDATIONS FOR A SEVEN-STORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA........................................................................................................... 3

5.1.1

Basic Information .................................................................................................................. 3

5.1.2

Design for Gravity Loads ...................................................................................................... 8

5.1.3

Design for Moment-Resisting Frame System ...................................................................... 11

5.1.4

Design for Concentrically Braced Frame System ............................................................... 16

5.1.5

Cost Comparison ................................................................................................................. 24

5.2

DEEP FOUNDATIONS FOR A 12-STORY BUILDING, SEISMIC DESIGN CATEGORY D ..................................................................................................................................................... 25

5.2.1

Basic Information ................................................................................................................ 25

5.2.2

Pile Analysis, Design and Detailing .................................................................................... 33

5.2.3

Other Considerations ........................................................................................................... 47

FEMA P-751, NEHRP Recommended Provisions: Design Examples This chapter illustrates application of the 2009 Edition of the NEHRP Recommended Provisions to the design of foundation elements. Example 5.1 completes the analysis and design of shallow foundations for two of the alternative framing arrangements considered for the building featured in Example 6.2. Example 5.2 illustrates the analysis and design of deep foundations for a building similar to the one highlighted in Chapter 7 of this volume of design examples. In both cases, only those portions of the designs necessary to illustrate specific points are included. The force-displacement response of soil to loading is highly nonlinear and strongly time dependent. Control of settlement is generally the most important aspect of soil response to gravity loads. However, the strength of the soil may control foundation design where large amplitude transient loads, such as those occurring during an earthquake, are anticipated. Foundation elements are most commonly constructed of reinforced concrete. As compared to design of concrete elements that form the superstructure of a building, additional consideration must be given to concrete foundation elements due to permanent exposure to potentially deleterious materials, less precise construction tolerances and even the possibility of unintentional mixing with soil. Although the application of advanced analysis techniques to foundation design is becoming increasingly common (and is illustrated in this chapter), analysis should not be the primary focus of foundation design. Good foundation design for seismic resistance requires familiarity with basic soil behavior and common geotechnical parameters, the ability to proportion concrete elements correctly, an understanding of how such elements should be detailed to produce ductile response and careful attention to practical considerations of construction. In addition to the Standard and the Provisions and Commentary, the following documents are either referenced directly or provide useful information for the analysis and design of foundations for seismic resistance: ACI 318

American Concrete Institute. 2008. Building Code Requirements and Commentary for Structural Concrete.

Bowles

Bowles, J. E. 1988. Foundation Analysis and Design. McGraw-Hill.

CRSI

Concrete Reinforcing Steel Institute. 2008. CRSI Design Handbook. Concrete Reinforcing Steel Institute.

ASCE 41

ASCE. 2006. Seismic Rehabilitation of Existing Buildings.

Kramer

Kramer, S. L. 1996. Geotechnical Earthquake Engineering. Prentice Hall.

LPILE

Reese, L. C. and S. T. Wang. 2009. Technical Manual for LPILE Plus 5.0 for Windows. Ensoft.

Rollins et al. (a)

Rollins, K. M., Olsen, R. J., Egbert, J. J., Jensen, D. H., Olsen, K. G.and Garrett, B. H. (2006). “Pile Spacing Effects on Lateral Pile Group Behavior: Load Tests.” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 132, No. 10, p. 1262-1271.

Rollins et al. (b)

Rollins, K. M., Olsen, K. G., Jensen, D. H, Garrett, B. H., Olsen, R. J.and Egbert, J. J. (2006). “Pile Spacing Effects on Lateral Pile Group Behavior: Analysis.”

5-2

Chapter 5: Foundation Analysis and Design Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 132, No. 10, p. 1272-1283. Wang & Salmon

Wang, C.-K. and C. G. Salmon. 1992. Reinforced Concrete Design . HarperCollins.

Several commercially available programs were used to perform the calculations described in this chapter. SAP2000 is used to determine the shears and moments in a concrete mat foundation; LPILE, in the analysis of laterally loaded single piles; and spColumn, to determine concrete pile section capacities. 5.1

SHALLOW FOUNDATIONS FOR A SEVEN-‐STORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA

This example features the analysis and design of shallow foundations for two of the three framing arrangements for the seven-story steel office building described in Section 6.2 of this volume of design examples. Refer to that example for more detailed building information and for the design of the superstructure. 5.1.1

Basic Information

5.1.1.1 Description. The framing plan in Figure 5.1-1 shows the gravity load-resisting system for a representative level of the building. The site soils, consisting of medium dense sands, are suitable for shallow foundations. Table 5.1-1 shows the design parameters provided by a geotechnical consultant. Note the distinction made between bearing pressure and bearing capacity. If the long-term, service-level loads applied to foundations do not exceed the noted bearing pressure, differential and total settlements are expected to be within acceptable limits. Settlements are more pronounced where large areas are loaded, so the bearing pressure limits are a function of the size of the loaded area. The values identified as bearing capacity are related to gross failure of the soil mass in the vicinity of loading. Where loads are applied over smaller areas, punching into the soil is more likely.

5-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples

1'-2"

25'-0"

1'-2"

25'-0" 25'-0"

25'-0"

127'-4"

25'-0"

1'-2"

177'-4"

1'-2"

Figure 5.1-1 Typical framing plan Because bearing capacities are generally expressed as a function of the minimum dimension of the loaded area and are applied as limits on the maximum pressure, foundations with significantly non-square loaded areas (tending toward strip footings) and those with significant differences between average pressure and maximum pressure (as for eccentrically loaded footings) have higher calculated bearing capacities. The recommended values are consistent with these expectations. Table 5.1-1 Geotechnical Parameters Parameter

Value Medium dense sand

Basic soil properties

(SPT) N = 20

γ = 125 pcf Angle of internal friction = 33 degrees

5-4

Chapter 5: Foundation Analysis and Design Table 5.1-1 Geotechnical Parameters Parameter

Value ≤ 4,000 psf for B ≤ 20 feet

Net bearing pressure (to control settlement due to sustained loads)

≤ 2,000 psf for B ≥ 40 feet (may interpolate for intermediate dimensions) 2,000B psf for concentrically loaded square footings 3,000B' psf for eccentrically loaded footings

Bearing capacity (for plastic equilibrium strength checks with factored loads)

where B and B' are in feet, B is the footing width and B' is an average width for the compressed area. Resistance factor, φ = 0.7 [This φ factor for cohesionless soil is specified in Provisions Part 3 Resource Paper 4; the value is set at 0.7 for vertical, lateral and rocking resistance.] Earth pressure coefficients:

Lateral properties

§ § §

Active, KA = 0.3 At-rest, K0 = 0.46 Passive, KP = 3.3

“Ultimate” friction coefficient at base of footing = 0.65 Resistance factor, φ = 0.7 The structural material properties assumed for this example are as follows: §

f'c = 4,000 psi

§

fy = 60,000 psi

5.1.1.2 Seismic Parameters. The complete set of parameters used in applying the Provisions to design of the superstructure is described in Section 6.2.2.1 of this volume of design examples. The following parameters, which are used during foundation design, are duplicated here. §

Site Class = D

§

SDS = 1.0

§

Seismic Design Category = D

5-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples 5.1.1.3 Design Approach. 5.1.1.3.1 Selecting Footing Size and Reinforcement. Most foundation failures are related to excessive movement rather than loss of load-carrying capacity. In recognition of this fact, settlement control should be the first issue addressed. Once service loads have been calculated, foundation plan dimensions should be selected to limit bearing pressures to those that are expected to provide adequate settlement performance. Maintaining a reasonably consistent level of service load-bearing pressures for all of the individual footings is encouraged since it will tend to reduce differential settlements, which are usually of more concern than are total settlements. Once a preliminary footing size that satisfies serviceability criteria has been selected, bearing capacity can be checked. It would be rare for bearing capacity to govern the size of footings subjected to sustained loads. However, where large transient loads are anticipated, consideration of bearing capacity may become important. The thickness of footings is selected for ease of construction and to provide adequate shear capacity for the concrete section. The common design approach is to increase footing thickness as necessary to avoid the need for shear reinforcement, which is uncommon in shallow foundations. Design requirements for concrete footings are found in Chapters 15 and 21 of ACI 318. Chapter 15 provides direction for the calculation of demands and includes detailing requirements. Section capacities are calculated in accordance with Chapters 10 (for flexure) and 11 (for shear). Figure 5.1-2 illustrates the critical sections (dashed lines) and areas (hatched) over which loads are tributary to the critical sections. For elements that are very thick with respect to the plan dimensions (as at pile caps), these critical section definitions become less meaningful and other approaches (such as strut-and-tie modeling) should be employed. Chapter 21 provides the minimum requirements for concrete foundations in Seismic Design Categories D, E and F, which are similar to those provided in prior editions of the Provisions. For shallow foundations, reinforcement is designed to satisfy flexural demands. ACI 318 Section 15.4 defines how flexural reinforcement is to be distributed for footings of various shapes. Section 10.5 of ACI 318 prescribes the minimum reinforcement for flexural members where tensile reinforcement is required by analysis. Provision of the minimum reinforcement assures that the strength of the cracked section is not less than that of the corresponding unreinforced concrete section, thus preventing sudden, brittle failures. Less reinforcement may be used as long as “the area of tensile reinforcement provided is at least one-third greater than that required by analysis.” Section 10.5.4 relaxes the minimum reinforcement requirement for footings of uniform thickness. Such elements need only satisfy the shrinkage reinforcement requirements of Section 7.12. Section 10.5.4 also imposes limits on the maximum spacing of bars. 5.1.1.3.2 Additional Considerations for Eccentric Loads. The design of eccentrically loaded footings follows the approach outlined above with one significant addition: consideration of overturning stability. Stability calculations are sensitive to the characterization of soil behavior. For sustained eccentric loads, a linear distribution of elastic soil stresses is generally assumed and uplift is usually avoided. If the structure is expected to remain elastic when subjected to short-term eccentric loads (as for wind loading), uplift over a portion of the footing is acceptable to most designers. Where foundations will be subjected to short-term loads and inelastic response is acceptable (as for earthquake loading), plastic soil stresses may be considered. It is most common to consider stability effects on the basis of statically applied loads even where the loading is actually dynamic; that approach simplifies the calculations at the expense of increased conservatism. Figure 5.1-3 illustrates the distribution of soil stresses for the various assumptions. Most textbooks on foundation design provide simple equations to describe the conditions 5-6

Chapter 5: Foundation Analysis and Design shown in Parts b, c and d of the figure; finite element models of those conditions are easy to develop. Simple hand calculations can be performed for the case shown in Part f. Practical consideration of the case shown in Part e would require modeling with inelastic elements, but that offers no advantage over direct consideration of the plastic limit. (All of the discussion in this section focuses on the common case in which foundation elements may be assumed to be rigid with respect to the supporting soil. For the interested reader, Chapter 4 of ASCE 41 provides a useful discussion of foundation compliance, rocking and other advanced considerations.)

Outside face of concrete column or line midway between face of steel column and edge of steel base plate (typical)

P M

(a) Critical section for flexure

(a) Loading

(e = M P) B

(b) Elastic, no uplift P ⎛ e ⎞ qm ax = ⎜1 + 6 ⎟ B L ⎝ L ⎠

e ≤ L6

extent of footing (typical)

(b) Critical section for one-way shear d

e = L6

(d) Elastic, after uplift 2P qma x = ⎛ L ⎞ 3 B ⎜ − e ⎟ ⎝ 2 ⎠ ⎛ L ⎞ Lʹ′ = 3 ⎜ − e ⎟ ⎝ 2 ⎠

L 621 kips

5-8

Chapter 5: Foundation Analysis and Design For use in subsequent calculations, the factored bearing pressure qu = 621 kips/(11 ft)2 = 5.13 ksf. 5.1.2.3 Footing Thickness. Once the plan dimensions of the footing are selected, the thickness is determined such that the section satisfies the one-way and two-way shear demands without the addition of shear reinforcement. Demands are calculated at critical sections, shown in Figure 5.1-2, which depend on the footing thickness. Check a footing that is 26 inches thick: For the W14 columns used in this building, the side dimensions of the loaded area (taken halfway between the face of the column and the edge of the base plate) are approximately 16 inches. Accounting for cover and expected bar sizes, d = 26 - (3 + 1.5(1)) = 21.5 in. One-way shear:

⎛ 11 − 16 21.5 ⎞ 12 Vu = 11⎜ − ⎟ ( 5.13) = 172 kips 12 ⎠ ⎝ 2

φVn = φVc = ( 0.75) 2 4,000 (11×12 )( 21.5)

( ) = 269 kips > 172 kips 1 1,000

Two-way shear: 2

Vu = 621 − ( 16+1221.5 ) (5.13) = 571 kips

φVn = φVc = ( 0.75) 4 4,000 ⎣⎡4 × (16 + 21.5)⎤⎦ ( 21.5)

( ) = 612 kips > 571 kips 1 1,000

5.1.2.4 Footing Reinforcement. Footing reinforcement is selected considering both flexural demands and minimum reinforcement requirements. The following calculations treat flexure first because it usually controls: 2

⎛ 11 − 16 ⎞ 1 12 M u = (11) ⎜ ⎟ (5.13) = 659 ft-kips 2 ⎝ 2 ⎠ Try nine #8 bars each way. The distance from the extreme compression fiber to the center of the top layer of reinforcement, d = t - cover - 1.5db = 26 - 3 - 1.5(1) = 21.5 in. T = As fy = 9(0.79)(60) = 427 kips Noting that C = T and solving the expression C = 0.85 f'c b a for a produces a = 0.951 in. 1 = 673 ft-kips > 659 ft-kips φ M n = φT ( d − a2 ) = 0.90 ( 427 ) ( 21.5 − 0.951 2 ) ( 12 )

The ratio of reinforcement provided is ρ = 9(0.79)/[(11)(12)(26)] = 0.00207. The distance between bars spaced uniformly across the width of the footing is s = [(11)(12)-2(3+0.5)]/(9-1) = 15.6 in. According to ACI 318 Section 7.12, the minimum reinforcement ratio = 0.0018 < 0.00207

OK 5-9

FEMA P-751, NEHRP Recommended Provisions: Design Examples

and the maximum spacing is the lesser of 5 × 26 in. and 18 = 18 in. > 15.6 in.

5.1.2.5 Design Results. The calculations performed in Sections 5.1.2.2 through 5.1.2.4 are repeated for typical perimeter and corner footings. The footing design for gravity loads is summarized in Table 5.1-2; Figure 5.1-4 depicts the resulting foundation plan. Table 5.1-2 Footing Design for Gravity Loads Location

Interior

Perimeter

Corner

5-10

Loads

Footing Size and Reinforcement; Soil Capacity

D = 387 kip L = 98 kip

11'-0" × 11'-0" × 2'-2" deep 9-#8 bars each way

P = 485 kip Pu = 621 kip

Pallow = 484 kip φPn = 1863 kip

D = 206 kip L = 45 kip

8'-0" × 8'-0" × 1'-6" deep 9-#6 bars each way

P = 251 kip Pu = 319 kip

Pallow = 256 kip φPn = 716 kip

D = 104 kip L = 23 kip

6'-0" × 6'-0" × 1'-2" deep 6-#5 bars each way

P = 127 kip Pu = 162 kip

Pallow = 144 kip φPn = 302 kip

Critical Section Demands and Design Strengths One-way shear: Vu = 172 kip φVn = 269 kip Two-way shear: Vu = 571 kip φVn = 612 kip Flexure: Mu = 659 ft-kip φMn = 673 ft-kip Vu = 88.1 kip φVn = 123 kip Two-way shear: Vu = 289 kip φVn = 302 kip Flexure: Mu = 222 ft-kip φMn = 234 ft-kip One-way shear:

Vu = 41.5 kip φVn = 64.9 kip Two-way shear: Vu = 141 kip φVn = 184 kip Flexure: Mu = 73.3 ft-kip φMn = 75.2 ft-kip One-way shear:

Chapter 5: Foundation Analysis and Design

Corner: 6'x6'x1'-2" thick

Perimeter: 8'x8'x1'-6" thick

Interior: 11'x11'x2'-2" thick

Figure 5.1-4 Foundation plan

5.1.3

Design for Moment-‐Resisting Frame System

Framing Alternate A in Section 6.2 of this volume of design examples includes a perimeter momentresisting frame as the seismic force-resisting system. A framing plan for the system is shown in Figure 5.1-5. Detailed calculations are provided in this section for a combined footing at the corner and focus on overturning and sliding checks for the eccentrically loaded footing; settlement checks and design of concrete sections would be similar to the calculations shown in Section 5.1.2. The results for all footing types are summarized in Section 5.1.3.4.

5-11

FEMA P-751, NEHRP Recommended Provisions: Design Examples

7 at 25'-0"

5 at 25'-0"

Figure 5.1-5 Framing plan for moment-resisting frame system 5.1.3.1 Demands. A three-dimensional analysis of the superstructure, in accordance with the requirements for the equivalent lateral force (ELF) procedure, is performed using the ETABS program. Foundation reactions at selected grids are reported in Table 5.1-3. Table 5.1-3 Demands from Moment-Resisting Frame System Location Load Fx Fy Fz Mxx Myy D -203.8 L -43.8 A-5 Ex -13.8 4.6 3.8 53.6 -243.1 Ey 0.5 -85.1 -21.3 -1011.5 8.1 D -103.5 L -22.3 A-6 Ex -14.1 3.7 51.8 47.7 -246.9 Ey 0.8 -68.2 281.0 -891.0 13.4 Note: Units are kips and feet. Load Ex is for loads applied toward the east, including appropriately amplified counter-clockwise accidental torsion. Load Ey is for loads applied toward the north, including appropriately amplified clockwise accidental torsion.

5-12

Chapter 5: Foundation Analysis and Design Section 6.2.3.5 of this volume of design examples outlines the design load combinations, which include the redundancy factor as appropriate. A large number of load cases result from considering two senses of accidental torsion for loading in each direction and including orthogonal effects . The detailed calculations presented here are limited to two primary conditions, both for a combined foundation for columns at Grids A-5 and A-6: the downward case (1.4D + 0.5L + 0.3Ex + 1.0Ey) and the upward case (0.7D + 0.3Ex + 1.0Ey). Before loads can be computed, attention must be given to Standard Section 12.13.4. That Section states that “overturning effects at the soil-foundation interface are permitted to be reduced by 25 percent” where the ELF procedure is used and by 10 percent where modal response spectrum analysis is used. Because the overturning effect in question relates to the global overturning moment for the system, judgment must be used in determining which design actions may be reduced. If the seismic force-resisting system consists of isolated shear walls, the shear wall overturning moment at the base best fits that description. For a perimeter moment-resisting frame, most of the global overturning resistance is related to axial loads in columns. Therefore, in this example column axial loads (Fz) from load cases Ex and Ey are multiplied by 0.75 and all other load effects remain unreduced. 5.1.3.2 Downward Case (1.4D + 0.5L + 0.3Ex + 1.0Ey). In order to perform the overturning checks, a footing size must be assumed. Preliminary checks (not shown here) confirmed that isolated footings under single columns were untenable. Check overturning for a footing that is 9 feet wide by 40 feet long by 5 feet thick. Furthermore, assume that the top of the footing is 2 feet below grade (the overlying soil contributes to the resisting moment). (In these calculations the 0.2SDSD modifier for vertical accelerations is used for the dead loads applied to the foundation but not for the weight of the foundation and soil. This is the author’s interpretation of the Standard. The footing and soil overburden are not subject to the same potential for dynamic amplification as the dead load of the superstructure and it is not common practice to include the vertical acceleration on the weight of the footing and the overburden. Furthermore, for footings that resist significant overturning, this issue makes a significant difference in design.) Combining the loads from columns at Grids A-5 and A-6 and including the weight of the foundation and overlying soil produces the following loads at the foundation-soil interface: P = applied loads + weight of foundation and soil = 1.4(-203.8 - 103.5) + 0.5(-43.8 - 22.3) +0.75[0.3(3.8 + 51.8) + 1.0(-21.3 + 281)] - 1.2[9(40)(5)(0.15) + 9(40)(2)(0.125)] = -688 kips. Mxx = direct moments + moment due to eccentricity of applied axial loads = 0.3(53.6 + 47.7) + 1.0(-1011.5 - 891.0) + [1.4(-203.8) + 0.5(-43.8) + 0.75(0.3)(3.8) + 0.75(1.0)(-21.3)](12.5) + [1.4(-103.5) + 0.5(-22.3) + 0.75(0.3)(51.8) + 0.75(1.0)(281)](-12.5) = -6,717 ft-kips. Myy = 0.3(-243.1 - 246.9) + 1.0(8.1 + 13.4) = -126 ft-kips. (The resulting eccentricity is small enough to neglect here, which simplifies the problem considerably.) Vx = 0.3(-13.8 - 14.1) + 1.0(0.5 + 0.8) = -7.11 kips. Vy = 0.3(4.6 + 3.7) + 1.0(-85.1 -68.2) = -149.2 kips.

5-13

FEMA P-751, NEHRP Recommended Provisions: Design Examples Note that the above load combination does not yield the maximum downward load. Reversing the direction of the seismic load results in P = -1,103 kips and Mxx = 2,964 ft-kips. This larger axial load does not control the design because the moment is so much less that the resultant is within the kern and no uplift occurs. The following soil calculations use a different sign convention than that in the analysis results noted above; compression is positive for the soil calculations. The eccentricity is as follows: e = |M/P| = 6,717/688 = 9.76 ft Figure 5.1-3 shows the elastic and plastic design conditions and their corresponding equations. Where e is less than L/2, a solution to the overturning problem exists; however, as e approaches L/2, the bearing pressures increase without bound. Since e is greater than L/6 = 40/6 = 6.67 feet, uplift occurs and the maximum bearing pressure is:

qmax =

2P 2(688) = = 4.98 ksf ⎛ L ⎞ ⎛ 40 ⎞ 3B ⎜ − e ⎟ 3(9) ⎜ − 9.76 ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠

and the length of the footing in contact with the soil is:

⎛ L ⎞ ⎛ 40 ⎞ Lʹ′ = 3 ⎜ − e ⎟ = 3 ⎜ − 9.76 ⎟ = 30.7 ft ⎝ 2 ⎠ ⎝ 2 ⎠ The bearing capacity qc = 3,000B' = 3,000 × min(B, L'/2) = 3,000 × min(9, 30.7/2) = 27,000 psf = 27 ksf. (L'/2 is used as an adjustment to account for the gradient in the bearing pressure in that dimension.) The design bearing capacity φqc = 0.7(27 ksf) = 18.9 ksf > 4.98 ksf

The foundation satisfies overturning and bearing capacity checks. The upward case, which follows, will control the sliding check. 5.1.3.3 Upward Case (0.7D + 0.3Ex + 1.0Ey). For the upward case the loads are: P = -332 kips Mxx = -5,712 ft-kips Myy = -126 ft-kips (negligible) Vx = -7.1 kips Vy = -149 kips The eccentricity is: e = |M/P| = 5,712/332 = 17.2 feet

5-14

Chapter 5: Foundation Analysis and Design Again, e is greater than L/6, so uplift occurs and the maximum bearing pressure is:

qmax =

2(332) = 8.82 ksf ⎛ 40 ⎞ 3(10) ⎜ − 17.2 ⎟ ⎝ 2 ⎠

and the length of the footing in contact with the soil is:

⎛ 40 ⎞ Lʹ′ = 3 ⎜ − 17.2 ⎟ = 8.4 ft ⎝ 2 ⎠ The bearing capacity qc = 3,000 × min(9, 8.4/2) = 12,500 psf = 12.5 ksf. The design bearing capacity φqc = 0.7(12.5 ksf) = 8.78 ksf < 8.82 ksf.

Using an elastic distribution of soil pressures, the foundation fails the bearing capacity check (although stability is satisfied). Try the plastic distribution. Using this approach, the bearing pressure over the entire contact area is assumed to be equal to the design bearing capacity. In order to satisfy vertical equilibrium, the contact area times the design bearing capacity must equal the applied vertical load P. Because the bearing capacity used in this example is a function of the contact area and the value of P changes with the size, the most convenient calculation is iterative. By iteration, the length of contact area is L' = 4.19 feet. The bearing capacity qc = 3,000 × min(10, 4.19) = 12,570 psf = 12.57 ksf. (No adjustment to L' is needed as the pressure is uniform.) The design bearing capacity φqc = 0.7(12.6 ksf) = 8.80 ksf. (8.80)(4.19)(9) = 332 kips = 332 kips, so equilibrium is satisfied. The resisting moment, MR = P (L/2-L'/2) = 33 (40/2 - 4.19/2) = 5,944 ft-kip > 5,712 ft-kip.

Therefore, using a plastic distribution of soil pressures, the foundation satisfies overturning and bearing capacity checks. The calculation of demands on concrete sections for strength checks should use the same soil stress distribution as the overturning check. Using a plastic distribution of soil stresses defines the upper limit of static loads for which the foundation remains stable, but the extreme concentration of soil bearing tends to drive up shear and flexural demands on the concrete section. It should be noted that the foundation may remain stable for larger loads if they are applied dynamically; even in that case, the strength demands on the concrete section will not exceed those computed on the basis of the plastic distribution. For the sliding check, initially consider base traction only. The sliding demand is:

V = Vx2 + Vy2 = (−7.11)2 + (−149.2)2 = 149.4 kips As calculated previously, the total compression force at the bottom of the foundation is 332 kips. The design sliding resistance is: 5-15

FEMA P-751, NEHRP Recommended Provisions: Design Examples

φVc = φ × friction coefficient × P = 0.7(0.65)(332 kips) = 151 kips > 149.4 kips

If base traction alone had been insufficient, resistance due to passive pressure on the leading face could be included. Section 5.2.2.2 below illustrates passive pressure calculations for a pile cap. 5.1.3.4 Design Results. The calculations performed in Sections 5.1.3.2 and 5.1.3.3 are repeated for combined footings at middle and side locations. Figure 5.1-6 shows the results.

Corner: 9'x40'x5'-0" w/ top of footing 2'-0" below grade

Middle: 5'x30'x4'-0"

Side: 8'x32'x4'-0"

Figure 5.1-6 Foundation plan for moment-resisting frame system One last check of interest is to compare the flexural stiffness of the footing with that of the steel column, which is needed because the steel frame design was based upon flexural restraint at the base of the columns. Using an effective moment of inertia of 50 percent of the gross moment of inertia and also using the distance between columns as the effective span, the ratio of EI/L for the smallest of the combined footings is more than five times the EI/h for the steel column. This is satisfactory for the design assumption.

5.1.4

Design for Concentrically Braced Frame System

Framing Alternate B in Section 6.2 of this volume of design examples employs a concentrically braced frame system at a central core to provide resistance to seismic loads. A framing plan for the system is shown in Figure 5.1-7.

5-16

Chapter 5: Foundation Analysis and Design

Figure 5.1-7 Framing plan for concentrically braced frame system 5.1.4.1 Check Mat Size for Overturning. Uplift demands at individual columns are so large that the only practical shallow foundation is one that ties together the entire core. The controlling load combination for overturning has minimum vertical loads (which help to resist overturning), primary overturning effects (Mxx) due to loads applied parallel to the short side of the core and smaller moments about a perpendicular axis (Myy) due to orthogonal effects. Assume mat dimensions of 45 feet by 95 feet by 7 feet thick, with the top of the mat 3'-6" below grade. Combining the factored loads applied to the mat by all eight columns and including the weight of the foundation and overlying soil produces the following loads at the foundation-soil interface: §

P = -7,849 kips

§

Mxx = -148,439 ft-kips

§

Myy = -42,544 ft-kips

§

Vx = -765 kips

§

Vy = -2,670 kips

Figure 5.1-8 shows the soil pressures that result from application in this controlling case, depending on the soil distribution assumed. In both cases the computed uplift is significant. In Part a of the figure, the contact area is shaded. The elastic solution shown in Part b was computed by modeling the mat in SAP2000 with compression only soil springs (with the stiffness of edge springs doubled as recommended by Bowles). For the elastic solution, the average width of the contact area is 11.1 feet and the maximum soil pressure is 16.9 ksf. The bearing capacity qc = 3,000 × min(95, 11.1/2) = 16,650 psf = 16.7 ksf. The design bearing capacity φqc = 0.7(16.7 ksf) = 11.7 ksf < 16.9 ksf.

5-17

FEMA P-751, NEHRP Recommended Provisions: Design Examples

12.2 ksf

~ (a) Plastic solution

16 12 8 4 0

(b) Elastic solution pressures (ksf)

Figure 5.1-8 Soil pressures for controlling bidirectional case As was done in Section 5.1.3.3 above, try the plastic distribution. The present solution has an additional complication as the off-axis moment is not negligible. The bearing pressure over the entire contact area is assumed to be equal to the design bearing capacity. In order to satisfy vertical equilibrium, the contact area times the design bearing capacity must equal the applied vertical load P. The shape of the contact area is determined by satisfying equilibrium for the off-axis moment. Again the calculations are iterative. Given the above constraints, the contact area shown in Figure 5.1-8 is determined. The length of the contact area is 4.13 feet at the left side and 8.43 feet at the right side. The average contact length, for use in determining the bearing capacity, is (4.13 + 8.43)/2 = 6.27 feet. The distances from the center of the mat to the centroid of the contact area are as follows:

x = 5.42 ft y = 19.24 ft The bearing capacity is qc = 3,000 × min(95, 6.27) = 18,810 psf = 18.81 ksf. The design bearing capacity is φqc = 0.7(18.8 ksf) = 13.2 ksf. (13.2)(6.27)(95) = 7,863 kips ≈ 7,849 kips, confirming equilibrium for vertical loads. (7,849)(5.42) = 42,542 ft-kips ≈ 42,544 ft-kips, confirming equilibrium for off-axis moment. The resisting moment, M R, xx = P y = 7,849(19.24) = 151,015ft-kips >148,439 ft-kips.

5-18

Chapter 5: Foundation Analysis and Design So, the checks of stability and bearing capacity are satisfied. The mat dimensions are shown in Figure 5.1-9.

Mat: 45'x95'x7'-0" with top of mat 3'-6" below grade

Figure 5.1-9 Foundation plan for concentrically braced frame system 5.1.4.2 Design Mat for Strength Demands. As was previously discussed, the computation of strength demands for the concrete section should use the same soil pressure distribution as was used to satisfy stability and bearing capacity. Because dozens of load combinations were considered and hand calculations were used for the plastic distribution checks, the effort required would be considerable. The same analysis used to determine elastic bearing pressures yields the corresponding section demands directly. One approach to this dilemma would be to compute an additional factor that must be applied to selected elastic cases to produce section demands that are consistent with the plastic solution. Rather than provide such calculations here, design of the concrete section will proceed using the results of the elastic analysis. This is conservative for the demand on the concrete for the same reason that it was unsatisfactory for the soil: the edge soil pressures are high (that is, we are designing the concrete for a peak soil pressure of 16.9 ksf, even though the plastic solution gives 13.2 ksf). Standard Section 12.13.3 requires consideration of parametric variation for soil properties where foundations are modeled explicitly. This example does not illustrate such calculations. Concrete mats often have multiple layers of reinforcement in each direction at the top and bottom of their thickness. Use of a uniform spacing for the reinforcement provided in a given direction greatly increases the ease of construction. The minimum reinforcement requirements defined in Section 10.5 of ACI 318 were discussed in Section 5.1.1.3 above. Although all of the reinforcement provided to satisfy 5-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples Section 7.12 of ACI 318 may be provided near one face, for thick mats it is best to compute and provide the amount of required reinforcement separately for the top and bottom halves of the section. Using a bar spacing of 10 inches for this 7-foot-thick mat and assuming one or two layers of bars, the section capacities indicated in Table 5.1-4 (presented in order of decreasing strength) may be precomputed for use in design. The amount of reinforcement provided for Marks B, C and D are less than the basic minimum for flexural members, so the demands should not exceed three-quarters of the design strength where those reinforcement patterns are used. The amount of steel provided for Mark D is the minimum that satisfies ACI 318 Section 7.12. Table 5.1-4 Mat Foundation Section Capacities Mark A B C D

As (in.2 per ft)

φMn (ft-kip/ft)

3/4φMn (ft-kip/ft)

2 layers of #10 bars at 10 in. o.c. 2 layers of #9 bars at 10 in. o.c. 2 layers of #8 bars at 10 in. o.c.

3.05

1,012

Not used

2.40

Not used

601

1.90

Not used

477

#8 bars at 10 in. o.c.

0.95

Not used

254

Reinforcement

Note: Where the area of steel provided is less than the minimum reinforcement for flexural members as indicated in ACI 318 Sec. 10.5.1, demands are compared to 3/4 of φMn as permitted in Sec. 10.5.3. To facilitate rapid design, the analysis results are processed in two additional ways. First, the flexural and shear demands computed for the various load combinations are enveloped. Then the enveloped results are presented (see Figure 5.1-10) using contours that correspond to the capacities shown for the reinforcement patterns noted in Table 5.1-4.

5-20

Chapter 5: Foundation Analysis and Design

CL B

B C

C C

+ 881

B + 669

(a) M x positive

(b) M x negative

D C + 884

484

C C

D (c) M y positive

(d) M y negative

Figure 5.1-10 Envelope of mat foundation flexural demands Using the noted contours permits direct selection of reinforcement. The reinforcement provided within a contour for a given mark must be that indicated for the next higher mark. For instance, all areas within Contour B must have two layers of #10 bars. Note that the reinforcement provided will be symmetric about the centerline of the mat in both directions. Where the results of finite element analysis are used in the design of reinforced concrete elements, averaging of demands over short areas is appropriate. In Figure 5.1-11, the selected reinforcement is superimposed on the demand contours. Figure 5.1-12 shows a section of the mat along Gridline C.

5-21

FEMA P-751, NEHRP Recommended Provisions: Design Examples

4'-2"

CL B

4'-2"

(a) E-W bottom reinforcement

8'-4"

(b) E-W top reinforcement

10'-0"

3'-4"

5'-0"

7'-6"

2'-6" CL

(d) N-S top reinforcement

Figure 5.1-11 Mat foundation flexural reinforcement

5-22

4'-2"

Chapter 5: Foundation Analysis and Design

3" clear (typical)

Figure 5.1-12 Section of mat foundation Figure 5.1-13 presents the envelope of shear demands. The contours used correspond to the design strengths computed assuming Vs = 0 for one-way and two-way shear. In the hatched areas the shear stress exceeds φ 4 f cʹ′ and in the shaded areas it exceeds φ 2 f cʹ′ . The critical sections for two-way shear (as discussed in Section 5.1.1.3) also are shown. The only areas that need more careful attention (to determine whether they require shear reinforcement) are those where the hatched or shaded areas are outside the critical sections. At the columns on Gridline D, the hatched area falls outside the critical section, so closer inspection is needed. Because the perimeter of the hatched area is substantially smaller than the perimeter of the critical section for punching shear, the design requirements of ACI 318 are satisfied. One-way shears at the edges of the mat exceed the φ 2 f cʹ′ criterion. Note that the high shear stresses are not produced by loads that create high bearing pressures at the edge. Rather, they are produced by loads that create large bending stresses parallel to the edge. The distribution of bending moments and shears is not uniform across the width (or breadth) of the mat, primarily due to the torsion in the seismic loads and the orthogonal combination. It is also influenced by the doubled spring stiffnesses used to model the soil condition. However, when the shears are averaged over a width equal to the effective depth (d), the demands are less than the design strength. In this design, reinforcement for punching or beam shear is not required. If shear reinforcement cannot be avoided, standee bars may be used both to chair the upper decks of reinforcement and to provide resistance to shear in which case they may be bent thus:

5-23

FEMA P-751, NEHRP Recommended Provisions: Design Examples

(a) V x Critical section (typical)

(b) V y

Figure 5.1-13 Critical sections for shear and envelope of mat foundation shear demands

5.1.5

Cost Comparison

Table 5.1-5 provides a summary of the material quantities used for all of the foundations required for the various conditions considered. Corresponding preliminary costs are assigned. The gravity-only condition does not represent a realistic case because design for wind loads would require changes to the foundations; it is provided here for discussion. It is obvious that design for lateral loads adds cost as compared to a design that neglects such loads. However, it is also worth noting that braced frame systems usually have substantially more expensive foundation systems than do moment frame systems. This condition occurs for two reasons. First, braced frame systems are stiffer, which produces shorter periods

5-24

Chapter 5: Foundation Analysis and Design and higher design forces. Second, braced frame systems tend to concentrate spatially the demands on the foundations. In this case the added cost amounts to approximately $0.80/ft2, which is an increase of perhaps 4 or 5 percent to the cost of the structural system. Table 5.1-5 Summary of Material Quantities and Cost Comparison Design Condition

Concrete at Gravity Foundations

Concrete at Lateral Foundations

310 cy at $350/cy Gravity only (see Figure 5.1-4) = $108,600

Total Excavation

Total Cost

310 cy at $30/cy = $9,300

$117,900

233 cy at $350/cy Moment frame (see Figure 5.1-6) = $81,600

507 cy at $400/cy = $202,900

770 cy at $30/cy = $23,100

$307,600

Braced frame 233 cy at $350/cy (see Figure 5.1-9) = $81,600

1,108 cy at $400/cy = $443,300

1895 cy at $30/cy = $56,800

$581,700

5.2

DEEP FOUNDATIONS FOR A 12-‐STORY BUILDING, SEISMIC DESIGN CATEGORY D

This example features the analysis and design of deep foundations for a 12-story reinforced concrete moment-resisting frame building similar to that described in Chapter 7 of this volume of design examples.

5.2.1

Basic Information

5.2.1.1 Description. Figure 5.2-1 shows the basic design condition considered in this example. A 2×2 pile group is designed for four conditions: for loads delivered by a corner and a side column of a moment-resisting frame system for Site Classes C and E. Geotechnical parameters for the two sites are given in Table 5.2-1.

Figure 5.2-1 Design condition: Column of concrete moment-resisting frame supported by pile cap and cast-in-place piles

5-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Table 5.2-1 Geotechnical Parameters Depth

0 to 3 feet

Class E Site

Class C Site

Loose sand/fill

γ = 110 pcf Angle of internal friction = 28 degrees Soil modulus parameter, k = 25 pci

γ = 110 pcf Angle of internal friction = 30 degrees Soil modulus parameter, k = 50 pci

Neglect skin friction Neglect end bearing

Soft clay

3 to 30 feet

γ = 110 pcf Undrained shear strength = 430 psf Soil modulus parameter, k = 25 pci Strain at 50 percent of maximum stress, ε50 = 0.01 Skin friction (ksf) = 0.3 Neglect end bearing Medium dense sand

30 to 100 feet

γ = 120 pcf Angle of internal friction = 36 degrees Soil modulus parameter, k = 50 pci

Dense sand (one layer: 3- to 100-foot depth)

γ = 130 pcf Angle of internal friction = 42 degrees Soil modulus parameter, k = 125 pci Skin friction (ksf)* = 0.3 + 0.03/ft ≤ 2 End bearing (ksf)* = 65 + 0.6/ft ≤ 150

Skin friction (ksf)* = 0.9 + 0.025/ft ≤ 2 End bearing (ksf)* = 40 + 0.5/ft ≤ 100 Pile cap resistance

300 pcf, ultimate passive pressure

575 pcf, ultimate passive pressure

Resistance factor, φ

0.8 for vertical, lateral and rocking resistance of cohesive soil

0.7 for vertical, lateral and rocking resistance of cohesionless soil

Safety factor for settlement

2.5

*Skin friction and end bearing values increase (up to the maximum value noted) for each additional foot of depth below the top of the layer. (The values noted assume a minimum pile length of 20 ft.) The structural material properties assumed for this example are as follows: §

f'c = 3,000 psi

§

fy = 60,000 psi

5-26

Chapter 5: Foundation Analysis and Design 5.2.1.2 Seismic Parameters. §

Site Class = C and E (both conditions considered in this example)

§

SDS = 1.1

§

Seismic Design Category = D (for both conditions)

5.2.1.3 Demands. The unfactored demands from the moment frame system are shown in Table 5.2-2. Table 5.2-2 Gravity and Seismic Demands Location

Corner

Side

Load

-460.0

-77.0

Mxx

Myy

55.5

0.6

193.2

4.3

624.8

0.4

16.5

307.5

189.8

3.5

ATx

1.4

3.1

26.7

34.1

15.7

ATy

4.2

9.4

77.0

103.5

47.8

-702.0

-72.0

72.2

0.0

723.8

0.0

13.9

181.6

161.2

1.2

ATx

0.4

1.8

2.9

18.1

4.2

ATy

1.2

5.3

8.3

54.9

12.6

Note: Units are kips and feet. Load Vy is for loads applied toward the east. ATx is the corresponding accidental torsion case. Load Vx is for loads applied toward the north. ATy is the corresponding accidental torsion case. Using Load Combinations 5 and 7 from Section 12.4.2.3 of the Standard (with 0.2SDSD = 0.22D and taking ρ = 1.0), considering orthogonal effects as required for Seismic Design Category D and including accidental torsion, the following 32 load conditions must be considered. 1.42D + 0.5L ± 1.0Vx ± 0.3Vy ± max(1.0ATx, 0.3ATy) 1.42D + 0.5L ± 0.3Vx ± 1.0Vy ± max(0.3ATx, 1.0ATy) 0.68D ± 1.0Vx ± 0.3Vy ± max(1.0ATx, 0.3ATy) 0.68D ± 0.3Vx ± 1.0Vy ± max(0.3ATx, 1.0ATy) 5.2.1.4 Design Approach. For typical deep foundation systems, resistance to lateral loads is provided by both the piles and the pile cap. Figure 5.2-2 shows a simple idealization of this condition. The relative contributions of these piles and pile cap depend on the particular design conditions, but often both effects

5-27

FEMA P-751, NEHRP Recommended Provisions: Design Examples are significant. Resistance to vertical loads is assumed to be provided by the piles alone regardless of whether their axial capacity is primarily due to end bearing, skin friction, or both. Although the behavior of foundation and superstructure are closely related, they typically are modeled independently. Earthquake loads are applied to a model of the superstructure, which is assumed to have fixed supports. Then the support reactions are seen as demands on the foundation system. A similar substructure technique is usually applied to the foundation system itself, whereby the behavior of pile cap and piles are considered separately. This section describes that typical approach.

Passive resistance (see Figure 5.2-5)

Pile cap

Pile p-y springs (see Figure 5.2-4)

Figure 5.2-2 Schematic model of deep foundation system 5.2.1.4.1 Pile Group Mechanics. With reference to the free body diagram (of a 2×2 pile group) shown in Figure 5.2-3, demands on individual piles as a result of loads applied to the group may be determined as follows:

Vgroup − V passive 4

and M = V × ℓ, where ℓ is a characteristic length determined from analysis of a

laterally loaded single pile.

Pot =

Vgroup h + M group + 4M − hpV passive 2s

, where s is the pile spacing, h is the height of the pile cap

and hp is the height of Vpassive above Point O.

Pp =

5-28

Pgroup 4

and P = Pot + Pp

Chapter 5: Foundation Analysis and Design

P group

P group M group

M group

V group

V passive

V M Pp

M Pot

Pot

Figure 5.2-3 Pile cap free body diagram 5.2.1.4.2 Contribution of Piles. The response of individual piles to lateral loads is highly nonlinear. In recent years it has become increasingly common to consider that nonlinearity directly. Based on extensive testing of full-scale specimens and small-scale models for a wide variety of soil conditions, researchers have developed empirical relationships for the nonlinear p-y response of piles that are suitable for use in design. Representative p-y curves (computed for a 22-inch-diameter pile) are shown in Figure 5.2-4. The stiffness of the soil changes by an order of magnitude for the expected range of displacements (the vertical axis uses a logarithmic scale). The p-y response is sensitive to pile size (an effect not apparent in the figure, which is based on a single pile size); soil type and properties; and, in the case of sands, vertical stress, which increases with depth. Pile response to lateral loads, like the p-y curves on which the calculations are based, is usually computed using computer programs like LPILE.

5-29

FEMA P-751, NEHRP Recommended Provisions: Design Examples

100,000

Soil resistance, p (lb/in.)

10,000

1,000

100 Site Class C, depth = 30 ft

Site Class C, depth = 10 ft Site Class E, depth = 30 ft Site Class E, depth = 10 ft

1 0.0

0.1

0.2

0.3

0.4 0.5 0.6 Pile deflection, y (in.)

0.7

0.8

0.9

1.0

Figure 5.2-4 Representative p-y curves (note that a logarithmic scale is used on the vertical axis) 5.2.1.4.3 Contribution of Pile Cap. Pile caps contribute to the lateral resistance of a pile group in two important ways: directly as a result of passive pressure on the face of the cap that is being pushed into the soil mass and indirectly by producing a fixed head condition for the piles, which can significantly reduce displacements for a given applied lateral load. Like the p-y response of piles, the passive pressure resistance of the cap is nonlinear. Figure 5.2-5 shows how the passive pressure resistance (expressed as a fraction of the ultimate passive pressure) is related to the imposed displacement (expressed as a fraction of the minimum dimension of the face being pushed into the soil mass).

5-30

Chapter 5: Foundation Analysis and Design

1.0 0.9 0.8 0.7

P/Pult

0.6 0.5 0.4 0.3 0.2 0.1 0.0

0.01

0.02

0.03

0.04

0.05

0.06

δ /H

Figure 5.2-5 Passive pressure mobilization curve (after ASCE 41) 5.2.1.4.4 Group Effect Factors. The response of a group of piles to lateral loading will differ from that of a single pile due to pile-soil-pile interaction. (Group effect factors for axial loading of very closely spaced piles may also be developed but are beyond the scope of the present discussion.) Full-size and model tests show that the lateral capacity of a pile in a pile group versus that of a single pile (termed “efficiency”) is reduced as the pile spacing is reduced. The observed group effects are associated with shadowing effects. Various researchers have found that leading piles are loaded more heavily than trailing piles when all piles are loaded to the same deflection. The lateral resistance is primarily a function of row location within the group, rather than pile location within a row. Researchers recommend that these effects may be approximated by adjusting the resistance value on the single pile p-y curves (that is, by applying a p-multiplier). Based on full-scale testing and subsequent analysis, Rollins et al. recommend the following pmultipliers (fm), where D is the pile diameter or width and s is the center-to-center spacing between rows of piles in the direction of loading.

( D ) + 0.5 ≤ 1.0

First (leading) row piles:

f m = 0.26ln s

Second row piles:

f m = 0.52ln s

Third or higher row piles:

f m = 0.60ln s

( D ) ≤ 1.0 ( D ) − 0.25 ≤ 1.0

5-31

FEMA P-751, NEHRP Recommended Provisions: Design Examples Because the direction of loading varies during an earthquake and the overall efficiency of the group is the primary point of interest, the average efficiency factor is commonly used for all members of a group in the analysis of any given member. In that case, the average p-reduction factor is as follows:

fm =

1 n ∑ f mi n i =1 3

For a 2×2 pile group thus 4

1 2 with s = 3D, the group effect factor is calculated as follows:

For piles 1 and 2, in the leading row, f m = 0.26ln (3) + 0.5 = 0.79 . For piles 3 and 4, in the second row, f m = 0.52ln (3) = 0.57 . So, the group effect factor (average p-multiplier) is f m =

0.79 + 0.79 + 0.57 + 0.57 = 0.68 . 4

Figure 5.2-6 shows the group effect factors that are calculated for pile groups of various sizes with piles at several different spacings.

1.0

)r eli ipt l u m -p e ga re va ( r ot ca f tc ef fe p u or G

s = 7D

s = 5D

0.8

0.6

s = 3D

0.4 s = 2D 0.2

0.0 1

Pile group size (number of rows) Figure 5.2-6 Calculated group effect factors

5-32

Chapter 5: Foundation Analysis and Design

5.2.2

Pile Analysis, Design and Detailing

5.2.2.1 Pile Analysis. For this design example, it is assumed that all piles will be fixed-head, 22-inchdiameter, cast-in-place piles arranged in 2×2 pile groups with piles spaced at 66 inches center-to-center. The computer program LPILE Plus 5.0 is used to analyze single piles for both soil conditions shown in Table 5.2-1 assuming a length of 50 feet. Pile flexural stiffness is modeled using one-half of the gross moment of inertia because of expected flexural cracking. The response to lateral loads is affected to some degree by the coincident axial load. The full range of expected axial loads was considered in developing this example, but in this case the lateral displacements, moments and shears were not strongly affected; the plots in this section are for zero axial load. A p-multiplier of 0.68 for group effects (as computed at the end of Section 5.2.1.4) is used in all cases. Figures 5.2-7, 5.2-8 and 5.2-9 show the variation of shear, moment and displacement with depth (within the top 30 feet) for an applied lateral load of 15 kips on a single pile with the group reduction factor. It is apparent that the extension of piles to depths beyond 30 feet for the Class E site (or approximately 25 feet for the Class C site) does not provide additional resistance to lateral loading; piles shorter than those lengths would have reduced lateral resistance. The trends in the figures are those that should be expected. The shear and displacement are maxima at the pile head. Because a fixed-head condition is assumed, moments are also largest at the top of the pile. Moments and displacements are larger for the soft soil condition than for the firm soil condition.

Depth (ft)

25 Site Class C Site Class E 30 -5

5 Shear, V (kip)

Figure 5.2-7 Results of pile analysis-shear versus depth (applied lateral load is 15 kips)

5-33

FEMA P-751, NEHRP Recommended Provisions: Design Examples 0

Depth (ft)

Site Class C Site Class E

30 -1000

-500 0 Moment, M (in.-kips)

500

Figure 5.2-8 Results of pile analysis-moment versus depth (applied lateral load is 15 kips)

Depth (ft)

Site Class C Site Class E

30 -0.1

0.0

0.1

0.2

0.3

Displacement (in.)

Figure 5.2-9 Results of pile analysis-displacement versus depth (applied lateral load is 15 kips) 5-34

Chapter 5: Foundation Analysis and Design

The analyses performed to develop Figures 5.2-7 through 5.2-9 are repeated for different levels of applied lateral load. Figures 5.2-10 and 5.2-11 show how the moment and displacement at the head of the pile are related to the applied lateral load. It may be seen from Figure 5.2-10 that the head moment is related to the applied lateral load in a nearly linear manner; this is a key observation. Based on the results shown, the slope of the line may be taken as a characteristic length that relates head moment to applied load. Doing so produces the following: §

ℓ = 46 in. for the Class C site

§

ℓ = 70 in. for the Class E site

Head moment, M (in.-kip)

1600

1200

800

400

Site Class C Site Class E

Applied lateral load, V (kip)

Figure 5.2-10 Results of pile analysis – applied lateral load versus head moment

5-35

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Applied lateral load, V (kip)

30 25 20 15 10 Site Class C

Site Class E 0 0.0

0.2

0.4 0.6 Head displacement, Δ (inch)

0.8

Figure 5.2-11 Results of pile analysis – head displacement versus applied lateral load A similar examination of Figure 5.2-11 leads to another meaningful insight. The load-displacement response of the pile in Site Class C soil is essentially linear. The response of the pile in Site Class E soil is somewhat nonlinear, but for most of the range of response a linear approximation is reasonable (and useful). Thus, the effective stiffness of each individual pile is: §

k = 175 kip/in. for the Class C site

§

k = 40 kip/in. for the Class E site

5.2.2.2 Pile Group Analysis. The combined response of the piles and pile cap and the resulting strength demands for piles are computed using the procedure outlined in Section 5.2.1.4 for each of the 32 load combinations discussed in Section 5.2.1.3. Assume that each 2×2 pile group has a 9'-2" × 9'-2" × 4'-0" thick pile cap that is placed 1'-6" below grade. Check the Maximum Compression Case under a Side Column in Site Class C Using the sign convention shown in Figure 5.2-3, the demands on the group are as follows: §

P = 1,224 kip

§

Myy = 222 ft-kips

§

Vx = 20 kips

§

Myy = 732 ft-kips

§

Vy = 73 kips

From preliminary checks, assume that the displacements in the x and y directions are sufficient to mobilize 30 percent and 35 percent, respectively, of the ultimate passive pressure:

5-36

Chapter 5: Foundation Analysis and Design

⎛ 18 48 ⎞ ⎛ 48 ⎞⎛ 110 ⎞ 1 Vpassive, x = 0.30(575) ⎜ + ⎟ ⎜ ⎟⎜ ⎟ ( 1000 ) = 22.1 kips ⎝ 12 2(12) ⎠ ⎝ 12 ⎠⎝ 12 ⎠ and

⎛ 18 48 ⎞ ⎛ 48 ⎞⎛ 110 ⎞ 1 Vpassive, y = 0.35(575) ⎜ + ⎟ ⎜ ⎟⎜ ⎟ ( 1000 ) = 25.8 kips ⎝ 12 2(12) ⎠ ⎝ 12 ⎠⎝ 12 ⎠ and conservatively take hp = h/3 = 16 inches. Since Vpassive,x > Vx, passive resistance alone is sufficient for this case in the x direction. However, in order to illustrate the full complexity of the calculations, reduce Vpassive,x to 4 kips and assign a shear of 4.0 kips to each pile in the x direction. In the y direction, the shear in each pile is as follows:

73 − 25.8 = 11.8kips 4

The corresponding pile moments are: M = 4.0(46) = 186 in.-kips for x-direction loading and M = 11.8(46) = 543 in.-kips for y-direction loading The maximum axial load due to overturning for x-direction loading is:

Pot =

20(48) + 222(12) + 4(184) − 16(4) = 32.5kips 2(66)

and for y-direction loading (determined similarly), Pot = 106.4 kips. The axial load due to direct loading is Pp = 1224/4 = 306 kips. Therefore, the maximum load effects on the most heavily loaded pile are the following: Pu = 32.5 + 106.4 + 306 = 445 kips

M u = (184)2 + (543) 2 = 573in.-kips The expected displacement in the y direction is computed as follows:

δ = V/k = 11.8/175 = 0.067 in., which is 0.14 percent of the pile cap height (h) Reading Figure 5.2-5 with δ/H = 0.0014, P/Pult ≈ 0.34, so the assumption that 35 percent of Pult would be mobilized was reasonable.

5-37

FEMA P-751, NEHRP Recommended Provisions: Design Examples 5.2.2.3 Design of Pile Section. The calculations shown in Section 5.2.2.2 are repeated for each of the 32 load combinations under each of the four design conditions. The results are shown in Figures 5.2-12 and 5.2-13. In these figures, circles indicate demands on piles under side columns and squares indicate demands on piles under corner columns. Also plotted are the φP-φM design strengths for the 22-inchdiameter pile sections with various amounts of reinforcement (as noted in the legends). The appropriate reinforcement pattern for each design condition may be selected by noting the innermost capacity curve that envelops the corresponding demand points. The required reinforcement is summarized in Table 5.24, following calculation of the required pile length.

800

8-#7 8-#6

700

6-#6

600

6-#5 Side

500

Corner

400 )p i k ( 300 P , d a lo la 200 i x A 100 0 0

500

1000

1500

-100

2500 Moment, M (in.-kip)

-200 -300

Figure 5.2-12 P-M interaction diagram for Site Class C

5-38

2000

Chapter 5: Foundation Analysis and Design 800

8-#7 8-#6

700

6-#6

600

6-#5 Side

500

Corner

400 )p i k ( 300 P , d a lo la 200 i x A 100 0 0

500

1000

1500

2000

-100

2500 Moment, M (in.-kip)

-200 -300

Figure 5.2-13 P-M interaction diagram for Site Class E 5.2.2.4 Pile Length for Axial Loads. For the calculations that follow, recall that skin friction and end bearing are neglected for the top 3 feet in this example. The design is based on having 1’-6” of soil over a 4’-0” deep pile cap. 5.2.2.4.1 Length for Settlement. Service loads per pile are calculated as P = (PD + PL)/4. Check the pile group under the side column in Site Class C, assuming L = 52.5 feet – 5.5 feet = 47 feet: P = (752 + 114)/4 = 217 kips. Pskin = average friction capacity × pile perimeter × pile length for friction = 0.5[0.3 + 2.5(0.03) + 0.3 + 49.5(0.03)]π(22/12)(44) = 292 kips Pend = end bearing capacity at depth × end bearing area = [65 + 49.5(0.6)](π/4)(22/12)2 = 250 kips Pallow = (Pskin + Pend)/S.F. = (292 + 250)/2.5 = 217 kips = 217 kips (demand)

Check the pile group under the corner column in Site Class E, assuming L = 49 feet: P = (460 + 77)/4 = 134 kips

5-39

FEMA P-751, NEHRP Recommended Provisions: Design Examples Pskin = [friction capacity in first layer + average friction capacity in second layer] × pile perimeter = [24.5(0.3) + (24.5/2)(0.9 + 0.9 + 24.5[0.025])]π(22/12) = 212 kips Pend = [40 + 24.5(0.5)](π/4)(22/12)2 = 138 kips Pallow = (212 + 138)/2.5 = 140 kips > 134 kips

5.2.2.4.2 Length for Compression Capacity. All of the strength-level load combinations (discussed in Section 5.2.1.3) must be considered. Check the pile group under the side column in Site Class C, assuming L = 49 feet: As seen in Figure 5.1-12, the maximum compression demand for this condition is Pu = 394 kips. Pskin = 0.5[0.3 + 0.3 + 47(0.03)]π(22/12)(47) = 272 kips Pend = [65 + 47(0.6)](π/4)(22/12)2 = 246 kips

φPn = φ(Pskin + Pend) = 0.75(272 + 246) = 389 kips ≈ 390 kips

Check the pile group under the corner column in Site Class E, assuming L = 64 feet: As seen in Figure 5.2-13, the maximum compression demand for this condition is Pu = 340 kips. Pskin = [27(0.3) + (34/2)(0.9 + 0.9 + 34[0.025])]π(22/12) = 306 kips Pend = [40 + 34(0.5)](π/4)(22/12)2 = 150 kips

φPn = φ(Pskin + Pend) = 0.75(306 + 150) = 342 kips > 340 kips

5.2.2.4.3 Length for Uplift Capacity. Again, all of the strength-level load combinations (discussed in Section 5.2.1.3) must be considered. Check the pile group under side column in Site Class C, assuming L = 5 feet: As seen in Figure 5.2-12, the maximum tension demand for this condition is Pu = -1.9 kips. Pskin = 0.5[0.3 + 0.3 + 2(0.03)]π(22/12)(2) = 3.8 kips

φPn = φ(Pskin) = 0.75(3.8) = 2.9 kips > 1.9 kips

Check the pile group under the corner column in Site Class E, assuming L = 52 feet: As seen in Figure 5.2-13, the maximum tension demand for this condition is Pu = -144 kips. Pskin = [27(0.3) + (22/2)(0.9 + 0.9 + 22[0.025])]π(22/12) = 196 kips

φPn = φ(Pskin) = 0.75(196) = 147 kips > 144 kips

5-40

Chapter 5: Foundation Analysis and Design 5.2.2.4.4 Graphical Method of Selecting Pile Length. In the calculations shown above, the adequacy of the soil-pile interface to resist applied loads is checked once a pile length is assumed. It would be possible to generate mathematical expressions of pile capacity as a function of pile length and then solve such expressions for the demand conditions. However, a more practical design approach is to precalculate the capacity for piles for the full range of practical lengths and then select the length needed to satisfy the demands. This method lends itself to graphical expression as shown in Figures 5.2-14 and 5.215.

0 Compression

Tension

20 t)f 30 ( h t p e 40 d el i P 50 60 70 80 0

100

200

300

400

500

600

700

Design resistance (kip) Figure 5.2-14 Pile axial capacity as a function of length for Site Class C

5-41

FEMA P-751, NEHRP Recommended Provisions: Design Examples

0 Compression

Tension

20 )t 30 f( h t p e40 d el i P 50 60 70 80 0

100

200

300

400

500

600

700

800

Design resistance (kip) Figure 5.2-15 Pile axial capacity as a function of length for Site Class E 5.2.2.4.5 Results of Pile Length Calculations. Detailed calculations for the required pile lengths are provided above for two of the design conditions. Table 5.2-3 summarizes the lengths required to satisfy strength and serviceability requirements for all four design conditions. Table 5.2-3 Pile Lengths Required for Axial Loads Piles Under Corner Column Site Class Site Class C

Site Class E

5-42

Piles Under Side Column

Condition

Load

Min Length

Condition

Load

Min Length

Compression

369 kip

46 ft

Compression

394 kip

49 ft

Uplift

108 kip

32 ft

Uplift

13.9 kip

8 ft

Settlement

134 kip

27 ft

Settlement

217 kip

47 ft

Compression

378 kip

61 ft

Compression

406 kip

64 ft

Uplift

119 kip

42 ft

Uplift

23.6 kip

17 ft

Settlement

134 kip

48 ft

Settlement

217 kip

67 ft

Chapter 5: Foundation Analysis and Design 5.2.2.5 Design Results. The design results for all four pile conditions are shown in Table 5.2-4. The amount of longitudinal reinforcement indicated in the table is that required at the pile-pile cap interface and may be reduced at depth as discussed in the following section. Table 5.2-4 Summary of Pile Size, Length and Longitudinal Reinforcement Site Class Site Class C Site Class E

Piles Under Corner Column

Piles Under Side Column

22 in. diameter by 46 ft long

22 in. diameter by 49 ft long

8-#6 bars

6-#5 bars

22 in. diameter by 61 ft long

22 in. diameter by 67 ft long

8-#7 bars

6-#6 bars

5.2.2.6 Pile Detailing. Standard Sections 12.13.5, 12.13.6, 14.2.3.1 and 14.2.3.2 contain special pile requirements for structures assigned to Seismic Design Category C or higher and D or higher. In this section, those general requirements and the specific requirements for uncased concrete piles that apply to this example are discussed. Although the specifics are affected by the soil properties and assigned site class, the detailing of the piles designed in this example focuses on consideration of the following fundamental items: §

All pile reinforcement must be developed in the pile cap (Standard Sec. 12.13.6.5).

§

In areas of the pile where yielding might be expected or demands are large, longitudinal and transverse reinforcement must satisfy specific requirements related to minimum amount and maximum spacing.

§

Continuous longitudinal reinforcement must be provided over the entire length resisting design tension forces (ACI 318 Sec. 21.12.4.2).

The discussion that follows refers to the detailing shown in Figures 5.2-16 and 5.2-17. 5.2.2.6.1 Development at the Pile Cap. Where neither uplift nor flexural restraint are required, the development length is the full development length for compression. Where the design relies on head fixity or where resistance to uplift forces is required (both of which are true in this example), pile reinforcement must be fully developed in tension unless the section satisfies the overstrength load condition or demands are limited by the uplift capacity of the soil-pile interface (Standard Sec. 12.13.6.5). For both site classes considered in this example, the pile longitudinal reinforcement is extended straight into the pile cap a distance that is sufficient to fully develop the tensile capacity of the bars. In addition to satisfying the requirements of the Standard, this approach offers two advantages. By avoiding lap splices to field-placed dowels where yielding is expected near the pile head (although such would be permitted by the Standard), more desirable inelastic performance would be expected. Straight development, while it may require a thicker pile cap, permits easier placement of the pile cap’s bottom reinforcement followed by the addition of the spiral reinforcement within the pile cap. Note that embedment of the entire pile in the pile cap facilitates direct transfer of shear from pile cap to pile but is not a requirement of the Standard. (Section 1810.3.11 of the 2009 International Building Code requires that piles be embedded at least 3 inches into pile caps.)

5-43

FEMA P-751, NEHRP Recommended Provisions: Design Examples

4" pile embedment

(6) #5

6'-4"

#4 spiral at 4.5 inch pitch

Section A A (6) #5

23'-0"

#4 spiral at 9 inch pitch

Section B

(4) #5

21'-0"

#4 spiral at 9 inch pitch

Section C

Figure 5.2-16 Pile detailing for Site Class C (under side column)

5-44

Chapter 5: Foundation Analysis and Design

4" pile embedment

(8) #7

12'-4"

#5 spiral at 3.5 inch pitch

Section A A (6) #7

20'-0"

#5 spiral at 3.5 inch pitch

Section B B (4) #7

#4 spiral at 9 inch pitch 32'-0"

Section C C

Figure 5.2-17 Pile detailing for Site Class E (under corner column)

5-45

FEMA P-751, NEHRP Recommended Provisions: Design Examples

5.2.2.6.2 Longitudinal and Transverse Reinforcement Where Demands Are Large. Requirements for longitudinal and transverse reinforcement apply over the entire length of pile where demands are large. For uncased concrete piles in Seismic Design Category D, at least four longitudinal bars (with a minimum reinforcement ratio of 0.005) must be provided over the largest region defined as follows: the top one-half of the pile length, the top 10 feet below the ground, or the flexural length of the pile. The flexural length is taken as the length of pile from the cap to the lowest point where 0.4 times the concrete section cracking moment (see ACI 318 Section 9.5.2.3) exceeds the calculated flexural demand at that point. For the piles used in this example, one-half of the pile length governs. (Note that “providing” a given reinforcement ratio means that the reinforcement in question must be developed at that point. Bar development and cutoff are discussed in more detail in Chapter 7 of this volume of design examples.) Transverse reinforcement must be provided over the same length for which minimum longitudinal reinforcement requirements apply. Because the piles designed in this example are larger than 20 inches in diameter, the transverse reinforcement may not be smaller than 0.5 inch diameter. For the piles shown in Figures 5.2-16 and 5.2-17, the spacing of the transverse reinforcement in the top half of the pile length may not exceed the least of the following: 12db (7.5 in. for #5 longitudinal bars and 10.5 in. for #7 longitudinal bars), 22/2 = 11 in., or 12 in. Where yielding may be expected, even more stringent detailing is required. For the Class C site, yielding can be expected within three diameters of the bottom of the pile cap (3D = 3 × 22 = 66 in.). Spiral reinforcement in that region must not be less than one-half of that required in Section 21.4.4.1(a) of ACI 318 (since the site is not Class E, Class F, or liquefiable) and the requirements of Sections 21.4.4.2 and 21.4.4.3 must be satisfied. Note that Section 21.4.4.1(a) refers to Equation 10-5, which often will govern. In this case, the minimum volumetric ratio of spiral reinforcement is one-half that determined using ACI 318 Equation 10-5. In order to provide a reinforcement ratio of 0.01 for this pile section, a #4 spiral must have a pitch of no more than 4.8 inches, but the maximum spacing permitted by Section 21.4.4.2 is 22/4 = 5.5 inches or 6db = 3.75 inches, so a #4 spiral at 3.75-inch pitch is used. (Section 1810.3.2.1.2 of the 2009 International Building Code clarifies that ACI 318 Equation 10-5 need not be applied to piles.) For the Class E site, the more stringent detailing must be provided “within seven diameters of the pile cap and of the interfaces between strata that are hard or stiff and strata that are liquefiable or are composed of soft to medium-stiff clay” (Standard Sec. 14.2.3.2.1). The author interprets “within seven diameters of ... the interface” as applying in the direction into the softer material, which is consistent with the expected location of yielding. Using that interpretation, the Standard does not indicate the extent of such detailing into the firmer material. Taking into account the soil layering shown in Table 5.2-1 and the pile cap depth and thickness, the tightly spaced transverse reinforcement shown in Figure 5.2-17 is provided within 7D of the bottom of pile cap and top of firm soil and is extended a little more than 3D into the firm soil. Because the site is Class E, the full amount of reinforcement indicated in ACI 318 Section 21.6.4 must be provided. In order to provide a reinforcement ratio of 0.02 for this pile section, a #5 spiral must have a pitch of no more than 3.7 inches. The maximum spacing permitted by Section 21.6.4.3 is 22/4 = 5.5 inches or 6db = 5.25 inches, so a #5 spiral at 3.5-inch pitch is used. 5.2.2.6.3 Continuous Longitudinal Reinforcement for Tension. Table 5.2-3 shows the pile lengths required for resistance to uplift demands. For the Site Class E condition under a corner column (Figure 5.2-17), longitudinal reinforcement must resist tension for at least the top 42 feet (being developed at that point). Extending four longitudinal bars for the full length and providing widely spaced spirals at such bars is practical for placement, but it is not a specific requirement of the Standard. For the Site Class C condition under a side column (Figure5.2-16), design tension due to uplift extends only approximately 5 feet below the bottom of the pile cap. Therefore, a design with Section C of 5-46

Chapter 5: Foundation Analysis and Design Figure 5.2-16 being unreinforced would satisfy the Provisions requirements, but the author has decided to extend very light longitudinal and nominal transverse reinforcement for the full length of the pile. 5.2.3

Other Considerations

5.2.3.1 Foundation Tie Design and Detailing. Standard Section 12.13.5.2 requires that individual pile caps be connected by ties. Such ties are often grade beams, but the Standard would permit use of a slab (thickened or not) or calculations that demonstrate that the site soils (assigned to Site Class A, B, or C) provide equivalent restraint. For this example, a tie beam between the pile caps under a corner column and a side column is designed. The resulting section is shown in Figure 5.2-18. For pile caps with an assumed center-to-center spacing of 32 feet in each direction and given Pgroup = 1,224 kips under a side column and Pgroup = 1,142 kips under a corner column, the tie is designed as follows. As indicated in Standard Section 12.13.5.2, the minimum tie force in tension or compression equals the product of the larger column load times SDS divided by 10 = 1224(1.1)/10 = 135 kips. The design strength for six #6 bars is as follows

φAs fy = 0.9(6)(0.44)(60) = 143 kips > 135 kips

According to ACI 318 Section 21.12.3.2, the smallest cross-sectional dimension of the tie beam must not be less than the clear spacing between pile caps divided by 20 = (32'-0" - 9'-2")/20 = 13.7 inches. Use a tie beam that is 14 inches wide and 16 inches deep. ACI 318 Section 21.12.3.2 further indicates that closed ties must be provided at a spacing of not more than one-half the minimum dimension, which is 14/2 = 7 inches. Assuming that the surrounding soil provides restraint against buckling, the design strength of the tie beam concentrically loaded in compression is as follows:

φPn = 0.8φ[0.85f'c(Ag - Ast) + fyAst] = 0.8(0.65)[0.85(3){(16)(14) – 6(0.44)}+ 60(6)(0.44)] = 376 kips > 135 kips

2" clear at sides

(3) #6 top bars

#4 ties at 7" o.c.

(3) #6 bottom bars 3" clear at top and bottom

Figure 5.2-18 Foundation tie section

5-47

FEMA P-751, NEHRP Recommended Provisions: Design Examples

5.2.3.2 Liquefaction. For Seismic Design Categories C, D, E and F, Standard Section 11.8.2 requires that the geotechnical report address potential hazards due to liquefaction. For Seismic Design Categories D, E and F, Standard Section 11.8.3 further requires that the geotechnical report describe the likelihood and potential consequences of liquefaction and soil strength loss (including estimates of differential settlement, lateral movement, lateral loads on foundations, reduction in foundation soilbearing capacity, increases in lateral pressures on retaining walls and flotation of buried structures) and discuss mitigation measures. During the design of the structure, such measures (which can include ground stabilization, selection of appropriate foundation type and depths and selection of appropriate structural systems to accommodate anticipated displacements and forces) must be considered. Provisions Part 3, Resource Paper 12 contains a calculation procedure that can be used to evaluate the liquefaction hazard. 5.2.3.3 Kinematic Interaction. Piles are subjected to curvature demands as a result of two different types of behavior: inertial interaction and kinematic interaction. The term inertial interaction is used to describe the coupled response of the soil-foundation-structure system that arises as a consequence of the mass properties of those components of the overall system. The structural engineer’s consideration of inertial interaction is usually focused on how the structure loads the foundation and how such loads are transmitted to the soil (as shown in the pile design calculations that are the subject of most of this example) but also includes assessment of the resulting foundation movement. The term kinematic interaction is used to describe the manner in which the stiffness of the foundation system impedes development of free-field ground motion. Consideration of kinematic interaction by the structural engineer is usually focused on assessing the strength and ductility demands imposed directly on piles by movement of the soil. Although it is rarely done in practice, Standard Section 12.13.6.3 requires consideration of kinematic interaction for foundations of structures assigned to Seismic Design Category D, E, or F. Kramer discusses kinematic and inertial interaction and the methods of analysis employed in consideration of those effects and demonstrates “that the solution to the entire soil-structure interaction problem is equal to the sum of the solutions of the kinematic and inertial interaction analyses.” One approach that would satisfy the requirements of the Standard would be as follows: §

The geotechnical consultant performs appropriate kinematic interaction analyses considering free-field ground motions and the stiffness of the piles to be used in design.

§

The resulting pile demands, which generally are greatest at the interface between stiff and soft strata, are reported to the structural engineer.

§

The structural engineer designs piles for the sum of the demands imposed by the vibrating superstructure and the demands imposed by soil movement.

A more practical, but less rigorous, approach is to provide appropriate detailing in regions of the pile where curvature demands imposed directly by earthquake ground motions are expected to be significant. Where such a judgment-based approach is used, one must decide whether to provide only additional transverse reinforcement in areas of concern to improve ductility or whether additional longitudinal reinforcement should also be provided to increase strength. Section 18.10.2.4.1 of the 2009 International Building Code permits application of such deemed-to-comply detailing in lieu of explicit calculations and prescribes a minimum longitudinal reinforcement ratio of 0.005.

5-48

Chapter 5: Foundation Analysis and Design 5.2.3.4 Design of Pile Cap. Design of pile caps for large pile loads is a very specialized topic for which detailed treatment is beyond the scope of this volume of design examples. CRSI notes that “most pile caps are designed in practice by various short-cut rule-of-thumb procedures using what are hoped to be conservative allowable stresses.” Wang & Salmon indicates that “pile caps frequently must be designed for shear considering the member as a deep beam. In other words, when piles are located inside the critical sections d (for one-way action) or d/2 (for two-way action) from the face of column, the shear cannot be neglected.” They go on to note that “there is no agreement about the proper procedure to use.” Direct application of the special provisions for deep flexural members as found in ACI 318 is not possible since the design conditions are somewhat different. CRSI provides a detailed outline of a design procedure and tabulated solutions, but the procedure is developed for pile caps subjected to concentric vertical loads only (without applied overturning moments or pile head moments). Strut-and-tie models (as described in Appendix A of ACI 318) may be employed, but their application to elements with important three-dimensional characteristics (such as pile caps for groups larger than 2×1) is so involved as to preclude hand calculations. 5.2.3.5 Foundation Flexibility and Its Impact on Performance 5.2.3.5.1 Discussion. Most engineers routinely use fixed-base models. Nothing in the Provisions or Standard prohibits that common practice; the consideration of foundation flexibility and of soil-structure interaction effects (Standard Section 12.13.3 and Chapter 19) is “permitted” but not required. Such fixed-base models can lead to erroneous results, but engineers have long assumed that the errors are usually conservative. There are two obvious exceptions to that assumption: soft soil site-resonance conditions (e.g., as in the 1985 Mexico City earthquake) and excessive damage or even instability due to increased displacement response. Site resonance can result in significant amplification of ground motion in the period range of interest. For sites with a fairly long predominant period, the result is spectral accelerations that increase as the structural period approaches the site period. However, the shape of the general design spectrum used in the Standard does not capture that effect; for periods larger than T0, accelerations remain the same or decrease with increasing period. Therefore, increased system period (as a result of foundation flexibility) always leads to lower design forces where the general design spectrum is used. Site-specific spectra may reflect long-period site-resonance effects, but the use of such spectra is required only for Class F sites. Clearly, an increase in displacements, caused by foundation flexibility, does change the performance of a structure and its contents—raising concerns regarding both stability and damage. Earthquake-induced instability of buildings has been exceedingly rare. The analysis and acceptance criteria in the Standard are not adequate to the task of predicting real stability problems; calculations based on linear, static behavior cannot be used to predict instability of an inelastic system subjected to dynamic loading. While Provisions Part 2 Section 12.12 indicates that structural stability was considered in arriving at the “consensus judgment” reflected in the drift limits, such considerations were qualitative. In point of fact, the values selected for the drift limits were selected considering damage to nonstructural systems (and, perhaps in some cases, control of structural ductility demands). For most buildings, application of the Standard is intended to satisfy performance objectives related to life safety and collapse prevention, not damage control or post-earthquake occupancy. Larger design forces and more stringent drift limits are applied to structures assigned to Occupancy Category III or IV in the hope that those measures will improve performance without requiring explicit consideration of such performance. Although foundation flexibility can affect structural performance significantly, since all consideration of performance in the context of the Standard is approximate and judgment-based, it is difficult to define how such changes in performance should be characterized. Explicit consideration of performance measures also tends to increase engineering effort substantially, so mandatory performance checks often are resisted by the user community. 5-49

FEMA P-751, NEHRP Recommended Provisions: Design Examples

The engineering framework established in ASCE 41 is more conducive to explicit use of performance measures. In that document (Sections 4.4.3.2.1 and 4.4.3.3.1), the use of fixed-based structural models is prohibited for “buildings being rehabilitated for the Immediate Occupancy Performance Level that are sensitive to base rotations or other types of foundation movement.” In this case the focus is on damage control rather than structural stability. 5.2.3.5.2 Example Calculations. To assess the significance of foundation flexibility, one may compare the dynamic characteristics of a fixed-base model to those of a model in which foundation effects are included. The effects of foundation flexibility become more pronounced as foundation period and structural period approach the same value. For this portion of the example, use the Site Class E pile design results from Section 5.2.2.1 and consider the north-south response of the concrete moment frame building located in Berkeley (Section 7.2) as representative for this building. 5.2.3.5.2.1 Stiffness of the Structure. Calculations of the effect of foundation flexibility on the dynamic response of a structure should reflect the overall stiffness of the structure (e.g., that associated with the fundamental mode of vibration) rather than the stiffness of any particular story. Table 7-2 shows that the total weight of the structure is 43,919 kips. Table 7-3 shows that the calculated period of the fixed-base structure is 2.02 seconds and Table 7-7 indicates that 83.6 percent of the mass participates in that mode. Using the equation for the undamped period of vibration of a single-degree-of-freedom oscillator, the effective stiffness of the structure is as follows:

4π 2 M 4π = T2

((0.836)43,919 386.1) 2.022

= 920 kip/in.

5.2.3.5.2.2 Foundation Stiffness. As seen in Figure 7-1, there are 36 moment frame columns. Assume that a 2×2 pile group supports each column. As shown in Section 5.2.2.1, the stiffness of each pile is 40 kip/in. Neglecting both the stiffness contribution from passive pressure resistance and the flexibility of the beam-slab system that ties the pile caps, the stiffness of each pile group is 4 × 40 = 160 kip/in. and the stiffness of the entire foundation system is 36 × 160 = 5,760 kip/in. 5.2.3.5.2.3 Effect of Foundation Flexibility. Because the foundation stiffness is much greater than the structural stiffness, period elongation is expected to be minimal. To confirm this expectation, the period of the combined system is computed. The total stiffness for the system (springs in series) is as follows:

K combined =

K structure

1 + K fdn

1 = 793 kip/in. 1 1 + 920 5760

Assume that the weight of the foundation system is 4,000 kips and that 100 percent of the corresponding mass participates in the new fundamental mode of vibration. The period of the combined system is as follows:

T = 2π

M = 2π K

[(0.836)(43,919) + (1.0)(4000)] 386.1 = 2.29 sec 793

which is an increase of 13 percent over that predicted by the fixed-base model. For systems responding in the constant-velocity portion of the spectrum, accelerations (and thus forces) are a function of 1/T and relative displacements are a function of T. Therefore, with respect to the fixed-based model, the 5-50

Chapter 5: Foundation Analysis and Design combined system would have forces that are 12 percent smaller and displacements that are 13 percent larger. In the context of earthquake engineering, those differences are not significant.

5-51

6 Structural Steel Design Rafael Sabelli, S.E. and Brian Dean, P.E. Originally developed by James R. Harris, P.E., PhD, Frederick R. Rutz, P.E., PhD and Teymour Manouri, P.E., PhD

Contents 6.1

INDUSTRIAL HIGH-CLEARANCE BUILDING, ASTORIA, OREGON ................................. 3

6.1.1

Building Description .............................................................................................................. 3

6.1.2

Design Parameters ................................................................................................................. 6

6.1.3

Structural Design Criteria ...................................................................................................... 7

6.1.4

Analysis ............................................................................................................................... 10

6.1.5

Proportioning and Details .................................................................................................... 16

6.2

SEVEN-STORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA ................................ 40

6.2.1

Building Description ............................................................................................................ 40

6.2.2

Basic Requirements ............................................................................................................. 42

6.2.3

Structural Design Criteria .................................................................................................... 44

6.2.4

Analysis and Design of Alternative A: SMF ....................................................................... 46

6.2.5

Analysis and Design of Alternative B: SCBF ..................................................................... 60

6.2.6

Cost Comparison ................................................................................................................. 72

6.3

TEN-STORY HOSPITAL, SEATTLE, WASHINGTON .......................................................... 72

6.3.1

Building Description ............................................................................................................ 72

6.3.2

Basic Requirements ............................................................................................................. 76

6.3.3

Structural Design Criteria .................................................................................................... 78

6.3.4

Elastic Analysis ................................................................................................................... 80

6.3.5

Initial Proportioning and Details ......................................................................................... 86

6.3.6

Nonlinear Response History Analysis ................................................................................. 93

FEMA P-751, NEHRP Recommended Provisions: Design Examples

This chapter illustrates how the 2009 NEHRP Recommended Provisions (the Provisions) is applied to the design of steel framed buildings. The following three examples are presented: 1. An industrial warehouse structure in Astoria, Oregon 2. A multistory office building in Los Angeles, California 3. A mid-rise hospital in Seattle, Washington The discussion examines the following types of structural framing for resisting horizontal forces: §

Ordinary concentrically braced frames (OCBF)

§

Special concentrically braced frames

§

Intermediate moment frames

§

Special moment frames

§

Buckling-restrained braced frames, with moment-resisting beam-column connections

The examples cover design for seismic forces in combination with gravity they are presented to illustrate only specific aspects of seismic analysis and design—such as lateral force analysis, design of concentric and eccentric bracing, design of moment resisting frames, drift calculations, member proportioning detailing. All structures are analyzed using three-dimensional static or dynamic methods. ETABS (Computers & Structures, Inc., Berkeley, California, v.9.5.0, 2008) is used in Examples 6.1 and 6.2. In addition to the 2009 NEHRP Recommended Provisions, the following documents are referenced:

6-2

§

AISC 341

American Institute of Steel Construction. 2005. Seismic Provisions for Structural Steel Buildings, including Supplement No. 1.

§

AISC 358

American Institute of Steel Construction. 2005. Prequalified Connections for Special and Intermediate Steel Moment Frames for Seismic Applications.

§

AISC 360

American Institute of Steel Construction. 2005. Specification for Structural Steel Buildings.

§

AISC Manual

American Institute of Steel Construction. 2005. Manual of Steel Construction, 13th Edition.

§

AISC SDM

American Institute of Steel Construction. 2006. Seismic Design Manual.

§

IBC

International Code Council, Inc. 2006. 2006 International Building Code.

§

AISC SDGS-4

AISC Steel Design Guide Series 4. Second Edition. 2003. Extended EndPlate Moment Connections, 2003.

Chapter 6: Structural Steel Design §

SDI

Luttrell, Larry D. 1981. Steel Deck Institute Diaphragm Design Manual. Steel Deck Institute.

The symbols used in this chapter are from Chapter 11 of the Standard, the above referenced documents, or are as defined in the text. U.S. Customary units are used. 6.1

INDUSTRIAL HIGH-‐CLEARANCE BUILDING, ASTORIA, OREGON

This example utilizes a transverse intermediate steel moment frame and a longitudinal ordinary concentric steel braced frame. The following features of seismic design of steel buildings are illustrated: §

Seismic design parameters

§

Equivalent lateral force analysis

§

Three-dimensional analysis

§

Drift check

§

Check of compactness and spacing for moment frame bracing

§

Moment frame connection design

§

Proportioning of concentric diagonal bracing

6.1.1

Building Description

This building has plan dimensions of 180 feet by 90 feet and a clear height of approximately 30 feet. It includes a 12-foot-high, 40-foot-wide mezzanine area at the east end of the building. The structure consists of 10 gable frames spanning 90 feet in the transverse (north-south) direction. Spaced at 20 feet on center, these frames are braced in the longitudinal (east-west) direction in two bays at the east end. The building is enclosed by nonstructural insulated concrete wall panels and is roofed with steel decking covered with insulation and roofing. Columns are supported on spread footings. The elevation and transverse sections of the structure are shown in Figure 6.1-1. Longitudinal struts at the eaves and at the mezzanine level run the full length of the building and therefore act as collectors for the distribution of forces resisted by the diagonally braced bays and as weak-axis stability bracing for the moment frame columns. The roof and mezzanine framing plans are shown in Figure 6.1-2. The framing consists of a steel roof deck supported by joists between transverse gable frames. The mezzanine represents both an additional load and additional strength and stiffness. Because all the frames resist lateral loading, the steel deck functions as a diaphragm for distribution of the effects of eccentric loading caused by the mezzanine floor when the building is subjected to loads acting in the transverse direction. The mezzanine floor at the east end of the building is designed to accommodate a live load of 125 psf. Its structural system is composed of a concrete slab over steel decking supported by floor beams spaced at 10 feet on center. The floor beams are supported on girders continuous over two intermediate columns spaced approximately 30 feet apart and are attached to the gable frames at each end.

6-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples The member sizes in the main frame are controlled by serviceability considerations. Vertical deflections due to snow were limited to 3.5 inches and lateral sway due to wind was limited to 2 inches.

Ridge

West

East

Eave strut

35'-0"

Eave

Siding: 6" concrete insulated sandwich panels.

Collector

3'-9"

(a)

30'-6"

Roof moment-resisting steel frame. Concrete slab on grade

Ceiling

34'-3"

9'-0"

3'-0"

Mezzanine moment-resisting steel frame

32'-0"

(b)

(c)

Figure 6.1-1 Framing elevation and sections (1.0 ft = 0.3048 m; 1.0 in. = 25.4 mm)

Earthquake rather than wind governs the lateral design due to the mass of the insulated concrete panels. The panels are attached with long pins perpendicular to the concrete surface. These slender, flexible pins isolate the panels from acting as shear walls. The building is supported on spread footings based on moderately deep alluvial deposits (i.e., medium dense sands). The foundation plan is shown in Figure 6.1-3. Transverse ties are placed between the footings of the two columns of each moment frame to provide restraint against horizontal thrust from the moment frames. Grade beams carrying the enclosing panels serve as ties in the longitudinal direction as well as across the end walls. The design of footings and columns in the braced bays requires consideration of combined seismic loadings. The design of foundations is not included here. 6-4

Chapter 6: Structural Steel Design

182'-0" Mezzanine

11 2" type "B" 22 gage metal deck 1200 MJ12 C-joist

W21x62

90'-0"

3" embossed 20 gage deck

W14x43

Figure 6.1-2 Roof framing and mezzanine framing plan (1.0 ft = 0.3048 m; 1.0 in. = 25.4 mm)

6-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples

9 bays at 20'-0"=180'-0" 40'-0"

20'-0" (typical)

Mezzanine

11 4" dia. tie rod (or equal) at each frame. Embed in thickened slab

2 30'-0"

90'-0"

Building is symmetrical about center line

30'-0"

6" concrete slab with 6x6-W1.4x W1.4 wwf over 6" gravel Typical 3'-4"x 3'-4"x1'-0" footings

Mezzanine 6'-6"x6'-8"x 1'-4" footings

30'-0"

3 Mezzanine 5'-6"x5'-6"x 1'-4" footings

Figure 6.1-3 Foundation plan

(1.0 ft = 0.3048 m; 1.0 in. = 25.4 mm)

6.1.2

Design Parameters

6.1.2.1 Ground motion and system parameters. See Section 3.2 for an example illustrating the determination of design ground motion parameters. For this example the parameters are as follows. §

SDS = 1.0

§

SD1 = 0.6

§

Occupancy Category II

§

Seismic Design Category D

Note that Standard Section 12.2.5.6 permits an ordinary steel moment frame for buildings that do not exceed one story and 65 feet tall with a roof dead load not exceeding 20 psf. Intermediate steel moment frames with stiffened bolted end plates and ordinary steel concentrically braced frames are used in this example. §

North-south (N-S) direction: Moment-resisting frame system = intermediate steel moment frame (Standard Table 12.2-1) R = 4.5

6-6

Chapter 6: Structural Steel Design

Ω0 = 3 Cd = 4 §

East-west (E-W) direction: Braced frame system = ordinary steel concentrically braced frame (Standard Table 12.2-1) R = 3.25 Ω0 = 2 Cd = 3.25

6.1.2.2 Loads §

Roof live load (L), snow = 25 psf

§

Roof dead load (D) = 15 psf

§

Mezzanine live load, storage = 125 psf

§

Mezzanine slab and deck dead load = 69 psf

§

Weight of wall panels = 75 psf

Roof dead load includes roofing, insulation, metal roof deck, purlins, mechanical and electrical equipment that portion of the main frames that is tributary to the roof under lateral load. For determination of the seismic weights, the weight of the mezzanine will include the dead load plus 25 percent of the storage load (125 psf) in accordance with Standard Section 12.7.2. Therefore, the mezzanine seismic weight is 69 + 0.25(125) = 100 psf. 6.1.2.3 Materials §

Concrete for footings: fc' = 2.5 ksi

§

Slabs-on-grade: fc' = 4.5 ksi

§

Mezzanine concrete on metal deck: fc' = 3.0 ksi

§

Reinforcing bars: ASTM A615, Grade 60

§

Structural steel (wide flange sections): ASTM A992, Grade 50

§

Plates (except continuity plates): ASTM A36

§

Bolts: ASTM A325

§

Continuity Plates: ASTM A572, Grade 50

6.1.3

Structural Design Criteria

6.1.3.1 Building configuration. Because there is a mezzanine at one end, vertical weight irregularities might be considered to apply (Standard Sec. 12.3.2.2). However, the upper level is a roof and the

6-7

FEMA P-751, NEHRP Recommended Provisions: Design Examples Standard exempts roofs from weight irregularities. There also are no plan irregularities in this building (Standard Sec. 12.3.2.1). 6.1.3.2 Redundancy. In the N-S direction, the moment frames do not meet the requirements of Standard Section 12.3.4.2b since the frames are only one bay long. Thus, Standard Section 12.3.4.2a must be checked. A copy of the three-dimensional model is made, with the moment frame beam at Gridline A pinned. The structure is checked to make sure that an extreme torsional irregularity (Standard Table 12.3-1) does not occur:

1.4(

ΔK + Δ A ) ≥ ΔA 2

⎛ 4.17 in. + 6.1in. ⎞ 1.4 ⎜ ⎟ = 7.19 in. ≥ 6.1in. 2 ⎝ ⎠ where: ∆A = maximum displacement at knee along Gridline A, in. ∆K = maximum displacement at knee along gridline K, in. Thus, the structure does not have an extreme torsional irregularity when a frame loses moment resistance. Additionally, the structure must be checked in the N-S direction to ensure that the loss of moment resistance at Beam A has not resulted in more than a 33 percent reduction in story strength. This can be checked using elastic methods (based on first yield) as shown below, or using strength methods. The original model is run with the N-S load combinations to determine the member with the highest demandcapacity ratio. This demand-capacity ratio, along with the applied base shear, is used to calculate the base shear at first yield:

⎛ 1 Vyield = ⎜ ⎜ ( D / C ) max ⎝

⎞ ⎟Vbase ⎟ ⎠

⎛ 1 ⎞ Vyield = ⎜ ⎟ ( 223 kips ) = 250.5 kips ⎝ 0.89 ⎠ where: Vbase = base shear from Equivalent Lateral Force (ELF) analysis A similar analysis can be made using the model with no moment resistance at Frame A:

⎛ 1 ⎞ Vyield , MFremoved = ⎜ ⎟ ( 223 kips ) = 234.5 kips ⎝ 0.951 ⎠

Vyield , MFremoved Vyield 6-8

234.5 kips = 0.94 250.5 kips

Chapter 6: Structural Steel Design

Thus, the loss of resistance at both ends of a single beam only results in a 6 percent reduction in story strength. The moment frames can be assigned a value of ρ = 1.0. In the E-W direction, the OCBF system meets the prescriptive requirements of Standard Section 12.3.4.2a. As a result, no further calculations are needed and this system can be assigned a value of ρ = 1.0. 6.1.3.3 Orthogonal load effects. A combination of 100 percent seismic forces in one direction plus 30 percent seismic forces in the orthogonal direction must be applied to the columns of this structure in Seismic Design Category D (Standard Sec. 12.5.4). 6.1.3.4 Structural component load effects. The effect of seismic load (Standard Sec. 12.4.2) is:

E = ρ QE ± 0.2S DS D SDS = 1.0 for this example. The seismic load is combined with the gravity loads as shown in Standard Sec. 12.4.2.3, resulting in the following:

1.4 D + 1.0 L + 0.2S + ρ QE 0.7 D + ρ QE Note that 1.0L is for the storage load on the mezzanine; the coefficient on L is 0.5 for many common live loads. 6.1.3.5 Drift limits. For a building assigned to Occupancy Category II, the allowable story drift (Standard Table 12.12-1) is: §

Δa = 0.025hsx in the E-W direction

§

Δa/ρ = 0.025hsx/1.0 in the N-S direction

At the roof ridge, hsx = 34 ft-3 in. and Δ = 10.28 in. α

At the knee (column-roof intersection), hsx = 30 ft-6 in. and Δa = 9.15 in. At the mezzanine floor, hsx = 12 ft and Δa = 3.60 in. Footnote c in Standard Table 12.12-1 permits unlimited drift for single-story buildings with interior walls, partitions, etc., that have been designed to accommodate the story drifts. See Section 6.1.4.3 for further discussion. The main frame of the building can be considered to be a one-story building for this purpose, given that there are no interior partitions except below the mezzanine. (The definition of a story in building codes generally does not require that a mezzanine be considered a story unless its area exceeds one-third the area of the room or space in which it is placed; this mezzanine is less than one-third of the footprint of the building.) 6.1.3.6 Seismic weight. The weights that contribute to seismic forces are:

6-9

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Roof D = (0.015)(90)(180) = Panels at sides = (2)(0.075)(32)(180)/2 = Panels at ends = (2)(0.075)(35)(90)/2 = Mezzanine slab and 25% LL = Mezzanine framing = Main frames = Seismic weight =

E-W direction 243 kips 0 kips 224 kips 360 kips 35 kips 27 kips 889 kips

N-S direction 243 kips 437 kips 0 kips 360 kips 35 kips 27 kips 1,102 kips

The weight associated with the main frames accounts for only the main columns, because the weight associated with the remainder of the main frames is included in the roof dead load above. The computed seismic weight is based on the assumption that the wall panels offer no shear resistance for the structure but are self-supporting when the load is parallel to the wall of which the panels are a part. Additionally, snow load does not need to be included in the seismic weight per Standard Section 12.7.2 because it does not exceed 30 psf. 6.1.4

Analysis

Base shear will be determined using an ELF analysis. 6.1.4.1 Equivalent Lateral Force procedure. In the longitudinal direction where stiffness is provided only by the diagonal bracing, the approximate period is computed using Standard Equation 12.8-7:

(

)

Ta = Cr hnx = ( 0.02) 34.250.75 = 0.28 sec where hn is the height of the building, taken as 34.25 feet at the mid-height of the roof. In accordance with Standard Section 12.8.2, the computed period of the structure must not exceed the following:

Tmax = CuTa = (1.4 )( 0.28) = 0.39 sec The subsequent three-dimensional modal analysis finds the computed period to be 0.54 seconds. For purposes of determining the required base shear strength, Tmax will be used in accordance with the Standard; drift will be calculated using the period from the model. In the transverse direction where stiffness is provided by moment-resisting frames (Standard Eq. 12.8-7):

(

)

Ta = Cr hnx = ( 0.028) 34.250.8 = 0.47 sec and

Tmax = CuTa = (1.4 )( 0.47 ) = 0.66 sec Also note that the dynamic analysis finds a computed period of 1.03 seconds. As in the longitudinal direction, Tmax will be used for determining the required base shear strength. The seismic response coefficient (Cs) is computed in accordance with Standard Section 12.8.1.1. In the longitudinal direction:

6-10

Chapter 6: Structural Steel Design

CS =

S DS 1.0 = = 0.308 R / I 3.25 / 1.0

but need not exceed:

CS =

S D1 0.6 = = 0.473 ⎛ R ⎞ ⎛ 3.25 ⎞ T ⎜ ⎟ 0.39 ⎜ ⎟ ⎝ I ⎠ ⎝ 1.0 ⎠

Therefore, use Cs = 0.308 for the longitudinal direction. In the transverse direction:

CS =

S DS 1.0 = = 0.222 R / I 4.5 / 1

but need not exceed:

CS =

S DS 0.6 = = 0.202 T ( R / I ) ( 0.66 )( 4.5 / 1)

Therefore, use Cs = 0.202 for the transverse direction. In both directions the value of Cs exceeds the minimum value (Standard Eq. 12.8-5) computed as:

CS = 0.044ISDS ≥ 0.01 = ( 0.044 )(1)(1.0 ) = 0.044 The seismic base shear in the longitudinal direction (Standard Eq. 12.8-1) is:

V = CSW = ( 0.308)(889 kips ) = 274 kips The seismic base shear in the transverse direction is:

V = CSW = ( 0.202 )(1102 kips ) = 223 kips Standard Section 12.8.3 prescribes the vertical distribution of lateral force in a multilevel structure. Even though the building is considered to be one story for some purposes, it is clearly a two-level structure. Using the data in Section 6.1.3.6 of this example and interpolating the exponent k as 1.08 for the period of 0.66 second, the distribution of forces for the N-S analysis is shown in Table 6.1-1. Table 6.1-1 ELF Vertical Distribution for N-S Analysis Level Roof

Weight (wx) (kips) 707

Height (hx) (ft)

w xh x

32.375

30,231

Cvx =

wx hxk n

∑i=1wih

k x

0.84

Fx = CvxV (kips) 187

6-11

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 6.1-1 ELF Vertical Distribution for N-S Analysis Level Mezzanine Total

Weight (wx) (kips) 395

Height (hx) (ft) 12

1,102

w xh x

Cvx =

wx hxk n

∑i=1wihk x

5,782

Fx = CvxV (kips)

0.16

36,013

36 223

It is not immediately clear whether the roof (a 22-gauge steel deck with conventional roofing over it) will behave as a flexible, semi-rigid, or rigid diaphragm. For this example, a three-dimensional model was created in ETABS including frame and diaphragm stiffness. 6.1.4.2 Three-dimensional ELF analysis. The three-dimensional analysis is performed for this example to account for the following: §

The differing stiffness of the gable frames with and without the mezzanine level

§

The different centers of mass for the roof and the mezzanine

§

The flexibility of the roof deck

§

The significance of braced frames in controlling torsion due to N-S ground motions

The gabled moment frames, the tension bracing, the moment frames supporting the mezzanine and the diaphragm chord members are explicitly modeled using three-dimensional beam-column elements. The tapered members are approximated as short, discretized prismatic segments. Thus, combined axial bending checks are performed on a prismatic element, as required by AISC 360 Chapter H. The collector at the knee level is included, as are those at the mezzanine level in the two east bays. The mezzanine diaphragm is modeled using planar shell elements with their in-plane rigidity being based on actual properties and dimensions of the slab. The roof diaphragm also is modeled using planar shell elements, but their in-plane rigidity is based on a reduced thickness that accounts for compression buckling phenomena and for the fact that the edges of the roof diaphragm panels are not connected to the wall panels. SDI’s Diaphragm Design Manual is used for guidance in assessing the stiffness of the roof deck. The analytical model includes elements with one-tenth the stiffness of a plane plate of 22 gauge steel. The ELF analysis of the three-dimensional model in the transverse direction yields an important result: the roof diaphragm behaves as a rigid diaphragm. Accidental torsion is applied at the center of the roof as a moment whose magnitude is the roof lateral force multiplied by 5 percent of 180 feet (9 feet). A moment is also applied to the mezzanine level in a similar fashion. The resulting displacements are shown in Table 6.1-2. Table 6.1-2 ELF Analysis Displacements in N-S Direction

6-12

Grid

Roof Displacement (in.)

4.98

4.92

Chapter 6: Structural Steel Design Table 6.1-2 ELF Analysis Displacements in N-S Direction Grid

Roof Displacement (in.)

4.82

4.68

4.56

4.46

4.34

4.19

4.05

3.92

The average of the extreme displacements is 4.45 inches. The displacement at the centroid of the roof is 4.51 inches. Thus, the deviation of the diaphragm from a straight line is 0.06 inch, whereas the average frame displacement is approximately 75 times that. Clearly, then, the diaphragm flexibility is negligible and the deck behaves as a rigid diaphragm. The ratio of maximum to average displacement is 1.1, which does not exceed the 1.2 limit given in Standard Table 12.3-1 and torsional irregularity is not triggered. The same process needs to be repeated for the E-W direction. Table 6.1-3 ELF Analysis Displacements in N-S Direction Grid

Roof Displacement (in.)

0.88

2/3

0.82

0.75

The ratio of the maximum to average displacement is 1.07, well under the torsional irregularity threshold ratio of 1.2. The demands from the three-dimensional ELF analysis are combined to meet the orthogonal combination requirement of Standard Section 12.5.3 for the columns: §

E-W: (1.0)(E-W direction spectrum) + (0.3)(N-S direction spectrum)

§

N-S: (0.3)(E-W direction spectrum) + (1.0)(N-S direction spectrum)

6.1.4.3 Drift. The lateral deflection cited previously must be multiplied by Cd = 4 to find the transverse drift:

δx =

Cd δ e ( 4 )( 4.51) = = 18 in. 1.0 I 6-13

FEMA P-751, NEHRP Recommended Provisions: Design Examples

This exceeds the limit of 10.28 inches computed previously. However, there is no story drift limit for single-story structures with interior wall, partitions, ceilings and exterior wall systems that have been designed to accommodate the story drifts. Detailing for this type of design may be problematic. In the longitudinal direction, the lateral deflection is much smaller and is within the limits of Standard Section 12.12.1. The deflection computations do not include the redundancy factor. 6.1.4.4 P-delta effects. The P-delta effects on the structure may be neglected in analysis if the provisions of Standard Section 12.8.7 are followed. First, the stability coefficient maximum should be determined using Standard Equation 12.8-17. β may be assumed to be 1.0.

θ max =

0.5 ≤ 0.25 β Cd

θ max, N − S =

0.5 = 0.125 (1.0)( 4)

θ max, E −W =

0.5 = 0.154 (1.0 )(3.25)

Next, the stability coefficient is calculated using Standard Equation 12.8-16. The stability coefficient is calculated at both the roof and mezzanine levels in both orthogonal directions. For purposes of illustration, the roof level check in the N-S direction will be shown as:

θ=

Px Δ I Vx hsx Cd

Δ=

Px (δ e 2 − δ e1 ) Cd I Vx hsx Cd I

Cd (δ e 2 − δ e1 ) I Px (δ e 2 − δ e1 ) Vx hsx

Proof = Roof LL + Roof DL + Panels + Frames Proof = (180 ft × 90 ft )(15 psf + 25 psf ) + 437 kips + 224 kips + 27 kips

Proof = 1336 kips

θ=

(1,336 kips )( 4.51in.) 0.083 0.1 = < (187 kips )(32.375 ft )

The three other stability coefficients were all determined to be less than θmax, thus allowing P-delta effects to be excluded from the analysis. 6.1.4.5 Force summary. The maximum moments and axial forces caused by dead, live and earthquake loads on the gable frames are listed in Tables 6.1-3 and 6.1-4. The frames are symmetrical about their

6-14

Chapter 6: Structural Steel Design ridge and the loads are either symmetrical or can be applied on either side on the frame because the forces are given for only half of the frame extending from the ridge to the ground. The moments are given in Table 6.1-4 and the axial forces are given in Table 6.1-5. The moment diagram for the combined load condition is shown in Figure 6.1-4. The load combination is 1.4D + L + 0.2S + ρQE, which is used throughout the remainder of calculations in this section, unless specifically noted otherwise. The size of the members is controlled by gravity loads, not seismic loads. The design of connections will be controlled by the seismic loads. Forces in the design of the braces are discussed in Section 6.1.5.5. Table 6.1-4 Moments in Gable Frame Members Location

D (ft-kips)

L (ft-kips)

S (ft-kips)

1 - Ridge

128

112 (279)

2 - Knee

161

333

162

447 (726)

137

3 - Mezzanine 4 - Base

QE (ft-kips)

Combined* (ft-kips)

* Combined Load = 1.4D + L + 0.2S + ρQE (or 1.2D + 1.6S). Individual maxima are not necessarily on the same frame; combined load values are maximum for any frame. 1.0 ft = 0.3048 m, 1.0 kip = 1.36 kN-m.

Table 6.1-5 Axial Forces in Gable Frame Members Location

D (ft-kips)

L (ft-kips)

S (ft-kips)

ρQ E (ft-kips)

Combined* (ft-kips)

1 - Ridge

3.5

0.8

2 - Knee

4.5

7.0

3 - Mezzanine

127

4 - Base

127

* Combined Load = 1.4D + L + 0.2S + ρQE. Individual maxima are not necessarily on the same frame; combined load values are maximum for any frame. 1.0 ft = 0.3048 m, 1.0 kip = 1.36 kN-m.

6-15

FEMA P-751, NEHRP Recommended Provisions: Design Examples

447 ft - kips

40 ft - kips

112 ft - kips

1.4D + 0.2S + ρ QE

53 ft - kips 53 ft - kips

0.7D - ρ QE

Figure 6.1-4 Moment diagram for seismic load combinations (1.0 ft-kip = 1.36 kN-m)

6.1.5

Proportioning and Details

The gable frame is shown schematically in Figure 6.1-5. Using the load combinations presented in Section 6.1.3.4 and the loads from Tables 6.1-4 and 6.1-5, the proportions of the frame are checked at the roof beams and the variable-depth columns (at the knee). The mezzanine framing, also shown in Figure 6.1-1, was proportioned similarly. The diagonal bracing, shown in Figure 6.1-1 at the east end of the building, is proportioned using tension forces determined from the three-dimensional ELF analysis.

6-16

Chapter 6: Structural Steel Design 90'

W21x73

30'-6"

W24x94 (split)

12'-0"

Mezzanine (2 end bays)

Figure 6.1-5 Gable frame schematic: Column tapers from 12 in. at base to 36 in. at knee; roof beam tapers from 36 in. at knee to 18 in. at ridge; plate sizes are given in Figure 6.1-7 (1.0 in. = 25.4 mm)

Additionally, the bolted, stiffened, extended end-plate connections must be sized correctly to conform to the pre-qualification standards. AISC 358 Table 6.1 provides parametric limits on the beam and connection sizes. Table 6.1-6 shows these limits as well as the values used for design. Table 6.1-6 Parametric Limits for Moment Frame Connection Parameter

Minimum (in.)

As Designed (in.)

Maximum (in.)

3/4

1 1/4

2 1/2

pfi, pfo

1 3/4

3 1/2

3 3/4

18 1/2

tbf

19/32

5/8

bbf

7 3/4

12 1/4

6-17

FEMA P-751, NEHRP Recommended Provisions: Design Examples 6.1.5.1 Frame compactness and brace spacing. According to Standard Section 14.1.3, steel structures assigned to Seismic Design Category D, E, or F must be designed and detailed per AISC 341. For an intermediate moment frame (IMF), AISC 341, Part I, Section 1, “Scope,” stipulates that those requirements are to be applied in conjunction with AISC 360. Part I, Section 10 of AISC 341 itemizes a few additional items from AISC 360 for intermediate moment frames, but otherwise the intermediate moment frames are to be designed per AISC 360. AISC 341 requires IMFs to have compact width-thickness ratios per AISC 360, Table B4.1. All width-thickness ratios are less than the limiting λp from AISC 360, Table B4.1. All P-M ratios (combined compression and flexure) are less than 1.00. This is based on proper spacing of lateral bracing. Lateral bracing is provided by the roof joists. The maximum spacing of lateral bracing is determined using beam properties at the ends and AISC 341, Section 10.8:

Lb,max ≤ 0.17ry

E Fy

⎛ 29000 ksi ⎞ Lb,max ≤ 0.17 (1.46 in.) ⎜ ⎟ = 148 in. ⎝ 50 ksi ⎠ Lb is 48 inches; therefore, the spacing is OK. Also, the required brace strength and stiffness are calculated per AISC 360, Equations A-6-7 and A-6-8:

Pbr =

0.02 M r Cd ho

1 ⎛ 10M Cd ⎞ ⎟ ⎠

r βbr = ⎜ φ ⎝ Lb ho

where:

M r = Ry ZFy Cd = 1.0 ho = distance between flange centroids, in. Lb = distance between braces or Lp (from AISC 360 Eq. F2-5), whichever is greater, in.

(

)

M r = (1.1) 309 in.3 (50 ksi ) = 16,992 in.-kip = 1,416 ft-kip Pbr =

6-18

0.02 (16,992 in.-kip )(1.0 )

(36 in. − 5/8 in.)

Chapter 6: Structural Steel Design

Pbr = 9.61 kips L p = 1.76ry

E Fy

L p = 1.76 (1.46 in.)

βbr =

29,000 ksi = 62 in. 50 ksi

1 ⎛ 10 (16,992 in.-kip )(1.0 ) ⎞ ⎜ ⎟ ( 0.75) ⎜⎝ ( 62 in.)(35.375 in.) ⎟⎠

βbr = 104 kips/in. Adjacent to the plastic hinge regions, lateral bracing must have additional strength as defined in AISC 341 10.8

Pu =

0.06 M u ho

0.06 (16992 in ‐kip )

(35.375 in.)

Pu = 28.8 kips The C-joists used in this structure likely are not adequate to brace the moment frames. Instead, tube brace members will be used, but they are not analyzed in this example. At the negative moment regions near the knee, lateral bracing is necessary on the bottom flange of the beams and inside the flanges of the columns (Figure 6.1-6).

6-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples

L3 x3

1" dia. A490 (typical)

Gusset plate

Section "A" 2x2 X-brace

MC8 girt

Elevation Filler pad

L3x3

Section "B"

Figure 6.1-6 Arrangement at knee (1.0 in. = 25.4 mm)

6.1.5.2 Knee of the frame. The knee detail is shown in Figures 6.1-6 and 6.1-7. The vertical plate shown near the upper left corner in Figure 6.1-6 is a gusset providing connection for X-bracing in the longitudinal direction. The beam-to-column connection requires special consideration. The method of AISC 358 for bolted, stiffened end plate connections is used. Refer to Figure 6.1-8 for the configuration. Highlights from this method are shown for this portion of the example. Refer to AISC 358 for a discussion of the entire procedure.

6-20

Chapter 6: Structural Steel Design

Plate: 11 4x7x1'-05 8" t p : 11 4" b p : 9"

3'-0"

30°

Tapered beam

P f = 1.75"

Pb = 3.5"

Plate: 5 8"x7x1'-1" t s = 5 8"

Lst

1 3 8"

Varies

d b = 36"

h4 = 29.81" h3 = 33.21"

h2 = 37.49"

h1 = 40.94"

16"

18"-36"

1 3 8" 7

Bolts: 1" dia. A490 g = 5"

t p = 2"

W21x73: tbf = 0.740" bbf = 8.295" tbw = 0.455" Typical

Weld per AWS D1.1

2.5

Varies 3

12"-36"

Detail "1" Detail "1" 3

Figure 6.1-7 Bolted stiffened connection at knee (1.0 in. = 25.4 mm)

The AISC 358 method for bolted stiffened end plate connection requires the determination of the maximum moment that can be developed by the beam. The steps in AISC 358 for bolted stiffened end plates follow: Step 1.

Determine the maximum moment at the plastic hinge location. The end plate stiffeners at the top and bottom flanges increase the local moment of inertia of the beam, forcing the plastic hinge to occur away from the welds at the end of beam/face of column. The stiffeners should be long enough to force the plastic hinge to at least d/2 away from the end of the beam. With

6-21

FEMA P-751, NEHRP Recommended Provisions: Design Examples the taper of the section, the depth will be slightly less than 36 inches at the location of the hinge, but that reduction will be ignored here. The probable maximum moment, Mpe, at the plastic hinge is computed using AISC 358 Equation 6.9-2 as follows:

M pe = C pr Ry Fy Z x Where, per AISC 358 Equation 2.4.3-2:

C pr =

Fy + Fu 2 Fy

≤ 1.2

50 + 65 2 (50 )

C pr = 1.15 where: Ry = 1.1 from AISC 341 Table I-6-1 Ze = 309 in.3 Fy = 50 ksi Therefore:

(

)

M pe = (1.15)(1.1)(50 ksi ) 309 in.3 = 19,541 in.-kip = 1,628 ft-kip The moment at the column flange, Mf , which drives the connection design, is determined from AISC 358 Equation 6.9-2 as follows:

M f = M pe + Vu Sh where: Vu = shear at location of plastic hinge L’ = distance between plastic hinges Sh = distance from the face of the column to the plastic hinge, ft.

Sh = Lst + t p where: Lst = length of end-plate stiffener, as shown in AISC 358 Figure 6.2.

6-22

Chapter 6: Structural Steel Design tp = thickness of end plate, in.

Lst ≥

hst tan 30°

where: hst = height of the end-plate from the outside face of the beam flange to the end of the end-plate

Lst ≥

( 7 in.) tan 30°

Lst ≥ 12.1 in. Use Lst = 13 in.

Sh = 13 in. + 1.25 in. = 14.3 in.

L' = Lout − 2dc − 2Sh L' = ( 90 ft ) − 2 ( 36 in.) − 2 (14.3 in.) = 81.63 ft

Vu = Vu =

2M pe L'

+ Vgravity

2 (1628 ft ‐kip ) 81.63 ft

+ 18.9 kips = 58.8 kips

M f = 19,541 in.-kip + (58.8 kips )(14.3 in.) Step 2.

M f = 1,698 ft-kip = 20,379 in.-kip Find bolt size for end plates. For a connection with two rows of two bolts inside and outside the flange, AISC 358 Equation 6.9-7 indicates the following:

db req ' d =

2M f

πφn Fnt ( h1 + h2 + h3 + h4 )

where: Fnt = nominal tensile stress of bolt, ksi hi = distance from the centerline of the beam compression flange to the centerline of the ith tension bolt row, in. Try A490 bolts. See Figure 6.1-7 for bolt geometry. 6-23

FEMA P-751, NEHRP Recommended Provisions: Design Examples

db req'd =

2 ( 20,379 in.-kip )

π ( 0.9 )(113 ksi )( 29.81 in. + 33.31 in. + 37.44 in. + 40.94 in.)

= 0.95 in.

Use 1 in. diameter A490N bolts. Step 3.

Determine the minimum end-plate thickness from AISC 358 Equation 6.9-8.

t p req ' d =

1.11M f

φd FypYp

where: Fyp = specified minimum yield stress of the end plate material, ksi Yp = the end-plate yield line mechanism parameter from AISC 358 Table 6.4

φd = resistance factor for ductile limit states, taken as 1.0 From AISC 358 Table 6.4:

1 bp g 2

where: bp = width of the end plate, in. g = horizontal distance between bolts on the end plate, in.

1 2

(9 in.)(5 in.) = 3.35 in.

de = 7 in. (see Figure 6.1-7) Use Case 1 from AISC 358 Table 6.4, since de > s

bp ⎡ ⎛ 1 ⎢ h1 ⎜ 2 ⎢ ⎝ 2de ⎣ 2 ⎡ ⎛ + ⎢ h1 ⎜ de + g ⎣ ⎝

Yp =

⎛ 1 ⎞ ⎛ 1 ⎞ ⎞ ⎛ 1 ⎞ ⎤ ⎟ + h3 ⎜ ⎟ + h4 ⎜ ⎟ ⎥ ⎟ + h2 ⎜⎜ ⎟ ⎜ p fi ⎟ ⎝ s ⎠ ⎥⎦ ⎠ ⎝ p fo ⎠ ⎝ ⎠ pb 4

3 pb ⎞ ⎛ ⎟ + h2 ⎜ p fo + 4 ⎠ ⎝

3 pb pb ⎞ ⎞ ⎛ ⎛ ⎟ + h3 ⎜ p fi + 4 ⎟ + h4 ⎜ s + 4 ⎠ ⎝ ⎠ ⎝

⎞ 2 ⎤ ⎟ + pb ⎥ + g ⎠ ⎦

where: pfo = vertical distance between beam flange and the nearest outer row of bolts, in.

6-24

Chapter 6: Structural Steel Design pfi = vertical distance between beam flange and the nearest inner row of volts, in. pb = distance between the inner and our row of bolts, in.

Yp =

(9 in.) ⎡

⎛ 1 ⎞ ⎛ 1 ⎞ ⎢( 40.94 in.) ⎜ 3.35 in. ⎟ + ( 37.44 in.) ⎜ 1.75 in. ⎟ ⎝ ⎠ ⎝ ⎠ ⎣

⎛ 1 ⎞ ⎛ 1 ⎞ ⎤ + ( 33.31 in.) ⎜ ⎟ + (29.81 in.) ⎜ ⎟⎥ ⎝ 1.75 in. ⎠ ⎝ 3.35 in. ⎠ ⎦ +

⎛ ⎛ 3 ( 3.5 in.) ⎞ (3.5 in.) ⎞ + 2 ⎡ ⎢( 40.94 in.) ⎜⎜ 3.35 in. + ⎟⎟ ( 37.44 in.) ⎜⎜1.75 in. + ⎟⎟ (5 in.) ⎢⎣ 4 4 ⎝ ⎠ ⎝ ⎠ ⎛ 3 ( 3.5 in.) ⎞ 3.5 in. ⎞ 2 ⎤ ⎛ + ( 33.31 in.) ⎜1.75 in. + + (29.81 in.) ⎜⎜ 3.35 in. + ⎟ + ( 3.5 in.) ⎥ ⎟ ⎟ 4 ⎠ 4 ⎝ ⎝ ⎠ ⎦⎥

+ (5 in.) Yp = 499 in.

t p req ' d =

1.11( 20,379 in.-kip )

(1.0)(36 ksi )( 499 in.)

= 1.12 in.

Use 1.25-inch thick end-plates. Step 4.

Calculate the factored beam flange force from AISC 358 Equation 6.9-9.

Ffu = where:

Mf d − tbf

d = depth of the beam, in. tbf = thickness of beam flange, in.

F fu = Step 5.

20,379 in.-kip = 576 kips 36 in. − 5 / 8 in.

Determine the end-plate stiffener thickness from AISC 358 Equation 6.9-13.

⎛ Fyb ⎞ ts ≥ tbw ⎜ ⎟ ⎜ Fys ⎟ ⎝ ⎠ where: tbw = thickness of the beam web, in.

6-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples Fyb = specified minimum yield stress of beam material, ksi Fys = specified minimum yield stress of stiffener material, ksi

⎛ 50 ksi ⎞ ts ≥ ( 7 / 16 in.) ⎜ ⎟ = 0.61 in. ⎝ 36 ksi ⎠ Use 5/8-inch plates. The stiffener width-thickness ratio must also comply with AISC 358 Equation 6.9-14.

hst E ≤ 0.56 ts Fys

hst ≤ 0.56

( 29000 ksi ) (5 / 8 in.) (36 ksi )

hst ≤ 9.93 in. hst = 7 in. Step 6.

Check bolt shear rupture strength at the compression flange by AISC 358 Equation 6.9-15.

Vu < φn Rn = φn ( nb ) Fv Ab where:

φn = resistance factor for non-ductile limit states, taken as 0.9 nb = number of bolts at compression flange Fv = nominal shear stress of bolts from AISC 360 Table J3.2, ksi Ab = nominal bolt area, in.

⎛ 12 ⎞ ⎟ = 339 kips > 58.8 kips ⎝ 4 ⎠

φn Rn = ( 0.9 )(8)( 60 ) ⎜ π Step 7.

Check bolt bearing/tear-out of the end-plate and column flange by AISC 358 Equation 6.9-17.

Vu < φn Rn = φn ( ni ) rni + φn ( no ) rno where: ni = number of inner bolts

6-26

Chapter 6: Structural Steel Design

no = number of outer bolts

rni = 1.2 Lc tFu < 2.4dbtFu for each inner bolt rno = 1.2 Lc tFu < 2.4dbtFu for each outer bolt Lc = clear distance, in the direction of force, between the edge of the hole and the edge of the adjacent hole or edge of the material, in. t = end-plate or column flange thickness, in. Fu = specified minimum tensile strength of end-plate or column flange material, ksi db = diameter of bolt, in.

Lci = pb − d e

Lco = Le −

de 2

where: de = effective area of bolt hole, in. Le = edge spacing of the bolts, in.

Lci = 3.5 in. − 1 18 in. = 2.38 in.

Lco = 1.75 in. −

1 18 in. = 1.19 in. 2

rni = 1.2 ( 2.38 in.)(1.25 in.)(58 ksi ) < 2.4 (1 in.)(1.25 in.)(58 ksi ) rni = 207 kips < 174 kips rni = 174 kips rno = 1.2 Lc tFu < 2.4dbtFu

rno = 1.2 (1.19 in.)(1.25 in.)(58 ksi ) < 2.4 (1 in.)(1.25 in.)(58 ksi ) rno = 103 kips < 174 kips rno = 103 kips

6-27

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Vu < φn Rn = (1)( 4 )(174 kips ) + (1)( 4 )(103 kips )

φn Rn = 998 kips > 58.8 kips Step 8.

Check the column flange for flexural yielding by AISC 358 Equation 6.9-20.

tcf req ' d =

1.11M f

φd FycYc

≤ tcf

where: Fyc = specified minimum yield stress of column flange material, ksi Yc = stiffened column flange yield line from AISC 358 Table 6.6 tcf = column flange thickness, in.

Yc =

bcf ⎡ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎤ ⎢ h1 ⎜ ⎟ + h2 ⎜ ⎟ + h3 ⎜ ⎟ + h4 ⎜ ⎟ ⎥ 2 ⎣⎢ ⎝ s ⎠ ⎝ s ⎠ ⎦⎥ ⎝ pso ⎠ ⎝ psi ⎠

3 p ⎞ 3p pb ⎞ p ⎞ 2 ⎡ ⎛ ⎛ ⎛ ⎛ + h2 ⎜ pso + b ⎟ + h3 ⎜ psi + b ⎟ + h4 ⎜ s + b ⎢ h1 ⎜ s + ⎟ 4 ⎠ 4 ⎠ 4 ⎠ 4 g ⎣ ⎝ ⎝ ⎝ ⎝

where: bcf = column flange width, in. psi = distance from column stiffener to inner bolts, in. pso = distance from column stiffener to outer bolts, in.

6-28

1 bcf g 2

1 2

(8 in.)(5 in.) = 3.16 in.

⎞ 2 ⎤ ⎟ + pb ⎥ + g ⎠ ⎦

Chapter 6: Structural Steel Design

Yc =

(8 in.) ⎡

⎛ 1 ⎞ ⎛ 1 ⎞ ⎢( 40.94 in.) ⎜ 3.16 in. ⎟ + ( 37.44 in.) ⎜ 1.75 in. ⎟ ⎝ ⎠ ⎝ ⎠ ⎣

⎛ 1 ⎞ ⎛ 1 ⎞ ⎤ + ( 33.31 in.) ⎜ ⎟ + (29.81 in.) ⎜ ⎟⎥ ⎝ 1.75 in. ⎠ ⎝ 3.16 in. ⎠ ⎦ +

⎛ ⎛ 3 ( 3.5 in.) ⎞ (3.5 in.) ⎞ + 2 ⎡ ⎢( 40.94 in.) ⎜⎜ 3.16 in. + ⎟⎟ (37.44 in.) ⎜⎜1.75 in. + ⎟⎟ (5 in.) ⎣⎢ 4 4 ⎝ ⎠ ⎝ ⎠ ⎛ 3 ( 3.5 in.) ⎞ 3.5 in. ⎞ 2 ⎤ ⎛ + ( 33.31 in.) ⎜1.75 in. + + (29.81 in.) ⎜⎜ 3.16 in. + ⎟⎟ + (3.5 in.) ⎥ ⎟ 4 ⎠ 4 ⎝ ⎥⎦ ⎝ ⎠

+ (5 in.)

Yc = 497 in.

1.11( 20,379 in.-kip )

tcf req ' d =

(1)(50 ksi )( 497 in.)

≤ 2 in.

tcf req ' d = 0.95 in. ≤ 2 in. Column flange of 2 inches is OK. Step 9.

Determine the required stiffener force by AISC 358 Equation 6.9-21.

φd M cf = φd FycYctcf2 2

φd M cf = (1)(50 ksi )( 497 in.)( 2 in.) = 99,344 in.-kip The equivalent column flange design force used for stiffener design by AISC 358 Equation 6.9-22.

φd Rn =

φd M cf

(d − t ) bf

φd Rn =

(99,690 in.-kip ) = 2,808 kips (36 in.) − (5 / 8 in.)

2,808 kips > 576 kips

Step 10. Check local column web yielding strength of the unstiffened column web at the beam flanges by AISC 358 Equations 6.9-23 and 6.9-24.

φd Rn ≥ Ffu

6-29

FEMA P-751, NEHRP Recommended Provisions: Design Examples

φd Rn = φd Ct ( 6kc + tbf + 2t p ) Fyctcw where: Ct = 0.5 if the distance from the column top to the top of the beam flange is less than the depth of the column: otherwise 1.0 kc = distance from outer face of the column flange to web toe of fillet weld, in. tp = end-plate thickness, in. Fyc = specified yield stress of the column web material, ksi tcw = column web thickness, in. tbf = beam flange thickness, in.

φd Rn = (1.0)(0.5) ( 6 ( 2 in. + 516 in.) + (0.5 in.) + 2 (1.25 in.)) (50 ksi )(0.5 in.) φd Rn = 212 kips ≤ 576 kips The design is not acceptable. Column stiffeners need to be provided. Step 11. Check the unstiffened column web buckling strength at the beam compression flange by AISC 358 Equations 6.9-25 and 6.9-27.

φ Rn ≥ Ffu φ Rn = φ

3 12tcw EFyc

where: h = clear distance between flanges when welds are used for built-up shapes, in. 3

φ Rn = ( 0.75 ) 42 kips < 576 kips

12 ( 0.5 in.)

( 29,000 ksi )(50 ksi ) = 42 kips (36 in. − 2 in. − 0.75 in.) NOT OK

Step 12. Check the unstiffened column web crippling strength at the beam compression flange by AISC 358 Equation 6-30.

φ Rn ≥ Ffu

6-30

Chapter 6: Structural Steel Design

φ Rn = φ

2 0.40tcw

⎡ ⎢1 + 3 ⎛⎜ N ⎢ ⎝ dc ⎣

1.5

⎞ ⎛ tcw ⎞ ⎟ ⎜⎜ ⎟⎟ ⎠ ⎝ tcf ⎠

⎤ EF t yc cf ⎥ ⎥ tcw ⎦

where: N = thickness of beam flange plus 2 times the groove weld reinforcement leg size, in. dc = overall depth of the column, in.

⎡ ⎛ 3 ( 5 8 in.) ⎞ ⎛ 0.5 in. ⎞1.5 ⎤ Rn = ( 0.75 )( 0.80 )( 0.5 in.) ⎢1 + 3 ⎜⎜ ⎟⎟ ⎜ ⎟ ⎥ ⎢⎣ ⎝ 36 in. ⎠ ⎝ 2 in. ⎠ ⎥⎦ 2

( 29,000 ksi )(50 ksi )( 2 in.) 0.5 in.

Rn = 184 kips 184 kips < 576 kips

NOT OK

Step 13. Check the required strength of the stiffener plates by AISC 358 Equation 6-32.

Fsu = Ffu − min φ Rn = 576 kips − 42 kips = 534 kips where:

min φ Rn = the minimum design strength value from column flange bending check, column web yielding, column web buckling and column web crippling check Although AISC 358 says to use this value of 534 kips to design the continuity plate, a different approach will be used in this example. In compression, the continuity plate will be designed to take the full force delivered by the beam flange, Fsu. In tension, however, the compressive limit states (web buckling and web yielding) are not applicable and column web yielding will control the design instead. The tension design force can be taken as follows:

Fsu = Ffu − φ Rn,web yielding = 576 kips − 212 kips = 364 kips

Step 14. Design the continuity plate for required strength by AISC 360 Section J10. Find the cross-sectional area required by the continuity plate acting in tension:

As ,reqd =

Fsu φ Fy

As ,reqd =

364 kips = 8.1 in.2 ( 0.9)(50 ksi )

6-31

FEMA P-751, NEHRP Recommended Provisions: Design Examples

ts ,reqd =

As ,reqd bst

8.1 in.2 = 1.01 in. 8 in.

Use a 1-3/8-inch continuity plate. As it will be shown later, net section rupture (not gross yielding) will control the design of this plate. From AISC Section J10.8, calculate member properties using an effective length of 0.75h and a column web length of 12tw = 6 in.:

Ix =

3 (b − t ) t 3 tcw12tcw + st cw st 12 12

3 3 0.5 in.)( 6 in.) 8 in. − 0.5 in.)(1.375 in.) ( ( = + = 10.6 in.4

t b3 (12tcw − tst ) tcw I y = st st + 12 12 Iy =

J≅

(1.375 in.)(8 in.)3 + ( 6 in. − 1.375 in.)(0.5 in.)3 12

= 58.7 in.4

bst tst3 (12tcw − tst ) tcw + 3 3

3 3 8 in.)(1.375 in.) 6 in. − 1.375 in.)( 0.5 in.) ( ( J≅ +

= 7.13 in.4

A = bst tst + (12tcw − tst ) tcw A = (8 in.)(1.375 in.) + ( 6 in. − 1.375 in.)( 0.5 in.) = 13.3 in.2 ry =

ry =

Iy A

58.7 in.4 = 2.1 in. 13.3 in.2

(

L = 0.75h = 0.75 dc − t f 1 − t f 2 − 2tweld

)

L = 0.75 (36 in. − 2 in. − 0.75 in. − 2 (5 /16 in.)) = 24.5 in. 6-32

Chapter 6: Structural Steel Design

Check the continuity plate in compression from AISC 360 Equation J4.4:

KL (1.0 )( 24.5 in.) = = 11.65 ≤ 25 ry 2.1 in. Strength in the other direction does not need to be checked because the cruciform section will not buckle in the plane of the column web. Since KL/r is less than 25, use AISC 360 Equation J4-6 to determine compression strength:

φ Pn = φ Fy Ag

(

)

φ Pn = (0.9)(50 ksi ) 13.3 in.2 = 599 kips However, torsional buckling may control. Therefore, check flexural-torsional buckling using AISC 360 Equation E4-4:

Fe =

GJ Ix + I y

(11, 200 ksi ) ( 7.13 in.4 ) 10.6 in.4 + 58.7 in.4

Fy ⎡ Fcr = ⎢0.658 Fe ⎢ ⎣

= 1,150 ksi

⎤ ⎥ Fy ⎥ ⎦

50 ksi ⎡ ⎤ 1,150 ksi ⎥ (50 ksi ) = 49.1 ksi Fcr = ⎢0.658 ⎢⎣ ⎥⎦

(

)

φ Pn = (0.9)( 49.1 ksi ) 13.3 in.2 = 588 kips Check the continuity plate in tension. The continuity plate had been previously sized for adequacy to tensile yielding of the gross section. Now tensile rupture of the net section must be checked using AISC 360 Section D2-2. The critical section will be analyzed where the continuity plates are clipped adjacent to the k-region of the column.

Ae = (tst ) (bst − 2 (tweld + clip )) Ae = (1.375 in.) (8 in. − 2 (5 /16 in. + 0.5 in.)) = 8.77 in.2

φt Pn = φt Fu Ae = 381 kips > 364 kips

6-33

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Step 15. Check the panel zone for required strength per AISC 341 Equation J10-9.

Pc = Fy A where: A = column cross sectional area, in2.

Pc = (50 ksi )(38.6 in.) = 1,931 kips

Pr = Ffu = 576 kips Pr 576 kips = = 0.3 < 0.75 Pc 1,931 kips Therefore, use AISC 360 Equation J10-111. Note that panel zone flexibility was accounted for in the ETABS model.

⎛

φ Rn = φ 0.60 Fy dc tcw ⎜1 + ⎜ ⎝

3bcf tcf2 ⎞ ⎟ db d c tcw ⎟⎠

2 ⎛ ⎞ 3 (8 in.)( 2 in.) ⎟ φ Rn = ( 0.9 )( 0.60 )( 50 ksi )( 36 in.)( 0.5 in.) ⎜1 + ⎜ ( 36 in.)( 36 in.)( 0.5 in.) ⎟ ⎝ ⎠

φ Rn = 558 kips < 576 kips

NOT OK

The column web is not sufficient to resist the panel zone shear. Although doubler plates can be added to the panel zone to increase strength, this may be an expensive solution. A more economical solution would be to simply upsize the column web to a sufficient thickness, such as 5/8 inch.

⎛

φ Rn = φ 0.60 Fy dc tcw ⎜1 + ⎜ ⎝

3bcf tcf2 ⎞ ⎟ db d c tcw ⎟⎠

2 ⎛ ⎞ 3 (8 in.)( 2 in.) ⎟ φ Rn = ( 0.9 )( 0.60 )( 50 ksi )( 36 in.)( 5 / 8 in.) ⎜1 + ⎜ ( 36 in.)( 36 in.)( 5 / 8 in.) ⎟ ⎝ ⎠

φ Rn = 680 kips > 576 kips Note that changing the column member properties might affect the analysis results. In this example, this is not the case, although the slight difference in web thickness would result in

6-34

Chapter 6: Structural Steel Design marginally different values for some of the end-plate connection calculations. For simplicity, these changes are not undertaken in this example. 6.1.5.3 Frame at the ridge. The ridge joint detail is shown in Figure 6.1-8. An unstiffened bolted connection plate is selected.

12 W21

Figure 6.1-8 End plate connection at ridge This is an AISC 360 designed connection, not an AISC 358 designed connection because there should not be a plastic hinge forming in this vicinity. Lateral seismic forces produce no moment at the ridge until yielding takes place at one of the knees. Vertical accelerations on the dead load do produce a moment at this point; however, the value is small compared to all other moments and does not appear to be a concern. Once seismic loads produce a plastic hinge at one knee, further lateral displacement produces positive moment at the ridge. Under the condition on which the AISC 358 design is based (a full plastic moment is produced at each knee), the moment at the ridge will simply be the static moment from the gravity loads less the horizontal thrust times the rise from knee to ridge. Analyzing this frame under the gravity load case 1.2D + 0.2S, the static moment is 406 ft-kip and the reduction for the thrust is 128 ft-kip, leaving a net positive moment of 278 ft-kip, coincidentally close to the design moment for the factored gravity loads. 6.1.5.4 Design of mezzanine framing. The design of the framing for the mezzanine floor at the east end of the building is controlled by gravity loads. The concrete-filled 3-inch, 20-gauge steel deck of the mezzanine floor is supported on steel beams spaced at 10 feet and spanning 20 feet (Figure 6.1-2). The steel beams rest on three-span girders connected at each end to the portal frames and supported on two intermediate columns (Figure 6.1-1). The girder spans are approximately 30 feet each. Those lateral forces that are received by the mezzanine are distributed to the frames and diagonal bracing via the floor diaphragm. A typical beam-column connection at the mezzanine level is provided in Figure 6.1-9. The design of the end plate connection is similar to that at the knee, but simpler because the beam is horizontal and not tapered. Also note that demands on the end-plate connection will be less because this connection is not at the end of the column.

6-35

FEMA P-751, NEHRP Recommended Provisions: Design Examples

L3x3 strut

3" concrete slab 3" embossed 20 ga. deck

W14x43 Split W27x84

W21x62

MC8x18.7

(b)

Figure 6.1-9 Mezzanine framing (1.0 in. = 25.4 mm) 6.1.5.5 Braced frame diagonal bracing Although the force in the diagonal X-braces can be either tension or compression, only the tensile value is considered because it is assumed that the diagonal braces are capable of resisting only tensile forces. See AISC 341 Section 14.2 for requirements on braces for OCBFs. The strength of the members and connections, including the columns in this area but excluding the brace connections, must be based on Standard Section 12.4.2.3: 1.4D + 1.0L + 0.2S + ρQE 0.7D + ρQE Recall that a 1.0 factor is applied to L when the live load is greater than 100 psf (Standard Sec. 2.3.2). For the case discussed here, the “tension only” brace does not carry any live or dead load, so the load factor does not matter. For simplicity, we can assume that the lateral force is equally divided among the roof level braces and is slightly amplified to account for torsional effects. Thus the brace force can be approximated using the following equation:

6-36

Chapter 6: Structural Steel Design

Pu = 0.55V × 1 2 ×

1 cos θ

Pu = 0.55 ( 211 kips ) × 1 2 ×

1 = 85 kips cos 47°

All braces at this level will have the same design. Choose a brace member based on tensile yielding of the gross section by AISC 360 Equation D2-1:

φt Pn = φt Fy Ag Ag ,reqd =

85 kips = 2.62 in.2 ( 0.9)(36 ksi )

This also needs to be checked for tensile rupture of the net section. Demand will be taken as either the expected yield strength of the brace or the amplified seismic load. Try a 2L3½x3x 7/16, which is the smallest seismically compact angle shape available. Ag = 5.34 in.2 The Kl/r requirement of AISC 341 Section 14.2 does not apply because this is not a V or an inverted V configuration. Check net rupture by AISC 360 Equation D2-2 and D3-1:

φt Pn = φt Fu Ae Ae = AnU Determine the shear lag factor, U, from AISC 360, Table D3.1, Case 2. In ordered to calculate U, the weld length along the double angles needs to be determined.

U =1− x

Brace connection demand is given as the expected yield strength of the brace in tension per AISC 341 Section 14.4.

(

)

Ry Fy Ag = (1.5)(36 ksi ) 5.34 in.2 = 288 kips Expected yield strength of the brace is 288 kips. However, AISC 341 Section 14.4b limits the brace connection design force to the amplified seismic load.

Ω0 PQE = ( 2 )(85 kips ) = 170 kips

6-37

FEMA P-751, NEHRP Recommended Provisions: Design Examples Use four fillet welds, two on each angle. Try 1/4-inch welds using AISC 360 Equation J2-3:

φ Rn = φ Fw Aw

φ Rn = φ ( 0.60FEXX ) ( 0.707tw ) ( L ) L=

170 kips = 7.63 in. 4 ( 0.75)( 0.6)(70 ksi ) (0.707)(0.25 in.)

Use four 1/4-inch fillet welds 8 inches long. Check the base metal:

φ Rn = φ FBM ABM Shear yielding from AISC 360 Equation J4-3:

φ Rn = φ 0.6 Fy Ag φ Rn = (1.0)( 0.6)(36 ksi )( 0.25 in.)(8 in.) = 173 kips

Shear rupture from AISC 360 Equation J4-4:

φ Rn = φ 0.6 Fu Anv

φ Rn = ( 0.75)( 0.6)(58 ksi )( 0.25 in.)(8 in.)

Calculate the shear lag factor and the effective net area:

U = 1 − (0.846 in.)

(

(9 in.)

= 0.89

)

Ae = 5.34 in.2 (0.89) = 4.78 in.2 Calculate the tensile rupture strength:

(

)

φt Pn = (0.9)(58 ksi ) 6.1in.2 = 207 kips > 170 kips

Additionally, the capacity of the eave strut at the roof must be checked. The eave strut, part of the braced frame, also acts as a collector element and must be designed using the overstrength factor per Standard Section 12.10.2.1. 6.1.5.6 Roof deck diaphragm. Figure 6.1-10 shows the in-plane shear force in the roof deck diaphragm for both seismic loading conditions. There are deviations from simple approximations in both directions. In the E-W direction, the base shear is 274 kips (Section 6.1.4.2) with 77 percent or 211 kips at the roof. Torsion is not significant, so a simple approximation is to take half the force to each side and divide by 6-38

Chapter 6: Structural Steel Design the length of the building, which yields (211,000/2)/180 feet = 586 plf. The plot shows that the shear in the edge of the diaphragm is significantly higher in the two braced bays. This is a shear lag effect; the eave strut in the three-dimensional model is a HSS 6x6x1/4. In the N-S direction, the shear is generally highest in the bay between the mezzanine frame and the first frame without the mezzanine. This is expected given the significant change in stiffness. There is no simple approximation to estimate the shear here without a three-dimensional model. The shear is also high at the longitudinal braced bays because they tend to resist the horizontal torsion. However, the shear at the braced bays is lower than observed for the E-W motion.

Figure 6.1-10 Shear force in roof deck diaphragm; upper diagram is for E-W motion and lower is for N-S motion

6-39

FEMA P-751, NEHRP Recommended Provisions: Design Examples 6.2

SEVEN-‐STORY OFFICE BUILDING, LOS ANGELES, CALIFORNIA

Two alternative framing arrangements for a seven-story office building are illustrated. 6.2.1

Building Description

6.2.1.1 General description. This seven-story office building of rectangular plan configuration is 177 feet, 4 inches long in the E-W direction and 127 feet, 4 inches wide in the N-S direction (Figure 6.2-1). The building has a penthouse. It is framed in structural steel with 25-foot bays in each direction. The typical story height is 13 feet, 4 inches; the first story is 22 feet, 4 inches high. The penthouse extends 16 feet above the roof level of the building and covers the area bounded by Gridlines C, F, 2 5 in Figure 6.2-1. Floors consist of 3-1/4-inch lightweight concrete over composite metal deck. The elevators and stairs are located in the central three bays. 6.2.1.2 Alternatives. This example includes two alternatives—a steel moment-resisting frame and a concentrically braced frame: §

Alternative A: Seismic force resistance is provided by special moment frames (SMF) with prequalified Reduced Beam Section (RBS) connections located on the perimeter of the building (on Gridlines A, H, 1 6 in Figure 6.2-1, also illustrated in Figure 6.2-2). There are five bays of moment frames on each line.

§

Alternative B: Seismic force resistance is provided by four special concentrically braced frames (SCBF) in each direction. They are located in the elevator core walls between Columns 3C and 3D, 3E and 3F, 4C and 4D 4E and 4F in the E-W direction and between Columns 3C and 4C, 3D and 4D, 3E and 4E 3F and 4F in the N-S direction (Figure 6.2-1). The braced frames are in a two-story X configuration. The frames are identical in brace size and configuration, but there are some minor differences in beam and column sizes. Braced frame elevations are shown in Figures 6.2-10 through 6.2-12.

6.2.1.3 Scope. The example covers: §

Seismic design parameters (Sec. 6.2.2.1)

§

Analysis of perimeter moment frames (Sec. 6.2.4.1)

§

Beam and column proportioning (Sec. 6.2.4.2.3)

§

Moment frame connection design (Sec. 6.2.4.2.5)

§

Analysis of concentrically braced frames (Sec. 6.2.5.1)

§

Proportioning of braces (Sec. 6.2.5.2.1)

§

Braced frame connection design (Sec. 6.2.5.2.5)

6-40

Chapter 6: Structural Steel Design

177'-4" 1'-2"

1'-2"

25'-0"

1'-2"

25'-0"

127'-4"

25'-0"

5 N

1'-2"

16'-0"

PH roof Roof 7

6 at 13'-4"

6 5 4 3

22'-4"

Figure 6.2-1 Typical floor framing plan and building section (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)

6-41

FEMA P-751, NEHRP Recommended Provisions: Design Examples

7 at 25'-0"

5 at 25'-0"

Figure 6.2-2 Framing plan for special moment frame (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)

6.2.2

Basic Requirements

6.2.2.1 Provisions parameters. See Section 3.2 for an example illustrating the determination of design ground motion parameters. For this example, the parameters are as follows §

SDS = 1.0

§

SD1 = 0.6

§

Occupancy Category II

§

Seismic Design Category D

For Alternative A, Special Steel Moment Frame (Standard Table 12.2-1) §

R=8

§

Ω0 = 3

§

Cd = 5.5

6-42

Chapter 6: Structural Steel Design For Alternative B, Special Steel Concentrically Braced Frame (Standard Table 12.2-1): §

R=6

§

Ω0 = 2

§

Cd = 5

6.2.2.2 Loads. §

Roof live load (L): 25 psf

§

Penthouse roof dead load (D): 25 psf

§

Exterior walls of penthouse: 25 psf of wall

§

Roof DL (roofing, insulation, deck beams, girders, fireproofing, ceiling, mechanical, electrical plumbing): 55 psf

§

Exterior wall cladding: 25 psf of wall

§

Penthouse floor D: 65 psf

§

Penthouse Equipment: 39 psf

§

Floor L: 50 psf

§

Floor D (deck, beams, girders, fireproofing, ceiling, mechanical electrical, plumbing, partitions): 68 psf

§

Floor L reductions: per the IBC

6.2.2.3 Basic gravity loads. §

Penthouse roof: Roof slab = (0.025 ksf)(25 ft)(75 ft) Walls = (0.025 ksf)(8 ft)(200 ft) Columns = (0.110 ksf)(8 ft)(8 ft) Total

§

= = = =

47 kips 40 kips 7 kips 94 kips

Lower roof: Roof slab = (0.055 ksf)[(127.33 ft)(177.33 ft) - (25 ft)(75 ft) Penthouse floor = (0.065 ksf)(25 ft)(75 ft) Walls = 40 kips + (0.025 ksf)(609 ft)(6.67 ft) Columns = 7 kips + (0.170 ksf)(6.67 ft)(48 ft) Equipment = (0.039 ksf)(25 ft)(75 ft) Total

= 1139 kips = 122 kips = 142 kips = 61 kips = 73 kips = 1,537 kips

6-43

FEMA P-751, NEHRP Recommended Provisions: Design Examples §

Typical floor: Floor = (0.068 ksf)(127.33 ft)(177.33 ft) Walls = (0.025 ksf)(609 ft)(13.33 ft) Columns = (0.285 ksf)(13.33 ft)(48 ft) Total

Total weight of building = 94 kips + 1,537 kips + 6 (1,920 kips)

= 1,535 kips = 203 kips = 182 kips = 1,920 kips = 13,156 kips

6.2.2.4 Materials §

Concrete for floors: fc' = 3 ksi, lightweight (LW)

§

All other concrete: fc' = 4 ksi, normal weight (NW)

§

Structural steel: Wide flange sections: ASTM A992, Grade 50 HSS: ASTM A500, Grade B Plates: ASTM A36

6.2.3

Structural Design Criteria

6.2.3.1 Building configuration. The building has no vertical irregularities despite the relatively tall height of the first story. The exception of Standard Section 12.3.2.2 is taken, in which the drift ratio of adjacent stories are compared rather than the stiffness of the stories. In the three-dimensional analysis, the first story drift ratio is less than 130 percent of that for the story above. Because the building is symmetrical in plan, plan irregularities would not be expected. Analysis reveals that Alternative B is torsionally irregular, which is not uncommon for core-braced buildings. 6.2.3.2 Orthogonal load effects. A combination of 100 percent of the seismic forces in one direction with 30 percent of the seismic forces in the orthogonal direction is required for structures in Seismic Design Category D for certain elements—namely, the shared columns in the SCBF (Standard Sec. 12.5.4). In using modal response spectrum analysis (MRSA), the bidirectional case is handled by using the square root of the sum of the squares (SRSS) of the orthogonal spectra. 6.2.3.3 Structural component load effects. The effect of seismic load is defined by Standard Section 12.4.2 as:

E = ρ QE + 0.2S DS D Using Standard Section 12.3.4.2, ρ is 1.0 for Alternative A and 1.3 for Alternative B. (For simplicity, ρ is taken as 1.3; the design does not comply with the prescriptive requirements of Standard Sec. 12.3.4.2. It is assumed that the design would fail the calculation-based requirements of Standard Sec. 12.3.4.2.) Substitute for ρ (and for SDS = 1.0). §

For Alternative A: E = QE ± 0.2D

§

6-44

Alternative B:

Chapter 6: Structural Steel Design

E = 1.3QE ± 0.2D 6.2.3.4 Load combinations Load combinations from ASCE 7-05 are as follows: §

1.4D

§

1.2D + 1.6L + 0.5Lr

§

1.2D + L + 1.6Lr

§

(1.2 + 0.2SDS)D + 0.5L + ρQE

§

(0.9 – 0.2 SDS)D + ρQE

For each of these load combinations, substitute E as determined above, showing the maximum additive and minimum subtractive. QE acts both east and west (or north and south): §

Alternative A: 1.4D 1.2D + 1.6L + 0.5Lr 1.2D + L + 1.6Lr 1.4D + 0.5L + QE 0.7D + QE

§

Alternative B" 1.4D 1.2D + 1.6L + 0.5Lr 1.2D + L + 1.6Lr 1.4D + 0.5L + 1.3QE 0.7D + 1.3QE

For both cases, six scaled response spectrum cases are used: 1) Spectrum in X direction 2) Spectrum in X direction with 5 percent eccentricity 3) Spectrum in Y direction 4) Spectrum in Y direction with 5 percent eccentricity 5) SRSS combined spectra in X and Y directions 6) SRSS combined spectra in X and Y directions with 5 percent eccentricity. 6.2.3.5 Drift limits. The allowable story drift per Standard Section 12.12.1 is Δa = 0.02hsx. 6-45

FEMA P-751, NEHRP Recommended Provisions: Design Examples

The allowable story drift for the first floor is Δa = (0.02)(22.33 ft)(12 in./ft) = 5.36 in. The allowable story drift for a typical story is Δa = (0.02)(13.33 ft)(12 in./ft) = 3.20 in. Adjust calculated story drifts by the appropriate Cd factor from Standard Table 12.2-1. 6.2.4

Analysis and Design of Alternative A: SMF

6.2.4.1 Modal Response Spectrum Analysis. Determine the building period (T) per Standard Equation 12.8-7:

Ta = Ct hnx = ( 0.028 ) (102.3)0.8 = 1.14 sec where hn, the height to the main roof, is conservatively taken as 102.3 feet. The height of the penthouse will be neglected since its seismic mass is negligible. CuTa, the upper limit on the building period, is determined per Standard Table 12.8-1:

T = CuTa = (1.4 )(1.14 ) = 1.596 sec It is assumed that the calculated period will exceed CuTa; this is verified after member selection. The seismic response coefficient (Cs) is determined from Standard Equation 12.8-2 as follows:

Cs =

S DS 1 = = 0.125 R / I 8 /1

However, Standard Equation 12.8-3 indicates that the value for Cs need not exceed:

Cs =

S D1 0.6 = = 0.047 T ( R / I ) (1.596 sec)(8 / 1)

and the minimum value for Cs per Standard Equation 12.8-5 is:

Cs = 0.044ISDS ≥ 0.01 = ( 0.044)(1)(1) = 0.044 Therefore, use Cs = 0.047. Seismic base shear is computed per Standard Equation 12.8-1 as:

V = CSW = ( 0.047 )(13,156 kips ) = 618 kips where W is the seismic mass of the building as determined above. In evaluating the building in ETABS, twelve modes are analyzed, resulting in a total modal mass participation of 97 percent. The code requires at least 90 percent participation for strength. A scaling factor is used to take the response spectrum to 85 percent of the base shear, with a minimum scale factor for strength calculations of I/R. Typical software utilizes a spectrum presented as a coefficient of g, thus

6-46

Chapter 6: Structural Steel Design requiring scaling by g, thus the scaling factor used here is g/(R/I) = 386/(8/1) = 48.3. For drift, results are scaled by Cd/(R/I); for a spectrum using a coefficient times g, this factor is gCd/(R/I). 6.2.4.2 Size members. The method used is as follows: 1. Select preliminary member sizes 2. Check deflection and drift (Standard Sec. 12.12) 3. Check the column-beam moment ratio rule (AISC 341 Sec. 9.6) 4. Check beam strength 5. Check connection design (AISC 341 Sec. 9.7) 6. Check shear requirement at panel-zone (AISC 341 Sec. 9.3; AISC 358 Sec. 5.4) After the weight and stiffness have been modified by changing member sizes, the response spectrum must be rescaled for strength. The most significant criteria for the design are drift limits, relative strengths of columns and beams the panel-zone shear. Member strength must be checked but rarely governs for this system. 1. Select Preliminary Member Sizes: The preliminary member sizes are shown for the moment frame in the X-direction in Figure 6.2-3 and in the Y direction in Figure 6.2-4. These sections are selected from AISC SDM Table 1-2, ensuring that they are seismically compact. Members are sized to meet the prequalification limits of AISC 358 Section 5.3 for span-depth ratios, weight flange thickness. Members are also sized for drift limitations and to satisfy strong column–weak beam requirements by using a target ratio of:

∑Zc ≥ 1.25 ∑Zb This proportioning does not guarantee compliance with AISC 341 Section 9.6, but is a useful target that makes conformance likely. Using a ratio of 2.0 may save on detailing costs, such as continuity plates, doublers bracing. The software used accounts explicitly for the increase in beam flexibility due to the RBS cuts. For every beam, RBS parameters were chosen as follows:

a = 0.625b f b = 0.75db c = 0.20b f In accordance with AISC 341 Table I-8-1, beam flange slenderness ratios are limited to 0.3 E / Fy (7.22 for Fy = 50 ksi) beam web height-to-thickness ratios are limited to 2.45 E / Fy (59.0 for Fy = 50 ksi). Since all members selected are seismically compact per AISC SDM Table 1-2, they conform to these limits. For columns in special steel moment frames such as this example, AISC 341 Table I-8-1 Footnote b requires that where the ratio of column moment strength to beam moment strength is less than or

6-47

FEMA P-751, NEHRP Recommended Provisions: Design Examples equal to 2.0, the more stringent λp requirements apply for b/t (given above) when Pu/φbPy is greater than or equal to 0.125, the more stringent h/t requirements apply. Per AISC 341 Table I-8-1, consider the W14x132 column at Gridline B:

h / tw ≤ 1.49 E / Fy

= 22.8 ≤ 35.9

Therefore, the column is seismically compact. Strength checks are performed using ETABS; all members are satisfactory for strength

W14x53

W24x55

7 W24x55

W21x44 RBS

W24x55

W14x53

W21x44

W14x74

W24x76

W21x62 RBS

6 W14x74

W14x74

W21x50 RBS

W14x82

W14x74

W24x76

W21x62 RBS

W14x132

1 W24x146

W24x207 RBS W24x146

W24x146

W21x73 RBS

W24x146

W14x74 W14x132

W14x132

W14x82

W21x62 RBS

Figure 6.2-3 SMRF frame in E-W direction (penthouse not shown) 2. Check Drift: Check drift is in accordance with Standard Section 12.12.1. The building is modeled in three dimensions using ETABS. Displacements at the building corners under the 5 percent accidental torsion load cases are used here. Calculated story drifts, response spectrum scaling factors Cd amplification factors are summarized in Table 6.2-1 below. P-delta effects are included.

6-48

Chapter 6: Structural Steel Design

W14x53

W24x55

7 W24x55

W21x44 RBS

W24x55

W14x53

All story drifts are within the allowable story drift limit of 0.020hsx per Standard Section 12.12 and Section 6.2.3.6 of this chapter.

W14x74

W24x62

W21x44 RBS

6 W14x62

W14x74

W21x44 RBS

W14x82

W24x76

W21x50 RBS

W14x132

1 W24x146

W24x207 RBS

W24x146

W21x62 RBS

W24x146

W14x132

W14x82

W21x44 RBS

Figure 6.2-4 SMRF frame in N-S direction (penthouse not shown)

6-49

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 6.2-1 Alternative A (Moment Frame) Story Drifts under Seismic Loads Elastic Displacement Expected Design Story Drift Ratio at Building Corner, Displacement (=δeCd) From Analysis

Allowable Story Drift Ratio

δe E-W (in.)

δe N-S (in.)

δ E-W (in.)

δ N-S (in.)

Δ E-W/h (%)

Δ N-S/h (%)

Δ/h (%)

Level 7

2.92

3.18

16.0

17.5

1.2

2.0

Level 6

2.66

2.89

14.7

15.9

1.4

1.7

2.0

Level 5

2.33

2.47

12.8

13.6

1.6

2.0

Level 4

1.91

1.95

10.5

10.7

1.9

2.0

Level 3

1.41

1.40

7.76

7.70

1.8

2.0

Level 2

0.90

0.88

4.96

4.85

1.2

2.0

Level 1

0.55

0.52

3.04

2.89

1.1

2.0

Level

1.0 in. = 25.4 mm.

3. Check the Column-Beam Moment Ratio: Check the column-beam moment ratio per AISC 341 Section 9.6. The expected moment strength of the beams is projected from the plastic hinge location to the column centerline per the requirements of AISC 341 Section 9.6. This is illustrated in Figure 6.2-5. For the columns, the moments at the location of the beam flanges are projected to the column-beam intersection as shown in Figure 6.2-6.

Center line of column

Center line of column Location of RBS Center

Center line of beam

M pr

M*Pb

Sh 25'-0"

Figure 6.2-5 Projection of expected moment strength of beam (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)

6-50

Chapter 6: Structural Steel Design

The column-beam strength ratio calculation is illustrated for the lower level in the E-W direction, Level 2, at Gridline D (W24x146 column and W21x73 beam). For the beams:

d ⎛ M pb = M pr + Ve ⎜ Sh + c 2 ⎝

dc ⎞ ⎞ ⎛ ⎟ ± Vg ⎜ Sh + 2 ⎟ ⎠ ⎝ ⎠

where: Mpr = CprRyFy Ze = (1.15)(1.1)(50) (122) = 7,361 in.-kips Ry = 1.1 for Grade 50 steel Ze = Zx -2ctbf (d - tbf) = 172 – 2(1.659)(0.74)(21.24-0.74) = 122 in.3 Sh = Distance from column face to centerline of plastic hinge (see Figure 6.2-9) = a + b/2 = 13.2 in. for the RBS

Ve = 2M pr / L' Vg = wu L' / 2 L’ = Distance between plastic hinges = 248.8 in. wu = Factored uniform gravity load along beam = 1.4D + 0.5L = 1.4[(0.068 ksf)(12.5 ft)+(0.025)(13.3 ft)] + 0.5(0.050 ksf)(12.5 ft) = 2.42 klf

6-51

V*c

Assume inflection point at mid-height M tf = M bf = Z c (F y - P A ) c

M*pc = Z c (F y - P A ) + V*c db c 2 V*c =

M tf,i+ M tf,i+1 hc

Figure 6.2-6 Moment in the column The shear at the plastic hinge (Figure 6.2-7) is computed as:

V p = Ve + Vg where: Vp = Shear at plastic hinge location

6-52

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Chapter 6: Structural Steel Design Mp

Plastic hinges

Figure 6.2-7 Free body diagram bounded by plastic hinges

M pr Vp

M pr

RBS Locations

Figure 6.2-8 Forces at beam-column connection

6-53

FEMA P-751, NEHRP Recommended Provisions: Design Examples

W24x146

W21x73

Sh Sh = a + b 2

a = 0.625b f, = 0.625(8.3) = 51 4" b = 0.75db, = 0.75(21.5) = 153 4" c

c = 0.20b f, = 0.20(8.3) = 13 4"

Figure 6.2-9 Reduced beam section dimensions (1.0 in. = 25.4 mm)

Therefore:

Ve = 2M pr / L' = 2 (7361in ‐kips ) / ( 248.8 in.) = 59.2 kips

Vg = wu L' / 2 = ( 2.42 klf ) (1/12)(248.8 in.) / 2 = 25.1 kips

6-54

Chapter 6: Structural Steel Design

Vp = 59.2 kips + 25.1 kips = 84.3 kips For the beam on the right, with gravity moments adding to seismic:

d ⎞ d ⎞ ⎛ ⎛ M pb,r = M pr + Ve ⎜ Sh + c ⎟ + Vg ⎜ Sh + c ⎟ 2 ⎠ ( 24.74 ⎝⎛ ⎝ ) ⎞ 2 ⎠ ⎛ ( 24.74 ) ⎞ = 9,517 in.-kips = ( 7,361) + (59.2 ) ⎜⎜ (13.2 ) + ⎟⎟ + ( 25.1) ⎜⎜ (13.2 ) + ⎟ 2 ⎠ 2 ⎟⎠ ⎝ ⎝ For the beam on the left, with gravity moments subtracting from seismic:

d ⎞ d ⎞ ⎛ ⎛ M pb,r = M pr + Ve ⎜ Sh + c ⎟ − Vg ⎜ Sh + c ⎟ 2 ⎠ ( 24.74 ⎝⎛ ⎝ ) ⎞ 2 ⎠ ⎛ ( 24.74 ) ⎞ = 8,233 in.-kips = ( 7,361) + (59.2 ) ⎜⎜ (13.2 ) + ⎟⎟ − ( 25.1) ⎜⎜ (13.2 ) + ⎟ 2 ⎠ 2 ⎟⎠ ⎝ ⎝

∑M pb = M pb,r + M pb,l = 9,517 + 8,233 = 17,749 in.-kips Note that in most cases, the gravity moments cancel out and can be ignored for this check. For the columns, the sum of the moments at the top and bottom flanges of the beam is:

⎛

P ⎞

⎝

∑M BF = ∑Zc ⎜⎜ Fyc − Auc ⎟⎟ ⎡

⎛

⎠

∑M BF = 2 ⎢418 in.3 ⎜⎝ 50 ksi − ⎣

228 kips ⎞ ⎤ ⎟ ⎥ = 37,367 in.-kips 43 in.2 ⎠ ⎦

where: MBF = column moment at beam flange elevation Referring to Figure 6.2-6, the moment at the beam centerline is:

∑M pc = ∑M BF + Vc

db 2

where:

Vc = ⎡ M BFi + M BFi+1 ⎤ /hc, based on the expected yielding of the spliced column assuming an ⎣ ⎦

inflection point at column mid-height (e.g., a portal frame) and not the expected shear when the mechanism forms, which is:

6-55

FEMA P-751, NEHRP Recommended Provisions: Design Examples

1 ⎡ 1 ⎤ Vc = ⎢ ∑M pbi + ∑M pbi+1 ⎥ / h , where h is the story height 2 ⎣ 2 ⎦ hc = clear column height between beams = (13.33 ft)(12 in./ft) – 21.24 in. = 139 in.

Vc =

(18,683 in.-kips) + (7,964 in.-kips) = 192 kips (139 in.)

1 1 ⎡ 1 ⎤ ⎡ 1 ⎤ Vc = ⎢ ∑M pbi + ∑M pbi+1 ⎥ / h = ⎢ (17749) + (14976) ⎥ / ⎡⎣(13.33) (12) ⎤⎦ 2 2 ⎣ 2 ⎦ ⎣ 2 ⎦ =102 kips Thus:

∑M pc = 37,367 in.-kips + (192 kips )

( 21.24 in.) = 39, 400 in.-kips 2

The ratio of column moment strengths to beam moment strengths is computed as:

M pc 39, 400 in.-kips ∑ Ratio = = = 2.22 > 1 ∑M pb 17,749 in.-kips

Since the ratio is greater than 2, bracing is only required at the top flange per AISC 341 Section 9.7a. 4. Check the Beam Strength: Per AISC 358 Equation 5.8-4, the beam strength at the reduced section is:

(

)

φ M pr = φ Fy Ze = (0.9) (50 ksi ) 122 in.3 = 5,490 in.-kips From analysis, Mu = 4072 in-kips. Therefore, φMpr ≥ Mu; the beam has adequate strength. The moment at the column face is:

M f = M pr + Ve Sh ± Vg Sh M f ,r = 7,361 in.-kips + ( 59.2 kips )(13.2 in.) + ( 25.1 kips ) (13.2 in.) = 8, 474 in.-kips M f ,r = 8, 474 in.-kips ≤ φd Ry Fy Z x = (1.0 )(1.1)(50 )(172 ) = 9, 460 in.-kips To check the shear in the beam, first the appropriate equation must be selected:

2.24

6-56

E (29,000 ksi) = 2.24 = 53.9 Fyw (50 ksi)

Chapter 6: Structural Steel Design

= 46.6 ≤ 53.9

Therefore:

Vn = 0.6Fy AwCv where Cv = 1.0.

Vn = 0.6 (50 ksi )( 21.2 in.)( 0.455 in.)(1.0 ) = 289 kips Comparing this to Vp:

φVn = 289 ≥ 84.3 = Vp

Check the beam lateral bracing. Per AISC 341 Section 9.8, the maximum spacing of the lateral bracing is:

Lb ≤ 0.086ry E / Fy = 0.086 (1.81 in.)( 29,000 ksi ) / ( 50 ksi ) = 90 in. = 7 '-6" The braces near the plastic hinges are required to have a minimum strength of:

Pbr =

0.06 M u ho

(

0.06 (1.1)(50 ksi ) 172 in.3 2

21.24 in. − 0.74 in.

) = 27.7 kips

where: M u = R yF yZ ho = the distance between flange centroids The required brace stiffness is:

β br =

10 M u Cd φ Lb ho

(

)

10 (1.1)( 50 ksi ) 172 in.3 (1.0) 0.75 ( 6.39 ft )(12 in. / ft ) (21.24 in.2 − 0.74 in.2 )

= 80.2 kips/in.

Lb is taken as Lp. These values are for the typical lateral braces. No supplemental braces are required at the reduced section per AISC 358 Section 5.3.1. 5. Check Connection Design:

6-57

FEMA P-751, NEHRP Recommended Provisions: Design Examples Check the need for continuity plates. Continuity plates are required per AISC 358 Section 2.4.4 unless:

⎛ Fyb Ryb tcf ≥ 0.4 1.8bbf tbf ⎜ ⎜ Fyc Ryc ⎝

⎞ ⎟ ⎟ ⎠

⎛ ( 50 ksi )(1.1) ⎞ ≥ 0.4 1.8 (8.30 in.)( 0.74 in.) ⎜ = 1.33 in. ⎜ ( 50 ksi )(1.1) ⎟⎟ ⎝ ⎠ And:

tcf ≥

bbf 6

8.30 in. = 1.38 in. 6

Since tcf = 1.09 inches, continuity plates are required. See below for the design of the plates. Checking web crippling per AISC 360 Section J10.3:

Ru =

M f ,r db − t f

Rn = 0.40tw2

8,474 in.-kips = 413 kips ( 21.24 in.) − (0.74 in.)

1.5 ⎡ ⎛ tw ⎞ ⎤ EFywt f N ⎛ ⎞ ⎢1 + 3 ⎥ ⎜ d ⎟ ⎜⎜ t ⎟⎟ ⎥ ⎢ tw ⎝ ⎠ ⎝ f ⎠ ⎣ ⎦ 1.5

⎡ ⎛ ( 0.74 + 5 / 16 ) ⎞ ⎛ (0.65) ⎞ Rn = 0.80(0.65) 2 ⎢1 + 3 ⎜⎜ ⎟⎟ ⎜ ⎟ ⎢⎣ ⎝ (24.74) ⎠ ⎝ (1.09) ⎠

⎤ ⎥ ⎥⎦

( 29,000 )(50 ) (1.09) (0.65)

φ Rn = 0.75 (558 kips ) = 419 kips ≥ 413 kips = Ru

= 558 kips

Checking web local yielding per Specification Section J10.2:

Ru = 413 kips

φ Rn = φ ( 5k + N ) Fywtw

φ Rn = (1.00) (5 (1.59 in.) + (0.74 in. + 516 in.)) (50 ksi ) (0.65 in.) = 293 kips

Therefore, since φRn ≤ Ru, as well as due to the check above, continuity plates are required. The force that the continuity plates must take is 413 - 293 = 120 kips. Therefore, each plate takes 60 kips. The minimum thickness of the plates is the thickness of the beam flanges, 0.74 inch. The minimum width of the plates per AISC 341 Section 7.4 is:

6-58

Chapter 6: Structural Steel Design

bpl = b f ,b − 2(k1,b + 1 2 in.) = 8.31 in. − 2 ( 0.875 in. + 0.5 in.) = 5.56 in. Checking the strength of the plate with minimum dimensions:

φ Rn = φ t pl bpl Fy = (1.0)( 0.74 in.)(5.56 in.)(50 ksi ) = 206 kips Therefore, since φRn = 206 kips > 60 kips, the minimum continuity plates have adequate strength. Alternatively, a W24x192 section will work in lieu of adding continuity plates. 6. Check Panel Zone: The Standard defers to AISC 341 for the panel zone shear calculation. The panel zone shear calculation for Story 2 of the frame in the E-W direction at Grid C (column: W24x176; beam: W21x73) is from AISC 360 Section J10.6. Check the shear requirement at the panel zone in accordance with AISC 341 Section 9.3. The factored shear Ru is determined from the flexural strength of the beams connected to the column. This depends on the style of connection. In its simplest form, the shear in the panel zone (Ru) is as follows for W21x73 beams framing into each side of a W24x146 column (such as Level 2 at Grid C):

Ru = ∑

Mf db − t fb

16,285 = 794 kips 21.24 − 0.74

Mf is the moment at the column face determined by projecting the expected moment at the plastic hinge points to the column faces (see Figure 6.2-5):

M f = M pr + Ve Sh ± Vg Sh M f ,r = 7,361 in.-kips + ( 59.2 kips )(13.2 in.) + ( 25.1 kips ) (13.2 in.) = 8, 474 in.-kips M f ,l = 7361in ‐kips + ( 59.2 kips )(13.2 in.) − ( 25.1kips ) (13.2 in.) = 7811in ‐kips Note that in most cases, the gravity moments cancel out and can be ignored for this check. The total moment at the column face is:

∑M f

= M f ,r + M f ,l = 8, 474 in.-kips + 7,811 in.-kips = 16, 285 in.-kips

The shear transmitted to the joint from the story above, Vc, opposes the direction of Ru and may be used to reduce the demand. Previously calculated, this is 102 kips at this location. Thus the frame Ru = 794 - 102 = 692 kips. The column axial force (Load Combination: 1.2D + 0.5L + ΩoE) is Pr = 228 kips.

6-59

FEMA P-751, NEHRP Recommended Provisions: Design Examples

(

)

0.75 Pc = 0.75Fy Ag = 0.75(50 ksi ) 43 in.2 = 1,613 kips Since Pr ≤ 0.75 Pc, using AISC 360 Equation J10-11:

⎛ 3bcf tcf2 ⎞ Rn = 0.60 Fy dc tw ⎜1 + ⎟ ⎜ db dc tw ⎟ ⎝ ⎠ 2 ⎛ ⎞ 3 (12.90 )(1.09 ) ⎟ = 547 kips Rn = 0.60 ( 50 )( 24.74 )( 0.65 ) ⎜1 + ⎜ ( 21.24 )( 24.74 )( 0.65 ) ⎟ ⎝ ⎠

Since φv is 1, φvRn = 547 kips.

φv Rn = 547 kips < 692 kips = Ru Therefore, doubler plates are required. The required additional strength from the doubler plates is 692 - 547 = 145 kips. The strength of the doubler plates is:

φv Rn = 0.6tdoub dc Fy Therefore, to satisfy the demand the doubler plate must be at least 1/4 inch thick. Plug welds are required as:

t = 0.25 in. < (d z + wz ) / 90 = ⎡⎣21.24 + 24.74 − 2 (1.09 )⎤⎦ / 90 = 0.49 in. Use four plug welds spaced 12 inches apart. Alternatively, the use of a W24x192 column will not require doubler plates (φvRn = 737 kips).

6.2.5

Analysis and Design of Alternative B: SCBF

6.2.5.1 Modal Response Spectrum Analysis. As with the SMF, find the approximate building period (Ta) using Standard Equation 12.8-7:

Ta = Ct hnx = ( 0.02 ) (102.3)0.75 = 0.64 sec CuTa, the upper limit on the building period, is determined per Standard Table 12.8-1:

T = CuTa = (1.4)( 0.64) = 0.896 sec It is assumed that the calculated period will exceed CuTa; this is verified after member selection. The seismic response coefficient (Cs) is determined from Standard Equation 12.8-2 as follows:

Cs =

6-60

S DS 1 = = 0.167 R / I 6 /1

Chapter 6: Structural Steel Design However, Standard Equation 12.8-3 indicates that the value for Cs need not exceed: S D1 0.6 Cs = = = 0.112 T ( R / I ) (0.896 sec)(6 / 1) and the minimum value for Cs per Standard Equation 12.8-5 is:

Cs = 0.044ISDS ≥ 0.01 = ( 0.044)(1)(1) = 0.044

Use Cs = 0.112. Seismic base shear is computed using Standard Equation 12.8-1 as:

V = CSW = ( 0.112)(13,156 kips ) = 1,473 kips

where W is the seismic mass of the building as determined above. In evaluating the building in ETABS, twelve modes are analyzed, resulting in a total modal mass participation of 99 percent. The Standard Sec. 12.9.1 requires at least 90 percent participation. As before with Alternative A, strength is scaled to 85 percent of the equivalent lateral force base shear and drift is scaled by gCd/(R/I). 6.2.5.2 Size members. The method used to size members is as follows: 1. Select brace sizes based on strength 2. Select column sizes based on special seismic load combinations (Standard Sec. 12.4.3.2) 3. Select beam sizes based on the load imparted by the expected strength of the braces 4. Check drift (Standard Sec. 12.12) 5. Design the connection Reproportion member sizes as necessary after each check. After the weight and stiffness have been modified by changing member sizes, the response spectrum must be rescaled. Torsional amplification is a significant consideration in this alternate. 1. Select Preliminary Member Sizes and Check Strength: The preliminary member sizes are shown for the braced frame in the E-W direction (seven bays) in Figure 6.2-10 and in the N-S direction (five bays) in Figures 6.2-11 and 6.2-12.

6-61

FEMA P-751, NEHRP Recommended Provisions: Design Examples

W18x50

W18x35

W18x50

8 Brace Sizes: HSS8x8x1 2

SCBF Lines 3 & 4

W18x46

W14x53

W18x35

W14x53

W18x35

W14x53

W18x35

7 HSS51 2x5 1 2 x 5 16

W18x46

HSS6x6x5 8

W18x40

4 W14x132

W14x120

W18x40

W18x50

W14x120

W14x132

W18x50

W18x86

W14x61

W18x86

W14x61

HSS6x6x1 2

HSS6x6x5 8

3 HSS7x7x1 2

W18x50

2 HSS8x8x5 8

1 W14x233

W14x211

W27x114

W14x211

W14x233

W27x114

Figure 6.2-10 Braced frame in E-W direction

6-62

HSS9x9x5 8

Chapter 6: Structural Steel Design

W18x35 SCBF, Lines C & F

W18x35

W14x53

HSS51 2x51 2x3 8

7 HSS51 2x5 1 2x516

W18x35

W18x55

W14x61

HSS6x6x1 2

5 HSS6x6x5 8

W18x35

W14x132

HSS6x6x5 8

3 HSS7x7x1 2

W18x35

2 HSS8x8x5 8

1 W14x233

W14x233

W27x102

HSS9x9x5 8

Figure 6.2-11 Braced frame in N-S direction on Gridlines C and F

6-63

FEMA P-751, NEHRP Recommended Provisions: Design Examples

W18x35 SCBF, Lines D & E

W18x35

W14x53

HSS51 2x51 2x3 8

7 HSS51 2x5 1 2x5 16

W18x35

W18x55

W14x61

HSS6x6x1 2

5 HSS6x6x5 8

W18x35

W14x120

HSS6x6x5 8

3 HSS7x7x1 2

W18x35

2 HSS8x8x5 8

1 W14x211

W14x211

W27x102

HSS9x9x5 8

Figure 6.2-12 Braced frame in N-S direction on Gridlines D and E Check slenderness and width-to-thickness ratios—the geometrical requirements for local stability. In accordance with AISC 341 Section C13.2a, bracing members must satisfy the following:

kl ≤ 200 r

6-64

Chapter 6: Structural Steel Design All members are seismically compact for SCBF per AISC SDM Table 1-2, thus satisfying slenderness requirements. Columns: Wide flange members must comply with the width-to-thickness ratios contained in AISC 341 Table I-8-1. Flanges must satisfy the following:

b ≤ 0.30 E / Fy = 7.23 t Webs in combined flexural and axial compression (where Pu/φbPy = 0.385 > 0.125) must satisfy the following:

⎛ hc P ≤ 1.12 E / Fy ⎜ 2.33 − u ⎜ φb Py tw ⎝

⎞ ⎟ = 52.5 ⎟ ⎠

Braces: Rectangular HSS members must satisfy the following:

b ≤ 0.64 E / Fy = 16.1 t Using a redundancy factor of 1.3 on the earthquake loads, the braces are checked for strength using ETABS and found to be satisfactory. 2. Select Column Sizes: Columns are checked using special seismic load combinations; ρ does not apply in these combinations (see Standard Sec. 12.3.4.1 Item 6). The columns are then checked for strength using ETABS and found to be satisfactory. 3. Select Beam Sizes: The beams are sized to be able to resist the expected plastic and post-buckling capacity of the braces. In the computer model, the braces are removed and replaced with forces representing their capacities. These loads are applied for four cases reflecting earthquake loads applied both left and right in the two orthogonal directions (T1x, T2x, T1y, T2y). For instance, in T1x, the earthquake load is imagined to act left to right; the diagonal braces expected to be in tension under this loading are replaced with the force RyFyAg and the braces expected to be in compression are replaced with the force 0.3Pn. For T2x, the tension braces are now in compression and vice versa. T1y and T2y apply in the other orthogonal direction. The load cases applied are as follows:

(1.2 + 0.2SDS ) D + 0.5L + T (0.9 − 0.2SDS ) D + T

(four combinations; use all four T’s)

Beam strength is checked for each of these eight load combinations using ETABS and found to be satisfactory. 4. Check Story Drift: After designing the members for strength, the ETABS model is used to determine the design story drift. The results are summarized in Table 6.2-2.

6-65

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Table 6.2-2 Alternative B Story Drifts under Seismic Load Elastic Displacement Expected Design Story Drift Ratio at Building Corner, Displacement (=δeCd) From Analysis

Allowable Story Drift Ratio

δe E-W (in.)

δe N-S (in.)

δ E-W (in.)

δ N-S (in.)

Δ E-W/h (%)

Δ N-S/h (%)

Δ/h (%)

Level 7

1.63

1.75

8.14

8.76

0.72

0.93

2.0

Level 6

1.41

1.48

7.07

7.38

0.74

0.94

2.0

Level 5

1.19

1.20

5.97

5.99

0.76

0.84

2.0

Level 4

0.96

0.94

4.80

4.72

0.81

0.85

2.0

Level 3

0.71

0.69

3.56

3.43

0.72

0.71

2.0

Level 2

0.49

0.47

2.44

2.33

0.60

0.59

2.0

Level 1

0.30

0.28

1.49

1.40

0.56

0.52

2.0

Level

1.0 in. = 25.4 mm.

All story drifts are within the allowable story drift limit of 0.020hsx in accordance with Standard Section 12.12 and the allowable deflections for this building from Section 6.2.3.6 above. As shown in the table above, the drift is far from being the governing design consideration. 5. Design the Connection: Figure 6.2-13 illustrates a typical connection design at a column per AISC 341 Section 13.

6-66

Chapter 6: Structural Steel Design

PL 1 2"x33" PL 1"

3. 11 "

W18x35

20.8"

17 "

Sym

14 1/2

Typical

5/16

HSS6x6x5 8 5/8

W14x132

Figure 6.2-13 Bracing connection detail (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) The connection designed in this example is at the fourth floor on Gridline C. The required strength of the connection is to be the nominal axial tensile strength of the bracing member. For an HSS6x6x5/8, the expected axial tensile strength is computed using AISC 341 Section 13.3a: Ru = RyFyAg = (1.4)(46 ksi)(11.7 in.2) = 753 kips The area of the gusset is determined using the plate thickness and section width (based on geometry). See Figure 6.2-13 for the determination of this dimension. The thickness of the gusset is chosen to be 1 inch. For tension yielding of the gusset plate:

φTn = φFyAg = (0.90)(50 ksi)(1 in. × 17 in.) = 765 kips > 753 kips

6-67

FEMA P-751, NEHRP Recommended Provisions: Design Examples For fracture in the net section:

φTn = φFuAn = (0.75)(65 ksi )(1 in. × 17 in.) = 829 kips > 753 kips

For a tube slotted to fit over a connection plate, there will be four welds. The demand in each weld will be 753 kips/4 = 188 kips. The design strength for a fillet weld per AISC 360 Table J2.5 is: φFw = φ(0.6Fexx) = (0.75)(0.6)(70 ksi) = 31.5 ksi For a 1/2-inch fillet weld, the required length of weld is determined to be:

Lw =

188 = 16.9 in. ( 0.707 )(0.5 in.) (31.5 ksi)

Therefore, use 17 inches of weld. In accordance with the exception of AISC 341 Section 13.3b, the design of brace connections need not consider flexure if the gusset can accommodate the inelastic rotation associated with brace postbuckling deformations. This is typically done by providing a “hinge zone"; the gusset plate is detailed such that it can form a plastic hinge over a distance of 2t (where t = thickness of the gusset plate) from the end of the brace. The gusset plate must be permitted to flex about this hinge, unrestrained by any other structural member. See also AISC 341 Section C13.3b. With such a pinned-end condition, the compression brace tends to buckle out-of-plane. During an earthquake, there will be alternating cycles of compression and tension in a single bracing member and its connections. Proper detailing is imperative so that tears or fractures in the steel do not initiate during the cyclic loading. While the gusset is permitted to hinge, it must not buckle. To prevent buckling, the gusset compression strength must exceed the expected brace strength in compression per AISC 341 Section 13.3c. Determine the nominal compressive strength of the brace member. The effective brace length (kL) is the distance between the hinge zones on the gusset plates at each end of the brace member. This length is somewhat dependent on the gusset design. For the brace being considered, kL = 161 inches the expected compressive strength is determined using expected (not specified minimum) material properties per AISC 360 Section E3:

Pn = Fcr Ag where: Ag = gross area of the brace Fcr = flexural buckling stress, determined as follows When:

kL ≤ 5.18 E / Ry Fy = 119 r

6-68

Chapter 6: Structural Steel Design Ry Fy ⎡ ⎢ Fcr ≤ 0.692 Fe ⎢ ⎣

⎤ ⎥ Ry Fy ⎥ ⎦

Otherwise,

Fcr ≤ Fe where:

Fe =

π 2E ⎛ kL ⎞ ⎜ ⎟ ⎝ r ⎠

The equations have been recalibrated to use the expected stress rather than the specified minimum yield stress. Note that the 0.877 factor, which represents out-of-straightness, is not used here in order to calculate an upper bound brace strength and thereby ensure adequate gusset compression strength. Here, kL/r = (1)(161)/(2.17) = 74.2, thus:

Fe =

π 2 (29,000 ksi)

( 74.2 )2

= 52.0 ksi

(1.4 )( 46 ksi ) ⎤ ⎡ 52 ksi ) ⎥ Fcr = ⎢0.692 ( (1.4) ( 46 ksi ) = 40.8 ksi ⎢ ⎥ ⎣ ⎦

(

)

Pn = ( 40.8 ksi ) 11.7 in.2 = 478 kips Now, using the expected compressive load from the brace of 449 kips, check the buckling capacity of the gusset plate using the section above. By this method, illustrated by Figure 6.2-13, the compressive force per unit length of gusset plate is (478 kips/23.5 in.) = 20.3 kips/in. Try a plate thickness of 1 inch: fa = P/A = 20.3 kips/(1 in. × 1 in.) = 20.3 ksi The gusset plate is modeled as a 1-inch-wide by 1-inch-deep rectangular section, fixed at both ends. The length, from geometry, is 17.2 inches. The effective length factor, k, for this “column” is 1.2 per AISC 360 Table C-C2.2. The radius of gyration, r, for a plate is t / 12 . Per AISC 360 Section E3:

kL (1.2 ) (17.2 in.) = = 71.2 r (0.29 in.)

6-69

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Fe =

π 2 (29,000 ksi)

( 71.2 )2

= 56.5 ksi

(50 ksi ) ⎤ ⎡ 56.5 ksi ) ⎥ ( Fcr = ⎢0.658 (50 ksi ) = 34.5 ksi ⎢ ⎥ ⎣ ⎦

Fcr = 34.5 ksi > 20.3 ksi = f a

Next, check the reduced section of the tube, which has a 1-1/8-inch-wide slot for the gusset plate (the thickness of the gusset plus an extra 1/8 inch for ease of construction). The reduction in HSS6x6x5/8 section due to the slot is (0.581 in. × 1.125 in. × 2) = 1.31 in.2 the net section, Anet = (11.7 - 1.31) = 10.4 in.2 To ensure gross section yielding governs, reinforcement is added over the area of the slot. The shear lag factor is computed per AISC 360 Table D3.1:

U =1− x

where: 2 B 2 + 2BH (6 in.) + 2 ( 6 in.) (6 in.) x= = = 2.25 in. 4( B + H ) 4(6 in. + 6 in.)

and l is the length of the weld as determined above.

U =1−

(2.25 in.) = 0.867 (17 in.)

Thus, the effective area of the section is:

(

)

Ae = UAnet = ( 0.867 ) 10.4 in.2 = 9.02 in.2 Try a reinforcing plate 1/2 inch thick and 3-1/2 inches wide on each side of the brace. (The necessary width can be computed from the effective area, but that calculation is not performed here.) Grade 50 material is used in order to match or exceed the brace material strength, thus allowing for treatment of the material as hom*ogenous. The area of the section is (2 × 0.5 in. × 3.5 in.) = 3.5 in.2. The distance of its center of gravity from the center of gravity of the slotted brace is:

B treinf (6 in.) (0.5 in.) + = + = 3.25 in. 2 2 2 2

Thus, the area of the reinforced section is:

A = An + Areinf = 10.4 in.2 + 3.5 in.2 = 13.9 in.2

6-70

Chapter 6: Structural Steel Design

The weighted average of the x’s is 2.53 inches. Thus, the shear lag factor for the reinforced section is:

U =1−

(2.53 in.) = 0.850 (16.9 in.)

Thus, the effective area of the section is:

(

)

Ae = UAnet = ( 0.850) 13.9 in.2 = 11.8 in.2 Now, check the effective area of the reinforced section against the original section of the brace per AISC 341 Section 13.2b:

Ag Ae

(11.7 in.2 ) = 0.99 ≤ 1 (11.8 in.2 )

The reinforcement is attached to the brace such that its expected yield strength is developed.

Ru = Areinf Ry Fy = 3.5 in.2 (1.1)(50 ksi ) = 193 kips The plate will be developed with two 5/16-inch fillet welds, 14 inches long:

Rn = 2φ 0.6Fexx 2

sL = 2(0.75)(0.6) ( 70 ksi ) 2

in.)(14 in.) = 195 kips

The force must be developed into the plate, carried past the reduced section developed out of the plate. To accomplish this, the reinforcement plate will be 33 inches: 14 inches on each side of the reduced section, 2 inches of anticipated over slot, plus 1 inch to provide erection tolerance. The complete connection design includes the following checks (which are not demonstrated here): §

Attachment of reinforcement to brace

§

Brace shear rupture

§

Brace shear yield

§

Gusset block shear

§

Gusset yield, tension rupture, shear rupture weld at both the column and the beam

§

Web crippling and yielding for both the column and the beam

§

Gusset edge buckling

§

Beam-to-column connection

6-71

FEMA P-751, NEHRP Recommended Provisions: Design Examples 6.2.6

Cost Comparison

For each case, the total structural steel was estimated. The takeoffs are based on all members, but do not include an allowance for plates and bolts at connections. The result of the material takeoffs are as follows: §

Alternative A, Special Steel Moment Resisting Frame: 640 tons

§

Alternative B, Special Steel Concentrically Braced Frame: 646 tons

The higher weight of the systems with bracing is primarily due to the placement of the bracing in the core, where resistance to torsion is poor. Torsional amplification and drift limitations both increased the weight of steel in the bracing. The weight of the moment-resisting frame is controlled by drift and the strong column rule. 6.3

TEN-‐STORY HOSPITAL, SEATTLE, WASHINGTON

This example features a buckling-restrained braced frame (BRBF) building. The example covers: §

Analysis issues specific to buckling-restrained braced frames

§

Proportioning of buckling-restrained braces

§

Capacity design principles

§

Nonlinear response history analysis

§

Buckling-restrained brace connections

6.3.1

Building Description

This ten-story hospital includes a two-story podium structure beneath an eight-story tower, as shown in Figures 6.3-1 and 6.3-2. The podium is 211.3 feet by 121.3 feet in plan, while the tower’s floorplate is square with 91.3-foot sides. Story heights are 18 feet in the podium and reduce to 15 feet throughout the tower, bringing the total building height to 156 feet. As the tower is centered horizontally on the podium below, the entire building is symmetric about a single axis. Both the podium and the tower have large roof superimposed dead loads due to heavy HVAC equipment located there.

6-72

Chapter 6: Structural Steel Design

W8x10 W16x26

W21x44

W16x26 W16x26

W16x26

W21x50

W16x26

W16x26 W16x26

W16x26

W16x26 W16x26

W21x50

W16x26

W21x50

W16x26 W16x26

W16x26

W16x26 W16x26

W21x44

W16x26 W16x26

W21x50

W21x62 W21x44

W21x44

W16x26

W21x50

W21x44

W16x26

W8x10

W21x50

W16x26

W21x44

W21x50 W16x26 W8x10

W8x10

W16x26

W21x44

W8x10

W21x62

W21x44

W8x10

Figure 6.3-1 Typical tower plan

W16x26

W16x31

W16x31 W16x31

W16x26 W16x26

W16x31 W16x31

W21x57

W16x26

W21x68

W16x26

W21x50

W16x31

W21x50

W16x31

W21x50

W16x31

W21x57

W16x26

W16x31 W16x31

W16x31

W16x26

W16x31

W16x31 W16x31

W16x26 W16x26

W16x31 W16x31

W21x83

W16x26

W21x83

W16x26

W21x50

W16x26

W21x50

W16x31

W21x83

W16x31

W16x31 W16x31

W16x31

W21x57

W16x31

W8x10 W16x31

W16x31 W16x31 W21x44

W21x57

W21x68

W16x31

W21x55

W16x31

W16x31 W16x31

W21x44 W21x44

W16x31

W16x26

W21x44

W16x31

W16x31 W16x31

W21x44

W16x31

W21x83

W16x26

W16x31 W21x83

W21x50

W21x83

W16x26

W8x10 W21x44

W16x26

W21x55

W21x44

W16x26

W16x26 W16x26

W16x31

W21x57

W21x44

W16x31

W21x50

W16x26 W16x26

W8x10 W16x26 W21x57

W8x10

W21x44

W16x26 W16x26

W16x31

W21x83

W21x44

W16x31

W8x10

W8x10 W21x50

W16x31

W21x57

W21x44

W16x31

W21x50

W21x44

W16x31

W8x10 W16x26 W21x57

W8x10

W21x83

W21x44

Figure 6.3-2 Level 3 podium plan The structure exemplifies a common situation for hospital facilities. The combination of a stiff podium structure beneath a more flexible tower results in significant force transfer at the floor level between them.

6-73

FEMA P-751, NEHRP Recommended Provisions: Design Examples The vertical-load-carrying system consists of lightweight concrete fill on steel deck floors supported by steel beams and girders that span to steel columns. The bay spacing is 30 feet each way. There are three floor beams per bay. All beams and girders are composite. BRBFs have been selected for this building because they provide high stiffness paired with a high degree of ductility and stable hysteretic properties. The building has a thick mat foundation. The foundation soil is representative of Site Class C conditions identical to those discussed in Section 3.2. The design of foundations is not included here. 6.3.1.1 Design method. Seismic forces, rather than wind forces, govern the building’s lateral design (in part due to the mass of the thick concrete-filled decks). The lateral force-resisting system throughout the tower consists of BRBFs in the middle bay along each side of the perimeter—Gridlines 3, 6, A D, as can be seen in the representative elevation of Figure 6.3-3. These BRBFs deliver lateral loads to the collectors and diaphragm at the third floor where both in-plane and out-of-plane discontinuities exist. This transfer occurs in-plane along Gridlines A and D to two braced bays nearer the ends of the podium and out-of-plane from a single braced bay in the tower along Gridlines 3 and 6 to braces in the two adjacent bays along Gridlines 2 and 7 in the podium below. The podium bracing configuration is illustrated in Figure 6.3-2. A typical bracing elevation in the transverse direction of the podium (illustrating the out-of-plane offset) is shown in Figure 6.3-4.

W14x30 W14x30

Story 9

W14x43

Story 8

W14x43

Story 7

W14x48

Story 6

W14x48

Story 5

W14x61

Story 4

n2 6i

Figure 6.3-3 Longitudinal elevation at Gridline D

6-74

W10x39

W16x31

W14x132

W16x31

W10x39

W16x31 W14x132

W16x40

W14x132

W14x61

Story 3

W16x31

W14x132

W14x48 W14x48 W14x193 W14x193 W14x132 W14x132 W14x132 W14x132

W21x57

Story 10

W16x31

W14x257

W16x31

W14x257

W14x48 W14x48 4 in 2 W14x193 W14x193 W14x132 W14x132 W14x132 W14x132 6 5. 4. 6. 4. 4 in 2 5 5 5 5 in 2 in 2 in 2 in 2 in 2

W21x44

W14x257

W14x43 W14x43 W14x48 W14x48 W14x61

W21x50 6.

W14x61

W21x44

W21x50 6

W14x132

W21x44

W14x132

W21x44

W14x132

W21x50 5.

W21x44

W14x132

W10x39

W21x50 4.

W10x39

W21x44

W16x31

W21x57

W21x44

W21x50 4.

W21x44

W21x50 in

W21x44

W21x50

W16x31

W21x44

in 2

W16x31

W16x40

W21x44

4.5

W16x31

W21x44

W14x30

W21x44

Story 2 Story 1 Base

Chapter 6: Structural Steel Design

W14x30

W14x48

W14x30 W14x43 W14x43 W14x48 W14x48

W21x44

W14x61

W21x44

6. 5

W14x61

W21x62

W14x132

W21x50

W14x132

W14x257

W16x50

W21x83

W14x257

W21x50

W14x257

W14x48 4 in 2

W14x48 W14x193 W14x193 W14x132 W14x132 W14x132 W14x132

W21x44

W14x257

W21x62 6

W14x193 W14x193 W14x132 W14x132 W14x132 W14x132 6 5 4. 6. 4. 4 in 2 in 2 5 5 5 in 2 in 2 in 2 in 2

W14x43 W14x43 W14x48 W14x48

W14x61

W21x62 5

W14x61

W21x44

W14x132

W21x44

W14x132

W10x33

W21x62 4.

W10x33

W21x44

W21x50

W21x83

W21x57

W21x44

W21x62 4.

W21x44

W21x62 4

W21x44

W21x62 2

W21x44

in 2

W14x30

W21x44

4 .5

W21x57

W21x62 W14x48

W14x30

W21x44

Story 10 Story 9 Story 8 Story 7 Story 6 Story 5 Story 4 Story 3 Story 2 Story 1 Base

Gridlines 3 & 5

W14x82

in 2

W14x82

W14x74

5 .5 6i n2

W14x74

W14x74 W14x74

5 .5

in 2

W14x132

6i n2

Story 1

W14x132

W21x83 n 6i

n 6i

W10x33

W21x57

Story 2 in

W21x83

W21x83 5 .5

W21x57

W21x83 5 .5

W10x33

W21x57

Base

Gridlines 2 & 6

Figure 6.3-4 Transverse elevations An ELF analysis is first performed to scale the base shear for the subsequent MRSA used for strength design of the buckling-restrained braces (BRB). Each BRB is designed for its share of 100 percent of the horizontal component of the earthquake lateral load without considering additional tributary vertical loads. This is done to encourage distributed yielding of braces up the height of the structure and is justified because the braces will shed any gravity load upon first yield and transfer it to the connecting beams and columns, which are designed to accommodate gravity loads without support provided by the braces. Beams, columns collectors are preliminarily sized using capacity design principles considering plastic mechanisms that develop based on the brace sizes determined using elastic MRSA. Finally, a 6-75

FEMA P-751, NEHRP Recommended Provisions: Design Examples nonlinear response history analysis (NRHA) is executed to verify BRB strains remain at acceptable levels, check that story drifts do not exceed allowable limits possibly reduce column sizes from what the plastic mechanism analyses require. The details of this design procedure are summarized in Table 6.3-1. Table 6.3-1 Design Philosophy Element BRBs Columns Diaphragms Collector beams, bracedbay beams, etc.

6.3.2

Action

Analysis Method

Software

Acceptance Criteria

Strength

MRSA

ETABS

AISC 341

Deformation

NRHA

PERFORM

ASCE 41

Strength

NRHA

PERFORM

AISC 360

Rotation

NRHA

PERFORM

ASCE 41

Story Drift

NRHA

PERFORM

ASCE 7

Strength

MRSA

ETABS

AISC 360

Basic Requirements

6.3.2.1 Provisions parameters. Section 3.2 illustrates the determination of design ground motion parameters for this example. They are as follows: §

SDS = 0.859

§

SD1 = 0.433

§

Occupancy Category IV

§

Seismic Design Category D

For BRBFs (Standard Table 12.2-1): §

R=8

§

Ωo = 2.5

§

Cd = 5

These values are representative of a seismic force-resisting system that is a unification of two previously different BRBF system classifications: those with non-moment-resisting beam-column connections and those with moment-resisting beam-column connections. Modifications to Section 15.7 of AISC 341 now require the beam-to-column connections for a building frame system either to meet the requirements for fully restrained (FR) moment connections as specified in AISC 341 Section 11.2a or to possess sufficient rotation capacity to accommodate the rotation required to achieve a story drift of 2.5 percent. The later compliance path is selected for this design example. This requirement effectively amounts to a braced frame with simple beam-to-column connections per AISC 360 Section B3.6a with rotation specified at 2.5 percent. A standard detail illustrating this connection is presented in Figure 6.3-5.

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Chapter 6: Structural Steel Design

Figure 6.3-5 “Pinned” beam-to-column connection 6.3.2.2 Loads. §

Roof live load, Lr: 25 psf

§

Roof dead load, D: 135 psf

§

Exterior wall cladding: 300 plf of spandrel beams

§

Floor live load, L: 60 psf

§

Partitions: 10 psf

§

Floor dead load, D: 104 psf

§

Floor live load reductions: per the IBC

Roof dead load includes roofing, insulation, lightweight concrete-filled metal deck, concrete ponding allowance, framing, mechanical and electrical equipment, ceiling fireproofing. Floor dead load includes lightweight concrete-filled metal deck, ponding allowance, framing, mechanical and electrical equipment, ceiling and fireproofing. Due to potential for rearrangement, partition loads are considered live loads per Standard Section 4.2.2 but are also included in the effective seismic weight in accordance with Standard Section 12.7.2. Therefore, the seismic weight of a typical tower floor, whose footprint is a square 8,342 ft2 in area, is 104 + 10 + 300(30)(3)(4)/8,342 = 127 psf.

6-77

FEMA P-751, NEHRP Recommended Provisions: Design Examples 6.3.2.3 Materials §

Concrete for drilled piers: fc' = 5 ksi, normal weight (NW)

§

Concrete for floors: fc' = 3 ksi, lightweight (LW)

§

All other concrete: fc' = 4 ksi, NW

§

Structural steel: Wide flange sections: ASTM A992, Grade 50 Plates: ASTM A36

6.3.3

Structural Design Criteria

6.3.3.1 Building configuration. The hospital building does not possess any stiffness, strength, or weight irregularities despite the relatively tall height of the podium stories. At the podium levels, the two braced bays corresponding to each line of single-bay chevron bracing in the tower above provide more than enough additional strength to compensate for the slight increase in floor-to-floor height. The story drift ratio increases up the full height of the structure, meeting the exception of Standard Section 12.3.2.2 for assessing vertical stiffness and weight irregularities. However, the structure does possess both a vertical geometric irregularity (Type 3) and an in-plane discontinuity in vertical lateral force-resisting element irregularity (Type 4) since the lateral force-resisting system transitions from a single chevron braced bay in the tower to two chevron braced bays at the podium levels. The two chevron braced bays in the podium occur two bays away from the tower braced bay in the longitudinal direction. The Type 4 vertical irregularity triggers an increase in certain design forces per Standard Sections 12.3.3.3 and 12.3.3.4. Together, the Type 3 and Type 4 vertical irregularities preclude the use of an equivalent lateral force analysis as defined in Standard Section 12.8 based on the permissions in Standard Table 12.6-1. Note that this analysis prohibition is also triggered by the flexibility of the structure, as its fundamental period (see Sec. 6.3.4.1) exceeds 3.5Ts = 3.5(SD1/SDS) seconds = 3.5 × (0.433/ 0.859) seconds = 1.76 seconds. Nevertheless, the design base shear still must be determined using the equivalent lateral force analysis procedures to ensure that the design base shear for a modal response spectrum analysis meets the requirements of Standard Section 12.9.4. Due to the building’s symmetry and the strong torsional resistance provided by the layout of the vertical lateral force-resisting elements, numerous plan irregularities are not expected. Analysis reveals that the structure is torsionally regular the only horizontal structural irregularity present is an out-of-plane offset irregularity (Type 4) triggered by the shift in the vertical lateral force-resisting system from Gridlines 3 and 6 in the tower to Gridlines 2 and 7 in the podium structure below. The only additional provisions triggered by the Type 4 horizontal structural irregularity relate to three-dimensional modeling requirements. 6.3.3.2 Redundancy. The limited number of braced bays in each direction of the tower require the redundancy factor (ρ) to be taken as 1.3 per Standard Section 12.3.4.2 Item a and Table 12.3-3. Because there are only two BRBF chevrons in each direction throughout the tower, removal of a single brace would dramatically increase flexural demands in the beam at that location and would certainly result in at least a 33 percent reduction in story strength even if the resulting system does not have an extreme torsional irregularity. The 1.3 redundancy factor (ρ) is incorporated as a load factor on the seismic loads used in the design of the braces.

6-78

Chapter 6: Structural Steel Design 6.3.3.3 Orthogonal load effects. Standard Section 12.5.4 stipulates a combination of 100 percent of the seismic forces in one direction plus 30 percent of the seismic forces in the orthogonal direction, at a minimum, for structures in Seismic Design Category D. However, it has been shown (Wilson, 2004) that use of the 100/30 percentage combination rule can result in member designs that are not equally resistant to earthquake ground motions originating from different directions. Instead, a SRSS combination of seismic forces from two full-magnitude response spectra analyses conducted along each principal axis of the building is performed to ensure the design forces remain independent of the selected reference coordinate system (in this case, the building’s main orthogonal axes). In the context of NRHA, orthogonal pairs of ground motion acceleration histories are applied simultaneously in accordance with the requirements of Standard Section 12.5.4 for structures in Seismic Design Category D. 6.3.3.4 Structural component load effects. The effect of seismic load as defined by Standard Section 12.4.2 is as follows: E = ρQE ± 0.2SDSD In this example, SDS = 0.859. The seismic load is combined with the gravity loads in elastic analyses as shown in Standard Section 12.4.3.2, resulting in the following load combinations: 1.37D + 0.5L + 0.2S + ρQE 0.73D + 1.6H + ρQE The 0.5 coefficient on L is permitted for all occupancies in which Lo in Standard Table 4.1 is less than or equal to 100 psf per Exception 1 to Standard Section 2.3.2. The braces are designed without considering additional tributary vertical loads to encourage distributed yielding up the height of the structure. However, the surrounding beams and columns that are part of the lateral force-resisting system are designed for the above gravity loads in conjunction with the earthquake effect as specified in AISC 341 Section 16.5b. Again, the redundancy factor, ρ, is taken as 1.3 for design of the braces themselves. In a NRHA, the structure is analyzed for the effects of the scaled pairs of ground motions simultaneously with the effects of dead load and 25 percent of the required live loads per Standard Section 16.2.3. 6.3.3.5 Drift limits. For a building assigned to Occupancy Category IV, the allowable story drift (Standard Sec. 12.12.1 and Table 12.12-1) is Δa = 0.010hsx. The allowable story drift for a typical podium floor is Δa = (0.01)(18 ft)(12 in./ft) = 2.16 in. The allowable story drift for a typical tower floor is Δa = (0.01)(15 ft)(12 in./ft) = 1.80 in. The calculated design story drifts are amplified by the appropriate Cd factor from Standard Table 12.2-1 in elastic analysis procedures that employ seismic response coefficients reduced by the appropriate response modification factor, R. Standard Section 16.2.4.3 permits the allowable story drift obtained from a nonlinear response history analysis to be increased by 25 percent relative to the drift limit specified in Section 12.12.1.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples The maximum allowable value of story drifts summed to the roof of the ten-story hospital building (156 feet) obtained from an elastic analysis is 18.72 inches. This same figure extracted from a nonlinear response history analysis cannot exceed 1.25(18.72 in.) = 23.40 in. 6.3.3.6 Seismic weight. The area of the tower floorplate is approximately equal to [(3)(30 ft) + (2)(8 in.)(1 ft/12 in.)]2 = 8,342 ft2, while the area of the podium floorplate is approximately [(7)(30 ft) + (2)(8 in.)(1 ft/12 in.)] × [(4)(30 ft) + (2)(8 in.)(1 ft/12 in.)] = 25,642 ft2. Thus, the weights that contribute to seismic forces are as follows: §

Tower roof: Roof D = (0.135)(8,342) Cladding = (4)(3)(30)(0.300) Total

§

Tower floor: Floor D = (0.104)(8,342) Partitions = (0.010)(8,342) Cladding = (4)(3)(30)(0.300) Total

§

= 868 kips = 83 kips = 108 kips = 1,059 kips

Podium roof: Roof D = (0.135)(25,642 - 8,342) Floor D = (0.104)(8,342) Partitions = (0.010)(8,342) Cladding = (2)(11)(30)(0.300) Total

§

= 1,126 kips = 108 kips = 1,234 kips

= 2,336 kips = 868 kips = 83 kips = 198 kips = 3,485 kips

Podium floor: Floor D = (0.104)(25,642) Partitions = (0.010)(25,642) Cladding = (2)(11)(30)(0.300) Total

= 2,667 kips = 256 kips = 198 kips = 3,121 kips

Total effective seismic weight of building = 1,234 + 7(1,059) + 3,485 + 3,121 = 15,253 kips 6.3.4

Elastic Analysis

The base shear is determined using an ELF analysis; the base shear so computed is needed later when evaluating the scaling of the base shears obtained from the modal response spectrum analysis. In a subsequent section (Section 6.3.6.3.3), columns are designed using forces obtained from nonlinear response history analyses that are intended to represent the maximum force that can develop in these elements per the exception to Standard Section 12.4.3.1. Compliance with story drift limits is also evaluated using the results of the nonlinear response history analyses.

6-80

Chapter 6: Structural Steel Design 6.3.4.1 Equivalent Lateral Force procedure. First, the ELF base shear will be determined, followed by its vertical distribution up the height of the building. 6.3.4.1.1 ELF base shear. Compute the approximate building period, Ta, using Standard Equation 12.8-7:

(

)

Ta = Ct hnx = (0.03) 1560.75 =1.32 sec In accordance with Standard Section 12.8.2, the building period used to determine the design base shear must not exceed the following:

Tmax = CuTa = (1.4 )(1.32 ) =1.85 sec The subsequent three-dimensional modal analysis finds the computed period to be approximately 2.30 seconds in each principal direction. Thus the upper limit on the fundamental period Tmax applies. The seismic response coefficient, Cs, is computed in accordance with Standard Section 12.8.1.1. Equation 12.8-2 provides the value of Cs that generally governs at short periods:

Cs =

S DS 0.859 = = 0.161 R / I 8 / 1.5

However, Standard Equation 12.8-3 indicates that the value for Cs need not exceed the following:

Cs =

S D1 0.433 = = 0.044 T ( R / I ) (1.85) (8 / 1.5)

and the minimum value for Cs per Standard Equation 12.8-5 is:

Cs = 0.044ISDS ≥ 0.01 = ( 0.044 )(1.5)(0.859 ) = 0.057 Therefore, use Cs = 0.057. The seismic base shear is computed per Standard Equation 12.8-1 as follows:

V = CsW = ( 0.057 )(15,253) = 865 kips The redundancy factor (ρ) is accounted for by setting the coefficient on the horizontal seismic load effect to 1.3 in all earthquake load combinations used for strength design of the BRBs. The redundancy factor is not applicable to the determination of deflections. 6.3.4.1.2 Vertical distribution of ELF seismic forces. Standard Section 12.8.3 prescribes the vertical distribution of lateral force in a multilevel structure. The floor force, Fx, is calculated using Standard Equation 12.8-11 as:

Fx = CvxV

6-81

FEMA P-751, NEHRP Recommended Provisions: Design Examples where (per Standard Eq. 12.8-12):

Cvx =

wx hxk

∑ wi hik i −1

Using the data in Section 6.3.3.5 of this example and interpolating the exponent k as 1.68 for the period of 1.85 seconds, the vertical distribution of forces for the ELF analysis is shown in Table 6.3-2. The seismic design shear in any story is computed as follows (per Standard Eq. 12.8-13): n

Vx = ∑Fi i= x

Table 6.3-2 ELF Vertical Seismic Load Distribution Weight (wx)

Height (hx)

w xh xk

Cvx

Roof

1,234 kips

156 ft

5,818,337

0.24

210 kips

Story 10

1,059 kips

141 ft

4,215,392

0.18

152 kips

362 kips

Story 9

1,059 kips

126 ft

3,491,537

0.15

126 kips

488 kips

Story 8

1,059 kips

111 ft

2,823,657

0.12

102 kips

590 kips

Story 7

1,059 kips

96 ft

2,214,116

0.09

80 kips

670 kips

Story 6

1,059 kips

81 ft

1,665,744

0.07

60 kips

730 kips

Story 5

1,059 kips

66 ft

1,182,040

0.05

43 kips

772 kips

Story 4

1,059 kips

51 ft

767,496

0.03

28 kips

800 kips

Story 3

3,485 kips

36 ft

1,409,320

0.06

51 kips

851 kips

Story 2

3,121 kips

18 ft

395,253

0.02

14 kips

865 kips

23,982,891

1.00

865 kips

Level

Total

15,253 kips

1.0 kip = 4.45 kN 1.0 ft = 30.5 cm

All floor decks, including the roofs, are constructed of 3¾ in. lightweight concrete over 3 in. metal deck. Standard Sec. 12.3.1.2 allows for such diaphragms to be modeled as rigid as long as their span-to-depth ratios do not exceed three. Since the floor span-to-depth ratio is a maximum of 1.75 at the podium levels, the hospital diaphragms meet these conditions. However, Standard Sec. 12.3.1.2 also requires the structure to have no horizontal irregularities for its diaphragms to be modeled as rigid. Due to the out-ofplane offsets irregularity (Type 4) in the transverse lateral frames at the podium-to-tower interface, the hospital does not meet this restriction. As such, the effect on the vertical lateral force distribution of explicitly considering the stiffness of the diaphragm at the podium roof level was examined in a threedimensional computer model of the structure and found to be insignificant (i.e., results closely matching a model with rigid diaphragms) due to the stiff nature of the thick concrete floor. Thus, it was deemed acceptable to model all diaphragms as rigid for subsequent analyses. The rigid diaphragm assumption is especially helpful in the context of nonlinear response history analysis, where the additional degrees of freedom needed to model diaphragm stiffness explicitly can render analysis times prohibitive.

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Chapter 6: Structural Steel Design Assessment of load transfer in the level 3 diaphragm would require a separate, explicit analysis. Another reasonable approach to the primary model would be to include the level 3 diaphragm explicitly and model all other diaphragms as rigid. 6.3.4.2 Three-dimensional static and Modal Response Spectrum Analysis. The three-dimensional analysis is performed for this example to accurately account for the following: §

The different centers of mass for the podium and tower levels

§

The varying stiffness of the braced frames as they transition from their wide configuration at the podium levels to a single-bay arrangement throughout the tower

§

The effects of bi-directional frame interaction on the columns engaged by orthogonal braced frames in the podium

§

The ability of the braced frames to control torsion

The braced frames and diaphragm chords and collectors, together with all gravity system beams and columns, are explicitly modeled using three-dimensional beam-column elements. The floor diaphragms are modeled as rigid. As mentioned previously, the ELF analysis procedure of Standard Section 12.8 is not admissible for this structure due to the restrictions on fundamental period in Standard Table 12.6-1. However, the ELF analysis of the three-dimensional model is still useful in assessing whether torsional irregularities are present. The ELF seismic forces derived in Table 6.3-2 above are applied to each diaphragm at 5 percent eccentricity orthogonal to the direction of loading. The maximum and average story drifts along an edge transverse to the direction of loading for the critical direction of eccentricity at each level are then compared. This ratio of δmax/δavg never exceeds 1.11, which is below the 1.2 limit that defines torsional irregularity. For this torsionally regular structure, the accidental torsion amplification factor, Ax, is equal to 1.0. A three-dimensional modal response spectrum analysis is performed per Standard Section 12.9 using the three-dimensional computer model. The design response spectrum is based on Standard Section 11.4.5 and is shown in Figure 6.3-6.

6-83

FEMA P-751, NEHRP Recommended Provisions: Design Examples 1

Spectral acceleration, Sa (g)

SDS = 0.859

0.5

SD1 = 0.433

2 Period, T (s)

Figure 6.3-6 Design response spectrum, 5 percent damped Within this model, the first twelve modes of vibration and the corresponding mode shapes of the structure were determined. Twelve modes provide more than enough participation to capture 90 percent of the actual mass in each direction of response as required by Standard Section 12.9.1. The design value for modal base shear, Vt, is determined by combining the individual modal values for base shear after dividing the design response spectrum by the quantity R/I = 8/1.5 = 5.33 as prescribed by Standard Section 12.9.2. The complete quadratic combination (CQC) modal combination rule was selected for this task to account for coupling of closely-spaced modes that are likely present in symmetrical structures. Five percent modal damping in all modes is specified for the response spectrum analysis to match the assumption used in deriving the design response spectrum of Figure 6.3-6. Base shears thus obtained from the model having an effective seismic weight of 15,253 kips are as follows: §

Longitudinal: Vt = 627 kips

§

Transverse: Vt = 637 kips

In accordance with Standard Section 12.9.4, the design values of modal base shear are compared to the base shear determined by the ELF method. If the design value for modal base shear is less than 85 percent of the ELF base shear calculated using a period of CuTa (see Sec. 6.3.4.1.1 above), a factor greater than unity must be applied to the design forces to raise the modal base shear up to this minimum ELF comparison value. Accordingly: §

Multiplier: 0.85 (V/Vt)

§

Longitudinal multiplier: 0.85 (865 kips / 627 kips) = 1.17

§

Transverse multiplier: 0.85 (865 kips / 637 kips) = 1.15

In a typical elastic analysis, it is recommended to examine lateral displacements early in the design process, as seismic (or wind) drift often controls the design of taller structures. For this building, 6-84

Chapter 6: Structural Steel Design compliance with drift limits will ultimately be checked using a nonlinear response history analysis. However, lateral displacements are still examined in the elastic analysis to ensure they remain reasonably close to the limits derived in Section 6.3.3.5 (that is, in order to prevent wasting analysis effort on a design that is unlikely to meet the drift limits). This check is illustrated in Section 6.3.4.3 below. To obtain elastic design forces for the BRBs, the results from the two orthogonal MRSAs in the threedimensional model are combined via SRSS, exceeding the requirements of Standard Section 12.5.4. 6.3.4.3 Preliminary drift assessment. Seismic drift is examined in accordance with Standard Section 12.12.1. The design story drift in each translational direction was extracted from the threedimensional ETABS model corresponding to the response spectrum case, including 5 percent accidental torsion, exciting that same direction. Although only strictly required for structures possessing torsional irregularities as defined by Standard Table 12.3-1, story drift was nonetheless examined at the building corners rather than the centers of mass for this structure because the location of cladding attachment is the most critical location for this check. The lateral deflections obtained from the response spectrum analysis must be multiplied by Cd/I(I/R) = Cd//R = 5/8 = 0.625 to find the design story drift. However, the response spectrum used in the analysis has already been scaled twice. The spectrum was first scaled by R/I = 8 / 1.5 = 5.33 to obtain design-level forces; thus the resulting displacements can be amplified by Cd/I = 5 / 1.5 = 3.33 to obtain expected drifts. The second scaling was by a factor (in each direction) to ensure that the design base shear forces in each direction to meet the minimum 85 percent of ELF base shear. This latter scaling does not apply to drifts, per Standard Section 12.9.4. Thus, the 1.17 and 1.15 scale factors applied in the longitudinal and transverse directions, respectively, must be divided back out of the drifts extracted from the model used to obtain design forces for the braces. The resulting scale factors applied to the results of the scaled spectra are 3.33/1.17 = 2.85 and 3.33/1.15 = 2.90 applied in the longitudinal and transverse directions, respectively. (It would also be possible to simply perform an additional response spectrum analysis with the design spectrum multiplied by 0.625 and use the resulting story drifts directly.) Story drifts in all ten stories of the hospital building are within the allowable story drift limit of 0.010hsx per Standard Section 12.12.1 and Section 6.3.5.5 of this chapter. Although story drifts calculated using MRSA reach a maximum value at the roof level that is just 89 percent of the 0.010hsx limit (and hence acceptable), a nonlinear response history analysis is nevertheless used to confirm that all story drifts indeed remain within prescribed limits. A comparison of story drift ratios also confirms that no story drift ratio is more than 130 percent of that for the story above, as required to prove certain vertical irregularities are not present in the structure via the exception to Standard Section 12.3.2.2. 6.3.4.4 Second-order (P-delta) effects. AISC 360 requires consideration of second order effects. Such effects were investigated by conducting a three-dimensional P-delta analysis, which determined that secondary P-delta effects on the frame accounted for less than 10 percent of the primary demand. Furthermore, Standard Section 12.8.7 gives a different means of determining the significance of P-delta effects through the stability coefficient, θ, defined in Standard Equation 12.8-6. In either case, P-delta effects were found to be insignificant for this particular braced frame structure. 6.3.4.5 Brace design force summary. The maximum axial forces in each level’s individual BRBs caused by horizontal earthquake loads are listed in Table 6.3-3. Again, each brace is designed for its share of 100 percent of the horizontal earthquake load effect times the redundancy factor (ρ) of 1.3 without considering additional vertical loads to encourage distributed yielding of braces up the height of

6-85

FEMA P-751, NEHRP Recommended Provisions: Design Examples the structure. Nonetheless, the braces are also checked to ensure they do not yield under maximum live load (i.e., the load combination of 1.2D + 1.6L + 0.5Lr). Because the length of the yielding segment of a BRB is significantly less than its workpoint-to-workpoint length (see Sec. 6.5.3.1.1 below), the axial stiffness of the brace elements in the three-dimensional elastic analysis model must be adjusted to account for the non-prismatic nature of these elements. The modulus of elasticity of the steel in the brace elements was increased by a factor of 1.51 for single-diagonal and chevron bracing throughout the tower and 1.45 for chevron or V bracing configurations in the podium to match the true elastic stiffness of these elements as they are defined in the nonlinear response history analysis. The sizes of the BRBF members are controlled by seismic loads, always bi-directional and with eccentricity, rather than wind loads. Standard Section 12.8.4.2 only requires that the 5 percent displacement of the center of mass associated with accidental torsion be applied in the direction that generates the greater effect when earthquake forces are applied simultaneously in two orthogonal directions. However, due to the intricacies of SRSS directional combination of response spectra in ETABS, the 5 percent offset is applied in both orthogonal directions at the same time, which is slightly conservative for torsional response (and is not a significant penalty for this particular building due to its regular nature in plan). The design of connections will be governed by the seismic requirements of AISC 341. Table 6.3-3 Design Axial Forcesa in Buckling-Restrained Bracing Membersb Location

Gridline A (kips)

Gridline D (kips)

Gridline 2, 3, 6, or 7 (kips)

Roofc

139

144

142

Story 10

117

127

126

Story 9

135

136

133

Story 8

142

141

Story 7

152

151

Story 6

167

175

170

Story 5

193

195

197

Story 4

212

214

210

Story 3

152

197

187

Story 2

162

233

195

Individual maxima are not necessarily on the same frame; values are maximum for any frame. All braces are oriented in the chevron configuration except for single diagonalc or Vd. 1.0 kip = 4.45 kN b

6.3.5

Initial Proportioning and Details

The BRBFs occur on Gridlines 3, 6, A D in the tower and transfer their loads at the third floor to two BRBFs per line on Gridlines 2, 7, A D in the podium. These frames are shown schematically in plan in Figures 6.3-1 and 6.3-2 and in elevation in Figures 6.3-3 and 6.3-4. Using the horizontal component of the seismic load (amplified by the redundancy factor) as determined by response spectrum analysis and the loads from Table 6.3-3, the proportions of the braces are checked for adequacy. Then, initial sizes for 6-86

Chapter 6: Structural Steel Design the lateral columns, beams collectors are determined from the three-dimensional elastic analysis model using capacity design principles and relevant plastic mechanism analyses. In the preliminary elastic design stage, generic BRB properties are used to derive expected brace strengths. This allows for a specific BRB supplier to be selected further downstream in the project schedule, as is usually done in traditional project delivery methods. Design forces for columns are obtained from a summation of the vertical component of the adjusted brace strengths above the level of interest, while those for horizontal elements are derived from two different plastic mechanisms, always using adjusted brace strengths in tension and compression as required by AISC 341 Section 16.5b. All lateral columns are then subject to resizing according to the force and displacement demands determined using nonlinear response history analysis. 6.3.5.1 Buckling-Restrained Brace sizes 6.3.5.1.1 Buckling-Restrained Brace mechanics. The mechanics of BRBs are such that compression buckling need not be considered in their selection. Their required strength is controlled by yielding of the steel core material only. A BRB consists of a steel core that resists imposed axial stresses together with a mortar-filled sleeve that resists buckling. The steel core has both a yielding portion and two non-yielding portions at its ends where the cross-section enlarges to facilitate connection to a gusset plate. A debonding agent, often proprietary, decouples the axial behavior of the core from the buckling behavior of the sleeve. In compression, a BRB acts as a sleeved column—the steel core is able to achieve the full magnitude of its squash load while, at the same time, the sleeve can provide its full Euler buckling resistance without taking on any axial load. From a performance standpoint, such a component produces very desirable balanced hysteretic behavior that exhibits both isotropic and kinematic (cyclic) strainhardening. Unlike conventional bracing, BRB behavior is much more symmetric with respect to tension and compression and is not subject to strength and stiffness degradation. 6.3.5.1.2 Steel core area. According to AISC 341 Section 16.2a, the steel core must resist the entire axial force in the brace. This force is tabulated in Table 6.3-3. The brace design axial strength, φPysc, in either tension or compression, as controlled by the limit state of yielding, is equal to the following:

φ Pysc = φ Fysc Asc where:

φ = 0.90 Fysc = specified minimum yield stress (or actual from coupon test) of the steel core Asc = net area of steel core Setting φPysc equal to the Pu values in Table 6.3-3 and rearranging terms, the required net area of steel core can be expressed as follows:

Asc =

Pu φ Fysc

This required area, together with the actual steel core area provided, is shown for each brace in Table 6.3-4. Rarely do designers know the actual yield stress of the steel core during the design phase;

6-87

FEMA P-751, NEHRP Recommended Provisions: Design Examples hence, a minimum yield stress (Fysc) of the steel core equal to 38 ksi is assumed for this example. This minimum yield stress value would be specified on the design drawings. Table 6.3-4 Steel Core Areas for Buckling-Restrained Bracing Membersa Location

Gridline A (kips)

Gridline D (kips)

Gridline 2, 3, 6, or 7 (kips)

Asc req’d (in.2)

Asc (in.2)

Asc req’d (in.2)

Asc (in.2)

Asc req’d (in.2)

Asc (in.2)

Roofb

4.06

4.5

4.21

4.5

4.15

4.5

Story 10

3.42

3.5

3.71

4.0

3.68

4.0

Story 9

3.95

4.0

3.98

4.0

3.89

4.0

Story 8

4.15

4.5

4.12

4.5

4.12

4.5

Story 7

4.44

4.5

4.42

4.5

4.42

4.5

Story 6

4.88

5.0

5.12

5.5

4.97

5.0

Story 5

5.64

6.0

5.70

6.0

5.76

6.0

Story 4

6.20

6.5

6.26

6.5

6.14

6.5

6.0

5.47

5.5

7.0

5.70

6.0

Story 3

4.44

4.5

5.76

Story 2

4.74

5.0

6.81

All braces are oriented in the chevron configuration except for single diagonal or V 1.0 in = 25.4 mm

6.3.5.2 Lateral force-resisting columns. To design the frame containing the BRBs, unless using a nonlinear analysis, the designer should assume a plastic mechanism in which all BRBs are yielding in tension or compression and have reached their strain-hardened adjusted strengths, including all sources of overstrength. These adjusted brace strengths per AISC 341 Section 16.2d are as follows: §

Compression: βωRyFyscAsc

§

Tension: ωRyFyscAsc

The adjusted brace strength values represent the yield strength of the steel core adjusted for material overstrength (Ry), strain-hardening (ω) compression overstrength (β). Whereas conventional bracing usually buckles in compression well before reaching its yield strength, BRBs are often slightly stronger in compression than in tension. The strain-hardening (ω) and compression overstrength (β) factors traditionally are provided by BRB manufacturers and are calculated from cyclic sub-assemblage testing to a brace deformation equivalent to twice the design story drift per AISC 341 Appendix T. For the initial proportioning of braced-frame columns and beams using the three-dimensional elastic analysis model, generic values of β = 1.05, ω = 1.36 Ry = 1.21 are assumed. A material overstrength factor, Ry, of 1.21 is selected to bring the design yield strength of the steel core, Fysc = 38 ksi, up to the typical maximum specified steel core yield strength of 46 ksi. Note that all of these values are subject to revision for use in the nonlinear response history analysis once the BRB calibration has been performed in Section 6.3.6.2.

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Chapter 6: Structural Steel Design This capacity design methodology can easily be implemented to design lateral columns in a BRBF once the three-dimensional analysis model is constructed. The designer simply needs to generate an axial force in each brace corresponding to its adjusted brace strength as defined above, either by deleting the braces from the model and replacing them with their associated forces directly or by some other method that achieves the same result (for example, hand calculations or spreadsheets). Two different load cases must be examined for each principal direction: one with all braces in either tension or compression based on lateral load originating from one side of the frame a second with the brace forces determined by lateral load originating from the other side of the frame. Appropriate consideration should be given to bi-directional combination of the resulting brace loads on columns engaged by two orthogonal frames. While AISC 341 is unwavering in its requirement to design columns for the full adjusted brace strengths, the displacement corresponding to the adjusted brace strength should remain constant in any direction. Thus, such a displacement imposed at 45 degrees to the principal building axes will cause yielding of all braced frames, but not full strain hardening. This reduction factor for bi-directional loading is dependent on the brace’s post-yield behavior and will not be much less than one for a ductile system such as a BRBF. The 100%/30% orthogonal combination procedure defined in Standard Section 12.5.3 is not applicable in the context of capacity design as mandated for columns by AISC 341 Section 16.5b. To complete the preliminary column design, the column axial loads resulting from the maximum expected brace forces defined above are substituted for the earthquake load effect and combined with vertical loads as specified in Section 5.3.3.4. The translation and twist of all diaphragms should be locked when performing the column design for stability of the model. This will ensure each column is designed for the axial force equal to the summation of the vertical components of the adjusted brace strengths of all braces above it. Column flexural forces are not considered in their design, consistent with AISC 341 Section 8.3 (1). Such an approximation is valid because localized flexural yielding of a column at locations where it receives a brace is deemed acceptable from a performance standpoint. The preliminary column designs can be seen in Figures 6.3-3 and 6.3-4. To illustrate this process, a detailed calculation of the column design forces for the column at D4 can be seen in Table 6.3-5. Because column strengths are governed by compression buckling rather than yielding, brace actions that induce compression on the columns are considered critical. For the uppermost column below the tower roof, the 228-kip design force is equal to the sum of the design load due to gravity/vertical earthquake effect of 102 kips and the vertical component of the adjusted brace strength in tension at that level, equal to 126 kips. The adjusted brace strength in tension is used here because tension in the single diagonal brace at the roof will impose compressive forces on the column below, as can be seen in the elevation of Figure 6.3-3. At lower levels with chevron bracing, the design axial load in the column is calculated as follows: 1. Start with the vertical component of the roof brace in tension (126 kips). 2. Add the associated design gravity and vertical earthquake effect loading at that level. 3. Add the sum of the vertical components of the adjusted brace strengths in compression of all chevron bracing at levels above. 4. Subtract half of the sum of the unbalanced vertical loads (difference in vertical components of adjusted brace strengths in compression and tension) on the beams intersecting all chevron bracing at that level and above.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples This procedure recognizes that for levels with chevron bracing, the adjusted brace strength in compression will always control over that in tension and will enter the column below at the base of that level. Additionally, the unbalanced vertical load from the chevron transmits shear to the beam above and ultimately the column that works to alleviate the downward gravity and brace compression forces. Table 6.3-5 Determination of Column Design Forces for Column at Gridline D4 Vertical Component of Adjusted Brace Strengths Tensionc Compressiond

Column Required Strength Pu

Level

Bracea Area

(1.2+0.2SDS)D + 0.5L + 0.2S

Brace Angle αb

Roof

4.5 in2

102 kips

26.6°

126 kips

132 kips

228 kips

4.0 in

196 kips

45°

177 kips

186 kips

318 kips

4.0 in

293 kips

45°

177 kips

186 kips

596 kips

4.5 in

390 kips

45°

199 kips

209 kips

873 kips

4.5 in

486 kips

45°

199 kips

209 kips

1174 kips

5.5 in

583 kips

45°

243 kips

255 kips

1474 kips

6.0 in

681 kips

45°

265 kips

279 kips

1821 kips

Story 4

6.5 in

780 kips

45°

288 kips

302 kips

2191 kips

Story 3

none

947 kips

2660 kips

Story 2

none

1106 kips

2819 kips

Story 10 Story 9 Story 8 Story 7 Story 6 Story 5

All braces are oriented in the chevron configuration except for single diagonal at the roof level. Measured from the horizontal. c ωRyFyscAscsinα d βωRyFyscAscsinα 1.0 kip = 4.45 kN 1.0 in = 25.4 mm b

Columns are spliced at every other level to simplify erection. Column sizes are subject to revision due to results from nonlinear response history analysis. While tension forces in the columns may not control their design, tension demands can certainly affect the design of base plates, anchor rods drilled piers. The design tension force at the base of this same column calculated in a similar manner (using the appropriate load combination) is equal to 1,222 kips. 6.3.5.3 Lateral force-resisting beams. The braced frame beams were designed for gravity loads corresponding to 1.37D + 0.5L + 0.2S, without accounting for any mid-span support provided by chevron bracing, together with earthquake loads extracted from two different plastic mechanism analyses. The first of these mechanisms assumes both the brace(s) above and below the beam of interest have reached their full adjusted brace strengths. The beam is then designed for its share (depending on its location along the line of framing) of the horizontal component of the resulting story force together with the largest drag force from the brace(s) above. In the second mechanism, the brace(s) below the beam of interest is assumed to have reached their full adjusted brace strengths at the same time the diaphragm reaches its design force (this mechanism requires a rough diaphragm analysis to derive collector forces). The beam must resist the same share of the horizontal component of the diaphragm force at that line of framing plus the largest drag force from the brace(s) above, determined by the difference between the force in the brace(s) below minus the diaphragm force at that line of framing. In either mechanism, the

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Chapter 6: Structural Steel Design

c F

As y

F y

As y

As Fy

O0E

R βω

O0E

R βω

F 2C O Sθ

As y

As Fy

Sθ CO F 2

R βω

braced frame beam is also designed for any unbalanced upward component of the BRBs that would arise in a chevron or V-bracing configuration. These mechanisms and their associated forces are illustrated in Figure 6.3-7.

Mechanism 1

Mechanism 2

(a)

(b)

Figure 6.3-7 BRB plastic mechanism As an example, consider the beam engaged in a chevron bracing configuration along Gridline D above the ninth story. The shear and moment in the beam corresponding to the load combination of 1.37D + 0.5L + 0.2S, calculated by neglecting the mid-span support provided by the chevron bracing below, are equal to 18 kips and 128 kip-ft, respectively. The brace above is a 4.5 in2 single diagonal while two 4 in.2 braces frame into the midpoint of the beam from below in a chevron configuration. Hence the upward force at midspan resulting from the unbalanced upward component of the chevron braces below reaching their adjusted strengths is given by the following: ωRyFyscAsc(β - 1)sinα = (1.36)(1.21)(38)(4)(1.05 - 1)(sin45°) = 9 kips This unbalanced upward component reduces the shear demand in the beam by P/2 = 9/2 = 5 kips and the moment by PL/4 = (9)(30)/4 = 66 kip-ft. Hence the moment demand that will eventually be combined with the critical axial demand determined by plastic mechanism analysis is: 128 kip-ft - 66 kip-ft = 62 kip-ft The first plastic mechanism shown in Figure 6.3-7 considers both the brace(s) above and below the beam of interest to have reached their adjusted strengths. Hence, the plastic story shear below the tenth floor is equal to: ωRyFyscAsc(1 + β)cosα = (1.36)(1.21)(38)(4)(1 + 1.05)(cos45°) = 363 kips The plastic story shear above the tenth floor is: βωRyFyscAsccosα = (1.05) (1.36)(1.21)(38)(4.5)(cos26.6°) = 264 kips (The brace at the tenth story is a single-diagonal.)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

The story force corresponding to this yielding mechanism is equal to the difference between these two values, or 99 kips. Since the chevron braces below the tenth floor beam receive the lateral force at the midpoint of the associated line of framing, half of this story force (attributed to inertial mass) is presumed to come from each half of the braced frame beam, or approximately 50 kips. This 50-kip force is added to the 264-kip horizontal component from the yielding single-diagonal brace above that must be dragged through the braced frame beam to the chevron braces below, resulting in a design axial force of 314 kips for the first plastic mechanism. The second plastic mechanism illustrated in Figure 6.3-7 requires estimation of the force entering the braced frame beam from the diaphragm at that floor in accordance with Standard Section 12.10. As shown in Figure 6.3-7 this is a collector force thus the value of Fpx calculated using Standard Equation 12.10-1 requires amplification by the overstrength value, Ω0, of 2.5; this gives 2.5 × 142 kips = 355 kips. This value cannot be taken as less than the minimum of 0.2SDSIwpx, which gives a value of 270 kips. (Note that the overstrength factor does not apply to this minimum, even for collectors.) Thus, the 355 kips governs the corresponding frame force can be taken as 55 percent of this force (that is taking 1/2 adding 10 percent to account for accidental eccentricity): 195 kips. The chevron braces below the tenth-floor braced frame beam are still assumed to have reached their yield strength, resulting in the same 363-kip frame shear at that level. Hence the statically consistent force in the (now elastic) single-diagonal brace above the tenth floor is equal to the difference between these two values, or 168 kips. Just as is done in the calculation of the design axial force resulting from the first plastic mechanism, half of the 195-kip story force (the maximum from the diaphragm) is added to the 168-kip horizontal component from the single-diagonal brace above that must be dragged through the braced frame beam to the chevron braces below, resulting in a design axial force of 265 kips for the second plastic mechanism. The 314 kips obtained from the first plastic mechanism (both braces above and below the beam at their adjusted strengths) controls. Thus, the braced frame beam along Gridline D above the ninth story must be designed for a 62 kip-ft moment in combination with a 314 kip axial force. The axial strength is typically determined without accounting for the benefits of composite action with the concrete-filled deck above. Although some minor benefit can be obtained from considering the composite contribution to flexural strength, this is often neglected for simplicity; flexural forces due to brace unbalanced loading are typically small in the inverted-V configuration they oppose gravity forces. The axial capacity of braced frame beams is often controlled by flexural-torsional buckling (as opposed to buckling about the weak axis). 6.3.5.4 Third floor/low roof collector forces. Collector elements that transfer forces between the single bay of chevron bracing in the tower to the multiple, offset bays of chevron bracing in the podium are sized in a manner identical to the lateral force-resisting beams. The same two plastic mechanisms—one involving braces above and below the level of interest reaching their full adjusted strengths the second involving the braces below the level of interest reaching their full adjusted strengths in conjunction with the diaphragm delivering its maximum force to the framing in line with the braces—are assumed the forces are traced from the single bays of chevron bracing above through the collector lines to the chevron bracing below. Story forces accumulate in the collectors and braced frame beams based on the fraction of the full length of the line of framing represented by the particular beam section of interest. In the case of the out-of-plane offset that occurs between Gridlines 2 and 3 (and 6 and 7), the horizontal component of the adjusted chevron brace strengths above the third floor must be distributed into the diaphragm via the adjacent collector elements, then collected by collector elements along the outer line of framing (at Gridlines 2 or 7) to be channeled to the chevron braces in the podium levels below. As with the braced frame beams, the axial capacity of these collector elements is based on that of the bare steel section and usually is governed by flexural-torsional buckling. It is acceptable to calculate the flexural capacity of the collector elements considering composite action with the concrete-filled deck above. 6-92

Chapter 6: Structural Steel Design

6.3.5.5 Connection design. According to AISC 341 Section 16.3, gussets and beam-column connections must be designed for 1.1 × Cmax, where Cmax is the adjusted brace strength in compression as defined in AISC 341 Section 16.2d. Connection design is not illustrated here since this topic is more thoroughly treated elsewhere and is not unique to BRBF systems. The connection of the 4.5 in.2 single diagonal brace to the frame beam below it at the tenth floor would need to be designed for 1.1βωRyFyscAsc = 1.1(1.05)(1.36)(1.21)(38)(4.5) = 325 kips in tension and compression based the full expected and strainhardened brace capacity. However, it should be mentioned that a designer might consider using NRHA to design for potentially reduced connection force demands; such a reduction would likely be limited to establishing a lower value for the strain-hardening factor ω for each brace. 6.3.6

Nonlinear Response History Analysis

After completing a preliminary design using three-dimensional modal response spectrum analysis, nonlinear response history analysis is performed to: §

Establish brace deformation demands (and verify the adequacy of specified braces for the application)

§

Determine expected drifts

§

Re-examine the required strength of column members in the BRBF

The braced frames and diaphragm chords and collectors, together with all primary gravity system beams and columns, are explicitly modeled using three-dimensional beam-column elements. Secondary gravity framing is omitted from the nonlinear model for simplicity. The floor diaphragms are still modeled as rigid. The specific goals of the nonlinear response history analysis are threefold. First, even though the original elastic design is found to comply with the drift limitations of Standard Section 12.12.1, the nonlinear response history analysis can more accurately predict results such as story drift. Hence, the acceptability of the design in satisfying story drift requirements is evaluated using the procedures of Standard Section 16.2.4. Second, the ability of the BRBs to perform at the Immediate Occupancy (IO) performance level in a design basis earthquake (DBE) event will be verified explicitly. Third, the required strength of column elements in the BRBF system is re-evaluated according to the “maximum force that can be developed by the system” as permitted by AISC 341-05 Section 16.5b. The use of a nonlinear response history analysis to determine this maximum force for individual elements is justified in the provisions of Standard Section 12.4.3.1. Specifically, the exception to this section permits the determination of “the maximum force that can develop in the element … by a rational, plastic mechanism analysis or nonlinear response analysis utilizing realistic expected values of material strengths.” Standard Section 16.2 then defines the requirements for nonlinear response history analysis. In this design example, only the columns in the BRBF system are examined. This is due to the limited savings potential of economizing the relatively small number of BRBF beam and collector elements in the structure that have already been reasonably sized (attributable to the absence of large BRB elements) using a rational plastic mechanism analysis. The designer of a taller building with bulky BRB elements should probably consider using the same procedure to potentially reduce design force demands on the BRBF beams and collectors as well as the columns.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples A second, separate three-dimensional building model is assembled in the PERFORM program its similarity to the ETABS model used for the elastic design is confirmed by comparing fundamental periods and loads. 6.3.6.1 Design ground motions. In Chapter 3, risk-targeted maximum considered earthquake (MCE) response spectra are determined in accordance with Standard Section 11.4 using NGA attenuation relations. Seven pairs of time histories are selected and scaled to be consistent with the event magnitudes, fault distances source mechanisms controlling the MCE spectrum for the Seattle, Washington, hospital site. The base ground motions in the suite are scaled to the MCE response spectrum so as to satisfy the requirements of Standard Section 16.1.3.2 for periods between 0.18 and 4.95 seconds. The translational structural periods for the hospital facility are found to be approximately 2.3 seconds, so the period range of interest is from 0.2 × 2.3 = 0.46 seconds to 1.5 × 2.3 = 3.45 seconds, which is narrower than the preliminary range used in the ground motion selection and scaling. Design-level ground motions are obtained by multiplying MCE motions by 2/3 per Standard Section 16.2.3. 6.3.6.2 Basis of nonlinear design. In keeping with the intent of the building code to protect essential facilities in a seismic event, the hospital should perform at the IO performance level under ground shaking corresponding to the DBE. As is traditionally the case in elastic designs, an explicit performance check at the MCE is not done here. The building is assumed to meet Life Safety (LS) performance objectives at the MCE if it meets IO performance criteria at the DBE. At the present time, there are three international providers of BRBs: CoreBrace and StarSeismic in the United States Nippon Steel in Japan. In the preliminary design stage, it is often the case that any one of these three brace manufacturers may ultimately be chosen as the supplier. Thus, generic BRB properties may be assumed until the later stages of a project. However, a specific BRB supplier must be selected to accurately model actual brace behavior in the PERFORM NRHA model. BRBs in the NRHA are modeled using expected properties based on test results provided by CoreBrace; acceptance criteria are based on Section 2.8.3 of ASCE 41 for deformation-controlled elements. There are three different types of brace-to-gusset connections used for BRBs: bolted, welded pinned. However, generic “ACME” braces are used throughout the structure hence the nature of the brace-to-gusset connection is not considered. All other structural elements are modeled using expected properties. Because the lateral beams and columns are connected by simple pin connections in this example (see Section 6.3.2.1), the beams will not require consideration of inelastic behavior. Unless some column element sizes are reduced based on NRHA results, inelastic column actions are not likely, although it is possible that uneven (or higher-mode) story drifts will result in inelastic flexural demands. The initial gravity load condition is 1.0D + 0.25L per Standard Section 16.2.3. 6.3.6.3 Buckling-Restrained Brace calibration. In order to capture the nonlinear BRB behavior as accurately as possible in the NRHA computer model, brace properties are calibrated to match test results provided by CoreBrace. Typically, a number of calibrations are performed on a range of different brace sizes. Critical modeling parameters are then interpolated between (or extrapolated from) those established for a brace or braces of similar size. For illustration purposes, one calibration is shown and modeling parameters are matched to corresponding values for the flat core plate test specimen. (Note that the specific series of BRBs tested possess a post-yield modulus of elasticity that is unusually stiff.) Data to which critical brace parameters are calibrated is obtained from a standard cyclic testing protocol that is in line with the requirements of AISC 341 Appendix T and ASCE 41 Section 2.8.3. Elastic displacement components in both the connection region and the non-yielding brace segments (the larger “elastic bar” segment in PERFORM) of the BRB specimens are subtracted out of measured displacements 6-94

Chapter 6: Structural Steel Design in the test data. As a result, the inelastic core plate’s force-displacement behavior is isolated to facilitate calibration. The result of the calibration is a backbone curve that reasonably envelopes the observed cyclic test hysteretic behavior. BRB elements typically are modeled in nonlinear computer analysis software as three components in series: a stiff connection region, an elastic bar segment an inelastic yielding segment. The workpoint-toworkpoint length of the BRB element in the three-dimensional computer model must be divided reasonably into these three components. Thus, the stiff connection zone size and relative stiffness are determined by a rough gusset plate design based on a representative brace size and geometry in the building. The elastic bar segment’s length and cross-sectional area are set such that their proportions relative to the inelastic yielding segment remain identical to those of the test specimen. The remainder of this section is devoting to developing modeling parameters for the inelastic yielding portion of the BRB element. One key modeling parameter for the inelastic portion of the BRB elements is its initial elastic stiffness, Ko. This value is simply set to equal AscE / Ly, where Ly is the length of the yielding segment set to equal the same proportion of the brace length outside the connection region as the test specimen. Next, the designer must select a post-yield stiffness, Kf, for the inelastic portion of the backbone curve. This value is chosen such that the post-yield slope of the hysteresis loops obtained from the computer model is similar to that in the test data is typically a percentage of the initial stiffness, Ko. For the hospital building in this example, Kf = 1.5 percent of Ko to match the observed hysteretic behavior of the test specimen. Other key modeling inputs are the strain-hardening and compression overstrength factors that define the ultimate strength of the BRB elements. Each brace in the hospital computer analysis model is assigned strain-hardening (ω) and compression overstrength (β) factors equal to 1.63 and 1.05, respectively. These values are selected to match the full hardened strength of the test specimen in tension and compression and are different from the generic values assumed earlier for the plastic mechanism analysis used to design beams in columns in the BRBF. Because the compression overstrength factor does not equal 1.0, the BRB behavior is not symmetric in tension and compression. The material strength of the braces in the model is set to equal the minimum specified strength for this project (i.e., RyFy = 38 ksi). Alternatively, one could use the actual material strength as determined by coupon test of BRB specimens tested specifically for that particular project. All nonlinear analysis software programs require an ultimate deformation or strain value corresponding to the BRB having reached its fully-hardened strength as an input in the inelastic BRB component’s properties. This figure was set such that the strain matched that observed in the test at full compression hardening. Additionally, the program requires information about the rate of isotropic hardening between cycles. In the nonlinear analysis software used for this design example, the rate of isotropic hardening is captured by inputting the maximum brace deformation corresponding to the average of the initial BRB yield strength and its strength after full hardening (this cyclic hardening parameter can also be defined in terms of accumulated deformations but is not done so here). To match the vertical progression of hysteretic behavior (i.e., hardening between progressive loading cycles) observed in the test specimen, this hardening parameter was set to 2/3 of the deformation from initial yield to fully-hardened strength. Finally, the software must know at what point the deformation in the nonlinear BRB component has exceeded its capacity. Since the BRBs in this example are expected to perform well below their capacity and this value was never reached during the standard testing protocol, an artificially high number was selected for this input individual brace performance was subject to review as outlined in Section 6.3.6.3.1. The above parameters are sufficient to define a bilinear elastic-plastic backbone curve for the BRB elements in the computer analysis model. The hysteretic behavior of the BRB component matching the test specimen extracted from the computer model is compared with the experimental test data in 6-95

FEMA P-751, NEHRP Recommended Provisions: Design Examples Figure 6.3-8. One can see that the backbone curve modeling parameters accurately capture the observed experimental hysteresis properties for this size component. With some additional effort, a trilinear backbone curve can also be calibrated to the test data to better capture the “rounding” of the actual BRB hysteretic loops. The trilinear curve considers a higher component stiffness value just after yield before the ultimate post-yield stiffness value, Kf, prevails. This third stiffness value, together with the forcedeformation point at which the ultimate post-yield stiffness (Kf) “takes over,” must be calibrated to the test data as well. Figure 6.3-9 shows the inelastic BRB component’s hysteretic behavior as modeled using the trilinear backbone curve together with the experimental test data to which it is calibrated. Subsequent analysis results are based on BRB inelastic components modeled using the trilinear backbone curve because the additional accuracy of the tri-linear backbone curve can be realized without causing excessive analysis run times for this example building. 800

F (kips)

600 400 200 0 -2.5

-2

-1.5

-1

-0.5

0.5

1.5

-200 -400 -600 -800

Figure 6.3-8 Bilinear BRB calibration (Asc = 12 in2) (1.0 in. = 25.4 mm; 1.0 kip = 4.45 kN)

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2.5 Δ (inch)

Chapter 6: Structural Steel Design 800

F (kips)

600 400 200 0 -2.5

-2

-1.5

-1

-0.5

0.5

1.5

-200

2.5 Δ (inch)

-400 -600 -800

Figure 6.3-9 Trilinear BRB calibration (Asc = 12 in2) (1.0 in. = 25.4 mm; 1.0 kip = 4.45 kN)

6.3.6.4 Results. 6.3.6.4.1 BRB strains. One goal of the NRHA is to confirm the ability of the BRBs to perform at the IO performance level in a DBE event. The acceptance criteria for BRB strain, as dictated by ASCE 41 Section 2.8.3, can be seen in Figure 6.3-10 (overlaid on the corresponding test results). No permanent, visible damage was observed during the standard experimental testing protocol used to calibrate the BRB elements the test was terminated at or near Point 2 as defined in the Type 1 and Type 2 component force versus deformation curves for deformation-controlled actions in ASCE 41 Section 2.4.4.3. Consequently, the IO, LS Collapse Prevention (CP) acceptance criteria are equal to 0.67 × 0.75, 0.75 1.0 times the deformation at Point 2 on the component force versus deformation curves, respectively. Observe that the limiting BRB strains for IO performance in tension and compression are 6.64Δy / Ly = 6.64 × 0.00131 = 0.008704 and 6.47Δy / Ly = 6.47 × 0.00131 = 0.008482, in turn.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 2

P/Pye CP

1.5

IO 1 0.5 tension

Δ /Δ Δ by

0 -15

-10

-5

compression

-0.5 -1 -1.5 -2

Figure 6.3-10 BRB strain acceptance criteria The ratio of maximum inelastic deformation demand Ψ = Δ/Δy observed along a subset of the 92 total BRB elements during each of the seven ground motion time histories to the relevant IO performance point in tension or compression is shown in Table 6.3-6. Braces for which results are presented are chosen to represent BRB elements in all ten levels of the structure. As permitted by Standard Section 16.2.4 for analyses including at least seven ground motion pairs, the average BRB Ψmax /ΨIO value across the seven earthquake records is calculated for each brace. In assessing the performance of the entire structure, the maximum of these 92 average BRB inelastic deformation demand values is extracted and compared with the relevant IO performance point in tension or compression. Table 6.3-6 shows that this critical Ψmax/ΨIO value for the hospital structure is equal to 0.566, indicating acceptable performance of the BRB elements in the DBE event according to the methodology set forth in Standard Section 16.2.4. The procedure set forth in the Standard may lead to designs that fail a criterion in some element or measure for every ground motion but still pass the criteria on average. Table 6.3-6 BRB ID

6-98

Ratio of Maximum BRB Inelastic Deformation Demand Ψ to Immediate Occupancy (IO) Performance Limit (Ψmax /ΨIO) Record ID

Average

0.245

0.350

0.173

0.159

0.125

0.250

0.486

0.255

0.331

0.472

0.179

0.267

0.201

0.282

0.732

0.351

. . . 16

. . . 0.187a

. . . 0.189a

. . . 0.154

. . . 0.185

. . . 0.266

. . . 0.294

. . . 0.215

. . . 0.212

Chapter 6: Structural Steel Design Table 6.3-6 BRB ID

Ratio of Maximum BRB Inelastic Deformation Demand Ψ to Immediate Occupancy (IO) Performance Limit (Ψmax /ΨIO) Record ID

Average

0.574

0.258

0.319

0.251

0.350

0.450

0.314

0.359

0.770

0.314

0.321

0.217

0.313

0.369

0.316

0.374

0.956

0.537

0.542

0.245

0.343

0.615

0.373

0.515

1.082

0.532

0.588

0.268

0.329

0.753

0.310

0.551

1.001

0.392

0.574

0.351

0.231

0.609

0.465

0.518

0.875

0.402

0.520

0.264

0.357

0.718

0.534

0.524

0.545

0.414

0.371

0.258

0.399

0.644

0.455

0.440

. . . 89

. . . 0.642

. . . 1.064a

. . . 0.572a

. . . 0.254a

. . . 0.695

. . . 0.478

. . . 0.263

. . . 0.566b

0.605

0.959a

0.328a

0.180

0.425

0.262

0.215

0.424

0.518

0.592

0.382

0.371

0.347

0.212

0.181

0.372

0.241

0.366

0.364

0.631

0.368

0.248

0.410

0.375

Max

1.086

1.286

0.834

0.631

0.857

0.759

0.792

0.566b

All Ψmax /ΨIO values for the individual braces are controlled by the compression strain limit except those denoted by the superscript a (for which the tension strain limit controls). b Note that the 0.566 structure “usage ratio” is equal to the maximum of the average Ψmax /ΨIO values across the seven ground motion pairs for each BRB element, not the average of the maximum Ψmax /ΨIO values for each of the seven ground motion pairs across all 92 BRB elements.

6.3.6.4.2 Drift assessment. Although seismic drift was preliminarily examined in Section 6.3.4.3 to ensure the design possessed reasonable stiffness before modeling its nonlinear behavior, conformance with story drift limits is assessed using the procedures of Standard Section 16.2. The maximum story drift ratio in each translational direction was extracted from the three-dimensional PERFORM model for each of the seven ground motion pairs. As before, story drift was examined at the building corners rather than the center of mass. Once the maximum story drift ratio at every story and corner of the floorplate is identified in each of the seven time histories, the resulting seven values for each story are averaged (as allowed by Standard Section 16.2.4) the maximum average story drift ratio is identified to assess compliance with the allowable story drift, which Standard Section 16.2.4.3 permits to be increased by 25 percent relative to the drift limit specified in Section 12.12.1 in the context of nonlinear response history analysis. Thus, the relevant story drift limit for nonlinear response history analysis is (1 + 0.25)(0.010hsx) = 0.0125hsx. Figure 6.3-11 shows, for each story, the story drift ratios in the longitudinal direction for each time history, the average of all seven the allowable story drift ratio. (Also shown are the analysis results and drift limit for MRSA, which are discussed below.) The controlling story drift for the NRHA occurs in the seventh story, where it is about 93 percent of the allowable story drift. Table 6.3-7 illustrates how this value is calculated using drift ratios at the critical corner of the floorplate. Drifts in the transverse direction are somewhat smaller.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Story Roof 10

9 8 7

MRSA

6 5 4 NRHA avg

2 MRSA limit 0

NRHA limit

0.01 Story drift ratio, Δ /hsx

0.02

Figure 6.3-11 Longitudinal story drift ratios (and drift limits) Table 6.3-7 Maximum Longitudinal Story Drift Ratio at Critical Corner of Building Floorplate Level

Record ID

Average

Roof

0.0084

0.0088

0.0076

0.0108

0.0062

0.0060

0.0083

0.0080

Story 10

0.0064

0.0079

0.0108

0.0067

0.0052

0.0062

0.0073

Story 9

0.0095

0.0164

0.0082

0.0073

0.0077

0.0056

0.0062

0.0087

Story 8

0.0109

0.0212

0.0108

0.0069

0.0084

0.0063

0.0055

0.0100

Story 7

0.0110

0.0225

0.0145

0.0090

0.0116

0.0085

0.0045

0.0117a

Story 6

0.0096

0.0166

0.0110

0.0070

0.0109

0.0072

0.0041

0.0095

Story 5

0.0084

0.0098

0.0068

0.0049

0.0108

0.0066

0.0042

0.0074

Story 4

0.0063

0.0062

0.0047

0.0050

0.0131

0.0047

0.0039

0.0063

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Chapter 6: Structural Steel Design Table 6.3-7 Maximum Longitudinal Story Drift Ratio at Critical Corner of Building Floorplate Level

Record ID

Average

Story 3

0.0057

0.0045

0.0054

0.0058

0.0132

0.0038

0.0050

0.0062

Story 2

0.0048

0.0032

0.0051

0.0047

0.0084

0.0035

0.0054

0.0050

Note that the 0.0117 structure story drift ratio is equal to the maximum of the average story drift ratio values across the seven ground motion pairs for each story, not the average of the maximum story drift ratio values for each of the seven ground motion pairs across all ten stories.

The maximum story drift calculated earlier using MRSA reached a maximum value at the roof level equal to 89 percent of the 0.010hsx limit prescribed in Standard Section 12.12.1. The magnitude of this maximum was similar in both principal building axes. The design story drifts obtained from the MRSA were also found to increase from story to story up the height of the structure. Contrast the preliminary elastic MRSA drift results with those just presented from the NRHA. First, the maximum story drift no longer always occurs at the uppermost level of the building; the specific location of the maximum varies depending on the ground motion. Second, the maximum story drift is no longer consistent between the two primary structural axes. This is an important point and is a direct result of the way the ground motion pairs were applied to the structure. While not required by the Standard, some engineers elect to re-analyze the structure with the ground motions rotated, in order to investigate sensitivity to ground-motion orientation; this results in a total of 14 analyses, with a corresponding increase in effort. However, a detailed discussion of this and other ground motion issues is beyond the scope of this design example. Finally perhaps most importantly, the maximum story drift from the NRHA, 0.0117hsx, is 31 percent higher than that from the MRSA, 0.89 × 0.010hsx = 0.0089hsx. The results in this section highlight an important misconception of NRHA in general. There is no guarantee of economizing a design with respect to the required strength or stiffness of a frame simply by performing a NRHA. Rather, when executed correctly, a NRHA simply assures a more accurate representation of actual structural performance in a particular seismic event. This increase in the accuracy of seismic response parameters can actually increase the required frame strength or stiffness in some instances. 6.3.6.4.3 Column design forces. All BRBF columns were initially designed in Section 6.3.5.2 to resist the vertical component of the adjusted strengths of any braces above, using capacity design principles and generic values of the brace material overstrength (Ry), strain-hardening (ω) compression overstrength (β) parameters. Thus, the lateral columns are expected to remain nominally elastic in the DBE event. More realistic expected BRB behavior specific to a particular BRB product line and supplier is modeled in Section 6.3.6.2 based on experimental test data. The required strengths of the columns in the BRBF as specified in AISC 341 Section 16.5b are subject to revision based on results from the NRHA, as is permitted in the exception to Standard Section 12.4.3.1. To this end, Table 6.3-8 shows the maximum compressive axial force attained at every story in the BRBF column at Gridline D/4 during each of the seven time history analyses. This force represents the summation of the gravity load prescribed in Standard Section 16.2.3 (1.0D + 0.25L) plus the additional force imposed by the relevant earthquake time history pair. As above, compression forces are the most critical for column design because column strengths are governed by compression buckling rather than yielding.

6-101

FEMA P-751, NEHRP Recommended Provisions: Design Examples The model included column potential hinges with axial-moment interaction relationships determined from ASCE 41. Inelastic rotations were limited to IO values per that standard. However, virtually no inelastic rotation was recorded in the analyses. Table 6.3-8 also identifies the overall maximum compression force in the column at every story across all seven ground motion pairs, together with the column design force determined by plastic mechanism analysis in Section 6.3.5.2. While Standard Section 16.2.4 permits the use of average member forces in determining design values with at least seven ground motions, maximum member force values are selected for design of the columns due to their critical role in sustaining the vertical load-carrying capacity of the structure. Even when using maximum values of member forces extracted from the time histories, substantial savings in column design forces and hence steel tonnage are facilitated by NRHA in this example. Table 6.3-8

Comparison of Column Design Forces from NRHA and Plastic Analysis for Column at Gridline D/4 Maximum Compression Force (kips) Record ID

Level

Design Axial Force QE Percent Reduction (kips) in Design NRHAa Plasticb Forcec

Roof

152

154

152

156

148

155

168

228

26%

Story 10

214

216

219

211

217

231

318

27%

Story 9

404

412

393

382

397

416

596

30%

Story 8

603

616

589

567

571

578

590

616

873

29%

Story 7

812

842

800

743

774

768

781

842

1174

28%

Story 6

986

1068

1018

948

996

948

972

1068

1474

28%

Story 5

1198

1320

1264

1183

1214

1113

1145

1320

1821

28%

Story 4

1414

1566

1504

1430

1421

1276

1264

1566

2191

29%

Story 3

1750

1851

1780

1707

1651

1486

1421

1851

2660

30%

Story 2

1861

1965

1894

1818

1764

1598

1535

1965

2819

30%

Maximum of individual maxima from the seven time histories. As determined in Table 6.3-5. c Relative to that determined using plastic analysis. 1.0 kip = 4.45 kN1.0 in = 25.4 mm b

In addition to these significant reductions in column design forces, the same procedure reveals a design tension uplift force at this same column of 336 kips, which is just 28 percent of the comparable 1,222-kip force obtained from plastic analysis. This force reduction enables tremendous savings in the design of foundation elements such as base plates, anchor rods drilled piers. With the possible exception of the foundation elements just mentioned, despite the fact that the column design forces have been sharply reduced relative to those obtained from the plastic mechanism analysis in Section 6.3.5.2, none of the column sizes are actually reduced in this example. This is because the structure is currently very close to the allowable story drift limit. In other words, the NRHA reveals that stiffness, not strength, governs the design of the BRBF examined here. A prudent designer might consider enlarging select braces in conjunction with reducing column sizes as allowed by the NRHA

6-102

Chapter 6: Structural Steel Design design forces in Table 6.3-8 to increase cost savings on the project, depending on the relative prices of BRBs and structural steel. However, the tradeoff between BRB and column stiffness would have to preserve the overall stiffness of the frame in order to ensure the allowable drift limit is still satisfied. In particular, it may be possible to increase the brace size in Story 7 (and possibly Stories 6 and 8) while reducing the column size considerably. 6.3.6.4.4 Brace-to-gusset connection design. As mentioned earlier, brace-to-gusset connections for BRBs take on three different forms: bolted, welded pinned. The specific nature of this connection is not considered in this design example. However, the possible additional benefit of economizing on material by using NRHA to determine connection design forces will be demonstrated. As shown in Section 6.3.5.5, the connection of the single diagonal brace below the roof to its gusset would need to be designed for 325 kips in tension and compression. Using the maximum BRB force results from NRHA (obtained in the same manner that column design forces were determined), this design value can be reduced to 210 kips in tension and 213 kips in compression, representing a reduction of approximately 35 percent in both cases. This corresponds to utilizing a lower strain-hardening factor, ω, corresponding to a more refined method of establishing deformation demands. It should be noted that this reduction is not explicitly allowed by AISC 341; therefore, it would constitute “alternate means and methods” and be subject to approval by the building official. 6.3.6.4.5 Summary of NRHA goals. Before concluding, the exact extent to which NRHA was used in this design example merits emphasis one last time. Based on the fundamental period of the structure, the minimum level of sophistication required for its seismic lateral analysis is an elastic MRSA. Thus, in keeping with code requirements, a three-dimensional model of the structure was created the BRBs were designed to accommodate force demands determined by MRSA. BRBF beams, columns collectors were then designed using a rational plastic mechanism analysis with the assumption that any earthquake load effect is determined from the full adjusted brace strengths in tension and compression. This example then goes beyond elastic analysis and relies on a NRHA for three additional aspects of the design. First, NRHA is used to verify that the strains in the BRBs designed to MRSA forces indeed satisfy the IO performance requirements in DBE-level shaking using the methodology in ASCE 41 Section 2.8.3. Second, compliance with the allowable story drift limit was evaluated in the context of NRHA and the provisions of Standard Section 16.2.4. Finally, as a demonstration, NRHA was used to establish reduced design axial forces for the BRBF columns using the exception to Standard Section 12.4.3.1 to justify the use of NRHA in determining their required strength as stipulated in AISC 341 Section 16.5b. However, because the structure was found to be at the allowable drift limit as designed using plastic mechanism analysis, the potential savings offered by these reduced column design forces went unrealized. Designers could elect to place an even greater emphasis on NRHA during the design of such a structure. For example, design forces for BRBF beams and collectors might also be set using NRHA, as was illustrated for the columns, rather than by plastic mechanism analysis. Depending on the relative prices of BRBs and structural steel, designers could economize the design by enlarging select BRB elements (relative to what MRSA finds is necessary) in order to reduce some column sizes. Justifying such a design would require iterative NRHA runs to ensure the structure’s overall stiffness is such that allowable drift limits are not exceeded. Finally, the entire structure might be designed using NRHA exclusively. BRB sizes would be established not by MRSA but instead by NRHA and hence a different force distribution that takes into account the building’s inelastic characteristics.

6-103

7 Reinforced Concrete By Peter W. Somers, S.E. Originally developed by Finley A. Charney, PhD, P.E.

7.1

SEISMIC DESIGN REQUIREMENTS ........................................................................................ 7

7.1.1

Seismic Response Parameters................................................................................................ 7

7.1.2

Seismic Design Category ....................................................................................................... 8

7.1.3

Structural Systems ................................................................................................................. 8

7.1.4

Structural Configuration ........................................................................................................ 9

7.1.5

Load Combinations ................................................................................................................ 9

7.1.6

Material Properties............................................................................................................... 10

7.2

DETERMINATION OF SEISMIC FORCES ............................................................................. 11

7.2.1

Modeling Criteria................................................................................................................. 11

7.2.2

Building Mass ...................................................................................................................... 12

7.2.3

Analysis Procedures............................................................................................................. 13

7.2.4

Development of Equivalent Lateral Forces ......................................................................... 13

7.2.5

Direction of Loading............................................................................................................ 19

7.2.6

Modal Analysis Procedure ................................................................................................... 20

7.3

DRIFT AND P-DELTA EFFECTS ............................................................................................. 21

7.3.1

Torsion Irregularity Check for the Berkeley Building ........................................................ 21

7.3.2

Drift Check for the Berkeley Building ................................................................................ 23

7.3.3

P-delta Check for the Berkeley Building ............................................................................. 27

7.3.4

Torsion Irregularity Check for the Honolulu Building ........................................................ 29

7.3.5

Drift Check for the Honolulu Building ................................................................................ 29

7.3.6

P-Delta Check for the Honolulu Building ........................................................................... 31

7.4

STRUCTURAL DESIGN OF THE BERKELEY BUILDING .................................................. 32

7.4.1

Analysis of Frame-Only Structure for 25 Percent of Lateral Load ..................................... 33

7.4.2

Design of Moment Frame Members for the Berkeley Building .......................................... 37

7.4.3

Design of Frame 3 Shear Wall............................................................................................. 60

FEMA P-751, NEHRP Recommended Provisions: Design Examples 7.5

7–2

STRUCTURAL DESIGN OF THE HONOLULU BUILDING ................................................. 66

7.5.1

Compare Seismic Versus Wind Loading ............................................................................. 66

7.5.2

Design and Detailing of Members of Frame 1 .................................................................... 69

Chapter 7: Reinforced Concrete In this chapter, a 12-story reinforced concrete office building with some retail shops on the first floor is designed for both high and moderate seismic loading. For the more extreme loading, it is assumed that the structure will be located in Berkeley, California and for the moderate loading, in Honolulu, Hawaii. The basic structural configuration for both locations is shown in Figures 7-1 and 7-2, which show a typical floor plan and building section, respectively. The building has 12 stories above grade and one basem*nt level. The typical bays are 30 feet long in the north-south (N-S) direction and either 40 or 20 feet long in the east-west (E-W) direction. The main gravity framing system consists of seven continuous 30-foot spans of pan joists. These joists are spaced at 36 inches on center and have an average web thickness of 6 inches and a depth below slab of 16 inches. Due to fire code requirements, a 4-inch-thick floor slab is used, giving the joists a total depth of 20 inches. The joists are supported by concrete beams running in the E-W direction. The building is constructed of normal-weight concrete. Concrete walls are located around the entire perimeter of the basem*nt level. For both locations, the seismic force-resisting system in the N-S direction consists of four 7-bay momentresisting frames. At the Berkeley location, these frames are detailed as special moment-resisting frames. Due to the lower seismicity and lower demand for system ductility, the frames of the Honolulu building are detailed as intermediate moment-resisting frames as permitted by ASCE 7. In the E-W direction, the seismic force-resisting system for the Berkeley building is a dual system composed of a combination of moment frames and frame-walls (walls integrated into a moment-resisting frame). Along Grids 1 and 8, the frames have five 20-foot bays. Along Grids 2 and 7, the frames consist of two exterior 40-foot bays and one 20-foot interior bay. At Grids 3, 4, 5 and 6, the interior bay consists of shear walls infilled between the interior columns. The exterior bays of these frames are similar to Grids 2 and 7. For the Honolulu building, the structural walls are not necessary, so E-W seismic resistance is supplied by the moment frames along Grids 1 through 8. The frames on Grids 1 and 8 are five-bay frames and those on Grids 2 through 7 are three-bay frames with the exterior bays having a 40foot span and the interior bay having a 20-foot span. Hereafter, frames are referred to by their gridline designation (e.g., Frame 1 is located on Grid 1). The foundation system is not considered in this example, but it is assumed that the structure for both the Berkeley and Honolulu locations is founded on very dense soil (shear wave velocity of approximately 2,000 feet per second).

7–3

FEMA P-751, NEHRP Recommended Provisions: Design Examples

D 1

Figure 7-2B

3 Figure 7-2A

7 at 30'-0"

212'-6"

8 5 at 20'-0" 102'-6"

Figure 7-1 Typical floor plan of the Berkeley building; the Honolulu building is similar but without structural walls (1.0 ft = 0.3048 m)

7–4

B 20'-0"

C' 40'-0"

A. Section at Wall (Grids 3-6)

40'-0"

11 at 13'-0"

Level

Story

20'-0"

A' 20'-0"

B 20'-0"

C 20'-0"

C' 20'-0"

B. Section at Exterior Frame (Grids 1 & 8)

Level R

11 at 13'-0"

Story

Chapter 7: Reinforced Concrete

18'-0" 15'-0"

Figure 7-2 Typical elevations of the Berkeley building; the Honolulu building is similar but without structural walls (1.0 ft = 0.3048 m)

7–5

FEMA P-751, NEHRP Recommended Provisions: Design Examples The calculations herein are intended to provide a reference for the direct application of the design requirements presented in the 2009 NEHRP Recommended Provisions (hereafter, the Provisions) and its primary reference document ASCE 7-05 Minimum Design Loads for Buildings and Other Structures (hereafter, the Standard) and to assist the reader in developing a better understanding of the principles behind the Provisions and ASCE 7. Because a single building configuration is designed for both high and moderate levels of seismicity, two different sets of calculations are required. Instead of providing one full set of calculations for the Berkeley building and then another for the Honolulu building, portions of the calculations are presented in parallel. For example, the development of seismic forces for the Berkeley and Honolulu buildings are presented before structural design is considered for either building. The design or representative elements then is given for the Berkeley building followed by the design of the Honolulu building. Each major section (development of forces, structural design, etc.) is followed by discussion. In this context, the following portions of the design process are presented in varying amounts of detail for each structure: 1. Seismic design criteria 2. Development and computation of seismic forces 3. Structural analysis and drift checks 4. Design of structural members including typical beams, columns and beam-column joints in Frame 1; and for the Berkeley building only, the design of the shear wall on Grid 3 In addition to the Provisions and the Standard, ACI 318-08 is the other main reference in this example. Except for very minor exceptions, the seismic force-resisting system design requirements of ACI 318 have been adopted in their entirety by the Provisions. Cases where requirements of the Provisions, the Standard and ACI 318 differ are pointed out as they occur. In addition to serving as a reference standard for seismic design, the Standard is also cited where discussions involve gravity loads, live load reduction, wind loads and load combinations. The following are referenced in this chapter: ACI 318

American Concrete Institute. 2008. Building Code Requirements and Commentary for Structural Concrete.

ASCE 7

American Society of Civil Engineers. 2005. Minimum Design Loads for Buildings and Other Structures.

ASCE 41

American Society of Civil Engineers. 2006. Seismic Rehabilitation of Existing Buildings, including Supplement #1.

Moehle

Moehle, Jack P., Hooper, John D and Lubke, Chris D. 2008. “Seismic design of reinforced concrete special moment frames: a guide for practicing engineers,” NEHRP Seismic Design Technical Brief No. 1, produced by the NEHRP Consultants Joint Venture, a partnership of the Applied Technology Council and the Consortium of Universities for Research in Earthquake Engineering, for the National Institute of Standards and Technology, Gaithersburg, MD., NIST GCR 8-917-1

7–6

Chapter 7: Reinforced Concrete The structural analysis for this chapter was carried out using the ETABS Building Analysis Program, version 9.5, developed by Computers and Structures, Inc., Berkeley, California. Axial-flexural interaction for column and shear wall design was performed using the PCA Column program, version 3.5, created and developed by the Portland Cement Association.

For Berkeley, California, the short period and one-second period spectral response acceleration parameters SS and S1 are 1.65 and 0.68, respectively. For the very dense soil conditions, Site Class C is appropriate as described in Standard Section 20.3. Using SS = 1.65 and Site Class C, Standard Table 11.4-1 lists a short period site coefficient, Fa, of 1.0. For S1 > 0.5 and Site Class C, Standard Table 11.4-2 gives a velocity based site coefficient, Fv, of 1.3. Using Standard Equation 11.4-1 and 11.4-2, the adjusted maximum considered spectral response acceleration parameters for the Berkeley building are: SMS = FaSS = 1.0(1.65) = 1.65 SM1 = FvS1 = 1.3(0.68) = 0.884 The design spectral response acceleration parameters are given by Standard Equation 11.4-3 and 11.4-4: SDS = 2/3 SMS = 2/3 (1.65) = 1.10 SD1 = 2/3 SM1 = 2/3 (0.884) = 0.589 The transition period, Ts, for the Berkeley response spectrum is:

Ts =

S D1 0.589 = = 0.535 sec S DS 1.10

Ts is the period where the horizontal (constant acceleration) portion of the design response spectrum intersects the descending (constant velocity or acceleration inversely proportional to T) portion of the spectrum. It is used later in this example as a parameter in determining the type of analysis that is required for final design. For Honolulu, the short-period and one-second period spectral response acceleration parameters are 0.61 and 0.18, respectively. For Site Class C soils and interpolating from Standard Table 11.4-1, the Fa is 1.16 and from Standard Table 11.4-1, the interpolated value for Fv is 1.62. The adjusted maximum considered spectral response acceleration parameters for the Honolulu building are: SMS = FaSS = 1.16(0.61) = 0.708 SM1 = FvS1 = 1.62(0.178) = 0.288 and the design spectral response acceleration parameters are: SDS = 2/3 SMS = 2/3 (0.708) = 0.472 SD1 = 2/3 SM1 = 2/3 (0.288) = 0.192

7–7

FEMA P-751, NEHRP Recommended Provisions: Design Examples

The transition period, Ts, for the Honolulu response spectrum is:

Ts =

S D1 0.192 = = 0.407 sec S DS 0.472

According to Standard Section 1.5, both the Berkeley and the Honolulu buildings are classified as Occupancy Category II. Standard Table 11.5-1 assigns an occupancy importance factor, I, of 1.0 to all Occupancy Category II buildings. According to Standard Tables 11.6-1 and 11.6-2, the Berkeley building is assigned to Seismic Design Category D and the Honolulu building is assigned to Seismic Design Category C.

The seismic force-resisting systems for both the Berkeley and the Honolulu buildings consist of momentresisting frames in the N-S direction. E-W loading is resisted by a dual frame-wall system in the Berkeley building and by a set of moment-resisting frames in the Honolulu building. For the Berkeley building, assigned to Seismic Design Category D, Standard Table 12.2-1 requires that all concrete moment-resisting frames be designed and detailed as special moment frames. Similarly, Standard Table 12.2-1 requires shear walls in dual systems to be detailed as special reinforced concrete shear walls. For the Honolulu building assigned to Seismic Design Category C, Standard Table 12.2-1 permits the use of intermediate moment frames for all building heights. Standard Table 12.2-1 provides values for the response modification coefficient, R, the system overstrength factor, Ω0 and the deflection amplification factor, Cd, for each structural system type. The values determined for the Berkeley and Honolulu buildings are summarized in Table 7-1. Table 7-1 Response Modification, Overstrength and Deflection Amplification Coefficients for Structural Systems Used Response Cd Location Building Frame Type R Ω0 Direction Berkeley N-S Special moment frame 8 3 5.5 Dual system incorporating special moment E-W 7 2.5 5.5 frame and special shear wall Honolulu N-S Intermediate moment frame 5 3 4.5 E-W Intermediate moment frame 5 3 4.5

For the Berkeley building dual system, Standard Section 12.2.5.1 requires that the moment frame portion of the system be designed to resist at least 25 percent of the total seismic force. As discussed below, this requires that a separate analysis of a frame-only system be carried out for loading in the E-W direction.

7–8

Chapter 7: Reinforced Concrete

Based on the plan view of the building shown in Figure 7-1, the only potential horizontal irregularity is a Type 1a or 1b torsional irregularity (Standard Table 12.3-1). While the actual presence of such an irregularity cannot be determined without analysis, it appears unlikely for both the Berkeley and the Honolulu buildings because the lateral force-resisting elements of both buildings are distributed evenly over the floor. However, this will be determined later. As for the vertical irregularities listed in Standard Table 12.3-2, the presence of a soft or weak story cannot be determined without analysis. In this case, however, the first story is suspect, because its height of 18 feet is well in excess of the 13-foot height of the story above. However, it is assumed (but verified later) that a vertical irregularity does not exist.

The combinations of loads including earthquake effects are provided in Standard Section 12.4. Load combinations for other loading conditions are in Standard Chapter 2. For the Berkeley structure, the basic strength design load combinations that must be considered are: 1.2D + 1.6L (or 1.6Lr) 1.2D + 0.5L ± 1.0E 0.9D ± 1.0E In addition to the combinations listed above, for the Honolulu building wind loads govern the design of a portion of the building (as determined later), so the following strength design load combinations should also be considered: 1.2D + 0.5L ± 1.6W 0.9D ± 1.6W The load combination including only 1.4 times dead load will not control for any condition in these buildings. In accordance with Standard Section 12.4.2 the earthquake load effect, E, be defined as:

E = ρ QE + 0.2 S DS D where gravity and seismic load effects are additive and

E = ρ QE − 0.2 S DS D where the effects of seismic load counteract gravity. The earthquake load effect requires the determination of the redundancy factor, ρ, in accordance with Standard Section 12.3.4. For the Honolulu building (Seismic Design Category C), ρ = 1.0 per Standard Section 12.3.4.1.

7–9

FEMA P-751, NEHRP Recommended Provisions: Design Examples

For the Berkeley building, ρ must be determined in accordance with Standard Section 12.3.4.2. For the purpose of the example, the method in Standard Section 12.3.4.2, Method b, will be utilized. Based on the preliminary design, it is assumed that ρ = 1.0 because the structure has a perimeter moment frame and is assumed to be regular based on the plan layout. As discussed in the previous section, this will be verified later. For the Berkeley building, substituting E and with ρ taken as 1.0, the following load combinations must be used for seismic design: (1.2 + 0.2SDS)D + 0.5L ± QE (0.9 - 0.2 SDS)D ± QE Finally, substituting 1.10 for SDS, the following load combinations must be used: 1.42D + 0.5L ± QE 0.68D ± QE For the Honolulu building, substituting E and with ρ taken as 1.0, the following load combinations must be used for seismic design: (1.2 + 0.2SDS)D + 0.5L ± QE (0.9 - 0.2SDS)D ± QE Finally, substituting 0.472 for SDS, the following load combinations must be used: 1.30D + 0.5L ± QE 0.80D ± QE The seismic load combinations with overstrength given in Standard Section 12.4.3.2 are not utilized for this example because there are no discontinuous elements supporting stiffer elements above them and collector elements are not addressed.

For the Berkeley building, normal-weight concrete of 5,000 psi strength is used everywhere (except as revised for the lower floor shear walls as determined later). All reinforcement has a specified yield strength of 60 ksi. As required by ACI 318 Section 21.1.5.2, the longitudinal reinforcement in the moment frames and shear walls either must conform to ASTM A706 or be ASTM A615 reinforcement, if the actual yield strength of the steel does not exceed the specified strength by more than 18 ksi and the ratio of actual ultimate tensile stress to actual tensile yield stress is greater than 1.25. The Honolulu building also uses 5,000 psi concrete and ASTM A615 Grade 60 reinforcing steel. ASTM 706 reinforcing is not required for an intermediate moment frame.

7 – 10

Chapter 7: Reinforced Concrete

The determination of seismic forces requires an understanding of the magnitude and distribution of structural mass and the stiffness properties of the structural system. Both of these aspects of design are addressed in the mathematical modeling of the structure.

Both the Berkeley and Honolulu buildings will be analyzed with a three-dimensional mathematical model using the ETABS software. Modeling criteria for the seismic analysis is covered in Standard Section 12.7. This section covers how to determine the effective seismic weight (addressed in the next section) and provides guidelines for the modeling of the building. Of most significance in a concrete building is modeling realistic stiffness properties of the structural elements considering cracked sections in accordance with Standard Section 12.7.3, Item a. Neither the Standard nor ACI 318 provides requirements for modeling cracked sections for seismic analysis, but the typical practice is to use a reduced moment of inertia for the beams, columns and walls based on the expected level of cracking. This example utilizes the following effective moment of inertia, Ieff, for both buildings:

Beams: Ieff = 0.3Igross

Columns: Ieff = 0.5Igross

Walls: Ieff = 0.5Igross

The effective stiffness of the moment frame elements is based on the recommendations in Moehle and ASCE 41 and account for the expected axial loads and reinforcement levels in the members. The value for the shear walls is based on the recommendations in ASCE 41 for cracked concrete shear walls. The following are other significant aspects of the mathematical model that should be noted: 1. The structure is modeled with 12 levels above grade and one level below grade. The perimeter basem*nt walls are modeled as shear panels as are the main structural walls at the Berkeley building. The walls are assumed to be fixed at their base, which is at the basem*nt level. 2. All floor diaphragms are modeled as infinitely rigid in plane and infinitely flexible out-of-plane, consistent with common practice for a regular-shaped concrete diaphragm (see Standard Section 12.3.1.2). 3. Beams, columns and structural wall boundary members are represented by two-dimensional frame elements. The beams are modeled as T-beams using the effective slab width per ACI 318 Section 8.10, as recommended by Moehle. 4. The structural walls of the Berkeley building are modeled as a combination of boundary columns and shear panels with composite stiffness. 5. Beam-column joints are modeled in accordance with Moehle, which references the procedure in ASCE 41. Both the beams and columns are modeled with end offsets based on the geometry, but the beam offset is modeled as 0 percent rigid, while the column offset is modeled as 100 percent rigid. This provides effective stiffness for beam-column joints consistent with the expected behavior of the

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FEMA P-751, NEHRP Recommended Provisions: Design Examples joint: strong column-weak beam condition. (While the recommendations in Moehle are intended for special moment frames, the same joint rigidities are used for Honolulu for consistency.) 6. P-delta effects are neglected in the analysis for simplicity. This assumption is verified later in this example. 7. While the base of the model is located at the basem*nt level, the seismic base for determination of forces is assumed to be at the first floor, which is at the exterior grade.

Before the building mass can be determined, the approximate size of the different members of the seismic force-resisting system must be established. For special moment frames, limitations on beam-column joint shear and reinforcement development length usually control. An additional consideration is the amount of vertical reinforcement in the columns. ACI 318 Section 21.4.3.1 limits the vertical steel reinforcing ratio to 6 percent for special moment frame columns; however, 3 to 4 percent vertical steel is a more practical upper-bound limit. Based on a series of preliminary calculations (not shown here), it is assumed that for the Berkeley building all columns and structural wall boundary elements are 30 inches by 30 inches, beams are 24 inches wide by 32 inches deep and the panel of the structural wall is 16 inches thick. It has already been established that pan joists are spaced at 36 inches on center, have an average web thickness of 6 inches and, including a 4-inch-thick slab, are 20 inches deep. For the Berkeley building, these member sizes probably are close to the final sizes. For the Honolulu building (which does not have the weight of concrete walls and ends up with slightly smaller frame elements: 28- by 28-inch columns and 20- by 30inch beams), the masses computed from the Berkeley member sizes are slightly high but are used for consistency. In addition to the building structural weight, the following superimposed dead loads are assumed:

Roofing = 10 psf

Partition = 10 psf (see Standard Section 12.7.2, Item 2)

Ceiling and M/E/P = 10 psf

Curtain wall cladding = 10 psf (on vertical surface area)

Based on the above member sizes and superimposed dead load, the individual story weights and masses are listed in Table 7-2. These masses are used for the analysis of both the Berkeley and the Honolulu buildings. Note from Table 7-2 that the roof and lowest floor have masses slightly different from the typical floors. It is also interesting to note that the average density of this building is 12.4 pcf, which is in the range of typical concrete buildings with relatively high floor-to-floor heights.

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Chapter 7: Reinforced Concrete Table 7-2 Story Weights and Masses Level Weight (kips) Roof 3,352 12 3,675 11 3,675 10 3,675 9 3,675 8 3,675 7 3,675 6 3,675 5 3,675 4 3,675 3 3,675 2 3,817 Total 43,919

Mass (kips-sec2/in.) 8.675 9.551 9.551 9.551 9.551 9.551 9.551 9.551 9.551 9.551 9.551 9.879 113.736

(1.0 kip = 4.45 kN, 1.0 in. = 25.4 mm)

In the ETABS model, these masses are applied as uniform distributed masses across the extent of the floor diaphragms in order to provide a realistic distribution of mass in the dynamic model as described below. The structural framing is modeled utilizing massless elements since their mass is included with the floor mass. Note that for relatively heavy cladding systems, it would be more appropriate to model the cladding mass linearly along the perimeter in order to more correctly model the mass moment of inertia. This has little impact in relatively light cladding systems as is the case here, so the cladding masses are distributed across the floor diaphragms for convenience.

The selection of analysis procedures is in accordance with Standard Table 12.6-1. Based on the initial review, it appears that the Equivalent Lateral Force (ELF) procedure is permitted for both the Berkeley and Honolulu buildings. However, as we shall see, the analysis demonstrates that the Berkeley building is torsionally irregular, meaning that the Model Response Spectrum Analysis (MRSA) procedure is required. Regardless of irregularities, it is common practice to use the MRSA for buildings in regions of high seismic hazard since the more rigorous analysis method tends to provide lower seismic forces and therefore more economical designs. For the Honolulu building, located in a region of lower seismic hazard and with wind governing in some cases, the ELF will be used. However, a dynamic model of the Honolulu building is used for determining the structural periods. It should be noted that even though the Berkeley building utilizes the MRSA, the ELF must be used for at least determining base shear for scaling of results as discussed below.

This section covers the ELF procedure for both the Berkeley and Honolulu buildings. Since the final analysis of the Berkeley building utilizes the MRSA procedure, the ELF is illustrated for determining base shear only. The complete ELF procedure is illustrated for the Honolulu building.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 7.2.4.1 Period Determination. Requirements for the computation of building period are given in Standard Section 12.8.2. For the preliminary design using the ELF procedure, the approximate period, Ta, computed in accordance with Standard Equation 12.8.7 can be used:

Ta = Ct hnx The method for determining approximate period will generally result in periods that are lower (hence, more conservative for use in predicting base shear) than those computed from a more rigorous mathematical model. If a more rigorous analysis is carried out, the resulting period may be too high due to a variety of possible modeling simplifications and assumptions. Consequently, the Standard places an upper limit on the period that can be used for design. The upper limit is T = CuTa where Cu is provided in Standard Table 12.8-1. For the N-S direction of the Berkeley building, the structure is a reinforced concrete moment-resisting frame and the approximate period is calculated according to Standard Equation 12.8-7 using Ct = 0.016 and x = 0.9 per Standard Table 12.8-2. For hn = 161 feet, Ta = 1.55 seconds and SD1 > 0.40 for the Berkeley building, Cu = 1.4 and the upper limit on the analytical period is T = 1.4(1.55) = 2.17 seconds. For E-W seismic activity in Berkeley, the structure is a dual system, so Ct = 0.020 and x =0.75 for “other structures.” The approximate period, Ta = 0.90 second and the upper limit on the analytical period is 1.4(0.90) = 1.27 seconds. For the Honolulu building, the Ta = 1.55 second period computed above for concrete moment frames is applicable in both the N-S and E-W directions. For Honolulu, SD1 is 0.192 and, from Standard Table 12.8-1, Cu can be taken as 1.52. The upper limit on the analytical period is T = 1.52(1.55) = 2.35 seconds. For the detailed period determination at both the Berkeley and Honolulu buildings, computer models were developed based on the criteria in Section 7.2.1. A summary of the Berkeley analysis is presented in Section 7.2.6, but the fundamental periods are presented here. The computed N-S period of vibration is 2.02 seconds. This is between the approximate period, Ta = 1.55 seconds and CuTa = 2.17 seconds. In the E-W direction, the computed period is 1.42 seconds, which is greater than both Ta = 0.90 second and CuTa = 1.27 seconds. Therefore, the periods used for the ELF procedure are 2.02 seconds in the N-S direction and 1.27 seconds in the E-W direction. For the Honolulu building, the computed periods in the N-S and E-W directions are 2.40 seconds and 2.33 seconds, respectively. The N-S period is similar to the Berkeley building because there are no walls in the N-S direction of either building, but the Honolulu period is higher due to the smaller framing member sizes. In the E-W direction, the increase in period from 1.42 seconds at the Berkeley building to 2.33 seconds indicates a significant reduction in stiffness due to the lack of the walls in the Honolulu building. For both the E-W and the N-S directions, Ta for the Honolulu building is 1.55 seconds and CuTa is 2.35 seconds. Therefore, for the purpose of computing ELF forces, the periods are 2.35 seconds and 2.33 seconds in the N-S and E-W directions, respectively.

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Chapter 7: Reinforced Concrete A summary of the approximate and computed periods is given in Table 7-3. Table 7-3 Comparison of Approximate and Computed Periods (in seconds) Berkeley Honolulu Method of Period Computation N-S E-W N-S Approximate Ta 1.55 0.90 1.55 Approximate × Cu 2.17 1.27 2.35 ETABS 2.02 1.42 2.40 * Bold values should be used in the ELF analysis.

E-W 1.55 2.35 2.33

7.2.4.2 Seismic Base Shear. For the ELF procedure, seismic base shear is determined using the short period and 1-second period response acceleration parameters, the computed structural period and the system response modification factor (R). Using Standard Equation 12.8-1, the design base shear for the structure is: V = C sW where W is the total effective seismic weight of the building and Cs is the seismic response coefficient computed in accordance with Standard Section 12.8.1.1. The seismic design base shear for the Berkeley is computed as follows: For the moment frame system in the N-S direction with W = 43,919 kips (see Table 7-2), SDS = 1.10, SD1 = 0.589, R = 8, I = 1 and T = 2.02 seconds:

Cs ,max = Cs =

S DS 1.10 = = 0.1375 R / I 8 /1

S D1 0.589 = = 0.0364 T ( R / I ) 2.02(8 / 1)

Cs,min = 0.044S DS I = 0.044(1.1)(1) = 0.0484 Cs,min = 0.01 Cs,min = 0.0484 controls and the design base shear in the N-S direction is V = 0.0484 (43,919) = 2,126 kips. In the E-W direction with the dual system, Cs,max and Cs,min are as before, T = 1.27 seconds and

Cs =

S D1 0.589 = = 0.0670 T ( R / I ) 1.27(7 / 1)

In this case, Cs = 0.0670 controls and V = 0.0670 (43,919) = 2,922 kips.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples For the Honolulu building, base shears are computed in a similar manner and are nearly the same for the N-S and the E-W directions. With W = 43,919 kips, SDS = 0.474, SD1 = 0.192, R = 5, I = 1 and T = 2.35 seconds in the N-S direction:

S DS 0.472 = = 0.0944 R/I 5 /1

Cs ,max = Cs =

S D1 0.192 = = 0.0163 T ( R / I ) 2.35(5 / 1)

Cs,min = 0.044SDS I = 0.044(0.472)(1.0) = 0.0207 Cs,min = 0.01 Cs = 0.0207 controls and V = 0.0207 (43,919) = 908 kips. Due to rounding, the E-W base shear is also 908 kips. A summary of the Berkeley and Honolulu seismic design parameters are provided in Table 7-4. Table 7-4 Comparison of Periods, Seismic Shears Coefficients and Base Shears for the Berkeley and Honolulu Buildings Response V (kips) Location Building Frame Type T (sec) Cs Direction Berkeley N-S Special moment frame 2.02 0.0485 2,126 Dual system incorporating special moment E-W 1.27 0.0670 2,922 frame and structural wall Honolulu N-S Intermediate moment frame 2.35 0.0207 908 E-W Intermediate moment frame 2.33 0.0207 908 (1.0 kip = 4.45 kN)

7.2.4.3 Vertical Distribution of Seismic Forces. The vertical distribution of seismic forces for the ELF is computed from Standard Equations 12.8-11 and 12.8-12.: Fx = CvxV k

C vx =

wx h x n

∑w h i

k i

i=1

where: k = 1.0 for T < 0.5 second k = 2.0 for T > 2.5 seconds k = 0.75 + 0.5T for 1.0 < T < 2.5 seconds

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Chapter 7: Reinforced Concrete Based on the equations above, the seismic story forces, shears and overturning moments are easily computed using a spreadsheet. Since the analysis of the Berkeley building utilizes the MRSA procedure, the vertical force distribution for the ELF procedure will not be used for the design and are not shown here. The vertical force distribution computations for the Honolulu building are shown in Table 7-5. The table is presented with as many significant digits to the left of the decimal as the spreadsheet generates but that should not be interpreted as real accuracy; it is just the simplest approach. Table 7-5 Vertical Distribution of N-S and E-W Seismic Forces for the Honolulu Building* Story Overturning Force Fx Height h Weight W Whk/Σ Level Whk Shear Vx Moment Mx (ft) (kips) (kips) (kips) (ft-k) R 161.00 3,352 59,048,176 0.196 177.8 177.8 12 148.00 3,675 55,053,755 0.183 165.8 343.6 2,312 11 135.00 3,675 46,128,207 0.153 138.9 482.5 6,779 10 122.00 3,675 37,963,112 0.126 114.3 596.9 13,052 9 109.00 3,675 30,564,359 0.101 92.0 688.9 20,811 8 96.00 3,675 23,938,555 0.079 72.1 761.0 29,767 7 83.00 3,675 18,093,222 0.060 54.5 815.5 39,660 6 70.00 3,675 13,037,074 0.043 39.3 854.8 50,262 5 57.00 3,675 8,780,453 0.029 26.4 881.2 61,374 4 44.00 3,675 5,336,045 0.018 16.1 897.3 72,830 3 31.00 3,675 2,720,196 0.009 8.2 905.5 84,494 2 18.00 3,817 992,774 0.003 3.0 908.5 96,265 Total 43,919 301,655,927 1.000 908 112,617 * Table based on T = 2.35 sec and k = 1.92. (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m)

The computed seismic story shears for the Honolulu buildings are shown graphically in Figure 7-3. Also shown in this figure are the wind load story shears determined in accordance with the Standard based on a 3-second gust of 105 mph and Exposure Category B. The wind shears have been multiplied by the 1.6 load factor to make them comparable to the strength design seismic loads (with a 1.0 load factor). As can be seen, the N-S seismic shears are significantly greater than the corresponding wind shears, but the E-W seismic and wind shears are closer. In the lower stories of the building, wind controls the strength demands and, in the upper levels, seismic forces control the strength demands. (A somewhat more detailed comparison is given later when the Honolulu building is designed.) With regards to detailing the Honolulu building, all of the elements must be detailed for inelastic deformation capacity as required by ACI 318 for intermediate moment frames.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

160

140

120

100 t)f ( h ,t h 80 ige H

60 N-S wind

E-W wind

20 Seismic

0 0

300

600

900

1200

1500

Story Shear, V (kips)

Figure 7-3 Comparison of wind and seismic story shears for the Honolulu building (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN) As expected, wind loads do not control the design of the Berkeley building based on calculations not presented here. (Note that the comparison between wind and seismic forces should be based on more than just the base shear values. For buildings where the wind and seismic loads are somewhat similar, it is possible that overturning moment for wind could govern even where the seismic base shear is greater, in which case a more detailed analysis of specific member forces would need to be performed to determine the controlling load case.) 7.2.4.4 Horizontal Force Distribution and Torsion. The story forces are distributed to the various vertical elements of the seismic force-resisting system based on relative rigidity using the ETABS model. As described previously, the buildings are modeled using rigid diaphragms at each floor. Since the structures are symmetric in both directions and the distribution of mass is assumed to be uniform, there is no inherent torsion (Standard Section 12.8.4.1) at either building. However, accidental torsion needs to be considered in accordance with Standard Section 12.8.4.2.

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Chapter 7: Reinforced Concrete For this example, accidental torsion is applied to each level as a moment equal to the story shear multiplied by 5 percent of the story width perpendicular to the direction of loading. The applied moment is based on the ELF forces for both the Berkeley building (analyzed using the MRSA) and Honolulu building (ELF). The computation of the accidental torsion moments for the Honolulu building is shown in Table 7-6. Table 7-6 Accidental Torsion for the Honolulu Building N-S E-W N-S Torsion Force Fx Building Building Level (kips) (ft-kips) Width (ft) Width (ft) R 177.8 103 911 216 12 165.8 103 850 216 11 138.9 103 712 216 10 114.3 103 586 216 9 92.0 103 472 216 8 72.1 103 369 216 7 54.5 103 279 216 6 39.3 103 201 216 5 26.4 103 136 216 4 16.1 103 82 216 3 8.2 103 42 216 2 3.0 103 15 216

E-W Torsion (ft-kips) 1,915 1,787 1,499 1,234 995 780 590 426 287 175 90 33

(1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m)

Amplification of accidental torsion, which needs to be considered for buildings with torsional irregularities in accordance with Standard Section 12.8.4.3, will be addressed if required after the irregularities are determined.

For the initial analysis, the seismic loading is applied in two directions independently as permitted by Standard Section 12.5. This assumption at the Berkeley building will need to be verified later since Standard Section 12.5.4 requires consideration of multi-directional loading (the 100 percent-30 percent procedure) for columns that form part of two intersection systems and have a high seismic axial load. Note that rather than checking whether or not multi-directional loading needs to be considered, some designers apply the seismic forces using the 100 percent-30 percent rule (or an SRSS combination of the two directions) as common practice when intersecting systems are utilized since today’s computer analysis programs can make the application of multi-directional loading easier than checking each specific element. Since multi-directional loading is not a requirement of the Standard, the Berkeley building will not be analyzed in this manner unless required for specific columns. The Honolulu building, in Seismic Design Category C, does not require consideration of multi-directional loading since it does not contain the nonparallel system irregularity (Standard Section 12.5.3).

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

The Berkeley building will be analyzed using the MRSA procedure of Standard Section 12.9 and the ETABS software. The building is modeled based on the criteria discussed in Section 7.2.1 and analyzed using a response spectrum generated by ETABS based on the seismic response parameters presented in Section 7.1.1. The modal parameters were combined using the complete quadratic combination (CQC) method per Standard Section 12.9.3. The computed periods and the modal response characteristics of the Berkeley building are presented in Table 7-7. In order to capture higher mode effects, 12 modes were selected for the analysis and with 12 modes, the accumulated modal mass in each direction is more than 90 percent of the total mass as required by Standard Section 12.9.1. Table 7-7 Periods and Modal Response Characteristics for the Berkeley Building % of Effective Mass Represented by Mode* Period Description Mode (sec) N-S E-W 1 2.02 83.62 (83.62) 0.00 (0.00) First Mode N-S 2 1.46 0.00 (83.62) 0.00 (0.00) First Mode Torsion 3 1.42 0.00 (83.62) 74.05 (74.05) First Mode E-W 4 0.66 9.12 (92.74) 0.00 (74.05) Second Mode N-S 5 0.38 2.98 (95.72) 0.00 (74.05) Third Mode N-S 6 0.35 0.00 (95.72) 16.02 (90.07) Second Mode E-W 7 0.25 1.36 (97.08) 0.00 (90.07) Fourth Mode N-S 8 0.18 0.86 (97.94) 0.00 (90.07) Fifth Mode N-S 9 0.17 0.00 (97.94) 0.09 (90.16) Second Mode Torsion 10 0.15 0.00 (97.94) 5.28 (95.44) Third Mode E-W 11 0.10 0.59 (98.53) 0.00 (95.44) Sixth Mode N-S 12 0.08 0.00 (98.53) 3.14 (98.58) Fourth Mode E-W * Accumulated modal mass in parentheses. One of the most important aspects of the MRSA procedure is the scaling requirement. In accordance with Standard Section 12.9.4, the results of the MRSA cannot be less than 85 percent of the results of the ELF. This is commonly accomplished by running the MRSA to determine the modal base shear. If the modal base shear is more than 85 percent of the ELF base shear in each direction, then no scaling is required. However, if the model base shear is less than the ELF base shear, then the response spectrum is scaled upward so that the modal base shear is equal to 85 percent of the ELF base shear. This is illustrated in Table 7-8.

Table 7-8 Scaling of MRSA results for the Berkeley Building Direction

VELF (kips)

N-S E-W

2,126 2,922

(1.0 kip = 4.45 kN)

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0.85VELF (kips) 1,807 2,483

VMRSA (kips) 1,462 2,296

Scale Factor 1.24 1.08

Chapter 7: Reinforced Concrete Therefore, the response spectrum functions for the Berkeley analysis will be scaled by 1.24 and 1.08 in the N-S and E-W directions, respectively, which will result in the modal base shears being equal to 85 percent of the static base shears. As discussed previously, the accidental torsion requirement for the model analysis will be satisfied by applying the torsional moments computed for the ELF procedure as a static load case that will be combined with the dynamic load case for the MRSA forces.

$ # ! The checks of story drift and P-delta effect are contained in this section, but first, deflection-related configuration checks are performed for each building. As discussed previously, these structures could contain torsional or soft-story irregularities. The output from the drift analysis will be used to determine if either of these irregularities is present in the buildings. However, the presence of a soft story irregularity impacts only the analysis procedure limitations for the Berkeley building and has no impact on the design procedures for the Honolulu building. $#"

In Section 7.1.4 it was mentioned that torsional irregularities are unlikely for the Berkeley building because the elements of the seismic force-resisting system were well distributed over the floor area. This will now be verified by comparing the story drifts at each end of the building in accordance with Standard Table 12.3-1. For this check, drifts are computed using the ETABS program using the ELF procedure (to avoid having to obtain modal combinations of drifts at multiple points) and including accidental torsion with Ax = 1.0. Note that since this check is only for relative drifts, the Cd factor is not included. The drift computations and torsion check for the E-W direction are shown in Table 7-9. The drift values are shown only for one direction of accidental torsion (positive torsion moment) since the other direction is the opposite due to symmetry. Table 7-9 Torsion Check for Berkeley Building Loaded in the E-W Direction Story Drift Story Drift Average Story Max Drift / Story North End (in) South End(in) Drift (in) Average Drift Roof 0.180 0.230 0.205 1.12 12 0.184 0.246 0.215 1.14 11 0.188 0.261 0.225 1.16 10 0.192 0.276 0.234 1.18 9 0.193 0.288 0.241 1.20 8 0.192 0.296 0.244 1.21 7 0.188 0.298 0.243 1.23 6 0.178 0.292 0.235 1.24 5 0.164 0.278 0.221 1.26 4 0.144 0.254 0.199 1.28 3 0.117 0.218 0.167 1.30 2 0.119 0.229 0.174 1.32 (1.0 in = 25.4 mm)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples As can be seen from the table, a torsional irregularity (Type 1a) does exist at Story 8 and below because the ratio of maximum to average drift exceeds 1.2. This is counterintuitive for a symmetric building but can happen for a building in which the lateral elements are located towards the center of a relatively long floor plate, as occurs here. This configuration results in a relatively large accidental torsion load but relatively low torsional resistance. For loading in the N-S direction, similar computations (not shown here) demonstrate that the structure is torsionally regular. The presence of the torsional irregularity in the E-W direction has several implications for the design:

The qualitative determination for using the redundancy factor, ρ, equal to 1.0 is not applicable per Standard Section 12.3.4.2, Item b, as previously assumed in Section 7.1.5. For the purposes of this example, we will assume ρ = 1.0 based on Standard Section 12.3.4.2, Item a. Due to the number of shear walls and moment frames in the E-W direction, the loss of individual wall or frame elements would still satisfy the criteria of Standard Table 12.3-3. This would have to be verified independently and if those criteria were not met, then analysis would have to be revised with ρ = 1.3.

The ELF procedure is not permitted per Standard Table 12.6-1. This does not change the analysis since we are utilizing the MRSA procedure.

The amplification of accidental torsion needs to be considered per Standard Section 12.8.4.3. The Ax factor is computed for each floor in this direction and the analysis is revised. See below.

Story drifts need to be checked at both ends of the building rather than at the floor centroid, per Standard Section 12.12.1. This is covered in Section 7.3.2 below.

The initial determination of accidental torsion was based on Ax = 1.0. Due to the torsional irregularity, accidental torsion for the E-W direction of loading needs to be computed again with the amplification factor. This is shown in Table 7-10. Note that while the determination of the torsional irregularity is based on story drifts, the computation of the torsional amplification factor is based on story displacements.

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Chapter 7: Reinforced Concrete Table 7-10 Accidental Torsion for the Berkeley Building E-W E-W Force Fx Max Displ Ave Displ Level Building Torsion (kips) (in) (in) Width (ft) (ft-k) Roof 474.8 213 5,045 3.17 2.60 12 463.3 213 4,923 2.94 2.40 11 408.0 213 4,335 2.69 2.18 10 354.7 213 3,769 2.43 1.96 9 303.5 213 3,225 2.15 1.72 8 254.6 213 2,706 1.87 1.48 7 208.2 213 2,213 1.57 1.24 6 164.5 213 1,748 1.27 1.00 5 123.8 213 1,316 0.98 0.76 4 86.6 213 920 0.70 0.54 3 53.3 213 567 0.45 0.34 2 26.1 213 278 0.23 0.17

Ax 1.03 1.04 1.06 1.07 1.08 1.10 1.11 1.13 1.15 1.17 1.19 1.20

E-W Torsion, AxMta (ft-k) 5,186 5,128 4,575 4,029 3,493 2,970 2,463 1,976 1,511 1,075 674 333

(1.0 kip = 4.45 kN, .0 ft = 0.3048 m, 1.0 ft-kip = 1.36 kN-m)

With the revised accidental torsion values for the E-W direction of loading, the ETABS model is rerun for the drift checks and member design in subsequent sections.

Story drifts are computed in accordance with Standard Section 12.9.2 and then checked for acceptance based on Standard Section 12.12.1. According to Standard Table 12.12-1, the story drift limit for this Occupancy Category II building is 0.020hsx, where hsx is the height of story x. This limit may be thought of as 2 percent of the story height. Quantitative results of the drift analysis for the N-S and E-W directions are shown in Tables 7-11a and 7-11b, respectively. The story drifts are taken directly from the modal combinations in ETABS. Due to the torsional irregularity in the E-W direction, drifts are checked at both ends of the structure, while N-S drifts are checked at the building centroid. In neither case does the computed drift ratio (amplified story drift divided by hsx) exceed 2 percent of the story height. Therefore, the story drift requirement is satisfied. A plot of the total deflection in both the N-S and E-W directions is shown in Figure 7-4 and a plot of story drifts is in Figure 7-5.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples An example calculation for drift in Story 5 loaded in the N-S direction is given below. Note that the relevant row is highlighted in Table 7-11a. Story drift = Δ5e = 0.208 inch Deflection amplification factor, Cd = 5.5 Importance factor, I = 1.0 Amplified story drift = Δ5 = Cd Δ5e/I = 5.5(0.208)/1.0 = 1.14 inches Amplified drift ratio = Δ5/h5 = (1.14/156) = 0.00733 = 0.733% < 2.0%

amplified (Cd = 5.5)

elastic

160

E-W

N-S

E-W

N-S

140

120

100 )t (f h ,t h 80 gi e H

0 0

5 Total deflection, Ξ (inches)

Figure 7-4 Deflected shape for Berkeley building (1.0 ft = 0.3048 m, 1.0 in = 25.4 mm)

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Chapter 7: Reinforced Concrete

N-S

160

E-W

allowable

140

120

100 t)f ( h ,t h 80 ige H

0 0

0.5

1 1.5 Design story drift, ≅ (inches)

Figure 7-5 Drift profile for Berkeley building (1.0 ft = 0.3048 m)

7 – 25

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 7-11a Drift Computations for the Berkeley Building Loaded in the N-S Direction Story Story Drift (in) Story Drift × Cd * (in) Drift Ratio** (%) Roof 0.052 0.29 0.184 12 0.083 0.46 0.293 11 0.112 0.61 0.393 10 0.134 0.74 0.474 9 0.153 0.84 0.540 8 0.168 0.93 0.593 7 0.182 1.00 0.641 6 0.195 1.07 0.688 5 0.208 1.14 0.733 4 0.221 1.21 0.778 3 0.232 1.28 0.819 2 0.287 1.58 0.732 * Cd = 5.5 for loading in this direction. ** Story height = 156 inches for Stories 3 through roof and 216 inches for Story 2. (1.0 in = 25.4 mm)

Table 7-11b Drift Computations for the Berkeley Building Loaded in the E-W Direction Story Drift Story Drift Max Story Drift × Max Drift Ratio** Story North End (in) South End (in) Cd * (in) (%) Roof 0.163 0.163 0.90 0.576 12 0.177 0.177 0.97 0.623 11 0.188 0.188 1.03 0.663 10 0.199 0.199 1.10 0.702 9 0.208 0.208 1.14 0.734 8 0.214 0.214 1.18 0.755 7 0.217 0.217 1.19 0.763 6 0.215 0.215 1.18 0.756 5 0.207 0.207 1.14 0.729 4 0.192 0.192 1.05 0.675 3 0.167 0.168 0.92 0.591 2 0.167 0.167 0.92 0.425 * Cd = 5.5 for loading in this direction. ** Story height = 156 inches for Stories 3 through roof and 216 inches for Story 2. (1.0 in = 25.4 mm)

The story deflection information will be used to determine whether or not a soft story irregularity exists. As indicated previously, a soft story irregularity (Vertical Irregularity Type 1a) would not impact the design since we are utilizing the MRSA. However, an extreme soft story irregularity (vertical irregularity Type 1b) is prohibited in Seismic Design Category D building per Standard Section 12.3.3.1.

7 – 26

Chapter 7: Reinforced Concrete However, Standard Section 12.3.2.2 lists an exception: Structural irregularities of Types 1a, 1b, or 2 in Table 12.3-2 do not apply where no story drift ratio under design lateral load is less than or equal to 130 percent of the story drift ratio of the next story above…. The story drift ratios of the top two stories of the structure are not required to be evaluated. To determine whether the exception applies to the Berkeley building, the ratio of the drift ratios are reported in Table 7-11c. Table 7-11c Drift Ratio Comparisons for Stiffness Irregularity Check North-South Ratio to East-West Ratio to Story Drift Ratio Story Above Drift Ratio Story Above Roof 0.184 0.184 12 0.293 0.293 1.59 1.09 11 0.393 0.393 1.34 1.06 10 0.474 0.474 1.21 1.06 9 0.540 0.540 1.14 1.05 8 0.593 0.593 1.10 1.03 7 0.641 0.641 1.08 1.01 6 0.688 0.688 1.07 0.99 5 0.733 0.733 1.07 0.96 4 0.778 0.778 1.06 0.93 3 0.819 0.819 1.05 0.88 2 0.732 0.732 0.89 0.72 As can be seen the vertical irregularity does not apply in the E-W direction since the ratio is less than 1.3 at all stories. In the N-S direction, however, the ratio exceeds 1.3 at the two upper stories. While the top stories are excluded from this check, the ratio of 1.34 at Story 11 means that the story stiffness’s need to be evaluated to determine whether there is a stiffness irregularity based on Standard Table 12.3-2. Since this controlling ratio of drift ratios is at an upper floor and just exceeds the 1.3 limit, it could be reasonable to conclude that a stiffness irregularity does not exist. For the purposes of this example, as long as an extreme stiffness irregularity is not present (which seems highly unlikely given the relative drift ratios), the presence of a non-extreme stiffness irregularity does not have a substantive impact on the design since this example utilizes the MRSA procedure anyway. In accordance with Standard Table 12.6-1, the ELF procedure would not be permitted if there were to be a stiffness irregularity. Therefore, the required stiffness checks for the N-S direction are not shown in this example.

In accordance with Standard Section 12.8.7 (as referenced by Standard Section 12.9.6 for the MRSA), Pdelta effects need not be considered in the analysis if the stability coefficient, θ, is less than 0.10 for each story. However, the Standard also limits θ to a maximum value determined by Standard Equation 12.817 as:

θ max =

0.5 ≤ 0.25 βC d 7 – 27

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Taking β as 1.0 (see Standard Section 12.8.7), the limit on stability coefficient for both directions is 0.5/(1.0)5.5 = 0.091. The P-delta analysis for each direction of loading is shown in Tables 7-12a and 7-12b. For this P-delta analysis a story live load of 20 psf (50 psf for office occupancy reduced to 40 percent per Standard Section 4.8.1) was included in the total story load calculations. Deflections and story shears are based on the MRSA with no upper limit on period in accordance with Standard Sections 12.9.6 and 12.8.6.2. As can be seen in the last column of each table, θ does not exceed the maximum permitted value computed above and P-delta effects can be neglected for both drift and strength analyses. An example P-delta calculation for the Story 5 under N-S loading is shown below. Note that the relevant row is highlighted in Table 7-12a. Amplified story drift = Δ5 = 1.144 inches Story shear = V5 = 1,240 kips Accumulated story weight P5 = 36,532 kips Story height = hs5= 156 inches I = 1.0 Cd = 5.5 θ = (P5IΔ5/(V5hs5Cd) = (36,532)(1.0)(1.144)/(6.5)(1,240)(156) = 0.0393 < 0.091

Table 7-12a P-Delta Computations for the Berkeley Building Loaded in the N-S Direction Story Dead Story Live Total Story Accum. Story Stability Story Drift Story Shear Story (in) (kips) Load (kips) Load (kips) Load (kips) Load (kips) Coeff, θ Roof 0.287 261 3,352 420 3,772 3,772 0.0048 12 0.457 495 3,675 420 4,095 7,867 0.0085 11 0.613 672 3,675 420 4,095 11,962 0.0127 10 0.740 807 3,675 420 4,095 16,057 0.0172 9 0.842 914 3,675 420 4,095 20,152 0.0216 8 0.926 1,003 3,675 420 4,095 24,247 0.0261 7 1.000 1,083 3,675 420 4,095 28,342 0.0305 6 1.073 1,161 3,675 420 4,095 32,437 0.0349 5 1.144 1,240 3,675 420 4,095 36,532 0.0393 4 1.214 1,322 3,675 420 4,095 40,627 0.0435 3 1.278 1,400 3,675 420 4,095 44,722 0.0476 2 1.581 1,462 3,817 420 4,237 48,959 0.0446 (1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)

7 – 28

Chapter 7: Reinforced Concrete Table 7-12b P-Delta Computations for the Berkeley Building Loaded in the E-W Direction Story Drift Story Shear Story Dead Story Live Total Story Accum. Story Stability Story (in) (kips) Load (kips) Load (kips) Load (kips) Load (kips) Coeff, θ Roof 0.899 463 3,352 420 3,772 3,772 0.0085 12 0.972 843 3,675 420 4,095 7,867 0.0106 11 1.035 1,104 3,675 420 4,095 11,962 0.0131 10 1.096 1,275 3,675 420 4,095 16,057 0.0161 9 1.145 1,396 3,675 420 4,095 20,152 0.0193 8 1.177 1,512 3,675 420 4,095 24,247 0.0220 7 1.191 1,645 3,675 420 4,095 28,342 0.0239 6 1.180 1,787 3,675 420 4,095 32,437 0.0250 5 1.137 1,927 3,675 420 4,095 36,532 0.0251 4 1.054 2,073 3,675 420 4,095 40,627 0.0241 3 0.921 2,215 3,675 420 4,095 44,722 0.0217 2 0.918 2,296 3,817 420 4,237 48,959 0.0165 (1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)

A test for torsional irregularity for the Honolulu building can be performed in a manner similar to that for the Berkeley building. Based on computations not shown here, the Honolulu building is not torsionally irregular. This is the case because the walls, which draw the torsional resistance towards the center of the Berkeley building, do not exist in the Honolulu building. Therefore, the torsional amplification factor, Ax = 1.0 for all levels and the accidental torsion moments used for the analysis do not need to be revised.

The story drift computations for the Honolulu building deforming under the N-S and E-W seismic loading are shown in Tables 7-13a and 7-13b. These tables show that the story drift at all stories is less than the allowable story drift of 0.020hsx (Standard Table 12.12-1). Even though it is not pertinent for Seismic Design Category C buildings, a soft first story does not exist for the Honolulu building because the ratio of first story drift to second story drift does not exceed 1.3.

7 – 29

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 7-13a Drift Computations for the Honolulu Building Loaded in the N-S Direction Story Drift × Cd * Story Total Drift (in) Story Drift (in) Drift Ratio (%) (in) Roof 1.938 0.057 0.259 0.166 12 1.880 0.087 0.391 0.251 11 1.793 0.115 0.517 0.331 10 1.678 0.138 0.623 0.399 9 1.540 0.157 0.708 0.454 8 1.382 0.172 0.773 0.496 7 1.210 0.182 0.821 0.526 6 1.028 0.190 0.854 0.547 5 0.838 0.194 0.873 0.559 4 0.644 0.195 0.881 0.565 3 0.449 0.198 0.889 0.570 2 0.251 0.247 1.113 0.515 * Cd = 4.5 for loading in this direction; total drift is at top of story, story height = 156 inches for Stories 3 through roof and 216 inches for Story 2. (1.0 in. = 25.4 mm)

Table 7-13b Drift Computations for the Honolulu Building Loaded in the E-W Direction Story Drift × Cd * Drift Ratio (%) Story Total Drift (in) Story Drift (in) (in) Roof 2.034 0.051 0.230 0.147 12 1.983 0.083 0.376 0.241 11 1.899 0.115 0.518 0.332 10 1.784 0.142 0.639 0.410 9 1.642 0.164 0.736 0.473 8 1.478 0.181 0.814 0.522 7 1.297 0.194 0.874 0.559 6 1.103 0.203 0.915 0.586 5 0.900 0.209 0.942 0.604 4 0.691 0.213 0.958 0.614 3 0.478 0.216 0.970 0.622 2 0.262 0.261 1.173 0.543 Cd = 4.5 for loading in this direction; total drift is at top of story, story height = 156 inches for Levels 2 through roof and 216 inches for Level 1. (1.0 in = 25.4 mm)

7 – 30

Chapter 7: Reinforced Concrete A sample calculation for Story 5 of Table 7-13b (highlighted in the table) is as follows: Deflection at top of story = δ5e =0.900 inches Deflection at bottom of story = δ4e = 0.691 inch Story drift = Δ5e = δ5e - δ4e = 0.900 - 0.0691 = 0.209 inch Deflection amplification factor, Cd = 4.5 Importance factor, I = 1.0 Amplified story drift = Δ5 = Cd Δ5e/I = 4.5(0.209)/1.0 = 0.942 inch Amplified drift ratio = Δ5 / h5 = (0.942/156) = 0.00604 = 0.604% < 2.0%

Therefore, story drift satisfies the drift requirements.

Calculations for P-delta effects are shown in Tables 7-14a and 7-14b for N-S and E-W loading, respectively.

Table 7-14a P-Delta Computations for the Honolulu Building Loaded in the N-S Direction Story

Story Drift (in)

Story Shear (kips)

Story Dead Load (kips)

Roof 12 11 10 9 8 7 6 5 4 3 2

0.259 0.391 0.517 0.623 0.708 0.773 0.821 0.854 0.873 0.881 0.889 1.113

177.8 343.6 482.5 596.9 688.9 761.0 815.5 854.8 881.2 897.3 905.5 908.5

3,352 3,675 3,675 3,675 3,675 3,675 3,675 3,675 3,675 3,675 3,675 3,817

Story Live Total Story Accum. Story Stability Load (kips) Load (kips) Load (kips) Coeff, θ 420 420 420 420 420 420 420 420 420 420 420 420

3,772 4,095 4,095 4,095 4,095 4,095 4,095 4,095 4,095 4,095 4,095 4,237

3,772 7,867 11,962 16,057 20,152 24,247 28,342 32,437 36,532 40,627 44,722 48,959

0.0069 0.0123 0.0183 0.0245 0.0307 0.0370 0.0432 0.0494 0.0556 0.0618 0.0682 0.0650

(1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)

7 – 31

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 7-14b P-Delta Computations for the Honolulu Building Loaded in the E-W Direction Story Drift Story Shear Story Dead Story Live Total Story Accum. Story Stability Story (in) (kips) Load (kips) Load (kips) Load (kips) Load (kips) Coeff, θ Roof 0.230 177.8 3,352 420 3,772 3,772 0.0079 12 0.376 343.6 3,675 420 4,095 7,867 0.0128 11 0.518 482.5 3,675 420 4,095 11,962 0.0183 10 0.639 596.9 3,675 420 4,095 16,057 0.0239 9 0.736 688.9 3,675 420 4,095 20,152 0.0296 8 0.814 761.0 3,675 420 4,095 24,247 0.0351 7 0.874 815.5 3,675 420 4,095 28,342 0.0407 6 0.915 854.8 3,675 420 4,095 32,437 0.0462 5 0.942 881.2 3,675 420 4,095 36,532 0.0515 4 0.958 897.3 3,675 420 4,095 40,627 0.0568 3 0.970 905.5 3,675 420 4,095 44,722 0.0626 2 1.173 908.5 3,817 420 4,237 48,959 0.0617 (1.0 in = 25.4 mm, 1.0 kip = 4.45 kN)

The stability ratio at Story 5 from Table 7-14b is computed as follows: Amplified story drift = Δ5 = 0.942 inch Story shear = V5 = 881.2 = kips Accumulated story weight P5 = 36,532 kips Story height = hs5 = 156 inches Cd = 4.5 θ = [P5 (Δ5/Cd)]/(V5hs5) = 36,532(0.942/4.5)/(881.2)(156) = 0.0515 The requirements for maximum stability ratio (0.5/Cd = 0.5/4.5 = 0.111) are satisfied. Because the stability ratio is less than 0.10 at all floors, P-delta effects need not be considered (Standard Section 12.8.7).

Frame-wall interaction plays an important role in the behavior of the structure loaded in the E-W direction. This behavior has the following attributes: 1. For frames without walls (Frames 1, 2, 7 and 8), the shears developed in the beams (except for the first story) do not differ greatly from story to story. This allows for uniformity in the design of the beams. 2. For frames containing structural walls (Frames 3 through 6), the overturning moments in the structural walls are reduced as a result of interaction with the remaining frames (Frames 1, 2, 7 and 8). 3. For the frames containing structural walls, the 40-foot-long girders act as outriggers further reducing the overturning moment resisted by the structural walls.

7 – 32

Chapter 7: Reinforced Concrete 4. A significant load reversal occurs at the top of frames with structural walls. This happens because the structural wall pulls back on (supports) the top of Frame 1. The deflected shape of the structure loaded in the E-W direction also shows the effect of frame-wall interaction because the shape is neither a cantilever mode (wall alone) nor a shear mode (frame alone). It is the “straightening out” of the deflected shape of the structure that causes the story shears in the frames without walls to be relatively equal. Some of these attributes are shown graphically in Figure 7-6, which illustrates the total story force resisted by Frames 1, 2 and 3.

160

140

120

100 t)f ( h ,t h 80 gi e H

Frame 2 Frame 3 (frame & wall)

Frame 1

0 0

100

200

300

400

500

Story shear, V (kips)

Figure 7-6 Story shears in the E-W direction (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)

Where a dual system is utilized, Standard Section 12.2.5.1 requires that the moment frames themselves are designed to resist at least 25 percent of the total base shear. This provision ensures that the dual system has sufficient redundancy to justify the increase from R = 6 for a special reinforced concrete

7 – 33

FEMA P-751, NEHRP Recommended Provisions: Design Examples structural wall to R = 7 for a dual system (see Standard Table 12-2). This 25 percent analysis was carried out using the ETABS program with the mathematical model of the building being identical to the previous version except that the panels of the structural walls were removed. The boundary elements of the walls were retained in the model so that behavior of the interior frames (Frames 3, 4, 5 and 6) would be analyzed in a rational way. (It could be argued that keeping the boundary columns in the 25 percent model violates the intent of the provision since they are an integral part of the shear walls. However, in this condition, the columns are needed for the moment frames adjacent to the walls and those in longitudinal direction (which resist a small amount of torsion). Since these eight boundary columns resist only a small portion (just over 15 percent) the total base shear for the 25 percent model, the intent of the dual system requirements is judged to be satisfied. It should be noted that it is not the intent of the Standard to allow dual systems of co-planar and integral moment frames and shear walls.) The seismic demands for this frame-only analysis were scaled such that the spectra base shear is equal to 25 percent of the design base shear for the dual system. In this case, the response spectrum was scaled such that the frame-only base shear is equal to (0.25)(0.85)VELF. While this may not result in story forces exactly equal to 25 percent of the story forces from the MRSA of the dual system, the method used is assumed to meet the intent of this provision of the Standard. The results of the analysis are shown in Figures 7-7, 7-8 and 7-9 for the frames on Grids 1, 2 and 3, respectively. The frames on Grids 6, 7 and 8 are similar by symmetry and Grids 4 and 5 are similar to Grid 3. In these figures, the original analysis (structural wall included) is shown by a heavy line and the 25 percent (frame-only) analysis is shown by a light, dashed line. As can be seen, the 25 percent rule controls only at the lower level of the building. Therefore, for the design of the beams and columns at the lower two levels (not part of this example), the greater of the dual system and frame-only analysis should be used. For the purposes of this example, which includes representative designs for the framing at a middle level, design forces from the dual system analysis will satisfy the 25 percent requirement.

7 – 34

Chapter 7: Reinforced Concrete

25% V

160

Frame 1

140

120

100 t)f ( h ,t h 80 ige H

0 0

100

200

Story shear, V (kips)

Figure 7-7 25 percent story shears, Frame 1 E-W direction (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)

7 – 35

FEMA P-751, NEHRP Recommended Provisions: Design Examples

25% V

160

Frame 2

140

120

100 t)f ( h ,t h 80 ige H

0 0

100

200

Story shear, V (kips)

Figure 7-8 25 percent story shears, Frame 2 E-W direction (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)

7 – 36

Chapter 7: Reinforced Concrete

25% V

160

Frame 3

140

120

100 t)f ( h ,t h 80 ige H

0 0

100

200

Story shear, V (kips)

Figure 7-9 25 percent story shear, Frame 3 E-W direction (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN)

For this part of the example, the design and detailing of five beams and one interior column along Grid 1 on Level 5 are presented in varying amounts of detail. The beams are designed first because the flexural capacity of the as-designed beams is a factor in the design and detailing of the column and the beamcolumn joint. Before continuing with the example, it should be mentioned that the design of ductile reinforced concrete moment frame members is controlled by the flexural reinforcement in the beams. The percentage and placement of beam flexural reinforcement governs the flexural rebar cutoff locations, the size and spacing of beam shear reinforcement, the cross-sectional characteristics of the column, the column flexural reinforcement and the column shear reinforcement. The beam reinforcement is critical because the basic concept of ductile frame design is to force most of the energy-dissipating deformation to occur through inelastic rotation in plastic hinges at the ends of the beams.

7 – 37

FEMA P-751, NEHRP Recommended Provisions: Design Examples In carrying out the design calculations, three different flexural strengths are used for the beams. These capacities are based on the following:

Design strength: φ = 0.9, tensile stress in reinforcement at 1.00 fy

Nominal strength: φ = 1.0, tensile stress in reinforcement at 1.00 fy

Probable strength: φ = 1.0, tensile stress in reinforcement at 1.25 fy

Various aspects of the design of the beams and other members depend on the above capacities are as follows:

Beam rebar cutoffs: Design strength

Beam shear reinforcement: Probable strength of beam

Beam-column joint strength: Probable strength of beam

Column flexural strength: 6/5 × nominal strength of beam

Column shear strength: Probable strength of column or beam

In addition, beams in ductile frames will always have top and bottom longitudinal reinforcement throughout their length. In computing flexural capacities, only the tension steel will be considered. This is a valid design assumption because reinforcement ratios are quite low, yielding a depth to the neutral axis similar to the depth of the compression reinforcement. Finally, a sign convention for bending moments is required in flexural design. In this example, where the steel at the top of a beam section is in tension, the moment is designated as a negative moment. Where the steel at the bottom is in tension, the moment is designated as a positive moment. All moment diagrams are drawn using the reinforced concrete or tension-side convention. For beams, this means negative moments are plotted on the top and positive moments are plotted on the bottom. For columns, moments are drawn on the tension side of the member. 7.4.2.1 Preliminary Calculations. Before the quantity and placement of reinforcement is determined, it is useful to establish, in an overall sense, how the reinforcement will be distributed. The preliminary design established that the moment frame beams would be 24 inches wide by 32 inches deep and the columns would be 30 inches by 30 inches. Note that the beam widths were selected to consider the beamcolumn joints “confined” per ACI 318 Section 21.7.4.1, which requires beam widths of at least 75 percent of the column width. In order to determine the effective depth used for the design of the beams, it is necessary to estimate the size and placement of the reinforcement that will be used. In establishing this depth, it is assumed that #8 bars will be used for longitudinal reinforcement and that hoops and stirrups will be constructed from #4 bars. In all cases, clear cover of 1.5 inches is assumed. Since this structure has beams spanning in two orthogonal directions, it is necessary to layer the flexural reinforcement as shown in Figure 7-10. The reinforcement for the E-W spanning beams was placed in the upper and lower layers because the strength demand for these members is somewhat greater than that for the N-S beams.

7 – 38

Chapter 7: Reinforced Concrete

1.5" cover

32"

29.5"

28.5"

#8 bar

North-south spanning beam

#4 hoop

East-west spanning beam

Column

30"

Figure 7-10 Layout for beam reinforcement (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) Given Figure 7-10, compute the effective depth for both positive and negative moment as follows: Beams spanning in the E-W direction, d = 32 - 1.5 - 0.5 - 1.00/2 = 29.5 inches Beams spanning in the N-S direction, d = 32 - 1.5 - 0.5 - 1.0 - 1.00/2 = 28.5 inches For negative moment bending, the effective width is 24 inches for all beams. For positive moment, the slab is in compression and the effective T-beam width varies according to ACI 318 Section 8.12. The effective widths for positive moment are as follows (with the parameter controlling effective width shown in parentheses): 20-foot beams in Frames 1 and 8: b = 24 + 20(12)/12 = 44 inches (span length) 20-foot beams in Frames 2 and 7: b = 20(12)/4 = 60 inches (span length) 40-foot beams in Frames 2 through 7: b = 24 + 2[8(4)] = 88 inches (slab thickness) 30-foot beams in Frames A and D: b = 24 + [6(4)] = 48 inches (slab thickness) 30-foot beams in Frames B and C: b = 24 + 2[8(4)] = 88 inches (slab thickness) ACI 318 Section 21.5.2 controls the longitudinal reinforcement requirements for beams. The minimum reinforcement to be provided at the top and bottom of any section is as follows:

As ,min =

200bw d 200(24)(29.5) = = 2.36 in 2 fy 60,000 7 – 39

FEMA P-751, NEHRP Recommended Provisions: Design Examples

This amount of reinforcement can be supplied by three #8 bars with As = 2.37 in2. Since the three #8 bars will be provided continuously top and bottom, reinforcement required for strength will include these #8 bars. Before getting too far into member design, it is useful to check the required tension development length for hooked bars since the required length may control the dimensions of the exterior columns and the boundary elements of the structural walls. From Equation 21-6 of ACI 318 Section 21.7.5.1, the required development length is as follows:

ldh =

f y db 65 f c′

For normal-weight (NW) concrete, the computed length cannot be less than 6 inches or 8db. For straight typical bars, ld = 2.5ldh and for straight “top” bars, ld = 3.25ldh (ACI 318 Sec. 21.7.5.2). These values are applicable only where the bars are anchored in well-confined concrete (e.g., column cores and plastic hinge regions with confining reinforcement). The development length for the portion of the bar extending into unconfined concrete must be increased by a factor of 1.6 per ACI 318 Section 12.7.5.3. Development length requirements for hooked and straight bars are summarized in Table 7-15. Where hooked bars are used, the hook must be 90 degrees and be located within the confined core of the column or boundary element. For bars hooked into 30-inch-square columns with 1.5 inches of cover and #4 ties, the available development length is 30 - 1.50 - 0.5 = 28.0 inches. With this amount of available length, there will be no problem developing hooked bars in the columns. Table 7-15 is applicable to bars anchored in joint regions only. For development of bars outside of joint regions, ACI 318 Chapter 12 should be used. Table 7-15 Tension Development Length Requirements for Hooked Bars and Straight Bars in 5,000 psi NW Concrete Bar Size db (in) ldh hook (in) ld typ (in) ld top (in) #4 0.500 6.5 16.3 21.2 #5 0.625 8.2 20.4 26.5 #6 0.750 9.8 24.5 31.8 #7 0.875 11.4 28.6 37.1 #8 1.000 13.1 32.6 42.4 #9 1.128 14.7 36.8 47.9 #10 1.270 16.6 41.4 53.9 #11 1.410 18.4 46.0 59.8 (1.0 in = 25.4 mm)

Another requirement to consider prior to establishing column sizes is ACI 381 Section 21.7.2.3 which sets a minimum ratio of 20 for the column width to the diameter of the largest longitudinal beam bar passing through the joint. This requirement is easily satisfied for the 30-inch columns in this example.

7 – 40

Chapter 7: Reinforced Concrete 7.4.2.2 Design of Representative Frame 1 Beams. The preliminary design of the beams of Frame 1 was based on members with a depth of 32 inches and a width of 24 inches. The effective depth for positive and negative bending is 29.5 inches and the effective widths for positive and negative bending are 44 and 24 inches, respectively. This assumes the stress block in compression is less than the 4.0-inch flange thickness. The layout of the geometry and gravity loading on the three eastern-most spans of Level 7 of Frame 1 as well as the unfactored gravity and seismic moments are illustrated in Figure 7-11. The seismic and gravity moments are taken directly from the ETABS program output. The seismic forces are from the E-W spectral load case plus the controlling accidental torsion case (the torsional moment where translation and torsion are additive). Note that all negative moments are given at the face of the column and that seismic moments are considerably greater than those due to gravity. Factored bending moment envelopes for all five spans are shown in Figure 7-11. Negative moment at the supports is controlled by the 1.42D + 0.5L + 1.0E load combination and positive moment at the support is controlled by 0.68D - 1.0E. Mid-span positive moments are based on the load combination 1.2D + 1.6L.

CL sym

wL = 1.06 kips/ft wD = 2.31 kips/ft

(a) Span layout and loading

17'-6" 20'-0"

3,665

3,692

3,383

602

550

594

278

273

4,171

4,672

218 4,290

605

3,665 599

277

275

4,664

4,653

869 2,768

3,288

275 206

206

918

599 449

449

474 251

(b) Earthquake moment (in.-kips)

3,666

3,177

3,009

20'-0"

3,258

(d) Unfactored LL moment (in.-kips)

4,653

868 3,255

(e) Required strength envelopes (in.-kips)

3,258

1.42D + 0.5L + E 1.2D + 1.6L - midspan 0.68D - E

Figure 7-11 Bending moments for Frame 1 (1.0 ft = 0.3048 m, 1.0 in-kip = 0.113 kN-m)

7 – 41

FEMA P-751, NEHRP Recommended Provisions: Design Examples 7.4.2.2.1 Longitudinal Reinforcement. The design process for determining longitudinal reinforcement is illustrated as follows for Span A-A’.

1. Design for Negative Moment at the Face of the Exterior Support (Grid A): Mu = 1.42(-550) + 0.5(-251) + 1.0(-3,383) = -4,290 inch-kips Try one #8 bar in addition to the three #8 bars required for minimum steel: As = 4(0.79) = 3.16 in2 fc' = 5,000 psi fy = 60 ksi Width b for negative moment = 24 inches d = 29.5 in. Depth of compression block, a = Asfy/0.85fc'b a = 3.16 (60)/[0.85 (5) 24] = 1.86 inches Design strength, φ Mn = φ Asfy(d - a/2) φ Mn = 0.9(3.16)60(29.5 – 1.86/2) = 4,875 inch-kips > 4,290 inch-kips

2. Design for Positive Moment at Face of Exterior Support (Grid A): Mu = [-0.68(550)] + [1.0(3,383)] = 3,008 inch-kips Try the three #8 bars required for minimum steel: As = [3(0.79)] = 2.37 in 2 Width b for positive moment = 44 inches d = 29.5 inches a = [2.37(60)]/[0.85(5)44] = 0.76 inch φ Mn = 0.9(2.37) 60(29.5 – 0.76/2) = 3,727 inch-kips > 3,008 inch-kips

3. Positive Moment at Midspan: Mu = [1.2(474)] + [1.6(218)] = 918.1 inch-kips Minimum reinforcement (three #8 bars) controls by inspection.

4. Design for Negative Moment at the Face of the Interior Support (Grid A’): Mu = 1.42(-602) + 0.5(-278) + 1.0(-3,177) = -4,172 inch-kips Try one #8 bars in addition to the three #8 bars required for minimum steel:

φ Mn = 4,875 inch-kips > 4,172 inch-kips

5. Design for Positive Moment at Face of Interior Support (Grid A’): Mu = [-0.68(602)] + [1.0(3,177)] = 2,767 inch-kips Three #8 bars similar to the exterior support location are adequate by inspection.

7 – 42

Chapter 7: Reinforced Concrete

Similar calculations can be made for the Spans A'-B and B-C and then the remaining two spans are acceptable via symmetry. A summary of the preliminary flexural reinforcing is shown in Table 7-16. In addition to the computed strength requirements and minimum reinforcement ratios cited above, the final layout of reinforcing steel also must satisfy the following from ACI 318 Section 21.5.2: Minimum of two bars continuous top and bottom

OK (three #8 bars continuous top and bottom)

Positive moment strength greater than 50 percent negative moment strength at a joint

OK (at all joints)

Minimum strength along member greater than 0.25 maximum strength

OK (As provided = three #8 bars is more than 25 percent of reinforcement provided at joints)

The preliminary layout of reinforcement is shown in Figure 7-12. The arrangement of bars is based on the above computations and Table 7-16 summary of the other spans. Note that a slightly smaller amount of reinforcing could be used at the top of the exterior spans, but #8 bars are selected for consistency. In addition, the designer could opt to use four #8 bars continuous throughout the span for uniformity and ease of placement.

(1) #8 (TYP)

2'-8"

(3) #8

2'-6"

20'-0" Note: 1. Drawing not to scale. 2. Splices not shown.

(typical)

Figure 7-12 Preliminary rebar layout for Frame 1 (1.0 ft = 03.048 m) As mentioned above, later phases of the frame design will require computation of the design strength and the maximum probable strength at each support. The results of these calculations are shown in Table 7-16.

7 – 43

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 7-16 Design and Maximum Probable Flexural Strength For Beams in Frame 1 Location*

Item Negative Moment

Positive Moment

Moment Demand (inch-kips)

4,290

4,672

4,664

4,672

4,290

Reinforcement

four #8

Design Strength (inch-kips) Probable Strength (inch-kips) Moment Demand (inch-kips)

4,875

7,042

3,009

3,288

3,255

3,289

3,009

three #8

Reinforcement

Design Strength 3,727 3,727 3,727 3,727 3,727 (inch-kips) Probable Strength 5,159 5,159 5,159 5,159 5,159 (inch-kips) *Moment demand is taken as the larger of the beam moments on each side of the column.

3,727 5,159

(1.0 in-kip = 0.113 kN-m)

As an example of computation of probable strength, consider the case of four #8 top bars plus the portion of slab reinforcing within the effective beam flange width computed above, which is assumed to be 0.002(4 inches)(44-24)=0.16 square inches. (The slab reinforcing, which is not part of this example, is assumed to be 0.002 for minimum steel.) As = 4(0.79) + 0.16 = 3.32 in2 Width b for negative moment = 24 inches d = 29.5 inches Depth of compression block, a = As(1.25fy)/0.85fc'b a = 3.32(1.25)60/[0.85(4)24] = 2.44 inches Mpr = 1.0As(1.25fy)(d - a/2) Mpr = 1.0(3.32)1.25(60)(29.6 – 2.44/2) = 7,042 inch-kips For the case of three #8 bottom bars: As = 3(0.79) = 2.37 in2 Width b for positive moment = 44 inches d = 29.5 inches a = 2.37(1.25)60/[0.85(5)44] = 0.95 inch Mpr = 1.0(2.37)1.25(60)(29.5 – 0.95/2) = 5,159 inch-kips At this point in the design process, the layout of reinforcement has been considered preliminary because the quantity of reinforcement placed in the beams has a direct impact on the magnitude of the stresses developed in the beam-column joint. If the computed joint stresses are too high, the only remedies are increasing the concrete strength, increasing the column area, changing the reinforcement layout, or 7 – 44

Chapter 7: Reinforced Concrete increasing the beam depth. The complete check of the beam-column joint is illustrated in Section 7.4.2.3 below, but preliminary calculations indicate that the joint is adequate, so the design can progress based on the reinforcing provided. Because the arrangement of steel is acceptable from a joint strength perspective, the cutoff locations of the various bars may be determined (see Figure 7-12 for a schematic of the arrangement of reinforcement). The three #8 bars (top and bottom) required for minimum reinforcement are supplied in one length that runs continuously across the two end spans and are spliced in the center span. An additional #8 top bar is placed at each column. To determine where added top bars should be cut off in each span, it is assumed that theoretical cutoff locations correspond to the point where the continuous top bars develop their design flexural strength. Cutoff locations are based on the members developing their design flexural capacities (fy = 60 ksi and φ = 0.9). Using calculations similar to those above, it has been determined that the design flexural strength supplied by a section with only three #8 bars is 3,686 inch-kips for negative moment. Sample cutoff calculations are given for Span B-C. To determine the cutoff location for negative moment, both the additive and counteractive load combinations must be checked to determine the maximum required cutoff length. In this case, the 1.42D + 0.5L ± QE load combination governs. Loading diagrams for determining cutoff locations are shown in Figure 7-13. For negative moment cutoff locations, refer to Figure 7-14, which is a free body diagram of the left end of the member in Figure 7-13. Since the goal is to develop a negative moment capacity of 3,686 inch-kips in the continuous #8 bars, summing moments about Point A in Figure 7-14 can be used to determine the location of this moment demand. The moment summation is as follows:

4,653 +

0.318x 2 − 68.2 x = 3,686 2

In the above equation, 4,653 (inch-kips) is the negative moment demand at the face of column, 0.318 (kips/inch) is the factored gravity load, 68.2 kips is the end shear and 3,686 inch-kips is the design strength of the section with three #8 bars. Solving the quadratic equation results in x = 14.7 inches. ACI 318 Section 12.10.3 requires an additional length equal to the effective depth of the member or 12 bar diameters (whichever is larger). Consequently, the total length of the bar beyond the face of the support is 14.7 + 29.5 = 44.2 inches and a 3’-9” extension beyond the face of the column could be used at this location. Similar calculations should be made for the other beam ends.

7 – 45

FEMA P-751, NEHRP Recommended Provisions: Design Examples

W L = 1.06 klf

W D = 2.31 klf

Face of support

17'-6" 4,653 3,686 E + 1.42D + 0.5L Bending moment (in.-kips)

Figure 7-13 Loading for determination of rebar cutoffs (1.0 ft = 0.3048 m, 1.0 klf = 14.6 kN/m, 1.0 in-kip = 0.113 kN-m)

1.42wD + 0.5wL = 0.318 kip/in. 4,653 in.-kips

3,686 in.-kips A

x E + 1.42D + 0.5L = 68.2 kips

Figure 7-14 Free body diagram (1.0 kip = 4.45kN, 1.0 klf = 14.6 kN/m, 1.0 in-kip = 0.113 kN-m) As shown in Figure 7-15, another requirement in setting cutoff length is that the bar being cut off must have sufficient length to develop the strength required in the adjacent span. From Table 7-15, the required development length of the #8 top bars in tension is 42.4 inches if the bar is anchored in a confined joint region. The confined length in which the bar is developed is shown in Figure 7-15 and consists of the column depth plus twice the depth of the beam (the length for beam hoops per ACI 318 Section 21.5.3.1). This length is 30 + 32 + 32 = 94 inches, which is greater than the 42.4 inches required. The column and beam are considered confined because of the presence of closed hoop reinforcement as required by ACI 318 Sections 21.5.3 and 21.6.4.

7 – 46

Chapter 7: Reinforced Concrete

B Must also check for force F. Required length = 3.25 ldh = 42.4"

db = 2'-8"

#8 bar

Confined region Cut off length based on moments in span B-C dc = 2'-6"

3'-9" 7'-10"

Figure 7-15 Development length for top bars (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) The continuous top bars are spliced at the center of Span B-C and the bottom bars at Spans A’-B and C-C’ as shown in Figure 7-16. The splice length is taken as Class B splice length for #8 bars. These splice locations satisfy the requirements of ACI 318 Section 21.5.2.3 for permitted splice locations. The splice length is determined in accordance with ACI 318 Section 12.15, which indicates that the splice length is 1.3 times the development length. From ACI 318 Section 12.2.2, the development length, ld, is computed as:

⎛ ⎜ ⎜ 3 fy ψ tψ eψ s ld = ⎜ ' ⎜ 40 λ f c ⎛⎜ cb + K tr ⎜ d ⎜ b ⎝ ⎝

⎞ ⎟ ⎟ ⎟d b ⎞⎟ ⎟⎟ ⎟ ⎠⎠

using ψt = 1.3 (top bar), ψe =1.0 (uncoated), ψs = 1.0 (#8 bar), λ = 1.0 (NW concrete) and taking (c + Ktr) / db as 2.5 (based on clear cover and confinement), the development length for one #8 top bar is:

⎛ 3 ⎛ 60,000 ⎞ (1.3)(1.0)(1.0) ⎞ ⎟(1.0) = 33.1 inches ⎟ ld = ⎜ ⎜ ⎜ 40 ⎜ (1.0) 5,000 ⎟ ⎟ ( ) 2 . 5 ⎠ ⎝ ⎝ ⎠ The splice length = 1.3(33.1) = 43 inches. Therefore, use a 43-inch contact splice for the top bars. Computed in a similar manner, the required splice length for the #8 bottom bars is 34 inches. According to ACI 318 Section 21.5.2.3, the entire region of the splice must be confined by closed hoops spaced at the smaller of d/4 or 4 inches.

7 – 47

FEMA P-751, NEHRP Recommended Provisions: Design Examples The final bar placement and cutoff locations for all five spans are shown in Figure 7-16.

3'-9" 3'-7" (3) #8

(3) #8

(1) #8, (TYP)

(3) #8

2'-10" Note: 1. Hoop spacing (from each end): (4) #4 leg 1 at 2", 9 at 7" 2. Hoop spacing (midspan): (2) #4 leg at 7" 3. Hoop spacing (at splice): (2) #4 leg at 4"

Figure 7-16 Final bar arrangement (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) 7.4.2.2.2 Transverse Reinforcement. The requirements for transverse reinforcement in special moment frame beams, include shear strength requirements (ACI 318 Sec. 21.5.4) covered here first and then detailing requirements (ACI 318 Sec. 21.5.3). To avoid nonductile shear failures, the shear strength demand is computed as the sum of the factored gravity shear plus the maximum earthquake shear. The maximum earthquake shear is computed based on the maximum probable beam moments described and computed previously. The probable moment strength at each support is shown in Table 7-16. Figure 7-17 illustrates the development of the design shear strength envelopes for Spans A-A', A'-B and B-C. In Figure 7-17a, the maximum probable earthquake moments are shown for seismic forces acting to the east (solid lines) and to the west (dashed lines). The moments shown occur at the face of the supports. The earthquake shears produced by the maximum probable moments are shown in Figure 7-17b. For Span B-C, the values shown in the figure are:

VE =

− + M pr + M pr

lclear

where lclear = 17 feet-6 inches = 210 inches Note that the magnitude of the earthquake shear can vary with direction (if the beam moment capacities are different at each end). However, in this case the shears are the same in both directions and are computed as: VE = (7,042 + 5,159) / 210 = 58.1 kips

7 – 48

Chapter 7: Reinforced Concrete

B 7,042

7,042

C 7,042

(a) Seismic moment (tension side) in.-kips

5,519

5,519 210" 240"

15"

5,519

15"

58.1

58.1 (b) Seismic shear positive kips

58.1

33.8 32.9

58.1

33.3 33.3

91.4

91.9 24.8

25.2

24.8

24.3 91.0

91.4

(d) Design shear seismic + gravity positive kips

Figure 7-17 Shear forces for transverse reinforcement (1.0 in. = 25.4 mm, 1.0 kip = 4.45kN, 1.0 in-kip = 0.113 kN-m)

7 – 49

FEMA P-751, NEHRP Recommended Provisions: Design Examples The gravity shears shown in Figure 7-17c are taken from the ETABS model: Factored gravity shear = VG = 1.42Vdead + 0.5Vlive Vdead = 20.2 kips Vlive = 9.3 kips VG = 1.42(20.2) + 0.5(9.3) = 33.3 kips Total design shears for each span are shown in Figure 7-17d. The strength envelope for Span B-C is shown in detail in Figure 7-18, which indicates that the maximum design shears is 58.1 + 33.3 = 91.4 kips. While this shear acts at one end, a shear of 58.1 – 33.3 = 24.8 kips acts at the opposite end of the member. In the figure the sloping lines indicate the shear demands along the beam and the horizontal lines indicate the shear capacities at the end and center locations.

φ Vs = 151.7 kips

(4) Legs, s = 7"

91.4 kips

φ Vs = 79.5 kips

91.4 kips

(2) Legs, s = 7"

24.8 kips

15"

64"

41"

Figure 7-18 Detailed shear force envelope in Span B-C (1.0 in. = 25.4 mm, 1.0 kip = 4.45kN) In designing shear reinforcement, the shear strength can consist of contributions from concrete and from steel hoops or stirrups. However, according to ACI 318 Section 21.5.4.2, the design shear strength of the concrete must be taken as zero where the axial force is small (Pu/Agf’c < 0.05) and the ratio VE/Vu is greater than 0.5. From Figure 7-17, this ratio is VE/Vu = 58.1/91.4 = 0.64, so concrete shear strength must be taken as zero. Compute the required shear strength provided by reinforcing steel at the face of the support: Vs = 91.4 kips Vu = φ Vs = Avfyd/s For reasons discussed below, assume four #4 vertical legs (Av = 0.8 in2), fy = 60 ksi and d = 29.5 inches and compute the required spacing as follows: s=φ

7 – 50

Avfyd/Vu = 0.75[4(0.2)](60)(29.5/91.4) = 11.6 inches

Chapter 7: Reinforced Concrete At midspan, the design shear Vu = (91.4 + 24.8)/2 = 58.1 kips, which is the same as the earthquake shear since gravity shear is nominally zero. Compute the required spacing assuming two #4 vertical legs: s = 0.75[2(0.2)](60)(29.5/58.1) = 9.13 inches In terms of detailing requirements, ACI 318 Section 21.5.3.1 states that closed hoops at a tighter spacing are required over a distance of twice the member depth from the face of the support and ACI 318 Section 21.5.3.4 indicates that stirrups are permitted away from the ends. Therefore, the shear strength requirements at this transition point should be computed. At a point equal to twice the beam depth, or 64 inches from the support, the shear is computed as: Vu = 91.4 - (64/210)(91.4 – 24.8) = 71.1 kips Compute the required spacing assuming two #4 vertical legs: s = 0.75[2(0.2)](60)(29.5/71.1) = 7.4 inches Before the final layout can be determined, the detailing requirements need to be considered. The first hoop must be placed 2 inches from the face of the support and the maximum hoop spacing at the beam ends is per ACI 318 Section 21.5.3.2 as follows: d/4 = 29.5/4 = 7.4 inches 8db = 8(1.0) = 8.0 inches 24dh = 24(0.5) = 12.0 inches Outside of the region at the beam ends, ACI 318 Section 21.5.3.4 permits stirrups with seismic hooks to be spaced at a maximum of d/2. Therefore, at the beam ends, overlapped close hoops with four legs will be spaced at 7 inches and in the middle, closed hoops with two legs will be spaced at 7 inches. This satisfies both the strength and detailing requirements and results in a fairly simple pattern. Note that hoops are being used along the entire member length. This is being done because the earthquake shear is a large portion of the total shear, the beam is relatively short and the economic premium is negligible. This arrangement of hoops will be used for Spans A-A', B-C and C'-D. In Spans A'-B and C-C', the bottom flexural reinforcement is spliced and hoops must be placed over the splice region at d/4 or a maximum of 4 inches on center per ACI 318 Section 21.5.2.3. One additional requirement at the beam ends is that where hoops are required (the first 64 inches from the face of support), longitudinal reinforcing bars must be supported as specified in ACI 318 Section 7.10.5.3 as required by ACI 318 Section 21.5.3.3. Hoops should be arranged such that every corner and alternate longitudinal bar is supported by a corner of the hoop assembly and no bar should be more than 6 inches clear from such a supported bar. This will require overlapping hoops with four vertical legs as assumed previously. Details of the transverse reinforcement layout for all spans of Level 5 of Frame 1 are shown in Figure 7-16. 7.4.2.3 Check Beam-Column Joint at Frame 1. Prior to this point in the design process, preliminary calculations were used to check the beam-column joint, since the shear force developed in the beamcolumn joint is a direct function of the beam longitudinal reinforcement. These calculations are often done early in the design process because if the computed joint shear is too high, the only remedies are 7 – 51

FEMA P-751, NEHRP Recommended Provisions: Design Examples increasing the concrete strength, increasing the column area, changing the reinforcement layout, or increasing the beam depth. At this point in the design, the joint shear is checked for the final layout of beam reinforcing. The design of the beam-column joint is based on the requirements of ACI 318 Section 21.7. While ACI 318 provides requirements for joint shear strength, it does not specify how to determine the joint shear demand, other than to indicate that the joint forces are computed using the probable moment strength of the beam (ACI 318 Sec. 21.7.2.1). This example utilizes the procedure for determining joint shear demand contained in Moehle. The shear in the joint is a function of the shear in the column and the tension/compression couple contributed by the beam moments. The method for determining column shear is illustrated in Figure 7-19. In this free-body diagram, the column shear, Vcol, is determined from equilibrium as follows:

Vcol

h⎤ ⎡ ⎢(M pr , L + M pr , R ) + (Ve, L + Ve, R ) 2 ⎥ ⎦ =⎣ lc V col

V e,L lc

M pr,R

V e,R

M pr,L

V col

Figure 7-19 Column shear free body diagram The determination of the forces in the joint of the column on Grid C of Frame 1 is based on Figure 7-16a, which shows how plastic moments are developed in the various spans for equivalent lateral forces acting to the east. An isolated sub-assemblage from the frame showing moments is shown in Figures 7-20b. The beam shears shown in Figure 7-20c are based on the probable moment strengths shown in Table 7-16.

7 – 52

Chapter 7: Reinforced Concrete

For forces acting from west to east, compute the earthquake shear in Span B-C as follows: VE = (Mpr- + Mpr+ )/lclear = (7,042 + 5,159)/(240 - 30) = 58.1 kips For Span C-C', the earthquake shear is the same since the probable moments are equal and opposite.

(a) Plastic mechanism

C' 7,042

7,042

(b) Plastic moments (in.-kips) in spans B-C and C-C' 5,159

5,159

89.4 kips

58.1 kips

7,042 ft-kips

5,159 ft-kips

58.1 kips

89.4 kips

Figure 7-20 Diagram for computing column shears (1.0 ft = 0.3048 m, 1.0 kip = 4.45kN, 1.0 in-kip = 0.113 kN-m)

7 – 53

FEMA P-751, NEHRP Recommended Provisions: Design Examples With h = 30 inches and lc = 156 inches, the column shear is computed as follows:

Vcol

30 ⎤ ⎡ ⎢(7,042 + 5,159) + (58.1 + 58.1) 2 ⎥ ⎦ = 89.4 kips =⎣ 156

With equal spans, gravity loads do not produce significant column shears, except at the end column, where the seismic shear is much less. Therefore, gravity loads are not included in this computation. The forces in the beam reinforcement for negative moment are based on four #8 bars at 1.25 fy: T = C = 1.25(60)[(4(0.79)] = 237.0 kips For positive moment, three #8 bars also are used, assuming C = T, C = 177.8 kips. As illustrated in Figure 7-21, the joint shear force Vj is computed as follows: Vj = T + C – Vcol = 237.0 + 177.8 – 89.4 = 325.4 kips

V col = 89.4 kips T = 237.0 kips C = 177.8 kips V j = 237.0 + 177.8 - 89.4 = 325.4 kips

C = 237.0 kips T = 177.8 kips

Figure 7-21 Computing joint shear stress (1.0 kip = 4.45kN) For joints confined on three faces or on two opposite faces, the nominal shear strength is based on ACI 318 Section 21.7.4 as follows:

Vn = 15 fc′ Aj = 15 5,000(30)2 = 954.6 kips = 0.85, so φ Vn = 0.85(954.6 For joints of special moment frames, ACI 318 Section 9.3.4 permits φ kips) = 811.4 kips, which exceeds the computed joint shear, so the joint is acceptable. Joint stresses would be checked for the other columns in a similar manner.

7 – 54

Chapter 7: Reinforced Concrete

ACI 318 Section 21.7.3.1 specifies the amount of transverse reinforcement required in the joint. Since the joint is not confined on all four sides by a beam, the total amount of transverse reinforcement required by ACI 318 Section 21.6.4.4 will be placed within the depth of the joint. As shown later, this reinforcement consists of four-leg #4 hoops at 4 inches on center. 7.4.2.4 Design of a Typical Interior Column of Frame 1. This section illustrates the design of a typical interior column on Gridline A'. The column, which supports Level 7 of Frame 1, is 30 inches square and is constructed from 5,000 psi concrete and 60 ksi reinforcing steel. An isolated view of the column is shown in Figure 7-22. The flexural reinforcement in the beams framing into the column is shown in Figure 7-16. Using simple tributary area calculations (not shown), the column supports an unfactored axial dead load of 367 kips and an unfactored axial reduced live load of 78 kips. The ETABS analysis indicates that the maximum axial earthquake force is 33.7 kips, tension or compression. The load combination used to compute this force consists of the full earthquake force in the E-W direction plus amplified accidental torsion. Since this column is not part of a N-S moment frame, orthogonal effects need not be considered per Standard Section 12.5.4. Hence, the column is designed for axial force plus uniaxial bending.

13'-0"

32"

P L = 78 kips Includes P D = 367 kips level 7

See Figure 7-16 for girder reinforcement Level 7

32"

30"

20'-0"

Level 6

20'-0"

Figure 7-22 Layout and loads on column of Frame A’ (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm, 1.0 kip = 4.45kN) 7.4.5.3.1 Longitudinal Reinforcement. To determine the axial design loads, use the controlling basic load combinations: 1.42D + 0.5L + 1.0E 0.68D - 1.0E

7 – 55

FEMA P-751, NEHRP Recommended Provisions: Design Examples The combination that results in maximum compression is: Pu = 1.42(367.2) + 0.5(78.0) + 1.0(33.7) = 595 kips (compression) The combination for minimum compression (or tension) is: Pu = 0.68(367.2) - 1.0(33.7) = 216 kips (compression) The maximum axial compression force of 595 kips is greater than 0.1fc'Ag = 0.1(5)(302) = 450 kips, so the design is based on ACI 318 Section 21.6 for columns (see ACI 318 Sec. 21.6.1). According to ACI 318 Section 21.6.2, the sum of nominal column flexural strengths at the joint must be at least 6/5 of the sum of nominal flexural strength of the beams framing into the column. Beam moments at the face of the support are used for this computation. These capacities are provided in Table 7-16. Nominal (negative) moment strength at end A' of Span A-A' = 4,875/0.9 = 5,417 inch-kips Nominal (positive) moment strength at end A' of Span A' B = 3,727/0.9 = 4,141 inch-kips Sum of beam moment at the joint = 5,417 + 4,141 = 9,558 inch-kips Required sum of column design moments = 6/5 × 9,558 = 11,469 inch-kips. Individual column design moment = 11,469/2 = 5,735 inch-kips Knowing the factored axial load and the required design flexural strength, a column with adequate capacity must be selected. Figure 7-23 shows a P-M interaction curve for a 30- by 30-inch column with longitudinal reinforcing consisting of twelve #8 bars (1.05 percent steel). Computed using PCA Column, factor of 1.0 as required for nominal strength. At axial forces of 595 kips and the curve is based on a φ 216 kips, solid horizontal lines are drawn. The dots on the lines represent the required average nominal flexural strength (5,735 inch-kips) at each axial load level. These dots must lie to the left of the curve representing the nominal column strengths. Since the dots are within the capacity curve for both design and nominal moments strengths at both the minimum and maximum axial forces, this column design is clearly adequate.

7 – 56

Chapter 7: Reinforced Concrete

4,500 4,000 nominal

3,500 3,000 )s ip k( ,P d a ol l ia x A

2,500 design

2,000 1,500 1,000 demands 500 0 0

5,000

10,000

-500

15,000 Moment, M (inch-kips)

-1,000

Figure 7-23 Design interaction diagram for column on Gridline A' (1.0 kip = 4.45kN, 1.0 ft-kip = 1.36 kN-m) 7.4.2.4.2 Transverse Reinforcement. The design of transverse reinforcement for columns of special moment frames must consider confinement requirements (ACI 318 Sec. 21.6.4) and shear strength requirements (ACI 318 Sec. 21.6.5). The confinement requirements are typically determined first. Based on ACI 318 Section 21.6.4.1, tighter spacing of confinement is generally required at the ends of the columns, over a distance, lo, equal to the larger of the following: Column depth = 30 inches One-sixth of the clear span = (156-32)/6 = 20.7 inches 18 inches There are both spacing and quantity requirements for the reinforcement. ACI 318 Section 21.6.4.3 specifies the spacing as the minimum of the following: One-fourth the minimum column dimension = 30/4=7.5 inches Six longitudinal bar diameters = 6(1.0) = 6.0 inches Dimension so = 4 + (14 - hx) / 3, where so is between 4 inches and 6 inches and hx is the maximum horizontal spacing of hoops or cross ties. For the column with twelve #8 bars and #4 hoops and cross ties, hx = 8.833 inches and so = 5.72 inches, which controls the spacing requirement.

7 – 57

FEMA P-751, NEHRP Recommended Provisions: Design Examples ACI 318 Section 21.6.4.4 gives the requirements for minimum transverse reinforcement in terms of cross sectional area. For rectangular sections with hoops, ACI 318 Equations 21-4 and 21-5 are applicable:

⎛ sb f ′ ⎞ ⎛ Ag ⎞ Ash = 0.3 ⎜ c c ⎟ ⎜ − 1⎟ ⎜ f yt ⎟ ⎝ Ach ⎠ ⎝ ⎠ ⎛ sb f ′ ⎞ Ash = 0.09 ⎜ c c ⎟ ⎜ f yt ⎟ ⎝ ⎠ The first of these equations controls when Ag/Ach > 1.3. For the 30- by 30-inch columns: Ach = (30 - 1.5 - 1.5)2 = 729 in2 Ag = 30 (30) = 900 in2 Ag/Ach = 900/729 = 1.24 Therefore, ACI 318 Equation 21-5 controls. Try hoops with four #4 legs: bc = 30 - 1.5 - 1.5 = 27.0 inches s = [4 (0.2)(60,000)]/[0.09 (27.0)(5,000)] = 3.95 inches This spacing controls the design, so hoops consisting of four #4 bars spaced at 4 inches will be considered acceptable. ACI 318 Section 21.6.4.5 specifies the maximum spacing of transverse reinforcement in the region beyond the lo zones. The maximum spacing is the smaller of 6.0 inches or 6db, which for #8 bars is also 6 inches. Hoops and crossties with the same details as those placed in the critical regions of the column will be used. 7.4.2.4.3 Check Column Shear Strength. The amount of transverse reinforcement computed in the previous section is the minimum required for confinement. The column also must be checked for shear strength in based on ACI 318 Sec. 21.6.5.1. According to that section, the column shear is based on the probable moment strength of the columns, but need not be more than what can be developed into the column by the beams framing into the joint. However, the design shear cannot be less than the factored shear determined from the analysis. The shears computed based on the probable moment strength of the column can be conservative since the actual column moments are limited by the moments that can be delivered by the beams. For this example, however, the shear from the column probable moments will be checked first and then a determination will be made if a more detailed limit state analysis should be used. As determined from PCA Column, the maximum probable moment of the column in the range of factored axial load is 14,940 in.-kips. With a clear height of 124 inches, the column shear can be determined as 2(14,940)/124 = 241 kips. This shear will be compared to the capacity provided by the 4-leg #4 hoops spaced at 6 inches on center. If this capacity is in excess of the demand, the columns will be acceptable for shear.

7 – 58

Chapter 7: Reinforced Concrete For the design of column shear capacity, the concrete contribution to shear strength may be considered because the minimum Pu > Agf’c/20. The design shear strength contributed by concrete and reinforcing steel are as follows:

Vc = 2 f c′bd = 2 5,000(30)(27.5) = 116.7 kips

Vs = Av f y d / s = (4)( 0.2)( 60)( 27.5) / 6 = 220 .0 kips φ

Vn = φ

(Vc + Vs) = 0.75(116.7 + 220.0) = 252.5 kips > 241 kips

The column with the minimum transverse steel is therefore adequate for shear. The final column detail with both longitudinal and transverse reinforcement is given in Figure 7-24. The spacing of reinforcement through the joint has been reduced to 4 inches on center. This is done for practical reasons only. Column bar splices, where required, should be located in the center half of the column and must be proportioned as Class B tension splices.

7 at 4"

32"

Level 7

(12) #8 bars

7 at 6"

30"

#4 hoops

7 at 4"

Level 6

32"

2" 7 at 4"

30"

Figure 7-24 Details of reinforcement for column (1.0 in. = 25.4 mm)

7 – 59

FEMA P-751, NEHRP Recommended Provisions: Design Examples

This section addresses the design of a representative shear wall. The shear wall includes the 16-inch wall panel in between two 30- by 30-inch columns. The design includes shear, flexure-axial interaction and boundary elements. The factored forces acting on the structural wall of Frame 3 are summarized in Table 7-17. The axial compressive forces are based on the self-weight of the wall, a tributary area of 1,800 square feet of floor area for the entire wall (includes column self-weight), an unfactored floor dead load of 139 psf and an unfactored (reduced) floor live load of 20 psf. Based on the assumed 16-inch wall thickness, the wall between columns weighs (1.33 feet)(17.5 feet)(13 feet)(150 pcf) = 45.4 kips per floor. The total axial force for a typical floor is: Pu = 1.42D + 0.5L = 1.42[1,800(0.139) + 45,400] + 0.5[1,800(0.02)] = 456 kips for maximum compression Pu = 0.68D = 0.68[1,800(0.139) + 45,400] = 201 kips for minimum compression The bending moments come from the ETABS analysis, using a section cut to combine forces in the wall panel and end columns. Note that the gravity moments and the earthquake axial loads on the shear wall are assumed to be negligible given the symmetry of the system, so neither of these load effects are considered in the shear wall design. Table 7-17 Design Forces for Grid 3 Shear Wall Supporting Level R 12 11 10 9 8 7 6 5 4 3 2 1

Axial Compressive Force Pu (kips) 1.42D + 0.5L

0.68D

Shear Vu (kips)

Moment Mu (inchkips)

420 876 1,332 1,788 2,243 2,699 3,155 3,611 4,067 4,523 4,979 5,435 5,891

201 402 603 804 1,005 1,206 1,408 1,609 1,810 2,011 2,212 2,413 2,614

173.2 133.9 156.7 195.7 221.8 252.4 294.6 344.9 400.7 467.5 546.0 663.3 580.3 (use 663.3)

35,375 50,312 63,337 73,993 81,646 86,298 90,678 102,405 132,941 178,321 241,021 366,136 258,851

(1.0 kip = 4.45 kN, 1.0 inch-kip = 0.113 kN-m)

7 – 60

Chapter 7: Reinforced Concrete 7.4.3.1 Design for Shear Loads. First determine the required shear reinforcement in the wall panel and then design the wall for combined bending and axial force. The nominal shear strength of the wall is given by ACI 318 Equation 21-7:

Vn = Acv (αc λ fc′ + ρt f y ) where αc = 2.0 because hw/lw = 161/22.5 = 7.15 > 2.0, where the 161 feet is the wall height and 22.5 feet is the overall wall length from the edges of the 30-inch boundary columns. Using fc' = 5,000 psi, fy = 60 ksi, λ = 1.0, Acv = (22.5)(12)(16) = 4,320 in2, the required amount of shear Vn = Vu. In accordance with ACI 318 Section 9.3.4, reinforcement, ρt , can be determined by setting φ the φ factor for shear is 0.60 for special structural walls unless the wall is specifically designed to be governed by flexure yielding. If the walls were designed to be flexure-critical, then the φ factor for shear would be 0.75, consistent with typical shear design. Unlike special moment frames, shear-critical special shear walls are permitted (with the reduced φ ), although it should be noted that in areas of high seismic hazard many practitioners recommend avoiding shear-critical shear walls where practical. In this = 0.60 will be used for design. case, φ The required reinforcement ratio for strength is determined as:

⎛ 663,300 ⎞ ⎟ − 2 5,000 (4,320 ⎜ 0.60 ⎠ ⎝ ρt = = 0.0019 4,320(60,000)

(

)

Since this is less than the minimum ratio of 0.0025 required by ACI 318 Section 21.9.2.1, that minimum will apply to all levels of the wall. (This is a good indication that the actual wall thickness can be reduced, but this example will proceed with the 16-inch wall thickness.) Assuming two curtains of #5 bars spaced at 15 inches on center, ρt = 0.0026 and φ Vn = 768 kips, which exceeds the required shear capacity at all levels. Vertical reinforcing will be the same as the horizontal reinforcing based on the minimum reinforcing ratio requirements of ACI 318 Section 21.9.2.1 7.4.3.2 Design for Flexural and Axial Loads. The flexural and axial design of special shear walls includes two parts: design of the wall for flexural and axial loads and the design of boundary elements where required. This section covers the design loads and the following section covers the boundary elements. The wall analysis was performed using PCA Column and considers the wall panel plus the boundary columns. For axial and flexural loads, φ = 0.65 and 0.90, respectively. Figure 7-25 shows the interaction diagram for the wall section below Level 2, considering the range of possible factored axial loads. The wall panel is 16 inches thick and has two curtains of #5 bars at 15 inches on center. The boundary columns are 30 by 30 inches with twelve #9 bars at this location. The section is clearly adequate because the interaction curve fully envelopes the design values.

7 – 61

FEMA P-751, NEHRP Recommended Provisions: Design Examples

16,000 14,000

12,000 10,000 )s ip k( 8,000 P , d a ol l 6,000 ai x A 4,000

1.42D + 0.5L

0.68D

2,000

0 0 -2,000

200,000

400,000

600,000 800,000 Moment, M (inch-kips)

Figure 7-25 Interaction diagram for structural wall (1.0 kip = 4.45kN, 1.0 in-kip = 0.113 kN-m) 7.4.3.3 Design of Boundary Elements. An important consideration in the ductility of special reinforced concrete shear walls is the determination of where boundary elements are required and the design of them where they are required. ACI 318 provides two methods for this. The first approach, specified in ACI 318 Section 21.9.6.2, uses a displacement based procedure. The second approach, described in ACI 318 Section 21.9.6.3 uses a stress-based procedure and will be illustrated for this example. In accordance with ACI 318 Section 21.9.6.3, special boundary elements are required where the maximum extreme fiber compressive stress exceeds 0.2f’c and they can be terminated where the stress is less than 0.15f’c. The stresses are determined based on the factored axial and flexure loads as shown in Table 7-18. The stresses are determined using a wall area of 5,160 in2, a section modulus of 284,444 in3 and f’c = 5,000 psi.

7 – 62

Chapter 7: Reinforced Concrete Table 7-18 Grid 3 Shear Wall Boundary Element Check Maximum stress

Supporting Level

Axial Force Pu (kips)

Moment Mu (inch-kips)

(ksi)

(× fc’)

R 12 11 10 9 8 7 6 5 4 3 2 1

420 876 1,332 1,788 2,243 2,699 3,155 3,611 4,067 4,523 4,979 5,435 5,891

35,375 50,312 63,337 73,993 81,646 86,298 90,678 102,405 132,941 178,321 241,021 366,136 258,851

0.206 0.347 0.481 0.607 0.722 0.827 0.930 1.060 1.256 1.503 1.812 2.340 2.052

0.04 0.07 0.10 0.12 0.14 0.17 0.19 0.21 0.25 0.30 0.36 0.47 0.41

Boundary Element Required? No No No No No Yes Yes Yes Yes Yes Yes Yes Yes

(1.0 kip = 4.45 kN, 1.0 in-kip = 0.113 kN-m)

As can be seen, special boundary elements are required at the base of the wall and can be terminated above Level 8. Where they are required, the detailing of the special boundary element is based on ACI 318 Section 21.9.6.4. According to ACI 318 Section 21.9.6.4 Item (a), the special boundary elements must have a minimum plan length equal to the greater of c - 0.1lw, or c/2, where c is the neutral axis depth and lw is the wall length. The neutral axis depth is a function of the factored axial load and the nominal (φ = 1.0) flexural capacity of the wall section. This value is obtained from the PCA Column analysis for the wall section and range of axial loads. For the Level 2 wall with twelve #9 vertical bars at each boundary column and two curtains of #5 bars at 15 inches at vertical bars, the computed neutral axis depths are 32.4 inches and 75.3 inches for axial loads of 5,434 and 2,513 kips, respectively. For the governing case of 75.3 inches and a wall length of 270 inches, the boundary element length is the greater of 75.3 - 0.1(270) = 48.3 inches and the second is 75.3/2 = 37.7 inches. It is clear, therefore, that the special boundary element needs to extend beyond the 30-inch edge columns at least at the lower levels. For the wall below Level 5 where the maximum factored axial load is 4,067 kips, c = 53.6 inches and the required length is 27 inches, which fits within the boundary column. For the walls from the basem*nt to below Level 4, the boundary element can be detailed to extend into the wall panel, or the concrete strength could be increased. Based on the desire to simply the reinforcing, the wall concrete could be increased to fc' = 7,000 psi and the required boundary element length below Level 2 is 26.9 inches. Figure 7-26 illustrates the variation in neutral axis depth based on factored axial load and concrete strength. Although there is a cost premium for the higher strength concrete, this is still in the range of commonly supplied concrete and will save costs by allowing the column rebar cage to serve as the boundary element and have only distributed reinforcing in the wall panel itself. The use of 7,000 psi concrete at the lower levels will impact the calculations for maximum extreme fiber stress per

7 – 63

FEMA P-751, NEHRP Recommended Provisions: Design Examples Table 7-18, but since the 7,000 psi concrete extends up to Level 4, not Level 8, the vertical extent of the boundary elements is unchanged. It is expected that the increase in concrete strength (and thus the modulus of elasticity) at the lower floors will have a slight impact on the overall building stiffness, but this will not impact the overall design. However, this should be verified. 7,000 f c ' = 7 ksi

f c ' = 5 ksi

6,000 )s ip k ( u P , d a o l l ia x a ed r o tc a F

5,000 4,000 3,000 2,000 1,000 0 0

60 80 100 Neutral axis depth (inches)

Figure 7-26 Variations of neutral axis depth (1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN) Where special boundary elements are required, transverse reinforcement must conform to ACI 318 Section 21.9.6.4(c), which refers to ACI 318 Sections 21.6.4.2 through 21.6.4.4. In addition, this section indicates that ACI 318 Equation 21-4 need not apply and the transverse reinforcing spacing limit of ACI 318 Section 21.6.4.3(a) can be one-third of the least dimension of the element. Similar to columns of special moment frames, there are requirements for spacing and total area of transverse reinforcing. The spacing is determined as follows: One-third of least dimension = 30/3 = 10 inches Six longitudinal bar diameters = 6(1.125) = 6.75 inches Dimension so = 4 + (14 - hx) / 3, where so is between 4 inches and 6 inches and hx is the maximum horizontal spacing of hoops or cross ties. Where hoops are used, the transverse reinforcement must satisfy ACI 318 Equation 21-5:

⎛ sb f ′ ⎞ Ash = 0.09 ⎜ c c ⎟ ⎜ f yt ⎟ ⎝ ⎠ If #4 hoops with two crossties in each direction are used similar to the moment frame columns, Ash = 0.80 in2 and bc = 27 inches. For fc' = 7,000 psi and fyt = 60 ksi, s = [(0.8)(60,000)]/[0.09(27.0)(7,000)] = 2.82 inches 7 – 64

Chapter 7: Reinforced Concrete

which is impractical. Therefore, use #5 hoops and cross ties for the 7,000 psi concrete below Level 4, so Ash = 4(0.31) = 1.24 in2 and s = 4.4 inches. Where the concrete strength is 5,000 psi above Level 4, use #4 hoops and cross ties and the spacing, s = 3.95 inches. Therefore, for the special boundary elements, use hoops with two cross ties spaced at 4 inches. The hoops and cross ties are #5 below Level 4 and #4 above Level 4. ACI 318 Section 21.9.6.4(d) also requires that the boundary element transverse reinforcement be extended beyond the base of the wall a distance equal to the tension development length of the longitudinal reinforcement in the boundary elements unless there is a mat or footing, in which case the transverse reinforcement extends down at least 12 inches. Details of the boundary element and wall panel reinforcement are shown in Figures 7-27 and 7-28, respectively. The vertical reinforcement in the boundary elements will be spliced as required using either Class B lap splices or Type 2 mechanical splices at all locations. According to Table 7-15 (prepared for 5,000 psi concrete), there should be no difficulty in developing the horizontal wall panel steel into the 30by 30-inch boundary elements.

(12) #9 #5 x developed in wall

4" #5 at 4" o.c. Alternate location of 90° bend

4" #5

at 4" o.c.

* #4 Hoops and Cross Ties above Level 4.

Figure 7-27 Details of structural wall boundary element (1.0 in. = 25.4 mm)

7 – 65

FEMA P-751, NEHRP Recommended Provisions: Design Examples

B R

#4 at 4" #5 at 15" EF

B 8

f 'c= 5.0ksi (LW) #5 at 15" EF

#5 at 15" EF

#5 at 15" EF #5 at 15" EF

#4 at 4"

#5 at 15" EF

#5 at 15" EF #5 at 15" EF

#5 at 15" EF

#4 at 4"

#5 at 15" EF

#5 at 4"

#5 at 15" EF #5 at 15" EF

#5 at 15" EF

G #4 at 4"

#5 at 15" EF

#5 at 4" #5 at 15" EF (12) #9

#5 at 15" EF

#4 at 4"

#5 at 15" EF

#5 at 4"

#5 at 15" EF

Class B

#4 at 12" EF

(12) #9

f 'c = 7.0ksi (NW)

(12) #8

#4 at 4"

#5 at 15" EF

Class B

(12) #8

#5 at 15" EF (12) #8

#4 at 4"

#5 at 15" EF

Class B

#4 at 4"

Class B

(12) #8

#5 at 4"

#5 at 15" EF

Extend hoops 12" into pile cap.

Figure 7-28 Overall details of structural wall (1.0 in. = 25.4 mm)

$!# The structure illustrated in Figures 7-1 and 7-2 is now designed and detailed for the Honolulu building. Because of the relatively moderate level of seismicity, the lateral load-resisting system will consist of a series of intermediate moment-resisting frames in both the E-W and N-S directions. This is permitted for Seismic Design Category C buildings in accordance with Standard Table 12.2-1. Design guidelines for the reinforced concrete framing members are provided in ACI 318 Section 21.3. As noted previously, the beams are assumed to be 30 inches deep by 20 inches wide and the columns are 28 inches by 28 inches. These are slightly smaller than the Berkeley building, reflecting the lower seismicity. $!#!"

As has been discussed and as illustrated in Figure 7-3, wind forces appear to govern the strength requirements of the structure at the lower floors and seismic forces control at the upper floors. The seismic and wind shears, however, are so close at the middle levels of the structure that a careful

7 – 66

Chapter 7: Reinforced Concrete evaluation must be made to determine which load governs for strength. This determination requires consideration of several load cases for both wind and seismic loads. Because the Honolulu building is in Seismic Design Category C and does not have a Type 5 horizontal irregularity (Standard Table 12.3-1); orthogonal loading effects need not be considered per Standard Section 12.5.3. However, as required by Standard Section 12.8.4.2, accidental torsion must be considered. Torsional amplification is not required per Provisions Section 12.8.4.3 because the building does not have a torsional irregularity as determined previously. For wind, the Standard requires that buildings over 60 feet in height be checked for four loading cases under the Method 2 Analytical Procedure of Standard Section 6.5. The required load cases are shown in Figure 7-29, which is reproduced directly from Standard Figure 6-9. In Cases 1 and 2, load is applied separately in the two orthogonal directions, but Case 2 adds a torsional component. Cases 3 and 4 involve wind loads in two directions simultaneously and Case 4 adds a torsional component.

0.75 PWY

PWY

0.75 PWX

PWX

PLX

P LY

0.75 P LX

0.75 P LY

Case 1

Case 3 BY

BY 0.563 PWY

0.75 PWY

MT 0.75PWX

MT 0.75PLX

0.75 P LY

0.563 PWX

0.563 PLX 0.563 P LY

M T = 0.75 (PWX + PLX ) B X e X

M T = 0.75 (PWY + P LY ) BY eY

e X = ±0.15 B X

eY = ±0.15 BY

Case 2

M T = 0.563 (PWX + P LX ) B X e X + 0.563 (PWY + P LY ) BY eY e X = ±0.15 B X

eY = ±0.15 BY

Case 4

Figure 7-29 Wind loading requirements from ASCE 7

7 – 67

FEMA P-751, NEHRP Recommended Provisions: Design Examples In this example, only loading in the E-W direction is considered. Hence, the following lateral load conditions are applied to the ETABS model: E-W seismic with accidental torsion Wind Case 1 applied in E-W direction only Wind Case 2 applied in E-W direction only Wind Case 3 Wind Case 4 All cases with torsion are applied in such a manner as to maximize the shears in the elements of Frame 1, for whose members the design is illustrated in the following section. A simple method for determining which load case is likely to govern is to compare the beam shears for each story. For the five load cases indicated above, the beam shears produced from seismic effects control at the sixth level, with the next largest forces coming from direct E-W wind Case 1. This is shown graphically in Figure 7-30, where the beam shears at the center bay of Frame 1 are plotted versus story height. Note that this comparison is based on 1.0 times seismic loads and 1.6 times wind loads consistent with the strength design load combinations. Wind controls load at the lower four stories and seismic controls for all other stories. This is somewhat different from that shown in Figure 7-3, wherein the total story shears are plotted and where wind controlled for the lower five stories. A basic difference between Figures 7-3 and 7-30 is that Figure 7-30 includes torsion effects.

7 – 68

Chapter 7: Reinforced Concrete

160

1.0E (with accidental torsion) 140

120

100 t)f ( h ,t h 80 ige H

1.6W (case 1)

0 0

Girder shear, V (kips)

Figure 7-30 Wind versus seismic shears in center bay of Frame 1 (1.0 ft = 0.3048 m, 1.0 kip = 4.45kN)

In this section, the beams and a typical interior column of Level 6 of Frame 1 are designed and detailed. 7.5.2.1 Initial Calculations. The girders of Frame 1 are 30 inches deep and 20 inches wide. For positive moment bending, the effective width of the compression flange is taken as 20 + 20(12)/12 = 40.0 inches. Assuming 1.5-inch cover, #4 stirrups and #9 longitudinal reinforcement, the effective depth for computing flexural and shear strength is 30 - 1.5 - 0.5 - 1.125 / 2 = 27.4 inches. 7.5.2.2 Design of Representative Beams. ACI 318 Section 21.3.4 provides the minimum requirements for longitudinal and transverse reinforcement in the beams of intermediate moment frames. The requirements for longitudinal steel are as follows: 1. The positive moment strength at the face of a joint shall be at least one-third of the negative moment strength at the same joint.

7 – 69

FEMA P-751, NEHRP Recommended Provisions: Design Examples 2. Neither the positive nor the negative moment strength at any section along the length of the member shall be less than one-fifth of the maximum moment strength supplied at the face of either joint. The second requirement has the effect of requiring top and bottom reinforcement along the full length of the member. The minimum reinforcement ratio at any section is taken from ACI 318 Section 10.5.1 as 200/fy or 0.0033 for fy = 60 ksi. However, according to ACI 318 Section 10.5.3, the minimum reinforcement provided need not exceed 1.33 times the amount of reinforcement required for strength. The gravity loads and design moments for the first three spans of Frame 1 are shown in Figure 7-31. The seismic and gravity moments are determined from ETABS analysis, similar to the Berkeley building. All moments are given at the face of the support. The gravity moments shown in Figures 7-31c and 7-31d are slightly different from those shown for the Berkeley building (Figure 7-11) because the beam self-weight is less and the clear span is longer due to the reduction in column size.

CL sym

wL = 1.06 kips/ft wD = 2.31 kips/ft

(a) Span layout and loading

17.67' 20'-0"

2,634

2,659

2,557

628

567

615

290

283

218 3,424

624

2,634 620

286

284

3,600

918

3,601

3,582

869 1,911

2,167

284 206

206 3,374

620 449

449

474 259

(b) Earthquake moment (in.-kips)

2,659

2,413

2,103

20'-0"

2,138

(d) Unfactored LL moment (in.-kips)

3,582

868 2,148

2,138

(e) Required strength envelopes (in.-kips) 1.3D + 0.5L + E 1.2D + 1.6L - midspan 0.8D - E

Figure 7-31 Bending moment envelopes at Level 6 of Frame 1 (1.0 ft = 0.3048 m, 1.0 kip/ft = 14.6 kN/m, 1.0 in-kip = 0.113 kN-m) 7.5.2.2.1 Longitudinal Reinforcement. Based on a minimum amount of longitudinal reinforcing of 0.0033bwd = 0.0033(20)(27.4)=1.83 in2, provide two #9 bars continuous top and bottom as a starting point and provide additional reinforcing as required. 7 – 70

Chapter 7: Reinforced Concrete

1. Design for Negative Moment at the Face of the Exterior Support (Grid A) Mu = -1.3 (567) - 0.5 (259) - 1.0 (2,557) = -3,423 inch-kips Try two #9 bars plus one #7 bar. As = 2 (1.00) + 0.60 = 2.60 in2 Depth of compression block, a = [2.6 (60)]/[0.85 (5) 20] = 1.83 inches Nominal strength, Mn = [2.60 (60)] [27.4 - 1.83/2] = 4,131 inch-kips Design strength, φ Mn = 0.9 (4,131) = 3,718 inch-kips > 3,423 inch-kips

This reinforcement also will work for negative moment at all other supports.

2. Design for Positive Moment at the Face of the Exterior Support (Grid A) Mu = -0.8 (567) + 1.0 (2,557) = 2,114 inch-kips Try the minimum of two #9 bars. As = 2 (1.00) = 2.00 in2 a = 2.00 (60)/[0.85 (5) 40] = 0.71 inch Mn = [2.00 (60)] [27.4 – 0.71/2] = 3,246 inch-kips φ Mn = 0.9 (3,246) = 2,921 inch-kips > 2,114 inch-kips

This reinforcement also will work for positive moment at all other supports. The layout of flexural reinforcement layout is shown in Figure 7-32. The top short bars are cut off 5 feet-0 inch from the face of the support. The bottom bars are spliced in Spans A'-B and C-C' with a Class B lap length of 37 inches. Unlike special moment frames, there are no requirements that the spliced region of the bars in intermediate moment frames be confined by hoops over the length of the splice. Note that the steel clearly satisfies the detailing requirements of ACI 318 Section 21.3.4.1.

5'-0" (2) #9

(2) #9

(1) #7 (TYP)

2'-6"

(2) #9

(2) #9 20'-0"

2'-4"

(2) #9 3'-1"

(typical)

#4x hoops spaced from each support: 1 at 2", 10 at 6", balance at 10" (typical each span).

Figure 7-32 Longitudinal reinforcement layout for Level 6 of Frame 1 (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)

7 – 71

FEMA P-751, NEHRP Recommended Provisions: Design Examples

7.5.2.2.2 Transverse Reinforcement. The requirements for transverse reinforcement in intermediate moment frames are somewhat different from those in special moment frames, both in terms of detailing and shear design. The shear strength requirements will be covered first, followed by the detailing requirements. In accordance with ACI 318 Section 21.3.3, the design earthquake shear for the design of intermediate moment frame beams must be larger than the smaller of the following: a. The sum of the shears associated with the nominal moment strength at the ends of the members. Nominal moment strengths are computed with a flexural reinforcement tensile strength of 1.0fy and a flexural φ factor of 1.0. b. Two times the factored earthquake shear force determined from the structural analysis. In either case, the earthquake shears are combined with the factored gravity shears to determine the total design shear. Consider the end span between Grids A and A’. For determining earthquake shears per Item a above, the nominal strengths at the ends of the beam were computed earlier as 3,246 inch-kips for positive moment at Support A’ and 4,131 inch-kips for negative moment at Support A. Compute the design earthquake shear VE:

VE =

3,246 + 4,131 = 34.8 kips 212

where 212 inches is the clear span of the member. The shear is the same for earthquake forces acting in the other direction. For determining earthquake shears per Item b above, the shear is taken from the ETABS analysis as 23.4 kips. The design earthquake shear for this method is 2(23.4 kips) = 46.8 kips. Since the design shear using Item a is the smaller value, it is used for computing the design shear. The gravity load shears are taken from the ETABS model. Since the gravity shears at Grid A’ are similar but slightly larger than those at Grid A, Grid A’ will be used for the design. From the ETABS analysis, VD = 20.7 kips and VL = 9.5 kips. The factored design shear Vu = 1.3(20.7) + 0.5(9.5) + 1.0(46.8) = 66.5 kips. This shear force applies for earthquake forces coming from either direction as shown in the shear strength design envelope in Figure 7-33. The design shear force is resisted by a concrete component, Vc and a steel component, Vs. Note that the concrete component may be used regardless of the ratio of earthquake shear to total shear. The required design strength is: Vu ≤ φ

Vc + φ

where φ = 0.75 for shear.

7 – 72

Chapter 7: Reinforced Concrete

Vc =

2 5,000 (20)(27.4) = 77.5 kips 1,000

The shear to be resisted by reinforcing steel, assuming two #4 vertical legs (Av = 0.4) and fy = 60 ksi is:

Vs =

Vu − φVc

66.5 − 0.75(77.5) = 11.2 kips 0.75

Using Vs = Av fyd/s:

(0.4)(60)(27.4) = 58.7 inches 11.2

Minimum transverse steel requirements are given in ACI 318 Section 21.3.4.2. At the ends of the beam, hoops are required. The first hoop must be placed 2 inches from the face of the support and within a distance 2h from the face of the support, the spacing should be not greater than d/4, eight times the smallest longitudinal bar diameter, 24 times the hip bar diameter, or 12 inches. For the beam under consideration d/4 controls minimum transverse steel, with the maximum spacing being 27.4/4 = 6.8 inches, which is less than what is required for shear strength. In the remainder of the span, stirrups are permitted and must be placed at a maximum of d/2 (ACI 318 Sec. 21.3.4.3). Because the earthquake shear (at midspan where the gravity shear is essentially zero) is greater than 50 percent of the shear strength provided by concrete alone, the minimum requirements of ACI 318 Section 11.4.6.1 must be checked:

s max =

Av f yt ' c

0.75 f bw

0.4(60,000) 0.75 5,000 (20)

= 20.8 inches

This spacing does not control over the d/2 requirement. The layout of transverse reinforcement for the beam is shown in Figure 7-32. This spacing is used for all other spans as well.

7 – 73

FEMA P-751, NEHRP Recommended Provisions: Design Examples

4,131

4,131 (a) Seismic moment (tension side) in.-kips

3,246

14"

212" 240"

34.8

(b) Seismic shear positive kips

34.8

34.8 31.7

30.8

34.8 31.2

31.2

66.5 4.0

31.2

66.0 3.6

(d) Design shear seismic + gravity positive kips

3.1 65.6

3.6 66.0

Figure 7-33 Shear strength envelopes for Span A-A' of Frame 1 (1.0 in. = 25.4 mm, 1.0 kip = 4.45kN, 1.0 in-kip = 0.113 kN-m)

7 – 74

Chapter 7: Reinforced Concrete 7.5.2.3 Design of Representative Column of Frame 1. This section illustrates the design of a typical interior column on Gridline A'. The column, which supports Level 6 of Frame 1, is 28 inches square and is constructed from 5,000 psi concrete and 60 ksi reinforcement. An isolated view of the column is shown in Figure 7-34. The column supports an unfactored axial dead load of 506 kips and an unfactored axial live load (reduced) of 117 kips. The ETABS analysis indicates that the axial earthquake force is ±19.9 kips, the earthquake shear force is ±37.7 kips and the earthquake moments at the top and the bottom of the column are ±2,408 and ±2,340 inch-kips, respectively. Moments and shears due to gravity loads are assumed to be negligible.

30"

P L = 117 kips P D = 500 kips

Includes level 6

Level 6

30"

13'-0"

28"

Level 5

See Figure 7-23 for girder reinforcement 20'-0"

20'-0"

Figure 7-34 Isolated view of Column A' (1.0 ft = 0.3048 m, 1.0 kip = 4.45kN) 7.5.2.3.1 Longitudinal Reinforcement. The factored gravity force for maximum compression (without earthquake) is: Pu = 1.2(506) + 1.6(117) = 794 kips This force acts with no significant gravity moment. The factored gravity force for maximum compression (including earthquake) is: Pu = 1.3(506) + 0.5(117) + 19.9 = 736 kips The factored gravity force for minimum compression (including earthquake) is: Pu = 0.8(506) – 19.9 = 385 kips

7 – 75

FEMA P-751, NEHRP Recommended Provisions: Design Examples Before proceeding with the flexural strength calculations, first determine whether or not slenderness effects need to be considered. For a frame that is unbraced against sideway, ACI 318 Section 10.10.1 allows slenderness effects to be neglected where klu/r < 22. For a 28- by 28-inch column with a clear unbraced length, lu = 126 inches, r = 0.3(28) = 8.4 inches (ACI 318 Sec. 10.10.1.2) and lu/r = 126/8.4 = 15.0. Therefore, as long as the effective length factor k for this column is less than 22/15.0 = 1.47, then slenderness effects can be ignored. It is reasonable to assume that k is less than 1.47 and this can be confirmed using the commentary to ACI 318 Section 10.10.1. Continuing with the design, an axial-flexural interaction diagram for a 28- by 28-inch column with 12 #8 bars (ρ = 0.0121) is shown in Figure 7-35. The column clearly has the strength to support the applied loads (represented as solid dots in the figure).

2,500

2,000

1,500 )s ip k( 1,000 P , d a lo l ai 500 x A 0 0 -500

2,000

4,000

6,000

8,000

10,000

12,000

Moment, M (inch-kips)

-1,000

Figure 7-35 Interaction diagram for column (1.0 kip = 4.45kN, 1.0 ft-kip = 1.36 kN-m) 7.5.2.3.2 Transverse Reinforcement. The design earthquake shear for columns in determined in the same manner as for beams in accordance with ACI 318 Section 21.3.3 as described in Section 7.5.2.2.2. Assuming two times the shear from analysis will produce the smaller design shear, the ETABS analysis indicates that the shear force is 37.7 kips and the design shear is 2.0(37.7) = 75.4 kips. The concrete supplies a capacity of:

φVc = 0.75(2 5,000 (28)(25.6)) = 76.0 kips > 75.4 kips

7 – 76

Chapter 7: Reinforced Concrete

Therefore, steel reinforcement is not required for strength, but shear reinforcement is required per ACI 318 Section 11.4.6.1 since the design shear exceeds one-half of the design capacity of the concrete alone. First, however, determine the detailing requirements for transverse reinforcement in intermediate moment frame columns in accordance ACI 318 Section 21.3.5. Within a region lo from the face of the support, the tie spacing must not exceed: 8db = 8(1.0) = 8.0 inches (using #8 longitudinal bars) 24dtie = 24 (0.5) = 12.0 inches (using #4 ties) 1/2 the smallest dimension of the frame member = 28/2 = 14 inches 12 inches The 8-inch maximum spacing controls. Ties at this spacing are required over a length lo of: 1/6 clearspan of column = 126/6 = 21 inches Maximum cross section dimension = 28 inches 18 inches Given the above, a four-legged #4 tie spaced at 8 inches over a depth of 28 inches will be used. The top and bottom ties will be provided at 4 inches from the beam soffit and floor slab. Beyond the end regions, ACI 318 Section 21.3.5.4 requires that tie spacing satisfy ACI 318 Sections 7.10 and 11.4.5.1, but the minimum shear reinforcing requirement of ACI 318 Section 11.4.6.1 also applies. Of the above requirements, ACI 318 Section 11.4.5.1, which requires ties at d/2 maximum spacing, governs. Therefore, use a 12-inch tie spacing in the middle region of the column. The layout of the transverse reinforcing for the subject column is shown in Figure 7-36

7 – 77

FEMA P-751, NEHRP Recommended Provisions: Design Examples

30"

3 at 8"

Level 7

28"

3 at 8" o.c. 3 at 8" o.c. 8 at 12" o.c.

(12) #8 bars

28"

30"

Level 6

28"

Figure 7-36 Column reinforcement (1.0 in. = 25.4 mm) 7.5.2.4 Design of Beam-Column Joint. Joint reinforcement for intermediate moment frames is addressed in ACI 318 Section 21.3.5.5, which refers to ACI 318 Section 11.10. ACI 318 Section 11.10 requires that all beam-column connections have a minimum amount of transverse reinforcement through the beam-column joints. The only exception is in non-seismic frames where the column is confined on all four sides by beams framing into the column. The amount of reinforcement required is given by ACI 318 Equation 11-13:

Av ,min = 0.75 f c′

7 – 78

bw s f yt

Chapter 7: Reinforced Concrete This is the same equation used to proportion minimum transverse reinforcement in beams. Assuming Av is supplied by four #4 ties and fy = 60 ksi:

4(0.2)(60,000) 0.75 5,000 (28)

= 32.4 inches

This essentially permits no ties to be located in the joint. Since it is good practice to provide transverse reinforcing in moment frame joints, ties will be provided at the same 8-inch spacing as at the ends of the columns. The arrangement of ties within the beam-column joint is shown in Figure 7-36.

7 – 79

8 Precast Concrete Design Suzanne Dow Nakaki, S.E. Originally developed by Gene R. Stevens, P.E. and James Robert Harris, P.E., PhD

Contents 8.1

HORIZONTAL DIAPHRAGMS .................................................................................................. 4

8.1.1

Untopped Precast Concrete Units for Five-Story Masonry Buildings Located in Birmingham, Alabama and New York, New York ............................................................... 4

8.1.2

Topped Precast Concrete Units for Five-Story Masonry Building Located in Los Angeles, California (see Sec. 10.2) .................................................................................................... 18

8.2

THREE-STORY OFFICE BUILDING WITH INTERMEDIATE PRECAST CONCRETE SHEAR WALLS ......................................................................................................................... 26

8.2.1

Building Description ............................................................................................................ 27

8.2.2

Design Requirements ........................................................................................................... 28

8.2.3

Load Combinations .............................................................................................................. 29

8.2.4

Seismic Force Analysis........................................................................................................ 30

8.2.5

Proportioning and Detailing ................................................................................................ 33

8.3

ONE-STORY PRECAST SHEAR WALL BUILDING ............................................................. 45

8.3.1

Building Description ............................................................................................................ 45

8.3.2

Design Requirements ........................................................................................................... 48

8.3.3

Load Combinations .............................................................................................................. 49

8.3.4

Seismic Force Analysis........................................................................................................ 50

8.3.5

Proportioning and Detailing ................................................................................................ 52

8.4

SPECIAL MOMENT FRAMES CONSTRUCTED USING PRECAST CONCRETE ............. 65

8.4.1

Ductile Connections............................................................................................................. 65

8.4.2

Strong Connections .............................................................................................................. 67

FEMA P-751, NEHRP Recommended Provisions: Design Examples This chapter illustrates the seismic design of precast concrete members using the NEHRP Recommended Provisions (referred to herein as the Provisions) for buildings in several different seismic design categories. Over the past several years there has been a concerted effort to coordinate the requirements in the Provisions with those in ACI 318, so that now there are very few differences between the two. Very briefly, the Provisions set forth the following requirements for precast concrete structural systems. §

Precast seismic systems used in structures assigned to Seismic Design Category C must be intermediate or special moment frames, or intermediate precast or special structural walls.

§

Precast seismic systems used in structures assigned to Seismic Design Category D must be special moment frames, or intermediate precast (up to 40 feet) or special structural walls.

§

Precast seismic systems used in structures assigned to Seismic Design Category E or F must be special moment frames or special structural walls.

§

Prestress provided by prestressing steel resisting earthquake-induced flexural and axial loads in frame members must be limited to 700 psi or f’c/6 in plastic hinge regions. These values are different from the ACI 318 limitations, which are 500 psi or f’c/10.

§

An ordinary precast structural wall is defined as one that satisfies ACI 318 Chapters 1-18.

§

An intermediate precast structural wall must meet additional requirements for its connections beyond those defined in ACI 318 Section 21.4. These include requirements for the design of wall piers that amplify the design shear forces and prescribe wall pier detailing and requirements for explicit consideration of the ductility capacity of yielding connections.

§

A special structural wall constructed using precast concrete must satisfy the acceptance criteria defined in Provisions Section 9.6 if it doesn’t meet the requirements for special structural walls constructed using precast concrete contained in ACI 318 Section 21.10.2.

Examples are provided for the following concepts:

8-2

§

The example in Section 8.1 illustrates the design of untopped and topped precast concrete floor and roof diaphragms of the five-story masonry buildings described in Section 10.2 of this volume of design examples. The two untopped precast concrete diaphragms of Section 8.1.1 show the requirements for Seismic Design Categories B and C using 8-inch-thick hollow core precast, prestressed concrete planks. Section 8.1.2 shows the same precast plank with a 2-1/2-inch-thick composite lightweight concrete topping for the five-story masonry building in Seismic Design Category D described in Section 10.2. Although untopped diaphragms are commonly used in regions of low seismic hazard, their design is not specifically addressed in the Provisions, the Standard, or ACI 318.

§

The example in Section 8.2 illustrates the design of an intermediate precast concrete shear wall building in a region of low or moderate seismicity, which is where many precast concrete seismic force-resisting systems are constructed. The precast concrete walls in this example resist the seismic forces for a three-story office building located in southern New England (Seismic Design Category B). The Provisions have a few requirements beyond those in ACI 318 and these requirements are identified in this example. Specifically, ACI 318 requires that in connections that are expected to yield, the yielding be restricted to steel elements or reinforcement. The Provisions also require that the deformation capacity of the connection be compared to the deformation demand on the connection unless Type 2 mechanical splices are used. There are

Chapter 8: Precast Concrete Design additional requirements for intermediate precast structural walls relating to wall piers; however, due to the geometry of the walls used in this design example, this concept is not described in the example. §

The example in Section 8.3 illustrates the design of a special precast concrete shear wall for a single-story industrial warehouse building in Los Angeles. For buildings assigned to Seismic Design Category D, the Provisions require that the precast seismic force-resisting system be designed and detailed to meet the requirements for either an intermediate or special precast concrete structural wall. The detailed requirements in the Provisions regarding explicit calculation of the deformation capacity of the yielding element are shown here.

§

The example in Section 8.4 shows a partial example for the design of a special moment frame constructed using precast concrete per ACI 318 Section 21.8. Concepts for ductile and strong connections are presented and a detailed description of the calculations for a strong connection located at the beam-column interface is presented.

Tilt-up concrete wall buildings in all seismic zones have long been designed using the precast wall panels as concrete shear walls for the seismic force-resisting system. Such designs usually have been performed using design force coefficients and strength limits as if the precast walls emulated the performance of cast-in-place reinforced concrete shear walls, which they usually do not. Tilt-up buildings assigned to Seismic Design Category C or higher should be designed and detailed as intermediate or special precast structural wall systems as defined in ACI 318. In addition to the Provisions, the following documents are either referred to directly or are useful design aids for precast concrete construction: ACI 318

American Concrete Institute. 2008. Building Code Requirements for Structural Concrete.

AISC 360

American Institute of Steel Construction. 2005. Specification for Structural Steel Buildings.

AISC Manual

American Institute of Steel Construction. 2005. Manual of Steel Construction, Thirteen Edition.

Moustafa

Moustafa, Saad E. 1981 and 1982. “Effectiveness of Shear-Friction Reinforcement in Shear Diaphragm Capacity of Hollow-Core Slabs.” PCI Journal, Vol. 26, No. 1 (Jan.-Feb. 1981) and the discussion contained in PCI Journal, Vol. 27, No. 3 (May-June 1982).

PCI Handbook

Precast/Prestressed Concrete Institute. 2004. PCI Design Handbook, Sixth Edition.

PCI Details

Precast/Prestressed Concrete Institute. 1988. Design and Typical Details of Connections for Precast and Prestressed Concrete, Second Edition.

SEAA Hollow Core

Structural Engineers Association of Arizona, Central Chapter. Design and Detailing of Untopped Hollow-Core Slab Systems for Diaphragm Shear.

8-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples The following style is used when referring to a section of ACI 318 for which a change or insertion is proposed by the Provisions: Provisions Section xxx (ACI 318 Sec. yyy) where “xxx” is the section in the Provisions and “yyy” is the section proposed for insertion into ACI 318. 8.1

HORIZONTAL DIAPHRAGMS

Structural diaphragms are horizontal or nearly horizontal elements, such as floors and roofs, that transfer seismic inertial forces to the vertical seismic force-resisting members. Precast concrete diaphragms may be constructed using topped or untopped precast elements depending on the Seismic Design Category. Reinforced concrete diaphragms constructed using untopped precast concrete elements are not addressed specifically in the Standard, in the Provisions, or in ACI 318. Topped precast concrete elements, which act compositely or noncompositely for gravity loads, are designed using the requirements of ACI 318 Section 21.11. 8.1.1

Untopped Precast Concrete Units for Five-‐Story Masonry Buildings Located in Birmingham, Alabama and New York, New York

This example illustrates floor and roof diaphragm design for five-story masonry buildings located in Birmingham, Alabama, on soft rock (Seismic Design Category B) and in New York, New York (Seismic Design Category C). The example in Section 10.2 provides design parameters used in this example. The floors and roofs of these buildings are to be untopped 8-inch-thick hollow core precast, prestressed concrete plank. Figure 10.2-1 shows the typical floor plan of the diaphragms. 8.1.1.1 General Design Requirements. In accordance with ACI 318, untopped precast diaphragms are permitted only in Seismic Design Categories A through C. Static rational models are used to determine shears and moments on joints as well as shear and tension/compression forces on connections. Dynamic modeling of seismic response is not required. Per ACI 318 Section 21.1.1.6, diaphragms in Seismic Design Categories D through F are required to meet ACI 318 Section 21.11, which does not allow untopped diaphragms. In previous versions of the Provisions, an appendix was presented that provided a framework for the design of untopped diaphragms in higher Seismic Design Categories in which diaphragms with untopped precast elements were designed to remain elastic and connections designed for limited ductility. However, in the 2009 Provisions, that appendix has been removed. Instead, a white paper describing emerging procedures for the design of such diaphragms has been included in Part 3 of the Provisions. The design method used here is that proposed by Moustafa. This method makes use of the shear friction provisions of ACI 318 with the friction coefficient, µ, being equal to 1.0. To use µ = 1.0, ACI 318 requires grout or concrete placed against hardened concrete to have clean, laitance free and intentionally roughened surfaces with a total amplitude of approximately 1/4 inch (peak to valley). Roughness for formed edges is provided either by sawtooth keys along the length of the plank or by hand roughening with chipping hammers. Details from the SEAA Hollow Core reference are used to develop the connection details. Note that grouted joints with edges not intentionally roughened can be used with µ = 0.6. The terminology used is defined in ACI 318 Section 2.2. 8.1.1.2 General In-Plane Seismic Design Forces for Untopped Diaphragms. For Seismic Design Categories B through F, Standard Section 12.10.1.1 defines a minimum diaphragm seismic design force.

8-4

Chapter 8: Precast Concrete Design For Seismic Design Categories C through F, Standard Section 12.10.2.1 requires that collector elements, collector splices and collector connections to the vertical seismic force-resisting members be designed in accordance with Standard Section 14.4.3.2, which amplifies design forces by means of the overstrength factor, Ωo. For Seismic Design Categories D, E and F, Standard Section 12.10.1.1 requires that the redundancy factor, ρ, be used on transfer forces only where the vertical seismic force-resisting system is offset and the diaphragm is required to transfer forces between the elements above and below, but need not be applied to inertial forces defined in Standard Equation 12.10-1. Parameters from the example in Section 10.2 used to calculate in-plane seismic design forces for the diaphragms are provided in Table 8.1-1. Table 8.1-1 Design Parameters from Example 10.2 Design Parameter

Birmingham 1

New York City

1.0

Ωo

2.5

0.12

0.156

wi (roof)

861 kips

869 kips

wi (floor)

963 kips

978 kips

SDS

0.24

0.39

1.0

1.0 kip = 4.45 kN.

8.1.1.3 Diaphragm Forces for Birmingham Building 1. The weight tributary to the roof and floor diaphragms (wpx) is the total story weight (wi) at Level i minus the weight of the walls parallel to the direction of loading. Compute diaphragm weight (wpx) for the roof and floor as follows: §

Roof: Total weight Walls parallel to force = (45 psf)(277 ft)(8.67 ft / 2) wpx

§

= 861 kips = -54 kips = 807 kips

Floors: Total weight Walls parallel to force = (45 psf)(277 ft)(8.67 ft) wpx

= 963 kips = -108 kips = 855 kips

8-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples Compute diaphragm demands in accordance with Standard Equation 12.10-1: n

Fpx =

∑ Fi i= x n

∑ wi

w px

i=x

Calculations for Fpx are provided in Table 8.1-2. Table 8.1-2 Birmingham 1 Fpx Calculations n

∑ wi

Level

wi (kips)

(kips)

Fi (kips)

Roof 4 3 2 1

861 963 963 963 963

861 1,824 2,787 3,750 4,713

175 156 117 78 39

i= x

∑ Fi = Vi (kips)

wpx (kips)

Fpx (kips)

175 331 448 526 565

807 855 855 855 855

164 155 137 120 103

i=x

1.0 kip = 4.45 kN.

The values for Fi and Vi used in Table 8.1-2 are listed in Table 10.2-2. The minimum value of Fpx = 0.2SDSIwpx

= 0.2(0.24)1.0(807 kips) = 0.2(0.24)1.0(855 kips)

= 38.7 kips (roof) = 41.0 kips (floors)

The maximum value of Fpx = 0.4SDSIwpx

= 2(38.7 kips) = 2(41.0 kips)

= 77.5 kips (roof) = 82.1 kips (floors)

Note that the calculated Fpx in Table 8.1-2 is substantially larger than the specified maximum limit value of Fpx. This is generally true at upper levels if the R factor is less than 5. To simplify the design, the diaphragm design force used for all levels will be the maximum force at any level, 82 kips. 8.1.1.4 Diaphragm Forces for New York Building. The weight tributary to the roof and floor diaphragms (wpx) is the total story weight (wi) at Level i minus the weight of the walls parallel to the force.

8-6

Chapter 8: Precast Concrete Design

Compute diaphragm weight (wpx) for the roof and floor as follows: §

Roof: Total weight Walls parallel to force = (48 psf)(277 ft)(8.67 ft / 2) wpx

§

= 870 kips = -58 kips = 812 kips

Floors: Total weight Walls parallel to force = (48 psf)(277 ft)(8.67 ft) wpx

= 978 kips = -115 kips = 863 kips

Calculations for Fpx are provided in Table 8.1-3. Table 8.1-3 New York Fpx Calculations n

∑ wi

Level

wi (kips)

(kips)

Fi (kips)

Roof 4 3 2 1

870 978 978 978 978

870 1,848 2,826 3,804 4,782

229 207 155 103 52

i= x

∑ Fi = Vi (kips)

wpx (kips)

Fpx (kips)

229 436 591 694 746

812 863 863 863 863

214 204 180 157 135

i=x

1.0 kip = 4.45 kN.

The values for Fi and Vi used in Table 8.1-3 are listed in Table 10.2-7. The minimum value of Fpx = 0.2SDSIwpx

= 0.2(0.39)1.0(870 kips) = 0.2(0.39)1.0(978 kips)

= 67.9 kips (roof) = 76.3 kips (floors)

The maximum value of Fpx = 0.4SDSIwpx

= 2(67.9 kips) = 2(76.3 kips)

= 135.8 kips (roof) = 152.6 kips (floors)

As for the Birmingham example, note that the calculated Fpx given in Table 8.1-3 is substantially larger than the specified maximum limit value of Fpx. To simplify the design, the diaphragm design force used for all levels will be the maximum force at any level, 153 kips. 8.1.1.5 Static Analysis of Diaphragms. The balance of this example will use the controlling diaphragm seismic design force of 153 kips for the New York building. In the transverse direction, the loads will be distributed as shown in Figure 8.1-1.

8-7

FEMA P-751, NEHRP Recommended Provisions: Design Examples

F 40'-0"

3 at 24'-0" = 72'-0" 152'-0"

F 40'-0"

Figure 8.1-1 Diaphragm force distribution and analytical model (1.0 ft = 0.3048 m) The Standard requires that structural analysis consider the relative stiffness of the diaphragms and the vertical elements of the seismic force-resisting system. Since a pretopped precast diaphragm doesn’t satisfy the conditions of either the flexible or rigid diaphragm conditions identified in the Standard, maximum in-plane deflections of the diaphragm must be evaluated. However, that analysis is beyond the scope of this document. Therefore, with a rigid diaphragm assumption, assuming the four shear walls have the same stiffness and ignoring torsion, the diaphragm reactions at the transverse shear walls (F as shown in Figure 8.1-1) are computed as follows: F = 153 kips/4 = 38.3 kips The uniform diaphragm demands are proportional to the distributed weights of the diaphragm in different areas (see Figure 8.1-1). W1 = [67 psf (72 ft) + 48 psf (8.67 ft)4](153 kips / 863 kips) = 1,150 lb/ft W2 = [67 psf (72 ft)](153 kips / 863 kips) = 855 lb/ft Figure 8.1-2 identifies critical regions of the diaphragm to be considered in this design. These regions are:

8-8

§

Joint 1: Maximum transverse shear parallel to the panels at panel-to-panel joints

§

Joint 2: Maximum transverse shear parallel to the panels at the panel-to-wall joint

§

Joint 3: Maximum transverse moment and chord force

§

Joint 4: Maximum longitudinal shear perpendicular to the panels at the panel-to-wall connection (exterior longitudinal walls) and anchorage of exterior masonry wall to the diaphragm for out-ofplane forces

§

Joint 5: Collector element and shear for the interior longitudinal walls

Chapter 8: Precast Concrete Design

8.1 7

72'-0"

8.1 3

8.1 6

8.1 8

8.1 5

8.1 4

36'-0"

24'-0" 4'-0"

Figure 8.1-2 Diaphragm plan and critical design regions (1.0 ft = 0.3048 m) Joint forces are as follows: §

Joint 1 – Transverse forces: Shear, Vu1 = 1.15 kips/ft (36 ft) = 41.4 kips Moment, Mu1 = 41.4 kips (36 ft / 2) = 745 ft-kips Chord tension force, Tu1 = M/d = 745 ft-kips / 71 ft = 10.5 kips

§

Joint 2 – Transverse forces: Shear, Vu2 = 1.15 kips/ft (40 ft) = 46 kips Moment, Mu2 = 46 kips (40 ft / 2) = 920 ft-kips Chord tension force, Tu2 = M/d = 920 ft-kips / 71 ft = 13.0 kips

§

Joint 3 – Transverse forces: Shear, Vu3 = 46 kips + 0.86 kips/ft (24 ft) – 38.3 kips = 28.3 kips Moment, Mu3 = 46 kips (44 ft) + 20.6 kips (12 ft) - 38.3 kips (24 ft) = 1,352 ft-kips Chord tension force, Tu3 = M/d = 1,352 ft-kips / 71 ft = 19.0 kips

§

Joint 4 – Longitudinal forces: Wall force, F = 153 kips / 8 = 19.1 kips Wall shear along wall length, Vu4 = 19.1 kips (36 ft)/(152 ft / 2) = 9.0 kips Collector force at wall end, Tu4 = Cu4 = 19.1 kips - 9.0 kips = 10.1 kips

8-9

FEMA P-751, NEHRP Recommended Provisions: Design Examples §

Joint 4 – Out-of-plane forces:

The Standard has several requirements for out-of-plane forces. None are unique to precast diaphragms and all are less than the requirements in ACI 318 for precast construction regardless of seismic considerations. Assuming the planks are similar to beams and comply with the minimum requirements of Standard Section 12.14 (Seismic Design Category B and greater), the required outof-plane horizontal force is: 0.05(D+L)plank = 0.05(67 psf + 40 psf)(24 ft / 2) = 64.2 plf According to Standard Section 12.11.2 (Seismic Design Category B and greater), the minimum anchorage for masonry walls is: 400(SDS)(I) = 400(0.39)1.0 = 156 plf According to Standard Section 12.11.1 (Seismic Design Category B and greater), bearing wall anchorage must be designed for a force computed as: 0.4(SDS)(Wwall) = 0.4(0.39)(48 psf)(8.67 ft) = 64.9 plf Standard Section 12.11.2.1 (Seismic Design Category C and greater) requires masonry wall anchorage to flexible diaphragms to be designed for a larger force. Due to its geometry, this diaphragm is likely to be classified as rigid. However, the relative deformations of the wall and diaphragm must be checked in accordance with Standard Section 12.3.1.3 to validate this assumption. Fp = 0.85(SDS)(I)(Wwall) = 0.85(0.39)1.0[(48 psf)(8.67 ft)] = 138 plf (Note that since this diaphragm is not flexible, this load is not used in the following calculations.) The force requirements in ACI 318 Section 16.5 will be described later. §

Joint 5 – Longitudinal forces: Wall force, F = 153 kips / 8 = 19.1 kips Wall shear along each side of wall, Vu5 = 19.1 kips [2(36 ft) / 152 ft]/2 = 4.5 kips Collector force at wall end, Tu5 = Cu5 = 19.1 kips - 2(4.5 kips) = 10.1 kips

§

Joint 5 – Shear flow due to transverse forces: Shear at Joint 2, Vu2 = 46 kips Q=Ad A = (0.67 ft) (24 ft) = 16 ft2 d = 24 ft Q = (16 ft2) (24 ft) = 384 ft3 I = (0.67 ft) (72 ft)3 / 12 = 20,840 ft4 Vu2Q/I = (46 kip) (384 ft3) / 20,840 ft4 = 0.847 kip/ft maximum shear flow Joint 5 length = 40 ft Total transverse shear in joint 5, Vu5 = 0.847 kip/ft) (40 ft)/2 = 17 kips

ACI 318 Section 16.5 also has minimum connection force requirements for structural integrity of precast concrete bearing wall building construction. For buildings over two stories tall, there are force 8-10

Chapter 8: Precast Concrete Design requirements for horizontal and vertical members. This building has no vertical precast members. However, ACI 318 Section 16.5.1 specifies that the strengths “... for structural integrity shall apply to all precast concrete structures.” This is interpreted to apply to the precast elements of this masonry bearing wall structure. The horizontal tie force requirements for a precast bearing wall structure three or more stories in height are: §

1,500 pounds per foot parallel and perpendicular to the span of the floor members. The maximum spacing of ties parallel to the span is 10 feet. The maximum spacing of ties perpendicular to the span is the distance between supporting walls or beams.

§

16,000 pounds parallel to the perimeter of a floor or roof located within 4 feet of the edge at all edges.

ACI’s tie forces are far greater than the minimum tie forces given in the Standard for beam supports and anchorage of masonry walls. They do control some of the reinforcement provided, but most of the reinforcement is controlled by the computed connections for diaphragm action. 8.1.1.6 Diaphragm Design and Details. The phi factors used for this example are as follows: §

Tension control (bending and ties): φ = 0.90

§

Shear: φ = 0.75

§

Compression control in tied members: φ = 0.65

The minimum tie force requirements given in ACI 318 Section 16.5 are specified as nominal values, meaning that φ = 1.00 for those forces. Note that although buildings assigned to Seismic Design Category C are not required to meet ACI 318 Section 21.11, some of the requirements contained therein are applied below as good practice but shown as optional. 8.1.1.6.1 Joint 1 Design and Detailing. The design must provide sufficient reinforcement for chord forces as well as shear friction connection forces, as follows: §

Chord reinforcement, As1 = Tu1/φfy = (10.5 kips)/[0.9(60 ksi)] = 0.19 in2 (The collector force from the Joint 4 calculations at 10.1 kips is not directly additive.)

§

Shear friction reinforcement, Avf1 = Vu1/φµfy = (41.4 kips)/[(0.75)(1.0)(60 ksi)] = 0.92 in2

§

Total reinforcement required = 2(0.19 in2) + 0.92 in2 = 1.30 in2

§

ACI tie force = (1.5 kips/ft)(72 ft) = 108 kips; reinforcement = (108 kips)/(60 ksi) = 1.80 in 2

Provide four #5 bars (two at each of the outside edges) plus four #4 bars (two each at the interior joint at the ends of the plank) for a total area of reinforcement of 4(0.31 in2) + 4(0.2 in2) = 2.04 in2. Because the interior joint reinforcement acts as the collector reinforcement in the longitudinal direction for the interior longitudinal walls, the cover and spacing of the two #4 bars in the interior joints will be provided to meet the requirements of ACI 318 Section 21.11.7.6 (optional):

8-11

FEMA P-751, NEHRP Recommended Provisions: Design Examples

§

Minimum cover = 2.5(4/8) = 1.25 in., but not less than 2.00 in.

§

Minimum spacing = 3(4/8) = 1.50 in., but not less than 1.50 in.

Figure 8.1-3 shows the reinforcement in the interior joints at the ends of the plank, which is also the collector reinforcement for the interior longitudinal walls (Joint 5). The two #4 bars extend along the length of the interior longitudinal walls as shown in Figure 8.1-3.

(2) #4 (collector bars) #3 x 4'-0" (behind) at each joint between planks

31 2"

21 2" 2"

33 4"

Figure 8.1-3 Interior joint reinforcement at the ends of plank and collector reinforcement at the end of the interior longitudinal walls - Joints 1 and 5 (1.0 in. = 25.4 mm) Figure 8.1-4 shows the extension of the two #4 bars of Figure 8.1-3 into the region where the plank is parallel to the bars (see section cut on Figure 8.1-2). The bars will need to be extended the full length of the diaphragm unless supplemental plank reinforcement is provided. This detail makes use of this supplement plank reinforcement (two #4 bars or an equal area of strand) and shows the bars anchored at each end of the plank. The anchorage length of the #4 bars is calculated using ACI 318 Chapter 12:

⎛ f ψ ψ y t e ld = ⎜ ⎜ 25λ f ' c ⎝

⎞ ⎛ 60,000 psi (1.0 )(1.0 ) ⎞ ⎟ db = ⎜ ⎟ d = 37.9db ⎜ 25 (1.0 ) 4,000 psi ⎟ b ⎟ ⎝ ⎠ ⎠

Using #4 bars, the required ld = 37.9(0.5 in.) = 18.9 in. Therefore, use ld = 4 ft, which is the width of the plank.

8-12

Chapter 8: Precast Concrete Design

11 2" 21 2"

(2) #4 anchored 4'-0" into plank at ends.

Figure 8.1-4 Anchorage region of shear reinforcement for Joint 1 and collector reinforcement for Joint 5 (1.0 in. = 25.4 mm) 8.1.1.6.2 Joint 2 Design and Detailing. The chord design is similar to the previous calculations: §

Chord reinforcement, As2 = Tu2/φfy = (13.0 kips)/[0.9(60 ksi)] = 0.24 in2

The shear force may be reduced along Joint 2 by the shear friction resistance provided by the supplemental chord reinforcement (2Achord - As2) and by the four #4 bars projecting from the interior longitudinal walls across this joint. The supplemental chord bars, which are located at the end of the walls, are conservatively excluded here. The shear force along the outer joint of the wall where the plank is parallel to the wall is modified as follows:

(

)

= Vu 2 − ⎣⎡φ f y µ ( A4#4 )⎦⎤ = 46 − ⎡0.75 ( 60 ksi )(1.0 ) 4 × 0.2 in 2 ⎤ = 36.0 kips VuMod 2 ⎣ ⎦ This force must be transferred from the planks to the wall. Using the arrangement shown in Figure 8.1-5, the required shear friction reinforcement (Avf2) is computed as:

Avf 2 =

VuMod 2

φ f y ( µ sin α f + cos α f

)

36.0 kips = 0.60 in 2 0.75 (1.0sin 26.6°+ cos 26.6° )

Use two #3 bars placed at 26.6 degrees (2-to-1 slope) across the joint at 6 feet from the ends of the plank (two sets per plank). The angle (αf) used above provides development of the #3 bars while limiting the grouting to the outside core of the plank. The total shear reinforcement provided is 6(0.11 in 2) = 0.66 in2. Note that the spacing of these connectors will have to be adjusted at the stair location. The shear force between the other face of this wall and the diaphragm is: Vu2-F = 46-38.3 = 7.7 kips The shear friction resistance provided by #3 bars in the grout key between each plank (provided for the 1.5 klf requirement of ACI 318) is computed as:

φAvffyµ = (0.75)(10 bars)(0.11 in2)(60 ksi)(1.0) = 49.5 kips

8-13

FEMA P-751, NEHRP Recommended Provisions: Design Examples

The development length of the #3 bars will now be checked. For the 180 degree standard hook, use ACI 318 Section 12.5, ldh times the factors of ACI 318 Section 12.5.3, but not less than 8db or 6 inches. Side cover exceeds 2-1/2 inches and cover on the bar extension beyond the hook is provided by the grout and the planks, which is close enough to 2 inches to apply the 0.7 factor of ACI 318 Section 12.5.3. For the #3 hook:

⎛ 0.02ψ f e y ldh = 0.7 ⎜ ⎜ fc' ⎝

⎞ ⎛ 0.02 (1.0 )( 60,000 psi ) ⎞ ⎟ db = 0.7 ⎜ ⎟⎟ 0.375 = 4.98 in. (≤ 6 in. minimum) ⎜ ⎟ 4,000 psi ⎝ ⎠ ⎠

The available distance for the perpendicular hook is approximately 5-1/2 inches. The bar will not be fully developed at the end of the plank because of the 6-inch minimum requirement. The full strength is not required for shear transfer. By inspection, the diagonal #3 hook will be developed in the wall as required for the computed diaphragm-to-shear-wall transfer. The straight end of the #3 bar will now be checked. The standard development length of ACI 318 Section 12.2 is used for ld.

ld =

f y db 25 f cʹ′

60,000 ( 0.375) 25 4,000

= 14.2 in.

Figure 8.1-5 shows the reinforcement along each side of the wall on Joint 2.

8-14

Chapter 8: Precast Concrete Design

" 2'-2

standard hooks

2'-2 "

#3 x

#3x 2'-6"

embedded in grouted edge cell of plank. Provide 2 sets for each plank.

standard hook grouted into each key joint

2" cover

71 2"

(2) #5 in masonry bond beam

(1) #5 continuous in joint to anchor hooks

Vertical reinforcement in wall

Figure 8.1-5 Joint 2 transverse wall joint reinforcement (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) 8.1.1.6.3 Design and Detailing at Joint 3. Compute the required amount of chord reinforcement at Joint 3 as: As3 = Tu3/φfy = (19.0 kips)/[0.9(60 ksi)] = 0.35 in2 Use two #4 bars, As = 2(0.20) = 0.40 in2 along the exterior edges (top and bottom of the plan in Figure 8.1-2). Require cover for chord bars and spacing between bars at splices and anchorage zones per ACI 318 Section 21.11.7.6 (optional). §

Minimum cover = 2.5(4/8) = 1.25 in., but not less than 2.00 in.

§

Minimum spacing = 3(4/8) = 1.50 in., but not less than 1.50 in.

Figure 8.1-6 shows the chord element at the exterior edges of the diaphragm. The chord bars extend along the length of the exterior longitudinal walls and act as collectors for these walls in the longitudinal direction (see the Joint 4 collector reinforcement calculations and Figure 8.1-7).

8-15

FEMA P-751, NEHRP Recommended Provisions: Design Examples

4"Ø spiral of 1 4" wire with 2" pitch over each lap splice may be required depending on geometry of specific voids in plank.

Artificially roughened surfaces of void as required

2"±

3"±

(2) #5 bars (chord bars)

Prestressed hollow core plank

Splice bars Contact lap splice

Grouted chord / collector element along exterior edge of precast plank

Figure 8.1-6 Joint 3 chord reinforcement at the exterior edge (1.0 in. = 25.4 mm) Joint 3 must also be checked for the minimum ACI tie forces. The chord reinforcement obviously exceeds the 16 kip perimeter force requirement. To satisfy the 1.5 kips per foot requirement, a 6 kip tie is needed at each joint between the planks, which is satisfied with a #3 bar in each joint (0.11 in2 at 60 ksi = 6.6 kips). This bar is required at all bearing walls and is shown in subsequent details. 8.1.1.6.4 Joint 4 Design and Detailing. The required shear friction reinforcement along the wall length is computed as: Avf4 = Vu4/φµfy = (9.0 kips)/[(0.75)(1.0)(60 ksi)] = 0.20 in2 Based upon the ACI tie requirement, provide #3 bars at each plank-to-plank joint. For eight bars total, the area of reinforcement is 8(0.11) = 0.88 in2, which is more than sufficient even considering the marginal development length, which is less favorable at Joint 2. The bars are extended 2 feet into the grout key, which is more than the development length and equal to half the width of the plank. The required collector reinforcement is computed as: As4 = Tu4/φfy = (10.1 kips)/[0.9(60 ksi)] = 0.19 in2 The two #4 bars, which are an extension of the transverse chord reinforcement, provide an area of reinforcement of 0.40 in2. The reinforcement required by the Standard for out-of-plane force (156 plf) is far less than the ACI 318 requirement. Figure 8.1-7 shows this joint along the wall.

8-16

Chapter 8: Precast Concrete Design

Vertical wall reinforcement beyond

#3x 2'-6" standard hook grouted into each key joint 2" cover

(2) #5 bars in joint (chord bars)

(2) #5 in bond beam

Figure 8.1-7 Joint 4 exterior longitudinal walls to diaphragm reinforcement and out-of-plane anchorage (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) 8.1.1.6.5 Joint 5 Design and Detailing. The required shear friction reinforcement along the wall length is computed as: Avf5 = Vu5/φµfy = (16.9 kips)/[(0.75)(1.0)(0.85)(60 ksi)] = 0.44 in2 Provide #3 bars at each plank-to-plank joint for a total of 8 bars. The required collector reinforcement is computed as: As5 = Tu5/φfy = (10.1 kips)/[0.9(60 ksi)] = 0.19 in2 Two #4 bars specified for the design of Joint 1 above provide an area of reinforcement of 0.40 in2. Figure 8.1-8 shows this joint along the wall.

8-17

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Vertical wall reinforcement beyond

#3 x 4'-8" grouted into each key joint

(2) #4 bars in joint (collector bars) (2) #5 in bond beam

Figure 8.1-8 Wall-to-diaphragm reinforcement along interior longitudinal walls - Joint 5 (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)

8.1.2

Topped Precast Concrete Units for Five-‐Story Masonry Building Located in Los Angeles, California (see Sec. 10.2)

This design shows the floor and roof diaphragms using topped precast units in the five-story masonry building in Los Angeles, California. The topping thickness exceeds the minimum thickness of 2 inches as required for composite topping slabs by ACI 318 Section 21.11.6. The topping is lightweight concrete (weight = 115 pcf) with a 28-day compressive strength (f'c) of 4,000 psi and is to act compositely with the 8-inch-thick hollow-core precast, prestressed concrete plank. Design parameters are provided in Section 10.2. Figure 10.2-1 shows the typical floor and roof plan. 8.1.2.1 General Design Requirements. Topped diaphragms may be used in any Seismic Design Category. ACI 318 Section 21.11 provides design provisions for topped precast concrete diaphragms. Standard Section 12.10 specifies the forces to be used in designing the diaphragms.

8-18

Chapter 8: Precast Concrete Design 8.1.2.2 General In-Plane Seismic Design Forces for Topped Diaphragms. The in-plane diaphragm seismic design force (Fpx) is calculated using Standard Equation 12.10-1 but must not be less than 0.2SDSIwpx and need not be more than 0.4SDSIwpx. Vx must be added to Fpx calculated using Equation 12.10-1 where: SDS =

the spectral response acceleration parameter at short periods

occupancy importance factor

wpx =

the weight tributary to the diaphragm at Level x

Vx =

the portion of the seismic shear force required to be transferred to the components of the vertical seismic force-resisting system due to offsets or changes in stiffness of the vertical resisting member at the diaphragm being designed

For Seismic Design Category C and higher, Standard Section 12.10.2.1 requires that collector elements, collector splices and collector connections to the vertical seismic force-resisting members be designed in accordance with Standard Section 12.4.3.2, which combines the diaphragm forces times the overstrength factor (Ω0) and the effects of gravity forces. The parameters from the example in Section 10.2 used to calculate in-plane seismic design forces for the diaphragms are provided in Table 8.1-4. Table 8.1-4 Design Parameters from Section 10.2 Design Parameter

Value

Ωo

2.5

wi (roof)

1,166 kips

wi (floor)

1,302 kips

SDS

1.0

Seismic Design Category

1.0 kip = 4.45 kN.

8.1.2.3 Diaphragm Forces. As indicated previously, the weight tributary to the roof and floor diaphragms (wpx) is the total story weight (wi) at Level i minus the weight of the walls parallel to the force. Compute diaphragm weight (wpx) for the roof and floor as follows: §

Roof: Total weight Walls parallel to force = (60 psf)(277 ft)(8.67 ft / 2) wpx

= 1,166 kips = -72 kips = 1,094 kips

8-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples §

Floors: Total weight Walls parallel to force = (60 psf)(277 ft)(8.67 ft) wpx

= 1,302 kips = -144 kips = 1,158 kips

Compute diaphragm demands in accordance with Standard Equation 12.10-1: n

Fpx =

∑ Fi i= x n

∑ wi

w px

i=x

Calculations for Fpx are provided in Table 8.1-5. The values for Fi and Vi are listed in Table 10.2-17. Table 8.1-5 Fpx Calculations from Section 10.2 n

∑ wi

Level

wi (kips)

(kips)

Fi (kips)

Roof 4 3 2 1

1,166 1,302 1,302 1,302 1,302

1,166 2,468 3,770 5,072 6,374

564 504 378 252 126

i= x

∑ Fi = Vi (kips)

wpx (kips)

Fpx (kips)

564 1,068 1,446 1,698 1,824

1,094 1,158 1,158 1,158 1,158

529 501 444 387 331

i=x

1.0 kip = 4.45 kN.

The minimum value of Fpx = 0.2SDSIwpx

= 0.2(1.0)1.0(1,094 kips) = 0.2(1.0)1.0(1,158 kips)

= 219 kips (roof) = 232 kips (floors)

The maximum value of Fpx = 0.4SDSIwpx

= 2(219 kips) = 2(232 kips)

= 438 kips (roof) = 463 kips (floors)

The value of Fpx used for design of the diaphragms is 463 kips, except for collector elements where forces will be computed below. 8.1.2.4 Static Analysis of Diaphragms. The seismic design force of 463 kips is distributed as in Section 8.1.1.6 (Figure 8.1-1 shows the distribution). The force is three times higher than that used to design the untopped diaphragm for the New York design due to the higher seismic demand. Figure 8.1-2 shows critical regions of the diaphragm to be considered in this design. Collector elements will be designed for 2.5 times the diaphragm force based on the overstrength factor (Ω0).

8-20

Chapter 8: Precast Concrete Design Joint forces taken from Section 8.1.1.5 times 3.0 are as follows: §

Joint 1 – Transverse forces: Shear, Vu1 = 41.4 kips × 3.0 = 124 kips Moment, Mu1 = 745 ft-kips × 3.0 = 2,235 ft-kips Chord tension force, Tu1 = M/d = 2,235 ft-kips / 71 ft = 31.5 kips

§

Joint 2 – Transverse forces: Shear, Vu2 = 46 kips × 3.0 = 138 kips Moment, Mu2 = 920 ft-kips × 3.0 = 2,760 ft-kips Chord tension force, Tu2 = M/d = 2,760 ft-kips / 71 ft = 38.9 kips

§

Joint 3 – Transverse forces: Shear, Vu3 = 28.3 kips × 3.0 = 84.9 kips Moment, Mu2 = 1,352 ft-kips × 3.0 = 4,056 ft-kips Chord tension force, Tu3 = M/d = 4,056 ft-kips / 71 ft = 57.1 kips

§

Joint 4 – Longitudinal forces: Wall force, F = 19.1 kips × 3.0 = 57.3 kips Wall shear along wall length, Vu4 = 9 kips × 3.0 = 27.0 kips Collector force at wall end, Ω0Tu4 = 2.5(10.1 kips)(3.0) = 75.8 kips

§

Joint 4 – Out-of-plane forces: Just as with the untopped diaphragm, the out-of-plane forces are controlled by ACI 318 Section 16.5, which requires horizontal ties of 1.5 kips per foot from floor to walls.

§

Joint 5 – Longitudinal forces: Wall force, F = 463 kips / 8 walls = 57.9 kips Wall shear along each side of wall, Vu4 = 4.5 kips × 3.0 = 13.5 kips Collector force at wall end, Ω0Tu4 = 2.5(10.1 kips)(3.0) = 75.8 kips

§

Joint 5 – Shear flow due to transverse forces: Shear at Joint 2, Vu2 = 138 kips Q=Ad A = (0.67 ft) (24 ft) = 16 ft2 d = 24 ft Q = (16 ft2) (24 ft) = 384 ft3 I = (0.67 ft) (72 ft)3 / 12 = 20,840 ft4 Vu2Q/I = (138 kip) (384 ft3) / 20,840 ft4 = 2.54 kips/ft maximum shear flow Joint 5 length = 40 ft Total transverse shear in joint 5, Vu5 = 2.54 kips/ft) (40 ft)/2 = 50.8 kips

8.1.2.5 Diaphragm Design and Details

8-21

FEMA P-751, NEHRP Recommended Provisions: Design Examples 8.1.2.5.1 Minimum Reinforcement for 2.5-inch Topping. ACI 318 Section 21.11.7.1 references ACI 318 Section 7.12, which requires a minimum As = 0.0018bd for grade 60 welded wire reinforcement. For a 2.5-inch topping, the required As = 0.054 in2/ft. WWR 10×10 - W4.5×W4.5 provides 0.054 in2/ft. The minimum spacing of wires is 10 inches and the maximum spacing is 18 inches. Note that the ACI 318 Section 7.12 limit on spacing of five times thickness is interpreted such that the topping thickness is not the pertinent thickness. 8.1.2.5.2 Boundary Members. Joint 3 has the maximum bending moment and is used to determine the boundary member reinforcement of the chord along the exterior edge. The need for transverse boundary member reinforcement is reviewed using ACI 318 Section 21.11.7.5. Calculate the compressive stress in the chord with the ultimate moment using a linear elastic model and gross section properties of the topping. It is conservative to ignore the precast units, but this is not necessary since the joints between precast units are grouted. As developed previously, the chord compressive stress is: 6Mu3/td2 = 6(4,056×12)/(2.5)(72×12)2 = 157 psi The chord compressive stress is less than 0.2f'c = 0.2(4,000) = 800 psi. Transverse reinforcement in the boundary member is not required. The required chord reinforcement is: As3 = Tu3/φfy = (57.1 kips)/[0.9(60 ksi)] = 1.06 in2 8.1.2.5.3 Collectors. The design for Joint 4 collector reinforcement at the end of the exterior longitudinal walls and for Joint 5 at the interior longitudinal walls is the same. As4 = As5 = Ω0Tu4/φfy = (75.8 kips)/[0.9(60 ksi)] = 1.40 in2 Use two #8 bars (As = 2 × 0.79 = 1.58 in2) along the exterior edges, along the length of the exterior longitudinal walls and along the length of the interior longitudinal walls. Provide cover for chord and collector bars and spacing between bars per ACI 318 Section 21.11.7.6. §

Minimum cover = 2.5(8/8) = 2.5 in., but not less than 2.0 in.

§

Minimum spacing = 3(8/8) = 3.0 in., but not less than 1.5 in.

Figure 8.1-9 shows the diaphragm plan and section cuts of the details and Figure 8.1-10 shows the boundary member and chord/collector reinforcement along the edge. Given the close margin on cover, the transverse reinforcement at lap splices also is shown.

8-22

Chapter 8: Precast Concrete Design

8.1 10

8.1 13

8.1 12

8.1 11

Figure 8.1-9 Diaphragm plan and section cuts

Splice bars Artificially roughened edge

WWF bend down into chord

21 2" min (concrete topping)

(2) #8 bars (chord bars)

Prestressed hollow core plank with roughened top surface

41 2"Ø spiral of 1 4" wire with 2" pitch over each lap splice. Contact lap splice

31 2"

21 2"

Grouted chord / collector element along exterior edge of precast plank

Figure 8.1-10 Boundary member and chord and collector reinforcement (1.0 in. = 25.4 mm) Figure 8.1-11 shows the collector reinforcement for the interior longitudinal walls. The side cover of 2-1/2 inches is provided by casting the topping into the cores and by the stems of the plank. A minimum space of 1 inch is provided between the plank stems and the sides of the bars.

8-23

FEMA P-751, NEHRP Recommended Provisions: Design Examples

2" min topping

WWF (2) #8 (collector bars)

21 2"

Figure 8.1-11 Collector reinforcement at the end of the interior longitudinal walls - Joint 5 (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) 8.1.2.5.4 Shear Resistance. In thin composite and noncomposite topping slabs on precast floor and roof members, joints typically are tooled during construction, resulting in cracks forming at the joint between precast members. Therefore, the shear resistance of the topping slab is limited to the shear friction strength of the reinforcing crossing the joint. ACI 318 Section 21.11.9.1 provides an equation for the shear strength of the diaphragm, which includes both concrete and reinforcing components. However, for noncomposite topping slabs on precast floors and roofs where the only reinforcing crossing the joints is the field reinforcing in the topping slab, the shear friction capacity at the joint will always control the design. ACI 318 Section 21.11.9.3 defines the shear strength at the joint as follows:

φVn = φAvffyµ = 0.75(0.054 in2/ft)(60 ksi)(1.0)(0.85) = 2.07 kips/ft Note that µ = 1.0λ is used since the joint is assumed to be pre-cracked. The shear resistance in the transverse direction is: 2.07 kips/ft (72 ft) = 149 kips which is greater than the Joint 1 shear (maximum transverse shear) of 124 kips. At the plank adjacent to Joint 2, the shear strength of the diaphragm in accordance with ACI 318 Section 21.11.9.1 is:

(

)

(

)

φVn = φ Acv 2λ fc' + ρt f y = 0.75 ( 2.5 × 72 ×12) 2 (1.0) 4,000 + 0.0018 × 60,000 = 348 kips

8-24

Chapter 8: Precast Concrete Design Number 3 dowels are used to provide continuity of the topping slab welded wire reinforcement across the masonry walls. The topping is to be cast into the masonry walls as shown in Figure 8.1-12 and the spacing of the #3 bars is set to be modular with the CMU.

Vertical reinforcement 1" clear

Cut out alternate face shells (16" o.c. each side) and place topping completely through wall and between planks #3x4'-0" at 16" to lap with WWF

WWF 10 x 10 W4.5 x W4.5

(2) #8 collector bars

(2) #5 in masonry bond beam

Figure 8.1-12 Wall-to-diaphragm reinforcement along interior longitudinal walls - Joint 5 (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) The required shear reinforcement along the exterior longitudinal wall (Joint 4) is: Avf4 = Vu4/φµfy = (27.0 kips)/[(0.75)(1.0)(0.85)(60 ksi)] = 0.71 in2 The required shear reinforcement along the interior longitudinal wall (Joint 5) is: Avf5 = Vu5/φµfy = (50.8 kips)/[(0.75)(1.0)(0.85)(60 ksi)] = 1.32 in2 Number 3 dowels spaced at 16” o.c. provide Av = (0.11 in2) (40 ft x 12 in/ft) / 16 in = 3.30 in2 8.1.2.5.5 Check of Out-of-Plane Forces. At Joint 4, the out-of-plane forces are checked as follows: Fp = 0.85 SDS I Wwall = 0.85(1.0)(1.0)(60 psf)(8.67 ft) = 442 plf With bars at 4 feet on center, Fp = 4 ft (442 plf) = 1.77 kips. The required reinforcement, As = 1.77 kips/(0.9)(60ksi) = 0.032 in2. Provide #3 bars at 4 feet on center, which provides a nominal strength of 0.11×60/4 = 1.7 klf. This detail satisfies the ACI 318 Section 16.5

8-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples required tie force of 1.5 klf. The development length was checked in the prior example. Using #3 bars at 4 feet on center will be adequate and the detail is shown in Figure 8.1-13. The detail at Joint 2 is similar.

Cut out face shells @ 4'-0" and place topping into wall

Vertical wall reinforcement beyond 2"

WWF 10 x10 W4.5 x W4.5

#3x STD HK 2'-6" at 4'-0" o.c. (2) #8 (collector bars)

(2) #5 in masonry bond beam

Figure 8.1-13 Exterior longitudinal wall-to-diaphragm reinforcement and out-of-plane anchorage - Joint 4 (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m).

8.2

THREE-STORY OFFICE BUILDING WITH INTERMEDIATE PRECAST CONCRETE SHEAR WALLS

This example illustrates the seismic design of intermediate precast concrete shear walls. These walls can be used up to any height in Seismic Design Categories B and C but are limited to 40 feet for Seismic Design Categories D, E and F. ACI 318 Section 21.4.2 requires that yielding between wall panels or between wall panels and the foundation be restricted to steel elements. However, the Provisions are more specific in their means to accomplish the objective of providing reliable post-elastic performance. Provisions Section 21.4.3 (ACI 318 Sec. 21.4.4) requires that connections that are designed to yield be capable of maintaining 80 percent of their design strength at the deformation induced by the design displacement. Alternatively, they can use Type 2 mechanical splices. Additional requirements are contained in the Provisions for intermediate precast walls with wall piers (Provisions Sec. 14.2.2.4 [ACI 318 Sec. 21.4.5]); however, these requirements do not apply to the solid wall panels used for this example.

8-26

Chapter 8: Precast Concrete Design

8.2.1

Building Description

This precast concrete building is a three-story office building (Occupancy Category II) in southern New England on Site Class D soils. The structure utilizes 10-foot-wide by 18-inch-deep prestressed double tees (DTs) spanning 40 feet to prestressed inverted tee beams for the floors and the roof. The DTs are to be constructed using lightweight concrete. Each of the above-grade floors and the roof are covered with a 2-inch-thick (minimum), normal-weight cast-in-place concrete topping. The vertical seismic forceresisting system is to be constructed entirely of precast concrete walls located around the stairs and elevator/mechanical shafts. The only features illustrated in this example are the rational selection of the seismic design parameters and the design of the reinforcement and connections of the precast concrete shear walls. The diaphragm design is not illustrated. As shown in Figure 8.2-1, the building has a regular plan. The precast shear walls are continuous from the ground level to 12 feet above the roof. The walls of the elevator/mechanical pits are cast-in-place below grade. The building has no vertical irregularities. The story height is 12 feet.

150'-0" 25'-0"

25'-0"

26 IT 28 precast beams

15'-0" 8'-0"

18" DT roof and floor slabs (10 DT 18)

8'-0"

40'-0"

120'-0"

40'-0"

25'-0"

40'-0"

8" precast shear walls

Figure 8.2-1 Three-story building plan (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) The precast walls are estimated to be 8 inches thick for building mass calculations. These walls are normal-weight concrete with a 28-day compressive strength, f'c, of 5,000 psi. Reinforcing bars used at the ends of the walls and in welded connectors are ASTM A706 (60 ksi yield strength). The concrete for the foundations and below-grade walls has a 28-day compressive strength, f'c, of 4,000 psi.

8-27

FEMA P-751, NEHRP Recommended Provisions: Design Examples 8.2.2

Design Requirements

8.2.2.1 Seismic Parameters. The basic parameters affecting the design and detailing of the building are shown in Table 8.2-1. Table 8.2-1 Design Parameters Design Parameter

Value

Occupancy Category II

I = 1.0

0.266

0.08

Site Class

1.59

2.4

SMS = FaSS

0.425

SM1 = FvS1

0.192

SDS = 2/3 SMS

0.283

SD1 = 2/3 SM1

0.128

Seismic Design Category

Basic Seismic Force-Resisting System

Bearing Wall System

Wall Type

Intermediate Precast Shear Walls

Ω0

2.5

A Bearing Wall System is defined in the Standard as “a structural system with bearing walls providing support for all or major portions of the vertical loads.” In the 2006 International Building Code, this requirement is clarified by defining a concrete Load Bearing Wall as one which “supports more than 200 pounds per linear foot of vertical load in addition to its own weight.” While the IBC definition is much more stringent, this interpretation is used in this example. Note that if a Building Frame Intermediate Precast Shear Wall system were used, the design would be based on R=5, Ωo=2 ½ and Cd=4½. Note that in Seismic Design Category B an ordinary precast shear wall could be used to resist seismic forces. However, the design forces would be 33 percent higher since they would be based on R = 3, Ωo = 2.5 and Cd = 3. Ordinary precast structural walls need not satisfy any provisions in ACI 318 Chapter 21. 8.2.2.2 Structural Design Considerations 8.2.2.2.1 Precast Shear Wall System. This system is designed to yield in bending at the base of the precast shear walls without shear slip at any of the joints. The remaining connections (shear connectors

8-28

Chapter 8: Precast Concrete Design and flexural connectors away from the base) are then made strong enough to ensure that the inelastic action is forced to the intended location. Although it would be desirable to force yielding to occur in a significant portion of the connections, it frequently is not possible to do so with common configurations of precast elements and connections. The connections are often unavoidable weak links. Careful attention to detail is required to assure adequate ductility in the location of first yield and to preclude premature yielding of other connections. For this particular example, the vertical bars at the ends of the shear walls (see Figure 8.2-6) act as flexural reinforcement for the walls and are selected as the location of first yield. The yielding will not propagate far into the wall vertically due to the unavoidable increase in flexural strength provided by unspliced reinforcement within the panel. The issue of most significant concern is the performance of the shear connections (see Figure 8.2-7) at the same joint. The connections are designed to provide the necessary shear resistance and avoid slip without providing increased flexural strength at the connection since such an increase would also increase the maximum shear force on the joint. At the base of the panel, welded steel angles are designed to be flexible for uplift but stiff for in-plane shear. 8.2.2.2.2 Building System. No height limits are imposed (Standard Table 12.2-1). For structural design, the floors are assumed to act as rigid horizontal diaphragms to distribute seismic inertial forces to the walls parallel to the motion. The building is regular both in plan and elevation, for which, according to Standard Table 12.6-1, use of the Equivalent Lateral Force (ELF) procedure (Standard Sec. 12.8) is permitted. Orthogonal load combinations are not required for this building (Standard Sec. 12.5.2). Ties, continuity and anchorage must be considered explicitly when detailing connections between the floors and roof and the walls and columns. This example does not include consideration of nonstructural elements. Collector elements are required due to the short length of shear walls as compared to the diaphragm dimensions, but they are not designed in this example. Diaphragms need to be designed for the required forces (Standard Sec. 12.10), but that design is not illustrated here. The bearing walls must be designed for a force perpendicular to their plane (Standard Sec. 12.11), but design for that requirement is not shown for this building. The drift limit is 0.025hsx (Standard Table 12.12-1), but drift is not computed here. ACI 318 Section 16.5 requires minimum strengths for connections between elements of precast building structures. The horizontal forces were described in Section 8.1; the vertical forces will be described in this example. 8 . 2 . 3 Load Combinations The basic load combinations require that seismic forces and gravity loads be combined in accordance with the factored load combinations presented in Standard Section 12.4.2.3. Vertical seismic load effects are described in Standard Section 12.4.2.2.

8-29

FEMA P-751, NEHRP Recommended Provisions: Design Examples

According to Standard Section 12.3.4.1, ρ = 1.0 for structures in Seismic Design Categories A, B and C, even though this seismic force-resisting system is not particularly redundant. The relevant load combinations from ASCE 7 are as follows: (1.2 + 0.2SDS)D ± ρQE + 0.5L (0.9 - 0.2SDS)D ± ρQE Into each of these load combinations, substitute SDS as determined above: 1.26D + QE + 0.5L 0.843D - QE These load combinations are for loading in the plane of the shear walls. 8.2.4

Seismic Force Analysis

8.2.4.1 Weight Calculations. For the roof and two floors: 18-inch double tees (32 psf) + 2-inch topping (24 psf) Precast beams at 40 feet 16-inch square columns Ceiling, mechanical, miscellaneous Exterior cladding (per floor area) Partitions Total

= 56.0 psf = 12.5 psf = 4.5 psf = 4.0 psf = 5.0 psf = 10.0 psf = 92.0 psf

Note that since the design snow load is 30 psf, it can be ignored in calculating the seismic weight (Standard Sec. 12.7.2). The weight of each floor including the precast shear walls is: (120 ft)(150 ft)(92 psf / 1,000) + [(15 ft)4 + (25 ft)2](12 ft)(0.10 ksf) = 1,788 kips Considering the roof to be the same weight as a floor, the total building weight is W = 3(1,788 kips) = 5,364 kips. 8.2.4.2 Base Shear. The seismic response coefficient, Cs, is computed using Standard Equation 12.8-2:

S DS 0.283 = 0.0708 = R/I 41

except that it need not exceed the value from Standard Equation 12.8-3 computed as:

CS =

8-30

S D1 0.128 = 0.110 = T ( R / I ) 0.29(4 / 1)

Chapter 8: Precast Concrete Design where T is the fundamental period of the building computed using the approximate method of Standard Equation 12.8-7:

Ta = Cr hnx = (0.02)(36)0.75 = 0.29 sec Therefore, use Cs = 0.0708, which is larger than the minimum specified in Standard Equation 12.8-5: Cs = 0.044(SDS)(I) ≥ 0.01 = 0.044(0.283)(1.0) = 0.012 The total seismic base shear is then calculated using Standard Equation 12.8-1 as: V = CsW = (0.0708)(5,364) = 380 kips Note that this force is substantially larger than a design wind would be. If a nominal 20 psf were applied to the long face and then amplified by a load factor of 1.6, the result would be less than half this seismic force already reduced by an R factor of 4. 8.2.4.3 Vertical Distribution of Seismic Forces. The seismic lateral force ,Fx, at any level is determined in accordance with Standard Section 12.8.3:

Fx = CvxV where:

Cvx =

wx hxk

∑ wi hik i =1

Since the period, T, is less than 0.5 seconds, k = l in both building directions. With equal weights at each floor level, the resulting values of Cvx and Fx are as follows: §

Roof: Cvr = 0.50; Fr = 190 kips

§

Third Floor: Cv3 = 0.333; F3 = 127 kips

§

Second Floor: Cv2 = 0.167; F2 = 63 kips

8.2.4.4 Horizontal Shear Distribution and Torsion 8.2.4.4.1 Longitudinal Direction. Design each of the 25-foot-long walls at the elevator/mechanical shafts for half the total shear. Since the longitudinal walls are very close to the center of rigidity, assume that torsion will be resisted by the 15-foot-long stairwell walls in the transverse direction. The forces for each of the longitudinal walls are shown in Figure 8.2-2.

8-31

FEMA P-751, NEHRP Recommended Provisions: Design Examples

12'-0"

95 kips

12'-0"

63.5 kips

12'-0"

31.5 kips

Grade V = ∑F = 190 kips 25'-0"

Figure 8.2-2 Forces on the longitudinal walls (1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m) 8.2.4.4.2 Transverse Direction. Design the four 15-foot-long stairwell walls for the total shear including 5 percent accidental torsion (Standard Sec. 12.8.4.2). A rough approximation is used in place of a more rigorous analysis considering all of the walls. The maximum force on the walls is computed as follows: V = 380/4 + 380(0.05)(150)/[(100 ft moment arm) × (2 walls in each set)] = 109 kips Thus: Fr = 109(0.50) = 54.5 kips F3 = 109(0.333) = 36.3 kips F2 = 109(0.167) = 18.2 kips Seismic forces on the transverse walls of the stairwells are shown in Figure 8.2-3.

8-32

Chapter 8: Precast Concrete Design

12'-0"

54.5 kips

12'-0"

36.3 kips

12'-0"

18.2 kips

Grade V= 15'-0"

∑F

= 109 kips

Figure 8.2-3 Forces on the transverse walls (1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m)

8.2.5

Proportioning and Detailing

The strength of members and components is determined using the strengths permitted and required in ACI 318 Chapters 1 through 19, plus Sections 21.1.2 and 21.4. 8.2.5.1 Overturning Moment and End Reinforcement. Design shear panels to resist overturning by means of reinforcing bars at each end with a direct tension coupler at the joints. A commonly used alternative is a threaded post-tensioning (PT) bar inserted through the stack of panels, but the behavior is different than assumed by ACI 318 Section 21.4 since the PT bars don’t yield. If PT bars are used, the system should be designed as an Ordinary Precast Shear Wall (allowed in SDC B.) For a building in a higher seismic design category, a post tensioned wall would need to be qualified as a Special Precast Structural Wall Based on Validation Testing per 14.2.4. 8.2.5.1.1 Longitudinal Direction. The free-body diagram for the longitudinal walls is shown in Figure 8.2-4. The tension connection at the base of the precast panel to the below-grade wall is governed by the seismic overturning moment and the dead loads of the panel and supported floors and roof. In this example, the weights for an elevator penthouse, with a floor and equipment load at 180 psf between the shafts and a roof load at 20 psf, are included. The weight for the floors includes double tees, ceiling and partitions (total load of 70 psf) but not beams and columns. Floor live load is 50 psf, except 100 psf is used in the elevator lobby. Roof snow load is 30 psf. (The elevator penthouse is so small that it was ignored in computing the gross seismic forces on the building, but it is not ignored in the following calculations.)

8-33

12'-0"

FEMA P-751, NEHRP Recommended Provisions: Design Examples

12'-0"

95 kips D

12'-0"

63.5 kips D

12'-0"

31.5 kips D

12'-0" 23'-6"

C 9"

Figure 8.2-4 Free-body diagram for longitudinal walls (1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m) At the base: ME = (95 kips)(36 ft) + (63.5 kips)(24 ft) + (31.5 kips)(12 ft) = 5,320 ft-kips ∑D = wall + exterior floors/roof + lobby floors + penthouse floor + penthouse roof = (25 ft)(48 ft)(0.1 ksf) + (25 ft)(48 ft / 2)(0.070 ksf)(3) + (25 ft)(8 ft / 2)(0.070 ksf)(2) + (25 ft)(8 ft / 2)(0.18 ksf) + (25ft )(24 ft / 2)(0.02 ksf) = 120 + 126 + 14 + 18 + 6 = 284 kips ∑L = (25 ft)(48 ft / 2)(0.05 ksf)(2) + (25 ft)(8 ft / 2)(0.1 ksf) = 60 + 10 = 70 kips ∑S = (25ft)(48 ft + 24 ft)(0.03 ksf)/2 = 27 kips Using the load combinations described above, the vertical loads for combining with the overturning moment are computed as: Pmax = 1.26D + 0.5L + 0.2S = 397 kips Pmin = 0.843D = 239 kips The axial load is quite small for the wall panel. The average compression Pmax/Ag = 0.165 ksi (3.3 percent of f'c). Therefore, the tension reinforcement can easily be found from the simple couple shown in Figure 8.2-4.

8-34

Chapter 8: Precast Concrete Design The effective moment arm is: jd = 25 - 1.5 = 23.5 ft and the net tension on the uplift side is:

Tu =

M Pmin 5,320 239 = 107 kips − = − jd 2 23.5 2

The required reinforcement is: As = Tu/φfy = (107 kips)/[0.9(60 ksi)] = 1.98 in2 Use two #9 bars (As = 2.0 in2) at each end with Type 2 couplers for each bar at each panel joint. Since the flexural reinforcement must extend a minimum distance, d, (the flexural depth) beyond where it is no longer required, use both #9 bars at each end of the panel at all three levels for simplicity. Note that if it is desired to reduce the bar size up the wall, the design check of ACI 318 Section 21.4.3 must be applied to the flexural strength calculation at the upper wall panel joints. At this point a check per ACI 318 Section 16.5 will be made. Bearing walls must have vertical ties with a nominal strength exceeding 3 kips per foot and there must be at least two ties per panel. With one tie at each end of a 25-foot panel, the demand on the tie is: Tu = (3 kip/ft)(25 ft)/2 = 37.5 kips The two #9 bars are more than adequate for the ACI requirement. Although no check for confinement of the compression boundary is required for intermediate precast shear walls, it is shown here for interest. Using the check from ACI 318 Section 21.9.6, the depth to the neutral axis is: §

Total compression force, As fy + Pmax = (2.0)(60) + 397 = 517 kips

§

Compression block, a = (517 kips)/[(0.85)(5 ksi)(8 in. width)] = 15.2 in.

§

Neutral axis depth, c = a/(0.80) = 19.0 in.

The maximum depth (c) with no boundary member per ACI 318 Equation 21-8 is:

c≤

l 600 (δ u / hw )

where the term (δu/hw) shall not be taken as less than 0.007. Once the base joint yields, it is unlikely that there will be any flexural cracking in the wall more than a few feet above the base. An analysis of the wall for the design lateral forces using 50 percent of the gross moment of inertia, ignoring the effect of axial loads and applying the Cd factor of 4 to the results gives a ratio (δu/hw) far less than 0.007. Therefore, applying the 0.007 in the equation results in a distance, c, of 71 inches, far in excess of the 19 inches required. Thus, ACI 318 would not require transverse

8-35

FEMA P-751, NEHRP Recommended Provisions: Design Examples reinforcement of the boundary even if this wall were designed as a special reinforced concrete shear wall. For those used to checking the compression stress as an index:

σ=

P M 389 6(5,320) = 694 psi + = + A S 8 ( 25)12 8 ( 25)2 (12)

The limiting stress is 0.2f'c, which is 1,000 psi, so no transverse reinforcement is required at the ends of the longitudinal walls.

12'-0"

8.2.5.1.2 Transverse Direction. The free-body diagram of the transverse walls is shown in Figure 8.2-5. The weight of the precast concrete stairs is 100 psf and of the roof over the stairs is 70 psf.

12'-0"

54.5 kips D

12'-0"

36.3 kips D

12'-0"

18.2 kips D

V 9"

C 7'-0" 13'-6"

Figure 8.2-5 Free-body diagram of the transverse walls (1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m) The transverse wall is similar to the longitudinal wall. At the base: ME = (54.5 kips)(36 ft) + (36.3 kips)(24 ft) + (18.2 kips)(12 ft) = 3,052 ft-kips ∑D = (15 ft)(48 ft)(0.1 ksf) + 2(12.5 ft / 2)(10 ft / 2)(0.07 ksf)(3) + (15 ft)(8 ft / 2)[(0.1 ksf)(3) + (0.07 ksf)] = 72 + 13 + 18 + 4 = 107 kips ∑L = 2(12.5 ft / 2)(10 ft / 2)(0.05 ksf)(2) + (15 ft)(8 ft / 2)(0.1 ksf)(3) = 6 + 18 = 24 kips 8-36

Chapter 8: Precast Concrete Design

∑S = [2(12.5 ft / 2)(10 ft / 2) + (15 ft)(8 ft / 2)](0.03 ksf) = 3.7 kips Pmax = 1.26(107) + 0.5(24) + 0.2(4) = 148 kips Pmin = 0.843(107) = 90.5 kips jd = 15 - 1.5 = 13.5 ft Tu = (Mnet/jd) - Pmin/2 = (3,052/13.5) - 90.5/2 = 181 kips As = Tu/φfy = (181 kips)/[0.9(60 ksi)] = 3.35 in2 Use two #10 and one #9 bars (As = 3.54 in2) at each end of each wall with a Type 2 coupler at each bar for each panel joint. All three bars at each end of the panel will also extend up through all three levels for simplicity. Following the same method for boundary member check as on the longitudinal walls: §

Total compression force, As fy + Pmax = (3.54)(60) + 148 = 360 kips

§

Compression block, a = (360 kips)/[(0.85)(5 ksi)(8 in. width)] = 10.6 in.

§

Neutral axis depth, c = a/(0.80) = 13.3 in.

Even though this wall is more flexible and the lateral loads will induce more flexural cracking, the computed deflections are still small and the minimum value of 0.007 is used for the ratio (δu/hw). This yields a maximum value of c = 42.9 inches, thus confinement of the boundary would not be required. The check of compression stress as an index gives:

σ=

P M 140 6(2,930) = 951 psi + = + A S 8 (15)12 8 (15)2 (12)

Since σ < 1,000 psi, no transverse reinforcement is required at the ends of the transverse walls. Note how much closer to the criterion this transverse wall is by the compression stress check. The overturning reinforcement and connection are shown in Figure 8.2-6.

8-37

FEMA P-751, NEHRP Recommended Provisions: Design Examples

8" Direct tension coupler-(typical) 1" shim and drypack (typical) 8" precast wall (2) #9 ea. end, full height of 25' longitudinal wall panel (2) #10 & (1) #9 ea. end, full ht. of 15' transverse wall panel

Transverse Wall

Longitudinal Wall

25" min for #9 3" min

28" min for 10"

Reinforced foundation not designed in the example

Standard hook to develop overturning reinforcement

Development at Foundation

Figure 8.2-6 Overturning connection detail at the base of the walls (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m) ACI 318 Section 21.4.3 requires that elements of the connection that are not designed to yield develop at least 1.5Sy. This requirement applies to the anchorage of the coupled bars. The bar in the panel is made continuous to the roof; therefore, no calculation of development length is necessary in the panel. The dowel from the foundation will be hooked; otherwise the depth of the foundation would be more than required for structural reasons. The size of the foundation will provide adequate cover to allow the 0.7 factor on ACI’s standard development length for hooked bars. For the #9 bar:

1.5ldh =

8-38

1.5 ( 0.7 )(1, 200 ) db f c'

1, 260(1.128) 4,000

= 22.5 in.

Chapter 8: Precast Concrete Design

Similarly, for the #10 bar, the length is 25.3 inches. Like many shear wall designs, this design does concentrate a demand for overturning resistance on the foundation. In this instance the resistance may be provided by a large footing (on the order of 20 feet by 28 feet by 3 feet thick) under the entire stairwell or by deep piers or piles with an appropriate cap for load transfer. Refer to Chapter 4 for examples of design of each type of foundation, although not for this particular example. Note that the Standard permits the overturning effects at the soil-foundation interface to be reduced under certain conditions. 8.2.5.2 Shear Connections and Reinforcement. Panel joints often are designed to resist the shear force by means of shear friction, but that technique is not used for this example because the joint at the foundation will open due to flexural yielding. This opening would concentrate the shear stress on the small area of the dry-packed joint that remains in compression. This distribution can be affected by the shims used in construction. With care taken to detail the grouted joint, shear friction can provide a reliable mechanism to resist this shear. Alternatively, the joint can be designed with direct shear connectors that will prevent slip along the joint. That concept is developed here. 8.2.5.2.1 Longitudinal Direction. The design shear force is based on the yield strength of the flexural connection. The flexural strength of the connection can be approximated as follows:

My Mu

As f y jd + Pmax ( jd / 2) ME

( 2.0 in ) (60 ksi )( 23.5 ft ) + (397 kip ) (23.5 ft/2) = 1.41 = 2

5,320 ft-kips

Therefore, the design shear, Vu, at the base is 1.5(1.41)(190 kips) = 402 kips. The base shear connection is shown in Figure 8.2-7 and is to be flexible vertically but stiff horizontally in the plane of the panel. The vertical flexibility is intended to minimize the contribution of these connections to overturning resistance, which would simply increase the shear demand.

8-39

FEMA P-751, NEHRP Recommended Provisions: Design Examples

#5,see (c) Welded wire reinforcement

Plate 3 8x4x1'-0" L4x3x516x0'-8" LLH

(b) Side elevation 3

4"Ø

H.A.S.

Plate 1 2x12x1'-6"

Drypack

C8x18.75

(a) Section through connection 1

Figure 8.2-7 Shear connection at base (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m) In the panel, provide an assembly with two face plates measuring 3/8" × 4" × 12" connected by a C8x18.75 and with diagonal #5 bars as shown in the figure. In the foundation, provide an embedded plate 1/2" × 12" × 1'-6" with six 3/4-inch-diameter headed anchor studs as shown. In the field, weld an L4×3×5/16 × 0'-8", long leg horizontal, on each face. The shear capacity of this connection is checked as follows: §

Shear in the two loose angles:

φVn = φ(0.6Fu)tl(2) = (0.75)(0.6)(58 ksi)(0.3125 in.)(8 in.)(2) = 130.5 kip §

Weld at toe of loose angles:

φVn = φ(0.6Fu)tel(2) = (0.75)(0.6)(70 ksi)(0.25 in. / √2)(8 in.)(2) = 89.1 kip

8-40

Chapter 8: Precast Concrete Design §

Weld at face plates, using Table 8-8 in AISC Manual (13th edition):

φVn = φCC1Dl (2 sides) φ =0.75 C1 = 1.0 for E70 electrodes L = 8 in. D = 4 (sixteenths of an inch) K = 2 in. / 8 in. = 0.25 a = eccentricity, summed vectorally: horizontal component is 4 in.; vertical component is 2.67 in.; thus, al = 4.80 in. and a = 4.8 in. / 8 in. = 0.6 from the table. By interpolation, C = 1.73 φVn = 0.75(1.73)(1.0)(4)(8)(2) = 83.0 kip Weld from channel to plate has at least as much capacity, but less demand. §

Bearing of concrete at steel channel: fc = φ(0.85f'c) = 0.65(0.85)(5 ksi) = 2.76 ksi The C8 has the following properties: tw = 0.487 in. bf = 2.53 in. tf = 0.39 in. (average) The bearing will be controlled by bending in the web (because of the tapered flange, the critical flange thickness is greater than the web thickness). Conservatively ignoring the concrete’s resistance to vertical deformation of the flange, compute the width (b) of flange loaded at 2.76 ksi that develops the plastic moment in the web: Mp = φFytw2/4 = (0.9)(50 ksi)(0.4872 in2)/4 = 2.67 in-kip/in. Mu = fc[(b-tw)2/2 - (tw/2)2/2] = 2.76[(b - 0.243 in.)2 - (0.243 in.)2]/2 setting the two equal results in b = 1.65 inches. Therefore, bearing on the channel is:

φVc = fc(2 - tw)(l) = (2.76 ksi)[(2(1.65) - 0.487 in.](6 in.) = 46.6 kip To the bearing capacity on the channel is added the four #5 diagonal bars, which are effective in tension and compression; φ = 0.75 for shear is used here:

φVs = φfyAscosα = (0.75)(60 ksi)(4)(0.31 in2)(cos45°) = 39.5 kip Thus, the total capacity for transfer to concrete is:

φVn = φVc + φVs = 46.6 + 39.6 = 86.1 kip

8-41

FEMA P-751, NEHRP Recommended Provisions: Design Examples The capacity of the plate in the foundation is governed by the headed anchor studs. ACI 318 Appendix D has detailed information on calculating the strength of headed anchor studs. ACI 318 Section D3.3 has additional requirements for anchors resisting seismic forces in Seismic Design Categories C through F. Capacity in shear for anchors located far from an edge of concrete, such as these and with sufficient embedment to avoid the pryout failure mode is governed by the capacity of the steel, which is required by ACI 318 Section D3.3.4:

φVsa = φ n Ase futa = (0.65)(6 studs)(0.44 in2 per stud)(60 ksi) = 103 kip In summary, the various shear capacities of the connection are as follows: §

Shear in the two loose angles: 130.5 kip

§

Weld at toe of loose angles: 89.1 kip

§

Weld at face plates: 83.0 kip

§

Transfer to concrete: 86.1 kip

§

Headed anchor studs at foundation: 103 kip

The number of embedded plates (n) required for a panel is: n =402/83.0 = 4.8 Use five connection assemblies, equally spaced along each side (4'-0" on center works well to avoid the end reinforcement). The plates are recessed to position the #5 bars within the thickness of the panel and within the reinforcement of the panel. It is instructive to consider how much moment capacity is added by the resistance of these connections to vertical lift at the joint. The vertical force at the tip of the angle that will create the plastic moment in the leg of the angle is: T = Mp/x = Fylt2/4 / (l-k) = (36 ksi)(8 in.)(0.31252 in2)/4]/(4 in. - 0.69 in.) = 2.12 kips There are five assemblies with two loose angles each, giving a total vertical force of 21 kips. The moment resistance is this force times half the length of the panel, which yields 265 ft-kips. The total demand moment, for which the entire system is proportioned, is 5,320 ft-kips. Thus, these connections will add approximately 5 percent to the resistance and ignoring this contribution is reasonable. If a straight plate measuring 1/4 inch by 8 inches (which would be sufficient) were used and if the welds and foundation embedment did not fail first, the tensile capacity would be 72 kips each, a factor of 42 increase over the angles and the shear connections would have the unintended effect of more than doubling the flexural resistance, which would require a much higher shear force to develop a plastic hinge at the wall base. Using ACI 318 Section 11.10, check the shear strength of the precast panel at the first floor:

φVc = φ 2 Acv fc' hd = 0.75 ( 2 ) 5,000 (8)( 23.5 )(12 ) = 239 kips

8-42

Chapter 8: Precast Concrete Design Because φVc ≥ Vu = 190 kips, the wall is adequate for shear without even considering the reinforcement. Note that the shear strength of the wall itself is not governed by the overstrength required for the connection. However, since Vu ≥ 0.5φVc = 120 kips, ACI 318 Section 11.9.8 requires minimum wall reinforcement in accordance with ACI 318 Section 11.9.9 rather than Chapters 14 or 16. For the minimum required ρh = 0.0025, the required reinforcement is: Av = 0.0025(8)(12) = 0.24 in2/ft As before, use two layers of welded wire reinforcement, WWF 4×4 - W4.0×W4.0, one on each face. The shear reinforcement provided is: Av = 0.12(2) = 0.24 in2/ft Next, compute the required connection capacity at Level 2. Even though the end reinforcing at the base extends to the top of the shear wall, the connection still needs to be checked for flexure in accordance with Provisions Section 21.4.3 (ACI 318 Sec. 21.4.4). At Level 2: ME = (95 kips)(24 ft) + (63.5 kips)(12 ft) = 3,042 ft-kips There are two possible approaches to the design of the joint at Level 2. First, if Type 2 couplers are used at the Level 2 flexural connection, then the connection can be considered to have been “designed to yield,” and no overstrength is required for the design of the flexural connection. In this case, the bars are designed for the moment demand at the Level 2 joint. Alternately, if a non-yielding connection is used at the Level 2 connection, then to meet the requirements of Provisions Section 21.4.4 (ACI 318 Sec. 21.4.3), the flexural strength of the connection at Level 2 must be 1.5Sy or: Mu = 1.5(1.41)ME = 1.5(1.41)(3,042 ft-kips) = 6,433 ft-kips At Level 2, the gravity loads on the wall are: ∑D = wall + exterior floors/roof + lobby floors + penthouse floor + penthouse roof = (25 ft)(36 ft)(0.1 ksf) + (25 ft)(48 ft / 2)(0.070 ksf)(2) + (25 ft)(8 ft / 2)(0.070 ksf)(1) + (25 ft)(8 ft / 2)(0.18 ksf) + (25 ft )(24 ft / 2)(0.02 ksf) = 90 + 84 + 7 + 18 + 6 = 205 kips ∑L = (25 ft)(48 ft / 2)(0.05 ksf)(1) + (25 ft)(8 ft / 2)(0.1 ksf) = 30 + 10 = 40 kips ∑S = (25ft)(48 ft + 24 ft)(0.03 ksf)/2 = 27 kips Pmax = 1.26(205) + 0.5(40) + 0.2(27) = 285 kips Pmin = 0.843(205) = 173 kips Note that since the maximum axial load was used to determine the maximum yield strength of the base moment connection, the maximum axial load is used here to determine the nominal strength of the Level 2 connection. For completeness, the base moment overstrength provided should be checked using the minimum axial load as well and compared to the moment strength at Level 2 using the minimum axial load. 8-43

FEMA P-751, NEHRP Recommended Provisions: Design Examples

(

)

φ M n = 0.9 ⎡⎣ As f y jd + Pmax ( jd / 2) ⎤⎦ = 0.9 ⎡ 2.0 in 2 ( 60 ksi )( 23.5 ft ) + ( 285 kips ) (23.5 ft/2) ⎤ = 5,552 ft-kips

⎣

⎦

Therefore, the non-yielding flexural connection at Level 2 must be strengthened. Provide:

Tu =

M u Pmin 6, 433 285 = 131 kips − = − jd 2 23.5 2

The required reinforcement is: As = Tu/φfy = (131 kips)/[0.9(60 ksi)] = 2.43 in2 In addition to the two #9 bars that extend to the roof, provide one #6 bar developed into the wall panel above and below the joint. Note that no increase on the development length for the #6 bar is required for this connection since the connection itself has been designed for the loads to promote base yielding per Provisions Section 21.4.4 (ACI 318 Sec. 21.4.3). Since the Level 2 connection is prevented from yielding, shear friction can reasonably be used to resist shear sliding at this location. Also, because of the lack of flexural yield at the joint, it is not necessary to make the shear connection flexible with respect to vertical movement should an embedded plate detail be desired. The design shear for this location is: Vu,Level 2 = 1.5(1.41)(95+63.5) = 335 kips Using the same recessed embedded plate assemblies in the panel as at the base, but welded with a straight plate, the number of plates, n, is 335/83.0 = 4.04. Use four plates, equally spaced along each side. Figure 8.2-8 shows the shear connection at the second and third floors of the longitudinal precast concrete shear wall panels.

8-44

Chapter 8: Precast Concrete Design

See Figure 8.2-7 for embedded plates

Shim and drypack

Horizontal and vertical edges

Plate 516x5"x0'-8"

Figure 8.2-8 Shear connections on each side of the wall at the second and third floors (1.0 in = 25.4 mm)

8.3

ONE-STORY PRECAST SHEAR WALL BUILDING

This example illustrates the design of a precast concrete shear wall for a single-story building in a region of high seismicity. For buildings assigned to Seismic Design Category D, ACI 318 Section 21.10 requires that special structural walls constructed of precast concrete meet the requirements of ACI 318 Section 21.9, in addition to the requirements for intermediate precast structural walls. Alternately, special structural walls constructed using precast concrete are allowed if they satisfy the requirements of ACI ITG-5.1, Acceptance Criteria for Special Unbonded Post-Tensioned Precast Structural Walls Based on Validation Testing (ACI ITG 5.1-07). Design requirements for one such type of wall have been developed by ACI ITG 5 and have been published by ACI as Requirements for Design of a Special Unbonded Post-Tensioned Precast Shear Wall Satisfying ACI ITG-5.1 (ACI ITG 5.2-09). ITG 5.1 and ITG 5.2 describe requirements for precast walls for which a self-centering mechanism is provided by post-tensioning located concentrically within the wall. More general requirements for special precast walls are contained in Provisions Section 14.2.4. Section 14.2.4 is an updated version of Section 9.6 of the 2003 Provisions, which formed the basis for ITG 5.1 and ITG 5.2. 8.3.1

Building Description

The precast concrete building is a single-story industrial warehouse building (Occupancy Category II) located in the Los Angeles area on Site Class C soils. The structure has 8-foot-wide by 12.5-inch-deep

8-45

FEMA P-751, NEHRP Recommended Provisions: Design Examples prestressed double tee (DT) wall panels. The roof is light gage metal decking spanning to bar joists that are spaced at 4 feet on center to match the location of the DT legs. The center supports for the joists are joist girders spanning 40 feet to steel tube columns. The vertical seismic force-resisting system is the precast/prestressed DT wall panels located around the perimeter of the building. The average roof height is 20 feet and there is a 3-foot parapet. Figure 8.3-1 shows the plan of the building, which is regular.

12 DT at 8'-0" = 96'-0"

24LH03 at 4'-0" o.c.

48'-0"

15 DT at 8'-0" = 120'-0"

Joist girder (typical)

48'-0"

24LH03 at 4'-0" o.c.

Steel tube columns

3 DT at 8'-0" =

16'-0"

24'-0"

O.H. door

5 DT at 8'-0" = 40'-0"

16'-0"

3 DT at 8'-0" =

O.H. door

24'-0"

Figure 8.3-1 Single-story industrial warehouse building plan (1.0 ft = 0.3048 m) The precast wall panels used in this building are typical DT wall panels commonly found in many locations but not normally used in southern California. For these wall panels, an extra 1/2 inch has been added to the thickness of the deck (flange). This extra thickness is intended to reduce cracking of the flanges and provide cover for the bars used in the deck at the base. The use of thicker flanges is addressed later. The wall panels are normal-weight concrete with a 28-day compressive strength of f'c = 5,000 psi. Reinforcing bars used in the welded connections of the panels and footings are ASTM A706 (60 ksi). The concrete for the foundations has a 28-day compressive strength of f'c = 4,000 psi.

8-46

Chapter 8: Precast Concrete Design In Standard Table 12.2-1 the values for special reinforced concrete shear walls are for both cast-in-place and precast walls. In Section 2.2, ACI 318 defines a special structural wall as “a cast-in-place or precast wall complying with the requirements of 21.1.3 through 21.1.7, 21.9 and 21.10, as applicable, in addition to the requirements for ordinary reinforced concrete structural walls.” ACI 318 Section 21.10 defines requirements for special structural walls constructed using precast concrete, including that the wall must satisfy all of the requirements of ACI 318 Section 21.9. Unfortunately, several of the requirements of ACI 318 Section 21.9 are problematic for a shear wall system constructed using DT wall panels. These include the following: 1. ACI 318 Section 21.9.2.1 requires reinforcement to be spaced no more than 18 inches on center and be continuous. This would require splices to the foundation along the DT flange. 2. ACI 318 Section 21.9.2.2 requires two curtains of reinforcement for walls with shear stress greater than 2λ√f'c. For low loads, this might not be a problem, but for high shear stresses, placing two layers of reinforcing in a DT flange would be a challenge. 3. While ACI 318 Section 21.1.5.3 allows prestressing steel to be used in precast walls, ACI 318 Commentary R21.1.5 states that the “capability of a structural member to develop inelastic rotation capacity is a function of the length of the yield region along the axis of the member. In interpreting experimental results, the length of the yield region has been related to the relative magnitudes of nominal and yield moments.” Since prestressing steel does not have a defined yield plateau, the ratio of nominal to yield moment is undefined. This limits the ability of the structural member to develop inelastic rotation capacity—a key assumption in the definition of the R value for a special reinforced concrete wall system. Therefore, these walls will be designed using the ACI category of intermediate precast structural walls.

8-47

FEMA P-751, NEHRP Recommended Provisions: Design Examples 8.3.2

Design Requirements

8.3.2.1 Seismic Parameters of the Provisions. The basic parameters affecting the design and detailing of the building are shown in Table 8.3-1. Table 8.3-1 Design Parameters Design Parameter

Value

Occupancy Category II

I = 1.0

1.5

0.60

Site Class

1.0

1.3

SMS = FaSS

1.5

SM1 = FvS1

0.78

SDS = 2/3 SMS

1.0

SD1 = 2/3 SM1

0.52

Seismic Design Category

Basic Seismic Force-Resisting System

Bearing Wall System

Wall Type

Intermediate Precast Structural Wall

Ω0

2.5

8.3.2.2 Structural Design Considerations 8.3.2.2.1 Intermediate Precast Structural Walls Constructed Using Precast Concrete. The intent of the intermediate precast structural wall requirements is to provide yielding in a dry connection for bending at the base of each precast shear wall panel while maintaining significant shear resistance in the connection. The flexural connection for a wall panel at the base is located in one DT leg while the connection at the other leg is used for compression. Per ACI 318 Section 21.4, these connections must yield only in steel elements or reinforcement and all other elements of the connection (including shear resistance) must be designed for 1.5 times the force associated with the flexural yield strength of the connection. Yielding will develop in the dry connection at the base by bending in the horizontal leg of the steel angle welded between the embedded plates of the DT and footing. The horizontal leg of this angle is designed in a manner to resist the seismic tension of the shear wall due to overturning and then yield and deform inelastically. The connections on the two legs of the DT are each designed to resist 50 percent of the shear. The anchorage of the connection into the concrete is designed to satisfy the 1.5Sy requirements of ACI 318 Section 21.4.3. Careful attention to structural details of these connections is required to ensure

8-48

Chapter 8: Precast Concrete Design tension ductility and resistance to large shear forces that are applied to the embedded plates in the DT and footing. 8.3.2.2.2 Building System. The height limit in Seismic Design Category D (Standard Table 12.2-1) is 40 feet. The metal deck roof acts as a flexible horizontal diaphragm to distribute seismic inertia forces to the walls parallel to the earthquake motion (Standard Sec. 12.3.1.1). The building is regular both in plan and elevation. The redundancy factor, ρ, is determined in accordance with Standard Section 12.3.4.2. For this structure, which is regular and has more than two perimeter wall panels (bays) on each side in each direction, ρ = 1.0. The structural analysis to be used is the ELF procedure (Standard Sec. 12.8) as permitted by Standard Table 12.6-1. Orthogonal load combinations are not required for flexible diaphragms in Seismic Design Category D (Standard Sec. 12.5.4). This example does not include design of the foundation system, the metal deck diaphragm, or the nonstructural elements. Ties, continuity and anchorage (Standard 12.11) must be considered explicitly when detailing connections between the roof and the wall panels. This example does not include the design of those connections, but sketches of details are provided to guide the design engineer. There are no drift limits for single-story buildings as long as they are designed to accommodate predicted lateral displacements (Standard Table 12.12-1, Footnote c). 8 . 3 . 3 Load Combinations The basic load combinations (Standard Sec. 12.4.2.3) require that seismic forces and gravity loads be combined in accordance with the following factored load combinations: (1.2 + 0.2SDS)D ± ρQE + 0.5L+ 0.2S (0.9 - 0.2SDS)D ± ρQE + 1.6H At this flat site, both S and H equal 0. Note that roof live load need not be combined with seismic loads, so the live load term, L, can be omitted from the equation. Therefore: 1.4D + ρQE 0.7D - ρQE These load combinations are for the in-plane direction of the shear walls.

8-49

FEMA P-751, NEHRP Recommended Provisions: Design Examples 8.3.4

Seismic Force Analysis

8.3.4.1 Weight Calculations. Compute the weight tributary to the roof diaphragm: Roofing Metal decking Insulation Lights, mechanical, sprinkler system, etc. Bar joists Joist girder and columns Total

= 2.0 psf = 1.8 psf = 1.5 psf = 3.2 psf = 2.7 psf = 0.8 psf = 12.0 psf

The total weight of the roof is computed as: (120 ft × 96 ft)(12 psf / 1,000) = 138 kips The exterior DT wall weight tributary to the roof is: (20 ft / 2 + 3 ft)[42 psf / 1,000](120 ft + 96 ft)2 = 236 kips Total building weight for seismic lateral load, W = 138+236 = 374 kips 8.3.4.2 Base Shear. The seismic response coefficient (Cs) is computed using Standard Equation 12.8-2 as:

CS =

S DS 1.0 = 0.25 = R/ I 4/ I

except that it need not exceed the value from Standard Equation 12.8-3, as follows:

CS =

S D1 0.52 = 0.69 = T ( R / I ) 0.189(4 / 1)

where T is the fundamental period of the building computed using the approximate method of Standard Equation 12.8-7: 0.75

Ta = Cr hnx = (0.02) ( 20.0)

= 0.189 sec

Therefore, use Cs = 0.25, which is larger than the minimum specified in Standard Equation 12.8-5: Cs = 0.044(SDS)(I) ≥ 0.01 = 0.044(1.0)(1.0) = 0.044 The total seismic base shear is then calculated using Standard Equation 12.8-1, as: V = CsW = (0.25)(374) = 93.5 kips 8.3.4.3 Horizontal Shear Distribution and Torsion. Torsion is not considered in the shear distribution in buildings with flexible diaphragms. The shear along each side of the building will be equal, based on a tributary area force distribution.

8-50

Chapter 8: Precast Concrete Design 8.3.4.3.1 Longitudinal Direction. The total shear along each side of the building is V/2 = 46.75 kips. The maximum shear on longitudinal panels (at the side with the openings) is: Vlu = 46.75/11 = 4.25 kips On each side, each longitudinal wall panel resists the same shear force as shown in the free-body diagram of Figure 8.3-2, where D1 represents roof joist reactions and D2 is the panel weight.

3'-0"

8'-0"

Vlu

20'-0"

2'-0"

DT leg

Foundation

Vlu T

Figure 8.3-2 Free-body diagram of a panel in the longitudinal direction (1.0 ft = 0.3048 m) 8.3.4.3.2 Transverse Direction. Seismic forces on the transverse wall panels are all equal and are: Vtu = 46.75/12 = 3.90 kips Figure 8.3-3 shows the transverse wall panel free-body diagram.

8-51

FEMA P-751, NEHRP Recommended Provisions: Design Examples Note the assumption of uniform distribution to the wall panels in a line requires that the roof diaphragm be provided with a collector element along its edge. The chord designed for diaphragm action in the perpendicular direction will normally be capable of fulfilling this function, but an explicit check should be made in the design.

3'-0"

8'-0"

Vtu

20'-0"

2'-0"

DT leg

Foundation

Vtu T

Figure 8.3-3 Free-body diagram of a panel in the transverse direction (1.0 ft = 0.3048 m)

8.3.5

Proportioning and Detailing

The strength of members and components is determined using the strengths permitted and required in ACI 318 including Chapter 21. 8.3.5.1 Tension and Shear Forces at the Panel Base. Design each precast shear panel to resist the seismic overturning moment by means of a ductile tension connector at the base of the panel. A steel angle connector will be provided at the connection of each leg of the DT panel to the concrete footing. The horizontal leg of the angle is designed to yield in bending as needed in an earthquake. ACI 318 8-52

Chapter 8: Precast Concrete Design Section 21.4 requires that dry connections at locations of nonlinear action comply with applicable requirements of monolithic concrete construction and satisfy both of the following: 1. Where the moment action on the connection is assumed equal to 1.5My, the co-existing forces on all other components of the connection other than the yielding element shall not exceed their design strength. 2. The nominal shear strength for the connection shall not be less than the shear associated with the development of 1.5My at the connection. 8.3.5.1.1 Longitudinal Direction. Use the free-body diagram shown in Figure 8.3-2. The maximum tension for the connection at the base of the precast panel to the concrete footing is governed by the seismic overturning moment and the dead loads of the panel and the roof. The weight for the roof is 11.2 psf, which excludes the joist girders and columns. §

At the base: ME = (4.25 kips)(20 ft) = 85.0 ft-kips

§

Dead loads:

⎛ 48 ⎞ D1 = (11.2 1,000 ) ⎜ ⎟ 4 = 1.08 kips ⎝ 2 ⎠ D2 = 0.042(23)(8) = 7.73 kips ΣD = 2(1.08) + 7.73 = 9.89 kips 1.4D = 13.8 kips 0.7D = 6.92 kips Compute the tension force due to net overturning based on an effective moment arm, d, of 4.0 feet (the distance between the DT legs). The maximum is found when combined with 0.7D: Tu = ME/d - 0.7D/2 = 85.0/4 - 6.92/2 = 17.8 kips 8.3.5.1.2 Transverse Direction. For the transverse direction, use the free-body diagram of Figure 8.3-3. The maximum tension for connection at the base of the precast panel to the concrete footing is governed by the seismic overturning moment and the dead loads of just the panel. No load from the roof is included, since it is negligible. At the base: ME = (3.90 kips)(20 ft) = 78.0 ft-kips The dead load of the panel (as computed above) is D2 = 7.73 kips and 0.7D = 5.41. The tension force is computed as above for d = 4.0 feet (the distance between the DT legs):

8-53

FEMA P-751, NEHRP Recommended Provisions: Design Examples Tu = 78.0/4 - 5.41/2 = 16.8 kips This tension force is less than that at the longitudinal wall panels. Use the tension force of the longitudinal wall panels for the design of the angle connections. 8.3.5.2 Size the Yielding Angle. The angle, which is the ductile element of the connection, is welded between the plates embedded in the DT leg and the footing. This angle is an L5×3-1/2×3/4 × 0'-6-1/2" with the long leg vertical. The steel for the angle and embedded plates will be ASTM A572, Grade 50. The horizontal leg of the angle needs to be long enough to provide significant displacement at the roof, although this is not stated as a requirement in either the Provisions or ACI 318. This will be examined briefly here. The angle and its welds are shown in Figure 8.3-4.

y y z

61 2"

t L5x31 2x3 4x61 2 (LLV)

Tu '

Vu '

CG Mx

Fillet weld "t"

Vu'

Fillet weld

Tu'

Tu '

Vu'

2516" Tu'

k = 1316" Location of plastic hinge

Figure 8.3-4 Free-body of the angle and the fillet weld connecting the embedded plates in the DT and the footing (elevation and section) (1.0 in = 25.4 mm) The location of the plastic hinge in the angle is at the toe of the fillet (at a distance, k, from the heel of the angle.) The bending moment at this location is: Mu = Tu(3.5 - k) = 17.8(3.5 - 1.1875) = 41.2 in.-kips

⎡ 6.5 ( 0.75)2 ⎤ ⎥ = 41.1 in-kips φb M n = 0.9 Fy Z = 0.9 ( 50 ) ⎢ 4 ⎢⎣ ⎥⎦ Providing a stronger angle (e.g., a shorter horizontal leg) will simply increase the demands on the remainder of the assembly. Using ACI 318 Section 21.4.3, the tension force for the remainder of this

8-54

Chapter 8: Precast Concrete Design connection and the balance of the wall design are based upon a probable strength equal to 150 percent of the yield strength. Thus: 2 M n (1.5) (50 )( 6.5) (0.75) / 4 = ×1.5 = 27.0 kips 3.5 − k 0.9(3.5 − 1.1875)

Tpr =

The amplifier, required for the design of the balance of the connection, is:

Tpr Tu

27.0 = 1.52 17.8

The shear on the connection associated with this force in the angle is:

V pr = VE

Tpr

= 4.25 × 1.52 = 6.46 kips

Check the welds for the tension force of 27.0 kips and a shear force 6.46 kips. The Provisions Section 21.4.4 (ACI 318 21.4.3) requires that connections that are designed to yield be capable of maintaining 80 percent of their design strength at the deformation induced by the design displacement. For yielding of a flat bar (angle leg), this can be checked by calculating the ductility capacity of the bar and comparing it to Cd. Note that the element ductility demand (to be calculated below for the yielding angle) and the system ductility, Cd, are only equal if the balance of the system is rigid. This is a reasonable assumption for the intermediate precast structural wall system described in this example. The idealized yield deformation of the angle can be calculated as follows:

Py =

2 M n 50 ( 6.5 ) (0.75 ) / 4 = 19.8 kips = L 2.25

Δ y ,idealized =

Py L3 3EI

19.83(2.313 ) = 0.012 in. 3 ( 29,000 ) (6.5 × 0.753 / 12)

It is conservative to limit the maximum strain in the bar to εsh = 15εy. At this strain, a flat bar would be expected to retain all its strength and thus meet the requirement of maintaining 80 percent of its strength. Assuming a plastic hinge length equal to the section thickness:

φp =

15 ε y d /2

15 ( 50 / 29,000 ) 0.75 / 2

= 0.06897

Lp ⎞ ⎛ 0.75 ⎞ ⎛ Δ sh = φ p Lp ⎜ L − ⎟ + Δ y = 0.06897 ( 0.75) ⎜ 2.31 − ⎟ + 0.012 = 0.112 in. 2 2 ⎝ ⎠ ⎝ ⎠

8-55

FEMA P-751, NEHRP Recommended Provisions: Design Examples Since the ductility capacity at strain hardening is 0.112/0.012 = 9.3 is larger than Cd = 4 for this system, the requirement of Provision Section 21.4.4 (ACI 318 Sec. 21.4.3) is met. 8.3.5.3 Welds to Connection Angle. Welds will be fillet welds using E70 electrodes. §

For the base metal, φRn = φ(Fy)ABM. For which the limiting stress is φFy = 0.9(50) = 45.0 ksi.

§

For the weld metal, φRn = φ(Fy)Aw = 0.75(0.6)70(0.707)Aw. For which the limiting stress is 22.3 ksi.

Size a fillet weld, 6.5 inches long at the angle to the embedded plate in the footing. Using an elastic approach: Resultant force =

Vpr2 + Tpr2 = 6.462 + 27.02 = 27.8 kips

Aw = 27.8/22.3 = 1.24 in2 t = Aw/l =1.24 in2 / 6.5 in. = 0.19 in. For a 3/4 inch angle leg, use a 5/16 inch fillet weld. Given the importance of this weld, increasing the size to 3/8 inch would be a reasonable step. With ordinary quality control to avoid flaws, increasing the strength of this weld by such an amount should not have a detrimental effect elsewhere in the connection. Now size the weld to the plate in the DT. Continue to use the conservative elastic method to calculate weld stresses. Try a fillet weld 6.5 inches long across the top and 4 inches long on each vertical leg of the angle. Using the free-body diagram of Figure 8.3-4 for tension and Figure 8.3-5 for shear, the weld moments and stresses are: Mx = Tpr(3.5) = 27.0(3.5) = 94.5 in-kips My = Vpr(3.5) = (6.46)(3.5) = 22.6 in-kips Mz = Vpr(yb + 1.0) = 6.46(2.77 + 1.0) = 24.4 in-kips

8-56

Chapter 8: Precast Concrete Design

X Z

Figure 8.3-5 Free-body of angle with welds, top view, showing only shear forces and resisting moments For the weld between the angle and the embedded plate in the DT as shown in Figure 8.3-5, the section properties for a weld leg (t) are: A = 14.5t in2 Ix = 25.0t in4 Iy = 107.4t in4 Ip = Ix + Iy = 132.4t in4 yb = 2.90 in. xL = 3.25 in. To check the weld, stresses are computed at all four ends (and corners). The maximum stress is at the lower right end of the inverted “U” shown in Figure 8.3-4.

σx =

Vpr A

σy = −

Tpr A

M z yb 6.46 ( 24.4 ) (2.90) 0.98 = + = ksi Ip t 14.5t 132.4t +

M z xL 27.0 ( 24.4 ) (3.25) −1.26 =− + = ksi Ip t 14.5t 132.4t 8-57

FEMA P-751, NEHRP Recommended Provisions: Design Examples

σz = −

M y xL A

−

M z yb (22.6)(3.25) (94.5) (2.90) −11.8 =− − = ksi Ip t 107.4t 25.0t

σ R = σ x2 + σ y2 + σ z2 =

1 11.9 0.982 + 1.262 + 11.82 = ksi t t

Thus, t = 11.9/22.3 = 0.53 inch, which can be taken as 9/16 inch. Field welds are conservatively sized with the elastic method for simplicity and to minimize construction issues. 8.3.5.4 Panel Reinforcement. Check the maximum compressive stress in the DT leg. Note that for an intermediate precast structural wall, ACI 318 Section 21.9.6 does not apply and transverse boundary element reinforcing is not required. However, the cross section must be designed for the loads associated with 1.5 times the moment that yields the base connectors. Figure 8.3-6 shows the cross section used. The section is limited by the area of dry-pack under the DT at the footing.

10"

43 4" average

23 8"

4'-0"

23 8"

21 2"

Figure 8.3-6 Cross section of the DT dry-packed at the footing (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m) The reason to limit the area of dry-pack at the footing is to locate the boundary elements in the legs of the DT, at least at the bottom of the panel. The flange between the legs of the DT is not as susceptible to cracking during transportation as are the corners of DT flanges outside the confines of the legs. The compressive stress due to the overturning moment at the top of the footing and dead load is: A = 227 in2 S = 3240 in3

8-58

Chapter 8: Precast Concrete Design

σz =

P M E 13,800 1.52 (85,000 ×12 ) = 539 ksi + = + 227 3,240 A S

Roof live loads need not be included as a factored axial load in the compressive stress check, but the force from the prestress steel will be added to the compression stress above because the prestress force will be effective a few feet above the base and will add compression to the DT leg. Each leg of the DT will be reinforced with one 1/2-inch-diameter strand and one 3/8-inch-diameter strand. Figure 8.3-7 shows the location of these prestressed strands.

21 2"

Deck mesh

(1) 1 2" dia. strand (1) 3 8" dia. strand

Leg mesh

43 4" average

Figure 8.3-7 Cross section of one DT leg showing the location of the bonded prestressing tendons or strand (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m) Next, compute the compressive stress resulting from these strands. Note that the moment at the height of strand development above the footing, about 26 inches for the effective stress (fse), is less than at the top of footing. This reduces the compressive stress by:

( 4.25) (26) 3,240

×1,000 = 34 psi

In each leg, use: P = 0.58fpu Aps = 0.58(270 ksi)[0.153 + 0.085] = 37.3 kips A = 168 in2 e = yb - CGStrand = 9.48 - 8.57 = 0.91 in.

8-59

FEMA P-751, NEHRP Recommended Provisions: Design Examples Sb = 189 in3

σ=

P Pe 37,300 0.91(37,300 ) = 402 psi + = + A S 168 189

Therefore, the total compressive stress is approximately 539 + 402 - 34 = 907 psi. Since yielding is restricted to the steel angle and the DT is designed to be 1.5 times stronger than the yield force in the steel angle, the full strength of the strand can be used to resist axial forces in the DT stem, without concern for yielding in the strand. D2 = (0.042)(20.83)(8) = 7.0 kips Pmin = 0.7(7.0 + 2(1.08)) = 6.41 kips ME = (1.52)(4.25)(17.83) = 115.2 ft-kips Tu,stem = ME/d - Pmin/2= 25.5 kips The area of tension reinforcement required is: Aps = Tu,stem/φfpy = (25.5 kips)/[0.9(270 ksi)] = 0.10 in2 The area of one 1/2-inch-diameter strand and one 3/8-inch-diameter strand is 0.153 in2 + 0.085 in2 = 0.236 in2. The mesh in the legs is available for tension resistance but is not required in this check. To determine the nominal shear strength of the concrete for the connection design, complete the shear calculation for the panel in accordance with ACI 318 Section 11.9. The demand on each panel is: Vu = Vpr = 6.46 kips Only the deck between the DT legs is used to resist the in-plane shear (the legs act like flanges, meaning that the area effective for shear is the deck between the legs). First, determine the minimum required shear reinforcement based on ACI 318 Section 11.9.

φVc = φ 2λ fc' hd = 0.75 ( 2 )(1.0 ) 5,000 ( 2.5 )( 48 ) = 12.7 kips Since Vu of 6.46 kips exceeds φVu/2 of 6.36 kips, provide minimum reinforcement per ACI 318 Section 11.9.9.2. Using welded wire reinforcement, the required areas of reinforcement are: Av = Avh = (0.0025)(2.5)(12) = 0.075 in2/ft Provide 6×6 – W4.0×W4.0 welded wire reinforcement. Asv = Ash = 0.08 in2/ft The prestress force and the area of the DT legs are excluded from the calculation of the nominal shear strength of the DT wall panel. The prestress force is not effective at the base, where the connection is and the legs are like the flanges of a channel, which are not effective in shear.

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Chapter 8: Precast Concrete Design

8.3.5.5 Tension and Shear at the Footing Embedment. Reinforcement to anchor the embedded plates is sized for the same tension and shear. Reinforcement in the DT leg and in the footing will be welded to embedded plates as shown in Figure 8.3-8. The welded reinforcement is sloped to provide concrete cover and to embed the bars in the central region of the DT leg and footing. The tension reinforcement area required in the footing is:

As , Sloped =

Tu , stem

φ f y cosθ

27.0 = 0.56 in2 ( 0.9 )( 60 ) cos 26.5°

Use two #5 bars (As = 0.62 in2) at each embedded plate in the footing. The shear bars in the footing will be two #4 bars placed on an angle of two-to-one. The resultant shear resistance is:

φVn = 0.75(0.2)(2)(60)(cos26.5°) = 16.1 kips

(2) #5x48" (See Fig 8.3-9) (2) #5 with standard hooks

(2) #4x24" (see Fig 8.3-9) Plate 1

x 6 x 0'-10"

1 9

61 2

(2) #3 with standard hooks

L5x31 2x3 4x61 2 (LLV) 16

weld on #4 61 2

Plate 6x41 2x1 2 2'-6 "

(2) #4x

C.I.P. concrete footing

L6x4x1 2x10"

2'-6

Interior slab

(2) #5

Figure 8.3-8 Section at the connection of the precast/prestressed shear wall panel and the footing (1.0 in = 25.4 mm) 8-61

FEMA P-751, NEHRP Recommended Provisions: Design Examples

8.3.5.6 Tension and Shear at the DT Embedment. The area of reinforcement for the welded bars of the embedded plate in the DT, which develops tension as the angle bends through cycles, is:

As =

Tu , stem

φ f y cosθ

27.0 = 0.503 in2 ( 0.9 )( 60 ) cos 6.3°

Two #5 bars are adequate. Note that the bars in the DT leg are required to extend upward the development length of the bar, which would be 22 inches. In this case, they will be extended 22 inches past the point of development of the effective stress in the strand, which totals approximately 48 inches. The same embedded plate used for tension will also be used to resist one-half the nominal shear. This shear force is 6.46 kips. The transfer of direct shear to the concrete is easily accomplished with bearing on the sides of the reinforcing bars welded to the plate. Two #5 and two #4 bars (explained later) are welded to the plate. The available bearing area is approximately Abr = 4(0.5 in.)(5 in.[available]) = 10 in2 and the bearing capacity of the concrete is φVn = (0.65)(0.85)(5 ksi)(10 in2) = 27.6 kips, which is greater than the 6.46 kip demand. The weld of these bars to the plate must develop both the tensile demand and this shear force. The weld is a flare bevel weld, with an effective throat of 0.2 times the bar diameter along each side of the bar. (Refer to the PCI Handbook.) Using the weld capacity for the #5 bar:

φVn = (0.75)(0.6)(70 ksi)(0.2)(0.625 in.)(2) = 7.9 kips/in The shear demand is prorated among the four bars as (6.46 kip)/4 = 1.6 kips. The tension demand is Tu,stem/2(13.5 kips). The vectorial sum of shear and tension demand is 13.6 kips. Thus, the minimum length of weld is 13.6/7.9 = 1.7 inches. 8.3.5.7 Resolution of Eccentricities at the DT Embedment. Check the twisting of the embedded plate in the DT for Mz. Use Mz = 24.4 in-kips.

As =

Mz 24.4 = 0.05 in2 = φ f y ( jd ) 0.9 ( 60 ) (9.0)

Use one #4 bar on each side of the vertical embedded plate in the DT as shown in Figure 8.3-9. This is the same bar used to transfer direct shear in bearing. Check the DT embedded plate for My (equal to 22.6 in-kips) and Mx (equal to 94.5 in-kips) using the two #4 bars welded to the back side of the plate near the corners of the weld on the loose angle and the two #3 bars welded to the back side of the plate near the bottom of the DT leg (as shown in Figure 8.3-9). It is relatively straightforward to compute the resultant moment magnitude and direction, assume a triangular compression block in the concrete and then compute the resisting moment. It is quicker to make a reasonable assumption as to the bars that are effective and then compute resisting moments about the X and Y axes. That approximate method is demonstrated here. The #5 bars are effective in resisting Mx and one each of the #3 and #5 bars are effective in resisting My. For My assume that the effective depth extends 1 inch beyond the edge of the angle (equal to twice the thickness of the plate). Begin by assigning one-half of the “corner” #5 bar to each component.

8-62

Chapter 8: Precast Concrete Design With Asx = 0.31 + 0.31/2 = 0.47 in2:

φMnx = φAs fy jd = (0.9)(0.47 in2)(60 ksi)(0.95)(5 in.) = 120 in-kips (> 94.5 in-kips) With Asy = 0.11 + 0.31/2 = 0.27 in2:

φMny = φAs fy jd = (0.9)(0.27 in2)(60 ksi)(0.95)(5 in.) = 69 in-kips (> 22.6 in-kips) Each component is strong enough, so the proposed bars are satisfactory.

#4 (2) #5 with standard hook

Plate 10"x6"x1 2"

(2) #3 with standard hook 10"

Plate 41 2"x6"x1 2" with 5 8" slot at center

Figure 8.3-9 Details of the embedded plate in the DT at the base (1.0 in = 25.4 mm)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Plate at each DT leg Deck straps as needed Metal deck #4 continuous weld to plates

Bar joist

L4x3x1 4 continuous

2'-0"

Figure 8.3-10 Sketch of connection of non-load-bearing DT wall panel at the roof (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)

DT Plate at each DT leg

L4x3x1 4x continuous Metal deck

Bar joists

DT corbel at each leg

Figure 8.3-11 Sketch of connection of load-bearing DT wall panel at the roof (1.0 in = 25.4 mm) 8.3.5.8 Other Connections. This design assumes that there is no in-plane shear transmitted from panel to panel. Therefore, if connections are installed along the vertical joints between DT panels to control the

8-64

Chapter 8: Precast Concrete Design out-of-plane alignment, they should not constrain relative movement in-plane. In a practical sense, this means the chord for the roof diaphragm should not be a part of the panels. Figures 8.3-10 and 8.3-11 show the connections at the roof and DT wall panels. These connections are not designed here. Note that the continuous steel angle would be expected to undergo vertical deformations as the panels deform laterally. Because the diaphragm supports concrete walls out of their plane, Standard Section 12.11.2.1 requires specific force minimums for the connection and requires continuous ties across the diaphragm. Also, it specifically prohibits use of the metal deck as the ties in the direction perpendicular to the deck span. In that direction, the designer may wish to use the top chord of the bar joists, with an appropriate connection at the joist girder, as the continuous cross ties. In the direction parallel to the deck span, the deck may be used, but the laps should be detailed accordingly. In precast DT shear wall panels with flanges thicker than 2-1/2 inches, consideration may be given to using vertical connections between the wall panels to transfer vertical forces resulting from overturning moments and thereby reduce the overturning moment demand. These types of connections are not considered here, since the uplift force is small relative to the shear force and cyclic loading of bars in thin concrete flanges is not always reliable in earthquakes. 8.4

SPECIAL MOMENT FRAMES CONSTRUCTED USING PRECAST CONCRETE

As for special concrete walls, the Standard does not distinguish between a cast-in-place and a precast concrete special moment frame in Table 12.2-1. However, ACI 318 Section 21.8 provides requirements for special moment frames constructed using precast concrete. That section provides requirements for designing special precast concrete frame systems using either ductile connections (ACI 318 Sec. 21.8.2) or strong connections (ACI 318 Sec. 21.8.3.) ACI 318 Section 21.8.4 also explicitly allows precast moment frame systems that meet the requirements of ACI 374.1, Acceptance Criteria for Moment Frames based on Structural Testing. 8.4.1

Ductile Connections

For moment frames constructed using ductile connections, ACI 318 requires that plastic hinges be able to form in the connection region. All of the requirements for special moment frames must still be met, plus there is an increased factor that must be used in developing the shear demand at the joint. It is interesting to note that while Type 2 connectors can be used anywhere (including in a plastic hinge region) in a cast-in-place frame, these same connectors cannot be used closer than h/2 from the joint face in a ductile connection. The rationale behind this requirement is that in a jointed system, a concentrated crack occurs at the joint between precast elements in a ductile connection. Thus the rotation is concentrated at this location. Type 2 connectors are actually strong connections, relative to the bar, as they are designed to develop the tensile strength of the bar. The objective of Type 2 connectors is that they relocate the yielding away from the connector, into the bar itself. If a Type 2 connector is used at the face of a column as shown in Figure 8.4-1 and the bar size is the same in both the column and the beam, yielding will occur at the joint at the face of the column but not be able to spread into the beam to develop a plastic hinge, due to the strength of the connector. This concentrates the yielding in the bar to the left of the connector and likely will fracture the bar when significant rotation is imposed on the beam.

8-65

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Type 2 coupler

Precast column

Figure 8.4-1 Type 2 coupler location in a strong connection (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m) In a ductile connection, frame yielding takes place at the connection. This is most easily accomplished by extending the reinforcement out of the precast column element and coupling this rebar at the end of the precast beam. Since the couplers have to be located a minimum distance of h/2 from the joint face (i.e., column face) the resulting gap between the precast beam and precast column is filled with cast-in-place concrete as shown in Figure 8.4-2.

Type 2 coupler Precast column

Figure 8.4-2 Type 2 coupler location in a ductile connection (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)

8-66

Chapter 8: Precast Concrete Design 8.4.2

Strong Connections

ACI 318 also provides design rules for strong connections used in special moment frames. The concept is to provide connections that are strong enough to remain elastic when a plastic hinge forms in the beam. Thus the frame behavior is the same as would occur if the connection were monolithic. Using the frame in Figure 8.4-3 (ignoring gravity forces for simplicity), design forces for the plastic hinge region and the associated forces on the precast connection are computed. Assuming inflection points at mid-height of the columns and a seismic shear force of Vcol on each column:

Vb = Vcol

Hc Lb

Lclr (3) #10 t&b

(3) #14 hooks t&b H col

lcoupler

hbeam

Plastic hinge location Type 2 coupler

hcol

Figure 8.4-3 Moment frame geometry (1.0 in = 25.4 mm, 1.0 ft = 0.3048 m)

8-67

FEMA P-751, NEHRP Recommended Provisions: Design Examples Under seismic loads alone, the shear is constant along the beam length. Therefore, the moment at the joint between the end of the beam and the column is:

M joint = Vb

Lb − hcol 2

The plastic hinge, however, will be relocated to the side of the Type 2 coupler away from the column. With a coupler length of lcoupler, the moment at the end of the coupler is:

M b = M joint − Vblcoupler In order to ensure that the hinge forms at the intended location (away from the precast connection), the connection needs to be designed to be stronger than the moment associated with the development of the plastic hinge. This is done by upsizing the bar that is anchored into the column. 8.4.2.1 Strong Connection Example. In the following numerical example, a single-bay frame is designed to meet the requirements of a precast frame using strong connections at the beam-column interface. Using Figure 8.4-3 and the following geometry: Hcol = 12 ft hcol = 36 in. Lb = 30 ft (column centerline to column centerline) lcoupler = 18 in. Lclr = Lb - hcol - 2lcoupler = 24 ft (distance between plastic hinge locations) hbeam = 42 in. Reinforcing the beam with three #10 bars top and bottom, the nominal moment strength of the beam is:

As Fy 0.85 fc'b

3(1.27 ) (60)

⎛

0.85 (5) (18) a ⎞

= 3.0 in.

⎛

3.0 ⎞

/ 12 = 540 ft-kips φ M n = φ As Fy ⎜ d − ⎟ = 0.9 ( 3)(1.27 )( 60 ) ⎜ 33 − 2 ⎠ 2 ⎟⎠ ⎝ ⎝ This is the moment strength at the plastic hinge location. The strong precast connection must be designed for the loads that occur at the connection when the beam at the plastic hinge location develops its probable strength. Therefore, the moment strength at the beam-column interface (which also is the precast joint location) must be at least:

M u , joint = M pr

8-68

Lb − hcol Lclr

Chapter 8: Precast Concrete Design

Where:

1.25

M pr = φ M n

= 540

1.25 = 750 ft-kips 0.9

Therefore, the design strength of the connection must be at least:

30 − 36 / 12 = 843 ft-kips 24

M u , joint = 750

Using #14 bars in the column side of the Type 2 coupler:

As Fy 0.85 fc'b

3( 2.25) (60) 0.85 (5) (18)

= 5.3 in.

a ⎞

⎛

5.3 ⎞

/ 12 = 921 ft-kips φ M n = φ As Fy ⎜ d − ⎟ = 0.9 ( 3)( 2.25 )( 60 ) ⎜ 33 − 2 ⎠ 2 ⎟⎠ ⎝ ⎝ which is greater than the load at the connection (843 ft-kips) when the plastic hinge develops. If column-to-column connections are required, ACI 318 Section 21.8.3(d) requires a 1.4 amplification factor, in addition to loads associated with the development of the plastic hinge in the beam. Locating the column splice near the point of inflection, while difficult for construction, can help to make these forces manageable. The beam shear, when the plastic hinge location reaches its nominal strength, is:

Vbeam =

φMn

Lclr / 2

540 = 20 kips 24 / 2

Assuming inflection points at the mid-span of the beam and mid-height of the column, the column shear is:

Vcol = Vbeam

Lb 30 = 50 kips = 20 H col 12

However, the column shear must be amplified to account for the development of the plastic hinge.

Vu = Vcol

M pr

φMn

= 50

750 = 69 kips 540

The column design moment is:

M u = Vu

( H col − hbeam ) = 69 12 − 42 / 12 2

= 293 ft-kips

8-69

FEMA P-751, NEHRP Recommended Provisions: Design Examples At the connection, this moment is amplified by 1.4 for a strong connection design moment of 410 ft-kips. This moment must be combined with the axial load on the connection from both gravity loads and amplified seismic forces. The balance of the design is the same as for a cast-in-place special moment frame.

8-70

9 Composite Steel and Concrete Clinton O. Rex, P.E., PhD Originally developed by James Robert Harris, P.E., PhD and Frederick R. Rutz, P.E., PhD

Contents 9.1

BUILDING DESCRIPTION ......................................................................................................... 3

9.2

PARTIALLY RESTRAINED COMPOSITE CONNECTIONS .................................................. 7

9.2.1

Connection Details................................................................................................................. 7

9.2.2

Connection Moment-Rotation Curves ................................................................................. 10

9.2.3

Connection Design............................................................................................................... 13

9.3

LOADS AND LOAD COMBINATIONS ................................................................................... 17

9.3.1

Gravity Loads and Seismic Weight ..................................................................................... 17

9.3.2

Seismic Loads ...................................................................................................................... 18

9.3.3

Wind Loads .......................................................................................................................... 19

9.3.4

Notional Loads..................................................................................................................... 19

9.3.5

Load Combinations .............................................................................................................. 20

9.4

DESIGN OF C-PRMF SYSTEM ................................................................................................ 21

9.4.1

Preliminary Design .............................................................................................................. 21

9.4.2

Application of Loading ........................................................................................................ 22

9.4.3

Beam and Column Moment of Inertia ................................................................................. 23

9.4.4

Connection Behavior Modeling ........................................................................................... 24

9.4.5

Building Drift and P-delta Checks ....................................................................................... 24

9.4.6

Beam Design ........................................................................................................................ 26

9.4.7

Column Design .................................................................................................................... 27

9.4.8

Connection Design............................................................................................................... 28

9.4.9

Column Splices .................................................................................................................... 29

9.4.10

Column Base Design ........................................................................................................... 29

FEMA P-751, NEHRP Recommended Provisions: Design Examples The 2009 NEHRP Recommended Provisions for the design of a composite building using a “Composite Partially Restrained Moment Frame” (C-PRMF) as the lateral force-resisting system is illustrated in this chapter by means of an example design. The C-PRMF lateral force-resisting system is recognized in Standard Section 12.2 and in AISC 341 Part II Section 8; and it is an appropriate choice for buildings in low to moderate Seismic Design Categories (SDC A to D). There are other composite lateral forceresisting systems recognized by the Standard and AISC 341; however, the C-PRMF is the only one illustrated in this set of design examples. The design of a C-PRMF is different from the design of a more traditional steel moment frame in three important ways. First, the design of a Partially Restrained Composite Connection (PRCC) differs in that the connection itself is not designed to be stronger than the beam it is connecting. Consequently, the lateral system typically will hinge within the connections and not within the associated beams or columns. Second, because the connections are neither simple nor rigid, their stiffness must be accounted for in the frame analysis. Third, because the connections are weaker than fully restrained moment connections, the lateral force-resisting system requires more frames with more connections, resulting in a highly redundant system. In addition to the 2009 NEHRP Recommended Provisions (referred to herein as the Provisions), the following documents are referenced throughout the example: ACI 318

American Concrete Institute. 2008. Building Code Requirements for Structural Concrete.

AISC 341

American Institute of Steel Construction. 2005. Seismic Provisions for Structural Steel Buildings, including Supplement No. 1.

AISC 360

American Institute of Steel Construction. 2005. Specification for Structural Steel Buildings.

AISC Manual

American Institute of Steel Construction. 2005. Steel Construction Manual. Thirteenth Edition.

AISC SDGS-8

American Institute of Steel Construction. 1996. Partially Restrained Composite Connections, Steel Design Guide Series 8. Chicago: AISC.

AISC SDM

American Institute of Steel Construction. 2006. Seismic Design Manual. First Edition.

Arum (1996)

Mayangarum, Arum, 12-5-1996. Design, Analysis and Application of Bolted SemiRigid Connections for Moment Resisting Frames, MS Thesis, Lehigh University.

ASCE TC

American Society of Civil Engineers Task Committee on Design Criteria for Composite Structures in Steel and Concrete. October 1998. “Design Guide for Partially Restrained Composite Connections,” Journal of Structural Engineering 124(10).

RCSC

Research Council on Structural Connections. 2004. Specification for Structural Joints Using ASTM A325 or A490 Bolts.

Standard

American Society of Civil Engineers, 2005, ASCE/SEI 7-05 Minimum Design Loads for Buildings and other Structures

9-2

Chapter 9: Composite Steel and Concrete

Yura (2006)

Yura, Joseph A and Helwig, Todd A. (2-8-2006) Notes from SSRC/AISC Short Course 2 on “Beam Buckling and Bracing”The short-form designations presented above for each citation are used throughout.

The PRCC used in the example has been subjected to extensive laboratory testing, resulting in the recommendations of AISC SDGS-8 and ASCE TC. ASCE TC is the newest of the two guidance documents and is referenced here more often; however, AISC SDGS-8 provides information not in ASCE TC, which is still pertinent to the design of this type of frame. While both of these documents provide guidance for design of PRCC, the method presented in this design example deviates from that guidance based on more recent code requirements for stability and on years of experience in designing C-PRMF systems. The structure is analyzed using three-dimensional, static, nonlinear methods. The SAP 2000 analysis program, v. 14.0 (Computers and Structures, Inc., Berkeley, California) is used in the example. The symbols used in this chapter are from Chapter 2 of the Standard or the above referenced documents, or are as defined in the text. U.S. Customary units are used.

9.1 BUILDING DESCRIPTION The example building is a four-story steel framed medical office building located in Denver, Colorado (see Figures 9.1-1 through 9.1-3). The building is free of plan and vertical irregularities. Floor and roof slabs are 4.5-inch normal-weight reinforced concrete on 0.6-inch form deck (total slab depth of 4.5 inches.). Typically slabs are supported by open web steel joists which are supported by composite steel girders. Composite steel beams replace the joists at the spandrel locations to help control cladding deflections. The lateral load-resisting system is a C-PRMF in accordance with Standard Table 12.2-1 and AISC 341 Part II Section 8. The C-PRMF uses PRCCs at almost all beam-to-column connections. A conceptual detail of a PRCC is presented in Figure 9.1-4. The key advantage of this type of moment connection is that it requires no welding. The lack of field welding results in erection that is quicker and easier than that for more traditional moment connections with CJP welding and the associated inspections.

9-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples

12'-6"

25'-0"

E 25'-0"

20K5 at 3'-11 2" o.c. (typical)

12'-6"

indicates partially restrained moment resisting connection

Figure 9.1-1 Typical floor and roof plan

9-4

F 25'-0"

1'-0" edge of slab (typical)

~ ~

W21x44 (typical for N-S beams)

W18x35 (typical for E-W beams)

25'-0"

W ( 1 A typi 0x8 & ca 8 F l lin @ es )

25'-0"

Records storage assumed in hatched bays

25'-0"

12'-6"

B 25'-0"

(ty W pi 10 ca x7 lu 7 .n .o .)

Chapter 9: Composite Steel and Concrete

B 25'-0"

C 25'-0"

25'-0"

F 25'-0" Roof

4 at 13'-0" = 52'-0"

W18x35 (typical)

4 3 2

Typical North Elevation

Figure 9.1-2 Building end elevation

7 12'-6"

6 12'-6"

5 25'-0"

4 25'-0"

3 25'-0"

2 12'-6"

1 12'-6"

4 at 13'-0" = 52'-0"

Roof W21x44 (typical)

East and West Side Elevation

Figure 9.1-3 Building side elevation The building is located in a relatively low seismic hazard region, but localized internal storage loading and Site Class E are used in this example to provide somewhat higher seismic design forces for purposes of illustration and to push the example building into Seismic Design Category C.

9-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Concrete

Headed stud Rebar

Girder

Column

Double angle web connection Seat angle

Figure 9.1-4 Conceptual partially restrained composite connection (PRCC) There are no foundations designed in this example. For this location and system, the typical foundation would be a drilled pier and voided grade beam system, which would provide flexural restraint for the strong axis of the columns at their base (very similar to the foundation for a conventional steel moment frame). The main purpose here is to illustrate the procedures for the PRCCs. The floor and roof slabs serve as horizontal diaphragms distributing the seismic forces and by inspection they are stiff enough to be considered as rigid. The typical bay spacing is 25 feet. Architectural considerations allowed an extra column at the end bay of each side in the north-south direction, which is useful in what is the naturally weaker direction. The exterior frames in the north-south direction have moment-resisting connections at all columns. The frames in each bay in the east-west direction have moment-resisting connections at all columns except the end columns. Composite connections to the weak axis of the column are feasible, but they are not used for this design. The PRCC connection locations are illustrated in Figure 9.1-1. Material properties in this example are as follows:

9-6

§

Structural steel beams and columns (ASTM A992): Fy = 50 ksi

§

Structural steel connection angles and plates (ASTM A36): Fy = 36 ksi

§

Concrete slab (4.5 inches thick on form deck, normal weight): fc' = 3,000 psi

§

Steel reinforcing bars (ASTM A615): Fy = 60 ksi

Chapter 9: Composite Steel and Concrete

9.2 PARTIALLY RESTRAINED COMPOSITE CONNECTIONS 9.2.1

Connection Details

The type of PRCC used for this example building consists of a reinforced composite slab, a double-angle bolted web connection and a bolted seat angle. In real partially restrained building design, it is advantageous to select and design the complete PRCC simply based on beam depth and element capacities. Generally it is impractical to “tune” connections to beam plastic moment capacities and/or lateral load demands. This allows the designer to develop an in-house suite of PRCC details and associated behavior curves for each nominal beam depth ahead of time. Slight adjustments can be made later to account for real versus nominal beam depth. It is considered good practice (particularly for capacity-based seismic design) to provide substantial rotation capacity at connections while avoiding non-ductile failure modes. This requirement for ductile rotation capacity is expressed in AISC 341 Part II Section 8.4 as a requirement for story drift of 0.04 radians. Because much of the drift in a partially restrained building comes from connection rotation, this story drift requirement implies a connection rotation ductility requirement. In short, connections must be detailed to allow ductile modes to dominate over non-ductile failure modes. Practical detailing is limited by commonly available components. For instance, the largest angle leg commonly available is 8 inches, which can reasonably accommodate four 1-inch-diameter bolts. As a result, the maximum shear that can be delivered from the beam flange to the seat angle is limited by shear in four A490-X bolts. Bolt shear failure is generally considered to be non-ductile, so the rest of the connection design and detailing aims to maximize moment capacity of the connection while avoiding this limit state. The connection details chosen for this example are illustrated in Figures 9.2-1, 9.2-2, 9.2-3 and 9.2-4.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

41 2"

8 - #4 Long bars see 9.2-3 for layout

23 4"

W18x35

One row 3/4" dia. studs spaced @ 6"

11 2"

53 4"

Place transverse reinf. below top of studs, See 9.2-3 for number and layout

Centerline beam Centerline connection

11 2"

Standard double angle shear connection w/ A36 2L4x4x1/4 w/ 4- 3/4" dia. A325-N-SC bolts

A36 L8x6x1/2x0'-9" (SLV) w/4- 1" dia. A490-X-SC bolts in horiz. leg & 2- 1" dia. A490-X-SC bolts in vert. leg

2" 2" 23 4" 23 4"

OVS holes in column at seat angle

Figure 9.2-1 Typical interior W18x35 PRCC

Place transverse reinf. below top of studs See 9.2-4 for number and layout

8#5 long. bars see 9.2-4 for layout

Drill hole in flange for bars

gap

41 2"

1'-0"

23 4"

W21x44 5/8" dia. x 24" long DBA match number and spacing of transverse reinf.

11 2"

31 4"

One row 3/4" dia. studs spaced @ 6" [2 rows @6" req'd @ 12.5 foot bays]

Centerline beam Centerline connection

11 2"

Standard double angle shear connection w/ A36 2L4x4x1/4 w/ 5-3/4" dia. A325-N-SC bolts

A36 L8x6x5/8x0'-9" (SLV) w/4- 1" dia. A490-X-SC bolts in horiz. leg & 2- 1" dia. A490-X-SC bolts in vert. leg.

OVS holes in column at seat angle

Figure 9.2-2 Typical spandrel W21x44 PRCC

9-8

2" 2" 23 4" 23 4"

Chapter 9: Composite Steel and Concrete

1 row 3/4" dia. H.A.S. at 6" W10 columns

Edge of concrete slab W18 spandrel girder 2-#5x8'-0" w/ double nuts at column flange 2 rows 3/4" dia. H.A.S. at 6" (12'-6" spans only)

3-#5 'U' bars in slab W21 spandrel beam lap with straight bars

Figure 9.2-3 Typical corner PRCC

8-#4 transverse

4-#5 longitudinal

36"

12"

4-#4 cont. longitudinal lap w/ 4#5 @ end bay

4-#5 cont. longitudinal

60"

4-#4 longitudinal

60"

36"

60"

#4x4'-0" @ 24" service

36"

4-#5 transverse 60"

12"

36"

12"

36"

Figure 9.2-4 Typical PRCC reinforcing plan

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 9.2.2

Connection Moment-‐Rotation Curves

Two connection moment-rotation curves are required for the design of partially restrained buildings: the nominal moment-rotation curve and the modified moment-rotation curve. The nominal moment-rotation curve, obtained from connection test data or from published momentrotation prediction models, is used for service-level load design. For this example, the published moment-rotation prediction model given in ASCE TC is used to define the moment-rotation curve for the PRCC. Negative moment-rotation behavior (slab in tension):

(

)

M c− = C1 1 − e−C2θ + C3θ

(ASCE TC, Eq. 4)

Where: C1 = 0.18(4 × ArbFyrb + 0.857AsaFya)(d + Y3), kip-in. C2 = 0.775 C3 = 0.007(Asa + Awa)Fya (d + Y3), kip-in. θ = connection rotation (mrad = radians × 1,000) d = beam depth, in. Y3 = distance from top of beam to the centroid of the longitudinal slab reinforcement, in. Arb = area of longitudinal slab reinforcement, in2 Asa = gross area of seat angle leg, in2 (For use in these equations, Asa is limited to a maximum of 1.5Arb) Awa = gross area of double web angles for shear calculations, in2 (For use in these equations, Awa is limited to a maximum of 2.0Asa) Fyrb = yield stress of reinforcing, ksi Fya = yield stress of seat and web angles, ksi Positive moment-rotation behavior (slab in compression):

(

)

M c+ = C1 1 − e−C2θ + (C3 + C4 )θ

(ASCE TC, Eq. 3)

Where: C1 = 0.2400[(0.48Awa) + Asa](d + Y3)Fya, kip-in. C2 = 0.0210(d + Y3/2) C3 = 0.0100(AwL + AL)(d + Y3)Fya, kip-in. C4 = 0.0065 AwL(d + Y3)Fya, kip-in. The modified moment-rotation curve is used for strength level load design. The Direct Analysis Method requires two modifications to the nominal moment-rotation curve: an elastic stiffness reduction and a strength reduction. AISC 360 Section 7.3(3) requires an elastic stiffness reduction of 0.8, which is accomplished by translating the connection rotation by an elastic stiffness reduction offset. This translation can be shown as follows:

θ cDAM = θ c +

9-10

Mc 4 × K ci

Chapter 9: Composite Steel and Concrete

Where: Mc = connection moment from the nominal moment-rotation curve, kip-in. Kci = connection initial stiffness, kip-in./mrad; because the moment-rotation curve is nonlinear, it is necessary to define how the initial stiffness will be measured. For this example, the initial stiffness will be taken as the secant stiffness to the moment-rotation curve at θ = 2.5 mrad as suggested in ASCE TC. Note that this will be different values for the positive and negative moment-rotation portions of the connection behavior.

K ci =

M c @2.5 mrad 2.5 mrad

The second modification to the nominal moment-rotation curve is a strength reduction associated with φ. ASCE TC recommends using φ equal to 0.85. The associated connection strength is given by: McDAM = 0.85 Mc From these equations, curves for M-θ can be developed for a particular connection. The moment-rotation curves for the typical connections associated with the W18x35 girder and the W21x44 spandrel beam are presented in Figures 9.2-5 and 9.2-6, respectively.

Connection moment (kip-ft)

400

300 nominal

200

modified 100

0 -20

-15

-10

-5

0 -100

10 15 20 Connection rotation (mrad)

-200

-300

-400

Figure 9.2-5 Typical interior W18x35 PRCC M-θ curves

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

400

Connection moment (kip-ft)

nominal

300

modified

200

100 0 -20

-15

-10

-5

Connection rotation (mrad)

-100 -200

-300 -400

Figure 9.2-6 Typical spandrel W21x44 PRCC M-θ curves Important key values from the above connection curves are summarized in Table 9.2-1 for reference in later parts of the example design. Table 9.2-1 Key Connection Values From Moment-Rotation Curves W18x35 PRCC

W21x44 PRCC

Kci (kip-in/rad) (nominal)

704,497

1,115,253

Kci+ (kip-in/rad) (nominal)

338,910

554,498

232/206

367/326

151/127

240/202

Mc @ 20 mrad (kip-ft) (nominal/modified) Mc+ @ 10 mrad (kip-ft) (nominal/modified)

These curves and the corresponding equations do not reproduce the results of any single test. Rather, they are averages fitted to real test data using numerical methods and they smear out the slip of bolts into bearing. Articles in the AISC Engineering Journal (Vol. 24, No.2; Vol. 24, No.4; Vol. 27, No.1; Vol. 27, No. 2; and Vol. 31, No. 2) describe actual test results. Those tests demonstrate clearly the ability of the connection to satisfy the rotation requirements of AISC 341 Part II Section 8.4.

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Chapter 9: Composite Steel and Concrete 9.2.3

Connection Design

This section illustrates the detailed design decisions and checks associated with the typical W21x44 spandrel beam connection. A complete design would require similar checks for each different connection type in the building. Design typically involves iteration on some of the chosen details until all the design checks are within acceptable limits. 9.2.3.1 Longitudinal Reinforcing Steel. The primary negative moment resistance derives from tensile yielding of slab reinforcing steel. Since ductile response of the connection requires that the reinforcing steel yield and elongate prior to failure of other connection components, providing too much reinforcing is not a good thing. The following recommendations are from ASCE TC. A minimum of six bars (three bars each side of column), #6 or smaller, should be used (eight #5 bars have been used in this example). The bars should be distributed symmetrically within a total effective width of seven column flange widths (36 inches at each side of the column has been used in this example). For edge beams, the steel should be distributed as symmetrically as possible, with at least one-third (minimum three bars) of the total reinforcing on the exterior side of the column. Bars should extend a minimum of one-fourth of the beam length or 24 bar diameters past the assumed inflection point at each side of the column. For seismic design a minimum of 50 percent of the reinforcing steel should be detailed continuously. Continuous reinforcing should be spliced with a Class B tension lap splice and minimum cover should be in accordance with ACI 318. 9.2.3.2 Transverse Reinforcing Steel. The purpose of the transverse reinforcing steel is to help promote the force transfer from the tension reinforcing to the column and to prevent potential shear splitting of the slab over the beams, thus allowing the beam studs to transfer the reinforcing tension force into the beam. ASCE TC recommends the following. Provide transverse reinforcement, consistent with a strut-and-tie model as shown in Figure 9.2-7. In the limit (maximum), this amount will be equal to the longitudinal reinforcement. The transverse reinforcing should be placed below the top of the studs to prevent a cone-type failure over the studs. The transverse bars should extend at least 12 bar diameters or 12 inches, whichever is larger, on either side of the outside longitudinal bars.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

~ ~ ~

Column

~ ~ ~

~ ~ ~ Longitudinal Reinforcment

Transverse Reinforcement

Figure 9.2-7 Force transfer mechanism from slab to column Concrete bearing stresses on the column flange should be limited to 1.8f cʹ′ per the ASCE TC recommendations. For the W21x44 PRCC, the sum of the positive and negative moment capacity is 607 kip-ft. The moment arm is approximately 22.95 inches (20.7 + 4.5/2). So the maximum possible transfer of force from the slab to the column, if each connection is at maximum and opposite strengths on each side of the column, is 607 ft-kip/22.95 inches = 317 kips. A W10x88 column has a 10.3-inch-wide flange. Assuming uniform bearing of the concrete on each flange, the bearing stress would be 317 kips / 2 flanges / 4.5-inch-thick slab / 10.3-inch-wide flange = 3.42 ksi, which is less than the recommended limit of 1.8f cʹ′ . It is also necessary to check this force against the flange local bending and web local yielding limit states given in Chapter J of AISC 360. It is important to have concrete filling the gap between column flanges; otherwise, the force must be transferred by a single column flange. 9.2.3.3 Connection Moment Capacity Limits. AISC 341 Part II Section 8.4 requires that the PRCC have a nominal strength that is at least equal to 50 percent of the nominal Mp for the connected beam ignoring composite action. ASCE TC recommends 75 percent as a good target, with 50 percent as a lower limit and 100 percent as an upper limit. ASCE TC also recommends using the moment capacity at 20 mrad for negative moment and 10 mrad for positive moment to determine the nominal connection moment capacity. From the W21x44 PRCC connection curve, the negative moment capacity at 20 mrad is 367 kip-ft and the positive moment capacity at 10 mrad is 240 kip-ft. With Mp of the beam being 398 kip-ft, the ratio of connection-to-beam moment capacity is 0.922 and 0.603 for negative and positive moments, respectively. 9.2.3.4 Seat Angle. The typical gage for the bolts attaching the seat angle to the column is 5.5 inches to allow sufficient room for bolt tightening on the inside of the column. For a 1-inch bolt diameter and a 1.75-inch minimum edge distance to a sheared edge, the minimum angle length is 9 inches. Per ASCE TC, the minimum area of the outstanding angle leg should be: Asamin = 1.33 × Fyrd × Arb / Fya = 5.497 in2 A 5/8-inch thick angle with the 9-inch angle length results in Asa equal to 5.625 in2.

9-14

Chapter 9: Composite Steel and Concrete The outstanding angle dimension is controlled by the number of bolts attaching the angle to the beam flange. As previously discussed, a minimum 8-inch dimension is desired here to allow room for four 1-inch-diameter bolts. The vertical angle dimension has to be sufficient both to allow room for bolts to the column flange and to permit yielding when the seat angle is in tension. The ductility of the connection, when in positive bending, is derived from angle hinging, as shown in Figure 9.2-8.

Tsa Angle yield @ end of fillet

Bsa

Qsa

Angle yield @ face of bolt head

Figure 9.2-8 Typical angle tension hinging mechanism This mechanism is based on research by Arum (1996). The following equations can be used to determine the associated angle tension, prying forces and bolt forces associated with the angle hinging mechanism. a’ = Lvsa – gsa + dbsa / 2 = 2.500 in. c’ = (Wsa – dbsa) / 2 = 0.313 in. b’ = Lvsa – a’ – c’ – ksa = 2.062 in. Mpsa = Fya × tsa2 × Lsa / 4 = 31.641 kip-in Tsa = 2 × Mpsa / b’ = 30.682 kips Qsa = Mpsa / a’ × (1 + 2×c’ / b’) = 16.491 kips Bsa = Tsa + Qsa = 47.173 kips The above equations were derived in the same fashion as the prying action equations currently given in Section 9 of the AISC Manual with the same limitations applied to a’. The nomenclature in the above equations is shown in Figure 9.2-9.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Lvsa

dbsa

Wsa

gsa

tsa

ksa

Gap

Lvsa

Figure 9.2-9 Seat angle nomenclature The author recommends that the ratio of tsa/b’ be limited to no more than 0.5, so that the angle can properly develop the assumed hinges. For the example detail, the ratio is 0.303. 9.2.3.5 Bolts in Vertical Seat Angle Leg. The bolts in the vertical seat angle leg are designed primarily to resist tension in the case of connection positive moment. To protect against premature tension failure, the bolt force calculated in the previous section should be magnified by Ry from AISC 341 Table I-6-1. Ry × Bsa = 1.5 × 47.173 kips = 70.76 kips The tension capacity for two 1-inch-diameter A490 bolts is 133 kips. 9.2.3.6 Bolts in Outstanding Seat Angle Leg. The bolts in the outstanding leg of the seat angle must be designed for the shear transfer between the beam flange and the seat angle. For positive moments, this force is limited by tension hinging of the seat angle as calculated previously. For negative moments, this force is the sum of tension from the reinforcing steel and tension developed from hinging of the web angles. In general, the later will be significantly more than the former. The tension hinging capacity of the web angles, Twa, is calculated in the same way as the tension hinging of the seat angle. Again, to protect against premature shear failure of bolts, the tension capacity of the web angle and the reinforcing steel is magnified by an appropriate Ry. ASCE TC recommends Ry = 1.25 for the reinforcing steel. Ry × Twa + Ry × Fyrd × Arb = 1.5 × 22.5 kips + 1.25 × 60 ksi × 2.48 in2 = 220 kips The published shear capacity for four 1-inch-diameter A490-X bolts is 177 kips; however, this capacity includes a 0.8 reduction to account for joint lengths up to 50 inches per the RCSC. The RCSC further states that this reduction does not apply in cases where the distribution of force is essentially uniform along the joint. When one increases the published shear capacity by 1/0.8, the revised shear capacity is 221 kips. Bolt bearing at the beam flange and at the seat angle should also be checked. 9.2.3.7 Double Angle Web Connection. The primary purpose of the double angle web connection is to resist shear. Therefore, it can be selected directly from the AISC Manual; the specific design limits will not be addressed here. The required shear force is determined by adding the seismic demand to the 9-16

Chapter 9: Composite Steel and Concrete gravity demand. The seismic demand for the W21x44 PRCC is the sum of the positive and negative moment capacity (607 kip-ft) divided by the appropriate beam length. For the typical 25-foot beam length, the seismic shear is approximately 25 kips.

9.3 LOADS AND LOAD COMBINATIONS 9.3.1

Gravity Loads and Seismic Weight

The design gravity loads and the associated seismic weights for the example building are summarized in Table 9.3-1. The seismic weight of the storage live load is taken as 50 percent of the design gravity load (a minimum of 25 percent is required by Standard Section 12.7.2). To simplify this design example, the roof design is assumed to be the same as the floor design and floor loads are used rather than considering special roof and snow loads. Table 9.3-1 Gravity Load and Seismic Weight

Non-Composite Dead Loads (Dnc) 4.5-in. Slab on 0.6-in. Form Deck (4.5-in. total thickness) plus Concrete Ponding Joist and Beam Framing Columns Total:

Gravity Load

Seismic Weight

58 psf

6 psf 2 psf

66 psf

4 psf 6 psf 2 psf

12 psf

800 plf

70 psf 200 psf

10 psf 100 psf

Composite Dead Loads (Dc) Fire Insulation Mechanical and Electrical Ceiling Total: Precast Cladding System Live Loads (L) Typical Area Live and Partitions (Reducible) Records Storage Area Live (Non-Reducible)

The reason for categorizing dead loads as non-composite and composite is explained in Section 9.4.2. Live loads are applied to beams in the analytical model, with corresponding live load reductions appropriate for beam design. Column live loads are adjusted to account for different live load reduction factors, including the 20 percent reduction on storage loads for columns supporting two or more floors per Standard Section 4.8.2.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 9.3.2

Seismic Loads

The basic seismic design parameters are summarized in Table 9.3-2 Table 9.3-2 Seismic Design Parameters Parameter Ss S1 Site Class Fa Fv SMS = FaSs SM1 = FvS1 SDS = 2/3SMS SD1 = 2/3SM1 Occupancy Category Importance Factor Seismic Design Category (SDC) Frame Type per Standard Table 12.2-1 R Ω0 Cd

Value 0.20 0.06 E 2.5 3.5 0.50 0.21 0.33 0.14 II 1.0 C Composite Partially Restrained Moment Frame 6 3 5.5

For Seismic Design Category C, the height limit is 160 feet, so the selected system is permitted for this 52-foot-tall example building. The building is regular in both plan and elevation; consequently, the Equivalent Lateral Force Procedure of Section 12.8 is permitted in accordance with Standard Table 12.6-1. The seismic weight, W, totals 7,978 kips. The approximate period is determined to be 0.66 seconds using Equation 12.8-7 and the steel moment-resisting frame parameters of Table 12.8-2. The coefficient for upper limit on calculated period, Cu, from Table 12.8-1 is 1.62, resulting in Tmax of 1.07 seconds for purposes of determining strength-level seismic forces. A specific value for PRCC stiffness must be selected in order to conduct a dynamic analysis to determine the building period. It is recommended that the designer use Kci of the negative moment-rotation behavior given in Section 9.2.2 above for this analysis. This should result in the shortest possible analytical building period and thus the largest seismic design forces. For the example building, the computed periods of vibration in the first modes are 2.13 and 1.95 seconds in the north-south and east-west directions, respectively. These values exceed Tmax, so strength-level seismic forces must be computed using Tmax for the period. The seismic response coefficient is then given by:

Cs =

S D1 0.14 = = 0.022 ⎛ R ⎞ ⎛ 6 ⎞ T ⎜ ⎟ 1.07 ⎜ ⎟ ⎝ I ⎠ ⎝ 1.0 ⎠

The total seismic forces or base shear is then calculated as: 9-18

Chapter 9: Composite Steel and Concrete

V = Cs W = (0.022)(7,978) = 174 kips

(Standard Eq. 12.8-1)

The distribution of the base shear to each floor (by methods similar to those used elsewhere in this volume of design examples) is: Roof (Level 4): Story 4 (Level 3): Story 3 (Level 2): Story 2 (Level 1): Σ:

77 kips 53 kips 31 kips 13 kips 174 kips

For Seismic Design Category C, the value of ρ is permitted to be taken as 1.0 per Standard Section 12.3.4.1, so the above story shears are applied as Eh without any additional magnification. 9.3.3

Wind Loads

From calculations not illustrated here, the gross service-level wind force following ASCE 7 is 83 kips (assuming 90 mph, 3-second-gust wind speed). Including the directionality effect and the strength load factor, the design wind force is less than the design seismic base shear. The wind force is not distributed in the same fashion as the seismic force, thus the story shears and the overturning moments for wind are considerably less than for seismic. The distribution of the wind base shear to each floor is: Roof (Level 4): 13 kips Story 4 (Level 3): 25 kips Story 3 (Level 2): 23 kips Story 2 (Level 1): 22 kips Σ: 83 kips Because the wind loads are substantially below the seismic loads, they are not considered in subsequent strength design calculations; however, wind drift is considered in the design. 9.3.4

Notional Loads

AISC 360 now requires that notional loads be included in the building analysis. As shown later, the example building qualifies for application of notional loads to gravity-only load combinations. The notional load at level i is Ni = 0.002Yi, where Yi is the gravity load applied at level i. For our example building, these values are as follows: NDnc = 4,258 kips × 0.002 = 8.516 kips / 4 floors = 2.13 kips/floor NDc = 2,393 kips × 0.002 = 4.786 kips / 4 floors = 1.20 kips/floor NL = 4,469 kips × 0.002 = 8.938 kips / 4 floors = 2.23 kips/floor The notional loads are applied in the same manner as the seismic and wind loads in each orthogonal direction of the building and they are factored by the same load factors that are applied to their corresponding source (such as 1.2 or 1.4 for dead loads). It is important to note that, in general, notional loads should be determined, at a minimum, on a column-by-column basis rather than for an entire floor as done above. This will allow the design to capture the effect of gravity loads that are not symmetric about

9-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples the center of the building. The example building happens to have gravity loads that are concentric with the center of the building, so it does not matter in this case. 9.3.5

Load Combinations

Three load combinations (from Standard Section 2.3.2) are considered in this design example. §

Load Combination 2: 1.2D + 1.6L

§

Load Combination 5: 1.2D + 0.5L + 1.0E

§

Load Combination 7: 0.9D + 1.0E

Expanding the combinations for vertical and horizontal earthquake effects, breaking D into Dnc and Dc (defined in Section 9.3.1) and including notional loads, results in: §

Load Combination 2: 1.2(Dnc + NDnc) + 1.2(Dc + NDc) + 1.6(L + NL)

§

Load Combination 5: 1.2Dnc + 1.2Dc + 0.5L + 1.0Eh +1.0Ev Ev = 0.2SDS (Dnc + Dc) = 0.2(0.33)(Dnc + Dc) = 0.067(Dnc + Dc) 1.267Dnc + 1.267Dc + 0.5L + 1.0Eh

§

Load Combination 7: 0.9 Dnc + 0.9 Dc + 1.0 Eh -1.0 Ev 0.833 Dnc + 0.833 Dc + 1.0 Eh

Dnc has to be applied separately to the columns and beams because of the two-stage connection behavior (discussed later). Dncc is for column loading and Dncb is for beam loading. This breakout of the loading results in the following combinations: §

Stage 1 Analysis: Load Combinations 2 and 5: 1.2 Dncb Load Combination 7: 0.9 Dncb

§

Stage 2 Analysis: Load Combination 2: 1.2(Dncc + NDnc) + 1.2(Dc + NDc) + 1.6(L + NL) Load Combination 5: 1.2Dncc + 0.067Dncb + 1.267Dc + 0.5L + 1.0Eh Load Combination 7: 0.9Dncc - 0.067Dncb + 0.833Dc + 1.0Eh

The columns are designed from the Stage 2 Analysis and the beams are designed from the linear combination of the Stage 1 and Stage 2 Analyses. Because partially restrained connection behavior is nonlinear, seismic and wind drift analyses must be carried out for each complete load combination, rather than for horizontal loads by themselves. Note that Standard Section 12.8.6.2 allows drifts to be checked using seismic loads based on the analytical building period.

9-20

Chapter 9: Composite Steel and Concrete

§

Seismic Drift: 1.0Dncc + 0.067Dncb + 1.0Dc + 0.5L + 1.0Eh

§

Wind Drift: 1.0Dncc + 1.0Dc + 0.5L + 1.0W

The typical permeations of the above combinations have to be generated for each orthogonal direction of the building; however, orthogonal effects need not be considered for Seismic Design Category C provided the structure does not have a horizontal structural irregularity (Standard Sec. 12.5.3).

9.4 DESIGN OF C-‐PRMF SYSTEM 9.4.1

Preliminary Design

The goal of an efficient partially restrained building design is to have a sufficient number of beams, columns and connections participating in the lateral system so that the forces developed in any of these elements from lateral loads is relatively small compared to the gravity design. In other words, design for gravity as if the connections are pinned; add the connections and check to see if any beams or columns must be upsized to handle the lateral loads. The author cautions designers against trying to reduce beam sizes below the initial gravity sizes unless a full inelastic, path-dependent analysis accounting for potential shakedown of the connections is conducted. At this time, such an analysis typically is relegated to academic study and is not applied in real building design. The analysis methods described below do not go to that level of detail. Once the building has been designed for gravity, a preliminary lateral analysis can be made to assess whether the proposed steel framing sizes may be suitable for lateral loads in combination with gravity loads. Typically this is done assuming all the PRCCs are rigid connections. Two basic checks can be based on this preliminary analysis. First, review connection moments that come from the lateral load cases alone (earthquake moments and wind moments) without gravity. If these moments (at strength levels) exceed approximately 75 percent of the negative moment capacity of the PRCC then either additional beams, columns and connections need to be added to the lateral system or existing beams need to be upsized to provide larger PRCCs with higher capacities. Second, perform a preliminary assessment of the building drift. While there is no simple, reliable relationship between rigid frame drift and C-PRMF drift, the author typically assumes that the partially restrained system will drift approximately twice as much as a fully rigid analysis indicates. Keep in mind that these preliminary checks are made to establish basic system proportions before extensive modeling efforts are made to include the real partially restrained behavior of the building. Using this preliminary design method, initial floor framing was selected. In accordance with the ASCE TC, the beams are designed to be 100 percent composite; no partial composite design is used. The W18x35 typical interior girder is determined from a simple beam design. This typical size would work for all locations with the exception of the girders that support storage load on both sides (Grids 4 and 5 between Grids C and D). For simplicity, the example design was not further refined. The W18x35 size would also work as the Grid Line A and F spandrel beams; however, a W21x44 spandrel beam is used to help control drift in the north-south direction and help equalize the building periods in both directions. Note that the W21x44 improves drift more due to the increase in beam depth, which increases PRCC moment-rotation stiffness, rather than because of the increase in the moment of inertia of the steel beam section.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples 9.4.2

Application of Loading

PRCC do not develop substantial beam end restraint until after the concrete has hardened (since the reinforcing steel cannot be mobilized without the concrete). At the time of concrete casting, the bare steel elements of the connection are all that are present to resist rotation at the beam ends. The degree of restraint provided by the bare steel connection varies depending on the details; however, for purposes of design, the connection stiffness prior to concrete hardening typically is assumed to be zero (a pinned beam end). Consequently, the connection actually has two stages of behavior that need to be accounted for in the analysis. These two stages are the pre-composite stage, when the connection is assumed to behave as a pin and the post-composite stage, when the connection is assumed to have the full momentrotation behavior determined in Section 9.2.2. In a building where the complete lateral system is provided by PRCCs, temporary bracing may be required to provide lateral stability prior to concrete hardening. The above two-stage connection behavior requires separation of dead load into portions consistent with each stage. This is why the dead loads in Section 9.3.1 are separated into Dnc and Dc. The Dnc load is placed on the beams during the Stage 1 analysis (when the connections are pins) but is not placed on the beams (other than the seismic fraction) during the Stage 2 analysis (when the connections have PRCC stiffness). In Stage 2 analysis, the Dnc loads are placed directly on the columns so that their destabilizing effects are accounted for properly in the nonlinear P-delta analysis. That is why Dnc loads are further broken down into Dncc and Dncb. The Stage 2 load combinations are presented graphically in Figures 9.4-1 and 9.4-2.

1.2 Dncc

1.267 Dc

1.0 Eh

0.067 Dncb

Figure 9.4-1 Stage 2 Load Combination 5

9-22

0.5 L

Chapter 9: Composite Steel and Concrete

0.9 Dncc

0.833 Dc

1.0 Eh

0.067 Dncb

Figure 9.4-2 Stage 2 Load Combination 7

9.4.3

Beam and Column Moment of Inertia

ASCE TC recommends that the beam moment of inertia used for frame analysis be increased to account for the stiffening effect that the composite slab has on the beam moment of inertia. The use of the increased moment of inertia is also required by AISC 341 Part II Section 8.3. The following equivalent moment of inertia is recommended: Ieq = 0.6ILB+ + 0.4ILB-

(Eq. 5, ASCE TC)

ILB+ and ILB- are the lower bound moments of inertia in positive and negative bending, respectively. ILB+ can be determined from Table 3-20 in the AISC Manual as 1,594 in4 for the W18x35 interior girder and 1,570 in4 for the W21x44 spandrel beam once composite beam design values are known. Note that the W21x44 spandrel 100 percent composite design is limited by the effective slab capacity, which is why its composite moment of inertia is so close to that of the W18x35 interior girder. ILB- can be assumed as the bare steel moment of inertia, as 510 in4 for the W18x35 interior girder and 843 in4 for the W21x44 spandrel beam. It is permitted to account for the transformed area of the reinforcing steel in calculating ILB-, but the bare steel beam property has been used in this example. The equivalent moment of inertia is then calculated as: §

W18x35 Interior Girder: Ieq = 0.6(1,594) + 0.4(510) = 1,160 in4

§

W21x44 Spandrel Beam: Ieq = 0.6(1,570) + 0.4(843) = 1,279 in4

The bare steel moment of inertia values in the building analysis are revised to these values, which are suitable for service-level limit state checks. Use of a 0.8 reduction factor on the beam moment of inertia is required by AISC 360 Section 7.3(3) for strength-level checks from direct analysis. The bare steel moment of inertia for the columns is appropriate for service-level checks. For strengthlevel checks, the same 0.8 reduction factor on the moment of inertia used on beams would apply to the columns. A further reduction on the column moment of inertia for strength-level checks is required if Pr/Py exceeds 0.5. A quick scan of the column loads from the building analysis results indicates that the only columns that exceed this value are the first-story columns at Grids C-4, C-5, D-4 and D-5 for Load Combination 2 only. The adjustment factor is calculated to be: τb = 4[Pr/Py(1-Pr/Py)] = 4[612 kips/1130 kips (1- 612 kips/1130 kips)] = 0.99

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

In the author’s judgment, the above reduction on so few columns will have little or no effect on the building analysis results and it is ignored for this example. 9.4.4

Connection Behavior Modeling

For each connection type (such as W18 PRCC or W21 PRCC), there are four different connection behavior models used, as developed in Section 9.2.2. First, the connection is modeled as a linear spring with nominal stiffness Kci. This is done for the dynamic analysis of the building needed to determine the building period. Second, a service-level analysis is conducted using the full nonlinear nominal service moment-rotation behavior. Third, a connection Stage 1 building analysis is done with the connections having no moment resistance (analytical pins) so the beam pre-composite loads can be applied. Finally, a Stage 2 building strength analysis under factored loads is performed with the full modified nonlinear moment-rotation behavior. The multi-linear elastic link option provided in SAP2000 is used to model the connection springs for all stages. This nonlinear spring model allows user-defined behavior for two types of analysis, linear and nonlinear, for each spring type. This is helpful to handle the various connection behaviors because the dynamic analysis and the Stage 1 pre-composite beam load analysis can both be linear analysis which automatically switches the connection spring to the defined linear behavior. Another important point is that this particular spring model stays on the defined connection curve in a nonlinear-elastic manner. That is, the analysis simply rides up and down always converging at moment-rotation points on the connection backbone curve. This allows what is known as a path independent analysis; the order of the loading does not matter. This is in contrast with a spring model with different connection unloading behavior, such as might be used to model the full hysteric connection behavior. If the connection unloading behavior is considered, the analysis is no longer path independent because the answer will depend on the sequence of loads that are applied. This path-dependent analysis is more accurate and allows consideration of connection shakedown to be captured in the model; however, it is also much more complicated when compared to the path-independent analysis. Since the simpler, path-independent connection modeling approach does not capture connection shakedown behavior, the author does not recommend reducing beam sizes from the pure simple pinned gravity design discussed in Section 9.4.1. 9.4.5

Building Drift and P-‐delta Checks

Drifts should be checked using the service moment-rotation curves along with the full moment of inertias for the beams and columns (no 0.8 reduction). Because of the nonlinear connection behavior, the analysis is nonlinear. Though optional, the author recommends including P-delta effects in the service drift checks for partially restrained building designs. Drifts are computed for the nonlinear load combinations developed in Section 9.3.5. 9.4.5.1 Torsional Irregularity Check. Standard Table 12.3-1 defines torsional irregularities. The story drift values at the each end of the example building and their average story drift values including P-delta are presented in Table 9.4-1. Since the ratio of maximum drift to average drift does not exceed 1.2, no torsional irregularity exists, accidental torsion need not be amplified and drift may be checked at the center of the building (rather than at the corners).

9-24

Chapter 9: Composite Steel and Concrete Table 9.4-1 Torsional Irregularity and Seismic Drift Checks North-south Direction (in.) Story

Displacement

East-west Direction (in.)

Story Drift

Displacement

Story Drift

A-1

F-1

A-1

F-1

avg

max/avg

F-1

F-8

F-1

F-8

avg

max/avg

0.40

0.45

0.40

0.45

0.43

1.06

0.31

0.37

0.31

0.37

0.34

1.08

0.91

1.03

0.51

0.58

0.55

1.06

0.72

0.84

0.41

0.47

0.44

1.07

1.32

1.49

0.41

0.46

0.43

1.06

1.05

1.22

0.33

0.38

0.35

1.08

1.55

1.76

0.23

0.27

0.25

1.06

1.23

1.44

0.19

0.22

0.20

1.08

9.4.5.2 Seismic Drift and P-delta Effect. The allowable seismic story drift is taken from Standard Table 12.12-1 as 0.025hsx = (0.025)(13 ft × 12 in./ft) = 3.9 in. With Cd of 5.5 and I of 1.0, this corresponds to a story drift limit of 0.71 inch under the equivalent elastic forces (see Standard Section 12.8.6 for story drift determination). Review of the average drift values in Table 9.4-1 shows that all drifts are within the 0.71-inch limit. Table 9.4-2 P-delta Effect Checks

Story 1 2 3 4

North-south Direction (in.) Displacement Story Drift w/o w/ w/o w/ P-Δ 0.38 0.86 1.25 1.48

P-Δ 0.43 0.97 1.41 1.66

P-Δ 0.38 0.48 0.39 0.24

P-Δ 0.43 0.55 0.43 0.25

PΔamp 1.14 1.14 1.10 1.06

East-west Direction (in.) Displacement Story Drift w/o w/ w/o w/ θ 0.12 0.12 0.09 0.06

P-Δ 0.30 0.70 1.02 1.22

P-Δ 0.34 0.78 1.13 1.33

P-Δ 0.30 0.40 0.32 0.19

P-Δ 0.34 0.44 0.35 0.20

P-Δamp 1.12 1.12 1.09 1.05

θ 0.10 0.10 0.08 0.04

Separate analyses are conducted to determine seismic drifts with and without P-delta effects. Due to the nonlinear connection behavior, all of the analyses are nonlinear. The ratio of these two drifts (P-Δamp) is compared to the 1.5 limit for ratio of second-order drift to first-order drift set forth in AISC 360 Section 7.3(2). Because the ratios are all below the 1.5 limit, it is permissible to apply the notional loads as a minimum lateral load for the gravity-only combination and not in combination with other lateral loads. The results of these analyses are given in Table 9.4-2. Provisions Section 12.8.7 now defines the stability coefficient (θ) as follows:

θ=

Px ΔI Vx hsx Cd

The story drift (Δ) is defined in Standard Section 12.8.6 as:

Δ=

Cd δ xe I

9-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples Replacing Δ in the stability coefficient equation results in:

θ=

Pxδ xe Vx hsx

This value of θ can also be calculated from the P-delta amplifier presented in Table 9.4-2 by the following:

θ =1−

1 PΔ amp

The stability coefficients presented in Table 9.4-2 were calculated in this manner. Review of the values shows that θ varies from 0.04 to 0.12. Provisions Section 12.8.7 now requires that θ not exceed 0.10 unless the building satisfies certain criteria when subjected to either nonlinear static (pushover) analysis or nonlinear response history analysis. Because θ for the building in the north-south direction exceeds 0.10 in the lower stories, the designer would have to either increase the building stiffness in that direction or conduct an approved nonlinear analysis. Such nonlinear analysis is beyond the scope of this example. 9.4.5.3 Wind Drift. A wind drift limit of hsx/400 was chosen based on typical office practice for this type of building. This gives a story drift limit of 13 × 12 / 400 = 0.39 inch. The wind drift values presented in Table 9.4-3 were determined for the 50-year return interval wind loads previously determined in Section 9.3.3 above. Review of the drift values indicates that all drifts are within the 0.39-inch limit. Table 9.4-3 Wind Drift Results Story 1 2 3

North-south Direction (in.) Displacement Story Drift 0.19 0.19 0.39 0.20 0.52 0.13

9.4.6

0.57

0.05

East-west Direction (in.) Displacement Story Drift 0.15 0.15 0.32 0.17 0.42 0.11 0.47

0.04

Beam Design

AISC 341 Part II Section 8.3 requires that composite beams be designed in accordance with AISC 360 Chapter I. The beams are designed for 100 percent composite action and sufficient shear studs to develop 100 percent composite action are provided between the end and midspan. They do not develop 100 percent composite action between the column and the inflection point, but it may be easily demonstrated that they are more than capable of developing the full force in the reinforcing steel within that distance. Composite beam design is not unique to this example; however, composite beams acting as part of the lateral load-resisting system is unique and deserves further attention. As a result of connection restraint, negative moments will develop at beam ends. These moments must be considered when checking beam strength. The inflection point cannot be counted on as a brace point, so it may be necessary to consider the full beam length as unbraced for checking lateral-torsional buckling and comparing that capacity to the negative end moments. Note that there are Cb equations in the

9-26

Chapter 9: Composite Steel and Concrete literature that do a better job (as compared to the standard Cb equation in AISC 360) of predicting the lateral-torsional buckling strength of beams that are continuously attached to a composite slab floor system (Yura, 2006) AISC 341 Part II Section 8 does not specifically address compactness criteria for beams; however, given that the beams are not being required to develop Mp, other than possibly under gravity loads, it is unlikely they would need to be seismically compact. The author recommends that they meet the compactness criteria of AISC 360. A quick check in Table 1-1 of the AISC Manual indicates that both W18x35 and W21x44 are compact for flexure. 9.4.7

Column Design

Requirements for column design are found in AISC 341 and AISC 360. AISC 341 Part II Section 8.2 requires that columns meet the requirements of AISC 341 Part I Sections 6 and 8. W10 columns of A992 steel meet all Section 6 material requirements. AISC 341 Part I Section 8.3 requires a special load combination if Pu/φcPn exceeds 0.4 for a column in a seismic load combination. The only columns that exceed this limit are the interior columns on Grid Lines C and D under the storage load. Because they are so close to the center of the building, the seismic axial force in these columns is very small. Consequently, including the overstrength factor of 3.0 on the seismic axial portion of the column load will not have a meaningful effect on the column loads and can be ignored in this example. The nominal strength of the columns is determined using K = 1.0 in accordance with AISC 360 Section 7.1. The associated column strength unity checks are presented in Table 9.4-4. The unity checks presented are for the first story of the center four columns in the building. Table 9.4-4 Column Strength Check for W10x77 Seismic Load Combination Axial force, Pu 370 kip Moment, Mu 55 ft-kip Interaction 0.606

Gravity Load Combination 612 kip 35 ft-kip 0.866

Part II Section 8 does not specifically address the required compactness criteria; however, given the high R value for the lateral load-resisting system, the author has assumed that the columns would need to meet the seismically compact criteria given in Part I Table I-9-1. A W10x77 column from the lower level of an interior bay with storage load is illustrated (the axial load from the seismic load combination is used): Column Flange:

§

λ ps = 0.3 bf 2t f

E 29,000 = 7.22 = 0.3 Fy 50

= 5.86 (AISC Manual) < 7.22

9-27

FEMA P-751, NEHRP Recommended Provisions: Design Examples §

Column Web:

Ca =

Pu 370 kips = 0.364 > 0.125 = φb Py 0.9 (50 ) 22.6

λ ps = 1.12

E E ( 2.33 − Ca ) ≥ 1.49 Fy Fy

λ ps = 1.12

29,000 29,000 = 35.88 ( 2.33 − 0.364 ) = 53.03 ≥ 1.49 50 50

h = 14.8 (AISC Manual) < 53.03 tw As an alternative to calculating the compactness criteria by hand, the designer can use the AISC SDM Table 1-2. A quick review of this table indicates that the W10x77 is compact for flexure (beam) and for axial loads (column). The dash in the table indicates that applied axial loads as large as Py still result in the column meeting the seismically compact criteria. The equivalent of the weak-beam–strong-column concept for the C-PRMF lateral system is a weak connection-strong column. This is not specifically addressed in AISC 341; however, ASCE TC recommends the following check:

⎛

P ⎞

⎝

∑M p,col ⎜⎜1 − Pu ⎟⎟ ≥ 1.25 ( M cu− + M cu+ ) ⎠

For the same lower level interior W10x77 one gets:

⎛ 370 kips ⎞ 2 × 50 × 97.6 ⎜1 − ⎟ = 547 ft-kips ≥ 1.25 ( 232 + 151) = 479 ft-kips ⎝ 50 ×22.6 ⎠ 9.4.8

Connection Design

There is really little to do with the connection design at this stage because the full nonlinear connection behavior is being used in the analysis. This means that the connection moments will never exceed the connection capacity during the analysis. This is in contrast to any analysis method that models the connections with linear behavior. When the connections are modeled with linear behavior, it is up to the designer to confirm that the final connection results are consistent with the expected connection behavior. This might be very easy for building designs where connection moments are small; however, when the connections are being pushed close to their capacity, that sort of independent connection check by the designer can be problematic. Although not entirely necessary, it is useful to check where the connections are along the expected behavior curves for any given analysis so one can see just how hard the connections are being pushed. The connection moment demand versus design capacities (including φ) are presented in Table 9.4-5. The demand values are from different load combinations. A quick check of this table indicates that this

9-28

Chapter 9: Composite Steel and Concrete building design is not being pushed particularly hard and that there is likely significant reserve capacity in the lateral system. Table 9.4-5 Connection Moment Demand vs. Capacity (kip-ft) W21 PRCC Demand Capacity Ratio

9.4.9

(-) M-θ 136 312 0.44

W18 PRCC (+) M-θ 87.0 204 0.43

(-) M-θ 126 197 0.64

(+) M-θ 37.0 128 0.29

Column Splices

Column splice design would be in accordance with AISC 341 Part I Section 8.4 but is not illustrated in this example. 9.4.10 Column Base Design Column base design would be in accordance with AISC 341 Part I Section 8.5 but is not illustrated in this example.

9-29

10 Masonry James Robert Harris, PE, PhD and Frederick R. Rutz, PE, PhD

Contents 10.1

WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES, CALIFORNIA ............................................................................................................................... 3

10.1.1

Building Description .............................................................................................................. 3

10.1.2

Design Requirements ............................................................................................................. 4

10.1.3

Load Combinations ................................................................................................................ 6

10.1.4

Seismic Forces ....................................................................................................................... 8

10.1.5

Side Walls .............................................................................................................................. 9

10.1.6

End Walls............................................................................................................................. 25

10.1.7

In-Plane Deflection – End Walls ......................................................................................... 44

10.1.8

Bond Beam – Side Walls (and End Walls) .......................................................................... 45

10.2

FIVE-STORY MASONRY RESIDENTIAL BUILDINGS IN BIRMINGHAM, ALABAMA; ALBUQUERQUE, NEW MEXICO; AND SAN RAFAEL, CALIFORNIA ............................. 45

10.2.1

Building Description ............................................................................................................ 45

10.2.2

Design Requirements .......................................................................................................... 48

10.2.3

Load Combinations .............................................................................................................. 50

10.2.4

Seismic Design for Birmingham 1 ...................................................................................... 51

10.2.5

Seismic Design for Albuquerque ......................................................................................... 69

10.2.6

Birmingham 2 Seismic Design ............................................................................................ 81

10.2.7

Seismic Design for San Rafael ............................................................................................ 89

10.2.8

Summary of Wall D Design for All Four Locations ......................................................... 101

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples This chapter illustrates application of the 2009 NEHRP Recommended Provisions (the Provisions) to the design of a variety of reinforced masonry structures in regions with different levels of seismicity. Example 10.1 features a single-story masonry warehouse building with tall, slender walls, and Example 10.2 presents a five-story masonry hotel building with a bearing wall system designed in areas with different seismicities. Selected portions of each building are designed to demonstrate specific aspects of the design provisions. Masonry is a discontinuous and heterogeneous material. The design philosophy of reinforced grouted masonry approaches that of reinforced concrete; however, there are significant differences between masonry and concrete in terms of restrictions on the placement of reinforcement and the effects of the joints. These physical differences create significant differences in the design criteria. All structures were analyzed using two-dimensional (2D) static methods using the RISA 2D program, V.5.5 (Risa Technologies, Foothill Ranch, California). Example 10.2 also uses the SAP 2000 program, V6.11 (Computers and Structures, Berkeley, California) for dynamic analyses to determine the structural periods. All examples are for buildings of concrete masonry units (CMU); neither prestressed masonry shear walls nor autoclaved aerated concrete masonry shear walls are included. In addition to the Provisions and the Standard, the following documents are referenced in this chapter: ACI 318

American Concrete Institute. 2008. Building Code Requirements for Structural Concrete.

TMS 402

The Masonry Society. 2008. Building Code Requirements for Masonry Structures, TMS 402/ACI 530/ASCE 5.

IBC

International Code Council. 2009. International Building Code.

NCMA

National Concrete Masonry Association. A Manual of Facts on Concrete Masonry, NCMA-TEK is an information series from the National Concrete Masonry Association, various dates. NCMA-TEK 14-1BA, Section Properties of Concrete Masonry Walls and NCMA-TEK 14-11B, Strength Design of Concrete Masonry Walls for Axial Load & Flexure, are referenced here.

USGS

United States Geological Survey. Seismic Design Maps web application.

The short form designations for each citation are used throughout. The citation to the IBC is because one of the designs employees a tall, slender wall that is partially governed by wind loads and the IBC provisions are used for that design. Regarding TMS 402: §

The 2005 edition of the Standard, in its Supplement 1, refers to the 2005 edition of TMS 402.

§

The 2010 edition of the Standard refers to the 2008 edition of TMS 402.

§

The examples herein are prepared according to the 2008 edition of TMS 402.

10-2

Chapter 10: Masonry

10.1 WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES, CALIFORNIA This example features a one-story building with reinforced masonry bearing walls and shear walls.

10.1.1 Building Description This simple rectangular warehouse is 100 feet by 200 feet in plan (Figure 10.1-1). The masonry walls are 30 feet high on all sides, with the upper 2 feet being a parapet. The wood roof structure slopes slightly higher towards the center of the building for drainage. The walls are 8 inches thick on the long side of the building, for which the slender wall design method is adopted, and 12 inches thick on both ends. The masonry is grouted in the cells containing reinforcement, but it is not grouted solid. The specified strength of masonry is 2,000 psi. Normal-weight CMU with Type S mortar are assumed.

5 bays at 40'-0" = 200'-0"

8" concrete masonry wall

12" concrete masory wall Open

5 bays at 20'-0" = 100'-0"

Typical glue-lam roof beam

Plywood roof sheathing

Figure 10.1-1 Roof plan (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) The long side walls are solid (no openings). The end walls are penetrated by several large doors, which results in more highly stressed piers between the doors (Figure 9.1-2); thus, the greater thickness for the end walls.

10-3

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

16'-0"

12" masonry Door opening (typical)

12'-0"

30'-0"

2'-0"

Roof line

4'-0"

12'-0"

Continuous stem wall and footing

8'-0" 100'-0"

Figure 10.1-2 End wall elevation (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) The floor is concrete slab-on-grade construction. Conventional spread footings are used to support the interior steel columns. The soil at the site is a dense, gravelly sand. The roof structure is wood and acts as a diaphragm to carry lateral loads in its plane from and to the exterior walls. The roofing is ballasted, yielding a total roof dead load of 20 psf. There are no interior walls for seismic resistance. This design results in a highly stressed diaphragm with large calculated deflections. The design of the wood roof diaphragm and the masonry wall-to-diaphragm connections is illustrated in Sec. 11.2. In this example, the following aspects of the structural design are considered: §

Design of reinforced masonry walls for seismic loads

§

Computation of P-delta effects.

10.1.2 Design Requirements This building could qualify for the simplified approach in Standard Section 12.14, although the “long” method per Standard Section 11.4-11.6 has been followed. 10.1.2.1 Seismic parameters. The ground motion response coefficients are found from USGS based upon latitude and longitude. The site class is taken from a site-specific geotechnical report and is typical for dense sands and gravels. The warehouse is not designated for hazardous materials and does not house any essential facility, thus the occupancy category is “all other”. Site Class = C SS = 2.14

10-4

Chapter 10: Masonry S1 = 0.74 Occupancy Category (Standard Sec. 1.5.1) = II The remaining basic parameters depend on the ground motion adjusted for site conditions. 10.1.2.2 Response parameter determination. The mapped spectral response factors must be adjusted for site class in accordance with Standard Section 11.4.3. The adjusted spectral response acceleration parameters are computed according to Standard Equations 11.4-1 and 11.4-2 for the short period and onesecond period, respectively, as follows: SMS = FaSS = 1.0(2.14) = 2.14 SM1 = FvS1 = 1.3(0.74) = 0.96 Where Fa and Fv are site coefficients defined in Standard Tables 11.4-1 and 11.4-2, respectively. The design spectral response acceleration parameters (Standard Sec. 11.4.4) are determined in accordance with Standard Equations 11.4-3 and 11.4-4 for the short-period and one-second period, respectively:

2 2 S DS = S MS = (2.14) = 1.43 3 3 2 2 S D1 = S M1 = (0.96) = 0.64 3 3 The Seismic Design Category may be determined by the design spectral acceleration parameters combined with the Occupancy Category. For buildings assigned to Seismic Design Category D, masonry shear walls must satisfy the requirements for special reinforced masonry shear walls in accordance with Standard Section 12.2. A summary of the seismic design parameters follows: §

Seismic Design Category (Standard Sec. 11.6): D

§

Seismic Force-Resisting System (Standard Table 12.2-1): Special Reinforced Masonry Shear Wall

§

Response Modification Factor, R: 5

§

Deflection Amplification Factor, Cd: 3.5

§

System Overstrength Factor, Ω0: 2.5

§

Redundancy Factor, ρ (Standard Sec. 12.3.4.2): 1.0

(Determination of ρ is discussed in Section 10.1.3 below.) 10.1.2.3 Structural design considerations. With respect to the lateral load path, the roof diaphragm supports approximately the upper 16 feet of the masonry walls (half the clear span plus the parapet) in the out-of-plane direction, transferring the lateral force to in-plane masonry shear walls. This is more precisely calculated in Section 10.1.4.1.

10-5

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Soil structure interaction is not considered. The building is of bearing wall construction. Other than the opening in the roof, the building is symmetric about both principal axes, and the vertical elements of the seismic force-resisting system are arrayed entirely at the perimeter. The opening is not large enough to be considered an irregularity (per Standard Table 12.3-1); thus, the building is regular, both horizontally and vertically. Standard Table 12.6-1 permits several analytical procedures to be used; the equivalent lateral force (ELF) procedure (Standard Sec. 12.8) is selected for use in this example. The direction of loading requirements of Standard Section 12.5 are for walls that act in both principal directions, which is not the case for this structure, as will be discussed in more detail. There is no inherent torsion because the building is symmetric. The effects of accidental torsion and its potential amplification, need not be included because the roof diaphragm is flexible (Standard Sec. 12.8.4.2). The masonry bearing walls also must be designed for forces perpendicular to their plane (Standard Sec. 12.11.1). For in-plane loading, the walls are treated as cantilevered shear walls. For out-of-plane loading, the walls are treated as simply supported at top and bottom. The assumption of a pinned connection at the base is deemed appropriate because the foundation is shallow and narrow, which permits rotation near the base of the wall. 10.1.3 Load Combinations The basic load combinations are the same as specified in Standard Section 2.3.2. The seismic load effect, E, is defined by Standard Equations 12.4-1, 12.4-3 and 12.4-4, as follows: E = Eh + Ev = ρQE ± 0.2SDSD = (1.0)QE ± 0.2(1.43)D = QE ± 0.286D This assumes ρ = 1.0 as will be confirmed in the following section. 10.1.3.1 Redundancy Factor. In accordance with Standard Section 12.3.4.2, the redundancy factor, ρ, applies to the in-plane load direction. In order to achieve ρ = 1.0, the two conditions in Standard Section 12.3.4.2 must be met. In the long direction there are no walls with height-to length ratios exceeding 1.0; thus ρ = 1.0 in the long direction. In the short direction the pier heights do exceed the length; thus their conditions must be checked. For our case, both are met. Although the calculation is not shown here, note that a single 8-foot-long pier carries approximately 23 percent (determined by considering the relative rigidities of the piers) of the in-plane load for each end wall. Thus, failure of a single pier results in less than 33 percent reduction in base shear resistance. Loss of a single pier will not result in extreme torsional irregularity because the diaphragm is flexible. Even if the diaphragm were rigid, an extreme torsional irregularity would not be created. The lateral deflection of end wall with all piers in place is approximately 0.018 inch (determined by RISA analysis). Lateral deflection of end wall with one pier removed is 0.024 inch. The larger deflection divided by the average of both deflections is less than 1.4: 10-6

Chapter 10: Masonry

0.024 = 1.14 < 1.4 ⎛ 0.018 + 0.024 ⎞ ⎜ ⎟ 2 ⎝ ⎠ Therefore, even if the diaphragm were rigid, there is no extreme torsional irregularity as per Standard Table 12.3-1. 10.1.3.2 Combination of load effects. Load combinations for the in-plane loading direction from Standard Section 2.3.2 are: 1.2D + 1.0E + 0.5L + 0.2S and 0.9D + 1.0E + 1.6H L, S and H do not apply for this example (roof live load, Lr, is not floor live load, L) so the load combinations become: 1.2D + 1.0E and 0.9D + 1.0E For this case, E = Eh ± Ev = ρQE ± 0.2 SDSD = (1.0)QE ± (0.2)(1.43)D = QE ± 0.286D Where the effect of the earthquake determined above, 1.2D + 1.0(QE ± 0.2D), is inserted in each of the load combinations, the controlling cases are 1.486D + QE when gravity and seismic are additive and 0.614D - QE when gravity and seismic counteract. These load combinations are for the in-plane direction of loading. Load combinations for the out-of-plane direction of loading are similar except that the redundancy factor, ρ, is not applicable. Thus, for this example (where ρ = 1.0), the load combinations for both the in-plane and the out-of-plane directions are 1.486D + QE and 0.614D - QE. The combination of earthquake motion (and corresponding loading) in two orthogonal directions as per Standard Section 12.5.3.a need not be considered. Standard Section 12.5.4 for Seismic Design Category D refers to Section 12.5.3 for Category C, which requires consideration of direction to produce maximum effect where horizontal irregularity Type 5 exists (“non-parallel systems”); this building does not have that irregularity. Standard Section 12.5.4 also requires consideration of direction for maximum effect for elements that are part of intersecting systems if those elements receive an axial load from seismic action that exceeds 20 percent of their axial strength; axial loads are less than that for this building. If a masonry control joint is provided at the corner, there are no elements acting in two directions. The short pier at the corner can be designed as an “L” shaped element, which means that it does participate in both directions. The vertical seismic force in that pier, generated by frame action, is small and easily less than 20 percent of its capacity. Therefore, no element of the seismic force-resisting system is required to

10-7

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples be checked for the direction of load that produces the maximum effect. Although it is not required, the typical pier in the end wall will be checked using the method of Standard Section 12.5.3.a. to illustrate the Standard’s method for design to account for orthogonal effects. 10.1.4 Seismic Forces Seismic base shear, diaphragm force and wall forces are discussed below. 10.1.4.1 Base Shear. Base shear is computed using the parameters determined previously. The Standard does not recognize the effect of long, flexible diaphragms on the fundamental period of vibration. The approximate period equations, which limit the computed period, are based only on the height. Since the structure is relatively short and stiff, short-period response will govern the design equations. According to Standard Section 12.8 (for short-period structures):

⎡ S ⎤ ⎡1.43 ⎤ V = C W = ⎢ DS ⎥ W = ⎢ ⎥ W = 0.286 W s ⎣ 5 / 1 ⎦ ⎣⎢ R / I ⎥⎦ The seismic weight for forces in the long direction is as follows: Roof = (20 psf)(100 ft)(200 ft) End walls = (103 psf) (2 walls)[(30 ft)(100 ft) – (5)(12 ft)(12 ft)](17.8 ft/28 ft)* Side walls = (65 psf) (30ft)(200ft)(2 walls) Total

= 400 kips = 299 kips = 780 kips = 1,479 kips

*Only the portion of the end walls that is distributed to the roof contributes to seismic weight in the long direction. (The initial estimates of 65 psf for 8-inch CMU and 103 psf for 12-inch CMU are slightly higher than normal-weight CMU with grouted cells at 24 inches on center. However, grouted bond beams at 4 feet on center will be included, as will certain additional grouted cells.) Note that the centroid of the end walls is determined to be 17.8 feet above the base, so the portion of the weight distributed to the roof is approximately the total weight multiplied by 17.8 feet/28 feet (weights and section properties of the walls are described subsequently). Therefore, the base shear to each of the long walls is as follows: Vu = (0.286)(1,479 kips)/2 = 211 kips The seismic weight for forces in the short direction is: Roof = (20 psf)(100 ft)(200 ft) Side walls = (65 psf)(2 walls)(30ft)(200ft)(15ft/28ft)* End walls = (103 psf)(2 walls)[(30ft)(100ft)-5(12ft)(12ft)] Total

= 400 kips = 418 kips = 470 kips = 1,288 kips

*Only the portion of the side walls that is distributed to the roof contributes to seismic weight in the short direction. The base shear to each of the short walls is as follows: 10-8

Chapter 10: Masonry

Vu = (0.286)(1,288 kips)/2 = 184 kips 10.1.4.2 Diaphragm force. See Section 11.2 for diaphragm forces and design. 10.1.4.3 Wall forces. Because the diaphragm is flexible with respect to the walls, shear is distributed to the walls on the basis of beam theory, ignoring walls perpendicular to the motion (this is the "tributary" basis). The building is symmetric. Given the previously explained assumption that accidental torsion need not be applied, the force to each wall becomes half the force on the diaphragm. All exterior walls are bearing walls and, according to Standard Section 12.11.1, must be designed for a normal (out-of-plane) force of 0.4SDSIWw where Ww = weight of wall. The out-of-plane design is shown in Section 10.1.5.2. 10.1.5 Side Walls The total base shear is the design force. Standard Section 14.4, which cites TMS 402, is the reference for design strengths. The compressive strength of the masonry, fm', is 2,000 psi. TMS 402 Section 1.8.2.2 gives Em = 900fm' = (900)(2 ksi) = 1,800 ksi. For 8-inch-thick CMU with vertical cells grouted at 24 inches on center and horizontal bond beams at 48 inches on center, the weight is conservatively taken as 65 psf (recall the CMU are normal weight) and the net bedded area is 51.3 in.2/ft based on tabulations in NCMA-TEK 14-1B. 10.1.5.1 Horizontal reinforcement – side walls. As determined in Section 10.1.4.1, the design base shear tributary to each longitudinal wall is 211 kips. Based on TMS 402 Section 1.17.3.6.2.1, the design shear strength must exceed either the shear corresponding to the development of 1.25 times the nominal flexure strength of the wall (which is very unlikely in this example due to the length of wall) or 2.5 times Vu , which in this case is 2.5(211) = 528 kips. From TMS 402 Section 3.3.4.1.2.1, the masonry component of the shear strength capacity for reinforced masonry is as follows:

⎡ ⎛ M ⎞ ⎤ Vnm = ⎢ 4.0 − 1.75 ⎜ u ⎟ ⎥ An f mʹ′ + 0.25 Pu ⎝ Vu dv ⎠ ⎥⎦ ⎣⎢ For a single-story cantilever wall, Mu/Vudv = h/d, which is (28/200) = 0.14 for this case. For the long walls and conservatively treating P as 0.614 times the weight of the wall only, without considering the roof weight contribution:

Vnm = [4.0 − 1.75(0.14)](51.3)(200) 2,000 + 0.25 [(0.614)(390)] = 1,783 kips and

φVm = 0.8(1,783) = 1,426 kips > 528 kips

where φ = 0.8 is the strength reduction factor for shear from TMS 402 Section 3.1.4.3.

10-9

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

Horizontal reinforcement therefore is not required for shear strength but is required if the wall is to qualify as a Special Reinforced Masonry Wall (TMS 402 Sec. 1.17.3.2.6.b). Standard Table 12.2-1 does not permit lower quality masonry walls in Seismic Design Category D. According to TMS 402 Section 1.17.3.2.6.c, minimum horizontal reinforcement is 0.0007Ag = (0.0007)(7.625 in.)(8 in.) = 0.043 in.2 per course, but the authors believe it prudent to use more horizontal reinforcement for shrinkage in this very long wall and then use minimum reinforcement in the vertical direction [this concept applies even though this wall requires far more than the minimum reinforcement (also 0.0007Ag) in the vertical direction due to its large height-to-thickness ratio]. Two #5 bars at 48 inches on center provide 0.103 in.2 per course. This amounts to 0.4 percent of the area of masonry plus the grout in the bond beams. The actual shrinkage properties of the masonry and the grout as well as local experience should be considered in deciding how much reinforcement to provide. For long walls that have no control joints, as in this example, providing more than minimum horizontal reinforcement is appropriate. 10.1.5.2 Out-of-plane flexure – side walls. The design demand for seismic out-of-plane flexure is 0.4SDSIww (Standard Sec. 12.11.1). For a wall weight of 65 psf for the 8-inch-thick CMU side walls, this demand is 0.4(1.43)(1)(65 psf) = 37 psf. Calculations for out-of-plane flexure become somewhat involved and include the following: 1. Select a trial design. 2. Investigate to ensure ductility (i.e., check maximum reinforcement limit). 3. Make sure the trial design is suitable for wind (or other nonseismic) lateral loadings using the wind provisions of the Standard (which satisfies the IBC). (This is not illustrated in this example).

4. Calculate mid-height deflection due to wind by TMS 402. (Not illustrated in this example). (Note

that while the Standard has story drift requirements, it does not impose a mid-height deflection limit for walls).

5. Calculate seismic demand. This computation requires consideration of P-delta effects because of the wall slenderness. (Seismic demand is greater than wind for this wall.)

6. Determine seismic resistance and compare to the demand determined in Step 5. Proceed with these steps as follows: 10.1.5.2.1 Trial design. A trial design of #7 bars at 24 inches on center is selected. See Figure 10.1-3.

10-10

Chapter 10: Masonry 8" concrete masonry unit

24" o.c.

Figure 10.1-3 Trial design for 8-inch-thick CMU wall (1.0 in. = 25.4 mm) 10.1.5.2.2 Investigate to ensure ductility. The critical strain condition corresponds to a strain in the extreme tension reinforcement (which is a single #7 centered in the wall in this example) equal to α times the strain at yield stress. α is the tension reinforcement strain factor (equal to 1.5 for out-of-plane flexure due to wind; see TMS 402 Commentary 3.3.3.5). Based on TMS 402 Section 3.3.3.5.1.a and Commentary 3.3.3.5 (where α is defined) for this case: t = 7.63 in. d = t/2 = 3.81 in. εm = 0.0025 εs = 1.5εy = 1.5(fy/Es) = 1.5(60 ksi / 29,000 ksi) = 0.0031

⎡ ε m ⎤ c = ⎢ ⎥ d = 1.70 in. ⎣⎢ (ε m + ε s ) ⎦⎥ a = 0.8c = 1.36 in. The Whitney compression stress block, a = 1.36 inches for this strain distribution, is greater than the 1.25-inch face shell width. Thus, the compression stress block is broken into two components: one for full compression against solid masonry (the face shell), and another for compression against the webs and grouted cells but accounting for the open cells. These are shown as C1 and C2 in Figure 10.1-4: C1 = 0.80fm' (1.25 in.)b = (0.80)(2 ksi)(1.25)(24) = 48 kips (for a 24-inch length) C2 = 0.80 fm' (a-1.25 in.)(8 in.) = (0.80)(2 ksi)(1.36-1.25)(8) = 1.41 kips (for a 24-inch length) The 8-inch dimension in the C2 calculation is for the combined width of the grouted cell and the adjacent mortared webs over a 24-inch length of wall. The actual width of one cell plus the two adjacent webs will vary with various block manufacturers and may be larger or smaller than 8 inches. The 8-inch value has the benefit of simplicity (and is correct for solidly grouted walls).

10-11

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

#7 at 24" o.c.

1.25"

1.25" 3.81" t = 7.63" P = (P f + Pw)

N.A.

d = 3.81" 1.70"

2.11"

ε m = 0.0025 1.08" 1.5ε y= 0.0031

0.40" C1

0.8 f 'm

0.11"

1.25"

N.A.

1.36"

Figure 10.1-4 Investigation of out-of-plane ductility for the 8-inch-thick CMU side walls (1.0 in. = 25.4 mm) T is based on FyAs (TMS 402 Sec. 3.3.3.5.1.c): T = FyAs = (1.0)(60 ksi)(0.60 in.2) = 36 kips (for a 24-inch length) P is based on the load combination of D + 0.75L +0.525QE (TMS 402 Sec. 3.3.3.5.1.d).

10-12

Chapter 10: Masonry QE is the effect of horizontal seismic motions, and P is a vertical force. QE produces overturning forces, but because this is such a long wall, the vertical force due to horizontal seismic motion is not significant, so the net total vertical force is taken as zero here. Therefore QE is zero in determining P for this wall. Conservatively neglecting the roof weight:

⎡ ⎛ 28 ft ⎞⎛ 24 in ⎞ ⎤ QE = Fp = 0.2S DS D = ( 0.2 )(1.43) ⎢( 65 psf ) ⎜ +2 ft ⎟⎜ ⎟ ⎥ = 595 lb/24 in. length ⎝ 2 ⎠⎝ 12 in ⎠ ⎦ ⎣

⎛ 28 ft ⎞⎛ 24 ft ⎞ P = ( 65 psf ) ⎜ + 2 ft ⎟⎜ ⎟ + ( 0.75 )( 0 ) ± ( 0.525 )( 0 lb ) = 2.08 kips/24 in. length ⎝ 2 ⎠⎝ 12 in ⎠ Check C1 + C2 > T + P (all for 24-inch length): T + P = 36 + 2.08 = 38.1 kips C1 + C2 = 49.4 kips > 38.1 kips

The compression capacity is greater than the tension capacity; therefore, the ductile failure mode criterion is satisfied. 10.1.5.2.3 Check for wind load. The wind design check is beyond the scope of this seismic example, so it is not presented here. Both strength and deflection need to be ascertained in accordance with a building code; most are based on the Standard, which we are using. For our example, a check on strength to resist wind was found to conform to the Standard and is not shown here. 10.1.5.2.4 Calculate mid-height deflection due to wind by Standard. Deflection due to wind was found to conform to the Standard and is not shown here.

10-13

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

h p = 2'

e = 7.32"

h = 28'

Pw = W w ( h2 + h p)

Pe 2

wh2 8

(Pw + P f δ)

plat

Figure 10.1-5 Basis for out-of-plane deflection calculation 10.1.5.2.5 Calculate seismic demand. For this case, the two load factors for dead load apply: 0.614D and 1.486D. Conventional wisdom holds that the lower dead load will result in lower moment-resisting capacity of the wall, so the 0.614D load factor would be expected to govern. However, the lower dead load also results in lower P-delta, so both cases should be checked. (As it turns out, the higher factor of 1.486D controls).

wu = 37 psf (from Sec. 10.1.5.3)

Check moment capacity for 0.614D: TMS 402 Section 3.3.5.3 requires consideration of the secondary moment from the axial force acting through the deflection. TMS 402 Section 3.3.5.4 gives an equation that is essentially bilinear (two straight lines joined at the point of cracking). NCMA TEK 14-1B illustrates that determining the final moment by this method requires iteration. Roof load, Pf = (20 psf)(10 ft) = 200 plf Wall load (at mid-height), Pw = (65 psf)(16 ft) = 1,040 plf P = Pf + Pw = 1,240 plf Puf = (0.614)(200 plf) = 123 plf Puw = (0.614)(1,040 plf) = 638 plf

10-14

Chapter 10: Masonry Pu = Puf + Puw = 761 plf Eccentricity, e = 7.32 in. (distance from wall centerline to roof reaction centerline) Modulus of elasticity, Em = 1,800,000 psi fm' = 2000 psi Modular ratio, n =

Es = 16.1 Em

The modulus of rupture, fr, is found from TMS 402 Table 3.1.8.2. The values given in the table are for either hollow CMU or fully grouted CMU. Values for partially grouted CMU are not given; Footnote a indicates that interpolation between these values must be performed. As illustrated in Figure 9.1-6, and shown below, the interpolated value for this example is based on relative areas of hollow and grouted cells:

⎡ (103 − 60 ) ⎤ fr = 63 + (163 − 63) ⎢ ⎥ = 98 psi ⎣⎢ (183 − 60 ) ⎦⎥ Another method for making the interpolation, while approximate, is simpler. It is based on 2/3 of the cells being hollow and 1/3 of the cells being grouted for the case of grouted cells at 24 inches on center:

⎛ 2 ⎞ ⎛ 1 ⎞ fr = 63 ⎜ ⎟ + 163 ⎜ ⎟ = 96 psi ⎝ 3 ⎠ ⎝ 3 ⎠ For this example, the value of fr = 98 psi will be used. From mechanics: Ig = 443 in.4/ft Sg = 116 in.3/ft From NCMA TEK 14-1B: In= 355 in.4/ft Sn = 93.2 in.3/ft An = 51.3 in.2/ft Mcr = Sn(fr + P/An) = 93.2(98 + 1240/51.3) = 11,386 in-lb/ft. Use Mcr = 11,400 in.-lb/ft. Note: this equation for Mcr is not in TMS 402; however, it is valid based on mechanics.

10-15

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

24"

CASE 1 All cells open A = 60 in.2 f r = 63 psi

CASE 2 (1) Cell grouted A = 103 in.2 f r = 98 psi *

24"

CASE 3 Fully grouted A = 183 in.2 f r = 163 psi

* By interpolation

Figure 10.1-6 Basis for interpolation of modulus of rupture, fr (1.0 in. = 25.4 mm, 1.0 psi = 6.89 kPa). Refer to Figure 10.1-7 for determining Icr. The neutral axis shown on the figure is not the conventional neutral axis by linear analysis; instead, it is the plastic centroid, which is simpler to locate, especially where the neutral axis position results in a T beam cross-section. (For this wall, the neutral axis does not produce a T section, but for the end wall in this building, a T section does result.) Cracked moments of inertia computed by this procedure are less than those computed by linear analysis but generally not so much less that the difference is significant. (This is the method used for computing the cracked section moment of inertia for slender walls in the standard for concrete structures, ACI 318.) Axial load does enter the computation of the plastic neutral axis and the effective area of reinforcement. Thus: T = (0.60 in.2/ 2 ft)(60 ksi) = 18.0 klf C = T + P = 19.24 klf a = C/(0.8 f'mb) = (19.24 klf)/(0.8(2.0 ksi)(12 in./ft) = 1.002 in. c = a/0.8 = 1.25 in. Ase = As + P/fy = 0.30 in.2/ft + 1.240 klf/60 ksi = 0.32 in.2/ft Note: TMS 402 uses the term As to mean the same thing as effective area of reinforcement (TMS 402 Sec. 1.5 and Commentary 3.3.5.4). Ase is used here to distinguish effective area from actual area, As. Icr = nAse(d-c)2 + bc3/3 = 16.1(0.32 in.2/ft)(3.81 in. - 1.25 in.)2 + (12 in./ft)(1.25 in.)3/3 = 42.1 in.4/ft Note that Icr could be recomputed for Pu = 0.614D and Pu = 1.486D, but that refinement is not pursued in this example. In the opinion of the authors, the most correct method for computing the cracked section properties is to use Pu. This will necessitate two sets of cracked section properties for this example. For purposes of illustration, one set of cracked section properties, with P = 1.0D, is computed.

10-16

N.A.

c d = 3.81"

1.25"

Chapter 10: Masonry

16"

8" 24"

bw = 8.32" inferred from NCMA tabulations bw = 8" used for convenience

Figure 10.1-7 Cracked moment of inertia (Icr) for 8-inch-thick CMU side walls (1.0 in. = 25.4 mm) The computation of the secondary moment in an iterative fashion is shown below: First iteration:

M u1 = wu h2 / 8 + Puf eu + ( Puf + Puw )δu M u1 =

(37 psf/12)(336 in.) 2 ⎛ 7.32 in. ⎞ + (123 plf) ⎜ ⎟ + (761 plf)(0) 8 ⎝ 2 ⎠

Mu1 = 43,512 + 450 + 0 = 43,962 in.-lb/ft > Mcr = 11,400 in.-lb/ft

δ s1 =

5M cr h2 5( M u1 − M cr )h2 + 48EI g 48EI cr

δ s1 =

5(11,400)(336)2 5(44,962 − 11,400)(336) 2 + = 0.168 + 5.05 = 5.07 in. 48(1,800,000) ( 443) 48(1,800,000)(42.1)

Second iteration:

M u 2 = 43,512 + 450 + (761)(5.07) = 47,820 in.-lb

δ s 2 = 0.165 +

5(47,820 − 11, 400)(336)2 = 0.165 + 5.652 = 5.82 in. 48(1,800,000)(42.1)

Third iteration

10-17

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

Mu3 = 43,512 + 450 + (761)(5.82) = 48,391 in.-lb/ft

δ s3 = 0.165 +

5 ( 48,391 − 11,400 ) (336)2 48(1,800,000)(42.1)

= 0.165 + 5.74 = 5.91 in.

Convergence check:

5.91 − 5.82 = 1.5% < 5% 5.82 Mu = 48,391 in.-lb/ft (for the 0.614D load case) Using the same procedure, find Mu for the 1.486D load case. The results are summarized below: First iteration: Pu = 1.486 (Pf + Pw) = 1.486(200 + 1,040) =297 + 1,545 = 1,843 plf Mu1 = 44,811 in.-lb/ft

δu1 = 5.39 in. Second iteration: Mu2 = 54,739 in.-lb/ft

δu2 = 6.93 in. Third iteration: Mu3 = 57,579 in.-lb/ft

δ3 = 7.37 in. Fourth iteration: Mu4 = 58,392 in.-lb/ft

δu4 = 7.50 in. Check convergence:

7.50 − 7.37 = 1.7% < 5% 7.50 Mu = 58,392 in.-lb/ft (for the 1.486D load case)

10-18

Chapter 10: Masonry The iterative method described above is consistent with NCMA TEK 14-11B. The authors note that ACI 318, the standard for concrete structures, includes provisions for the design of slender walls that are somewhat different. For the computation of deflection at nominal strength, 75 percent of the cracked stiffness is used. The 0.75 factor represents a margin for safety, because the required strength, Mu, depends on the computed deflection. The absence of the bilinear relation is much closer to deflection computations by other methods, such as given in TMS 402, Section 1.13.3.2. The absence of bilinear relations allows direct computation of the final deflection and moment, rather than iteration. For illustration, the method that predicts the secondary moment directly and upon which the ACI 318 slender wall direct calculation is based, is shown here:

⎛ ⎜ 1 M u = M u1 ⎜ ⎜ 1 − Pu ⎜ Pe ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

Where:

Pe =

π 2 EI cr h2

π 2 (1,800 ksi)(42.1 in.2 ) (28 ft x 12)2

= 6.62 k/ft

Therefore, for the 1.486D case:

⎛ ⎜ 1 M u = 44.8 in.-k/ft ⎜ 1.84 k/ft ⎜⎜ 1 − 6.62 k/ft ⎝

⎞ ⎟ ⎟ = 62.0 in.-k/ft ⎟⎟ ⎠

which is approximately 6 percent larger than Mu = 58.4 in.-k/ft by the iterative method above. In this calculation, the 0.75 factor on Pe used in ACI 318 has not been included. 10.1.5.2.6 Determine flexural strength of wall. Refer to Figure 10.1-8. As in the case for the ductility check, a strain diagram is drawn. Unlike the ductility check, the strain in the steel is not predetermined. Instead, as in conventional strength design of reinforced concrete, a rectangular stress block is computed first and then the flexural capacity is checked. T = Asfy = (0.30 in.2/ft)60 ksi = 18.0 klf The results for the two axial load cases are shown in Table 10.1-2 below. Table 10.1-2 Flexural Strength of Side Wall Load Case 0.614D + E Pu, klf 0.761 C = T + Pu, klf 18.76 a = C / (0.8f'mb), in. 0.978 Mn= C (d - a/2), in.-kip/ft 62.3 56.1 φMn= 0.9Mn, in.-kip/ft

1.486D + E 1.843 19.84 1.03 65.35 58.8

10-19

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Table 10.1-2 Flexural Strength of Side Wall Mu, in.-kip/ft 48.4 Acceptance OK

58.4* OK

*The Mu from the alternative direct computation is approximately 5% higher than the design strength.

#7 at 24" o.c.

1.25"

3.81" t = 7.63" Pu

c = a/0.80

N.A.

d = 3.81"

ε m = 0.0025 a/2

ε s > ε y = 0.00207

0.8 f 'm

T = As F y + P

N.A.

1.25"

Figure 10.1-8 Out-of-plane strength for 8-inch-thick CMU walls

10-20

Chapter 10: Masonry (1.0 in. = 25.4 mm) Note that either wind or earthquake may control the stiffness and strength out-of-plane; earthquake controls for this example. A careful reading of Standard Section 12.5 should be made to see if the orthogonal loading combination will be called for; as discussed earlier, the orthogonal combination is not required for this example (although an orthogonal combination check will be made for illustration purposes later). 10.1.5.3 In-plane flexure – side walls. In-plane calculations for flexure in masonry walls include two items per the Provisions: §

Ductility check

§

Strength check

It is recognized that this wall is very strong and stiff in the in-plane direction. Many engineers would not even consider these checks as necessary in ordinary design. The ductility check is illustrated here to show a method of implementing the requirement. 10.1.5.3.1 Ductility check. For this case, with Mu/Vudv < 1 and R > 5, TMS 402 Section 3.3.3.5.4 refers to Section 3.3.3.5.1, which stipulates that the critical strain condition corresponds to a strain in the extreme tension reinforcement equal to 1.5 times the strain associated with Fy. This calculation uses unfactored gravity loads. (See Figure 10.1-9.)

⎛ ε m c = ⎜ ⎝ ε m + ε s

⎞ 0.0025 ⎛ ⎞ ⎟ d = ⎜ ⎟ 200 ft = 89.29 ft 0.0025 0.0031 + ⎝ ⎠ ⎠

a = 0.8c = 71.43 ft Cm = 0.8f’mabavg = (0.8) (2 ksi)(71.43 ft)(51.3 in.2/lf) = 5,862 kips Where bavg is taken from the average area used earlier, 51.3 in.2/ft results; see Figure 10.1-9 for locations of tension steel and compression steel (the rebar in the compression zone will act as compression steel). From this it can be seen that:

⎛ ⎞ 73.80 (0.60) = 664 kips Ts1 = f y ⎜ ⎜ ( 2 )( 2 ft o.c.) ⎟⎟ ⎝ ⎠

⎛ 36.91 ⎞ Ts 2 = f y ⎜ ⎟ (0.60) = 664 kips ⎝ 2 ft ⎠ ⎛ 15.42 ⎞ Cs1 = f y ⎜ ⎟ (0.60) = 278 kips ⎝ 2 ft o.c. ⎠

10-21

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

⎛ 73.87 ⎞ Cs 2 = ( f y ) ⎜ ⎟ (0.60) = 665 kips ⎝ (2)(2 ft) ⎠

P = P f + Pw

= 0.0031

ε s = 1.5 ε y

εm = 0.0025

N.A.

10.71'

c = 89.29' Cm

110.71'

0.8 f 'm

53.58'

a = 71.43' C s1

17.86'

81.58'

73.80'

36.91'

C s2 49.25'

f y = 60ksi

15.42'

73.87'

f y = 60 ksi

49.20' T s2

N.A.

T s1 92.26'

Figure 10.1-9 In-plane ductility check for side walls (1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa)

10-22

Chapter 10: Masonry Some authorities would not consider the compression resistance of reinforcing steel that is not confined within ties. In one location (Section 3.1.8.3) TMS 402 clearly requires transverse reinforcement (ties) for any steel used in compression, while in another place (Section 3.3.3.5.1.e) it explicitly permits inclusion of compression reinforcement with or without lateral restraining reinforcement for checks on maximum flexural tensile reinforcement (i.e., ductility checks). TMS 402 Commentary 3.3.3.5 explains that confinement reinforcement is not required because the maximum masonry compressive strain will be less than ultimate values. This inconsistency does not usually have a significant effect on computed results. The authors have taken credit for unconfined compression reinforcement for strength and included it in ductility checks (but there is no objection to the practice of neglecting unconfined compression reinforcement used by some engineers). In the authors’ opinion, there are two approaches to the determination of P, one following TMS 402 and the other following the Standard: P is at the base of the wall rather than at the mid-height: §

TMS 402 Section 3.3.3.5.1.d: D + 0.75L + 0.525 QE Since QE represents the effect of horizontal seismic forces, which equals zero for our case, and roof live load is not combined with seismic loads, this reduces to D: P = Pw + Pf = [(0.065 ksf) (30 ft) + (0.02 ksf )(10 ft)](200 ft) = 430 kips

§

Standard Section 12.4.2.3: (1.2 + 0.2 SDS)D + ρQE + L + 0.2S which reduces to: [1.2 + (0.2)(1.43)]D + 0 + 0 + 0 Pu = (1.486)(Pf + Pw) = (1.486)(480 kips) = 713 kips

Continuing with the in-plane ductility check: ΣC > P + ΣT Cm + Cs1 + Cs2 > P + Ts1 + Ts2 And conservatively using the higher of the two values for P, 5,862 + 278 + 665 > 713 + 664 + 664

6,805 > 2,041

Therefore, there is enough compression capacity to ensure ductile failure. Note that either of the two values for P brings us to the same conclusion for this case. It should also be noted that even if the compression reinforcement were neglected, there would still be enough compression capacity to ensure ductile failure.

10-23

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples In the opinion of the authors, flexural yield is feasible for walls with Mu/Vudv in excess of 1.0; this criterion limits the compressive strain in the masonry, which leads to good performance in strong ground shaking. For walls with Mu/Vudv substantially less than 1.0, the wall will fail in shear before a flexural yield is possible. Therefore, the criterion does not affect performance. Well distributed and well developed reinforcement to control the shear cracks is the most important ductility attribute for such walls. 10.1.5.4.2 Strength check. The wall is so long with respect to its height that in-plane strength for flexure is acceptable by inspection. 10.1.5.5 Shear – side walls. 10.1.5.5.1 Out-of-plane shear in side walls. Compute out-of-plane shear at the base of a wall in accordance with Standard Section 12.11.1: Fp = 0.4SDSIww = (0.4)(1.43)(1.0)(65 psf)(28 ft/2) = (37 psf)(14 ft) = 521 plf. The “capacity design” requirement in TMS 402 Section 1.17.3.2.6.1 applies to behavior in-plane, not outof-plane. The capacity computed per TMS 402 Section 3.3.4.1.2.1 is as follows:

⎡ ⎛ M Vm = ⎢ 4.0 − 1.75 ⎜ u ⎢⎣ ⎝ Vu dv

⎞ ⎤ ⎟ ⎥ An f mʹ′ + 0.25 Pu ⎠ ⎥⎦

Mu/Vudv need not be taken larger than 1.0 (and Mu/Vudv does exceed 1.0 for a short distance above the base). An, as determined earlier, is taken as 51.3 in.2/ft. Note: If the dimensions from Figure 109.7 are used, An is taken as bwd = (8.32 in.)(3.81 in.) + (24 - 8.32 in.)(1.25 in.) = 51.3 in.2/ft, a similar value. The authors note that traditional practice in reinforced masonry has been to compute shear stress on the basis of areas equaling width times depth to reinforcement (bd). For the in-plane shear strength, the difference between bd and An is not too great, but for the out-of-plane shear strength of walls with one layer of reinforcement in their centers, the difference is very substantial. Therefore, the authors have substituted bwd (= 8 in. × 3.81in. = 30.5 in.2 ; see Figure 10.1-7) for An in the equation below. Because shear exists at both the bottom and the top of the wall, conservatively neglect the effect of P:

Vm = [4.0 − 1.75(1.0)](30.5in.2 / ft) 2,000 + 0 = 3.07 kips/24" = 1.53 klf φVm = (0.8)(1.53) = 1.23 klf > 0.52 klf

10.1.5.5.2 In-plane shear in side walls. As indicated in Sections 10.1.4.1 and 10.1.5.1, the in-plane demand at the base of the wall, Vu = 2.5(211 kips) = 528 kips and the shear capacity, φVm, is larger than (4.13 klf)(200 ft) = 826 kips. For the purpose of understanding likely behavior of the building somewhat better, Vn is estimated more accurately than simply limiting Mu/Vudv to 1 for these long walls: Mu/Vudv = h/l = 28/200 = 0.14

10-24

Chapter 10: Masonry

Pu = 0.614D = (0.614)(430 kips) = 264 kips Vm = [4.0 - 1.75(0.14)][200(51.3)](0.045) + 0.25(264) = 1,733 + 66 = 1,799 kips Vns = 0.5(Av/s)fyd = 0.5(0.62/4.0)(60)(200) = 930 kips (for 2-#5 in bond beams at 4 ft o.c.) Vn = 1,799 + 930 = 2,729 kips Maximum Vn = 6 f mʹ′ An = 6(0.045 ksi)(10,260 in.2) = 2,770 > 2,729 kips

φVn = 0.8(2,729) = 2,183 kip > 528 kips = Vu The calculated seismic force, VE = 211kips (from Sec. 10.1.4.1)

φVn/VE = 10.3 >> R used in design In other words, it is unlikely that the long masonry walls will yield in either in-plane shear or flexure at the design seismic ground motion. The walls will likely yield in out-of-plane response, and the roof diaphragm may also yield. The roof diaphragm for this building is illustrated in Section 11.2. 10.1.6 End Walls The transverse walls are designed in a manner similar to the longitudinal walls. Complicating the design of the transverse walls are the door openings, which leave a series of masonry piers between the doors. 10.1.6.1 Horizontal reinforcement – end walls. The minimum reinforcement, per TMS 402 Section 1.17.3.2.6, is 0.0007Ag = (0.0007)(11.625 in.)(8 in.) = 0.065 in.2 per course. The maximum spacing of horizontal reinforcement is 48 inches, for which the minimum reinforcement is 0.39 in.2. Two #4 in bond beams at 48 inches on center would satisfy the requirement. The large amount of vertical reinforcement would combine to satisfy the minimum total reinforcement requirement. However, given the 100-foot length of the wall, a larger amount is desired for control of restrained shrinkage as discussed in Section 10.1.5.1. Two #5 at 48 inches on center will be used. 10.1.6.2 Vertical reinforcement – end walls. The area for each bay subject to out-of-plane wind is 20 feet wide by 30 feet high because wind load applied to the doors is transferred to the masonry piers. However, the area per bay subject to both in-plane and out-of-plane seismic forces is reduced by the area of the doors. This is because the doors are relatively light compared to the masonry. See Figures 10.1-11 and 10.1-12. 10.1.6.3 Out-of-plane flexure – end walls. Out-of-plane flexure is considered in a manner similar to that illustrated in Section 10.1.5.2. The design of this wall must account for the effect of door openings between a row of piers. The steps are the same as identified previously and are summarized here for convenience: 1. Select a trial design. 2. Investigate to ensure ductility.

10-25

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples 3. Make sure the trial design is suitable for wind (or other non-seismic) lateral loadings using the wind provisions of the Standard. 4. If wind controls over seismic (it does not in this example), then calculate the mid-height deflection due to wind by TMS 402. 5. Calculate the seismic demand. 6. Determine the seismic resistance and compare to the demand determined in Step 5. 10.1.6.3.1 Trial design. A trial design of 12-inch-thick CMU reinforced with two #6 bars at 24 inches on center is selected. The self-weight of the wall, accounting for horizontal bond beams at 4 feet on center, is taken as 103 psf. Adjacent to each door jamb, the vertical reinforcement is placed into two cells. See Figure 10.1-10.

2'-0"

11.63"

2'-0"

6.88"

(2) #6

8'-0"

Figure 10.1-10 Trial design for piers on end walls (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) Next, determine the design load locations. The centroid for seismic loads, out-of-plane, is the centroid of the mass of the wall and, accounting for the door openings, is determined to be 17.8 feet above the base. See Figures 10.1-11 and 10.1-12. 10.1.6.3.2 Investigate to ensure ductility. The critical strain condition corresponds to a strain in the extreme tension reinforcement (which is a pair of #6 bars in the end cell in this example) equal to α times the strain at yield stress. As for the side walls, α = 1.5 for out-of-plane flexure due to wind (TMS 402 Section 3.3.3.5 and Commentary 3.3.3.5). See Figure 10.1-13. For this case: t = 11.63 in. d = 11.63 - 2.38 = 9.25 in. εm = 0.0025 (TMS Sec. 402 3.3.2.c) εs = 1.5 εy = 1.5 (fy/Es) = 1.5 (60 ksi /29,000 ksi) = 0.0031 (TMS 402 Sec. 3.3.3.5.1.a and Commentary 3.3.3.5)

10-26

Chapter 10: Masonry

⎡ ε m ⎤ c = ⎢ ⎥ d = 4.13 in. ⎣ (ε m + ε s ) ⎦ a = 0.8c = 3.30 in. (TMS 402 Sec. 3.3.3.5.1.b)

In-plane loads

20'-0" P f = 8 kips

2'-0'

V f + Vw (side wall)

17.8'

28'-0"

V w (end wall)

20'-0"

8'-0"

Area/ bay subject to wind (because doors transfer wind loads to masonry)

Area/ bay subject to seismic (because masonry walls are much heavier than doors)

Out-of-plane loads applied to bay

Figure 10.1-11 In-plane loads on end walls (1.0 ft = 0.3048 m)

10-27

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

wtop = (0.4) (1.43) (103 psf) (20') = 1200 plf/bay out-of-plane

Rtop = 17.5 kips/bay out-of-plane

16'

12'

17.8'

21'

Rbot = 9.8 kips /bay out-of-plane

wbot = (0.4) (1.43) (103) (8') = 480 plf/bay out-of-plane

Figure 10.1-12 Out-of-plane load diagram and resultant of lateral loads (1.0 ft = 0.3048 m, 1.0 lb = 4.45 N, 1.0 kip = 4.45 kN)

10-28

Chapter 10: Masonry

11.63"

1.50"

(6) #6

1.50"

(6) #6

P = (P f + Pw)

d = 9.25" c 4.13"

0.0025

N.A.

εm =

a = 0.8c = 3.30" C1 3.38"

Cres 2.76"

1.73"

1.5 ε y = 0.0031

1.6 ksi

0.8 f ' m

1.50"

1.80"

0.83"

5.12" T

Figure 10.1-13 Investigation of out-of-plane ductility for end wall (1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa) Note that the Whitney compression stress block, a = 3.30 inches deep, is greater than the 1.50-inch face shell thickness. Thus, the compression stress block is broken into two components: one for full compression against solid masonry (the face shell) and another for compression against the webs and grouted cells but accounting for the open cells. These are shown as C1 and C2 in Figure 10.1-14. The values are computed using TMS 402 Section 3.3.2.g: C1 = 0.80fm' (1.50 in.)b = (0.80)(2 ksi)(1.50)(96) = 230 kips (for full length of pier) C2 = 0.80fm' (a - 1.50 in.)(6(8 in.)) = (0.80)(2 ksi)(3.30 - 1.50)(48) = 138 kips

10-29

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples The 48-inch dimension in the C2 calculation is the combined width of grouted cell and adjacent mortared webs over the 96-inch length of the pier. T = FyAs = (60 ksi)(6 × 0.44 in.2) = 158 kips/pier P is computed at the head of the doors. The dead load component of P is: P = (Pf + Pw) = (0.020 ksf)(20 ft)(20 ft) + (0.103 ksf)(18 ft)(20 ft) = 8.0 + 37.1 P = 45.1 kips/pier From TMS 402 Section 3.3.3.5.1.d, axial forces are taken from the load combination of the following: P = D + 0.75L + 0.525QE with QE = Fp = 0.2SDSD = (0.2)(1.43)(45.1) = 12.9 kips/pier P = 45.1 kips/pier + (0.75)(0) + (0.525)(12.9 kips/pier) P = 51.9 kips/pier C1 + C2 > P + T 368 kip > 210 kips The compression capacity is greater than the tension capacity, so the ductility criterion is satisfied.

6.06" 1"

6.06"

bw = 8.31" 2'-0"

Figure 10.1-14 Cracked moment of inertia (Icr) for end walls. Dimension “c” depends on calculations shown for Figure 10.1-16. (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)

10-30

1.25"

N.A.

c = 1.66"

1.5"

N.A.

1.25"

d = 9.25"

11.63"

8.63"

1.5"

(2) #6 at 24"

10.1.6.3.3 Check for wind loading. Wind pressure per bay is over the full 20-foot-wide by 30-foot-high bay, as discussed above, and is based on the Standard. While both strength and deflection need to be ascertained per a building code (the IBC was used), the calculations are not presented here.

Chapter 10: Masonry

10.1.6.3.4 Calculate out-of-plane seismic demand. For this example, the load combination 0.614D has been used, and for this calculation, forces and moments over a single pier (width = 96 in.) are used. This does not violate the b > 6t rule (TMS 402 Sec. 3.3.4.3.3.d) because the pier is reinforced at 24 inches on center. The use of the full width of the pier instead of a 24-inch width is simply for calculation convenience. For this example, a P-delta analysis using RISA-2D was run, resulting in the following: Maximum moment, Mu = 95.6 ft-kips/bay = 95.6/20 ft = 4.78 klf Moment at top of pier, Mu = 89.3 ft-kips/pier = 89.3 / 8 ft = 11.2 klf

(does not control) (controls)

Shear at bottom of pier, Vu = 9.61 kips/pier Reaction at roof, Vu = 17.5 kips/bay Axial force at base, Ru = 31.2 kips/pier (includes load factor on D of 0.614) 10.1.6.3.6 Determine moment resistance at the top of the pier. See Figure 10.1-15. As = 6-#6 = 2.64 in.2/pier d = 9.25 in. T = 2.64(60) = 158.4 kip/pier C = T + P = 184.1 kip/pier (P is based on D of (0.614)(37.1 + 8 kip) = 27.7 kip/pier at top of pier) a = C / (0.8f'mb) = 184.1 / [(0.8)(2)96] = 1.20 in. Because a is less than the face shell thickness (1.50 in.), compute as for a rectangular beam. Moments are computed about the centerline of the wall. Mn = C (t/2 - a/2) + P (0) + T (d - t/2) = 184.1(5.81 - 1.20/2) + 158.4(9.25-5.81) = 1,504 in.-kip = 125.4 ft-kip φMn = 0.9(125.4) = 112.8 ft-kip Because moment capacity at the top of the pier, φMn = 112.8 ft-kips, exceeds the maximum moment demand at top of pier, Mu = 89.3ft-kips, the condition is acceptable.

10-31

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

11.63"

1.50"

Pu = (P f + Pw )

d = 9.25" 5.21"

=1.6 ksi

0.8 f ' m

4.61"

3.44"

1.20"

Figure 10.1-15 Out-of-plane seismic strength of pier on end wall (1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa) 10.1.6.4 In-plane flexure – end walls. There are several possible methods to compute the shears and moments in the individual piers of the end wall. For this example, the end wall was modeled using RISA2D. The horizontal beam was modeled at the top of the opening, rather than at its mid-height. The inplane lateral loads (see Figure 10.1-11) were applied at the 12-foot elevation and combined with joint moments representing transfer of the horizontal forces from their point of action down to the 12-foot elevation. Vertical load due to roof beams and the self-weight of the end wall were included. The input loads are shown in Figure 10.1-16. For this example: w = (18 ft)(103 psf) + (20 ft)(20 psf) = 2.254 klf H = (184 kip)/5 = 36.8 kip M = Cs[(Vf long + Vw long)hlong + (Vw short)(hshort)]

(refer to Fig. 10.1-11).

M = 0.286[(400 + 418)(28 ft – 12 ft) + 470(17.8 ft – 12 ft)] = 452 ft-kip

10-32

Chapter 10: Masonry

96'-0" w

M H

M/2 H/2

12'-0"

H/2

M/2

Figure 10.1-16 Input loads for in-plane end wall analysis (1.0 ft = 0.3048 m) The input forces at the end wall are distributed over all the piers to simulate actual conditions. The RISA2D frame analysis accounts for the relative stiffnesses of the 4-foot- and 8-foot-wide piers (continuity of the 4-foot-wide piers at the corners was not considered). The final distribution of forces, shears and moments for an interior pier is shown on Figure 10.1-17.

Ptop = 45.1 kip M top = 523 ft-kip El. 112'-0" = T.O. Pier

12'

V top = 41.0 kip

8' V bot = 43.6 kip M bot = 0 Rbot = 55.0 kip

Figure 10.1-17 In-plane design condition for 8-foot-wide pier (1.0 ft = 0.3048 m) Continuing with the trial design for in-plane pier design, use two #6 bars at 24 inches on center supplemented by adding two #6 bars in the cells adjacent to the door jambs (see Figures 10.1-10 and 10.1-18).

10-33

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

8'-0" = 96" d = 92"

P = (P f + Pn)

92"

17.3"

70.7"

9.3"

62.7" 38.7"

εm = 0.0025

0.0020

14.7" 0.0011 0.0017 0.0045

c = 21.3"

0.0073

ε s = 4ε y = 0.0083

12.8"

0.8 f 'm

a = 17.0"

f y = 60 ksi

C s1

4.3"

C s2

T s3

T s2

T s1 f y = 60ksi

14.7" 9.3" 17.3"

38.7" 70.7"

Figure 10.1-18 In-plane ductility check for 8-foot-wide pier (1.0 in. = 25.4 mm, 1.0 ksi = 6.89 MPa)

10-34

Chapter 10: Masonry The design values for in-plane design at the top of the pier are: Table 10.1-3 In-plane Design Values at Pier Top Unfactored 0.614D + 1.0E 1.486D + 1.0E P = 45.1 kips Pu = 41.2 kips Pu = 67.0 kips V = 43.6 kips Vu = 43.6 kips Vu = 43.6 kips M = 523 ft-kips Mu = 523 ft-kips Mu = 523 ft-kips Mu/Vudv 1.50 1.50 The ductility check is illustrated in Figure 10.1-18. Because Mu/Vudv > 1for this special reinforced masonry shear wall subject to in-plane loads, α = 4: εm = 0.0025 εs = 4εy = (4 )(60/29,000) = 0.0083 d = 92 in. From the strain diagram (Fig. 10.1-18), the strains at the rebar locations from left to right are: ε = 0.0020 ε = 0.0011 ε = 0.0017 ε = 0.0045 ε = 0.0073 ε = 0.0083 To check ductility, use unfactored loads (from Section 10.1.6.3.2): P = Pf + Pw = 8 kips + 37.1 kips = 45.1 kips a = 0.8c = 17.0 in. Cm = (0.8fm')ab = (1.6 ksi)(17.0 in.)(11.63 in.) = 315.5 kips Ts1 = Ts2 = FyAs = (60 ksi)(2 × 0.44 in.2) = 52.8 kips Ts3 = εΕAs = (0.0017)(29,000 ksi)(2 × 0.44 in.2) = 43.4 kip Cs1 = εΕAs = (0.0021)(29,000 ksi)(2 × 0.44 in.2) = 53.6 kip Cs2 = εΕAs = (0.0011)(29,000 ksi)(2 × 0.44 in.2) = 28.1 kip ΣC > ΣT + P Cm + Cs1 + Cs2 > Ts1 + Ts2 + Ts3 + P

10-35

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples 315.5 + 53.6 + 28.1 > 52.8 + 52.8 + 43.4 + 45.1 397 kips > 194 kips

Because compression capacity exceeds tension capacity, the requirement for ductile behavior is OK. Note that maximum P for the wall to remain ductile is Pmax = ΣC - ΣT = 248 kips. Thus, φPmax = 223 kips in order to assure ductility. For the strength check, see Figure 10.1-19.

10-36

Chapter 10: Masonry

P M

11.63"

96"

0.8 f ' m

48"

44"

42.35'

36" 12"

ε m = 0.0025

12" P = 0 Case

a = 11.3"

T s4

T s2

T s3

T s1

ε y= F y

c = 14.2" N.A.

E = 0.00207

Cm2

C ml

Cm3

0.8 f ' m

ε m = 0.0025 ε y = 0.00207

C s2 C s1

16"

Balanced Case

ε = 0.0019 *

16" a = 40.3"

8.3"

ε = 0.0017

T s2

T s1

2.3"

7.7"

48"

N.A.

c = 50.3"

44"

Center Line

Figure 10.1-19 In-plane seismic strength of pier. Strain diagram superimposed on strength diagram for both cases. Note that locations with low force in reinforcement, marked by *, are neglected. (1.0 in. = 25.4 mm)

10-37

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

To ascertain the strength of the pier, a φPn - φMn curve is developed. Only the portion below the “balance point” is examined since that portion is sufficient for the purposes of this example. (Ductile failures occur only at points on the curve that are below the balance point, so this is consistent with the overall approach). For the P = 0 case, assume all bars in tension reach their yield stress and neglect compression steel (a conservative assumption): Ts1 = Ts2 = Ts3 = Ts4 = (2)(0.44 in.2)(60 ksi) = 52.8 kips Cm = Σ Ts = (4)(52.8) = 211.2 kips Cm = 0.8f’mab = (0.8)(2 ksi)a(11.63 in.) = 18.6a Thus, a = 11.3 inches and c = a/0.8 = 11.3 / 0.8 = 14.2 inches. ΣMcl = 0 Mn = 42.35 Cm + 44Ts1 + 36Ts2 + 12Ts3 - 12Ts4 = 13,168 in.-kips

φMn = (0.9)(13,168) = 11,851 in.-kips = 988 ft-kips For the balanced case: d = 92 in. ε = 0.0025 εy = 60/29,000 = 0.00207

⎛ ε m c = ⎜ ⎜ ε + ε y ⎝ m

⎞ ⎟⎟ d = 50.3 in. ⎠

a = 0.8c = 40.3 in. Compression values are determined from the Whitney compression block adjusted for fully grouted cells or ungrouted cells: Cm1 = (1.6 ksi)(16 in.)(11.63 in.) = 297.8 kips Cm2 = (1.6 ksi)(16 in.)(2 × 1.50 in.) = 76.8 kips Cm3 = (1.6 ksi)(8.3 in.)(11.63 in.) = 154.4 kips Cs1 = (0.88 in.2)(60 ksi) = 52.8 kips Cs2 = (0.88 in.2)(60 ksi)(0.0019 / 0.00207) = 48.5 kips

10-38

Chapter 10: Masonry Ts1 = (0.88 in.2)(60 ksi) = 52.8 kips Ts2 = (0.88 in.2)(60 ksi)(0.0017 / 0.00207) = 43.4 kips Σ Fy = 0: Pn = ΣC - ΣT = 297.8 + 76.8 + 154.4 + 52.8 + 48.5 -52.8 - 43.4 = 534 kips

φPn = (0.9)(534) = 481 kips Σ Mcl = 0: Mn = 40Cm1 + 24Cm2 + 11.85Cm3 + 44Cs1 + 36Cs2 + 44Ts1 + 36Ts2 = 23,540 in.-kips

φMn = (0.9)(23,540) = 21,186 in.-kips = 1,765 ft-kips The two cases are plotted in Figure 10.1-20 to develop the φPn - φMn curve on the pier. The demand (Pu, - Mu) also is plotted. As can be seen, the pier design is acceptable because the demand is within the φPn - φMn curve. (See the Birmingham 1 example in Section 10.2 for additional discussion of φPn - φMn curves.) By linear interpolation, φMn at the minimum axial load is 1,096 ft- kip. The authors note that the use of φ = 0.9 on Pn at the balance point is consistent with TMS 402, but, because of the ductility requirement, the balance point will never be reached. The maximum Pn for this pier, as per the ductility requirement (from Sec. 10.1.6.4), would be (397 kips - 149 kips) = 248 kips (as discussed above), well below the 481 kips at Pb. To illustrate the point, this maximum expressed as φPnmax = 223 kips, is illustrated in Figure 10.1-20.

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

φ Pn

M u = 523 ft-kips

600 kips 500 kips

M u = 1096 ft-kips

φMn

Balance: (1765 ft-kips, 481 kips)

400 kips

φ Pn

Simplified φ Pn - φ M n curve

300 kips 200 kips 100 kips

Pu,min = 41 kips Pu,max = 67 kips

φ Pn max for ductility = 223 kips

P=0 (988 ft-kips, 0 kips) 500 ft-kips

1,000 ft-kips

φMn

1,500 ft-kips

2,000 ft-kips

Figure 10.1-20 In-plane φPn - φMn diagram for pier (1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m) 10.1.6.5 Combined loads. Although it is not required by the Standard, it is educational to illustrate the orthogonal combination of seismic loads for this pier (as if Standard Section 12.5.3.a were required), shown in Table 10.1-4: Table 10.1-4 Combined Loads for Flexure in End Wall Pier Out-of-Plane In-Plane Total 0.614D Case 1 1.0(87.7/112.8) + 0.3(523/1026) = 0.93 < 1.00 Case 2 0.3(87.7/112.8) + 1.0(523/1026) = 0.74 < 1.00 Values are in kips; 1.0 kip = 4.45 kN.

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OK OK

Chapter 10: Masonry

10.1.6.6 Shear – end walls. 10.1.6.6.1 In-plane shear at end wall piers. The in-plane shear at the base of the pier is 43.6 kips per bay. At the head of the opening where the moment demand is highest, the in-plane shear is slightly less (based on the weight of the pier). There, V = 43.6 kips - (0.286)(8 ft)(12 ft)(0.103 ksf) = 40.8 kips. Per TMS 402 Section 1.17.3.2.6.1.1, the design shear strength, φVn, must exceed the shear corresponding to the development of 1.25 times the nominal flexural strength, Mn, or 2.5Vu, whichever is smaller. Using the results in Figure 10.1-20, the 125 percent implies a factor on shear by analysis of:

⎛ φ M n 1.25 ⎜ ⎝ M u

⎞ ⎛ 1 ⎞ ⎛ 1096 ⎞⎛ 1 ⎞ ⎟ ⎜ ⎟ (Vu ) = 1.25 ⎜ ⎟⎜ ⎟Vu = 2.91 Vu ⎝ 523 ⎠⎝ 0.9 ⎠ ⎠ ⎝ φ ⎠

But 2.91Vu > 2.5Vu; therefore, 2.5Vu controls (TMS 402 Sec. 1.17.3.2.6.1.1). Therefore, the required shear capacities at the base and head of the pier are (2.5)(43.6 kips) = 109 kips and (2.5)(40.8) = 102 kips, respectively. The in-plane shear capacity is computed as follows where the net area, An, of the pier is the area of face shells plus the area of grouted cells and adjacent webs:

⎡ ⎛ M Vm = ⎢ 4.0 − 1.75 ⎜ u ⎢⎣ ⎝ Vu dv

⎞ ⎤ ⎟ ⎥ An f mʹ′ + 0.25Pu ⎠ ⎥⎦

As discussed previously, Mu/Vudv need not exceed 1.0 in the above equation. An = (96 in. × 1.50 in. × 2) + (6 cells × 8 in. × 8.63 in.) = 702 in.2 / bay Recall that horizontal reinforcement is 2-#5 at 48 inches in bond beams:

⎛ A ⎞ Vns = 0.5 ⎜ v ⎟ f y d v ⎝ s ⎠ ⎛ 0.62 in.2 = 0.5 ⎜ ⎝ 48 in. =37.2 kips/bay

⎞ ⎟ (60 ksi)(96 in.) ⎠

At the base of the pier: Vm = [4.0 - 1.75(0)](702 in.2)(0.0447 ksi) + (0.25)(0.614 × 55.0 kips) Vm = 79.0 kips/bay

φVn = (0.8)(79.0 + 37.2) = 116.2 kips/bay > 109 kips/bay = 2.5 Vu

At the head of the pier: Vm = [4.0 - 1.75(1.0)](702 in.2)(0.0447 ksi) + (0.25)(0.614 × 45.1 kips) = 77.5 kips/bay 10-41

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

φVn = (0.8)(77.5 + 37.2) = 91.8 kips/bay < 102 kips/bay = 2.5 Vu

N.G.

This non-ductile situation can be addressed by increasing the compression capacity. For this case, the other cells in the pier will be grouted, resulting in An = bwd = (11.63 in.)(92 in.) = 1070 in.2. (Note that while TMS 402 permits An = bwdv, the authors have elected to use the slightly more conservative bwd in the determination of area.) This results in Vm = 114.5 kips and φVn = 121 kips > 102 kips = 2.5 Vu which is OK. Note: The design of the piers in the end walls of this example will remain the same without iteration to reflect the additional grouted cells. Note also that there is no additional vertical reinforcement; only grout has been added to the cells.

V roof + side wall

(upper portion)

V end wall

30'

12' 12'

V base Pbase

P M V Ppier

V base M base Pbase

Figure 10.1-21 In-plane shear at end wall

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Chapter 10: Masonry 10.1.6.6.1 Out-of-Plane Shear at End Wall Piers. For out-of-plane shear, see Figure 10.1-12. Shear at the top of wall is 15.3 kips/bay, and shear at the base of the pier is 10.3 kips/bay. From the RISA-2D analysis, which included P-delta, the shear at the head of the opening is 4.57 kips. The same multiplier of 2.5 for development of 125 percent of flexural capacity will be applied to out-of-plane shear resulting in 38.25 kips at the top of the wall, 11.4 kips at the head of the opening (top of pier) and 25.8 kips at the base of the pier. Out-of-plane shear capacity is computed using the same equation. Σbwd is taken as the net area An. Note that Mu/Vudv is zero at the support because the moment is assumed to be zero; however, a few inches into the vertical span, Mu/Vudv will exceed 1.0, so the limiting value of 1.0 is used here. This is typically the case where considering out-of-plane loads on a wall. For computing shear capacity at the top of the wall: An = bwd = ((8 in./2 ft) × 20 ft)(9.25 in.) = 740 in.2 Vm = [4.0 - 1.75(1)](740 in.2)(0.0447 ksi) + (0.25)(0.614 × 8.0) = 75.7 kips/bay

φVm = (0.8)(75.7) = 60.5 kips/bay For computing shear capacity in the pier: An = (8 in./cell)(12 cells)(9.25 in.) = 888 in.2 Vm = [4.0 - 1.75(1)](888 in.2)(0.0447 ksi) + (0.25)(0.614 × 41.67) = 95.7 kips/bay

φVm = (0.8)(95.7) = 76.6 kips/bay At the top of wall:

φVn = φVm = 60.5 kips/bay > 15.3 kips/bay = Vu

At the pier:

φVn = φVm = 76.6 kips/bay > 10.3 kips/bay = Vu

10-43

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples 10.1.7 In-‐Plane Deflection – End Walls Deflection of the end wall (short wall) has two components as illustrated in Figure 10.1-22.

Δtot Δ2

V s (end wall)

12'

17.8'

Δ1

12'

28'

5.8'

16'

18'

V f + V (side wall)

Figure 10.1-22 In-plane deflection of end wall (1.0 ft = 0.3048 m) As obtained from the RISA-2D analysis of the piers, Δ1 = 0.047 in.:

Δ2 = ∑

αVL AG

where α is the form factor equal to 6/5 and: G = Em/2(1 + µ) = 1,800 ksi / 2(1 + 0.15) = 782 ksi A = An = area of face shells + area of grouted cells = (100 ft × 12 in./ft × 2 × 1.50 in.2) +(50)(8 in.)(8.63 in.) = 7,050 in.2 Note: Contribution to base shear of end walls (above the doors) is Cs (end wall weight) = (0.286)[(470 kips/2) - (103 psf)(5)(8 ft)(12 ft)] = 53 kips. Contribution to base shear of long walls plus roof is Cs (long wall + roof weight) = (0.286)[(400+418)/2] = 117 kips. Therefore:

⎛ 6 ⎞ (53)(5.8 ×12) ⎛ 6 ⎞ (117)(16 ×12) = 0.0008 + 0.0049 = 0.006 in. Δ 2 = ⎜ ⎟ + ⎜ ⎟ ⎝ 5 ⎠ (7,050)(782) ⎝ 5 ⎠ (7,050)(782) and Δtotal = Cd(0.047 + 0.006) = 3.5(0.053 in.) = 0.19 in. < 2.35 in. where (2.35 = 0.007hn = 0.01hsx) (TMS 402 Sec. 3.3.5.4).

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Chapter 10: Masonry

Note that the drift limits for masonry structures are smaller than for other types of structure. It is possible to interpret Standards Table 12.12-1 to give a limit of 0.007hn for this structure, but that limit also is easily satisfied. The real displacement in this structure is in the roof diaphragm; see Sec. 11.2.4.2.3. 10.1.8 Bond Beam – Side Walls (and End Walls) Reinforcement for the bond beam located at the elevation of the roof diaphragm can be used for the diaphragm chord. The uniform lateral load for the design of the chord is the lateral load from the long wall plus the lateral load from the roof and is equal to 1.17 klf. The maximum tension in rebar is equal to the maximum moment divided by the diaphragm depth:

(1.17 klf )(200ft)2 = 5,850 ft-kips 8

M/d = 5,850 ft-kips/100 ft = 58.5 kips The seismic load factor is 1.0. The required reinforcement is: Areqd = T/φFy = 58.5/(0.9)(60) = 1.081 in.2 This will be satisfied by two #7 bars, As = (2 × 0.60 in.2) = 1.20 in.2 In Sec. 11.2.4.2.2, the diaphragm chord is designed as a wood member utilizing the wood ledger member. Using either the wood ledger or the bond beam is considered acceptable.

10.2 FIVE-‐STORY MASONRY RESIDENTIAL BUILDINGS IN BIRMINGHAM, ALABAMA; ALBUQUERQUE, NEW MEXICO; AND SAN RAFAEL, CALIFORNIA 10.2.1 Building Description In plan, this five-story residential building has bearing walls at 24 feet on center (see Figures 10.2-1 and 10.2-2). All structural walls are of 8-inch-thick concrete masonry units (CMU). The floor is of 8-inchthick hollow core precast, prestressed concrete planks. To demonstrate the incremental seismic requirements for masonry structures, the building is partially designed for four locations: two sites in Birmingham, Alabama; a site in Albuquerque, New Mexico; and a site in San Rafael, California. The two sites in Birmingham have been selected to illustrate the influence of different soil profiles at the same location. The building is designed for Site Classes C and E in Birmingham.

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

40'-0"

24'-0"

40'-0"

4'-0"

24'-0"

Prestressed hollow core slabs

6'-0"

14'-0"

24'-0"

72'-0"

6'-8"

24'-0"

8" concrete masonry wall

152'-0"

Figure 10.2-1 Typical floor plan (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m)

80'-0"

36'-0"

5 at 8'-8" = 43'-4"

45'-4"

2'-0"

152'-0" 36'-0"

Figure 10.2-2 Building elevation (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) For the Albuquerque and both Birmingham sites, it is assumed that shear friction reinforcement in the joints of the diaphragm planks is sufficient to resist seismic forces, so no topping is used. For the San Rafael site, a cast-in-place 2½-inch-thick reinforced lightweight concrete topping is applied to all floors. The structure is free of irregularities both in plan and elevation. ACI 318, Sections 21.1.1.6 and 21.11.1, require reinforced cast-in-place toppings as diaphragms in Seismic Design Category D and higher. Thus, the Birmingham example in Site Class E / Seismic Design Category D would require a topping, although that is not included in this example. The design of an untopped diaphragm (for Seismic Design Categories A, B and C) is not addressed explicitly in ACI 318. The designs of both untopped and topped diaphragms for these buildings are

10-46

Chapter 10: Masonry described in Chapter 8 of this volume using ACI 318 for the topped diaphragm in the San Rafael building. The Provisions provide guidance for the design of untopped precast plank diaphragms in Part 3, RP10. For the purpose of determining the site class coefficient (Standard Sec. 11.4.2 and 20.3), a stiff soil profile with standard penetration test results of 15 < N < 50 is assumed for the San Rafael site resulting in a Site Class D for this location. The Birmingham 1 and Albuquerque sites have soft rock with N > 50, resulting in Site Class C. The Birmingham 2 site has soft clay with N < 15, which results in Site Class E. The two Birmingham sites are presented to illustrate how different soil conditions at the same location (same seismicity) can result in different Seismic Design Categories. No foundations are designed in this example. The foundation systems are assumed to be able to carry the superstructure loads including the overturning moments. The masonry walls in two perpendicular directions act as bearing and shear walls with different levels of axial loads. The geometry of the building in plan and elevation results in nearly equal lateral resistance in both directions. The walls are constructed of CMU and typically are minimally reinforced in all locations. Figure 10.2-3 illustrates the wall layout.

A 4

Wall length A 36'-0" B 34'-0' C 32'-8" D 32'-8" E 8'-0" F 8'-0" G 8'-0"

B E

D A

Figure 10.2-3 Plan of walls (1.0 ft = 0.3048 m) The floors serve as horizontal diaphragms distributing the seismic forces to the walls and are assumed to be stiff enough to be considered rigid. There is little information about the stiffness of untopped precast diaphragms. The design procedure in Section RP10 of Part 3 of the Provisions results in a diaphragm intended to remain below the elastic limit until the walls reach an upper bound estimate of strength; therefore, it appears that the assumption is reasonable. Material properties are as follows: §

The compressive strength of masonry, f’m, is taken as 2,000 psi, and the steel reinforcement has a yield limit of 60 ksi. 10-47

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

§

The design snow load (on an exposed flat roof) is less than the roof live load for all locations.

This example covers the following aspects of a seismic design: §

Determining the equivalent lateral forces

§

Design of selected masonry shear walls for their in-plane loads

§

Computation of drifts

The story heights are small enough that the design of the masonry walls for out-of-plane forces is nearly trivial. In-plane response governs both the reinforcement in the wall and the connections to the diaphragms. 10.2.2 Design Requirements 10.2.2.1 Seismic parameters. The basic parameters affecting the design and detailing of the buildings are shown in Table 10.2-1. The Seismic Design Category for Birmingham 2 deserves special comment. The value of SDS would imply a Seismic Design Category of C, while the value of SD1 would imply Seismic Design Category D, per Tables 11.6-1 and 11.6-2 of the Standard, where in Section 11.6 a provision permits the use of Table 11.6-1 alone if T < 0.8 SD1/SDS and the floor diaphragm is considered rigid or has a span of less than 40 feet. As will be shown for this building, Ta = 0.338 seconds and 0.8 SD1/SDS = 0.446. In the author’s opinion, the untopped diaphragm may not be sufficiently rigid and thus Table 11.6-2 is considered, resulting in Seismic Design Category D. 10.2.2.2 Structural design considerations. The floors act as horizontal diaphragms, and the walls parallel to the motion act as shear walls for all four buildings. The system is categorized as a bearing wall system (Standard Sec. 12.2). For Seismic Design Category D, the bearing wall system has a height limit of 160 feet and must comply with the requirements for special reinforced masonry shear walls. Note that the structural system is one of uncoupled shear walls. Crossing beams over the interior doorways (their design is not included in this example) will need to continue to support the gravity loads from the deck slabs above during the earthquake, but are not designed to provide coupling between the shear walls. The building is symmetric and appears to be regular both in plan and elevation. It will be shown, however, that the building is torsionally irregular. Standard Table 12.6-1 permits use of the ELF procedure in accordance with Standard Section 12.8 for Birmingham 1 and Albuquerque (Seismic Design Categories B and C). By the same table, the Seismic Design Category D buildings (Birmingham 2 and San Rafael) must use a dynamic analysis for design. A careful reading of Standard Table 12.6-1 for Seismic Design Category D reveals that all of the rows do not apply to our building except the last, “all other structures”; thus, ELF analysis is not permitted, but modal analysis is permitted. For this particular building arrangement, it will be shown that the modal response spectrum analysis does not identify any particular effect of the horizontal torsional irregularity, as will be illustrated; thus it is the authors’ opinion that ELF analysis would be sufficient.

10-48

Chapter 10: Masonry Table 10.2-1 Design Parameters Value for Value for Value for Design Parameter Birmingham 1 Birmingham 2 Albuquerque Ss (Map 1) 0.266 0.266 0.456 S1 (Map 2) 0.105 0.105 0.137 Site Class C E C Fa 1.2 2.45 1.2 Fv 1.7 3.49 1.66 SMS = FaSs 0.32 0.65 0.55 SM1 = FvS1 0.18 0.37 0.23 SDS = 2/3 SMS 0.21 0.43 0.37 SD1 = 2/3 SM1 0.12 0.24 0.15 Seismic Design B D C Category Diaphragm Topping req’d per No Yes* No ACI 318? Ordinary Special Intermediate Masonry Wall Type Reinforced Reinforced Reinforced Standard Design Coefficients (Table 12.2-1) R 2.0 5 3.5 Ω0 2.5 2.5 2.5 Cd 1.75 3.5 2.25

Value for San Rafael 1.5 0.6 D 1 1.5 1.5 0.9 1 0.6 D Yes Special Reinforced 5 2.5 3.5

*For this masonry example, Birmingham 2 is designed without topping on the precast planks. It is assumed that the precast planks at floors and roof have connections sufficiently rigid to permit the idealization of rigid horizontal diaphragms.

The type of masonry shear wall is selected to illustrate the various requirements as well as to satisfy Table 12.2-1 of the Standard. Note that “Ordinary Reinforced Masonry Shear Walls” could be used for Seismic Design Category C at this height. The orthogonal direction of loading combination requirement (Standard Sec. 12.5) needs to be considered for structures assigned to Seismic Design Category D. However, the arrangement of this building is not particularly susceptible to orthogonal effects; the walls are not subject to axial force from horizontal seismic motions, only bending and shear. The walls are all solid, and there are no significant discontinuities, as defined by Standard Section 12.3.2.2, in the vertical elements of the seismic force-resisting system. Ignoring the short walls at stairs and elevators, there are eight shear walls in each direction; therefore, the system appears to have adequate redundancy (Standard Sec. 12.3.4.2). The redundancy factor, however, will be computed. Tie and continuity requirements (Standard Sec. 12.11) must be addressed when detailing connections between floors and walls (see Chapter 8 of this volume). Nonstructural elements (Standard Chapter 13) are not considered in this example.

10-49

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Collector elements are required in the diaphragm for longitudinal response (Standard Sec. 12.10). Rebar in the longitudinal direction, spliced into bond beams, is used for this purpose (see Chapter 8 of this volume). Diaphragms must be designed for the required forces (Standard Sec. 12.10 and Provisions Part 3, Sec. RP10). The structural walls must be designed for the required out-of-plane seismic forces (Standard Sec. 12.11) in addition to out-of-plane wind on exterior walls and 5 psf differential air pressure on interior walls. Each wall acts as a vertical cantilever in resisting in-plane forces. The walls are classified as masonry cantilever shear wall structures in Standard Table 12.12-1, which limits story drift to 0.01 times the story height. 1 0 . 2 . 3 Load Combinations The basic load combinations are those in Standard Section 2.3.2. The seismic load effect, E, is defined by Standard Section 12.4, as follows: E = Eh + Ev = ρQE ± 0.2SDSD 10.2.3.1 Redundancy Factor. The Redundancy Factor, ρ, is a multiplier on design force effects and applies only to the in-plane direction of the shear walls. For structures in Seismic Design Categories A, B and C, ρ = 1.0 (Standard Sec. 12.3.4.1). For structures in Seismic Design Category D, ρ is determined per Standard Section 12.3.4.2. For a shear wall building assigned to Seismic Design Category D, ρ = 1.0 as long as it can be shown that failure of a shear wall or pier with a height-to-length ratio greater than 1.0 would not result in more than a 33 percent reduction in story strength or create an extreme torsional irregularity. The intent is that the aspect ratio is based on story height, not total height.

height 8' = = 0.24 < 1.0 length 32.67 ' Because no walls have a ratio exceeding 1.0, none have to be removed to check for redundancy and ρ = 1.0. If one were to consider the removal of one shear wall in either direction, 1/8 or 12.5 percent resistance would be removed. 12.5% < 33%, so ρ = 1.0. Therefore, for this example, the redundancy factor is 1.0 for the buildings assigned to Seismic Design Category D. 10.2.3.2 Combination of load effects. The seismic load effect, E, determined for each of the buildings is as follows: Birmingham 1: E = (1.0)QE ± (0.2)(0.21)D = QE ± 0.04D Birmingham 2: E = (1.0)QE ± (0.2)(0.43)D = QE ± 0.09D Albuquerque: E = (1.0)QE ± (0.2)(0.37)D = QE ± 0.07D San Rafael: E = (1.0)QE ± (0.2)(1.00)D = QE ± 0.20D

10-50

Chapter 10: Masonry The applicable load combinations from Standard Sections 2.3.2 and 12.4.2.3 are: 1.2D + 1.0E + 0.5L + 0.2S where the effects of gravity and seismic loads are additive, and 0.9D + 1.0E + 1.6H where the effects of gravity and seismic loads are counteractive. H is the effect of lateral pressures of soil and water in soil. The 0.5 factor on L is because L0 < 100 psf for these residential buildings. Per the Standard, corridors are “same as occupancy served”, except for the first floor. Load effect H does not apply for this design, and the snow load effect, S, does not exceed the minimum roof live load at any of the buildings. Consideration of snow loads is not required in the effective seismic weight, W, of the structure where the design snow load does not exceed 30 psf (Standard Sec. 12.7.2). The basic load combinations are combined with E as determined above, and the load combinations representing the extreme cases are as follows: Birmingham 1: 1.24D + QE +0.5L 0.86D - QE Birmingham 2: 1.29D + QE +0.5L 0.81D - QE Albuquerque:

1.27D + QE +0.5L +0.2S 0.83D - QE

San Rafael:

1.40D + QE +0.5L 0.70D - QE

These combinations are for the in-plane direction. Load combinations for the out-of-plane direction are similar except that the redundancy factor (1.0 in all cases for in-plane loading) is not applicable. 10.2.4 Seismic Design for Birmingham 1 10.2.4.1 Birmingham 1 weights. This site is assigned to Seismic Design Category B, and the walls are designed as ordinary reinforced masonry shear walls (Standard Table 12.2-1), which stipulates that the minimum reinforcement requirements of TMS 402 Section 1.17.3.2.3.1 be followed. Given the length of the walls, vertical reinforcement of #4 bars at 8 feet on center works well for detailing reasons and will be used here (10 feet is the maximum spacing per TMS 402). For this example, 45 psf will be used for the 8-inch-thick lightweight CMU walls. The 45 psf value includes grouted cells, as well as bond beams in the course just below the floor planks. 67 psf is used for 8-inch-thick, normal-weight hollow core plank plus the non-masonry partitions. 67 psf is also used for the roof plank plus roofing. Story weight, wi, is computed as follows.

10-51

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples For the roof: Roof slab (plus roofing) = (67 psf) (152 ft)(72 ft) = 733 kips Walls = (45 psf)(589 ft)(8.67 ft/2) + (45 psf)(4)(36 ft)(2 ft) = 128 kips Total = 861 kips Note that there is a 2-foot-high masonry parapet on four walls and the total length of masonry wall, including the short walls not used in the seismic force-resisting system, is 589 feet. For a typical floor: Slab (plus partitions) = 733 kips Walls = (45 psf)(589 ft)(8.67 ft) = 230 kips Total = 963 kips Total effective seismic weight, W = 861 + (4)(963) = 4,713 kips. This total excludes the lower half of the first story walls, which do not contribute to seismic loads that are imposed on CMU shear walls. 10.2.4.2 Birmingham 1 base shear calculation. The seismic response coefficient, Cs, is computed using Standard Section 12.8. Per Standard Equation 12.8-2:

S DS 0.21 = = 0.105 R/I 21

The value of Cs need not be greater than Standard Equation 12.8-3:

Cs =

S D1 0.12 = = 0.178 T ( R / I ) 0.338 ( 2 1)

where T is the fundamental period of the building approximated per Standard Equation 12.8-7 as follows:

T = C h x = (0.02)(43.330.75 ) = 0.338 sec a t n where Ct = 0.02 and x = 0.75 are from Standard Table 12.8-2 (the approximate period, based on building system and building height, is the same for all locations). The value for Cs is taken as 0.105 (the lesser of the two computed values). This value is larger than the minimum specified in Standard Equation 12.8-5 (Sup. 2): Cs = 0.044 ISDS ≥ 0.010

10-52

Chapter 10: Masonry

= ( 0.044)(1.0)( 0.21) = 0.00924 = 0.010

(0.105 controls)

The total seismic base shear is then calculated using Standard Equation 12.8-1 as follows: V = CsW = (0.105)(4,713) = 495 kips 10.2.4.3 Birmingham 1 vertical distribution of seismic forces. Standard Section 12.8.3 stipulates the procedure for determining the portion of the total seismic load assigned to each floor level. The story force, Fx, is calculated using Standard Equations 12.8-11 and 12.8-12 as follows: Fx = CvxV and

wx hxk

Cvx =

∑ wi hik

i =1

where Cvx is a vertical distribution factor which has the effect of distributing more of the base shear to the upper levels to mimic the dynamic response of the structure. For T = 0.338 sec < 0.5 sec, k = 1.0. The seismic design shear in any story is determined from Standard Equation 12.8-13: n

Vx = ∑ Fi i=x

Although not specified in the Standard or used in design, story overturning moment may be computed using the following equation: n

M x = ∑ Fi (hi − hx ) i=x

The application of these equations for this building is shown in Table 10.2-2. Table 10.2-2 Birmingham 1 Seismic Forces and Moments (i.e., Seismic Demand) by Level Level x

wx (kips)

hx (ft)

w xh xk (ft-kips)

Cvx

5 4 3 2 1 ∑

861 963 963 963 963 4,715

43.34 34.67 26.00 17.33 8.67

37,310 33,384 25,038 16,692 8,346 120,770

0.3089 0.2764 0.2073 0.1382 0.0691 1.0000

Fx (kips) 153 137 103 68 34 495

Vx (kips) 153 290 393 461 495

M(x-1) (ft-kips) 1,326 3,840 7,245 11,240 15,530

10-53

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples 1.0 kips = 4.45 kN, 1.0 ft = 0.3048 m.

Note that Fx, Vx and Mx are all factored loads. A note regarding locations of V and M: the vertical weight at the roof (fifth level), which includes the upper half of the wall above the fifth floor (fourth level), produces an inertial force that contributes to the shear, V, the walls supporting the fifth level. That shear in turn generates a moment that increases towards the level below (fourth level). Resisting this moment is the rebar in the wall combined with the wall weight above the fourth level. The story overturning moment is tabulated for the level below the level that receives the story force. This is illustrated in Figure 10.2-4.

Proof

V roof h

Contribution to weight concentrated at roof. Only upper half of walls out of plane contribute, but upper half of all walls used for convenience.

Dynamic response to ground motion results in lateral load applied at roof.

Moment at fifth floor M 5 = V roof h

P of roof slab plus entire height of wall helps to resist M 5.

Contribution to weight concentrated at all stories.

Dynamic response to round motion results in lateral load at all stories.

Moments are from ∑ Vh

Weight of entire building above ground floor helps to resist moments.

Figure 10.2-4 Location of moments due to story shears 10.2.4.4 Birmingham 1 horizontal distribution of forces. The wall lengths are shown in Figure 10.2-3. The initial grouting pattern is essentially the same for Walls A, B, C and D. Because of a low relative stiffness, the effects Walls E, F and G are ignored in this analysis. Walls A, B, C and D are so nearly the same length that their stiffnesses are assumed to be the same for this example.

10-54

Chapter 10: Masonry

Torsion is considered according to Standard Section 12.8.4. For a symmetric plan, as in this example, the only torsion to be considered is the accidental torsion, Mta, caused by an assumed eccentricity of the mass each way from its actual location by a distance equal to 5 percent of the dimension of the structure perpendicular to the direction of the applied loads. Dynamic amplification of the torsion need not be considered for Seismic Design Category B per Standard Section 12.8.4.3. For this example, the building is analyzed in the transverse direction only. The evaluation of Wall D is selected for this example. The rigid diaphragm distributes the lateral forces into walls in both directions. Two components of force must be considered: direct shear and shear induced by torsion. The direct shear force carried by each Wall D is one-eighth of the total story shear (eight equal walls). The torsional moment per Standard Section 12.8.4.2 is as follows: Mta = 0.05bVx = (0.05)(152 ft)Vx = 7.6Vx The torsional force per wall, Vt, is:

Vt =

M t Kd ∑ Kd 2

where K is the stiffness (rigidity) of each wall. Note that all the walls in both directions are included. Because all the walls in this example are assumed to be equally long, then they are equally stiff:

⎡ d ⎤ Vt = M t ⎢ 2 ⎥ ⎢⎣ ∑ d ⎥⎦ where d is the distance from each wall to the center of twisting. ∑d2 = 4(36)2 + 4(12)2 + 4(36)2 + 4(12)2 = 11,520 ft2 The maximum torsional shear force in Wall D, therefore, is:

Vt =

(7.6V )(36 ft) = 0.0238V 11,520 ft 2

The total shear in Wall D is:

Vtot = 0.125V + 0.0238V = 0.149V The total story shear and overturning moment may now be distributed to Wall D and the wall proportions checked. The wall capacity is checked before considering deflections. 10.2.4.5 Birmingham 1 transverse wall (Wall D). The Provisions and the Standard define the seismic load as a strength or limit state design level effect. TMS 402 Chapter 3 defines strength design for

10-55

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples masonry. Strength design of masonry, as defined in TMS 402, is illustrated here. It is also permissible to use the allowable stress design method of TMS 402 by factoring the seismic load effects, but that will not be illustrated here. The required strength is derived from the load combinations defined previously. 10.2.4.5.1 Birmingham 1 shear strength. TMS 402 Section 3.1.3 states that the design strength must be greater than the required strength. The design strength is equal to the nominal strength times a strength-reduction factor: Vu ≤ φVn The strength reduction factor, φ, is 0.8 (TMS 402 Sec. 3.1.4.3). The nominal shear strength, Vn, is: Vn = Vnm + Vns Likewise:

φVn = φ(Vnm + Vns) The shear strength provided by masonry (TMS 402 Sec. 3.3.4.1.2.1) is as follows:

⎡ ⎛ M Vnm = ⎢ 4.0 − 1.75 ⎜ u ⎢⎣ ⎝ Vu dv

⎞ ⎤ ⎟ ⎥ An f mʹ′ + 0.25 Pu ⎠ ⎥⎦

For grouted cells at 8 feet on center: An = (2 × 1.25 in. × 32.67 ft × 12 in./ft) + (8 in. × 5.13 in. × 5 cells) = 1,185 in.2 The shear strength provided by reinforcement (given by TMS 402 Sec. 3.3.4.1.2.3) is as follows:

⎛ A ⎞ Vns = 0.5 ⎜ v ⎟ Fy dv ⎝ s ⎠ The wall will have a bond beam with two #4 bars at each story to bear the precast floor planks and wire joint reinforcement at alternating courses. Common joint reinforcement with 9-gauge wires at each face shell will be used; each wire has a cross-sectional area of 0.017 in.2. With six courses of joint reinforcement and two #4 bars, the total area per story is 0.60 in.2 or 0.07 in.2/ft. Given that the story height is less than half the wall length, the authors believe that it is acceptable to treat the distribution of horizontal reinforcement as if being uniformly distributed for shear resistance. Vns = 0.5(0.07 in.2/ft)(60 ksi)(32.67 ft) = 68.3 kips The maximum nominal shear strength of the member (Wall D in this case) for M/Vdv > 1.00 is given by TMS 402 Section 3.3.4.1.2.1:

10-56

Chapter 10: Masonry

Vn (max) = 4 An f mʹ′ The coefficient 4 becomes 6 for M/Vdv < 0.25 (TMS 402 Sec. 3.3.4.1.2a). Interpolation between yields the following:

⎛ ⎛ M Vn (max) = ⎜ 6.67 − 2.67 ⎜ u ⎜ ⎝ Vu dv ⎝

⎞ ⎞ ' ⎟ ⎟⎟ An f m ⎠ ⎠

The shear strength of Wall D, based on the equations listed above, is summarized in Table 10.2-3. Note that Vx and Mx in this table are values from Table 10.2-2 multiplied by 0.149 (which represents the portion of direct and torsional shear assigned to Wall D). Pu is the dead load of the roof or floor times the tributary area for Wall D, taken as 0.86D for the minimum (conservative) Pu. (Note that there is a small load from the floor plank parallel to the wall.) Table 10.2-3 Shear Strength Calculations for Birmingham 1 Wall D Vx Mx Vu = Vx Pu Vnm Vns Story Mx/Vxdv (kips) (ft-kips) (kips) (kips) (kips) (kips) 5 22.8 198 0.266 22.8 35.3 196 68 4 43.2 572 0.405 43.2 76.5 193 68 3 58.6 1080 0.564 58.6 118 189 68 2 68.7 1675 0.746 68.7 158 182 68 1 73.8 2314 0.960 73.8 200 173 68

Vn (kips) 264 261 257 250 241

Vn (max) (kips) 316 296 274 248 218

φVn (kips) 211 209 206 198 174

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

Values shown in bold are the controlling values for Vn For all levels, φVn > Vu, so it is OK for this Ordinary Reinforced Masonry Shear Wall. 10.2.4.5.2 Birmingham 1 axial and flexural strength. All the walls in this example are bearing shear walls since they support vertical loads as well as lateral forces. In-plane calculations include: §

Strength check

§

Ductility check

10.2.4.5.2.1 Strength check. The wall demands, using the load combinations determined previously, are presented in Table 9.2-4 for Wall D. In the table, Load Combination 1 is 1.245D + QE + 0.5L and Load Combination 2 is 0.86D + QE.

10-57

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Table 10.2-4 Demands for Birmingham 1 Wall D Load Combination 1

Load Combination 2

Pu (kips) 61

Pu (kips) 42

129

572

147

195

1,080

126

1,080

196

260

1,675

168

1,675

245

324

2,314

210

2,314

Story

PD (kips)

PL (kips)

Mu (ft-kips) 198

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

PD and PL are based on floor tributary area of 540 ft2. PL has been reduced per Standard Section 4.8 using KLL = 2. Strength at the bottom story (where P, V and M are the greatest) is examined here. (For a real design, all levels should be examined). As will be shown, Load Combination 2 from Table 10.2-4 is the controlling case because it has the same lateral load as Load Combination 1, but with lower values of axial force. For the base of the shear walls:

Pumin

= 210 kips

Pumax

= 324 kips

Mu = 2,314 ft-kips Try one #4 bar in each end cell and a #4 bar at 8 feet on center for the interior cells. A curve of φPn - φMn , representing the wall strength envelope, are developed and used to evaluate Pu and Mu determined above. Three cases are analyzed and their results are used in plotting the φPn - φMn curve. In accordance with TMS 402 Section 3.3.2, the strength of the section is reached as the compressive strains in masonry reach their maximum usable value of 0.0025 for CMU. The force equilibrium in the section is attained by assuming an equivalent rectangular stress block of 0.8f’m over an effective depth of 0.8c, where c is the distance of the neutral axis from the fibers of maximum compressive strain. Stress in all steel bars is taken into account. The strains in the bars are proportional to their distance from the neutral axis. For strains above yield, the stress is independent of strain and is taken as equal to the specified yield strength, Fy. See Figure 10.2-5 for strains and stresses for all three cases selected. Case 1 (P = 0) Assume all tension bars yield (which can be verified later): Ts1 = (0.20 in.2)(60 ksi) = 12.0 kips Ts2 = (0.20 in.2)(60 ksi) = 12.0 kips each 10-58

Chapter 10: Masonry

Because the neutral axis is close to the compression end of the wall, compression steel, Cs1, is neglected (it would make little difference anyway) for Case 1: ΣFy = 0: Cm = ΣT Cm = (4)(12.0) = 48.0 kips The compression block is entirely within the first grouted cell: Cm = 0.8 f’mab 48.0 = (0.8)(2.0 ksi)a(7.625 in) a = 3.9 in. = 0.33 ft c = a/0.8 = 0.33/0.8 = 0.41 ft Thus, the neutral axis is determined to be 0.41 feet from the compression end on the wall, which is within the first grouted cell: ΣMcl = 0: (The math is a little easier if moments are taken about the wall centerline.) Mn = (16.33-0.33/2 ft)Cm + (16.00 ft)Ts1 + (0.00 ft)ΣTs2 + (0.00 ft)Pn Mn = (16.17)(48.0) + (16.00)(12) + 0 + 0 = 968 ft-kips

φMn = (0.9)(968) = 871 ft-kips

10-59

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

P M

32.66'

8.00' (typical)

16.33'

#4 at end #4 at 8'-0" o.c. 16.00'

0.33'

Cm tot 0.8 f 'm

ε m = 0.0025

P = 0 Case

ΣT s2

T sl a = 0.33' T s2

T s2

fy E = 0.00207

ε y=

N.A.

c = 0.41'

Cm cell

Cm shell

N.A.

0.8 f 'm

ε m = 0.0025 ε y= 0.00207

T s2*

T sl

a = 6.40' T s2

c = 8.00'

Cm cells

Cm shell

0.8 f 'm

T s2

ε m = 0.0025 C s1

Intermediate Case

Balanced Case

*C s2

T sl

a = 14.15'

*Ts2

c = 17.69' 16.33'

16.00' Center Line

Figure 10.2-5 Strength of Birmingham 1 Wall D. Strain diagram superimposed on strength diagram for the three cases. *The low force in the selected reinforcement is neglected in the calculations. (1.0 ft = 0.3048 m)

10-60

ε y= 0.00207

N.A.

C s1

ΣT

ε y= 0.00207

Chapter 10: Masonry

To summarize, Case 1:

φPn = 0 kips φMn = 871 ft-kips Case 2 (Intermediate case between P = 0 and Pbal) Let c = 8.00 feet (this is an arbitrary selection). Thus, the neutral axis is defined at 8 feet from the compression end of the wall: a = 0.8c = (0.8)(8.00) = 6.40 ft Cm shells = 0.8f’m(2 shells)(1.25 in. / shell)(6.40 ft (12 in./ft) = 307.2 kips Cm cells = 0.8 f’m(41 in.2) = 65.6 kips Cm tot = Cm shells + Cm cells = 307.2 + 65.6 = 373 kips Cs1 = (0.20 in.2)(60 ksi) = 12 kips (Compression steel is included in this case.) Ts1 = (0.20 in.2)(60 ksi) = 12 kips Ts2 = (0.20 in.2)(60 ksi) = 12 kips each Some authorities would interpret TMS 402 Section 3.1.8.3 to mean the compression resistance of reinforcing steel that is not enclosed within ties be neglected. The TMS 402 Commentary 3.3.3.5 allows inclusion of compression in the reinforcement where the maximum amount of reinforcement is limited to promote flexural ductility. The authors have chosen to follow the more specific example of the commentary. The difference in flexural resistance is not significant, but the difference for the maximum reinforcement requirement (i.e., ductility requirement) can be significant. ΣFy = 0: Cm tot + Cs1 = Pn + Ts1 + ΣTs2 373 + 12 = Pn + (3)(12.0) Pn = 349 kips

φPn = (0.9)(349) = 314 kips ΣMcl = 0: Mn = (13.13 ft)Cm shell + (16.00 ft)(Cm cell + Cs1) + (16.00 ft)Ts1 + (8.00 ft)Ts2 Mn = (13.13)(307.2) + (16.00)(65.6 + 12) + (16.00)(12.0) + (8.00 ft)(12.0) = 5,563 ft-kips

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

φMn = (0.9)(5,563) = 5,007 ft-kips To summarize Case 2:

φPn = 314 kips φMn = 5,007 ft-kips Case 3 (Balanced case) In this case, Ts1 just reaches its yield stress:

⎡ ⎤ 0.0025 c = ⎢ ⎥ (32.33 ft ) = 17.69 ft ⎣ (0.0025 + 0.00207) ⎦ a = 0.8c = (0.8)(17.69) = 14.15 ft Cm shells = 0.8f’m(2 shells)(1.25 in./shell)(14.15 ft) (12 in./ft) = 679.2 kips Cm cells =0.8f’m(2 cells)(41 in.2/cell) = 131.2 kips Cm tot = Cm shells + Cm cells = 810.4 kips Cs1 = (0.20 in.2)(60 ksi) = 12.0 kips Ts1 = (0.20 in.2)(60 ksi) = 12.0 kips Cs2 and Ts2 are neglected because they are small. ΣFy = 0: Pn = ΣC - ΣT Pn = Cm tot + Cs1 - Ts1 = 810.4 + 12.0 - 12.0 = 810.4 kips

φPn = (0.9)(810.4) = 729 kips ΣMcl = 0: Mn = 9.26 Cm shells + ((16 + 8)/2) Cm cells + 16 Cs1 + 16 Ts1 Mn = (9.26)(679.2) + (12.0)(131.2) + (16.00)(12.0) + (16.0)(12.0) = 8,248 kips

φMn = (0.9)(8,248) = 7,423 ft-kips

10-62

Chapter 10: Masonry To summarize Case 3:

φPn = 729 kips φMn = 7,423 ft-kips Using the results from the three cases above, the φPn - φMn curve shown in Figure 10.2-6 is plotted. Although the portion of the φPn - φMn curve above the balanced failure point could be determined, it is not necessary here. Thus, only the portion of the curve below the balance point is examined. This is the region of high moment capacity. Similar to reinforced concrete beam-columns, in-plane compression failure of the cantilevered shear wall will occur if Pu > Pbal and yield of tension steel will occur first if Pu < Pbal. A ductile failure mode is essential to the design, so the portion of the curve above the “balance point” is not useable. In fact, the ductility (maximum reinforcement) requirement prevents Pu from approaching Pbal. As can be seen, the points for (Pu min , Mu) and (Pu max , Mu) are within the φPn - φMn envelope; thus, the strength design is acceptable with the minimum reinforcement. Figure 10.2-6 shows two schemes for determining the design flexural resistance for a given axial load. One interpolates along the straight line between pure bending and the balanced load. The second makes use of intermediate points for interpolation. This particular example illustrates that there can be a significant difference in the interpolated moment capacity between the two schemes for axial loads midway between the balanced load and pure bending.

10-63

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

φ Pn

φMn

M u = 2314 ft-kips

1,000 kips

φ Pn

500 kips

Balance: (7423 ft-kips, 729 kips)

Actual φ Pn - φ M n curve Pu,max = 324 kips Pu,min = 210 kips

5,000 ft-kips P=0 (871 ft-kips, 0 kips)

3-Point φ Pn - φ M n curve Simplified φ Pn - φ M n curve Intermediate: (5007 ft-kips, 314 kips)

φMn

10,000 ft-kips

Figure 10.2-6 φPn - φMn Diagram for Birmingham 1 Wall D (1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m) 10.2.4.5.3 Ductility check. For this case, with Mu/Vudv < 1 and R > 1.5, TMS 402 Section 3.3.3.5.4 refers to Section 3.3.3.5.1, which stipulates that the critical strain condition correspond to a strain in the extreme tension reinforcement equal to 1.5 times the strain associated with Fy. This calculation uses unfactored gravity loads. Refer to Figure 10.2-7 and the following calculations which illustrate this use of loads at the bottom story (highest axial loads). Calculations for other stories are not presented in this example.

10-64

Chapter 10: Masonry

32'-8" P

0.33'

32.34'

ε m = 0.0025

17.90' N.A.

14.44' c

ε s = 1.5ε y

8.66'

60 ) 29,000 = 0.0031 = 1.5 (

0.8 f 'm = 1.6 ksi

2.89'

11.55' a

17.90' 9.90'

f y = 60 ksi

1.90' 31.37 ksi C s1

C s2 T s3

6.10'

T s1

9.55 ksi N.A.

14.10'

0.33'

T s2

49.78 ksi

f y = 60 ksi

(5) #4 @ 8'-0" o.c.

Figure 10.2-7 Ductility check for Birmingham 1 Wall D (1.0 ft = 0.3048 m, 1.0 ksi = 6.89 MPa) For Level 1 (bottom story), the unfactored axial loads are: P = 245 kips Refer to Figure 10.2-7:

10-65

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Cm = 0.8 f’m(ab + 2Acell) = (1.6 ksi)[(0.8 × 14.44 ft × 12 in./ft)(2)(1.25 in.) + (2)(41 in.2)] = 686 kips Cs1 = FyAs = (60 ksi)(0.20 in.2) = 12.0 kips Cs2 = (31.37 ksi)(0.20 in.2) = 6.3 kips Ts1 = (60 ksi)(0.20 in.2) = 12.0 kips Ts2 = (49.78 ksi)(0.20 in.2) = 10.0 kips Ts3 = (9.55 ksi)(0.20 in.2) = 1.9 kips. ∑C > ∑P + T Cm + Cs1 + Cs21 > P + Ts1 + Ts2 + Ts3 686 + 12.0 + 6.3 > 245.0 + 12.0 + 10.0 + 1.9 704 kips > 269 kips

There is more compression capacity than required, so the ductile failure condition controls. 10.2.4.6 Birmingham 1 deflections. The calculations for deflection involve many variables and assumptions, and it must be recognized that any calculation of deflection is approximate at best. The Standard requires that deflections be calculated and compared with the prescribed limits set forth by Standard Table 12.12-1. Furthermore, Standard Section 12.7.3 requires that the effect of cracking be considered in establishing the elastic stiffness of masonry elements. In contrast, TMS 402 has two provisions that contradict the Standard: Section 1.17.2.4 effectively dismisses the drift requirement for all masonry shear walls except Special Reinforced Masonry Shear Walls, and Section 1.9.2 permits the use of uncracked stiffness. This example follows the requirements of the Standard. Elastic deflections are calculated considering cracking and then increased by Cd to account for non-linear response during the design earthquake. Recognizing that P-delta effects are minor for the in-plane direction, we solve for δtotal = δflexural + δshear for elastic and increase that value by Cd. The story drift, Δ, is the difference between δtotal for adjacent stories. The following procedure is used for calculating deflections: 1. For purpose of illustration, moments and cracking moments in each story are computed and are shown in Table 10.2-5. 2. Cracking moment is determined from Mcr = S(fr + Pu min / An). 3. Compute deflection for each level. While Icr can be determined from principles of mechanics, the authors prefer to consider the following: §

Icr < Ig

§

For walls with no compression, the calculation for Icr is straightforward.

10-66

Chapter 10: Masonry §

For walls with compression, one can adjust As to account for the effect of compression, resulting in Ase.

§

ACI 318 permits Icr = 0.35Ig for cracked, reinforced concrete walls (ACI 318 Sec. 10.10.4.1).

§

Alternatively, a (complicated) equation for I can be used (ACI 318 Eq. 10-8).

§

TMS 402 Section 3.1.5.2 permits up to one-half of gross section properties for use in deflection calculations when considering effects of cracking on reinforced masonry members.

For this example, the effect of cracking is recognized by taking Ieff as 35 percent of the gross moment of inertia, as recommended for reinforced concrete walls in ACI 318. Other approximations can be used. In the authors’ opinion, the approximations pale in uncertainty in comparison to the approximation of nonlinear deformation using Cd. For the Birmingham 1 building: be = effective masonry wall width, averaged over the entire wall length be = [(2 × 1.25 in.)(32.67 ft × 12) + (5 cells)(41 in.2/cell)]/(32.67 ft × 12) = 3.02 in. S = be l2/6 = (3.02)(32.67 × 12)2/6 = 77,434 in.3 fr = (0.063 ksi )(11 cells/12 cells) + (0.163 ksi) (1 cell/12 cells) = 0.071 ksi (for CMU with every 12th cell grouted) An = be l = (3.02 in.)(32.67 ft × 12) = 1,185 in.2 Pu is calculated using 1.00D (see Table 10.2-4). 1.00D is considered to be a reasonable value for axial load for this admittedly approximate analysis. If greater conservatism is desired, Pu could be calculated using 0.86D. (Recall that the 0.86 factor accounts for Ev in the upward direction [i.e., 0.9 - 0.2 SDS], leading to a lower bound on Pu). The results are shown in Table 10.2-5. Table 10.2-5 Birmingham 1 Cracked Wall Determination Level 5 4 3 2 1

Pumin

(kips) 49 98 147 196 245

Mcr (ft-kips) 725 992 1,259 1,525 1,792

Mu (ft-kips) 198 572 1,080 1,675 2,314

Status Uncracked Uncracked Uncracked Cracked Cracked

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

10-67

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples For uncracked walls: In. = Ig = bl3/12 = (3.02 in.)(32.67 × 12)3 /12 = 1.52 × 106 in.4 Ieff = 0.35 Ig = 0.532 × 106 in.4 The calculation of δ considers flexural and shear deflections. For the final determination of deflection, a RISA-2D analysis is made. The result is summarized Table 10.2-6 below. Figure 10.2-8 illustrates the deflected shape of the wall.

δ5

δ4

δ3

δ2

δ1

be l

Figure 10.2-8 Shear wall deflections Table 10.2-6 Deflections, Birmingham 1 F Ieff δflexural Level (kips) (in.4) (in.) 5 22.8 1.52 × 107 0.201 4 20.4 1.52 × 107 0.150 3 15.3 1.52 × 107 0.099 7 2 10.1 0.053 × 10 0.051 1 5.1 0.053 × 107 0.014 1.0 kip = 4.45 kN, 1.0 in. = 25.4 mm. F = 0.149Fx , for Fx from Table 10.2-2. Δ = story drift.

δshear (in.) 0.032 0.028 0.024 0.017 0.009

δtotal (in.) 0.233 0.178 0.123 0.068 0.023

The maximum story drift occurs at Levels 3 and 4 (Standard Table 12.12-1):

10-68

Cd δtotal (in.) 0.408 0.312 0.215 0.119 0.040

Δ (in.) 0.096 0.097 0.096 0.079 0.040

Chapter 10: Masonry Δmax = 0.097 in. The drift limit = 0.01hn (TMS 402 Sec. 1.17.2.4 and Standard Table 12.12-1). Δmax = 0.097 in. < 1.04 in. = 0.01hn

10.2.4.7 Birmingham 1 out-of-plane forces. The Standard Section 12.11.1 requires that the bearing walls be designed for out-of-plane loads determined as follows: w = 0.40SDSIWw ≥ 0.1Ww w = (0.40)(0.24)(1)(45 psf) = 4.3 psf < 4.5 psf = 0.1Ww where: Ww = weight of wall The calculated seismic load, w = 4.5 psf, is much less than wind pressure for exterior walls and is also less than the 5 psf required by the IBC for interior walls. Thus, seismic loads do not control the design of any of the walls for loading in the out-of-plane direction. 10.2.4.8 Birmingham 1 orthogonal effects. Orthogonal effects do not have to be considered for Seismic Design Category B (Standard Section 12.5.2). This completes the design of Transverse Wall D. 10.2.4.9 Summary of Design for Birmingham 1 Wall D. §

8-inch CMU

§

f’m = 2,000 psi

§

Reinforcement: One vertical #4 bar at wall end cells. Vertical #4 bars at 8 feet on center at intermediate cells throughout. Bond beam with two #4 bars at each story just below the floor and roof slabs. Horizontal joint reinforcement at 16 inches.

§

Grout at cells with reinforcement and at bond beams.

10.2.5 Seismic Design for Albuquerque This example focuses on differences from the design for the Birmingham 1 site. The walls are designed as Intermediate Reinforced Masonry Shear Walls even though the Standard would permit Ordinary Masonry Shear Walls. While the maximum reinforcement, and thus the grout, is increased, the change in R factor is advantageous in that the required strength is less.

10-69

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples This site is assigned to Seismic Design Category C, and the walls will be designed as intermediate reinforced masonry shear walls (Standard Table 12.2-1. Intermediate reinforced masonry shear walls have a minimum of #4 bars at 4 feet on center (TMS 402 Sec. 1.17.3.2.5). 10.2.5.1 Albuquerque Weights As before, use 67 psf for 8-inch-thick normal-weight hollow core plank plus the non-masonry partitions. For this example, 48 psf will be assumed for the 8-inch CMU walls. The 48 psf value includes grouted cells as well as bond beams in the course just below the floor planks. It will be shown that this symmetric building, with a seemingly well distributed lateral force-resisting system, has “extreme torsional irregularity” by the Standard. Story weight, wi: §

Roof: Roof slab (plus roofing) = (67 psf) (152 ft)(72 ft) = 733 kips Walls = (48 psf)(589 ft)(8.67 ft/2) + (48 psf)(4)(36 ft)(2 ft) = 136 kips Total = 869 kips

There is a 2-foot-high masonry parapet on four walls, and the total length of masonry wall is 589 feet. §

Typical floor: Slab (plus partitions) = 733 kips Walls = (48 psf)(589 ft)(8.67 ft) = 245 kips Total = 978 kips

Total effective seismic weight, W = 869 + (4)(978) = 4,781 kips. This total excludes the lower half of the first-story walls, which do not contribute to seismic loads that are imposed on CMU shear walls. 10.2.5.2 Albuquerque base shear calculation. The seismic response coefficient, Cs, is computed from Standard Section 12.8:

S DS 0.37 = = 0.106 R / I 3.5 1

The value of Cs need not be greater than:

S D1 0.15 = = 0.127 T ( R / I ) 0.338 (3.5 1)

where T is the same as found in Section 10.2.4.2. The value for Cs is taken as 0.106 (the lesser of the two computed values). This value is still larger than the minimum specified in Standard Equation 12.8-5 (Sup. 2):

10-70

Chapter 10: Masonry

Cs = 0.044IS DS ≥ 0.01

= 0.044 (1.0)( 0.37 ) = 0.0163 ≥ 0.01

(0.106 controls)

Note that this is essentially the same as the value for Birmingham 1, even though SDS is 71 percent larger. This is because we are using a system with an R factor that is 75 percent larger. We continue with this example because we will find an unexpected result arising from a requirement which applies in Seismic Design Category C but not in Seismic Design Category B. The total seismic base shear is then calculated using Standard Equation 12.8-1: V = CsW = (0.106)(4,781) = 507 kips 10.2.5.3 Albuquerque vertical distribution of seismic forces The vertical distribution of seismic forces is determined in accordance with Standard Section 12.8.3, which was described in Section 10.2.4.3. Note that for the Standard, k = 1.0 because T is less than 0.5 seconds (similar to the Birmingham 1 building). The application of the Standard equations for this building is shown in Table 10.2-7: Table 10.2-7 Albuquerque Seismic Forces and Moments by Level Level wx hx w xh xk Cvx Fx (ft) x (kips) (ft-kips) (kips) 5 869 43.34 37,657 0.3076 156 4 978 34.67 33,904 0.2770 141 3 978 26.00 25,428 0.2077 105 2 978 17.33 16,949 0.1385 70 1 978 8.67 8,476 0.0692 35 ∑ 4,781 122,414 1.0000 507

VX (kips) 156 297 402 472 507

Mx (ft-kips) 1,350 3,930 7,410 11,500 15,900

1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m, 1.0 ft-kip = 1.36 kN-m.

10.2.5.4 Albuquerque horizontal distribution of forces The initial distribution is the same as Birmingham 1. See Section 10.2.4.4 and Figure 10.2-3 for wall designations. Total shear in Wall D:

Vtot = 0.125V + 0.0238V = 0.149V For Seismic Design Category C structures, Standard Section 12.8.4.3 requires a check of torsional irregularity using the ratio of maximum displacement at the end of the structure, including accidental torsion, to the average displacement of the two ends of the building. For this simple and symmetric structure, the actual displacements do not have to be computed to find the ratio. Relying on symmetry and the assumption of rigid diaphragm behavior used to distribute the forces, the ratio of the maximum

10-71

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples displacement of Wall D to the average displacement of the floor will be the same as the ratio of the wall shears with and without accidental torsion:

Fmax 0.149V = = 1.190 Fave 0.125V This can be extrapolated to the end of the rigid diaphragm:

δ max ⎛ 152 / 2 ⎞ = 1 + 0.190 ⎜ ⎟ = 1.402 δ ave ⎝ 36 ⎠ Standard Table 12.3-1 defines a building as having a “Torsional Irregularity” if this ratio exceeds 1.2 and as having an “Extreme Torsional Irregularity” if this ratio exceeds 1.4. Thus, an important result of the Seismic Design Category C classification is that the total torsion must be amplified by the factor (Standard Eq. 12.8-14): 2

⎛ δ ⎞ ⎛ 1.402 ⎞ Ax = ⎜ max ⎟ = ⎜ ⎟ = 1.365 ⎝ 1.2δ ave ⎠ ⎝ 1.2 ⎠ Therefore, the portion of the base shear for design of Wall D is increased to:

Vtot = 0.125V + 1.365(0.0238V ) = 0.158V which is a 6 percent increase from the fraction before considering torsional irregularity. The total story shear and overturning moment may now be distributed to Wall D and the wall proportions checked. The wall capacity will be checked before considering deflections. 10.2.5.5 Albuquerque Transverse Wall D The strength or limit state design concept is used in TMS 402 Chapter 3. 10.2.5.5.1 Albuquerque shear strength. Similar to the design for Birmingham 1, the shear wall design is governed by the following:

Vu ≤ φVn Vn = Vnm + Vns

Vn max = ( 4 to 6 ) An f m' depending on Mu/Vudv ⎡ ⎛ M Vnm = ⎢ 4-1.75 ⎜ u ⎢⎣ ⎝ Vu dv

⎛ A Vns = 0.5 ⎜ v ⎝ s 10-72

⎞ ⎟ f y dv ⎠

⎞ ⎤ ⎟ ⎥ An f mʹ′ + 0.25 Pu ⎠ ⎥⎦

Chapter 10: Masonry

where: An = (2 × 1.25 in. × 32.67 ft × 12 in.) + (41 in.2 × 9 cells) = 1,349 in.2 The shear strength of each Wall D, based on the aforementioned formulas and the strength reduction factor of φ = 0.8 for shear from TMS 402 Section 3.3.2, is summarized in Table 10.2-8. Note that Vx and Mx in this table are values from Table 10.2-7 multiplied by 0.158 (representing the portion of direct and indirect shear assigned to Wall D), and Pu is the dead load of the roof or floor times the tributary area for Wall D. Table 10.2-8 Albuquerque Shear Strength Calculation for Wall D Vx Mx 2.5 Vx Pu Vnm Vns Vn Story Mx/Vxdv (kips) (ft-kips) (kips) (kips) (kips) (kips) (kips) 5 24.6 213 0.266 24.6 35 222 68 290 4 46.9 621 0.405 47.0 75 217 68 285 3 63.5 1,171 0.564 63.5 115 211 68 279 2 74.6 1,817 0.746 74.6 156 202 68 270 1 80.1 2,512 0.960 80.1 196 189 68 257

Vn (max) (kips) 359 337 312 282 248

φVn (kips) 232 228 223 216 198

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

Values shown in bold are the controlling values for Vn For all levels, φVn > Vu, , so this Intermediate Reinforced Masonry shear wall is OK. 10.2.5.5.2 Albuquerque axial and flexural strength. The walls in this example are all load-bearing shear walls because they support vertical loads as well as lateral forces. In-plane calculations include the following: §

Strength check

§

Ductility check

10.2.5.5.2.1 Strength check. Wall demands, using load combinations determined previously, are presented in Table 9.2-9 for Wall D. In the table, Load Combination 1 is 1.27D + QE + 0.5L and Load Combination 2 is 0.83D + QE. Table 10.2-9 Demands for Albuquerque Wall D Load Combination 1 PD PL Level Pu Mu (kips) (kips) (kips) (ft-kips) 5 50 0 64 213 4 100 15 135 621 3 149 25 202 1,171 2 199 34 270 1,817

Load Combination 2 Pu Mu (kips) (ft-kips) 42 213 83 612 124 1,171 165 1,817

10-73

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Table 10.2-9 Demands for Albuquerque Wall D Load Combination 1 PD PL Level Pu Mu (kips) (kips) (kips) (ft-kips) 1 249 41 337 2,512

Load Combination 2 Pu Mu (kips) (ft-kips) 207 2,512

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

As in Section 10.2.4.5.2, the strength at the bottom story (where P, V and M are the greatest) is examined. The strength design considers Load Combination 2 from Table 10.2-9 to be the governing case because it has the same lateral load as Load Combination 1 but with lower values of axial force. Refer to Figure 10.2-9 for notation and dimensions.

10-74

Chapter 10: Masonry

P M

32.67'

(1) #4 #4 at 4.00' on center

16.33'

0.33'

16.00'

Cm tot 0.8 f'm ε m = 0.0025 ε y = 0.00207 C s1

P = 0 Case

Σ T s2

T s1

a = 0.66'

εy = 0.00207 N.A.

c = 0.82'

Cm shell

ε m = 0.0025 ε y= 0.00207 C s1

T s2*

T sl

a = 6.40' T s2

c = 8.00'

0.8 f 'm

Intermediate Case

ΣT

Cm cells Cm shell

ε m = 0.0025 ε y = 0.00207

T s2

ε y= 0.00207

N.A.

0.8 f 'm

N.A.

Cm cell

Balanced Case

C s1

T sl a = 14.15'

ε y= 0.00207

c = 17.69' Center Line

Figure 10.2-9 Strength of Albuquerque Wall D Strain diagram superimposed on strength diagram for the three cases. Low forces in the reinforcement are neglected in the calculations. (1.0 ft = 0.3048 m)

10-75

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Examine the strength of Wall D at Level 1: Pu min. = 207 kips Pu max = 337 kips Mu = 2,512 ft-kips Because intermediate reinforced masonry shear walls are used (Seismic Design Category C), vertical reinforcement is required at 4 feet on center in accordance with TMS 402 Section 1.17.3.2.5. Therefore, try one #4 bar in each end cell and #4 bars at 4 feet on center at all intermediate cells. The calculation procedure is similar to that for the Birmingham 1 building presented in Section 10.2.4.5.2. The results of the calculations (not shown) for the Albuquerque building are summarized below and shown in Figure 10.2-9. §

P = 0 case:

φPn = 0 φMn = 1,562 ft-kips §

Intermediate case: c = 8.0 ft

φPn = 349 kips φMn = 5,929 ft-kips §

Balanced case:

φPn = 854 φMn = 8,697 ft-kips With the intermediate case, it is simple to use the three points to make two straight lines on the interaction diagram. Use the simplified φPn - φMn curve shown in Figure 10.2-10. The straight line from pure bending to the balanced point is conservative and can easily be used where the design is not as close to the criterion. It is the nature of lightly reinforced and lightly loaded masonry walls that the intermediate point is frequently useful. Use one #4 bar in each end cell and one #4 bar at 4 feet on center throughout the remainder of the wall.

10-76

Chapter 10: Masonry

φ Pn

500 kips Pu,max = 260 kips Pu,min = 211 kips

P=0 (1562 ft-kips, 0 kips) P=0 (1926 ft-kips, 0 kips)

9) #4

φMn = 6250(±) ft-kips Intermediate: (5929 ft-kips, 349 kips)

φ Pn

Balance: (9857 ft-kips, 851 kips)

Balance: (8697 ft-kips, 854 kips)

1,000 kips

Birm #2, M u = 2111 ft-kips Albq, M u = 2512 ft-kips

φMn

rm Bi

Pu,max = 337 kips

5 )#

9 ,(

φ M n = 6000(±) ft-kips

Pu,min = 207 kips

1.25φ M n = 7500(±) ft-kips

Birm #2

Intermediate: (5650 ft-kips, 304 kips)

5,000 ft-kips

φMn

10,000 ft-kips

Figure 10.2-10 φPn - φMn Diagram for Albuquerque and Birmingham 2 Wall D (1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m) While Birmingham 2 has a lesser demand than Albuquerque, more robust reinforcement is required prescriptively because it is Seismic Design Category D. The greater flexural resistance will also necessitate a design to resist greater shear, a requirement that applies to special reinforced masonry shear walls (TMS 402, Sec. 1.17.3.2.6.1), which are required for Seismic Design Category D. 10.2.5.5.2.2 Ductility check. Refer to Section 10.2.4.5.2, Item 2, for explanation. The strain distribution is shown in Figure 10.2-11. If M/Vd equals or exceeds 1.0, the multiplier on steel yield strain for intermediated reinforced masonry walls (TMS 402 Sec. 333.5.2) is 3.0, not 1.5. For this design M/Vd = 0.96. For Level 1 (bottom story), the unfactored loads are as follows: P = 249 kips

10-77

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

Cm = 0.8 fm' [(a)(b) + Acells] where b = face shells = (2 × 1.25 in.) and Acell = 41 in.2 Cm = (1.6 ksi)[(11.55 ft × 12)(2.5 in.) + (3)(41)] = 751 kips Cs1 = FyAs = (60 ksi)(0.20 in.2) = 12 kips Cs2 = (51.9 ksi)(0.20 in.2) = 10.4 kips Cs3 = (31.4 ksi)(0.20 in.2) = 6.3 kips Cs4 and Ts5 are small, so are neglected Ts1 = Ts2 = = (60 ksi)(0.20 in.2 ) = 12 kips Ts3 = (49.8 ksi)(0.20 in.2) = 10.0 kips Ts4= (29.7 ksi)(0.20 sq. in.) = 5.9 kips ∑C>∑P+T Cm + Cs1 + Cs2 + Cs3 > P + Ts1 + Ts2 + Ts3 + Ts4 751 + 12 + 10.4 > 249 + 12 + 12 + 10.0 + 5.9 773 kips > 289 kips There is more compression capacity than required, so a ductile failure condition controls.

10-78

Chapter 10: Masonry

32'-8" P

0.33'

32.34'

ε m = 0.0025

17.90'

N.A.

14.44' c

ε s = 1.5ε y

8.66'

60 ) 29,000 = 0.0031 = 1.5 (

0.8 f 'm = 1.6 ksi

2.89'

11.55' a 72.51 ksi f y = 60 ksi

17.90'

51.9 ksi

C s1

C s2

C s3

31.4 ksi C s4 6.10'

10.10' 14.10'

T s5 T s4

T s3

T s2

T s1

29.7 ksi 5.90' 49.8 ksi 9.90' 13.90'

f y = 60 ksi 1.5f y = 90 ksi

17.90' 0.33'

(9) #4 @ 4'-0" o.c.

Figure 10.2-11 Ductility check for Albuquerque (1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m) 10.2.5.6 Albuquerque deflections. Refer to Section 10.2.4.6 for more explanation. For the Albuquerque building, the determination of whether the walls will be cracked is as follows:

10-79

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples be = effective masonry wall width be = [(2 × 1.25 in.)(32.67 ft × 12) + (9 cells)(41 in.2/cell)]/(32.67 ft × 12) = 3.44 in. An = be l = (3.44 in.)(32.67 × 12) = 1,349 in.2 S = be l2/6 = (3.44)(32.67 × 12)2 /6 = 88,100 in.3 fr = (0.063 ksi)(5 cells/6 cells) + (0.163 ksi)(1 cell/6 cells) = 0.080 ksi Pu is calculated using 1.00D (see Table 10.2-9 for values and refer to Sec. 10.2.4.6 for discussion). Table 10.2-10 summarizes these calculations. Table 10.2-10 Albuquerque Cracked Wall Determination Level

Pu (kips)

Mcr (ft-kips)

Mx (ft-kips)

Status

5 4 3 2 1

50 100 149 199 249

859 1,132 1,398 1,610 1,952

213 621 1,171 1,817 2,512

Uncracked Uncracked Uncracked Cracked Cracked

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

For the uncracked walls: In = Ig = bl3/12 = (3.44 in.)(32.67 × 12)3/12 = 1.73 × 107 in.4 For the cracked wall: Ieff = 0.35 Ig = 0.606 × 107 in.4 The calculation of δ should consider shear deflection in addition to flexural deflection. This example uses a RISA 2D analysis. The results are summarized in Table 10.2-11. Table 10.2-11 Albuquerque Deflections F Ieff δflexural Level (kips) (in.4) (in.) 5 23.2 1.73 × 107 0.181 4 21.0 1.73 × 107 1.134 7 3 15.6 1.73 × 10 0.089 2 10.4 0.61 × 107 0.046 7 1 5.2 0.61 × 10 0.013 1.0 kip = 4.45 kN, 1.0 in. = 25.4 mm. F = 0.149 Fx , for Fx from Table 10.2-7. Δ = story drift.

10-80

δshear (in.) 0.032 0.030 0.024 0.017 0.009

δtotal (in.) 0.213 0.164 0.113 0.063 0.022

Cd δtotal (in.) 0.479 0.369 0.254 0.142 0.050

Δ (in.) 0.110 0.115 0.112 0.092 0.052

Chapter 10: Masonry

The maximum story drift occurs between Levels 4 and 3: Δ4 = 0.115 in. < 1.04 in. = 0.01 hn (Standard Table 12.2-1)

10.2.5.7 Albuquerque out-of-plane forces. Standard Section 512.11.1 requires that bearing walls be designed for out-of-plane loads, determined as follows: w = 0.40 SDS IWw ≥ 0.1Ww w = (0.40)(0.37)(1)(48 psf) = 7.1 psf > 4.8 psf = 0.1W w, So the equivalent normal pressure due to the design earthquake is 7.1 psf. This is greater than the design differential air pressure of 5 psf. However, the lateral pressure is sufficiently low for this short wall that the authors consider it acceptable by inspection, without further calculation. So, seismic loads do not govern the design of Wall D for loading in the out-of-plane direction. 10.2.5.8 Albuquerque orthogonal effects. According to Standard Section 12.5.3, orthogonal interaction effects have to be considered for Seismic Design Category C where the ELF procedure is used (as it is here). However, the out-of-plane component of only 30 percent of 7.1 psf on the wall does not produce a significant effect where combined with the in-plane direction of loads, so no further calculation is made. This completes the design of the transverse Wall D for the Albuquerque building. 10.2.5.9 Summary of Albuquerque Wall D design §

8-inch CMU

§

f’m = 2,000 psi

§

Reinforcement: Vertical #4 bars at 4 feet on center throughout the wall. Bond beam with two #4 at each story just below the floor or roof slabs. Horizontal joint reinforcement at alternate courses.

10.2.6 Birmingham 2 Seismic Design The emphasis here is on differences from the previous two locations for the same building. Standard Table 12.6-1 requires that design of a Seismic Design Category D building with torsional irregularity be based on a dynamic analysis. Although it is not explicitly stated, the implication is that the analytical model should be three-dimensional in order to capture the torsional response. This example compares both the ELF procedure and the modal response spectrum analysis procedure and demonstrates that, as long as the torsional effects are accounted for, the static analysis (ELF method) could be considered adequate for design. 10.2.6.1 Birmingham 2 weights

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

The floor weight for this example uses the same 67 psf for 8-inch-thick, normal-weight hollow core plank plus roofing and the nonmasonry partitions as used in the prior examples (see Sec. 10.2.1). This site is assigned to Seismic Design Category D, and the walls are designed as special reinforced masonry shear walls (Standard Table 12.2-1). Special reinforced masonry shear walls have a maximum spacing of rebar at 4 feet on center both horizontally and vertically (TMS 402 Sec. 1.17.3.2.6). Also, the total area of horizontal and vertical reinforcement must exceed 0.002 times the gross area of the wall and neither direction may have a ratio of less than 0.0007. The vertical #4 bars at 48 inches used for the Albuquerque design yield a ratio of 0.00055, so it must be increased. #5 bars at 48 inches (yielding 0.00085) is selected. The latter is chosen in order to avoid unnecessarily increasing the shear demand. Therefore, the horizontal reinforcement must be (0.0020 - 0.00085)(7.625 in.)(12 in./ft) = 0.105 in.2/ft Two #5 bars in bond beams at 48 inches on center will be adequate. For this example, 56 psf weight for the 8-inch-thick CMU walls will be assumed. The 56 psf value includes grouted cells and bond beams. Story weight, wi: §

Roof: Roof slab (plus roofing) = (67 psf) (152 ft)(72 ft) = 733 kips Walls = (56 psf)(589 ft)(8.67 ft/2) + (56 psf)(4)(36 ft)(2 ft) = 159 kips Total = 892 kips

There is a 2-foot-high masonry parapet on four walls, and the total length of masonry wall is 589 feet. §

Typical floor: Slab (plus partitions) = 733 kips Walls = (56 psf)(589 ft)(8.67 ft) = 286 kips Total = 1,019 kips

Total effective seismic weight, W = 892 + (4)(1,019) = 4,968 kips. This total excludes the lower half of the first story walls, which do not contribute to seismic loads that are imposed on CMU shear walls. 10.2.6.2 Birmingham 2 base shear calculation. The ELF analysis proceeds as described for the previous locations. The seismic response coefficient, Cs, is computed using Standard Section 12.8:

S DS 0.43 = = 0.086 R/I 51

(Controls)

S D1 0.24 = = 0.142 T ( R / I ) 0.338 (5 1)

The fundamental period of the building, based on Standard Equation 12.8-7 is approximately 0.338 seconds as computed previously (the approximate period, based on building system and building

10-82

Chapter 10: Masonry height, is the same for all locations). The value for Cs is taken as 0.086 (the lesser of the two values). This value is still larger than the minimum specified in Standard (Sup. 2) Equation 12.8-5, which is: Cs = 0.044SD1I = (0.044)(0.24)(1) = 0.011 The total seismic base shear is calculated using Standard Equation 12.8-1: V = CsW = (0.086)(4,968) = 427 kips This is somewhat less than the 507 kips computed for the Albuquerque design, due to the larger R factor. A three-dimensional (3D) model is created in SAP2000 for the modal response spectrum analysis. The masonry walls are modeled as shell bending elements, and the floors are modeled as an assembly of beams and shell membrane elements. The beams have very little mass and a large flexural moment of inertia to avoid consideration of modes of vertical vibration of the floors. The flexural stiffness of the beams is released at the bearing walls in order to avoid a wall-slab frame that would inadvertently increase the torsional resistance. The mass of the floors is captured by the shell membrane elements. Table 10.2-12 shows data on the modes of vibration used in the analysis. Standard Section 11.4.5 is used to create the response spectrum for the modal analysis. The key points that define the spectrum are as follows: TS = SD1/SDS = 0.21/0.43 = 0.56 sec T0 = 0.2 TS = 0.11 sec at T = 0, Sa = 0.4 SDS/R = 0.034 g from T = T0 to TS, Sa = SDS/R = 0.086 g for T > TS, Sa = SD1/(RT) = 0.042/T The computed fundamental period is less than the approximate period. The transverse direction base shear from the SRSS combination of the modes is 293 kips, which is considerably less than that obtained using the ELF method. Standard Section 12.8.2 requires that the modal base shear be compared with the ELF base shear computed using a period no larger than CuTa. As shown in Section 10.2.4.2, Ta = 0.338 seconds. Per Standard Table 12.8-1, Cu = 1.46. Thus, CuTa = 0.49 seconds. However, the computed period, T, is only 0.2467 seconds (as shown in Table 10.2-12), which is less than CuTa so the ELF base shear must be computed at that period. Since T is less that TS/SDS., the ELF base shear for comparison is 427 kips as just computed. Because 85 percent of 427 kips = 363 kips, Standard Section 12.9.4 dictates that all the results of the modal analysis be multiplied by the following:

0.85VELF 363 = = 1.24 VModal 293 Both analyses are carried forward as discussed in the subsequent sections.

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Table 10.2-12 Birmingham 2 Periods, Mass Participation Factors and Modal Base Shears in the Transverse Direction for Modes Used in Analysis Mode Period, Individual Mode (percent) Cumulative Sum (percent) Trans. Number (seconds) Long. Trans. Vert. Long. Trans. Vert. Base Shear 1 0.2467 0.00 0.00 0.00 0.00 0.00 0.00 0.0 2 0.1919 0.00 70.18 0.00 0.00 70.18 0.00 288.7 3 0.1915 70.55 0.00 0.00 70.55 70.18 0.00 0.0 4 0.0579 0.00 18.20 0.00 70.55 88.39 0.00 47.3 5 0.0574 17.86 0.00 0.00 88.41 88.39 0.00 0.0 6 0.0535 0.00 4.09 0.00 88.41 92.48 0.00 10.3 7 0.0532 4.17 0.00 0.00 92.58 92.48 0.00 0.0 8 0.0413 0.00 0.01 0.00 92.58 92.48 0.00 0.0 9 0.0332 1.50 0.24 0.00 94.08 92.72 0.00 0.5 10 0.0329 0.30 2.07 0.00 94.38 94.79 0.00 4.5 11 0.0310 1.28 0.22 0.00 95.66 95.01 0.00 0.5 12 0.0295 0.22 1.13 0.00 95.89 96.14 0.00 2.4 13 0.0253 1.97 0.53 0.00 97.86 96.67 0.00 1.1 14 0.0244 0.53 1.85 0.00 98.39 98.52 0.00 3.8 15 0.0190 1.05 0.36 0.00 99.44 98.89 0.00 0.7 16 0.0179 0.33 0.94 0.00 99.77 99.82 0.00 1.8 17 0.0128 0.19 0.07 0.00 99.95 99.90 0.00 0.1 18 0.0105 0.03 0.10 0.00 99.99 99.99 0.00 0.2 1 kip = 4.45 kN.

10.2.6.3 Birmingham 2 vertical distribution of seismic forces. The dynamic analysis is revisited for the horizontal distribution of forces in the next section but, as demonstrated there, the ELF procedure is considered adequate to account for the torsional behavior in this example; the dynamic analysis certainly can be used to deduce the vertical distribution of forces. The purpose of this analysis is to study amplification of accidental torsion. Note that Mode 1 has no net base force in the longitudinal, transverse, or vertical directions. The mode shape confirms that it is purely torsional. The vertical distribution of seismic forces for the ELF analysis is determined in accordance with Standard Section 12.8.3, which was described in Section 10.2.4.3, in which k = 1.0 because T < 0.5 seconds (similar to the Birmingham 1 and Albuquerque buildings). It should be noted that the response spectrum analysis (modal analysis) may result in moments that are less than those calculated using the ELF method; however, because of its relative simplicity, the ELF is used in this example. Application of the Standard equations for this building is shown in Table 10.2-13: Table 10.2-13 Birmingham 2 Seismic Forces and Moments by Level Level x

10-84

wx (kips)

hx (ft)

w xh x (ft-kips)

Cvx

Fx (kips)

Vx (kips)

Mx (ft-kips)

Chapter 10: Masonry 5 4 3 2 1 ∑

892 1,019 1,019 1,019 1,019 4,968

43.34 34.67 26.00 17.33 8.67

38,659 35,329 26,494 17,659 8,835 126,976

0.3045 0.2782 0.2086 0.1391 0.0695 1.000

130 119 89 59 30 427

130 249 338 397 427

1,130 3,290 6,220 9,660 13,360

1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m, 1.0 ft-kip = 1.36 kN-m.

10.2.6.4 Birmingham 2 horizontal distribution of forces. The ELF analysis for Birmingham 2 is the same as that for the Albuquerque location; see Section 10.2.5.4. Total shear in Wall D:

Vtot = 0.125V + 1.365(0.0238)V = 0.158V = 67.4 kips The fact that the fundamental mode is torsional does confirm, to an extent, that the structure is torsionally sensitive. This modal analysis does not show any significant effect of the torsion, however, because of the symmetry. The pure symmetry of this structure is somewhat idealistic. Real structures usually have some real eccentricity between mass and stiffness and dynamic analysis then yields coupled modes, which contribute to computed forces. The Standard does not require that the accidental eccentricity be analyzed dynamically. For illustration, however, this is approximated by adjusting the mass of the floor elements to generate an eccentricity of 5 percent of the 152-foot length of the building. Table 10.2-14 shows the results of such an analysis. (Accidental torsion could also be considered using a linear combination of the dynamic results and a statically applied moment equal to the accidental torsional moment.) The transverse direction base shear from the SRSS combination of the modes with dynamic torsion is 258.4 kips, less than the 293 kips for the symmetric model. The amplification factor for this base shear is 363/258 = 1.41. This smaller base shear from modal analysis of a model with an artificially introduced eccentricity is normal for two primary reasons: First, the mass participates in more modes. The participation in the largest mode generally is less, and the combined result is dominated by the largest single mode. Second, the period for the fundamental mode generally increases, because there is more flexibility between the mass and the foundation. The increase in period will reduce the spectral response except for structures with short periods (such as this one). In order to demonstrate that the ELF method for addressing torsional effects per the Standard produces a conservative result, let us consider torsional effects based on modal analysis in greater detail than required by the Standard: The base shear in Wall D is computed by adding the in-plane reactions. For the symmetric model the result is 36.6 kips, which is 12.5 percent of the total of 293 kips, as would be expected. Amplifying this by the 1.24 factor (to bring the modal result to 85 percent of the ELF result) yields 45 kips Application of a static horizontal torsion equal to the 5 percent eccentricity times a base shear of 363 kips adds 8 kips, for a total of 53 kips. If the static horizontal torsion is amplified by 1.365, as found in the analysis for the Albuquerque location, the total becomes 56 kips, which is less than the 64 kips (0.149V) or 67 kips (0.158V) computed in the ELF analysis without and with, respectively, the amplification of accidental torsion. The Wall D base shear from the modal analysis with the eccentric model was 42 kips (SRSS); with the amplification of base shear equal to 1.41 (to reach 85 percent of the ELF), this becomes 59 kips. Note that this value is less than the shear from the ELF model including amplified static torsion (67 kips). The conclusion is that more careful consideration of torsional effects 10-85

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples than actually required by the Standard does not indicate any more penalty than already given by the procedures for the ELF in the Standard. Therefore the remainder of the example designs for this building are completed using the ELF. Table 10.2-14 Birmingham 2 Periods, Mass Participation Factors and Modal Base Shears in the Transverse Direction for Modes Used in Approximate Accidental Torsion Analysis Mode Period Individual Mode (percent) Cumulative Sum (percent) Trans. Number (sec) Long. Trans. Vert. Long. Trans. Vert. Base Shear 1 0.2507 0.0 8.8 0.0 0.0 8.8 0.0 36.0 2 0.1915 70.5 0.0 0.1 70.5 8.8 0.1 0.0 3 0.1867 0.0 61.4 0.0 70.5 70.2 0.1 252.7 4 0.0698 0.0 2.9 0.0 70.5 73.1 0.1 8.1 5 0.0613 1.1 0.0 23.0 71.6 73.1 23.1 0.0 6 0.0575 19.2 0.0 0.0 90.9 73.1 23.2 0.0 7 0.0570 0.0 13.7 0.0 90.9 86.8 23.2 35.5 8 0.0533 0.0 5.6 0.0 90.9 92.4 23.2 14.1 9 0.0480 1.2 0.0 12.8 92.0 92.4 35.9 0.0 10 0.0380 1.4 0.0 0.0 93.5 92.4 35.9 0.0 11 0.0374 0.0 0.4 0.0 93.5 92.8 35.9 0.8 12 0.0327 1.7 0.0 0.2 95.2 92.8 36.1 0.0 13 0.0322 0.0 3.1 0.0 95.2 95.9 36.1 6.7 14 0.0263 2.8 0.0 0.1 98.0 95.9 36.2 0.0 15 0.0243 0.0 3.0 0.0 98.0 98.8 36.2 6.1 16 0.0201 1.6 0.0 0.1 99.6 98.8 36.3 0.0 17 0.0164 0.0 1.1 0.0 99.6 100.0 36.3 2.2 18 0.0141 0.4 0.0 0.1 100.0 100.0 36.3 0.0 The “extreme torsional irregularity” has an additional consequence for Seismic Design Category D: Standard Section 12.3.3.4 requires that the design forces for connections between diaphragms, collectors and vertical elements (walls) be increased by 25 percent. For this example, the diaphragm of precast elements is designed using the different requirements of the Provisions, Part 3, RP10. 10.2.6.5 Birmingham 2 transverse wall (Wall D). The total story shear and overturning moment (from the ELF analysis) may now be distributed to Wall D and the wall proportions checked. The wall capacity is checked before considering deflections. The design demands are slightly smaller than for the Albuquerque design, largely due to an R of 5 instead of 3.5, yet there is more reinforcement, both vertical and horizontal in the walls, because of the prescriptive detailing requirements for Seismic Design Category D. This illustration will focus on those items where the additional reinforcement has special significance. 10.2.6.5.1 Birmingham 2 shear strength. Refer to Section 10.2.5.5.1 for most quantities. Compared to Albuquerque, the additional horizontal reinforcement raises Vs and the additional grouted cells raises An and therefore both Vnm and Vn(max). Av/s = (4)(0.31 in.2)/(8.67 ft) = 0.1431 in.2/ft 10-86

Chapter 10: Masonry

Vns = 0.5(0.1431)(60 ksi)(32.67 ft) = 140.2 kips An = (2 × 1.25 in. × 32.67 ft × 12 in.) + (41 in.2 × 9 cells) = 1,349 in.2 The shear strength of Wall D is summarized in Table 10.2-15 below. (Vx and Mx in this table are values from Table 10.2-13 multiplied by 0.158, the portion of direct and torsional shear assigned to the wall.) Clearly, the dynamic analysis would make it possible to design this wall for smaller forces, but the minimum configuration suffices. Note also that the format of Table 10.2-15 differs from that of its counterparts for Birmingham 1 and Albuquerque: a column for 2.5Vx is included here because, for special reinforced masonry shear walls, TMS 402 Section 1.17.3.2.6.1.1 requires the shear capacity to exceed the lesser of the shear corresponding to 1.25Mn or 2.5Vx. The intent is to require response controlled by flexure in most cases, but to permit non-ductile shear response if the shear capacity is 2.5 times the demand from analysis. Table 10.2-15 Birmingham 2 Shear Strength Calculation for Wall D Vx Mx 2.5Vx Pu Vnm Vns Vn Story Mx/Vxdv (kips) (ft-kips) (kips) (kips) (kips) (kips) (kips) 5 20.5 178 0.265 51.2 42 224 140 364 4 39.3 520 0.405 98.3 84 220 140 360 3 53.4 983 0.563 134 126 213 140 353 2 62.7 1,526 0.745 157 168 205 140 345 1 67.5 2,111 0.957 169 210 193 140 333

Vn(max) (kips) 361 337 312 282 248

φVn (kips) 289 270 249 226 199

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m. Values shown in bold are the controlling values for Vn

Vn(max) is less than Vn at all levels, so it controls in the determination of φVn. φVn > 2.5Vx for all levels, so the design is satisfactory for shear (without needing to check whether φVn is greater than the shear corresponding to 1.25Mn). 10.2.6.5.2 Birmingham 2 axial and flexural strength. Once again, the similarities to the design for the Albuquerque location are exploited. The in-plane calculations include the following: §

Strength check

§

Ductility check

10.2.6.5.2.1 Strength check. The wall demands, using the load combinations determined previously, are presented in Table 10.2-16 for Wall D. In the table, Load Combination 1 is 1.29D + QE + 0.5L and Load Combination 2 is 0.81D + QE. Table 10.2-16 Birmingham 2 Demands for Wall D Load Combination 1 PD PL Level Pu Mu (kips) (kips) (kips) (ft-kips) 5 53 0 68 178

Load Combination 2 Pu Mu (kips) (ft-kips) 43 178

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples Table 10.2-16 Birmingham 2 Demands for Wall D Load Combination 1 PD PL Level Pu Mu (kips) (kips) (kips) (ft-kips) 4 104 15 134 520 3 156 25 201 983 2 208 34 268 1526 1 260 41 335 2111

Load Combination 2 Pu Mu (kips) (ft-kips) 84 520 126 983 168 1526 211 2111

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

Strength at the bottom story (where P, V and M are the greatest) is less than required for the Albuquerque design. The demands for Birmingham 2 are plotted on Figure 10.2-10 along with those for Albuquerque (showing that the design for Albuquerque has sufficient axial and flexural capacity for this Birmingham 2 location). 10.2.6.5.2.2 Ductility check. The requirements for ductility are described in Sections 10.2.4.5.3 and 10.2.5.5.3. Because the wall reinforcement and loads are so similar to those for the Albuquerque building, the computations are not repeated here. A brief review of the Albuquerque ductility calculations (Sec. 10.2.5.5.3) reveals that the Birmingham 2 reinforcement satisfies the ductility provisions. 10.2.6.6 Birmingham 2 deflections. The calculations for deflection would be similar to that for the Albuquerque location. The calculation is not repeated here; refer to Sections 10.2.4.6 and 10.2.5.6. While the Cd factor is larger, 3.5 versus. 2.25, the resulting maximum story drift is still less than the 0.01 hn allowable and therefore is OK. 10.2.6.7 Birmingham 2 out-of-plane forces. Standard Section 12.11 requires that the bearing walls be designed for out-of-plane loads, determined as follows: w = 0.40 SDS IWw ≥ 0.1Ww w = (0.40)(0.43)(1)(56 psf) = 9.6 psf ≥ 0.1Ww The calculated seismic load, w = 9.6 psf, is less than wind pressure for exterior walls. This is larger than the design differential pressure of 5 psf across an interior wall (per the IBC). Given the story height for either interior or exterior walls, the out-of-plane seismic force is sufficiently low that it is considered acceptable by inspection without further calculation. 10.2.6.8 Birmingham 2 orthogonal effects. According to Standard Section 12.5.3, orthogonal interaction effects should be considered for Seismic Design Category D where the ELF procedure is used (as it is here). However, the out-of-plane component of only 30 percent of 9.6 psf on the wall does not produce a significant effect where combined with the in-plane direction of loads, so no further calculation is made. This completes the design of Transverse Wall D. 10.2.6.9 Summary of Birmingham 2 Wall D §

8-inch CMU

§

f'm = 2,000 psi

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Chapter 10: Masonry

§

Reinforcement: 9 vertical #5 bars per wall at 4’-0” on center. Two bond beams with two #5 at each story, at bearing for the planks and at 4 feet above each floor. Horizontal joint reinforcement at alternate courses is recommended, but not required.

10.2.7 Seismic Design for San Rafael Once again, the differences from the designs for the other locations are emphasized. As explained for the Birmingham 2 building, the Standard would require a dynamic analysis for the design of this building. For the reasons explained in Section 10.2.6.4, this design is illustrated using the ELF procedure. 10.2.7.1 San Rafael weights Use 91 psf for 8-inch-thick, normal-weight hollow core plank, 2.5-inch lightweight concrete topping (115 pcf), plus the non-masonry partitions. This building is in Seismic Design Category D, and the walls will be designed as special reinforced masonry shear walls (Standard Table 12.2-1), which requires prescriptive seismic reinforcement (TMS 402 Section 1.17.3.2.6). Special reinforced masonry shear walls have a minimum spacing of vertical reinforcement of 4 feet on center. The demand is considerably larger than that for the other Seismic Design Category D building (Birmingham 2), so more reinforcement is required. Trial reinforcement is selected as nine #7 bars at 4’-0” on center. For this example, a 60 psf weight for the 8-inch CMU walls is assumed. The 60 psf value includes grouted cells and bond beams in the course just below the floor planks and in the course 4 feet above the floors. (Note that the wall is 43.33 feet high, not 8 feet high, for purpose of determining the maximum spacing of vertical and horizontal reinforcement.) A typical wall section is shown in Figure 9.2-12.

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

8" concrete masonry wall

21 2"

8" precast concrete plank

See Figure 8.1-8 for grouted space detail.

4'-0"

Vertical #5 bar at 4'-0" o.c.

9 ga. horizontal joint reinforcement at 1'-4" o.c. (Recommended) Lightweight concrete topping

21 2"

4'-0"

7'-91 2"

Bond beam w/ (2) #5 (typ)

Figure 9.2-12 Typical wall section for the San Rafael location (1.0 in. = 25.4 mm, 1.0 ft = 0.3048 m) Story weight ,wi: §

Roof weight: Roof slab (plus roofing) = (91 psf) (152 ft)(72 ft) = 996 kips Walls = (60 psf)(589 ft)(8.67 ft/2) + (60 psf)(4)(36 ft)(2 ft) = 170 kips Total = 1,166 kips

There is a 2-foot-high masonry parapet on four walls, and the total length of masonry wall is 589 feet. §

Typical floor: Slab (plus partitions) = 996 kips Walls = (60 psf)(589 ft)(8.67 ft) =306 kips Total = 1,302 kips

Total effective seismic weight, W = 1,166 + (4)(1,302) = 6,374 kips. This total excludes the lower half of the first story walls, which do not contribute to the seismic loads that are not imposed on the CMU shear walls.

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Chapter 10: Masonry 10.2.7.2 San Rafael base shear calculation. The seismic response coefficient, Cs, is computed using Standard Section 12.8:

Cs =

S DS 1.00 = = 0.20 R/I 51

S D1 0.60 = = 0.355 T ( R / I ) 0.338 (5 1)

(Controls)

where T is the fundamental period of the building, which is 0.338 seconds as computed previously. The value for Cs is taken as 0.20 (the lesser of these two). This value is still larger than the minimum specified in Standard, Section 12.8-5, which is: Cs = 0.044SD1I = (0.044)(0.60)(1) = 0.026 The total seismic base shear is then calculated using Standard Equation 12.8-1: V = CsW = (0.20)(6,374) = 1,275 kips 10.2.7.3 San Rafael vertical distribution of seismic forces. The vertical distribution of seismic forces is determined in accordance with Standard Section 12.8.3, which is described in Section 10.2.4.3. Note that for the Standard, k = 1.0 because T = 0.338 seconds, which is less than 0.5 seconds (similar to the previous example buildings). The application of the Provisions equations for this building is shown in Table 10.2-17: Table 10.2-17 San Rafael Seismic Forces and Moments by Level Level x

wx (kips)

hx (ft)

w xh xk (ft-kips)

Cvx

Fx (kips)

Vx (kips)

Mx (ft-kips)

5 4 3 2 1 ∑

1,166 1,302 1,302 1,302 1,302 6,374

43.34 34.67 26.00 17.33 8.67

50,534 45,140 33,852 22,564 11,288 163,378

0.309 0.276 0.207 0.138 0.069 1.000

394 353 264 176 88 1,275

394 747 1,011 1,187 1,275

3,420 9,890 18,660 28,950 40,000

1.0 kip = 4.45 kN, 1.0 ft = 0.3048 m, 1.0 ft-kip = 1.36 kN-m

10.2.7.4 San Rafael horizontal distribution of forces This is the same as for the Birmingham 2 design; see Section 10.2.6.4. Total shear in Wall D: Vtot = 0.125V + 1.365(0.0238)V = 0.158V = 201.5 kips 10.2.7.5 San Rafael Transverse Wall D.

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

10.2.7.5.1 Shear strength. This design continues to illustrate ELF analysis and, as explained for the Birmingham 2 design, smaller demands could be derived from dynamic analysis. All other parameters are similar to those for Birmingham 2 except the following: An = (2 × 1.25 in. × 32.67 ft × 12 in.) + (41 in.2 × 9 cells) = 1,349 in.2 The shear strength of each Wall D, based on the aforementioned formulas and data, are summarized in Table 10.2-18. Table 10.2-18 San Rafael Shear Strength Calculations for Wall D Vx Mx 2.5Vx Pu Vnm Vns Story Mx/Vxdv (kips) (ft-kips) (kips) (kips) (kips) (kips) 5 62.3 540 0.265 156 46.3 225 163 4 118 1,563 0.405 295 92.6 222 163 3 160 2,948 0.564 400 161 222 163 2 188 4,574 0.745 470 185 209 163 1 201 6,320 0.962 503 231 197 163

Vn (kips) 388 385 385 372 360

Vn (max) (kips) 360 337 311 282 247

φVn (kips) 288 270 249 226 198

1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

The maximum on Vn controls over the sum of Vm and Vs at all stories. Since φVn does not exceed 2.5Vx except at the top story it is necessary to check the shear corresponding to 1.25φMn, (as discussed below in Section 10.2.7.5.3). It will be learned, once φMn is determined below, that an increase in shear capacity is required. However, as we are not there yet, let us proceed in a sequence similar to a real design and continue with the flexural design. 10.2.7.5.2 Axial and flexural strength. The basics of flexural design are demonstrated for the previous locations. The demand is much higher at this location, which introduces issues about the amount and distribution of reinforcement in excess of the minimum requirements. Therefore, both strength and ductility checks are examined. 10.2.7.5.2.1 Strength check. Load combinations, using factored loads, are presented in Table 10.2-19 for Wall D. In the table, Load Combination 1 is 1.4D + QE + 0.5L and Load Combination 2 is 0.7D + QE. Table 10.2-19 San Rafael Load Combinations for Wall D Load Combination 1 Level PD PL Pu Mu (kips) x (kips) (kips) (ft-kips) 5 66 0 92 540 4 132 15 185 1563 3 198 25 277 2948 2 265 34 371 4574 1 331 41 463 6320 1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

10-92

Load Combination 2 Pu Mu (kips) (ft-kips) 46 540 92 1563 139 2948 186 4574 232 6320

Chapter 10: Masonry

Strength at the bottom story (where P, V and M are the greatest) is examined. This example considers Load Combination 2 from Table 10.2.19 to be the governing case, because it has the same lateral load as Load Combination 1 but lower values of axial force. Refer to Figure 10.2-13 for notation and dimensions.

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

32.67'

#7 at 4.00' on center

16.33' Cm

0.33'

16.00'

15.33'

1.6 ksi

16.00'

2.00'

P = 0 Case

Cs ~ ˜ 28.8' 0.49'

a = 0.1.97'

ε y= 0.00207

c = 2.46' N.A.

13.87'

Σ Ts

1.60' 0.00130

c = 8.00'

T sl

T s2 22.6k

22.6k

a = 6.40'

Intermediate Case

Ts2*

C s2

0.00207 0.00130

C s1 3.6k

ε m = 0.0025

N.A.

Cm 1.6 ksi

ε y= 0.00207 c = 8.00'

8.33'

Σ T s1

Cms

1.6 ksi

ε m = 0.0025 ε y = 0.00207

10.62' Cms

C s1

8.00'

N.A.

7.07'

Balanced Case 10.00'

ε = 0.00132

C s2

ε = 0.00141

10.00' a = 14.15'

3.54'

ε y= 0.00207 ΣT

c = 17.69'

Figure 10.2-13 San Rafael: Strength of Wall D Strength diagrams superimposed on strain diagrams for the three cases. (1.0 ft = 0.3048 m)

10-94

Chapter 10: Masonry Examine the strength of Wall D at Level 1: § § §

Pumin

max

= 232 kips = 463 kips

Mu = 6,320 ft-kips

Because special reinforced masonry shear walls are used (Seismic Design Category D), vertical reinforcement at 4 feet on center and horizontal bond beams at 4 feet on center are prescribed (TMS 402, Sec. 1.14.2.2.5). For this bending moment, the #5 bars at 4’-0” on center used at Birmingham 2 will not suffice (refer to the φPn - φMn diagram for Birmingham 2 in Figure 10.2-10). It is desirable to keep the reinforcement to as small an amount as necessary in order to keep φMn relatively low, in order to keep the required shear capacity down when the check for shear corresponding to 1.25φMn is made. The calculation procedure is similar to that presented in Section 10.2.4.5.2. The strain and stress diagrams are shown in Figure 10.2-14 and the results are as follows: §

P = 0 case:

φPn = 0 φMn = 4,492 ft-kips §

Intermediate case (setting c = 8.0 ft):

φPn = 265 kips φMn = 7,261 ft-kips §

Balanced case:

φPn = 852 kips φMn = 10,364 ft-kips The simplified φPn - φMn curve is shown in Figure 10.2-14 and indicates that the design with nine #7 bars is satisfactory.

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FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

φ Pn

φMn

φ Pn

500 kips

Balance: (10,364 ft-kips, 852 kips)

M u = 6320 ft-kips

1,000 kips

φ M n = 8600(±) ft-kip 1.25φ M n = 10750(±) ft-kip

Pu,max = 463 kips

Intermediate: (7261 ft-kips, 265 kips)

Pu,min = 232 kips 3−Point φ Pn - φ M n curve

5,000 ft-kips P=0 (4492 ft-kips, 0 kips)

1.25φ M n = 8810(±) ft-kip M n = 7050(±) ft-kip Αppx. φ Pn - φ M n curve

φMn

10,000 ft-kips

Figure 10.2-14 φPn - φMn Diagram for San Rafael Wall D (1.0 kip = 4.45 kN, 1.0 kip-ft = 1.36 kN-m) 10.2.7.5.2.2 Ductility check. TMS 402 Section 3.3.3.5.4 is illustrated in the prior designs. Recall that this calculation uses factored gravity axial loads (based on the Standard) to result in the minimum Pu value instead of load combination D + 0.75L + 0.525QE per TMS 402 Section 3.3.3.5.1.d. Refer to Figure 10.2-15 and the following calculations which illustrate this using loads at the bottom story (highest axial loads). The extra grout required for shear is also ignored here. More grout gives higher compression capacity, which is conservative.

10-96

Chapter 10: Masonry

32'-8" P

0.33'

32.34'

ε m = 0.0025

17.90'

N.A.

14.44' c

ε s = 1.5ε y

8.66'

60 ) 29,000 = 0.0031 = 1.5 (

0.8 f 'm = 1.6 ksi

2.89'

11.55' a 72.51 ksi f y = 60 ksi

17.90'

51.9 ksi

C s1

C s2

C s3

31.4 ksi C s4 6.10'

10.10' 14.10'

T s5 Ts4

Ts3

Ts2

Ts1

29.7 ksi 5.90' 49.8 ksi 9.90' 13.90'

f y = 60 ksi 1.5f y = 90 ksi

17.90' 0.33'

(9) #7 @ 4'-0" o.c.

Figure 10.2-15 Ductility Check for San Rafael Wall D (1.0 ft = 0.3048 m, 1.0 ksi = 6.89 MPa) For Level 1 (the bottom story), the unfactored loads are as follows: P = 331 kips 10-97

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples

Cm = 0.8f’m[(2.5 in.)(11.55 ft)(12) + (3 cells)(41 in.2)] =751 kips Cs1 = (0.60 in.2)(60 ksi) = 36 kips Cs2 = (0.60 in.2)(51.9 ksi) = 31 kips Cs3 = (0.60 in.2)(31.4 ksi) = 19 kips Cs4 is neglected. ∑C = 837 kips Ts1 = Ts2 = (0.60 in.2)(60 ksi) = 36 kips Ts3 = (0.60 in.2)(49.8 ksi) = 30 kips Ts4 = (0.60 in.2)(18 ksi) = 18 kips Ts5 is neglected. ∑T = 120 kips ∑C > ∑P + T 837 kips > 451kips

The compression capacity is larger than the tension capacity, so ductile failure is assured. The maximum area of flexural tensile reinforcement requirement of TMS 402 Section 3.3.3.5 is satisfied. 10.2.7.5.3 Check for Shear Corresponding to 1.25φ Mn. From Figure 10.2-14, values for 1.25φMn can be obtained: §

Load Combination 1: Pu max = 463 kips 1.25φMn = 10,750 ft-kips

§

Load Combination 2: Pu min = 232 kips 1.25φMn = 8,810 ft-kips

Both cases need to be checked. As our example focuses on Load Combination 2, only that case is discussed below.

10-98

Chapter 10: Masonry

⎛ 8810 ⎞ ⎜ ⎟ M n ⎛ ⎞ ⎝ 0.9 ⎠ = 1.94 , which is less than the 2.5 upper bound. = 1.25 ⎜ 1.25 ⎟ 6320 ⎝ Mu ⎠ Therefore, the shear demand is 1.94 times the value from analysis. Referring to Table 10.2-18, Vu = (1.94)(201 kips) = 390 kips > 198 kips = φVn. There is more shear demand than allowed; this can be addressed by adding grouted cells.

⎛ ⎞ 390 − 198 Additional grouted cells = ⎜ ⎟ = 2.9 cells 2 ⎝ (1.6 ksi)(41 in. /cell) ⎠ If the two cells adjacent to the end cells of the wall are grouted (for a total of four additional grouted cells), the shear requirement is satisfied. The additional grout will add to the building weight slightly. The authors recommend that another design iteration be performed to address significant increases in building weight; however, another iteration is not presented here. Note that the above shear check is just for Load Combination 2. Load Combination 1 also needs to be checked; it may necessitate even more grouted cells. 10.2.7.6 San Rafael deflections. Recall the assertion that the calculations for deflection involve many variables and assumptions and that any calculation of deflection is approximate at best. The requirements and procedures for computing deflection are provided in Section 10.2.4.6. For the San Rafael building, the determination of whether the walls will be cracked is as follows: be = effective masonry wall width be = [(2 × 1.25 in.)(32.67 ft × 12) + (13 cells)(41 in.2/cell)]/32.67 ft × 12) = 3.86 in. An = be l = (3.86 in.)(32.67 × 12) = 1513 in.2 S = be l2/6 = (3.86)(32.67 × 12)2/6 = 98,877 in.3 fr = 0.063(36 cells/49 cells) + 1.63(13 cells/49 cells) = 0.090 ksi Pu is calculated using 1.00D (see Table 10.2-19 for values and refer to Section 10.2.4.6 for discussion). Table 10.2-20 provides a summary of these calculations. (The extra grout required for shear strength is also not considered here; the revision would reduce slightly the computed deflections by raising the cracking moment.) Table 10.2-20 San Rafael Cracked Wall Determination Level 5 4 3 2 1

Pumin

(kips) 66 132 198 265 331

Mcr (ft-kips)

Mx (ft-kips)

Status

1101 1460 1820 2185 2544

540 1563 2948 4574 6320

Uncracked Cracked Cracked Cracked Cracked 10-99

FEMA P-751, NEHRP Recommended Seismic Provisions: Design Examples 1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

For the uncracked wall: In. = Ig = bel3/12 = (3.86 in.)(32.67 × 12)3/12 = 1.94 × 107 in.4 As in the three previous examples, Icr will be taken as 0.35Ig for the wall deflection calculation. For this example, the deflection computation instead will use the cracked moment of inertia in the lower two stories and the gross moment of inertia in the upper three stories. The results from a RISA 2D analysis, in which both flexural and shear deflections are included, are shown in Table 10.2-21 and are approximately 50 percent higher than the use of Ieff over the full height. Table 10.2-21 San Rafael Deflections Level

F (kips)

Ieff (in.4)

δflexural (in.)

δshear (in.)

δtotal (in.)

Cd δtotal (in.)

Δ (in.)

5 4 3 2 1

62.3 55.8 41.7 27.8 13.9

1.94 × 107 0.679 × 107 0.679 × 107 0.679 × 107 0.679 × 107

0.431 0.321 0.212 0.110 0.030

0.067 0.062 0.050 0.035 0.019

0.498 0.382 0.262 0.145 0.049

1.743 1.337 0.917 0.508 0.172

0.406 0.420 0.409 0.336 0.172

kip = 4.45 kN, 1.0 in. = 25.4 mm. F = Fx for level (from Table 10.2-17) × 0.158

The maximum drift occurs at Level 4; per Provisions Table 5.2.8 it is: Δ = 0.420 in. < 1.04 in. = 0.01hn (Standard Table 12.12-1) 10.2.7.7 San Rafael out-of-plane forces Standard Section 12.11 requires that bearing walls be designed for out-of-plane loads determined as follows: w = 0.40 SDS IWw ≥ 0.1Ww w = (0.40)(1.00)(1)(60 psf) = 24psf ≥ 6.0 psf = 0.1Ww The out-of-plane bending moment, using the strength design method for masonry, for the pressure w =24 psf and considering the P-delta effect, is computed to be 2,232 in.-lb/ft. This compares to a computed strength of the wall of 30,000 in.-lb/ft, considering the #7 bars at 4 feet on center. Thus, the wall is loaded to approximately 7 percent of its capacity in flexure in the out-of-plane direction. (See Section 10.1.5.2.5 for a more detailed discussion of strength design of masonry walls, including the P-delta effect.)

10-100

Chapter 10: Masonry 10.2.7.8 San Rafael orthogonal effects According to Standard Section 12.5.3, orthogonal interaction effects have to be considered for Seismic Design Category D where the ELF procedure is used (as it is here). The out-of-plane effect is 7 percent of capacity, as discussed in Section 10.2.7.7. Where considering the 0.3 combination factor, the out-of-plane action adds approximately 2 percent overall to the interaction effect. For the lowest story of the wall, this could conceivably require a slight increase in capacity for inplane actions. In the authors’ opinion, this is on the fringe of requiring real consideration (in contrast to the end walls of Example 10.1). This completes the design of the transverse Wall D. 10.2.7.9 Summary of San Rafael Wall D §

8-inch CMU

§

f’m = 2,000 psi

§

Reinforcement: Vertical #7 bars at 4 feet on center at intermediate cells. Two bond beams with two #5 bars at each story, at floor bearing and at 4 feet above each floor. Horizontal joint reinforcement at alternate courses recommended but not required.

§

Grout: All cells with reinforcement and bond beams, plus grout at four additional cells.

10.2.8 Summary of Wall D Design for All Four Locations Table 10.2-22 compares the reinforcement and grout for Wall D designed for each of the four locations. Table 10.2-22 Variation in Reinforcement and Grout by Location for Wall D Birmingham 1 Albuquerque Birmingham 2 Vertical bars 5 - #4 9 - #4 9 - #5 Horizontal bars 10 - #4 + jt. reinf. 10 - #4 + jt. reinf. 20 - #5 Grout (cu. ft) 91 122 152

San Rafael 9 - #7 20 - #5 162

1 cu. ft = 0.0283 m3.

10-101

11 Wood Design Peter W. Somers, P.E., S.E. Contents 11.1 THREE-STORY WOOD APARTMENT BUILDING, SEATTLE, WASHINGTON ................ 3 11.1.1 Building Description .............................................................................................................. 3 11.1.2 Basic Requirements ............................................................................................................... 6 11.1.3 Seismic Force Analysis.......................................................................................................... 9 11.1.4 Basic Proportioning ............................................................................................................. 11 11.2 WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES, CALIFORNIA ............................................................................................................................. 30 11.2.1 Building Description ............................................................................................................ 30 11.2.2 Basic Requirements ............................................................................................................. 31 11.2.3 Seismic Force Analysis........................................................................................................ 33 11.2.4 Basic Proportioning of Diaphragm Elements ...................................................................... 34

FEMA P752, NEHRP Recommended Provisions: Design Examples This chapter examines the design of a variety of wood building elements. Section 11.1 features a threestory, wood-frame apartment building. Section 11.2 illustrates the design of the roof diaphragm and wallto-roof anchorage for the masonry building featured in Section 10.1. In both cases, only those portions of the designs necessary to illustrate specific points are included. Typically, the weak link in wood systems is the wood strength at the connections, but the desired ductility must be developed by means of these connections. Wood members have some ductility in compression (particularly perpendicular to grain) but little in tension. Nailed plywood shear panels develop considerable ductility through yielding of nails and crushing of wood adjacent to nails. Because wood structures are composed of many elements that must act as a whole, the connections must be considered carefully to ensure that the load path is complete. Tying the structure together is essential to good earthquake-resistant construction. Wood elements often are used in low-rise masonry and concrete buildings. The same basic principles apply to the design of wood elements, but certain aspects of the design (for example, wall-to-diaphragm anchorage) are more critical in mixed systems than in all-wood construction. Wood structural panel sheathing is referred to as “plywood” in this chapter. However, sheathing can include plywood and other products, such as oriented-strand board (OSB), that conform to the appropriate materials standards. The calculations herein are intended to provide a reference for the direct application of the design requirements presented in the 2009 NEHRP Recommended Provisions (hereafter, the Provisions) and its primary reference document, ASCE 7-05 Minimum Design Loads for Buildings and Other Structures (hereafter, the Standard) and to assist the reader in developing a better understanding of the principles behind the Provisions and the Standard. In addition to the Provisions, the documents below are referenced in this chapter. Although the Standard references the 2005 edition of the AF&PA SDPWS, this chapter utilizes the 2008 edition, which is the more recent, updated version. Note that the 2005 editions of the AF&PA NDS and AF&PA NDS Supplement are the latest versions. ACI 318

American Concrete Institute. 2008. Building Code Requirements and Commentary for Structural Concrete.

ACI 530

American Concrete Institute. 2005. Building Code Requirements for Masonry Structures.

ANSI/AITC A190.1

American Institute of Timber Construction. 2002. Structural GluedLaminated Timber.

ASCE 7

American Society of Civil Engineers. 2005. Minimum Design Loads for Buildings and Other Structures.

AF&PA Guideline

American Forest & Paper Association 1996. Manual for Engineered Wood Construction (LRFD), Pre-Engineered Metal Connectors Guideline.

AF&PA NDS

American Forest & Paper Association. 2005. National Design Specification.

11-2

Chapter 11: Wood Design AF&PA NDS

American Forest & Paper Association. 2005. National Supplement Design Specification, Design Values for Wood Construction.

AF&PA SDPWS

American Forest & Paper Association. 2008. Special Design Provisions for Wind and Seismic.

WWPA Rules

Western Wood Products Association. 2005. Western Lumber Grading Rules.

11.1 THREE-‐STORY WOOD APARTMENT BUILDING, SEATTLE, WASHINGTON This example features a wood-frame building with plywood diaphragms and shear walls. 11.1.1 Building Description This three-story wood-frame apartment building has a double-loaded central corridor. The building is typical stick-frame construction consisting of wood joists and stud bearing walls supported by a concrete foundation wall and strip footing system. The seismic force-resisting system consists of plywood floor and roof diaphragms and plywood shear walls. Figure 11.1-1 shows a typical floor plan and Figure 11.1-2 shows a longitudinal section and elevation. The building is located in a residential neighborhood a few miles north of downtown Seattle. The shear walls in the longitudinal direction are located on the exterior faces of the building and along the corridor. The entire solid (non-glazed) area of the exterior walls has plywood sheathing, but only a portion of the corridor walls will require sheathing. In the transverse direction, the end walls and one line of interior shear walls provide lateral resistance. It should be noted that while plywood sheathing generally is used at the exterior walls for reasons beyond just lateral load resistance, the interior longitudinal (corridor) and transverse shear walls could be designed using gypsum wallboard as permitted by AF&PA SDPWS Section 4.3.7.5. However, the corridor shear walls are not included in this example and the interior transverse walls are designed using plywood sheathing, largely due to the required shear capacity. The floor and roof systems consist of wood joists supported on bearing walls at the perimeter of the building, the corridor lines, plus one post-and-beam line running through each bank of apartments. Exterior walls are framed with 2×6 studs for the full height of the building to accommodate insulation. Interior bearing walls require 2×6 or 3×4 studs on the corridor line up to the second floor and 2×4 studs above the second floor. Apartment party walls are not load-bearing; however, they are double walls and are constructed of staggered 2×4 studs at 16 inches on center. Surfaced, dry (seasoned) lumber is used for all framing to minimize shrinkage. Floor framing members are assumed to be composed of Douglas FirLarch material and wall framing is Hem-Fir No. 2, as graded by the WWPA Rules. The material and grading of other framing members associated with the lateral design is as indicated in the example. The lightweight concrete floor fill is for sound isolation and is interrupted by the party walls, corridor walls and bearing walls. The building is founded on interior footing pads, continuous strip footings and concrete foundation walls (Figure 11.1-3). The depth of the footings and the height of the walls are sufficient to provide crawlspace clearance beneath the first floor.

11-3

FEMA P752, NEHRP Recommended Provisions: Design Examples

148'-0" 28'-0"

25'-0"

Post & beam lines

25'-0"

9'-0"

56'-0"

6'-0"

9'-0"

8'-0"

11 2" Lightweight concrete over plywood deck on joists at 16" o.c.

Typical apartment partitions

Figure 11.1-1 Typical floor plan (1.0 ft = 0.3048 m)

148'-0" Corridor walls 9'

Stairs

Doors

8'-0"

15'-0"

26'-0" Sheathed wall

30'-0"

26'-0"

30'-0" Glazed wall

Figure 11.1-2 Longitudinal section and elevation (1.0 ft = 0.3048 m)

11-4

13'-0"

Chapter 11: Wood Design

Pad footings Continuous footing and grade wall

Figure 11.1-3 Foundation plan 11.1.1.1 Scope. In this example, the structure is designed and detailed for forces acting in the transverse and longitudinal directions, including the following: §

Development of seismic loads using the Simplified Alternative Structural Design Criteria (herein referred to as the “simplified procedure”) contained in Standard Section 12.14.

§

Design and detailing of transverse plywood walls for shear and overturning moment.

§

Design and detailing of plywood floor and roof diaphragms.

§

Design and detailing of wall and diaphragm chord members.

§

Design and detailing of longitudinal plywood walls using the requirements for perforated shear walls.

The simplified procedure, new to the 2005 edition of the Standard, is permitted for relatively short, simple and regular structures utilizing shear walls or braced frames. The seismic analysis and design procedure is much less involved than a building utilizing a seismic force resisting system listed in Standard Section 12.2 and analyzed using one of the procedures listed in Standard Section 12.6. See Section 11.1.2.2 for a more detailed discussion of what is and is not required for the seismic design. In accordance with Standard Section 12.14.1.1, the subject building qualifies for the simplified procedure because of the following attributes: §

Residential occupancy

§

Three stories in height

§

Bearing wall lateral system

§

At least two lines of lateral force-resisting elements in both directions, at least one on each side of the center of mass

§

No cantilevered diaphragms or structural irregularities 11-5

FEMA P752, NEHRP Recommended Provisions: Design Examples

11.1.2 Basic Requirements 11.1.2.1 Seismic Parameters Table 11.1-1 Seismic Parameters Design Parameter

Value

Occupancy Category (Standard Sec. 1.5.1)

Short-Period Response, SS

1.34

Site Class (Standard Sec. 11.4.2)

Seismic Design Category (Standard Sec. 11.6)

Seismic Force-Resisting System (Standard Table 12.14-1)

Wood Structural Panel Shear Walls

Response Modification Coefficient, R

6.5

11.1.2.2 Structural Design Criteria 11.1.2.2.1 Ground Motion Parameter. Unlike the typical design procedures in Standard Chapter 12, the simplified procedure requires consideration of just one spectral response parameter, SDS. This is because the behavior of short, stiff buildings for which the simplified procedure is permitted will always be governed by short-period response. In accordance with Standard Section 12.14.8.1: SDS = 2/3FaSS The site coefficient, Fa, can be determined using Standard Section 12.14.8.1 with simple default values based on soil type or using Standard Table 11.4-1 if the site class is known. Since Standard Table 11.4-1 generally will result in more favorable value, that method is used for this example. Using SS = 1.34 and Site Class D, Standard Table 11.4-1 lists a short-period site coefficient, Fa, of 1.0. Therefore, in accordance with Standard Equation: SDS = 2/3(1.0)(1.34) = 0.89 11.1.2.2.2 Seismic Design Category (Standard Sec. 11.6). Where the simplified procedure is used, Standard Section 11.6 permits the Seismic Design Category to be determined based on Standard Table 11.6-1 only. Based on the Occupancy Category and the design spectral response acceleration parameter, the subject building is assigned to Seismic Design Category D. 11.1.2.2.3 Seismic Force-Resisting Systems (Standard Sec. 12.14.4). See Figure 11.1-4. For both directions, the load path for seismic loading consists of plywood floor and roof diaphragms and plywood shear walls. Because the lightweight concrete floor topping is discontinuous at each partition and wall, it is not considered to be a structural diaphragm. In accordance with Standard Table 12.14-1, building has a bearing wall system comprised of light-framed walls sheathed with wood structural panels. The response modification factor, R, is 6.5 for both directions.

11-6

Chapter 11: Wood Design

148'-0" 56'-0" 15'-0"

84'-0" 30'-0"

30'-0"

Perforated exterior wall

56'-0"

25'-0"

Some corridor walls are used as shear walls

25'-0"

Solid interior wall

15'-0"

30'-0"

Solid end wall

30'-0"

Figure 11.1-4 Load path and shear walls (1.0 ft = 0.3048 m) 11.1.2.2.4 Diaphragm Flexibility (Standard Sec. 12.14.5). Standard Section 12.14.5 defines a diaphragm comprised of wood structural panels as flexible. Because the lightweight concrete floor topping is discontinuous at each partition and wall, it is not considered to be a structural diaphragm. 11.1.2.2.5 Application of Loading (Standard Sec. 12.14.6). For the simplified procedure, seismic loads are permitted to be applied independently in two orthogonal directions. 11.1.2.2.6 Design and Detailing Requirements (Standard Sec. 12.14.7). The plywood diaphragms are designed for the forces prescribed in Standard Section 12.14.7.4. The design of foundations is per Standard Section 12.13 and wood design requirements are based on Standard Section 14.4 as discussed in greater detail below. This example does not require any collector elements (Standard Sec. 12.14.7.3). 11.1.2.2.7 Analysis Procedure (Standard Sec. 12.14.8). For the simplified procedure, only one analysis procedure is specified and it is described in greater detail in Section 11.1.3.1 below. 11.1.2.2.8 Drift Limits (Standard Sec. 12.14.8.5). Where the simplified procedure is used, there are not any specific drift limitations because the types of structures for which the simplified procedure is applicable are generally not drift-sensitive. As specified in Standard Section 12.14.8.5, if a determination of expected drift is required (for the design of cladding for example), then drift is permitted to be computed as 1 percent of the building height unless a more detailed analysis is performed. 11.1.2.2.9 Combination of Load Effects (Standard Sec. 12.14.3). The basic design load combinations are as stipulated in Standard Chapter 2 as modified by the Standard Sec. 12.14.3.1.3. Seismic load effects according to the Standard Equations 12.14-5 and 12.14-6 are as follows: E = QE + 0.2SDSD E = QE - 0.2SDSD

11-7

FEMA P752, NEHRP Recommended Provisions: Design Examples

where seismic and gravity are additive and counteractive, respectively. For SDS = 0.89, the design load combinations are as follows: (1.2 + 0.2SDS)D + 1.0QE + 0.5L + 0.2S = 1.38D + 1.0QE + 0.5L + 0.2S (0.9 - 0.2SDS)D - 1.0QE = 0.72D - 1.0QE Note that there is no redundancy factor for the simplified procedure. 11.1.2.3 Basic Gravity Loads §

Roof: Table 11.1-2 Roof Gravity Loads

§

Load Type

Value

Live/Snow Load (in Seattle, snow load governs over roof live load; in other areas this may not be the case)

25 psf

Dead Load (including roofing, sheathing, joists, insulation and gypsum ceiling)

15 psf

Floor: Table 11.1-3 Floor Gravity Loads

11-8

Load Type

Value

Live Load

40 psf

Dead Load (1-1/2-in. lightweight concrete, sheathing, joists and gypsum ceiling. At first floor, omit ceiling but add insulation.)

20 psf

Interior Partitions and Corridor Walls (8 ft high at 11 psf)

7 psf distributed floor load

Exterior Frame Walls (wood siding, plywood sheathing, 2×6 studs, batt insulation and 5/8-in. gypsum wallboard)

15 psf of wall surface

Exterior Double Glazed Window Wall

9 psf of wall surface

Party Walls (double-stud sound barrier)

15 psf of wall surface

Stairways

20 psf

Chapter 11: Wood Design Typical Footing (10 in. by 1 ft-6 in.) and Stem Wall (10 in. by 4 ft-0 in.)

690 plf

Applicable Seismic Weights at Each Level Wroof = Area (roof dead load + interior partitions + party walls) + End Walls + Longitudinal Walls

182.8 kips

W3 = W2 = Area (floor dead load + interior partitions + party walls) + End Walls + Longitudinal Walls

284.2 kips

Effective Total Building Weight, W

751 kips

For modeling the structure, the first floor is assumed to be the seismic base, because the short crawlspace with concrete foundation walls is stiff compared to the superstructure. 11.1.3 Seismic Force Analysis The analysis is performed manually following a step-by-step procedure for determining the base shear (Standard Sec. 12.14.8.1), vertical distribution of forces (Standard Sec. 12.14.8.2) and horizontal distribution of forces (Standard Sec. 12.14.8.3). For a building with flexible diaphragms, Standard Section 12.14.8.3.1 allows the horizontal distribution of forces to be based on tributary areas and accidental torsion need not be considered for the simplified procedure. 11.1.3.1 Base Shear Determination. According to Standard Equation 12.14-11:

FS DS W R

Where F = 1.2 for a three-story building, R = 6.5 and W = 751 kips as determined previously. Therefore, the base shear is computed as follows:

(1.2)(0.89) (751) = 123.4 kips (both directions) 6.5

11.1.3.2 Vertical Distribution of Forces. Forces are distributed as shown in Figure 11.1-5, where the story forces are calculated according to Standard Equation 12.14-12 as follows:

Fx =

wx V W

This results in a uniform vertical distribution of forces, where the story force is based on the relative seismic weight of the story with all stories at the same seismic acceleration (as opposed to the triangular or parabolic vertical distribution used in the Equivalent Lateral Force procedure of Standard Sec. 12.8)

11-9

FEMA P752, NEHRP Recommended Provisions: Design Examples

Froof F3rd F2nd

9'-0" 9'-0" 9'-0"

56'-0" Roof 3rd

floor

2nd

floor

Ground floor

Figure 11.1-5 Vertical shear distribution (1.0 ft = 0.3048 m) The story force at each floor is computed as: Froof = [182.8/751](123.4) F3rd = [284.2/751](123.4) F2nd = [284.2/751](123.4) Σ

= 30.0 kips = 46.7 kips = 46.7 kips = 123.4 kips

11.1.3.3 Horizontal Distribution of Shear Forces to Walls. Since the diaphragms are defined as flexible by Standard Section 12.14.5, the horizontal distribution of forces is based on tributary area to the individual shear walls in accordance with Standard Section 12.14.8.3.1. For this example, forces are distributed as described below. 11.1.3.3.1 Longitudinal Direction. In this direction, there are four lines of resistance, but only the exterior walls are considered in this example. The total story force tributary to the exterior wall is determined as follows: (25/2)/56Fx = 0.223Fx The distribution to each individual shear wall segment along this exterior line is discussed in Section 11.1.4.7 below. 11.1.3.3.2 Transverse Direction. Again, based on the flexible diaphragm assumption, force is to be distributed based on tributary area. As shown in Figure 11.1-4, there are three sets of two shear walls, each offset in plan by 8 feet. For the purposes of this example, each set of walls is assumed to be in alignment, resisting the same tributary width. The result is that the building is modeled with a diaphragm consisting of two simple spans, which provides a more reasonable horizontal distribution of force than a pure tributary area distribution. For a two-span, flexible diaphragm, the central walls will resist one-half of the total load, or 0.50Fx. The other walls resist story forces in proportion to the width of diaphragm between them and the central walls. The left set of walls in Figure 11.1-4 resists (60/2)/148Fx = 0.203Fx and the right set resists (88/2)/148Fx = 0.297Fx, where 60 feet and 88 feet represent the dimension from the ends of the building to the centroid of the two central walls. Note that this does not exactly match the existing diaphragm spans, but is a reasonable simplification to account for the three sets of offset shear walls at the ends and middle of the building.

11-10

Chapter 11: Wood Design 11.1.3.4 Diaphragm Design Forces. As specified in Standard Section 12.14.7.4, the design forces for floor and roof diaphragms are the same forces as computed for the vertical distribution in Section 11.1.3.2 above plus any force due to offset walls (not applicable for this example). The weight tributary to the diaphragm, wpx, need not include the weight of walls parallel to the force. For this example, however, since the shear walls in both directions are relatively light compared to the total tributary diaphragm weight, the diaphragm force is computed based on the total story weight, for convenience. Therefore, the diaphragm forces are exactly the same as the story forces shown above. 11.1.4 Basic Proportioning Designing a plywood diaphragm and plywood shear wall building principally involves the determination of sheathing thicknesses and nailing patterns to accommodate the applied loads. This is especially the case where the simplified procedure is utilized, since there are not any deflection checks and possible subsequent design iterations. In addition to the wall and diaphragm design, this design example features framing member and connection design for elements including shear wall end posts and hold-downs, foundation anchorage and diaphragm chords. Nailing patterns in diaphragms and shear walls have been established on the basis of tabulated requirements included in the AF&PA SDPWS. It is important to consider the framing requirements for a given nailing pattern and capacity as indicated in the notes following the tables. In addition to strength requirements, AF&PA SDPWS Section 4.2.4 places aspect ratio limits on plywood diaphragms (length/width must not exceed 4/1 for blocked diaphragms) and AF&PA SDPWS Section 4.3.4 places similar limits on shear walls (height/width must not exceed 2/1 for full design capacities). 11.1.4.1 Strength of Members and Connections. The Standard references the AF&PA NDS and AF&PA SDPWS for engineered wood structures. These reference standards support both Allowable Stress Design (ASD) and Load and Resistance Factor Design (LRFD) as permitted by the Standard. For this example, LRFD is utilized. The AF&PA NDS and AF&PA Supplement contains the material design values for framing members and connections, while the AF&PA SDPWS contains the diaphragm and shear wall tables as well as detailing requirements for shear wall and diaphragm systems. Throughout this example, the resistance of members and connections subjected to seismic forces, acting alone or in combination with other prescribed loads, is determined in accordance with the AF&PA NDS and AF&PA SDPWS. The methodology is somewhat different between the AF&PA NDS for framing members and connections and the AF&PA SDPWS for shear walls and diaphragms. For framing members and connections, the AF&PA NDS incorporates the notation Fb, Ft, Z, etc., for reference design values, which are then modified using standard wood adjustment factors, CM, Cr, CF, etc. (used for both ASD and LRFD) and then for LRFD are modified by a format conversion factor, KF, a resistance factor, φ and a time effect factor, λ, to compute an adjusted design resistance, Fb’, Ft’, Z’. These factors are defined in AF&PA NDS Appendix N. For shear walls and diaphragms, the AF&PA SDPWS contains tabulated unit shear values, vs, which are multiplied by a resistance factor, φD, equal to 0.8. This is the only modification to the tabulated design values since this building utilizes Douglas Fir Larch framing. Additional modification would be required for other species in accordance with the footnotes to the tabular values in the AF&PA SDPWS.

11-11

FEMA P752, NEHRP Recommended Provisions: Design Examples For pre-engineered connection elements, the AF&PA NDS does not contain a procedure for converting the manufacturer’s cataloged values (typically as ASD values) to LRFD. However, such a procedure is contained in a guideline published with the 1996 edition of the LRFD wood standard (AF&PA Guideline). The AF&PA Guideline contains a method for converting allowable stress design values for cataloged metal connection hardware (for example, tie-down anchors) into ultimate capacities for use with strength design. The procedure, which is used for this example, can generally be described as taking the catalog ASD value, multiplying by 2.88 and dividing the by the load duration factor on which the cataloged value is based (typically 1.33 or 1.60 for pre-engineered connection hardware often used for wind or seismic design). 11.1.4.2 Transverse Shear Walls. The design will focus on the more highly loaded interior walls; the end walls would be designed in a similar manner. 11.1.4.2.1 Load to Interior Transverse Walls. As computed in Section 11.1.3.3.2, the total story force resisted by the central walls is 0.50Fx. Since the both walls are the same length and material, each individual wall will resist one-half of the total or 0.25Fx. Therefore: Froof = 0.25(30.0) F3rd = 0.25(46.7) F2nd = 0.25(26.7) Σ

= 7.50 kips = 11.68 kips = 11.68 kips = 30.86 kips

The story forces and story shears resisted by the individual wall segment is illustrated in Figure 11.1-6.

v = 0.767 kip/ft v = 1.236 kip/ft

27'-0"

F 2nd = 11.68 kips

v = 0.300 kip/ft

F 3rd = 11.68 kips

F roof = 7.50 kips

25'-0"

22.09 kips

Figure 11.1-6 Transverse section: end wall (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN, 1.0 kip/ft = 14.6 kN/m) 11.1.4.2.2 Roof to Third Floor V = 7.50 kips v = 7.50/25 = 0.300 klf Try a 1/2-inch (15/32) plywood rated sheathing (not Structural I) on blocked 2× Hem-Fir members at 16 inches on center with 8d common nails at 6 inches on center at panel edges and 12 inches on center at intermediate framing members. From AF&PA SDPWS Table 4.3A, this shear wall assembly has a nominal unit shear capacity, vs, of 0.520 klf. However, according to Note 3 of AF&PA SDPWS Table 4.3A, the design shear resistance values are for Douglas Fir-Larch or Southern Pine and must be adjusted for Hem-Fir wall framing. The specific gravity adjustment factor equals 1-(0.5-SG) where SG is 11-12

Chapter 11: Wood Design the specific gravity of the framing lumber. From AF&PA NDS Table 11.3.2A, the SG for Hem-Fir is 0.43. Therefore, the adjustment factor is 1-(0.5-0.43) = 0.93. The adjusted shear capacity is computed as follows: 0.93φDvs = 0.93(0.8)(0.520) = 0.387 klf > 0.300 klf

While 3/8- or 7/16-inch plywood could be used at this level, 1/2-inch is used for consistency with the lower floors. 11.1.4.2.3 Third Floor to Second Floor V = 7.50 + 11.68 = 19.18 kips v = 19.18/25 = 0.767 klf Try 1/2-inch (15/32) plywood rated sheathing (not Structural I) on blocked 2× Hem-Fir members at 16 inches on center with 10d nails at 3 inches on center at panel edges and at 12 inches on center at intermediate framing members. From AF&PA SDPWS Table 4.3A, this shear wall assembly has a nominal unit shear capacity, vs, of 1.200 klf. The adjusted shear capacity is computed as follows: 0.93φDvs = 0.93(0.8)(1.200) = 0.893 klf > 0.767 klf

For this shear wall assembly, the width of framing at panel edges needs to be checked relative to AF&PA SDPWS Section 4.3.7.1. In accordance with Item 4 of that section, 3× framing is required at adjoining panel edges since the wall has 10d nails spaced at 3 inches or less and because the unit shear capacity exceeds 0.700 klf for a building assigned to Seismic Design Category D. However, an exception to this section permits double 2× framing to be substituted for the 3× member, provided that the 2× framing is adequately stitched together. Since the double 2× framing is often preferred over the 3× member, this procedure will be utilized for this example. The exception requires the double 2× members to be connected to “transfer the induced shear between members.” For the purposes of this example, the induced shear along the vertical plane between adjacent panels will assumed to be equal to the adjusted design shear of 0.893 klf. Using 16d common wire nails and 2× Hem-Fir framing, AF&PA NDS Table 11N specifies a lateral design value, Z, of 0.122 kips per nail. The adjusted design capacity is: Z’ = ZKFφλ = (0.122)(2.16/0.65)(0.65) = 0.264 kips per nail and the number of nails per foot is 0.893/0.264 = 3.4, so provide 4 nails per foot. Therefore, use double 2× framing at panel edges fastened with 16d at 3 inches on center and staggered (as required by the exception where the nail spacing is less than 4 inches). 11.1.4.2.4 Second Floor to First Floor 19.18 + 11.68 = 30.86 kips v = 30.86/25 = 1.236 klf Try 5/8-inch (19/32) plywood rated sheathing (not Structural I) on blocked 2-inch Hem-Fir members at 16 inches on center with 10d common nails at 2 inches on center at panel edges and 12 inches on center at 11-13

FEMA P752, NEHRP Recommended Provisions: Design Examples intermediate framing members. From AF&PA SDPWS Table 4.3A, this shear wall assembly has a nominal unit shear capacity, vs, of 1.740 klf. The adjusted shear capacity is computed as follows: 0.93φDvs = 0.93(0.8)(1.740) = 1.294 klf > 1.236 klf

This shear wall assembly also requires 3× or stitched double 2× framing at panel edges. In this case, 3× framing is recommended, since the tight nail spacing required to stitch the double 2× members could lead to splitting and bolts or lag screws would not be economical. Rather than increasing the plywood thickness at this level, adequate capacity could be achieved by using Douglas Fir-Larch framing members or using 1/2-inch plywood on both sides of the shear wall framing. 11.1.4.3 Transverse Shear Wall Anchorage. AF&PA SDPWS Section 4.3.6.4.2 requires tie-down (hold-down) anchorage at the ends of shear walls where net uplift is induced. Net uplift is computed as the combination of the seismic overturning moment and the dead load counter-balancing moment using the load combination 0.72D - 1.0QE. The design requirements for the shear wall end posts and tie-downs have evolved over the past several code cycles. The 2008 AF&PA SDPWS requires the tie-down devices (Sec. 4.3.6.4.2) and end posts (Sec. 4.3.6.1.1) to be designed for a tension or compression force equal to the induced unit shear multiplied by the shear wall height. It can be inferred from AF&PA SDPWS Figure 4E, that the shear wall height, h, refers to the sheathing height and not the story height, since the end post load is a function of the length of shear wall sheathing that engages the end post. 11.1.4.3.1 Tie-down Anchors at Third Floor. For the typical 25-foot interior wall segment, the overturning moment at the third floor is: M0 = 9(7.50) = 67.5 ft-kip = QE For the counter-balancing moment, it is assumed that the interior transverse walls will engage a certain length of exterior and corridor bearing wall for uplift resistance. The width of floor is taken as the length of solid wall panel at the exterior, or 10 feet. See Figures 11.1-1 and 11.1-13. For convenience, the same length is used for the longitudinal walls. The designer should take care to assume a reasonable amount of tributary dead loads that can be engaged considering the connections and stiffness of the cross wall elements. In this situation, considering that the exterior and corridor walls are plywood-sheathed shear walls, the assumption noted above is considered reasonable. The weight of interior wall, 11 psf, is used for both conditions. Shear wall self weight = (9 ft)(25 ft)(11 psf)/1,000 Tributary floor = (10 ft)(25 ft)(15 psf)/1,000 Tributary longitudinal walls = (9 ft)(10 ft)(11 psf)(2)/1,000 Σ

= 2.47 kips = 3.75 kips = 1.98 kips = 8.20 kips

0.72QD = 0.72(8.20)(12.5) = 73.8 ft-kip Since the dead load stabilizing moment exceeds the overturning moment, uplift anchorage is not required at the third floor. An end post for shear wall boundary compression is required, but since the design is similar to the second floor end post, it is not illustrated here. 11.1.4.3.2 Tie-down Anchors at Second Floor 11-14

Chapter 11: Wood Design

The overturning moment at the second floor is: M0 = 18(7.50) + 9(11.68) = 240 ft-kip The counter-balancing moment is computed using the same assumptions as for the third floor. Shear wall self weight = (18 ft)(25 ft)(11 psf)/1,000 Tributary floor = (10 ft)(25 ft)(15 psf)(2)/1,000 Tributary longitudinal walls = (18 ft)(10 ft)(11 psf)(2)/1,000 Σ

= 4.95 kips = 7.50 kips = 3.96 kips = 16.41 kips

0.72QD = 0.72(16.41)(12.5) = 148 ft-kips M0 (net) = 240 - 148 = 92 ft-kips As would be expected, uplift anchorage is required. As described above, the design uplift force is computed using a unit shear demand of 0.768 klf at the second floor and a net length of wall height equal to 8 feet. Note that 8 feet is appropriate for this calculation given the detailing for this structure. As shown in Figure 11.1-10, the plywood sheathing is not detailed as continuous across the floor framing, which results in a net sheathing height of approximately 8 feet. If the sheathing were detailed across the floor framing, then 9 feet would be the appropriate wall height for use in computing tie-down demands. Since there is no net uplift force at the third floor, the third floor load need not be considered. Therefore, the design uplift force at the second floor is: T = 0.768 klf (8 ft) = 6.14 kips Note that this uplift force exceeds the forces determined using the net overturning moment, which would be equal to 92 ft-kips / 25 ft = 3.68 kips, thus providing the intended added level of conservatism for the end posts and tie-downs. Use a double tie-down anchor to connect the end posts. For ease of construction, select a tie-down device that screws to the end post. See Figure 11.1-7. A tie-down with a 5/8-inch threaded rod and fourteen 1/4-inch screws has a cataloged ASD capacity of 5.645 kips for Douglas Fir-Larch framing based on a load duration factor of 1.6. Using the AF&PA Guideline procedure for pre-engineered connections described in Section 11.1.4.1 (KF = 2.88/1.60), the LRFD capacity is determined as follows: ZKFφλ = (5.645)(2.88/1.60)(0.65)(1.0) = 6.60 kips > 6.14 kips

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FEMA P752, NEHRP Recommended Provisions: Design Examples

Post and tie down Corner of wall Solid blocking at post Floor framing

Threaded rod connected to tie-down anchors screwed to studs

Figure 11.1-7 Shear wall tie down at suspended floor framing 11.1.4.3.3 Tie-down Anchors at First Floor. The overturning moment at the first floor is: M0 = 27(7.50) + 18(11.68) + 9(11.68) = 517 ft-kip = QE The counter-balancing moment is computed using the same assumptions as for the second floor. Shear wall self weight = (27 ft)(25 ft)(11 psf)/1,000 Tributary floor = (10 ft)(25 ft)(15 psf)(3)/1,000 Tributary longitudinal walls = (27 ft)(10 ft)(11 psf)(2)/1,000 Σ

= 7.42 kips = 11.25 kips = 5.94 kips = 24.61 kips

0.72QD = 0.72(24.61)12.5 = 222 ft-kip M0 (net) = 517-222 = 295 ft-kip As expected, uplift anchorage is required. The design uplift force is computed using a unit shear force of 1.236 klf at the first floor and a net wall height of 8 feet. Combined with the uplift force at the second floor, the total design uplift force at the first floor is: T = 6.14 kips + 1.236 klf (8 ft) = 16.0 kips Use a double tie-down anchor that extends down into the foundation with an anchor bolt. Tie-downs with a 7/8-inch threaded rod anchor and three 1-inch bolts through a 6×6 Douglas Fir-Larch end post have a

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Chapter 11: Wood Design cataloged capacity of 12.1 kips based on a load duration factor of 1.6. The LRFD capacity of two tie downs is computed as follows: 2ZKFφλ = 2(12.1)(2.88/1.60)(0.65)(1.0) = 28.3 kips > 16.0 kips

Next, check the LRFD capacity of the bolts in double shear. For the three bolts, the AF&PA NDS gives the following equation: 3ZKFφλ = 3(5.50)(2.16/0.65)(0.65)(1.0) = 35.6 kips > 16.0 kips

The strength of the end post, based on failure across the net section, must also be checked. A reasonable approach to preclude net tension failure from being a limit state would be to provide an end post whose nominal resistance exceeds the nominal strength of the tie-down device. The nominal strength of the first-floor double tie-down is 28.3/0.65 = 43.5 kips. Therefore, the nominal tension capacity at the net section should be greater than 43.5 kips. Try a 6×6 Douglas Fir-Larch No. 1 end post. Accounting for 1-1/16-inch bolt holes, the net area of the post is 24.4 in2. Using φ = 1.0 for nominal strength, according to the AF&PA NDS Supplement: Ft’ = FtKFφλ = (0.825)(2.16/0.8)(1.0)(1.0) = 2.228 ksi T’ = Ft’A = 2.228(24.4) = 44.5 kips > 54.4 kips

Not shown here but for a group of bolts, the row and/or group tear-out capacity must be checked for all bolted connections with multiple fasteners. Refer to AF&PA NDS Section 10.1.2 and Appendix E. For the maximum compressive load at the end post, combine the maximum gravity load plus the seismic overturning load. However, since the exterior and interior longitudinal walls are load-bearing stud walls, the gravity load demand on the shear wall end post is minimal. Therefore, without any significant gravity load, the compression force on the end post is the same as the tension force per AF&PA SDPWS Section 4.3.6.1.1 and equal to 16.0 kips at the first floor. Due to the relatively short clear height of the post, the governing condition is bearing perpendicular to the grain on the bottom plate. Check the bearing of the 6×6 end post on a 3×6 Douglas Fir-Larch No. 2 plate, per the AF&PA NDS Supplement: F’c┴ = Fc┴KFφλ = (0.625)(1.875/0.9)(0.9)(1.0) = 1.17 ksi C’ = F’c┴A = 1.17(5.5)(5.5) = 35.4 kips > 16.0 kips

11.1.4.3.4 Check Overturning at the Soil Interface. A summary of the overturning forces is shown in Figure 11.1-8. To compute the overturning at the soil interface, the overturning moment must be increased for the 4-foot foundation height: M0 = 517 + 30.9(4.0) = 640 ft-kip

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FEMA P752, NEHRP Recommended Provisions: Design Examples

F3rd = 11.68 kips

WD Provide 5 8" anchor bolts at 1'-4" o.c.

F roof = 7.50 kips

F 2nd = 11.68 kips

25'-0" Provide tie-down anchor at each end bolted to end post at each level.

Figure 11.1-8 Transverse wall: overturning (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN) However, it then may be reduced in accordance with Standard Section 12.14.8.4: M0 = 0.75(640) = 480 ft-kip To determine the total resistance, combine the weight above with the dead load of the first floor and foundation. Load from first floor = (25 ft) (10 ft) (20-4+1) psf / 1,000 = 4.25 kips where 4 psf is the weight reduction due to the absence of a ceiling and 1 psf is the weight of insulation. The length of the longitudinal foundation wall included is a conservative approximation of the amount that can be engaged assuming minimum nominal reinforcement in the foundation. Foundation weight = (690 plf [10 ft +10 ft + 25 ft])/1,000 First floor Structure above Σ

= 31.05 kips = 4.25 kips = 24.61 kips = 59.91 kips

Therefore, 0.72D - 1.0QE = 0.72(59.91)(12.5 ft) - 1.0(480) = 59.2 ft-kips, which is greater than zero, so the overturning check is acceptable. 11.1.4.3.5 Anchor Bolts for Shear. At the first floor, the unit shear demand, v, is 1.236 klf. Try 5/8-inch bolts in a 3×6 Douglas Fir-Larch sill plate, in single shear, parallel to the grain. In accordance with the AF&PA NDS: ZKFφλ = (1.11)(2.16/0.65)(0.65)(1.0) = 2.40 kips per bolt The required bolt spacing is 2.4/(1.236/12) = 23.3 in. Therefore, provide 5/8-inch bolts at 16 inches on center to match the joist layout. AF&PA SDPWS Section 4.3.6.4.3 requires plate washers at all shear wall anchor bolts and where the nominal unit shear capacity exceeds 400 plf, the plate washer needs to extend within 1/2 inch of the edge of the plate on the side with the sheathing, so provide 4.5-inch-square plate washers.

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Chapter 11: Wood Design Note that in addition to the capacity of the bolt in the wood sill, the bolt capacity in the concrete foundation wall should be checked based on ACI 318 Appendix D. 11.1.4.6 Remarks on Shear Wall Connection Details. In typical platform frame construction, details must be developed that will transfer the lateral loads through the floor system and, at the same time, accommodate normal material sizes and the cross-grain shrinkage in the floor system. The connections for wall overturning in Section 11.1.4.5 are an example of one of the necessary force transfers. The transfer of diaphragm shear to supporting shear walls is another important transfer, as is the transfer from a shear wall on one level to the level below. The floor-to-floor height is 9 feet with approximately 1 foot occupied by the floor framing. Using standard 8-foot-long plywood sheets for the shear walls, a gap occurs over the depth of the floor framing. It is common to use the floor framing to transfer the lateral shear force. Figures 11.1-9 and 11.1-10 depict this accomplished by nailing the plywood to the bottom plate of the shear wall, which is nailed through the floor plywood to the double 2×12 chord in the floor system.

Bottom plate nailing to develop capacity of shear wall Wall edge nailing Floor edge nailing

Joists at 16" o.c. toe-nailed to wall plate with 2-8d

2-2x12 chord for diaphragm Where shear forces are small, sheet metal framing clips may be replaced by toe-nailing the 2-2x12 chord to the wall top plate

2x12 blocking between joists toe-nailed to wall plate with 2-16d and to 2-2x12 chord with 2-16d Provide sheet metal framing clips (under blocking) to connect 2-2x12 chord to wall top plate

Figure 11.1-9 Bearing wall (1.0 in = 25.4 mm)

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FEMA P752, NEHRP Recommended Provisions: Design Examples

8d at 6" o.c. into blocking

Wall and floor nailing as in bearing wall detail (Figure 11.1-9)

2x12 joists

Sheet metal strap connecting blocks across bottom of joists

Framing clip or alternate toe nailing as in bearing wall detail (Figure 11.1-9)

2x12 blocking between joists at 4'-0" o.c. extending inward from wall to third joist and toe-nailed to each joist with 2-8d and to wall plate with 2-16d. Connect across joist with sheet metal strap and 2-10d each side. Remainder of blocking at plywood joists to be 2x3 lumber.

Figure 11.1-10 Nonbearing wall (1.0 in. = 25.4 mm) The top plate of the lower shear wall also is connected to the double 2×12 by means of sheet metal framing clips to the double 2×12 to transfer the force back out to the lower plywood. (Where the forces are small, toe nails between the double 2×12 and the top plate may be used for this connection.) This technique leaves the floor framing free for cross-grain shrinkage. The floor plywood is nailed directly to the framing at the edge of the floor, before the plate for the upper wall is placed. Also, the floor diaphragm is connected directly to framing that spans over the openings between shear walls. The axial strength and the connections of the double 2×12 chords, allows them to function as collectors to move the force from the full length of the diaphragm to the discrete shear walls. (According to Standard Sec. 12.14.7.3, the design of collector elements in wood shear wall buildings need not consider increased seismic demands due to overstrength.) The floor joist is toe nailed to the wall below for forces normal to the wall. Likewise, full-depth blocking is provided adjacent to walls that are parallel to the floor joists, as shown in Figure 11.1-10. (Elsewhere, the blocking for the floor diaphragm only need be small pieces, flat 2×4s for example.) The connections at the foundation are similar (see Figure 11.1-11).

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Chapter 11: Wood Design

Bottom plate nailing Wall edge nailing Flashing, sheathing, and finish siding not shown

2x12 joist toe-nailed to sill with 3-16d 2x12 blocking between joists. Connect to sill with sheet metal framing clips. 3x6 sill, dense, treated 5

8" dia. anchor bolt at 2'-8" or 1'-4" o.c. w/ plate washer

Figure 11.1-11 Foundation wall detail (1.0 in. = 25.4 mm) The particular combinations of nails and bent steel framing clips shown in Figures 11.1-9, 11.1-10 and 11.1-11 to accomplish the necessary force transfers are not the only possible solutions. A great amount of leeway exists for individual preference, as long as the load path has no gaps. Common carpentry practices often will provide most of the necessary transfers, but careful attention to detailing and inspection is an absolute necessity to ensure a complete load path. 11.1.4.5 Roof Diaphragm. While it has been common practice to design plywood diaphragms as simply supported beams spanning between shear walls, the diaphragm design for this example will consider some continuity at the central shear walls. The design will be based on the shears associated with the tributary area distribution of force to the shear walls and will account for moments at the diaphragm midspan as well as the central shear walls. (Note that this diaphragm assumption would result in a slightly different distribution of lateral loads to the shear walls, which is not accounted for in this example.) From Section 11.1.3.4, the diaphragm design force at the roof is the same as the roof story force, so FP,roof = 30.0 kips.

11-21

FEMA P752, NEHRP Recommended Provisions: Design Examples As discussed previously, the design force computed in this example includes the internal force due to the weight of the walls parallel to the motion. Particularly for one-story buildings, it is common practice to remove that portion of the design force. It is conservative to include it, as is done here. 11.1.4.5.1 Diaphragm Nailing. Idealizing the building as a two-span diaphragm with three sets of walls as described previously, the maximum diaphragm shear occurs at the ends of the 88-foot diaphragm span. Assuming a uniform distribution of the diaphragm force across the building, the maximum shear over the entire diaphragm width is computed as follows: V = (30.0)(88/148)/2 = 8.93 kips v = 8.93 / 56 ft = 0.160 klf Try 1/2-inch (15/32) plywood rated sheathing (not Structural I) on blocked 2-inch Douglas Fir-Larch members at 16 inch on center, with 8d nails at 6 inches on center at all boundaries and panel edges and 12 inches on center at intermediate framing members. From AF&PA SDPWS Table 4.2A, this diaphragm assembly has a nominal unit shear capacity, vs, of 0.540 klf. The adjusted shear capacity is computed as follows:

φDvs = 0.8(0.540) = 0.432 klf > 0.160 klf

11.1.4.5.2 Chord and Splice Connection. Diaphragm continuity is an important factor in the design of the chords. The design must consider the tension/compression forces, due to positive moment at the middle of the span as well as negative moment at the interior shear wall. It is reasonable (and conservative) to design the chord for the positive moment assuming a simply supported beam and for the negative moment accounting for continuity. The positive moment is wl2/8, where w is the unit diaphragm force and l is the length of the governing diaphragm span. For a continuous beam of two unequal spans, under a uniform load, the maximum negative moment is:

M− =

wl13 + wl23 8(l1 + l2 )

where w is the unit diaphragm force and l1 and l2 are the lengths of the two diaphragm spans. For w = 30.0 kips / 148 ft = 0.203 klf, the maximum positive moment is: 0.203(88)2 / 8 = 197 ft-kip and the maximum negative moment is:

0.203(88)3 + 0.203(60)3 = 154 ft-kips 8(88 + 60) The positive moment controls and the design chord force is 197/56 = 3.51 kips. Try a double 2×12 Douglas Fir-Larch No. 2 chord. Due to staggered splices, compute the tension capacity based on a single 2×12, with an area of An = 16.88 in2. According to the AF&PA NDS Supplement: Ft’ = FtKFφλ = (0.575)(2.16/0.8)(0.8)(1.0) = 1.537 ksi T’ = Ft’A = 1.537(16.88) = 25.9 kips > 3.51 kips

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Chapter 11: Wood Design

For chord splices, use 16d nails in the staggered chord members. According to the AF&PA NDS, the capacity of one 16d common wire nail in single shear with two 2× Douglas Fir-Larch members is 0.141 kips. The adjusted strength per nail is: Z’ = ZKFφλ = (0.141)(2.16/0.65)(0.65)(1.0) = 0.305 kips The number of required nails at the splice is 3.51/0.305 = 11.5, so use twelve 16d nails. Assuming a 4-foot splice length, provide two rows of six 16d nails at 8 inches on center. A typical chord splice connection is shown in Figure 11.1-12.

2-2x12 chord - Douglas Fir-Larch, No. 1

Stagger joints in opposite faces by 4'-0"

2 Rows of 6-16d nails @ 8" o.c.

Drive shim in joint

Figure 11.1-12 Diaphragm chord splice (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) 11.1.4.6 Second- and Third-Floor Diaphragm. The design of the second- and third-floor diaphragms follows the same procedure as for the roof diaphragm. From Section 10.1.3.4, the diaphragm design force for both floors is Fp,3rd = Fp,2nd = 46.7 kips. 11.1.4.6.1 Diaphragm Nailing. The maximum diaphragm shear is computed as follows: V = (46.7)(88/148)/2 = 13.88 kips v = 13.88 / 56 ft = 0.248 klf With an adjusted capacity of 0.432 klf, the same diaphragm as at the roof also works for the floors.

11-23

FEMA P752, NEHRP Recommended Provisions: Design Examples

11.1.4.6.2 Chord and Splice Connection. Computed as described above for the roof diaphragm, the maximum positive moment is 306 ft-kips and the design chord force is 5.48 kips. By inspection, a double 2×12 chord spliced with 16d nails similar to the roof level is adequate. The number of nails at the floors is 5.48/0.305 = 18 nails, so for the 4-foot splice length, provide two rows of nine 16d nails at 5 inches on center on each side of the splice joint. 11.1.4.7 Longitudinal Direction. Only one exterior shear wall section will be designed here. The design of the corridor shear walls would be similar to that of the transverse walls. For loads in the longitudinal direction, diaphragm stresses are negligible and the nailing provided for the transverse direction is more than adequate. The design of the exterior wall utilizes the provisions for perforated shear walls as defined in AF&PA SDPWS Sections 4.3.4.1 and 4.3.5.3. The procedure for perforated shear walls applies to walls with openings that have not been specifically designed and detailed for force transfer around the openings. Essentially, a perforated wall is treated in its entirety rather than as a series of discrete wall piers. The use of this design procedure is limited by several conditions as specified in AF&PA SDPWS Section 4.3.5.3. The main aspects of the perforated shear wall design procedure are as follows. The design shear capacity of the shear wall is the sum of the capacities of each segment (all segments must have the same sheathing and nailing) reduced by an adjustment factor that accounts for the geometry of the openings. Uplift anchorage (tie-down) is required only at the ends of the wall (not at the ends of all wall segments), but all wall segments must resist a specified tension force (using anchor bolts at the foundation and strapping or other means at upper floors). Requirements for shear anchorage and collectors (drag struts) across the openings are also specified. It should be taken into account that the design capacity of a perforated shear wall is less than that of a standard segmented wall with all segments restrained against overturning. However, the procedure is useful in eliminating interior hold downs for specific conditions and thus is illustrated in this example. The portion of the story force resisted by each exterior wall was computed previously as 0.223Fx. The exterior shear walls are composed of three separate perforated shear wall segments (two at 30 feet long and one at 15 feet long, all with the same relative length of full-height sheathing), as shown in Figure 11.1-2. This section will focus on the design of a 30-foot section. Assuming that load is distributed to the wall sections based on relative length of the shear panel, then the total story force to the 30-foot section is (30/75)0.223Fx = 0.089Fx per floor. The load per floor is: Froof = 0.089(30.0) F3rd = 0.089(46.7) F2nd = 0.089(46.7) Σ

= 2.67 kips = 4.16 kips = 4.16 kips = 10.99 kips

11.1.4.7.1 Perforated Shear Wall Resistance. The design shear capacity for perforated shear walls is computed as the factored shear resistance for the sum of the wall segments, multiplied by an adjustment factor that accounts for the percentage of full-height (solid) sheathing and the ratio of the maximum opening to the story height as described in AF&PA SDPWS Section 4.3.3.5. At each level, the design shear capacity, Vwall, is: Vwall = (vC0)ΣLi where: 11-24

Chapter 11: Wood Design

v = factored shear resistance (AF&PA SDPWS Table 4.3A) C0 = shear capacity adjustment factor (AF&PA SDPWS Table 4.3.3.5) ΣLi = sum of shear wall segment lengths For the subject wall, the widths of perforated shear wall segments are 4+10+4 = 18 feet, the percent of full-height sheathing is 18/30 = 0.60 and the maximum opening height is 4 feet. Therefore, per AF&PA SDPWS Table 4.3.3.5, C0 = 0.83.

4'-0"

6'-0"

10'-0"

6'-0"

4'-0"

F roof

Floor framing

2'-6" 2'-6" 4'-0"

F 3rd

F 2nd Perforated shear wall h = 8'-0"

Provide tie-down for uplift at ends

Perforated shear wall segment

Figure 11.1-13 Perforated shear wall at exterior (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) The wall geometry (and thus the adjustment factor and total length of wall segments) is the same at all three levels, as shown in Figure 11.1-13. Perforated shear wall plywood and nailing are determined below. §

Roof to third floor: V = 2.67 kips Required v = 2.67/0.83/18 = 0.179 klf

Try 1/2-inch (15/32) plywood rated sheathing (not Structural I) on blocked 2× Hem-Fir members at 16 inches on center with 8d nails at 6 inches on center at panel edges and at 12 inches on center at 11-25

FEMA P752, NEHRP Recommended Provisions: Design Examples intermediate framing members. From AF&PA SDPWS Table 4.3A, this shear wall assembly has a nominal unit shear capacity, vs, of 0.520 klf. Adjusting for framing material, the shear capacity is computed as follows: 0.93φDvs = 0.93(0.8)(0.520) = 0.387 klf > 0.179 klf §

Third floor to second floor: V = 2.67+4.16 = 6.83 kips Required v = 6.83/0.83/18 = 0.457 klf

Try 1/2-inch (15/32) plywood rated sheathing (not Structural I) on blocked 2× Hem-Fir members at 16 inches on center with 8d nails at 4 inches on center at panel edges and at 12 inches on center at intermediate framing members. From AF&PA SDPWS Table 4.3A, this shear wall assembly has a nominal unit shear capacity, vs, of 0.760 klf. Adjusting for framing material, the shear capacity is computed as follows: 0.93φDvs = 0.93(0.8)(0.760) = 0.565 klf > 0.457 klf §

Second floor to first floor: V = 6.83+4.16 = 10.99 kips Required v = 10.99/0.83/18 = 0.735 klf

Try 1/2-inch (15/32) plywood rated sheathing (not Structural I) on blocked 2× Hem-Fir members at 16 inches on center with 10d nails at 3 inch on center at panel edges and at 12 inch on center at intermediate framing members. From AF&PA SDPWS Table 4.3A, this shear wall assembly has a nominal unit shear capacity, vs, of 1.200 klf. Adjusting for framing material, the shear capacity is computed as follows: 0.93φDvs = 0.93(0.8)(1.200) = 0.893 klf > 0.735 klf

Note that the nominal unit shear capacity of 1.200 klf is less than the maximum permitted nominal shear capacity of 1.740 klf in accordance with AF&PA SDPWS Section 4.3.5.3, Item 3. 11.1.4.7.2 Perforated Shear Wall Tension Chord. According to AF&PA SDPWS Section 4.3.6.1.2, tension and compression chords and associated anchorage must be evaluated at the ends of the wall only. Uplift anchorage at each wall segment is treated separately as described later. The tension and compression forces at the wall ends are determined per AF&PA SDPWS Equation 4.3-8 as follows:

T =C =

Vh C0 ∑ Li

where: V = design shear force in the shear wall h = shear wall height (per floor) C0 = shear capacity adjustment factor ΣLi = sum of widths of perforated shear wall segments 11-26

Chapter 11: Wood Design

For this example, the tension chord and tie-down will be designed at the first floor only; the other floors would be computed similarly and tie-down devices, as shown in Figure 11.1-7, would be used. For h = 8 ft, C0 = 0.83 and ΣLi = 18 ft, the tension force is computed as follows: Third floor: T = 2.67(8)/(0.83×18) Second floor: T = (2.67 + 4.16)(8)/(0.83×18) First floor: T = (2.67 + 4.16 + 4.16)(8)/(0.83×18) Σ

= 1.42 kips = 3.66 kips = 6.42 kips = 11.50 kips

For the dead load to resist the tension chord uplift, assume a tributary floor width equal to the half the span of the window header at the end wall segment. The tributary width is 6.5 feet and the tributary joist span is 8 feet. The tributary weight is computed as follows: Exterior wall weight = (27 ft)(6.5 ft)(9 psf)/1,000 Tributary roof = (8 ft)(6.5 ft)(15 psf)/1,000 Tributary floor = (8 ft)(6.5 ft)(20 psf)(2)/1,000 Σ

= 1.58 kips = 0.78 kips = 2.08 kips = 4.44 kips

The net uplift is computed as follows: 0.72D - 1.0E = 0.72(4.44) - 11.5 = 8.30 kips Therefore, uplift anchorage is required per AF&PA SDPWS Section 4.3.6.4.2. Since the chord member resists the perforated shear wall compression load and supports the window header as well, use a 6×6 Douglas Fir-Larch No. 1, similar to the transverse walls. The post has ample tension capacity. For the anchorage, try a tie-down device with a 7/8-inch anchor bolt and twenty 1/4-inch screws into the post. Using the method described above for computing the strength of a pre-engineered tie-down, the capacity is computed as follows: ZKFφλ = (7.87)(2.88/1.60)(0.65)(1.0) = 9.2 kips > 8.3 kips

The design of the tie-downs at the second and third floors is similar. 11.1.4.7.3 Perforated Shear Wall Compression Chord. The force in the compression chord is the same as the tension chord equal to 11.5 kips at the first floor. Again, just the chord at the first floor will be designed here; the design at the upper floors would be similar. Although not explicitly required by AF&PA SDPWS Section 4.3.6.1.2, it is rational to combine the chord compression with gravity loading, using the load combination 1.4D + 1.0QE + 0.5L + 0.2S, in order to design the chord member. The dead load is as computed above and the live load and snow load are 4.16 kips and 1.30 kips, respectively. Therefore, the design compression force is as follows: 1.38(4.44) + 1.0(11.5) + 0.5(4.16) + 0.2(1.30) = 20.0 kips The bearing capacity on the bottom plate was computed previously as 35.4 kips, which is greater than 20.0 kips. Note that where end posts are loaded in both directions, orthogonal effects must be considered in accordance with Standard Section 12.5. 11.1.4.7.4 Anchorage at Shear Wall Segments. The anchorage at the base of a shear wall segment (bottom plate to floor framing or foundation wall) is designed per AF&PA SDPWS Section 4.3.6.4. This section requires two types of anchorage: in-plane shear anchorage (AF&PA SDPWS Sec. 4.3.6.4.1.1) 11-27

FEMA P752, NEHRP Recommended Provisions: Design Examples and distributed uplift anchorage (AF&PA SDPWS Sec. 4.3.6.4.1.2). While both types of anchorage need only be provided at the full-height sheathing, the shear anchorage is usually extended at least over the entire length of the perforated shear wall to simplify the detailing and reduce the possibility of construction errors. The in-plane shear anchorage is required to resist the following:

V C0 ∑ Li

where: V = design shear force in the shear wall C0 = shear capacity adjustment factor ΣLi = sum of widths of perforated shear wall segments This equation is the same as was previously used to compute unit shear demand on the wall segments. Therefore, the in-plane anchorage will be designed to meet the following unit, in-plane shear forces: §

Third floor: v = 0.179 klf

§

Second floor: v = 0.457 klf

§

First floor: v = 0.735 klf

The required distributed uplift force, t, is equal to the in-plane shear force, v. Per AF&PA SDPWS Section 4.3.6.4, this uplift force must be provided with a complete load path to the foundation. That is, the uplift force at each level must be combined with the uplift forces at the levels above (similar to the way overturning moments are accumulated down the building). At the foundation level, the unit in-plane shear force, v and the unit uplift force, t, are combined for the design of the bottom plate anchorage to the foundation wall. The design unit forces are as follows: §

Shear: v = 0.735 klf

§

Tension: t = 0.179+0.457+0.735 = 1.371 klf

Assuming that stresses on the wood bottom plate govern the design of the anchor bolts, the anchorage is designed for shear (single shear, wood-to-concrete connection) and tension (plate washer bearing on bottom plate). The interaction between shear and tension need not be considered in the wood design for this configuration of loading. Try a 5/8-inch bolt at 32 inches on center with a 4.5-inch square plate washer (AF&PA SDPWS Section 4.3.6.4.3 requires plate washer to extend within 1/2 inch of the 5.5-inch-wide bottom plate). As computed previously, the shear capacity of a 5/8-inch bolt in a 3×6 Douglas Fir-Larch sill plate is 2.40 kips. The demand per bolt is 0.735 klf (32/12) = 1.96 kips, so the 32-inch spacing is adequate for shear.

11-28

Chapter 11: Wood Design For anchor bolts at 32 inches on center, the tension demand per bolt is 1.371 klf (32/12) = 3.66 kips. Bearing capacity of the plate washer (using a Douglas Fir No. 2 bottom plate) is computed per AF&PA NDS Supplement as follows: F’c┴ = Fc┴KFφλ = (0.625)(1.875/0.9)(0.9)(1.0) = 1.17 ksi C’ = F’c┴A = 1.17(4.5)(4.5) = 23.7 kips > 3.66 kips

The anchor bolts themselves must be designed for combined shear and tension in accordance with ACI 318-08. In addition to designing the anchor bolts for uplift, a positive load path must be provided to transfer the uplift forces into the bottom plate. One method for providing this load path continuity is to use metal straps nailed to the studs and lapped around the bottom plate, as shown in Figure 11.1-14. Attaching the studs directly to the foundation wall (using embedded metal straps) for uplift and using the anchor bolts for shear only is an alternative approach.

Nail to stud Metal strap used to resist uplift, lapped under sill plate.

1 1 16x4 2x4 2 steel plate washer at anchor bolt

8" dia. anchor bolt at 2'-8" o.c. (Used to resist shear and tension)

Figure 11.1-14 Perforated shear wall detail at foundation (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) At the upper floors, the load transfer for in-plane shear is accomplished by using nailing or framing clips between the bottom plates, rim joists and top plates in a manner similar to that for standard shear walls. The uniform uplift force can be resisted either by using the nails in withdrawal (for small uplift demand) or by providing vertical metal strapping between studs above and below the level considered. This type of connection is shown in Figure 11.1-15. For this type of connection (and the one shown in

11-29

FEMA P752, NEHRP Recommended Provisions: Design Examples Figure 11.1-14) to be effective, shrinkage of the floor framing must be minimized using dry or manufactured lumber.

Nails to resist shear Nail to stud (typical)

Metal strap used to resist uplift

Sheet metal framing clips (Under blocking) used to resist shear.

Figure 11.1-15 Perforated shear wall detail at floor framing For example, consider the second floor. The required uniform uplift force, t = 0.244+0.496 = 0.740 klf. Place straps at every other stud, so the required strap force is 0.740(32/12) = 1.97 kips. Provide an 18-gauge strap with twelve 10d nails at each end.

11.2 WAREHOUSE WITH MASONRY WALLS AND WOOD ROOF, LOS ANGELES, CALIFORNIA This example features the design of the wood roof diaphragm and wall-to-diaphragm anchorage for the one-story masonry building described in Section 10.1 of this volume of design examples. Refer to that example for more detailed building information and the design of the masonry walls. 11.2.1 Building Description This is a very simple rectangular warehouse, 100 feet by 200 feet in plan (see Figure 11.2-1), with a roof height of 28 feet. The wood roof structure slopes slightly, but it is nominally flat. The long walls (side walls) are 8 inches thick and solid and the shorter end walls are 12 inches thick and penetrated by several large openings.

11-30

Chapter 11: Wood Design

5 bays at 40'-0" = 200'-0"

5 bays at 20'-0" = 100'-0"

Concrete masonry walls

Typ. articulated glulam beams

Typ. roof joists at 24" o.c. Steel column on spread footing

Open

Plywood roof sheathing

Figure 11.2-1 Building plan (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) Based on gravity loading requirements, the roof structure consists of wood joists, supported by 8-3/4-inch-wide by 24-inch-deep glued-laminated timber beams on steel columns. The joists span 20 feet and the beams span 40 feet, as an articulated system. Typical roof framing is assumed to be Douglas FirLarch No 1 as graded by the WWPA Rules. The glued-laminated timber beams meet the requirements of Combination 24F-V4 per AITC A190.1. The plywood roof deck acts as a diaphragm to carry lateral loads to the exterior walls. There are no interior walls for seismic resistance. The roof contains a large opening that interrupts the diaphragm continuity. The diaphragm contains continuous cross ties in both principal directions that serve as part of the wall anchorage system. The following aspects of the structural design are considered in this example: §

Development of diaphragm forces based on the Equivalent Lateral-Force Procedure used for the masonry wall design (Sec. 10.1)

§

Design and detailing of a plywood roof diaphragm with a significant opening

§

Computation of drift and P-delta effects

§

Anchorage of diaphragm and roof joists to masonry walls

§

Design of cross ties and subdiaphragms

11.2.2 Basic Requirements 11.2.2.1 Seismic Parameters

11-31

FEMA P752, NEHRP Recommended Provisions: Design Examples Table 11.2-1 Seismic Parameters Design Parameter

Value

1.50

0.60

Site Class (Standard Sec. 11.4.2)

Occupancy Category (Standard Sec. 1.5.1)

Seismic Design Category (Standard Sec. 11.6)

Seismic Force-Resisting System (Standard Table 12.2-1)

Special Reinforced Masonry Shear Walls

Response Modification Factor, R

System Overstrength Factor, Ω0

2.5

Deflection Amplification Factor, Cd

3.5

11.2.2.2 Structural Design Criteria. A complete discussion on the criteria for ground motion, seismic design category, load path, structural configuration, redundancy, analysis procedure and shear wall design is included in Section 10.1 of this volume of design examples. 11.2.2.2.1 Design and Detailing Requirements. Since this building has a wood structural panel diaphragm with masonry shear walls, the diaphragm can be considered flexible in accordance with Standard Section 12.3.1.1. There are not any irregularities (Standard Sec. 12.3.2) that would impact the diaphragm design and the diaphragm and wall anchorage system is permitted to be designed with the redundancy factor equal to 1.0 per Standard Section 12.3.4.1. The design of the diaphragm is based on Standard Section 12.10. The large opening in the diaphragm must be fitted with edge reinforcement (Standard Sec. 12.10.1). However, the diaphragm does not require any collector elements that would have to be designed for the special load combinations (Standard Sec. 12.10.2.1). The requirements for anchorage of masonry walls to flexible diaphragms (Standard Sec. 12.11.2) are of great significance in this example. 11.2.2.2.2 Seismic Load Effects and Combinations. The basic design load combinations for the seismic design, as stipulated in Standard Section 12.4.2.3, were computed in Section 10.1 of this volume of design examples, as follows: 1.4D + 1.0QE where gravity and earthquake are additive and 0.7D - 1.0QE where gravity and earthquake counteract. The roof live load, Lr, is not combined with seismic loads (see Standard Chapter 2) and the design snow load is zero for this Los Angeles location. 11-32

Chapter 11: Wood Design

11.2.2.2.3 Deflection and Drift Limits. In-plane deflection and drift limits for the masonry shear walls are considered in Section 10.1. As illustrated below, the diaphragm deflection is much greater than the shear wall deflection. According to Standard Section 12.12.2, in-plane diaphragm deflection must not exceed the permissible deflection of the attached elements. Because the walls are essentially pinned at the base and simply supported at the roof, they are capable of accommodating large deflections at the roof diaphragm. For illustrative purposes, story drift is determined and compared to the requirements of Standard Table 12.12-1. However, according to this table, there is essentially no drift limit for a single-story structure as long as the architectural elements can accommodate the drift (assumed to be likely in a warehouse structure with no interior partitions). As a further check on the deflection, P-delta effects (Standard Sec. 12.8.7) are evaluated. 11.2.3 Seismic Force Analysis Building weights and base shears are as computed in Section 10.1. (The building weights used in this example are based on a preliminary version of Example 10.1 and thus minor numerical differences may exist between the two examples). Standard Section 12.10.1.1 specifies that floor and roof diaphragms be designed to resist a force, Fpx, computed in accordance with Standard Equation 12.10-1 as follows: n

Fpx =

∑ Fi i=x n

∑ wi

w px

i=x

plus any force due to offset walls (not applicable for this example). For one-story buildings, the first term of this equation will be equal to the seismic response coefficient, Cs, which is 0.286. The effective diaphragm weight, wpx, is equal to the weight of the roof plus the tributary weight of the walls perpendicular to the direction of the motion. The tributary weights are as follows: §

Roof = 20(100)(200) = 400 kips

§

Side walls = 2(65)(28/2+2)(200) = 416 kips

§

End walls = 2(103)(28/2+2)(100) = 330 kips

The diaphragm design force is computed as: §

Transverse: Fp,roof = 0.286(400+416) = 233 kips

§

Longitudinal: Fp,roof = 0.286(400+330) = 209 kips

These forces exceed the minimum diaphragm design forces given in Standard Section 12.10.1.1, because Cs exceeds the minimum factor of 0.2SDS.

11-33

FEMA P752, NEHRP Recommended Provisions: Design Examples 11.2.4 Basic Proportioning of Diaphragm Elements The design of plywood diaphragms primarily involves the determination of sheathing sizes and nailing patterns to accommodate the applied loads. Large openings in the diaphragm and wall anchorage requirements, however, can place special requirements on the diaphragm capacity. Diaphragm deflection is also a consideration. Nailing patterns for diaphragms are established on the basis of tabulated requirements included in the AF&PA SDPWS. It is important to consider the framing requirements for a given nailing pattern and capacity as indicated in the notes following the tables. In addition to strength requirements, AF&PA SDPWS Section 4.2.4 places aspect ratio limits on plywood diaphragms (length-to-width must not exceed 4/1 for blocked diaphragms). However, it should be taken into consideration that compliance with this aspect ratio does not guarantee that drift limits will be satisfied. While there is no specific limitation on deflection for this example, the diaphragm has been analyzed for deflection as well as for shear capacity. In the calculation of diaphragm deflections, the chord splice slip factor can result in large additions to the total deflection. This chord splice slip, however, is often negligible where the diaphragm is continuously anchored to a bond beam in a masonry wall. Therefore, chord splice slip is assumed to be zero in this example. 11.2.4.1 Strength of Members and Connections. As described in more detail in Section 11.1.4.1, the Standard references the AF&PA NDS and AF&PA SDPWS for engineered wood structures. Diaphragm design is based on AF&PA SDPWS Section 4.2, which provides design criteria for both ASD and LRFD methods. This example utilizes LRFD as the design basis, so the diaphragm design is based on the tabulated unit shear values, vs, which are multiplied by a resistance factor, φD, equal to 0.80. Refer to Section 11.1.4.1 for a summary of the design methodology in the AF&PA NDS for framing members and connections. 11.2.4.2 Roof Diaphragm Design for Transverse Direction 11.2.4.2.1 Plywood and Nailing. The diaphragm design force is Fp,roof = 233 kips and the maximum end shear is 0.5Fp,roof = 116.5 kips. This corresponds to a unit shear force of v = (116.5/100) = 1.165 klf. (Note that per Standard Sec. 12.8.4.2, accidental torsion need not be considered for flexible diaphragms.) Due to the relatively high diaphragm shears, closely spaced nailing will be required, so in accordance with AF&PA SDPWS Section 4.2.7.1.1, Item 3, 3-inch nominal framing will be provided. Assuming 3-inch nominal framing, try blocked 1/2-inch (15/32) Structural I plywood rated sheathing with 10d common nails at 2 inches on center at diaphragm boundaries and continuous panel edges and at 3 inches on center at other panel edges. The use of 2×4 flat blocking at continuous panel edges satisfies the requirements for blocked diaphragms. From AF&PA SDPWS Table 4.2A:

φDvs = 0.80(1.640) = 1.31 klf > 1.165 klf

Because the diaphragm shear decreases towards the midspan of the diaphragm, the diaphragm capacity may be reduced towards the center of the building. A reasonable configuration for the interior of the building utilizes 2-inch nominal framing and 1/2-inch (15/32) Structural I plywood rated sheathing plywood with 10d at 4 inches on center at diaphragm boundaries and continuous panels edges and

11-34

Chapter 11: Wood Design 6 inches on center nailing at other panel edges. Determine the distance, X, from the end wall where the transition can be made, as follows: §

φDvs = 0.80(0.850) = 0.68 klf (AF&PA SDPWS Table 4.2A)

§

Shear capacity = 0.68(100) = 68.0 kips

§

Uniform diaphragm demand = 233/200 = 1.165 klf

§

X = (117-68)/1.165 = 42.1 ft (assumed as 50 ft from the diaphragm edge)

In a building of this size, it may be beneficial to further reduce the diaphragm nailing towards the middle of the roof. However, due to the requirements for subdiaphragms (see below) and diaphragm capacity in the longitudinal direction and for simplicity of design, no additional nailing pattern is used. Table 11.2-1 contains a summary of the diaphragm framing and nailing requirements (all nails are 10d common). See Figure 11.2-2 for designation of framing and nailing zones and Figure 11.2-3 for typical plywood layout. Table 11.2-2 Roof Diaphragm Framing and Nailing Requirements Nail Spacing (in.) Structural 1 Boundaries Intermediate Zone* Framing Other Panel Plywood and Cont. Framing Edges Panel Edges Members

Capacity (kip/ft)

3×12

15/32 in.

1.31

2×12

15/32 in.

0.68

1.0 in. = 25.4 mm, 1.0 kip/ft = 14.6 kN/m. * Refer to Figure 11.2-2 for zone designation.

11-35

100'-0"

FEMA P752, NEHRP Recommended Provisions: Design Examples

50'-0" Zone A

90'-0" Zone B

10'

50'-0" Zone A Extended Zone A due to diaphragm opening.

Figure 11.2-2 Diaphragm framing and nailing layout (1.0 ft = 0.3048 m)

Transverse

4'x8' plywood with grain to joists and end joints staggered

2x12 or 3x12 joists at 24" o.c. 2x4 flat blocking at edges of plywood panels

Figure 11.2-3 Typical diaphragm plywood layout (1.0 ft = 0.3048 m, 1.0 in = 25.4mm)

11-36

Longitudinal

Chapter 11: Wood Design 11.2.4.2.2 Chord Design. Although the bond beam at the masonry wall could be used as a diaphragm chord, this example illustrates the design of the wood ledger member as a chord. Chord forces are computed using a simply supported beam analogy, where the design force is the maximum moment divided by the diaphragm depth. §

Diaphragm moment, M = wL2/8 = Fp,roofL/8 = 233(200/8) = 5,825 ft-kips

§

Chord force, T = C = 5,825/(100 - 16/12) = 59.0 kips

Try a select structural Douglas Fir-Larch 4×12 for the chord. Assuming two 1-1/16-inch bolt holes (for 1-inch bolts) at splice locations, the net chord area is 31.9 in2. Tension strength (parallel to wood grain), per the AF&PA NDS, is as follows: Ft’ = FtCFKFφλ = (1,000 psi)(1.0)(2.16/0.8)(0.8)(1.0) = 2,160 psi T’ = Ft’A = 2,160(31.9)/1000 = 68.9 kips > 59.0 kips

Design the splice for the maximum chord force of 59.0 kips. Try bolts with steel side plates using 1-inch A307 bolts, with a 3-1/2-inch length in the main member. The capacity, according to the AF&PA NDS, is as follows: Z’ = ZKFφλ = (4.90)(2.16/0.65)(0.65)(1.0) = 10.6 kips per bolt The number of bolts required (at each side of the splice joint) is 59.0/10.6 = 5.6. Use two rows of three bolts. The edge distance, end distance and spacing meet the AF&PA NDS requirements to avoid capacity reductions and the reduction for multiple bolts (group action factor) is negligible. The net area of the 4×12 chord with two rows of 1-1/16-inch holes is 31.9 in2 as assumed above. Therefore, use six 1-inch A307 bolts on each side of the chord splice (see Figure 11.2-4). In addition to the bolt checks, the steel splice plates would need to be check for tension. Although it is shown for illustration, this type of chord splice may not be the preferred splice against a masonry wall since the bolts and side plate, would have to be recessed into the wall.

11-37

31 2"

FEMA P752, NEHRP Recommended Provisions: Design Examples

2 plates 4"x7"x2'-9"

4x12 Select Structural Doug. fir chord

6-1" A-307 bolts on each side of chord. Splice in tight hole. 31 2" 41 4" 31 2"

Drive shim in joint

Figure 11.2-4 Chord splice detail (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) 11.2.4.2.3 Diaphragm Deflection and P-delta Check. Based on the procedure for contained in AF&PA SDPWS Section 4.2.2, diaphragm deflection is computed as follows:

δ dia =

5vL3 0.25vL ∑ ( xΔ c ) + + 8EAW 1000Ga 2W

The equation produces the midspan diaphragm displacement in inches and the individual variables must be entered in the force or length units as described below. A small increase in diaphragm deflection due to the large opening is neglected and the effects of the variable nail spacing are neglected for simplicity. The chord slip deflection is assumed to be zero because the chord is connected to the continuous bond beam at the top of the masonry wall. The variables above and associated units used for computations are as follows: v = (233/2)/100 = 1,165 plf (shear per foot at boundary) L = 200 ft (diaphragm length) W = 100 ft (diaphragm width) A = effective area of 4×12 chord and two-#6 bars assumed to be in the bond beam = 39.38 in2 + 2(0.44)(29,000,000/1,900,000) = 52.81 in2 E = 1,900,000 psi (for Douglas Fir-Larch select structural chord) Ga = 1.2(18) = 21.6 kips/in. (apparent diaphragm shear stiffness from AF&PA SDPWS Table 4.2A accounting for nail slip and panel shear deformation, based on sheathing and nailing at the outer zone and increased by 1.2 per Footnote 3, assuming four-ply minimum sheathing)

11-38

Chapter 11: Wood Design Bending deflection = 5vL3/8EAW = 0.58 in. Shear/nail slip deflection = 0.25vL/1000Ga = 2.71 in. Deflection due to chord slip at splices = Σ(xΔc)/2W ≈ 0.00 in. (as noted above) Total for diaphragm: δdia = 0.58+2.71+0.00 = 3.29 in. End wall deflection = 0.037 in. (see Sec. 10.1 of this volume of design examples) Therefore, the total elastic deflection δxe = 3.29+0.037 = 3.29 in. Total deflection, δx = Cd δxe/I = 3.5(3.29)/1.0 = 11.53 in. The drift ratio at the center of the diaphragm = δx/hsx = 11.53/[28(12)] = 0.034. This exceeds the maximum drift ratio of 0.025 permitted for most low-rise buildings in Occupancy Category II (Standard Table 12.12-1). However, for one-story buildings, Standard Table 12.12-1, Footnote c permits unlimited drift, provided that the structural elements and finishes can accommodate the drift. The limit for masonry cantilever shear wall structures (0.007) should only be applied to the inplane movement of the end walls (0.13/h = 0.0004 < 0.007). The construction of the out-of-plane walls allows them to accommodate very large drifts. It is further expected that the building does not contain interior elements that are sensitive to drift. P-delta effects are computed according to Provisions Section 12.8.7, which modifies Standard Section 12.8.7 for determining the stability coefficient, θ, per Provisions Equation 12.8-16:

θ=

Px ΔI Vx hsx Cd

(Note that the Provisions adds the importance factor, I, that was missing in the Standard equation.) Because the midspan diaphragm deflection is substantially greater than the deflection at the top of the masonry end walls, it would be overly conservative to consider the entire design load at the maximum deflection. Therefore, the stability coefficient is computed by splitting the P-delta product into two terms: one for the diaphragm and one for the end walls. For the diaphragm, consider the weight of the roof and side walls at the maximum displacement. (This overestimates the P-delta effect. The computation could consider the average displacement of the total weight, which would lead to a reduced effective delta. Also, the roof live load need not be included.) P = 400+416 = 816 kips Δ= 11.53 in. V = 233 kips (diaphragm force) For the end walls, consider the weight of the end walls at the wall displacement:

11-39

FEMA P752, NEHRP Recommended Provisions: Design Examples P = 330 kips Δ = (3.5)(0.037) = 0.13 in. V = 264 kips (additional base shear for wall design) For story height, h = 28 feet, the stability coefficient is:

⎛ PΔ PΔ ⎞ ⎛ 816(11.53) 330(0.13) ⎞ + hCd = ⎜ + (28)(12)(3.5) = 0.034 ⎟ V ⎠ 233 264 ⎟⎠ ⎝ V ⎝

θ = ⎜

For θ < 0.10, P-delta effects need not be considered based on Provisions Section 12.8.7. Since the P-delta effects are not significant for this structure and the Standard does not impose drift limitations for this type of structure, the computed diaphragm deflections appear acceptable. 11.2.4.2.4 Detail at Opening. Consider diaphragm strength at the roof opening as required by Standard Section 12.10.1. The diaphragm nailing must be checked for the reduced total width of diaphragm sheathing and the chords must be checked for bending forces at the opening. Check diaphragm nailing for the shear in the diaphragm at edge of opening. The maximum shear at the exterior-side edge of the opening is computed as follows: Shear = 116.5 - [40(1.165)] = 69.9 kips v = 69.9/(100-20) = 0.874 klf Because the opening is centered in the width of the diaphragm, half the force to the diaphragm must be distributed on each side of the opening. Diaphragm capacity in this area is 0.680 klf as computed previously (see Table 11.2-1 and Figure 11.2-2). Because the diaphragm demand at the reduced section exceeds the capacity, the extent of the Zone A nailing and framing should be increased. For simplicity, extend the Zone A nailing to the interior edge of the opening (60 feet from the end wall). The diaphragm strength is now adequate for the reduced overall width at the opening. 11.2.4.2.5 Framing around Opening. The opening is located 40 feet from one end of the building and is centered in the other direction (Figure 11.2-5). This does not create any panels with very high aspect ratios. In order to develop the chord forces, continuity will be required across the glued-laminated beams in one direction and across the roof joists in the other direction.

11-40

Chapter 11: Wood Design

Open 20'

Glued-laminated beams

20'

40' to diaphragm boundary

Figure 11.2-5 Diaphragm at roof opening (1.0 ft = 0.3048 m) 11.2.4.2.6 Chord Forces at Opening. To determine the chord forces on the edge joists, split the diaphragm into smaller free-body sections, assume the inflection points will be at the midpoint of the elements (Figure 11.2-6) and compute the forces at the opening using a uniformly distributed diaphragm demand of 233/200 = 1.165 klf. For Element 1 (shown in Figure 11.2-7): w1 = 1.165/2 = 0.582 kips/ft (assuming half the diaphragm load on each side of the opening) V1B = 0.5[116.5-(40)(1.165)] = 35.0 kips (based on diaphragm unit shear on right side of opening) V1A = 35.0-20(0.582) = 23.3 kips (based on diaphragm unit shear on left side of opening) M1 = (1/2)[35.0(10) + 23.3(10)] = 291 ft-kips (assuming equal moments at each edge of the section) The chord force due to M1 = 291/40 = 7.28 kips. This is only 35 psi on the glued-laminated beam on the edge of the opening. This member is adequate by inspection. On the other side of this diaphragm element, the chord force is much less than the maximum global chord force (59.0 kips), so the ledger and ledger splice are adequate.

11-41

FEMA P752, NEHRP Recommended Provisions: Design Examples

20'-0"

M1 116.5 kips

40'-0"

Element 1

40'-0"

V 1B

V 1A

1.165 kip/ft

Figure 11.2-6 Chord forces and Element 1 free-body diagram (1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN, 1.0 kip/ft = 14.6 kN/m) For Element 3, analyze Element 2 (shown in Figure 11.2-7) in the same manner as Element 1: w2 = 1.165(40/100) = 0.466 kips/ft V3 = 116.5(40/100) = 46.6 kips V1B = 35.0 kips M1 = 291 ft-kips. T1B is the chord force due to moment on the total diaphragm: M = 116.5(40) - 1.165(402/2) = 3,728 ft-kips T1B = 3,728/100 = 37.3 kips ΣM0:M3 = M1 + 40V3 - 40T1B - w2402/2 = 291 ft-kips Therefore, the chord force on the roof joist = 291/40 = 7.26 kips

11-42

Chapter 11: Wood Design T 1B 2 V3

V 1B M1 O

M3 = M2

Figure 11.2-7 Free-body diagram for Element 2 Alternatively, the chord design should consider the wall anchorage force interrupted by the opening. As described in Section 11.2.4.4.1, the edge members on each side of the opening are used as continuous cross-ties, with maximum cross-tie force of 16.6 kips. Therefore, the cross-tie will adequately serve as a chord at the opening. 11.2.4.3 Roof Diaphragm Design for Longitudinal Direction Force = 209 kips Maximum end shear = 0.50(209) = 104.5 kips Diaphragm unit shear, v = 104.5/200 = 0.523 klf For this direction, the plywood layout is Case 3 in AF&PA SDPWS Table 4.2A. Using 1/2-inch Structural I plywood rated sheathing, blocked, with 10d common nails at 4 inches on center at diaphragm boundaries and continuous panel edges parallel to the load (ignoring the capacity of the extra nails in the outer zones), per AF&PA SDPWS Table 4.2A:

φDvs = 0.80(0.850) = 0.68 klf > 0.523 klf

Therefore, use the same nailing designed for the transverse direction. Compared with the transverse direction, the diaphragm deflection and P-delta effects will be satisfactory. 11.2.4.4 Masonry Wall Anchorage to Roof Diaphragm. As stipulated in Standard Section 12.11.2.1, masonry walls must be anchored to flexible diaphragms to resist out-of-plane forces computed per Standard Equation 12.11-1 as follows: FP = 0.8SDSIWp = 0.8(1.0)(1.0)Wp = 0.8 Wp Side walls, FP = 0.8(65psf)(2+28/2)/1,000 = 0.83 klf End walls, FP = 0.8(103psf)(2 + 28/2)/1,000 = 1.32 klf 11.2.4.4.1 Anchoring Joists Perpendicular to Walls (Side Walls). The roof joists are spaced at 2 feet on center, so as a preliminary design, consider a connection at every other joist that will develop 4(0.83) =

11-43

FEMA P752, NEHRP Recommended Provisions: Design Examples 3.32 kip/joist. Note that 4 feet is the maximum anchor spacing allowed without having to check the walls for resistance to bending between anchors (Standard Sec. 12.11.2). A common connection for this application is a metal tension tie-down or hold-down device that is anchored to the masonry wall with an embedded bolt and is either nailed, screwed, or bolted to the roof joist. Other types of anchors include metal straps that are embedded in the wall and nailed to the top of the joist. The ledger is not used for this force transfer because the eccentricity between the anchor bolt and the plywood creates tension perpendicular to the grain in the ledger (cross-grain bending), which is prohibited. Also, using the edge nails to resist tension perpendicular to the edge of the plywood is not permitted. Try a tension tie with a 3/4-inch headed anchor bolt, embedded in the bond beam and with 18 10d nails into the side of the joist (Figure 11.2-8). The cataloged ASD tension capacity of this connector is 3.61 kips based on a load duration factor of 1.60. Modifying the allowable values using the procedure in Section 11.1.4.5 results in a design LRFD capacity of: Z’KFφλ = (3.61)(2.88/1.60)(0.65)(1.0) = 4.22 kips per anchor > 3.32 kips

The joists anchored to the masonry wall must also be adequately connected to the diaphragm sheathing. Determine the adequacy of the typical nailing for intermediate framing members. The nail spacing is 12 inches and the joist length is 20 feet, so there are 20 nails per joist. From the AF&PA NDS, the LRFD capacity of a single 10d common nail in 1/2-inch plywood is: Z’KFφλ = (0.090)(2.16/0.65)(0.65) = 0.194 kips per nail 20(0.194) = 3.88 kips > 3.32 kips

The embedded anchor bolt also serves as the ledger connection, for both gravity loading and in-plane shear transfer at the diaphragm. Therefore, the strength of the anchorage to masonry and the strength of the bolt in the wood ledger must be checked. For the anchorage to masonry, check the combined tension and shear resulting from the out-of-plane seismic loading (3.32 kips per bolt) and the vertical gravity loading. Assuming 20 psf dead load (roof live load need not be combined with seismic loads), a 10-foot tributary roof width and ledger bolts at 2 feet on center (at tension ties and in between) the vertical load per bolt = (20 psf)(10 ft)(2 ft)/1,000 = 0.40 kip. Using the load combinations described previously, the design horizontal tension and vertical shear on the bolt are as follows: baf = 1.0QE = 3.32 kips bvf = 1.4D = 1.4(0.40) = 0.56 kip The anchor bolts in masonry are designed according to ACI 530 as adopted by the Standard (Sec. 14.4) and as modified by Standard Sections 14.4.7.6 and 14.4.7.7. Standard Section 14.4.7.6 requires the strength of the anchorage connecting diaphragms to other parts of the seismic force-resisting system to be governed by steel tensile or shear yielding unless the anchorage is designed for 2.5 times the required forces. For this example, the anchorage is proportioned such that the steel governs the capacity. Standard Section 14.4.7.7 modifies the shear strength requirements for anchorage, requiring that the shear capacity is not more than 2 times the strength due to masonry pry-out.

11-44

Chapter 11: Wood Design Using 3/4-inch headed anchor bolts with an effective embedment depth of 6 inches, both tensile strength, Ban and shear strength, Bvn, will be computed assuming the masonry strength, f’m, is 2,000 psi and the steel strength, fy, is 36,000 psi. Tensile strength per ACI 530 Section 3.1.6.2 is taken as the lesser of the following:

Ban = Ab f y = 0.44(36) = 15.8 kips Ban = 4 Apt f m' = 4(113) 2,000 = 20.2 kips where Apt is the projected area of the right cone and is equal to π(lb)2, where lb is the effective embedment depth. Therefore, Apt = π(6)2=113 in2. Since the steel strength governs, Standard Section 14.4.7.6 is met and φ = 0.9. Therefore the design strength in tension is 0.9(15.8) = 14.2 kips. Shear strength per ACI 530 Section 3.1.6.3 is taken as the lesser of the following:

Bav = 0.6 Ab f y = 0.6(0.44)(36) = 9.50 kips Bav = 4 Apv f m' = 4(56.5) 2,000 = 10.1 kips where Apv is one half of the projected area of the right cone and is equal to 113/2 = 56.5 in2. Since the steel strength governs, Standard Section 14.4.7.6 is met for shear and φ = 0.9. Therefore, the design strength in shear is 0.9(9.50) = 8.55 kips. Shear and tension are combined per ACI 530 Section 3.1.6.4 as:

ban b 3.32 0.56 + av = + = 0.30 < 1.0 ϕ Ban ϕ Bav 14.2 8.55

Figure 11.2-8 summarizes the details of the connection. In-plane seismic shear transfer (combined with gravity) and orthogonal effects are considered in a subsequent section.

11-45

FEMA P752, NEHRP Recommended Provisions: Design Examples

4" dia. bolt at tension tie (48" o.c.) and between ties (48" o.c.)

Tension tie (e.g. Simpson HTT4) at each joist

Bond beam at top

Structural sheathing (See plan for thickness and nailing) Bond beam with 2-#7 cont. for chord Vert reinf. to top 2x12 or 3x12 joist with joist hanger 4x12 ledger

Figure 11.2-8 Anchorage of masonry wall perpendicular to joists (1.0 in. = 25.4 mm) According to Standard Section 12.11.2.2.1, diaphragms must have continuous cross-ties to distribute the anchorage forces into the diaphragms. Although the Standard does not specify a maximum spacing, 20 feet is common practice for this type of construction and seismic design category. For cross-ties at 20 feet on center, the wall anchorage force per cross-tie is: (0.83 klf)(20 ft) = 16.6 kips Try a 3×12 Douglas Fir-Larch No. 1 as a cross-tie. Assuming one row of 1-1/8-inch bolt holes, the net area of the section is 25.3 in2. Tension strength (parallel to wood grain) per the AF&PA NDS Supplement is: F’T = FtKFφλ = (0.675)(2.16/0.8)(0.8) = 1.46 ksi T’ = F’tΑ = (1.46)(25.3) = 36.9 kips > 16.6 kips

However, the cross-tie must be checked for combined gravity and lateral loads. The governing case for combined loads is midspan where the maximum gravity moment is combined with seismic tension. The 3×12 cross-tie has the following properties: A = 28.1 in2 S = 52.7 in3 F’t = 1.46 ksi F’b = FbCrKFφλ = (1.000)(1.15)(2.16/0.85)(0.85) = 2.48 kips

11-46

Chapter 11: Wood Design The factored dead load moment is computed using the load combinations described above as: Mu = 1.4(20 psf)(2 ft)(20 ft)2/8 = 2.80 ft-kips The factored stresses are computed as: ft = 16.6/28.1 = 0.591 ksi fb = (2.80)(12)/52.7 = 0.638 ksi Combined stresses are checked in accordance with AF&PA NDS Section 3.9.1 as follows:

ft f 0.591 0.638 + bt = + = 0.66 < 1.0 ' 2.48 Ft Fb 1.46

At the splices, try a double tie-down device with three 1-inch bolts in double shear through the 3×12 member (Figure 11.2-9). Product catalogs provide design capacities for single tie-downs only; the design of double hold-downs requires two checks. First, consider twice the capacity of one tie-down and, second, consider the capacity of the bolts in double shear. For the double tie-down, use the procedure in Section 11.1.4.5 to modify the allowable values: 2ZKFφλ = 2(8.81)(2.88/1.60)(0.65)(1.0) = 20.6 kips > 16.6 kips

For the four bolts, the AF&PA NDS gives: 4ZKFφλ = 4(3.50)(2.16/0.65)(0.65)(1.0) = 30.2 kips > 16.6 kips

Plywood with boundary nailing along 3x12 7 " dia. threaded 8 rod at hold downs

3x12 with joist hanger

Tie-down on both sides of 3x12 with 3-1" dia. thru bolts. (Each side of beam)

3x12 with joist hanger Glulam beam

Figure 11.2-9 Chord tie at roof opening (1.0 in. = 25.4 mm) In order to transfer the wall anchorage forces into the cross-ties, the subdiaphragms between these ties must be checked per Standard Section 12.11.2.2.1. There are several ways to perform these 11-47

FEMA P752, NEHRP Recommended Provisions: Design Examples subdiaphragm calculations. One method is illustrated in Figure 11.2-10. The subdiaphragm spans between cross-ties and utilizes the glued-laminated beam and ledger as its chords. The 1-to-1 aspect ratio meets the requirement of 2.5 to 1 for subdiaphragms per Standard Section 12.11.2.2.1. For the typical subdiaphragm (Figure 11.2-10): Fp = 0.83 klf v = (0.83)(20/2)/20 = 0.415 klf. The subdiaphragm demand is less than the minimum diaphragm capacity (0.68 klf along the center of the side walls). In order to develop the subdiaphragm strength and boundary nailing must be provided along the cross-tie beams.

Typical joist, anchored to masonry wall

Subdiaphragm

Masonry wall

20'-0"

F p = 0.83 kips/ft

Glued laminated beam 20'-0"

20'-0"

Spliced continuous cross ties

Subdiaphragm

Figure 11.2-10 Cross tie plan layout and subdiaphragm free-body diagram for side walls (1.0 ft = 0.3048 m, 1.0 kip/ft = 14.6 kN/m) 11.2.4.4.2 Anchorage at Joists Parallel to Walls (End Walls). Where the joists are parallel to the walls, tied elements must transfer the forces into the main body of the diaphragm, which can be accomplished by using either metal strapping and blocking or metal rods and blocking. This example uses threaded rods that are inserted through the joists and coupled to the anchor bolt (Figure 11.2-11). Blocking is

11-48

Chapter 11: Wood Design added on both sides of the rod to transfer the force into the plywood sheathing. The tension force in the rod causes a compression force on the blocking through the nut and on the bearing plate at the innermost joist.

Bond beam at top

4" Structural I plywood

3 " dia. bolt 4 at 2'-0" o.c.

3 " dia. threaded rod 4 at 4'-0" o.c. extend through joists

Coupler Bond beam with chord reinf.

10d nails at 8" o.c. at blocking Plate washer and nut at last joist

Vertical reinforcement 3x12 joist at 24" o.c.

4x12 ledger

2x12 blocking on each side of threaded rod

12'-0"

Figure 11.2-11 Anchorage of masonry wall parallel to joists (1.0 ft = 0.3048 m, 1.0 in. = 25.4 mm) The anchorage force at the end walls is 1.32 klf. Space the connections at 4 feet on center so that the wall need not be designed for flexure. Thus, the anchorage force is 5.28 kips per anchor. Try a 3/4-inch headed anchor bolt, embedded into the masonry. In this case, gravity loading on the ledger is negligible and can be ignored and the anchor can be designed for tension only. (In-plane shear transfer and orthogonal effects are considered later.) As computed for 3/4-inch headed anchor bolts (with 6 inch embedment), the design axial strength is φBan = 14.2 kips > 5.28 kips. Therefore, the bolt is acceptable. Using couplers rated for 125 percent of the strength of the rod material, the threaded rods are then coupled to the anchor bolts and extend six joist spaces (12 feet) into the roof framing. (This length of 12 feet is required for the subdiaphragm force transfer discussed below.) Nailing the blocking to the plywood sheathing is determined using nail capacities from the AF&PA NDS. As computed previously, the LRFD capacity of a single 10d common nail, Z’KFφλ = 0.194 kips per nail. Thus, 28 nails are required (5.28/0.194). This corresponds to a nail spacing of approximately 10 inches for two 12-foot rows of blocking. Space nails at 8 inches for convenience. Use the glued-laminated timber beams (at 20 feet on center) to provide continuous cross-ties and check the subdiaphragms between the beams to provide adequate load transfer to the beams per Standard Section 12.11.2.2.1:

11-49

FEMA P752, NEHRP Recommended Provisions: Design Examples Design tension force on beam = (1.32 klf)(20 ft) = 26.4 kips The stress on the beam is ft = 26,400/[8.75(24)] = 126 psi, which is small. The beam is adequate for combined moment due to gravity loading and axial tension. At the beam splices, try 3/4-inch bolts with steel side plates. Per the AF&PA NDS: ZKFφλ = (3.34)(2.16/0.65)(0.65)(1.0) = 7.21 kips per bolt The number of bolts required (on each side of the splice joint) is 26.4/7.21 = 3.7. Use four bolts in a single row at mid-height of the beam, with 1/4-inch by 4-inch steel side plates. The reduction (group action factor) for multiple bolts is negligible. Although not included in this example, the steel side plates should be checked for tension capacity on the gross and net sections. There are preengineered hinged connectors for glued-laminated beams that could provide sufficient tension capacity for the splices. In order to transfer the wall anchorage forces into the cross-ties, the subdiaphragms between these ties must be checked per Provisions Section 12.11.2.2.1. The procedure is similar to that used for the side walls as described previously. The end wall condition is illustrated in Figure 11.2-12. The subdiaphragm spans between beams and utilizes a roof joist as its chord. In order to adequately engage the subdiaphragm, the wall anchorage ties must extend back to this chord. Since the maximum aspect ratio for subdiaphragms is 2.5 to 1, the minimum depth is 20/2.5 = 8 feet.

11-50

Chapter 11: Wood Design

Roof joist for subdiaphragm chord

12'-0"

Masonry wall

F P = 1.32 kips/ft

20'-0"

Subdiaphragm

Typical wall, anchor at 4'-0" o.c.

Glued - laminated beam, spliced as continuous cross tie

V Subdiaphragm

Figure 11.2-12 Cross tie plan layout and subdiaphragm free-body diagram for end walls (1.0 ft = 0.3048 m, 1.0 kip/ft = 14.6 kN/m) For the typical subdiaphragm (Figure 11.2-12): Fp = 1.32 klf v = (1.32)(20/2)/8 = 1.65 klf As computed previously (see Table 11.2-1 and Figure 11.2.2), the diaphragm strength in this area is 1.17 klf < 1.65 klf. Therefore, increase the subdiaphragm depth to 12 feet (six joist spaces): v = (1.32)(20/2)/12 = 1.10 klf > 1.17 klf

In order to develop the subdiaphragm strength, boundary nailing must be provided along the cross-tie beams. There are methods of refining this analysis using multiple subdiaphragms so that all of the tension anchors need not extend 12 feet into the building. 11.2.4.4.3 Transfer of Shear Wall Forces. The in-plane diaphragm shear must be transferred to the masonry wall by the ledger, parallel to the wood grain. The connection must have sufficient capacity for the diaphragm demands as follows: §

Side walls: 0.523 klf

11-51

FEMA P752, NEHRP Recommended Provisions: Design Examples §

End walls: 1.165 klf

For each case, the capacity of the bolted wood ledger and the capacity of the anchor bolts embedded into masonry must be checked. Because the wall connections provide a load path for both in-plane shear transfer and out-of-plane wall forces, the bolts must be checked for orthogonal load effects in accordance with Standard Section 12.5. That is, the combined demand must be checked for 100 percent of the lateral load effect in one direction (e.g., shear) and 30 percent of the lateral load effect in the other direction (e.g., tension). At the side walls, the wood ledger with 3/4-inch bolts (Figure 11.2-8) must be designed for gravity loading (0.56 kip per bolt as computed above) as well as seismic shear transfer. The seismic load per bolt (at 2 feet on center) is 0.523(2) = 1.05 kips. Combining gravity shear and seismic shear produces a resultant force of 1.19 kips at an angle of 28 degrees from the axis of the wood grain. The bolt capacity in the wood ledger can be determined using the formulas for bolts at an angle to the grain per the AF&PA NDS (either adjusting for dowel bearing strength per Section 11.3.3 or adjusting the tabulated bolt values per Appendix J). The resulting design value, Z = 1.41 kips and the LRFD capacity is determined as follows: ZKFφλ = (1.41)(2.16/0.65)(0.65)(1.0) = 3.05 kips > 1.19 kips

This bolt spacing also satisfies the load combination for gravity loading (dead and roof live) only. For the check of the embedded anchor bolts, the factored demand on a single bolt is 1.05 kips in horizontal shear (in-plane shear transfer), 3.32 kips in tension (out-of-plane wall anchorage) and 0.56 kip in vertical shear (gravity). Orthogonal effects are checked, using the following two equations:

0.3(3.32) 1.052 + 0.562 = 0.56 + 8.75 2.66 and

(0.3 × 1.05) 2 + 0.562 3.32 = 0.62 (controls) < 1.0 + 8.75 2.66

At the end walls, the ledger with 3/4-inch bolts (Figure 11.2-11) need only be checked for in-plane seismic shear because gravity loading is negligible. For bolts spaced at 4 feet on center, the demand per bolt is 1.165(4) = 4.66 kips parallel to the grain of the wood. Per the AF&PA NDS: ZKFφλ = (1.61)(2.16/0.65)(0.65)(1.0) = 3.48 kips < 4.66 kips

Therefore, add 3/4-inch headed bolts evenly spaced between the tension ties such that the bolt spacing is 2 feet on center and the demand per bolt is 1.165(2) = 2.33 kips. These added bolts are used for in-plane shear only and do not have coupled tension tie rods. For the check of the embedded bolts, the factored demand on a single bolt is 2.33 kips in horizontal shear (in-plane shear transfer), 5.28 kips in tension (out-of-plane wall anchorage), 0 kip in vertical shear (gravity is negligible). Orthogonal effects are checked using the following two equations:

11-52

Chapter 11: Wood Design

0.3(5.28) 2.33 = 1.06 (controls) > 1.0 + 8.75 2.66

5.28 0.3(2.33) + 8.75 2.66 = 0.86 Since one of the equations is slightly more than unity, the bolt capacity can be increased by using a larger bolt or more embedment depth, or more bolts can be added. With this minor revision, the wall connections satisfy the requirements for combined gravity and seismic loading, including orthogonal effects.

11-53

12 Seismically Isolated Structures Charles A. Kircher, P.E., Ph.D. Contents 12.1

BACKGROUND AND BASIC CONCEPTS ............................................................................... 4

12.1.1

Types of Isolation Systems .................................................................................................... 4

12.1.2

Definition of Elements of an Isolated Structure .................................................................... 5

12.1.3

Design Approach ................................................................................................................... 6

12.1.4

Effective Stiffness and Effective Damping ........................................................................... 7

12.2

CRITERIA SELECTION .............................................................................................................. 7

12.3

EQUIVALENT LATERAL FORCE PROCEDURE .................................................................... 9

12.3.1

Isolation System Displacement.............................................................................................. 9

12.3.2

Design Forces ...................................................................................................................... 11

12.4

DYNAMIC LATERAL RESPONSE PROCEDURE ................................................................. 15

12.4.1

Minimum Design Criteria .................................................................................................... 15

12.4.2

Modeling Requirements....................................................................................................... 16

12.4.3

Response Spectrum Analysis ............................................................................................... 18

12.4.4

Response History Analysis .................................................................................................. 18

12.5 EMERGENCY OPERATIONS CENTER USING DOUBLE-CONCAVE FRICTION PENDULUM BEARINGS, OAKLAND, CALIFORNIA ...................................................................... 21 12.5.1

System Description .............................................................................................................. 22

12.5.2

Basic Requirements ............................................................................................................. 25

12.5.3

Seismic Force Analysis........................................................................................................ 34

12.5.4

Preliminary Design Based on the ELF Procedure ............................................................... 36

12.5.5

Design Verification Using Nonlinear Response History Analysis ...................................... 51

12.5.6

Design and Testing Criteria for Isolator Units .................................................................... 61

FEMA P-751, NEHRP Recommended Provisions: Design Examples Chapter 17 of ASCE/SEI 7-05 (the Standard) addresses the design of buildings that incorporate a seismic isolation system. It defines load, design and testing requirements specific to the isolation system and interfaces with the other chapters of the Standard for design of the structure above the isolation system and of the foundation and structural elements below. The 2009 NEHRP Recommended Provisions (the Provisions) incorporates changes to ground motion criteria that significantly affect the analysis of isolated structures. Isolated structures typically are used for important or essential facilities that have functional performance objectives beyond life safety. Accordingly, more advanced methods of analysis are required for design of these structures. Site-specific hazard analysis is common and dynamic (response history) analysis is routine for design, or design verification, of isolated structures. The Standard uses the notation MCE for “maximum considered earthquake” ground motions. The Provisions have modified the definition and designation of MCE to MCER, which is defined as “risktargeted maximum considered earthquake” ground motions. In the corresponding text of the Standard where maximum considered earthquake (MCE) is stated the intent of the Provisions is that risk-targeted maximum considered earthquake (MCER) ground motions be used. Consistent with the Provisions, ASCE/SEI 7-10 replaced maximum considered earthquake (MCE) by risk-targeted maximum considered earthquake (MCER) ground motions. A discussion of background, basic concepts and analysis methods is followed by an example that illustrates the application of the Standard to the structural design of a building with an isolation system. The example building is a three-story emergency operations center (EOC) with a steel concentrically braced frame above the isolation system. The isolation system utilizes sliding friction pendulum bearings, a type of bearing commonly used for seismic isolation of buildings. Although the facility is hypothetical, it is of comparable size and configuration to actual base-isolated EOCs and is generally representative of base-isolated buildings. The example comprehensively describes the EOC’s configuration, defines appropriate criteria and design parameters and develops a preliminary design using the equivalent lateral force (ELF) procedure. It also includes a check of the preliminary design using dynamic analysis as required by the Standard and a discussion of isolation system design and testing criteria. The example EOC is assumed to be located in Oakland, California, a region of very high seismicity subject to particularly strong ground motions. Large seismic demands pose a challenge for the design of base-isolated structures in terms of the displacement capacity of the isolation system and the configuration of the structure above the isolation system. The isolation system will need to accommodate large lateral displacements, often in excess of 2 feet. The structure above the isolation system should be configured to resist lateral forces without developing large overturning loads that could cause excessive uplift displacement of isolators. The example addresses these issues and illustrates that isolated structures can be designed to meet the requirements of the Standard, even in regions of very high seismicity. Designing an isolated structure in a region of lower seismicity would follow the same approach. The isolation system displacement, overturning forces and so forth, would all be reduced; and therefore easier to accommodate using available isolation system devices. The isolation system of the building in the example uses a type of friction pendulum system (FPS) bearing that has a double concave configuration. These bearings are composed of large top and bottom concave plates with an articulated slider in between that permits lateral displacement of the plates with gravity acting as a restoring force. FPS bearings have been used for a number of building projects, including the 2010 Mills-Peninsula hospital in Burlingame, California. Using FPS bearings in this example should not be taken as an endorsem*nt of this particular type of isolator to the exclusion of others. The requirements of the Standard apply to all types of isolations systems and other types of 12-2

Chapter 12: Seismically Isolated Structures isolators (and supplementary damping devices) could have been used equally well in the example. Other common types of isolators include high-damping rubber (HDR) and lead-rubber (LR) bearings. In addition to the Standard and the Provisions, the following documents are either referenced directly, provide background, or are useful aids for the analysis and design of seismically isolated structures. §

ATC 1996

Applied Technology Council. 1996. Seismic Evaluation and Retrofit of Buildings, ATC40.

§

Constantinou

Constantinou, M. C., P. Tsopelas, A. Kasalanati and E. D. Wolff. 1999. Property Modification Factors for Seismic Isolation Bearings, Technical Report MCEER99-0012. State University of New York.

§

Sarlis

Sarlis, A. A. S. and M. C. Constantinou. 2010. “Modeling Triple Friction Pendulum Isolators in Program SAP2000,” State University of New York, Buffalo, New York, June 27, 2010.

§

ETABS

Computers and Structures, Inc. (CSI). 2009. ETABS Linear and Nonlinear Static and Dynamic Analysis and Design of Building Systems. CSI, Berkeley, California.

§

ASCE 41

American Society of Civil Engineers (ASCE). 2006. Seismic Rehabilitation of Existing Buildings, ASCE 41-06, ASCE, Washington, D.C.

§

FEMA 222A

Federal Emergency Management Agency (FEMA). 1995. NEHRP Recommended Provisions for Seismic Regulations for New Buildings, FEMA 222A.

§

FEMA P-695

Federal Emergency Management Agency (FEMA). 2009. Quantification of Building Seismic Performance Factors, FEMA P-695, Washington, D.C.

§

1991 UBC

International Conference of Building Officials. 1991. Uniform Building Code.

§

1994 UBC

International Conference of Building Officials. 1994. Uniform Building Code.

§

2006 IBC

International Code Conference. 2006. International Building Code.

§

Kircher

Kircher, C. A., G. C. Hart and K. M. Romstad. 1989. “Development of Design Requirements for Seismically Isolated Structures” in Seismic Engineering and Practice, Proceedings of the ASCE Structures Congress, American Society of Civil Engineers, May 1989.

§

PEER 2006

Pacific Earthquake Engineering Research (PEER) Center. 2006. PEER NGA Database, PEER, University of California, Berkeley, California, http://peer.berkeley.edu/nga/

§

SEAOC 1999

Seismology Committee, Structural Engineers Association of California. 1999. Recommended Lateral Force Requirements and Commentary, 7th Ed.

§

SEAONC Isolation

Structural Engineers Association of Northern California. 1986. Tentative Seismic Isolation Design Requirements.

12-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples §

USGS 2009

http://earthquake.usgs.gov/design maps/usapp/

12.1 BACKGROUND AND BASIC CONCEPTS Seismic isolation, commonly referred to as base isolation, is a design concept that presumes a structure can be substantially decoupled from potentially damaging earthquake ground motions. By decoupling the structure from ground shaking, isolation reduces response in the structure that would otherwise occur in a conventional, fixed-base building. Alternatively, base-isolated buildings may be designed for reduced earthquake response to produce the same degree of seismic protection. Isolation decouples the structure from ground shaking by making the fundamental period of the isolated structure several times greater than the period of the structure above the isolation system. The potential advantages of seismic isolation and the advancements in isolation system products led to the design and construction of a number of isolated buildings and bridges in the early 1980s. This activity, in turn, identified a need to supplement existing seismic codes with design requirements developed specifically for such structures. These requirements assure the public that isolated buildings are safe and they provide engineers with a basis for preparing designs and building officials with minimum standards for regulating construction. Initial efforts to develop design requirements for base-isolated buildings began with ad hoc groups of the Structural Engineers Association of California (SEAOC), whose Seismology Committee has a long history of contributing to codes. The northern section of SEAOC was the first to develop guidelines for the use of elastomeric bearings in hospitals. These guidelines were adopted in the late 1980s by the California Office of Statewide Health Planning and Development (OSHPD) and were used to regulate the first base-isolated hospital in California. Efforts to develop general requirements to govern the design of base-isolated buildings resulted in the publication of SEAONC Isolation and appendix in the 1991 UBC and an appendix in FEMA 222A. While technical changes have been made subsequently, most of the basic concepts for the design of seismically isolated structures found in the Standard can be traced back to the initial work by the northern section of SEAOC. Additional background may be found in the commentary to the SEAOC 1999 Blue Book. The isolation system requirements in ASCE Standard 41, Seismic Rehabilitation of Existing Buildings, are comparable to those for new buildings. 12.1.1 Types of Isolation Systems The Standard requirements are intentionally broad, accommodating all types of acceptable isolation systems. To be acceptable, the Standard requires the isolation system to: §

Remain stable for maximum earthquake displacements.

§

Provide increasing resistance with increasing displacement.

§

Have limited degradation under repeated cycles of earthquake load.

§

Have well-established and repeatable engineering properties (effective stiffness and damping).

The Standard recognizes that the engineering properties of an isolation system, such as effective stiffness and damping, can change during repeated cycles of earthquake response (or otherwise have a range of

12-4

Chapter 12: Seismically Isolated Structures values). Such changes or variability of design parameters are acceptable provided that the design is based on analyses that conservatively bound (limit) the range of possible values of design parameters. The first seismic isolation systems used in buildings in the United States were composed of either highdamping rubber (HDR) or lead-rubber (LR) elastomeric bearings. Other types of isolation systems now include sliding systems, such as the friction pendulum system, or some combination of elastomeric and sliding isolators. Some applications at sites with very strong ground shaking use supplementary fluidviscous dampers in parallel with either sliding or elastomeric isolators to control displacement. While generally applicable to all types of systems, certain requirements of the Standard (in particular, prototype testing criteria) were developed primarily for isolation systems with elastomeric bearings. Isolation systems typically provide only horizontal isolation and are rigid or semi-rigid in the vertical direction. An example of a rare exception to this rule is the full (horizontal and vertical) isolation of a building in southern California, isolated by large helical coil springs and viscous dampers. While the basic concepts of the Standard can be extended to full isolation systems, the requirements are only for horizontal isolation systems. The design of a full isolation system requires special analyses that explicitly include vertical ground shaking and the potential for rocking response. Seismic isolation is commonly referred to as base isolation because the most common location of the isolation system is at or near the base of the structure. The Standard does not restrict the plane of isolation to the base of the structure but does require the foundation and other structural elements below the isolation system to be designed for unreduced (RI = 1.0) earthquake forces. 12.1.2 Definition of Elements of an Isolated Structure The design requirements of the Standard distinguish between structural elements that are either components of the isolation system or part of the structure below the isolation system (e.g., foundation) and elements of the structure above the isolation system. The isolation system is defined by Chapter 17 of the Standard as follows: The collection of structural elements includes all individual isolator units, all structural elements that transfer force between elements of the isolation system and all connections to other structural elements. The isolation system also includes the wind-restraint system, energy-dissipation devices and/or the displacement restraint system if such systems and devices are used to meet the design requirements of this chapter. Figure 12.1-1 illustrates this definition and shows that the isolation system consists not only of the isolator units but also of the entire collection of structural elements required for the system to function properly. The isolation system typically includes segments of columns and connecting girders just above the isolator units because such elements resist moments (due to isolation system displacement) and their yielding or failure could adversely affect the stability of isolator units.

12-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Structure Above The Isolation System

Structural Elements That Transfer Force Between Isolator Units

Isolator Unit

Isolation Interface

Isolator Unit

Structure Below The Isolation System

Figure 12.1-1 Isolation system terminology The isolation interface is an imaginary boundary between the upper portion of the structure, which is isolated and the lower portion of the structure, which is assumed to move rigidly with the ground. Typically, the isolation interface is a horizontal plane, but it may be staggered in elevation in certain applications. The isolation interface is important for design of nonstructural components, including components of electrical and mechanical systems that cross the interface and must accommodate large relative displacements. The wind-restraint system is typically an integral part of isolator units. Elastomeric isolator units are very stiff at very low strains and usually satisfy drift criteria for wind loads and the static (breakaway) friction force of sliding isolator units is usually greater than the wind force. 12.1.3 Design Approach The design of isolated structures using the Standard has two objectives: achieving life safety in a major earthquake and limiting damage due to ground shaking. To meet the first performance objective, the isolation system must be stable and capable of sustaining forces and displacements associated with maximum considered earthquake (MCER) ground motions and the structure above the isolation system must remain essentially elastic when subjected to design earthquake (DE) ground motions. Limited ductility demand is considered necessary for proper functioning of the isolation system. If significant inelastic response were permitted in the structure above the isolation system, unacceptably large drifts could result due to the nature of long-period vibration. Limiting ductility demand on the superstructure has the additional benefit of meeting the second performance objective of damage control. The Standard addresses the performance objectives by requiring the following:

12-6

Chapter 12: Seismically Isolated Structures

§

Design of the superstructure for forces associated with the design earthquake, reduced by only a fraction of the factor permitted for design of conventional, fixed-base buildings (RI = 3/8R ≤ 2.0).

§

Design of the isolation system and elements of the structure below the isolation system (i.e., the foundation) for unreduced design earthquake forces (RI = 1.0).

§

Design and prototype testing of isolator units for forces (including effects of overturning) and displacements associated with the MCER.

§

Provision of sufficient separation between the isolated structure and surrounding retaining walls and other fixed obstructions to allow unrestricted movement during the MCER.

12.1.4 Effective Stiffness and Effective Damping The Standard utilizes the concepts of effective stiffness and effective damping to define key parameters of inherently nonlinear, inelastic isolation systems in terms of amplitude-dependent linear properties. Effective stiffness is the secant stiffness of the isolation system at the amplitude of interest. Effective damping is the amount of equivalent viscous damping described by the hysteresis loop at the amplitude of interest. Figure 12.1-2 shows the application of these concepts to both hysteretic isolator units (e.g., friction or yielding devices) and viscous isolator units and shows the Standard equations used to determine effective stiffness and effective damping from tests of prototypes. Ideally, the effective damping of velocity-dependent devices (including viscous isolator units) should be based on the area of hysteresis loops measured during cyclic testing of the isolation system at full-scale earthquake velocities. Tests of prototypes are usually performed at lower velocities (due to test facility limitations), resulting in hysteresis loops with less area for systems with viscous damping, which produce lower (conservative) estimates of effective damping. Conversely, testing at lower velocities can overestimate effective damping for certain systems (e.g., sliding systems with a coefficient of friction that decreases for repeated cycles of earthquake load).

12.2 CRITERIA SELECTION As specified in the Standard, the design of isolated structures must be based on the results of the ELF procedure, response spectrum analysis, or (nonlinear) response history analysis. Because isolation systems typically are nonlinear, linear methods (ELF procedure and response spectrum analysis) use effective stiffness and damping properties to model nonlinear isolation system components. The ELF procedure is intended primarily to prescribe minimum design criteria and may be used for design of a very limited class of isolated structures (without confirmatory dynamic analyses). The simple equations of the ELF procedure are useful tools for preliminary design and provide a means of expeditious review and checking of more complex calculations. The Standard also uses these equations to establish lower-bound limits on results of dynamic analysis that may be used for design. Table 12.2-1 summarizes site conditions and structure configuration criteria that influence the selection of an acceptable method of analysis for designing of isolated structures. Where none of the conditions in Table 12.2-1 applies, all three methods are permitted.

12-7

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Viscous Isolator Force

Force

Hysteretic Isolator

F+

F + F +

keff =

Eloop

β eff F−

(

Disp.

Δ−

⎡ Eloop 2 ⎢ = ⎢ π k Δ+ + Δ− ⎢⎣ eff

F+

Δ+ + Δ−

Disp.

Δ−

−

)

Eloop

⎤ ⎥ 2 ⎥ ⎥⎦

F−

Figure 12.2-1 Effective stiffness and effective damping Table 12.2-1

Acceptable Methods of Analysis*

Site Condition or Structure Configuration Criteria

ELF Procedure

Modal Response Spectrum Analysis

Seismic Response History Analysis

Site Conditions Near-source (S1 ≥ 0.6)

Soft soil (Site Class E or F)

Superstructure Configuration Flexible or irregular superstructure (height > 4 stories, height > 65 ft, or TM > 3.0 sec., or TD ≤ 3T)**

Nonlinear superstructure (requiring explicit modeling of nonlinear elements; Standard Sec. 17.6.2.2.1)

Isolation System Configuration Highly nonlinear isolation system or system that otherwise does not meet the criteria of Standard Sec. 17.4.1, Item 7

P indicates permitted and NP indicates not permitted by the Standard. ** T is the elastic, fixed-base period of the structure above the isolation system.

Seismic criteria are based on the same site and seismic coefficients as conventional, fixed-base structures (e.g., mapped value of S1 as defined in Standard Chapter 11). Additionally, site-specific design criteria

12-8

Chapter 12: Seismically Isolated Structures are required for isolated structures located on soft soil (Site Class E or F) or near an active source such that S1 is greater than or equal to 0.6, or when nonlinear response history analysis is used for design. 12.3

EQUIVALENT LATERAL FORCE PROCEDURE

The ELF procedure is a displacement-based method that uses simple equations to determine isolated structure response. The equations are based on ground shaking defined by 1-second spectral acceleration and the assumption that the shape of the design response spectrum at long periods is inversely proportional to period as shown in Standard Figure 11.4-1. There is also a 1/T2 portion of the spectrum at periods greater than TL. However, in most parts of the Unites States TL is longer than the period of typical isolated structures. Although the ELF procedure is considered a linear method of analysis, the equations incorporate amplitude-dependent values of effective stiffness and damping to account implicitly for the nonlinear properties of the isolation system. The equations are consistent with the nonlinear static procedure of ASCE 41 assuming the superstructure is rigid and lateral displacements occur primarily in the isolation system. 12.3.1 Isolation System Displacement The isolation system displacement for the design earthquake is determined using Standard Equation 17.51:

⎛ g ⎞ S T DD = ⎜ 2 ⎟ D1 D ⎝ 4π ⎠ BD where the damping factor, BD, is based on effective damping, βD, using Standard Table 17.5-1. This equation describes the peak (spectral) displacement of a single-degree-of-freedom (SDOF) system with period, TD and damping, βD, for the design earthquake spectrum defined by the seismic coefficient, SD1. SD1 corresponds to 5 percent damped spectral response at a period of 1 second. BD, converts 5 percent damped response to the level of damping of the isolation system. BD is 1.0 when effective damping, βD, is 5 percent of critical. Figure 12.3-1 illustrates the underlying concepts of Standard Equation 17.5-1 and the amplitude-dependent equations of the Standard for effective period, TD and effective damping, βD.

12-9

Mass × Spectral acceleration

FEMA P-751, NEHRP Recommended Provisions: Design Examples

TD = 2π

βD =

1 2π

Mass × Design earthquake spectral acceleration (W/g × SD1/T)

k Dmax

kDmin g ⎛ ∑ ED ⎜⎜ 2 ⎝ k Dmax DD

⎞ ⎟⎟ ⎠

k Dmin

Response reduction, BD (effective damping, β D > 5% of critical)

Vb Maximum stiffness curve

∑ ED

⎛ g ⎜ 2 ⎝ 4π

Minimum stiffness curve

⎞ ⎟ S D1TD ⎠

Spectral displacement

Figure 12.3-1 Isolation system capacity and earthquake demand The equations for maximum displacement, DM and design displacement, DD, reflect differences due to the corresponding levels of ground shaking. The maximum displacement is associated with MCE R ground motions (characterized by SM1) whereas the design displacement corresponds to design earthquake ground motions (characterized by SD1). In general, the effective period and the damping factor (TM and BM, respectively) used to calculate the maximum displacement are different from those used to calculate the design displacement (TD and BD) because the effective period tends to shift and effective damping may change with the increase in the level of ground shaking. As shown in Figure 12.3-1, the calculation of effective period, TD, is based on the minimum effective stiffness of the isolation system, kDmin, as determined by prototype testing of individual isolator units. Similarly, the calculation of effective damping is based on the minimum loop area, ED, as determined by prototype testing. Use of minimum effective stiffness and damping produces larger estimates of effective period and peak displacement of the isolation system. The design displacement, DD and maximum displacement, DM, represent peak earthquake displacements at the center of mass of the building without the additional displacement that can occur at other locations due to actual or accidental mass eccentricity. Equations for determining total displacement, including the effects of mass eccentricity as an increase in the displacement at the center of mass, are based on the plan dimensions of the building and the underlying assumption that building mass and isolation stiffness have a similar distribution in plan. The increase in displacement at corners for 5 percent mass eccentricity is approximately 15 percent if the building is square in plan and as much as 30 percent if the building is long in plan. Figure 12.3-2 illustrates design displacement, DD and maximum displacement, DM, at the center of mass of the building and total maximum displacement, DTM, at the corners of an isolated building.

12-10

Chapter 12: Seismically Isolated Structures

Total Maximum Displacement (maximum considered earthquake corner of building)

DTM

Maximum Displacement (maximum considered earthquake center of building)

Design Displacement (design earthquake center of building)

Figure 12.3-2 Design, maximum and total maximum displacement

12.3.2 Design Forces Forces required by the Standard for design of isolated structures are different for design of the superstructure and design of the isolation system and other elements of the structure below the isolation system (i.e., the foundation). In both cases, however, use of the maximum effective stiffness of the isolation system is required to determine a conservative value of design force. In order to provide appropriate overstrength, peak design earthquake response (without reduction) is used directly for design of the isolation system and the structure below. Design for unreduced design earthquake forces is considered sufficient to avoid inelastic response or failure of connections and other elements for ground shaking as strong as that associated with the MCER (i.e., shaking as much as 1.5 times that of the design earthquake). The design earthquake base shear, Vb, is given by Standard Equation 17.5-7: Vb = kDmaxDD where kDmax is the maximum effective stiffness of the isolation system at the design displacement, DD. Because the design displacement is conservatively based on minimum effective stiffness, Standard Equation 17.5-7 implicitly induces an additional conservatism of a worst-case combination mixing maximum and minimum effective stiffness in the same equation. Rigorous modeling of the isolation 12-11

FEMA P-751, NEHRP Recommended Provisions: Design Examples system for dynamic analyses precludes mixing of maximum and minimum stiffness in the same analysis (although separate analyses typically are required to determine bounding values of both displacement and force). Design earthquake response is reduced by a modest factor for design of the superstructure above the isolation interface, as given by Standard Equation 17.5-8:

Vs =

Vb k D max DD = RI RI

The reduction factor, RI, is defined as three-eighths of the R factor for the seismic force-resisting system of the superstructure, as specified in Standard Table 12.2-1, with an upper-bound value of 2.0. A relatively small RI factor is intended to keep the superstructure essentially elastic for the design earthquake (i.e., keeping earthquake forces at or below the true strength of the seismic force-resisting system). The Standard also imposes three limits on design forces that require the value of Vs to be at least as large as each of the following: 1. The shear force required for design of a conventional, fixed-base structure of the same effective seismic weight (and seismic force-resisting system) and period TD. 2. The shear force required for wind design. 3. A factor of 1.5 times the shear force required for activation of the isolation system. The first two limits seldom govern design but do reflect principles of good design. The third often governs design of very long period systems with substantial effective damping (e.g., the example EOC in Sec. 12.5) and is included in the Standard to ensure that the isolation system displaces significantly before lateral forces reach the strength of the seismic force-resisting system. For designs using the ELF procedure, the lateral forces, Fx, must be distributed to each story over the height of the structure, assuming an inverted triangular pattern of lateral load (Standard Eq. 17.5-9):

Fx =

Vs wx hx n

∑ wi hi i =n

Because the lateral displacement of the isolated structure is dominated by isolation system displacement, the actual pattern of lateral force in the isolated mode of response is distributed almost uniformly over height. Nevertheless, the Standard requires an inverted triangular pattern of lateral load to capture possible higher-mode effects that might be missed by not modeling superstructure flexibility and explicitly considering isolation system nonlinearity. Response history analysis that models superstructure flexibility and nonlinear properties of isolators would directly incorporate higher mode effects in the results. The ELF formulas may be used to construct plots of design displacement and base shear as a function of effective period, TD or TM, of the isolation system. For example, design displacement (DD), total maximum displacement (DTM) and design forces for the isolation system (Vb) and the superstructure (Vs) are shown in Figure 12.3-3 for a steel special concentrically braced frame (SCBF) superstructure and in

12-12

Chapter 12: Seismically Isolated Structures Figure 12.3-4 for a steel ordinary concentrically braced frame (OCBF) superstructure as functions of the effective period of the isolation system. 0.5

50 Vb

/W0.4 V r,a e hs ng is e 0.3 d de zi la m r o 0.2 N

40 DTM 30

DM Vs

0.1

.)n i( tn e m ec al ps i d m et s ys n oi ta lo Is

0.0

0 1.0

1.5

2.0

2.5 Effective period (s)

3.0

3.5

4.0

Figure 12.3-3 Isolation system displacement and shear force (SCBF) (R/I = 6.0/1.5, RI = 2.0) (1.0 in. = 25.4 mm) 0.5

50 Vb, Vs

0.4 W / V ,r ae hs n gi 0.3 se d de zi la m ro 0.2 N

40 DTM 30

20 DD

0.1

). in( t ne m ec la ps id m tes ys n oit la os I

0.0

0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Effective period (s)

Figure 12.3-4 Isolation system displacement and shear force (OCBF) (R/I = 3.25/1.5, RI = 1.0) (1.0 in. = 25.4 mm)

12-13

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Figures 12.3-3 and 12.3-4 illustrate design properties for a building located in a region of high seismicity, relatively close to an active fault, with a 1-second spectral acceleration parameter, S1, equal to 0.75 and site conditions corresponding to the C-D boundary (Site Class CD). Note: Seismic hazard and site conditions (and effective damping of the isolation system) were selected to be the same as those of the example EOC facility (Sec. 12.5). Table 12.3-1 summarizes the key response and design parameters that are used with various ELF formulas to construct the plots shown Figures 12.3-3 and 12.3-4. Table 12.3-1

Symbol S1 Ss Fv Fa βD BD βM BM DTM/DM R I RI I *

Summary of Key Response and Design Parameters Used to Construct Illustrative Plots of Design Displacements and Forces as a Function of Effective Period in Figures 12.3-3 and 12.3-4 Key Response or Design Parameter Description Value Ground Motion and Site Amplification Parameters 1-Second Spectral Acceleration (g) 0.75 Short Period Spectral Acceleration (g) 1.25 1-Second Site Coefficient (Site Class C-D) 1.4 Short-Period Site Coefficient (Site Class C-D) 1.0 Isolation System Design Parameters Effective Damping - Design Level Response 20% Damping Factor - Design Level Response 1.5 Effective Damping - MCER Level Response 13% Damping Factor - MCER Level Response 1.3 Torsional Response Amplification 1.1 Superstructure Design Parameters SCBF OCBF Response Modification Factor - Fixed-Base Structure 6.0 3.25 Importance Factor - Fixed-Base Structure 1.5 1.5 Response Modification Factor - Isolated Structure 2.0 1.0 Importance Factor - Isolated Structure 1.0 1.0

Chapter 17 of the Standard does not require use of occupancy importance factor to determine design loads on the superstructure of an isolated building (I = 1.0).

The plots in Figures 12.3-3 and 12.3-4 illustrate the fundamental trade-off between displacement and force as a function of isolation system displacement. As the period is increased, design forces decrease and design displacements increase linearly. Plots like those shown in Figures 12.3-3 and 12.3-4 can be constructed during conceptual design once site seismicity and soil conditions are known (or are assumed) to investigate trial values of effective stiffness and damping of the isolation system. In this particular example, an isolation system with an effective period of approximately 3.5 to 4.0 seconds would require approximately 30 inches of total maximum displacement capacity, which is near the practical limit of moat covers, flexible utility connections, etc., that must accommodate isolation system displacement. Design force, Vs, on the superstructure would be approximately 10 percent of the building weight for a steel SCBF system and approximately 17 percent for a steel OCBF superstructure (subject to other limits on Vs per Standard Section 17.5.4.3).

12-14

Chapter 12: Seismically Isolated Structures The Standard does not permit use of steel OCBFs as the superstructure of an isolated building that is an Essential Facility located in a region of high seismicity (close to an active fault). In contrast, Section 1613.6.2 of the 2006 International Building Code (IBC) permits steel OCBFs and ordinary moment frames (OMFs) to be used for structures assigned to Seismic Design Category D, E or F, provided that the following conditions are satisfied: §

The value of RI is taken as 1.0

§

Steel OMFs and OCBFs are designed in accordance with AISC 341.

The underlying concept of Section 1613.6.2 of the 2006 IBC is to trade the higher strength of steel OCBFs designed using RI = 1.0 for the larger inelastic response capacity of steel SCBFs. This trade-off is not unreasonable provided that the isolation system and surrounding structure are configured to not restrict displacement of the isolated structure in a manner that could cause large inelastic demands to occur in the superstructure. If an isolated building is designed with a superstructure system not permitted by the Standard (e.g., steel OCBFs), then the stability of the superstructure should be verified for MCE R ground motions using the response history procedure with explicit modeling of the stiffening effects of isolators at very large displacements (e.g., due to high rubber strains in an elastomeric bearing, or engagement of the articulated slider with the concave plates of a friction pendulum bearing) and the effects of nearby structures (e.g., possible impact with moat walls). 12.4

DYNAMIC LATERAL RESPONSE PROCEDURE

While the ELF procedure equations are useful tools for preliminary design of the isolations system, the Standard requires a dynamic analysis for most isolated structures. Even where not strictly required by the Standard, the use of dynamic analysis (usually response history analysis) to verify the design is common. 12.4.1 Minimum Design Criteria The Standard encourages the use of dynamic analysis but recognizes that along with the benefits of more complex models and analyses also comes an increased chance of design error. To avoid possible underdesign, the Standard establishes lower-bound limits on results of dynamic analysis used for design. The limits distinguish between response spectrum analysis (a linear, dynamic method) and response history analysis (a nonlinear, dynamic method). In all cases, the lower-bound limit on dynamic analysis is established as a percentage of the corresponding design parameter calculated using the ELF procedure equations. Table 12.4-1 summarizes the percentages that define lower-bound limits on dynamic analysis. Table 12.4-1 Summary of Minimum Design Criteria for Dynamic Analysis Design Parameter Response Response Spectrum History Procedure Procedure Total design displacement, DTD 90% DTD 90% DTD Total maximum displacement, DTM 80% DTM 80% DTM Design force on isolation system, Vb Design force on irregular superstructure, Vs Design force on regular superstructure, Vs

90% Vb 100% Vs 80% Vs

90% Vb 80% Vs 60% Vs

12-15

FEMA P-751, NEHRP Recommended Provisions: Design Examples The Standard permits more liberal drift limits where the design of the superstructure is based on dynamic analysis. The ELF procedure drift limits of 0.010hsx are increased to 0.015hsx for response spectrum analysis and to 0.020hsx for response history analysis (where hsx is the story height at level x). Usually a stiff system (e.g., braced frames) is selected for the superstructure (to limit damage to nonstructural components sensitive to drift) and drift demand typically is less than approximately 0.005hsx. Standard Section 17.6.4.4 requires an explicit check of superstructure stability at the MCER displacement if the design earthquake story drift ratio exceeds 0.010/RI. 12.4.2 Modeling Requirements As for the ELF procedure, the Standard requires the isolation system to be modeled for dynamic analysis using stiffness and damping properties that are based on tests of prototype isolator units. Additionally, dynamic analysis models are required to account for the following: §

Spatial distribution of individual isolator units.

§

Effects of actual and accidental mass eccentricity.

§

Overturning forces and uplift of individual isolator units.

§

Variability of isolation system properties (due to rate of loading, etc.).

The Standard requires explicit nonlinear modeling of elements if response history analysis is used to justify design loads less than those permitted for ELF or response spectrum analysis. This option is seldom exercised and the superstructure typically is modeled using linear elements and conventional methods. Special modeling concerns for isolated structures include two important and related issues: uplift of isolator units and P-delta effects on the isolated structure. Typically, isolator units have little or no ability to resist tension forces and can uplift when earthquake overturning (upward) loads exceed factored gravity (downward) loads. Local uplift of individual elements is permitted (Standard Sec. 17.2.4.7), provided the resulting deflections do not cause overstress or instability of the isolated structure. To calculate uplift effects, gap elements may be used in nonlinear models or tension may be released manually in linear models. The effects of P-delta loads on the isolation system and adjacent elements of the structure can be quite significant. The compression load, P, can be large due to earthquake overturning (and factored gravity loads) at the same time that large displacements occur in the isolation system. Computer analysis programs (most of which are based on small-displacement theory) may not correctly calculate P-delta moments at the isolator level in the structure above or in the foundation below. Figure 12.4-1 illustrates moments due to P-delta effects (and horizontal shear loads) for an elastomeric bearing isolation system and three configurations of a sliding isolation system. For the elastomeric system, the P-delta moment is split one-half up and one-half down. For the flat and single-dish sliding systems, the full P-delta moment is applied to the foundation below (due to the orientation of the sliding surface). A reverse (upside down) orientation of the flat and single-sided sliding systems would apply the full P-delta moment on the structure above. For the double-dish sliding system, P-delta moments are split one-half up and one-half down, in a manner similar to an elastomeric bearing, provided that the friction (and curvature) properties of the top and bottom concave dishes are the same.

12-16

Chapter 12: Seismically Isolated Structures

A P

P H3

MA = VH1 + PΔ/2

MB = VH2 + PΔ/2

P V

MF = VH6 + PΔ/2

P V

Double Dish Sliding Isolator

MD = VH4 + PΔ

Flat Sliding Isolator

Elastomeric Isolator

ME = VH5 + PΔ/2

MC = VH3

P V

MG = VH7 MH = VH8 + PΔ

Single Dish Sliding Isolator

Figure 12.4-1 Moments due to horizontal shear and P-delta effects

12-17

FEMA P-751, NEHRP Recommended Provisions: Design Examples 12.4.3 Response Spectrum Analysis Response spectrum analysis methods require that isolator units be modeled using amplitude-dependent values of effective stiffness and damping that are essentially the same as those of the ELF procedure, subject to the limitation that the effective damping of the isolated modes of response not exceed 30 percent of critical. Higher modes of response usually are assumed to have 2 to 5 percent damping, a value of damping appropriate for a superstructure that remains essentially elastic. As previously noted, maximum and minimum values of effective stiffness of the isolation system are used to calculate separately maximum displacement of the isolation system (using minimum effective stiffness) and maximum forces in the superstructure (using maximum effective stiffness). The Standard requires horizontal loads to be applied in two orthogonal directions and peak response of the isolation system and other structural elements is determined using the 100 percent plus 30 percent combination method. The Provisions now define ground motions in terms of maximum spectral response in the horizontal plane (where previous editions used average horizontal response). Consequently, at a given period of interest (for instance, the effective period of the isolation system), it may be overly conservative to combine 30 percent of the maximum spectral response load applied in the orthogonal direction with 100 percent of the maximum spectral response load applied in the horizontal direction of interest to determine peak spectral response of the isolation system and other structural elements. In the opinion of the author, it would be reasonable to not apply 30 percent of spectral response load at the fundamental (isolated) mode in the orthogonal direction when applying 100 percent of the maximum spectral response load in the horizontal direction of interest. However, the 100 percent plus 30 percent combination method would still be appropriate for all higher modes, since spectral response at higher-mode periods is, in general, independent of fundamental (isolated) mode spectral response. The design shear at any story, determined by RSA, cannot be taken as less than the story shear resulting from application of the ELF distribution of force over height (Standard Equation 17.5-9) where anchored to a value of base shear, Vs, determined by RSA in the direction of interest. This limit is intended to avoid underestimation of higher-mode response when isolators are modeled using effective stiffness and damping properties, rather than actual nonlinear properties. The value of Vs determined by RSA is typically less than the value of Vs prescribed by ELF using Standard Equation 17.5-8, which combines maximum effective stiffness, kDmax, with design displacement, DD, based on minimum effective stiffness, although the difference generally is small and the values of design shear determined by RSA are similar to those required by the ELF procedure. Standard Section 17.6.3.4 does not explicitly require the value of base shear, Vs, determined by RSA to comply with the Standard Section 17.5.4.2 limits on Vs. In the opinion of the author, the Standard Section 17.5.4.3 limits on base shear apply to all methods of analysis and should be complied with when RSA is used as the basis for design. It may be noted that Standard Section 17.6.4.2 requires Vs to comply with the Standard Section 17.5.4.3 limits on Vs when the design is based on the response history analysis procedure, which otherwise has more liberal design shear requirements than either the RSA or ELF procedure. As previously discussed (Sec. 12.3.2), the third limit of Standard Section 17.5.4.3 is included in the Standard to ensure that the isolation system displaces significantly before lateral forces reach the strength of the seismic force-resisting system. 12.4.4 Response History Analysis For response history analysis, nonlinear force-deflection characteristics of isolator units are modeled explicitly (rather than using effective stiffness and damping). For most types of isolators, force-deflection properties can be approximated by bilinear, hysteretic curves that can be modeled using commercially available nonlinear structural analysis programs. Such bilinear hysteretic curves should have

12-18

Chapter 12: Seismically Isolated Structures approximately the same effective stiffness and damping at amplitudes of interest as the true forcedeflection characteristics of isolator units (as determined by prototype testing). More sophisticated nonlinear models may be necessary to represent accurately response of isolators with complex configurations or properties (e.g., “triple pendulum” sliding bearings), to capture stiffening effects at very large displacements (e.g., of elastomeric bearings), or to model rate-dependent effects explicitly (Sarlis 2010).

Force

Figure 12.4-2 shows a bilinear idealization of the response of a typical nonlinear isolator unit. Figure 12.4-2 also includes simple equations defining the yield point (Dy, Fy) and end point (D, F) of a bilinear approximation that has the same effective stiffness and damping as the true curve (at displacement, D).

At the displacement of interest:

F ≈ keff D 4 ( D Fy − F Dy ) ≈ Eloop

F Fy

Eloop

Displacement

DW (using D in inches) F ⎛ D Fy − F D y ⎞ = 0.637 ⎜ ⎟ FD ⎝ ⎠

Teff = 0.32

β eff

Figure 12.4-2 Bilinear idealization of isolator unit behavior Response history analysis with explicit modeling of nonlinear isolator units is commonly used for the evaluation of isolated structures. Where at least seven pairs of ground motion acceleration components are employed, the values used in design for each response parameter of interest may be the average of the corresponding analysis maxima. Where fewer pairs are used (with three pairs of ground motion acceleration components being the minimum number permitted), the maximum value of each parameter of interest must be used for design. The response history method is not a particularly useful design tool due to the complexity of results, the number of analyses required (to account for different locations of eccentric mass), the need to combine different types of response at each point in time, etc. It should be noted that while Standard Chapter 16 does not require consideration of accidental torsion for either the linear or nonlinear response history procedures, Chapter 17 does require explicit consideration of accidental torsion, regardless of the analysis method employed. Response history analysis is most useful when used to verify a design by checking a

12-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples few key design parameters, such as isolation displacement, overturning loads and uplift and story shear force. The Provisions (Sections 17.3.2 and 16.1.3.2) now require ground motions to be scaled to match maximum spectral response in the horizontal plane (where previous editions defined spectral response in terms of average horizontal response). In concept, at a given period of interest, maximum spectral response of scaled records should, on average, be the same as that defined by the design spectrum of interest (DE or MCER). Neither the Standard nor the Provisions specify how the two scaled components of each record should be applied to a three-dimensional model (i.e., how the two components of each record should be oriented with respect to the axes of the model). This lack of guidance has sometimes caused users to perform an unnecessarily large number of response history analyses for design verification of an isolated structure. In the author’s opinion, the following steps describe an acceptable approach for scaling, orienting and applying ground motion components to a three-dimensional model of an isolated structure (when based on the new maximum definition of the ground motions). §

Step 1 - Selection and Scaling of Records. Select and scale ground motion records as required by Provisions Section 17.3.2. The records would necessarily be scaled differently to match the DE spectrum and the MCER spectrum, respectively. While the Provisions permits as few as three records for response history analysis (with design/verification based the most critical of the three), a set of at least seven records is recommended such that the average value of the response parameter of interest may be used for design or design verification.

§

Step 2 - Grouping of Stronger Components. Group the horizontal components of each record in terms of stronger and weaker components. The stronger component of each record is the component that has the larger spectral response at periods of interest. For isolated structures, periods of interest are approximately from TD to TM. The records may require rotation of horizontal axes before grouping to better distinguish between stronger and weaker components at the period of interest. Orient stronger horizontal components such that peak response occurs in the same (positive or negative) direction. In general, the same grouping of components can be used for both DE and MCER analyses (just scaled by a different factor), since TD and TM are typically about the same and response spectra do not vary greatly from period to period at long periods.

§

Step 3 - Verification of Component Grouping. Verify that the average spectral response of the set of stronger components is comparable to the design spectrum of interest (e.g., DE or MCE R spectrum) at periods of interest (i.e., TM, if checking MCER displacement). If the average spectrum of stronger components is not adequate, then the scaling factor should be increased accordingly. Note: Increasing ground motions to match average spectral response of stronger components with the design spectrum of interest is not required by Provisions Section 17.3.2 but is consistent with the “maximum” definition of ground motions.

§

Step 4 - Application of Scaled Records. In general, apply the set of scaled records to the threedimensional model of the isolated structure in four basic orientations with respect to the primary (orthogonal) horizontal axes of the superstructure: 1. Apply the set of scaled records with stronger components aligned in the positive direction of the first horizontal axis of the model.

12-20

Chapter 12: Seismically Isolated Structures 2. Apply the set of scaled records with stronger components aligned in the positive direction of the second horizontal axis of the model. 3. Apply the set of scaled records with stronger components aligned in the negative direction of the first horizontal axis of the model. 4. Apply the set of scaled records with stronger components aligned in the negative direction of the second horizontal axis of the model. For each of the four sets of analyses, find the average value of the response parameter of interest (if the set of ground motion contains at least seven records) or the maximum value of the response parameter (if the set of ground motions contains less than seven records). The more critical response of the four sets of analyses should be used for design or design verification. The four orientations of ground motions of Step 4 are required, in general, for design of elements, since the most critical positive or negative direction of peak response typically is not known for individual elements. However, for design verification of key global response parameters that are relatively insensitive to positive/negative orientation of ground motion components (e.g., maximum isolation system displacement, peak story shear, peak story displacement), only the first two sets of analyses are necessary. Although the above steps require some “homework” by the engineer to develop an appropriate set of scaled records, only two sets of response history analyses are typically required to verify the design of most isolated structures. The above response history analysis recommendations apply to each model of the structure, which necessarily include at least two models whose properties represent upper-bound and lower-bound forcedeflection properties of the isolation system, respectively. Different models could also be used to explicitly evaluate various locations of accidental mass eccentricity, as required by Standard Section 17.6.2.1.b. However, this approach would require multiple additional models to consider “the most disadvantageous location of accidental eccentric mass.” In the opinion of the author, these additional response history analyses are unnecessary and the effects of accidental mass eccentricity can be calculated by factoring the results of response history analyses (of a model that does not explicitly include accidental mass eccentricity). The amount by which the response parameter of interest should be factored may be determined by assuming that accidental torsion increases response in proportion to the increase in isolation system displacement, as prescribed by the ELF requirements of Standard Section 17.5.3.5. 12.5

EMERGENCY OPERATIONS CENTER USING DOUBLE-‐CONCAVE FRICTION PENDULUM BEARINGS, OAKLAND, CALIFORNIA

This example features the seismic isolation of a hypothetical EOC, assumed to be located in Oakland, California, approximately 6 kilometers from the Hayward fault. The isolation system incorporates double-concave friction pendulum sliding bearings, although other types of isolators could have been used in this example. Isolation is an appropriate design strategy for EOCs and other buildings where the goal is to limit earthquake damage and protect facility function. The example illustrates the following design topics: §

Determination of seismic design parameters.

§

Preliminary design of superstructure and isolation systems (using the ELF procedure).

§

Dynamic analysis of a seismically isolated structure. 12-21

FEMA P-751, NEHRP Recommended Provisions: Design Examples

§

Specification of isolation system design and testing criteria.

While the example includes development of the entire structural system, the primary focus is on the design and analysis of the isolation system. Examples in other chapters have more in-depth descriptions of the provisions governing detailed design of the superstructure above and the foundation below. 12.5.1 System Description This EOC is a three-story, steel-braced frame structure with a large, centrally located mechanical penthouse. Story heights are 14 feet at the first floor to accommodate computer access flooring and other architectural and mechanical systems and 12 feet at the second and third floors (and penthouse). The roof and penthouse roof decks are designed for significant live load to accommodate a helicopter landing pad and to meet other functional requirements of the EOC. Figure 12.5-1 shows the three-dimensional model of the structural system.

Figure 12.5-1 Three-dimensional model of the structural system The structure (which is regular in configuration) has plan dimensions of 100 feet by 150 feet at all floors except for the penthouse, which is approximately 50 feet by 100 feet in plan. Columns are spaced at 25 feet in both directions. Figures 12.5-2 and 12.5-3 are framing plans for the typical floor levels (1, 2, 3 and roof) and the penthouse roof.

12-22

Chapter 12: Seismically Isolated Structures

Y X

2 25'-0"

3 25'-0"

4 25'-0"

5 25'-0"

6 25'-0"

7 25'-0"

25'-0"

Figure 12.5-2 Typical floor framing plan (1.0 ft = 0.3048 m)

1 X

2 25'-0"

3 25'-0"

4 25'-0"

5 25'-0"

6 25'-0"

7 25'-0"

25'-0"

Figure 12.5-3 Penthouse roof framing plan (1.0 ft = 0.3048 m) The vertical load-carrying system consists of concrete fill on steel deck floors, supported by steel beams at 8.3 feet on center and steel girders at column lines. Isolator units support the columns below the first floor. The foundation is a heavy mat (although spread footings or piles could be used depending on the soil type, depth to the water table and other site conditions).

12-23

FEMA P-751, NEHRP Recommended Provisions: Design Examples The lateral system consists of a roughly symmetrical pattern of concentrically braced frames. These frames are located on Column Lines B and D in the longitudinal direction and on Column Lines 2, 4 and 6 in the transverse direction. Figures 12.5-4 and 12.5-5 show the longitudinal and transverse elevations, respectively. Braces are specifically configured to reduce the concentration of earthquake overturning and uplift loads on isolator units by: §

Increasing the number of bays with bracing at lower stories.

§

Locating braces at interior (rather than perimeter) column lines (providing more hold-down weight).

§

Avoiding common end columns for transverse and longitudinal bays with braces.

6 bays at 25'-0" = 150'-0" Penthouse Roof Roof Third Floor Second Floor First Floor Base Lines B and D

Figure 12.5-4 Longitudinal bracing elevation (Column Lines B and D)

12-24

Chapter 12: Seismically Isolated Structures

4 bays at 25'-0" = 100'-0"

4 bays at 25'-0" = 100'-0" Penthouse Roof Roof Third Floor Second Floor First Floor Base

(a) Lines 2 and 6

(b) Line 4

Figure 12.5-5 Transverse bracing elevations: (a) on Column Lines 2 and 6 and (b) on Column Line 4 The isolation system has 35 identical elastomeric isolator units, located below columns. The first floor is just above grade and the isolator units are approximately 3 feet below grade to provide clearance below the first floor for construction and maintenance personnel. A short retaining wall borders the perimeter of the facility and provides approximately 3 feet of “moat” clearance for lateral displacement of the isolated structure. Access to the EOC is provided at the entrances by segments of the first floor slab, which cantilever over the moat. Girders at the first-floor column lines are much heavier than the girders at other floor levels and have moment-resisting connections to columns. These girders stabilize the isolator units by resisting moments due to vertical (P-delta effect) and horizontal (shear) loads. Column extensions from the first floor to the top plates of the isolator units are stiffened in both horizontal directions, to resist these moments and to serve as stabilizing haunches for the beam-column moment connections. 12.5.2 Basic Requirements 12.5.2.1 Specifications. §

General: ASCE Standard ASCE 7-05 (Standard)

§

Seismic Loads: 2009 NEHRP Recommended Provisions (Provisions)

§

Other Loads and Load Combinations: 2006 International Building Code (2006 IBC)

12.5.2.2 Materials. §

Concrete: Strength (floor slabs): f c‘ = 3 ksi

12-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples Strength (foundations below isolators): f c‘ = 5 ksi Weight (normal): 150 pcf §

Steel: Columns: Fy = 50 ksi Primary first-floor girders (at column lines): Fy = 50 ksi Other girders and floor beams: Fy = 36 ksi Braces: Fy = 46 ksi

§

Steel deck: 3-inch-deep, 20-gauge deck

12.5.2.3 Gravity loads. §

Dead Loads: Main structural elements (slab, deck and framing): self weight Miscellaneous structural elements (and slab allowance): 10 psf Architectural facades (all exterior walls): 20 psf Roof parapets: 20 psf Partitions (all enclosed areas): 20 psf Suspended MEP/ceiling systems and supported flooring: 15 psf Mechanical equipment (penthouse floor): 50 psf Roofing: 10 psf

§

Reducible live loads: Floors (1-3): 100 psf Roof decks and penthouse floor: 50 psf

§

Live load reduction: 2006 IBC Section 1607.9 permits area-based live load reduction of not more than 50 percent for elements with live loads from a single story (girders) and not more than 60 percent for elements with live loads from multiple stories (axial component of live load on columns at lower levels and isolator units).

§

EOC weight (dead load) and live load (from ETABS model, Guide Sec. 12.5.3.1): Penthouse roof Roof (penthouse floor)

12-26

WPR = 794 kips WR = 2,251 kips

Chapter 12: Seismically Isolated Structures Third floor Second floor First floor Total EOC weight (updated guess - k)

W3 W2 W1 W

Live load (L) without reduction Reduced live load (L) on isolation system

L = 5,476 kips L = 2,241 kips

Table 12.5-1 Column line

= = = =

1,947 kips 1,922 kips 2,186 kips 9,100 kips

Summary of dead load (D) and reduced live load (L) on isolator units in kips (from ETABS model, Guide Sec. 12.5.3.1)* (D/L) 1

138 / 34

251 / 58

206 / 44

204 / 43

253 / 58

290 / 77

323 / 86

342 / 92

206 / 43

323 / 86

367 / 99

334 / 90

1.0 kip = 4.45 kN. * Loads at Column Lines 5, 6 and 7 (not shown) are similar to those at Column Lines 3, 2 and 1, respectively; loads at Column Lines D and E (not shown) are similar to those at Column Lines B and A, respectively.

12.5.2.4 Seismic design parameters. 12.5.2.4.1 Performance criteria (Standard Sec. 1.5.1). §

Designated Emergency Operation Center: Occupancy Category IV

§

Occupancy Importance Factor: I = 1.5 (conventional)

§

Occupancy Importance Factor (Standard Chapter 17): I = 1.0 (isolated) Note: Standard Chapter 17 does not require use of the occupancy importance factor to determine the design loads on the structural system of an isolated building (i.e., I = 1.0). However, the component importance factor is still required by Chapter 13 to determine seismic forces on nonstructural components of isolated structures (Ip = 1.5 for Occupancy Category IV facilities).

§

Seismic Design Category (Standard Section 11.6): Seismic Design Category F

12.5.2.4.2 Ground motions for Oakland EOC site (Provisions Chapters 11 and 21). §

Site Location, Hazard and Soil Conditions (assumed): Site latitude and longitude: 37.80°, -122.25° Source (fault) controlling hazard at the Oakland site: Hayward Maximum moment magnitude earthquake on controlling source: M7.3

12-27

FEMA P-751, NEHRP Recommended Provisions: Design Examples Closest distance from site to Hayward Fault (Joyner-Boore distance): 5.9 km Site soil type (assumed for preliminary design): Site Class C-D Site shear wave velocity (assumed for response history analysis): vs,30 = 450 m/s §

Short-Period Design Parameters (USGS web site): Short-period MCER spectral acceleration (USGS 2009): SS = 1.24 Site coefficient (Standard Table 11.4-1): Fa = 1.0 Short-period MCER spectral acceleration adjusted for site class (FaSS): SMS = 1.24 DE spectral acceleration (2/3SMS): SDS = 0.83

§

1-Second Design Parameters (USGS web site): 1-Second MCER spectral acceleration: S1 = 0.75 Site coefficient (Standard Table 11.4-2): Fv = 1.4 1-Second MCER spectral acceleration adjusted for site class (FvS1): SM1 = 1.05 DE spectral acceleration (2/3SM1): SD1 = 0.7

12.5.2.4.3 Design spectra (Provisions Section 11.4). Figure 12.5-6 plots DE and MCER response spectra as constructed in accordance with the procedure of Provisions Section 11.4 using the spectrum shape defined by Provisions Figure 11.4-1. Standard Section 17.3.1 requires a site-specific ground motion hazard analysis to be performed in accordance with Chapter 21 for sites with S1 greater than 0.6 (e.g., sites near active sources). Subject to other limitations of Provisions Section 21.4, the resulting sitespecific DE and MCER spectra may be taken as less than 100 percent but not less than 80 percent of the default design spectrum of Standard Figure 11.4-1.

12-28

Chapter 12: Seismically Isolated Structures

2.0

Spectral Acceleration, Sa (g)

1.5 MCER

0.8MCER

1.0

DE 0.8DE 0.5

0.0 0.0

1.0

2.0

3.0

4.0

5.0

Period, T (s)

Figure 12.5-6 DE and MCER spectra and 80 percent limits For this example, site-specific spectra for the design earthquake and the maximum considered earthquake were assumed to be 100 percent of the respective spectra shown in Figure 12.5-6. In general, site-specific spectra for regions of high seismicity, with well defined fault systems (like the Hayward fault), would be expected to be similar to the default design spectra of Standard Figure 11.4-1. 12.5.2.4.4 DE and MCER ground motion records (Standard Sec. 17.3.2). For response history analysis, Standard Section 17.6.3.4 requires at least three pairs of horizontal ground motion acceleration components to be selected from actual earthquake records and scaled to match either the DE or the MCE R spectrum and at least seven pairs if the average value of the response parameter of interest is used for design (which is typically the case). Selection and scaling of appropriate ground motions should be performed by a ground motion expert experienced in earthquake hazard of the region, considering site conditions, earthquake magnitudes, fault distances and source mechanisms that influence ground motion hazard at the building site. For this example, a set of seven ground motion records are selected from the near-field (NF) and far-field (FF) record sets of FEMA P-695. FEMA P-695 is a convenient source of the 50 strongest ground motion records of large magnitude earthquakes available from the Pacific Earthquake Engineering Research Center (PEER) NGA database (PEER 2006), the records most suitable for response history evaluation of structures in regions of high seismicity. The seven records selected from the FEMA P-695 ground motion sets have fault source and site characteristics that best match those of the example EOC, that is, records of magnitude M7.0 or greater earthquakes recorded close to fault rupture at soil sites (Site Class C or D). Table 12.5-2 lists these seven earthquake records and summarizes key properties. As shown in this table, the average magnitude (M7.37), the average site-source distance (5.2 km) and the average shear wave

12-29

FEMA P-751, NEHRP Recommended Provisions: Design Examples velocity (446 m/s) closely match the maximum magnitude of the Hayward fault (M7.3), the closest distance from the Hayward fault to the Oakland site (5.9 km) and the Oakland site conditions (Site Class C-D), respectively, as required by record selection requirements of Standard Section 17.3.2. Table 12.5-2 FEMA P-695 Record ID No.

Seven Earthquake Records Selected for Response History Analysis of Example Base-Isolated EOC Facility Earthquake

Year

Name

NF-8 FF-10 NF-25 FF-3 NF-14

1992 1999 1999 1999 1999

FF-4

1999

Landers Kocaeli Kocaeli Duzce Duzce Hector Mine

NF-28

2002

Denali

Mean Property of Seven Records

Source Characteristics Distance, Df Fault (km) Mechanis m JB Rupture

Site Conditions Site Class

vS,30 (m/s)

Strike-slip Strike-slip Strike-slip Strike-slip Strike-slip

C C D D D

685 523 297 326 276

11.7

Strike-slip

685

0.2

3.8

Strike-slip

329

5.2

9.8

Record Station

Mag. (MW)

Lucerne Arcelik Yarimca Bolu Duzce

7.3 7.5 7.5 7.1 7.1

2.2 10.6 1.4 12.0 0.0

15.4 13.5 5.3 12.0 6.6

Hector

7.1

10.4

TAPS PS#10

7.9 7.37

446

Standard Section 17.3.2 provides criteria for scaling earthquake records to match a target spectrum over the period range of interest, defined as 0.5TD to 1.25TM. In this example, TD and TM are 3.5 and 3.9 seconds, respectively, so the period range of interest is from 1.75 to 4.9 seconds. For each period in this range, the average of the square-root-of-the-sum-of-the-squares (SRSS) combination of each pair of horizontal components of scaled ground motion should be equal to or greater than the target spectrum. The target spectrum is defined as 1.0 times the design spectrum of interest (either the DE or the MCE R spectrum). Table 12.5-3 summarizes the factors used to scale the seven records to mach either the DE or MCE R spectrum, in accordance with Standard Section 17.3.2. The seven records were first “normalized” by their respective values of PGVPEER to reduce inappropriate amounts of record-to-record variability using the procedures of Section A.8 of FEMA P-695. As shown in Table 12.5-3, PGV normalization tends to increase the intensity of those records of smaller than average magnitude events or from sites farther than average from the source and to decrease the intensity of those records of larger than average magnitude events or from sites closer than average to the source, but has no net effect on the overall intensity of the record set (i.e., median value of normalization factors is 1.0 for the record set). Scaling factors were developed such that the average value of the response spectra of normalized records equals or exceeds the spectrum of interest (either the DE or MCER spectrum) over the period range of interest, 1.75 to 4.9 seconds. These scaling factors are given in Table 12.5-3 and reflect the total amount that each as-recorded ground motion is scaled for response history analysis. Table 12.5-3 shows that a median scaling factor of 1.04 is required to envelop the DE spectrum (records are increased slightly, on average, to match the DE spectrum) and a median scaling factor of 1.56 is required to envelop the MCER spectrum over the period range of interest.

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Chapter 12: Seismically Isolated Structures

Table 12.5-3 FEMA P695 Record ID No.

Summary of Factors Used to Scale Each of the Seven Records to Match the DE or MCER Spectrum Earthquake Year

Name

NF-8 FF-10 NF-25 FF-3 NF-14 FF-4

1992 1999 1999 1999 1999 1999

Landers Kocaeli Kocaeli Duzce Duzce Hector Mine

NF-28

2002

Denali

Mean Property of Seven Records

Record Station Lucerne Arcelik Yarimca Bolu Duzce Hector TAPS PS#10

Normalization and Scaling Factors PGV Oakland Site PGVPEER Normal (cm/s) DE MCER Factor 97.2 0.60 0.62 0.94 27.4 2.13 2.21 3.32 62.4 0.93 0.97 1.46 59.2 0.99 1.02 1.54 69.6 0.84 0.87 1.31 34.1 1.71 1.78 2.67 98.5

0.59

0.62

0.92

58.3

1.00

1.04

1.56

Figure 12.5-7 compares the MCER spectrum for the Oakland site with the average spectrum of the SRSS combination, the average spectrum of stronger components and the average spectrum of other (orthogonal) components of the seven scaled records. This figure shows: §

The average spectrum of the SRSS combination of scaled record to envelop the MCER spectrum from 1.75 seconds (0.5TD) to 4.9 seconds (1.25TM), as required by Standard Section 17.3.2.

§

The average spectrum of larger scaled components comparable to the MCER spectrum at response periods of interest (e.g., 3.9 seconds for MCER analysis).

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

2.0

Spectral Acceleration, Sa (g)

Average, SRSS

1.5

1.0

MCER 0.5

0.0 0.0

1.0

2.0

3.0

4.0

5.0

Period, T (s)

Figure 12.5-7 Comparison of the MCER spectrum with the average spectrum of the SRSS combination of the seven scaled records, the average spectrum of the seven larger components of the seven scaled records and the average spectrum of the other orthogonal components of the seven scaled records listed in Table 12.5-3 12.5.2.5 Structural design criteria. 12.5.2.5.1 Design basis. §

Seismic force-resisting system: Special steel concentrically braced frames (height < 100 feet)

§

Response modification factor, R (Standard Table 12.2-1): R = 6 (conventional)

§

Response modification factor for design of the superstructure, RI (Standard Sec. 17.5.4.2, 3/8R ≤ 2): RI = 2 (isolated)

§

Plan irregularity (of superstructure) (Standard Table 12.3-1): None

§

Vertical irregularity (of superstructure) (Standard Table 12.3-2): None

§

Lateral response procedure (Standard Sec. 17.4.1, S1 > 0.6): Dynamic analysis

§

Redundancy factor (Standard Sec. 12.3.4): ρ ≥ 1.0 (conventional); ρ = 1.0 (isolated)

Standard Section 12.3.4 requires the use of a calculated ρ value, which could be greater than 1.0 for a conventional structure with a brace configuration similar to the superstructure of the base-isolated EOC. 12-32

Chapter 12: Seismically Isolated Structures However, in the author’s opinion, the use of RI equal to 2.0 (rather than R equal to 6) as required by Standard Section 17.5.4.2 precludes the need to further increase superstructure design forces for redundancy. 12.5.2.5.2 Horizontal earthquake loads and effects (Standard Chapters 12 and 17). §

Design earthquake (acting in either the X or Y direction): DE (site specific)

§

Maximum considered earthquake (acting in either the X or Y direction): MCE (site specific)

§

Mass eccentricity - actual plus accidental: 0.05b = 5 ft (X direction); 0.05d = 7.5 ft (Y direction)

The superstructure is essentially symmetric about both primary horizontal axes, however, the placement of the braced frames results in a ratio of maximum corner displacement to average displacement of 1.25 including accidental eccentricity, exceeding the threshold of 1.2 per the definition of the Standard. If the building were not on isolators, the accidental torsional eccentricity would need to be increased from 5 percent to 5.4 percent of the building dimension. The input to the superstructure is controlled by the isolation system and it is the author’s opinion that the amplification of accidental torsion is not necessary for such otherwise regular structures. Future editions of the Standard should address this issue. Also refer to the discussion of analytical modeling of accidental eccentricities in Guide Chapter 4. §

Superstructure design (reduced DE response): QE = QDE/2 = DE/2.0

§

Isolation system and foundation design (unreduced DE response) : QE = QDE = DE/1.0

§

Check of isolation system stability (unreduced MCE response): QE = QMCE = MCE/1.0

12.5.2.5.3 Combination of horizontal earthquake load effects. Response due earthquake loading in the X and Y directions is as follows: QE = Max (1.0QEX+0.3QEY, 0.3QEX+1.0QEY) In general, the horizontal earthquake load effect, QE, on the response parameter of interest is influenced by only one direction of horizontal earthquake load and QE = QEX or QE = QEY. Exceptions include vertical load on isolator units due to earthquake overturning forces. 12.5.2.5.4 Combination of horizontal and vertical earthquake load effects. §

Design earthquake (QE + 0.2SDSD): E = QE + 0.17D

§

Maximum considered earthquake (QE + 0.2SMSD): E = QE + 0.25D

12.5.2.5.5 Superstructure design load combinations (2006 IBC, Sec. 1605.2.1, using RI = 2). §

Gravity loads (dead load and reduced live load): 1.2D + 1.6L

§

Gravity and earthquake loads (1.2D + 0.5L + 1.0E): 1.37D + 0.5L + QDE/2

§

Gravity and earthquake loads (0.9D - 1.0E): 0.73D - QDE/2

12.5.2.5.6 Isolation system and foundation design load combinations (2006 IBC, Sec. 1605.2.1). 12-33

FEMA P-751, NEHRP Recommended Provisions: Design Examples

§

Gravity loads (for example, long term load on isolator units): 1.2D + 1.6L

§

Gravity and earthquake loads (1.2D + 0.5L + 1.0E): 1.37D + 0.5L + QDE

§

Gravity and earthquake loads (0.9D - 1.0E): 0.73D - QDE

12.5.2.5.7 Isolation system stability load combinations (Standard Sec. 17.2.4.6). §

Maximum short term load on isolator units (1.2D + 1.0L + |E|): 1.45D + 1.0L + QMCE

§

Minimum short term load on isolator units (0.8D - |E|): 0.8D - QMCE

Note that in the above combinations, the vertical earthquake load (0.2SMSD) component of |E| is included in the maximum (downward) load combination but excluded from the minimum (uplift) load combination. It is the author’s opinion that vertical earthquake ground shaking is of a dynamic nature, changing direction too rapidly to affect appreciably uplift of isolator units and need not be used with the load combinations of Standard Section 17.2.4.6 for determining minimum (uplift) vertical loads on isolator units due to the MCER. 12.5.3 Seismic Force Analysis 12.5.3.1 Basic approach to modeling. To expedite calculation of loads on isolator units and other elements of the seismic-force-resisting system, a three-dimensional model of the EOC is developed and analyzed using the ETABS computer program (CSI, 2009). While there are a number of commercially available programs to choose from, ETABS is selected for this example since it permits the automated release of tension in isolator units subject to uplift and has built-in elements for modeling other nonlinear properties of isolator units. Arguably, all of the analyses performed by the ETABS program could be done by hand or by spreadsheet calculation (except for confirmatory response history analyses). The ETABS model is used to perform the following types of analyses and calculations: §

Gravity Load Evaluation. Calculate maximum long-term load (1.2D + 1.6L) on isolator units (Guide Table 12.5-1).

§

ELF Procedure (and RSA). Calculate gravity and reduced design earthquake load response for design of the superstructure (ignoring uplift of isolator units).

§

Nonlinear Static Analysis (with ELF loads). Calculate gravity and unreduced design earthquake load response for design of the isolation system and foundation (considering uplift of isolator units).

§

Nonlinear Static Analysis (with ELF loads). Calculate gravity and unreduced design earthquake load response to determine maximum short-term load (downward force) on isolator units (Guide Table 12.5-5) and minimum short-term load (downward force) of isolator units (Guide Table 12.5-6).

§

Nonlinear Static Analysis (with ELF loads). Calculate gravity and unreduced MCE R load response to determine maximum short-term load (downward force) on isolator units (Guide

12-34

Chapter 12: Seismically Isolated Structures Table 12.5-7) and minimum short-term load (uplift displacement) of isolator units (Guide Table 12.5-8) §

Nonlinear Response History Analysis. Calculate gravity and scaled design earthquake or MCE R ground motion response (average of seven records) for key response parameters: 1. Design earthquake and MCER displacement (including uplift) of isolator units (Guide Table 12.5-12). 2. Design earthquake and MCER peak story shears (Guide Table 12.5-14). 3. MCER short-term load (downward force) on isolator units (Guide Table 12.5-15).

The Standard requires modal response spectrum analysis or seismic response history analysis for the EOC (see Guide Table 12.2-1). In general, the modal response spectrum method of dynamic analysis is considered sufficient for facilities that are located at a stiff soil site, which have an isolation system meeting the criteria of Standard Section 17.4.1, Item 7. However, nonlinear static analysis is used for the design of the EOC, in lieu of modal response spectrum analysis, to permit explicit modeling of potential uplift of isolator units. For similar reasons, nonlinear seismic response history analysis is used to verify design parameters with explicit modeling of potential uplift of isolator units. Chapter 17 of the Standard does not define methods for nonlinear static analysis of base-isolated structures. For this example, nonlinear static loads are applied in one orthogonal direction at a time and the more critical value of the response of the parameter of interest is used for design. In the author’s opinion, uni-directional application of earthquake load (in lieu of a 100 percent, 30 percent combination) is considered appropriate for static loads which are based on the maximum direction of response (the new ground motion criterion of the Provisions) when used with a conservative distribution of force over height (i.e., ELF distribution of force as described by Standard Eq. 17.5-9). 12.5.3.2 Detailed modeling considerations. Rather than a complete description of the ETABS model, key assumptions and methods used to model elements of the isolation system and superstructure are described below. 12.5.3.2.1 Mass eccentricity. Standard Section 17.6.2.1 requires consideration of mass eccentricity. Because the building in the example is doubly symmetric, there is no actual eccentricity of building mass (but such would be modeled if the building were not symmetric). Modeling of accidental mass eccentricity would require several analyses, each with the building mass located at different eccentric locations (for example, four quadrant locations in plan). This is problematic, particularly for dynamic analysis using multiple ground motion inputs. In this example, only a single (actual) location of mass eccentricity is considered and calculated demands are increased moderately for the design of the seismic force-resisting system and isolation system to account for accidental eccentricity (e.g., peak displacements calculated by dynamic analysis are increased by 10 percent for design of the isolation system). 12.5.3.2.2 P-delta effects. P-delta moments in the foundation and the first-floor girders just above isolator units due to the large lateral displacement of the superstructure are modeled explicitly. For this example which uses a “double dish” isolator configuration, the model distributes half of the P-delta moment to the structure above and half of the P-delta moment to the foundation below the isolator units. ETABS permits explicit modeling of the P-delta moment, but certain computer programs may not. In such cases, the designer must separately calculate these moments and add them to other forces for the

12-35

FEMA P-751, NEHRP Recommended Provisions: Design Examples design of affected elements. The P-delta moments are quite significant, particularly at isolator units that resist large earthquake overturning loads along lines of lateral bracing. 12.5.3.2.3 Isolator unit uplift. Standard Section 17.2.4.7 permits local uplift of isolator units, provided the resulting deflections do not cause overstress of isolator units or other structural elements. Uplift of some isolator units is possible (for unreduced earthquake loads) due to the high seismic demand associated with the site. Accordingly, isolator units are modeled with gap elements that permit uplift when there is a net tension load on an isolator unit. 12.5.3.2.4 Bounding values of bilinear stiffness of isolator units. The design of elements of the seismic force-resisting system is usually based on a linear, elastic model of the superstructure. When such models are used, Standard Section 17.6.2.2.1 requires that the stiffness properties of nonlinear isolation system components be based on the maximum effective stiffness of the isolation system (since this assumption produces larger earthquake forces in the superstructure). Conversely, the Standard requires that calculation of isolation system displacements be based on the minimum effective stiffness of the isolation system (since this assumption produces larger isolation system displacements). The concept of bounding values, as discussed above, applies to all analysis methods. For the ELF procedure (and RSA), values of maximum effective stiffness, kDmax and kMmax, are used for calculating design forces and minimum values of effective stiffness, kDmin and kDmax, are used for calculating design displacements. Where (nonlinear) response history analysis is used, isolators are explicitly modeled as bilinear hysteretic elements with upper- or lower-bound stiffness curves, respectively. Upper-bound stiffness curves are used to verify the forces used for the design of the superstructure and lower-bound stiffness curves are used to verify design displacements of the isolation system. 12.5.4 Preliminary Design Based on the ELF Procedure 12.5.4.1 Design of the isolation system. Preliminary design of the isolation system begins with determination of isolation system properties (e.g., effective period and damping of the isolation system), which depend on the type and size of isolation bearings (e.g., friction pendulum or elastomeric bearings) and the type and size of supplementary dampers if such are also incorporated into the isolation system. The size of bearings is related to the amount of vertical load that must be supported and the maximum amount of lateral earthquake displacement that must be accommodated. Maximum earthquake displacement is a function of both the MCER ground motions at the building site and the effective period and damping of the isolation system. Thus, preliminary design tends to be an iterative process that involves selecting a bearing type and size that can adequately support vertical loads while accommodating maximum earthquake displacement. While some projects “fine tune” the isolation system by using bearings of different types and sizes (due to large variations in vertical load on bearings), this example uses a single bearing type and size for each of the 35 bearing locations. Vertical loads on bearings include load combinations representing both long-term gravity loads, factored dead and live load (1.2D + 1.6L) and short-term loads that include both gravity and seismic load effects. Seismic loads on bearings due to overturning are not known initially (they must be calculated using models of the superstructure). For the example EOC building, a trial size of bearings is based on a conservative estimate of gravity loads. The most heavily loaded column (Column C3, Guide Table 12.5-1) has a long-term load of approximately 600 kips and the rated capacity of bearings should be at least 600 kips. The size of bearings (the maximum lateral displacement capacity) depends on the type of system and products available, but informed choices regarding an appropriate value of effective period can be made

12-36

Chapter 12: Seismically Isolated Structures by constructing plots (based on ELF design formulas) such as those shown in Figure 12.3-3 for a steel SCBF superstructure. Consider, first, values of superstructure design shear, Vs. Figure 12.3-3 shows that values of superstructure design shear will be the same for all isolation systems that have an effective period, TD, of approximately 2.5 seconds or greater. Longer effective periods reduce lateral forces (and overturning loads), but superstructure strength will be governed by minimum base shear requirements. An effective period, TD, of approximately 2.5 seconds (and effective period, TM, of not more than approximately 3.0 seconds) would be appropriate for isolation system with elastomeric bearings that have inherent limits on rubber stiffness and strain capacity. Friction systems can have somewhat longer periods that would not reduce the level of force required for lateral design of the superstructure but would reduce overturning loads on columns, isolators and foundations. Longer effective periods reduce forces at the expense of increased displacement, which may be infeasible or could increase the cost of other structural elements (such as a moat wall), flexible utility connections and other nonstructural components that cross the isolation interface. The differing conditions and criteria of each project must be considered in selecting appropriate (optimal) properties for the isolation system. For this example, which incorporates friction pendulum bearings, the trial bearing size and properties (such as the curvature of the concave plates) are selected from products that are available from a specific manufacturer (EPS), which have a relatively long effective period, TM. A long effective period will minimize loads on the superstructure but could require MCER displacement capacity beyond practical limits. As shown in Figure 12.3-3, an effective period of 4 seconds (or greater) would require over 36 inches of displacement and the moat clearance would need to be at least 36 inches, which would be infeasible for most projects. The design shear and displacement plots in Figure 12.3-3 are constructed with values of effective damping that closely match friction pendulum bearing properties at the response amplitudes of interest; they are valid for this type and size of bearing but not necessarily other bearing types or sizes. Based on the preceding discussions, the double-concave friction pendulum bearing (FPT8844/12-12/8-6) shown in Figure 12.5-8 is selected for the EOC example. This bearing has concave plate radii, rP, of 88 inches (for both top and bottom concave plates), which produces an effective period in the range of 3.5 to 4.0 seconds. Articulated slider dimensions are shown in Figure 12.5-8. The inside diameter of the concave plates (44 in.) provides approximately 33 inches of displacement capacity before the articulated slider engages the boundary of the concave plates ( 44 inches [dish diameter] minus 12 inches [slider diameter] plus approximately 1 inch [due to slider articulation]).

12-37

FEMA P-751, NEHRP Recommended Provisions: Design Examples

47" 44" 12" 11" 8"

1.5"

Min.

12.8"

Bottom Concave Plate (R=88.0")

Articulated Slider

Seal

Top Concave Plate (R=88.0")

Seal

R=12"

Figure 12.5-8 Section view of the double-concave friction pendulum bearing (FPT8844/12-12/8-6) The friction pendulum bearing (FPT8844/12-12/8-6) has a rated vertical load capacity of 800 kips, which can adequately support long-term loads (1.2D + 1.6L) which are summarized in Guide Table 12.5-1 (the maximum long-term load is approximately 600 kips). The rated capacity of the bearing should also exceed short-term isolation system design loads (1.37D + 0.5L + QDE), the values of which are calculated later in this section (see Guide Table 12.5-5). The friction pendulum bearing does not resist uplift and may not function properly should uplift occur. For this example, uplift is not permitted for short-term isolation system design loads (0.73D - QDE), which is shown to be the case later in this section (see Guide Table 12.5-6). Standard Section 17.2.4.6 requires bearings (and other elements of the isolation system) to remain stable for the total maximum displacement of the isolation system and short-term MCER loads (1.45D + 1.0L + QMCE and 0.80D - QMCE). Bearing stability must be verified by prototype testing for both maximum downward loads (1.45D + 1.0L + QMCE), which are calculated later in this section (see Guide Table 12.5-7) and maximum uplift displacements (due to 0.80D - QMCE), which are also calculated later in this section (See Guide Table 12.5-8). In addition to basic configuration and curvature, key properties of the friction pendulum bearings include the amount of dynamic (sliding) friction and the level of static (breakaway) friction of the sliding surfaces. For double-concave bearings, sliding surfaces include (1) the surface between the top of the articulated slider and the top concave plate and (2) the surface between the bottom of the articulated slider and the bottom concave plate (and to a lesser degree, friction surfaces inside the articulated slider). For this example, bearing friction is taken to be nominally the same value for the top and bottom sliding surfaces. The sliding friction (based on the friction coefficient) is influenced by several factors, including the following: §

12-38

Vertical load (pressure on sliding surfaces): In general, the greater the vertical-load pressure, the lower the value of the friction coefficient.

Chapter 12: Seismically Isolated Structures §

Rate of lateral load (bearing velocity): In general, the greater the velocity, the higher the value of the friction coefficient (although the friction coefficient tends to be fairly constant at moderate and high earthquake velocities).

§

Bearing temperature (surface temperature of sliding surfaces): In general, the hotter the bearing, the lower the value of the friction coefficient. Bearings get hot due to repeated cycles of earthquake load. Bearing temperature is a function of the duration of earthquake shaking (the number of cycles of dynamic load) and the friction force on the sliding surface. The greater the number of cycles of dynamic load, the hotter the bearing. The greater the friction force (due to larger vertical load or friction coefficient), the hotter the bearing.

Other factors influencing friction and sliding bearing performance include manufacturing tolerances (not all bearings can be made exactly the same) and the effects of aging and possible contamination of sliding surfaces (which should be minimal for bearings with protective seals and not exposed to the environment). For the preliminary design of the isolation system (and subsequent analyses of the isolation system), nominal and bounding values of the siding friction coefficient are used to construct hysteresis loops that define effective stiffness, effective period and the effective damping of the isolations system (i.e., all 35 bearings acting together) as a function of isolation system displacement. For this example, the nominal value of the friction coefficient of top and bottom concave plates is µp,nom = 0.06, with a lower-bound value of µp,min = 0.04 and an upper-bound value of µp,max = 0.08 and the friction coefficient of the articulated slider (internal surfaces) is µs = 0.02. Static breakaway friction is assumed to be less than or equal to µp,max. A relatively large variation in the friction coefficient, a factor of 2, is used to bound all sources of friction coefficient variability, described above, including possible change in properties over the life of the bearing. As discussed later in Guide Section 12.5.6, isolation system properties based on assumed values must be verified by testing of isolator prototypes. The effective stiffness (normalized by building weight), keff,D/W, may be calculated approximately as a function of bearing displacement, D, in inches for the double-concave friction pendulum bearing, as follows:

⎛ ⎞ 1 keff , D / W = µ P + ⎜ ⎟ D ⎝ 2 rP − hs + 2rs − hc ⎠ where the term, µP, represents nominal, upper-bound or lower-bound value of the sliding friction coefficient, rp is the radius of the concave plates, hs is the height of the articulated slider, rs is the radius of the articulated slider and hc is the height of the core of the articulated slider. Similarly, the bi-linear concepts shown in Guide Figure 12.1-2 are used to calculate effective period, Teff,D and effective damping, βeff,D, as a function of bearing displacement, D, in inches for the double concave friction pendulum bearing, as follows:

Teff , D = 0.32

D keff , D / W

⎛ D Fy − Dy keff , D / W ⎜ D keff , D / W ⎝

βeff , D = 0.637 ⎜

⎞ ⎟ ⎟ ⎠

12-39

FEMA P-751, NEHRP Recommended Provisions: Design Examples where:

Dy = 2 ( µ P − µ S ) d S

⎛ 1 Ay = µ P + ⎜ ⎝ 2 rP − hs + 2rs − hc

⎞ ⎟ Dy ⎠

Figures 12.5-9, 12.5-10 and 12.5-11 are plots of effective stiffness, effective period and effective damping, respectively, based on the above formulas and the geometric and sliding friction properties of the double-concave friction pendulum bearing. These curves are useful design aids that are referred to in subsequent sections when determining amplitude-dependent properties of the isolation system.

Normalized bearing force, F/W

0.25

0.20

0.15

0.10

0.05

0.00 0

Bearing displacement (in.)

Figure 12.5-9 Effective stiffness as a function of the displacement of the double-concave friction pendulum bearing (FPT8844/12-12/8-6)

12-40

Chapter 12: Seismically Isolated Structures

4.0

Effective period (s)

3.5

3.0

2.5

2.0

1.5

1.0 0

Bearing displacement (in.)

Figure 12.5-10 Effective period as a function of the displacement of the double-concave friction pendulum bearing (FPT8844/12-12/8-6)

40% 35%

Effective damping

30% 25% 20% 15% 10% 5% 0% 0

10 15 20 Bearing displacement (inches)

Figure 12.5-11 Effective damping as a function of the displacement of the double-concave friction pendulum bearing (FPT8844/12-12/8-6)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

12.5.4.2 Calculation of design values. 12.5.4.2.1 Design displacements. Preliminary design begins with the engineer’s selection of the effective period (and effective damping) of the isolated structure and the calculation of the design displacements. This process is necessarily iterative, since the effective period (and effective damping) is amplitude dependent. In this example, the effective DE period of the EOC facility (i.e., effective period at design earthquake displacement) is estimated using lower-bound properties of Guide Figure 12.5-10 to be approximately TD = 3.5 seconds at a design displacement of DD = 16.0 inches, calculated using Standard Equation 17.5-1 as follows:

0.7 ( 3.5 ) ⎛ g ⎞ S T = 16.0 in. DD = ⎜ 2 ⎟ D1 D = ( 9.8 ) 1.5 ⎝ 4π ⎠ BD The 1.5 value of the damping coefficient, BD, is given in Standard Table 17.5-1 assuming 20 percent effective damping at 16.0 inches of isolation system displacement, which is consistent with lower-bound effective damping shown in Guide Figure 12.5-11. The same approach is used to estimate the MCER period, in this case TM = 3.9 seconds at a maximum displacement of approximately DM = 30 inches, calculated using Standard Equation 17.5-3 as follows:

1.05 ( 3.9 ) ⎛ g ⎞ S T = 30.9 in. DM = ⎜ 2 ⎟ M 1 M = ( 9.8 ) 1.3 ⎝ 4π ⎠ BM The 1.3 value of the damping coefficient, BM, is given in Standard Table 17.5-1 assuming 13 percent effective damping at approximately 30 inches of isolation system displacement, which is consistent with lower-bound effective damping shown in Guide Figure 12.5-11. It may be noted from this figure that the effective damping of the isolation system and related benefits of reduced response decrease significantly with response amplitude at large displacements. The total displacement of specific isolator units (considering the effects of torsion) is calculated based on the plan dimensions of the building, the total torsion (due to actual, plus accidental eccentricity) and the distance from the center of resistance of the building to the isolator unit of interest. Using Standard Equations 17.5-5 and 17.5-6, the total design displacement, DTD and the total maximum displacement, DTM, of isolator units located on Column Lines 1 and 7 are calculated for the critical (transverse) direction of earthquake load to be approximately 20 percent greater than DD and DM, respectively. Standard Equations 17.5-5 and 17.5-6 assume that mass is distributed in plan in proportion to isolation system stiffness, offset by 5 percent, providing no special resistance to rotation of the building above the isolation system. These assumptions are reasonable, in general, but are not appropriate for isolation systems that have inherent resistance to torsion. For example, systems which have less load on bearings at the perimeter (as is the case of most buildings) and/or more stiffness at the perimeter (due to stiffer bearings) would not experience this amount of additional response due to torsion. For sliding systems, friction forces are roughly proportional to applied load and the center of resistance would tend to move with any change in the actual center of mass, such as that associated with accidental mass eccentricity. Accordingly, additional displacement due to accidental mass eccentricity is taken as 10 percent, the minimum value of additional displacement permitted by Standard Section 17.5.3.3. Applying a 10 percent increase to the design and maximum displacement values produces total design and maximum displacement (i.e., at building corners) values of DTD = 17.6 inches (1.1 × 16.0) and

12-42

Chapter 12: Seismically Isolated Structures DTM = 34.0 inches (1.1 × 30.9), respectively, the latter of which is about the displacement capacity of the bearing. 12.5.4.2.2 Minimum and maximum effective stiffness. Standard Equation 17.5-2 expresses the effective period at the design displacement in terms of building weight (dead load) and the minimum effective stiffness of the isolation system, kDmin. Rearranging terms and solving for minimum effective stiffness results in the following:

⎛ 4π 2 ⎞ W ⎛ 1 ⎞ 9,100 k D min = ⎜ = 75.8 kips/in. ⎟ 2 = ⎜ ⎟ 2 ⎝ g ⎠ TD ⎝ 9.8 ⎠ 3.5 Similarly, the minimum effective stiffness at the maximum displacement is calculated as follows:

⎛ 4π 2 ⎞ W ⎛ 1 ⎞ 9,100 kM min = ⎜ = 61.1 kips/in. ⎟ 2 = ⎜ ⎟ 2 ⎝ g ⎠ TD ⎝ 9.8 ⎠ 3.9 Estimates of the maximum effective stiffness at the design displacement, DD = 16.0 inches and maximum displacement, DM = 30.9 inches, can be made using Guide Figure 12.5-9, which shows the upper-bound effective stiffness to be approximately 20 percent greater than the lower-bound effective stiffness at 16.0 inches and the upper-bound effective stiffness to be approximately 15 percent greater than lowerbound effective stiffness at 30.9 inches of displacement:

kD max = 1.2(75.8) = 91.0 kips / in. kM max = 1.15(61.1) = 70.3 kips / in. As noted earlier, the range of effective stiffness illustrated in Guide Figure 12.5-9 represents the isolation system as a whole rather than the range of effective stiffness associated with individual bearings and is intended to address a number of factors that influence potential variation in bearing properties, including the effects of aging and contamination, etc. A report, Property Modification Factors for Seismic Isolation Bearings (Constantinou, 1999), provides guidance for establishing a range of effective stiffness (and effective damping) properties that captures various sources of the variation over the design life of the isolator units. The range of effective stiffness of the isolation system has a corresponding range of effective periods (with different levels of spectral demand). As discussed in Guide Section 12.3, the longest effective period (corresponding to the minimum effective stiffness) of the range is used to define isolation system design displacement and the shortest effective period (corresponding to the maximum effective stiffness) of the range is used to define the design forces on the superstructure. 12.5.4.2.3 Lateral design forces. The lateral force required for the design of the isolation system, foundation and other structural elements below the isolation system is given by Standard Equation 17.5-7: Vb = kDmax DD = 91.0(16.0) = 1,456 kips The lateral force required for checking stability and ultimate capacity of elements of the isolation system may be calculated as follows: VMCE = kMmax DM = 70.3(30.9) = 2,172 kips

12-43

FEMA P-751, NEHRP Recommended Provisions: Design Examples The (unreduced) base shear of the design earthquake is approximately 16 percent of the weight of the EOC and the (unreduced) base shear of the MCE is approximately 24 percent of the weight. Subject to the limits of Standard Section 17.5.4.3, the design earthquake base shear, Vs, is reduced by the RI factor in accordance with Standard Equation 17.5-8:

Vs =

k D max DD 91.0(16.0) = = 728 kips RI 2.0

This force is only approximately 8 percent of the dead load weight of the EOC, which is less than the limits of Item 3 of Section 17.5.4.3 that require a minimum base shear of at least 1.5 times the “breakaway friction level of a sliding system.” In this example, breakaway friction is assumed to be less than or equal to the maximum value of sliding friction, µP,max = 0.08, such that the minimum design earthquake base shear, Vs, is calculated as follows:

Vs = 1.5 µP,max W = 1.5(0.08)9,100 = 1,092 kips which is substantially less than the lateral force that would be required for the design of a conventional, fixed-base, EOC building of the same size and height, seismic force-resisting system and site seismic conditions. Story shear forces on the superstructure are distributed vertically over the height of the structure in accordance with Standard Equation 17.5-9, as shown in Table 12.5-4. Table 12.5-4

Vertical Distribution of Reduced Design Earthquake Forces (Vs = 1,092 kips)

Floor level, x (Story)

Floor weight, wx (kips)

PH Roof

Cumulative Height above weight isolation (kips) system, hx (ft)

794 794 2,251

(Third) Third Floor Second Floor

1,947

First Floor (Isolation)

196

25%

432

628

21%

267

895

18%

158

1,053

15%

1,092

12%

30 4,992

1,922

(First)

196 42

3,045

(Second)

Cum. force divided by cumulative weight

(Penthouse) Roof

Story force, Cumulative Fx, (kips) story force (Standard (kips) Eq. 17.5-9)

18 6,914

2,186

4 9,100

1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN.

Standard Equation 17.5-9 distributes lateral seismic design forces (Vs = 1.092 kips) over the height of the building’s superstructure in an inverted triangular pattern, effectively doubling shear force at the roof level, as indicated by values of cumulative story force divided by weight above shown in Table 12.5-4. Because the superstructure is much stiffer laterally than the isolation system, it tends to move as a rigid 12-44

Chapter 12: Seismically Isolated Structures body, rather than in the first mode, with a pattern of lateral seismic forces that is typically more uniformly distributed over the height of the building. The use of a triangular load pattern for design is intended to account for higher-mode response that may be excited by superstructure flexibility and isolation system nonlinearity. Standard Equation 17.5-9 is also used to distribute forces over the height of the building for unreduced DE and MCER forces, as summarized in Table 12.5-5. Unreduced forces are required for design of isolator units. Table 12.5-5

Vertical Distribution of Unreduced Design Earthquake and MCER Forces Design Earthquake Forces

Story

Story force, Fx (kips) (Standard Eq. 17.5-9)

Cum. force Cumulative divided by story force cumulative (kips) weight

MCER Forces Story force, Fx (kips) (Standard Eq. 17.5-9)

Cum. force Cumulative divided by story force cumulative (kips) weight

Penthouse

261

33%

389

49%

Third

576

837

27%

859

1,248

41%

Second

356

1,192

24%

530

1,778

35%

First

211

1,403

20%

315

2,093

30%

Isolation

1,456

16%

2,172

24%

12.5.4.2.4 Design earthquake forces for isolator units. Tables 12.5-6 and 12.5-7 show the maximum and minimum downward forces for design of the isolator units. These forces result from the simultaneous application of gravity loads and unreduced design earthquake story forces, summarized in Table 12.5-5, to the ETABS model of the EOC. (See Guide Sec. 12.5.2.5 for the design load combinations.) As described in Guide Section 12.5.2.5, loads are applied separately in the two horizontal directions and results for both directions are reported in these tables (i.e., x direction / y direction). All values in Table 12.5-7 are positive, indicating that uplift does not occur at any bearing location for unreduced DE loads. Table 12.5-6

Maximum Downward Force (kips) for Isolator Design (1.37D + 0.5L + QDE)* Maximum downward force (kips)

Line

225 / 225

383 / 498

334 / 356

336 / 359

480 / 389

590 / 668

674 / 700

633 / 812

349 / 328

641 / 709

632 / 631

586 / 570

1.0 kip = 4.45 kN

* Forces at Column Lines 5, 6 and 7 (not shown) are similar to those at Column Lines 3, 2 and 1, respectively; loads at Column Lines D and E (not shown) are similar to those at Column Lines B and A, respectively.

12-45

FEMA P-751, NEHRP Recommended Provisions: Design Examples Minimum Downward Force (kips) for Isolator Design (0.73D - QDE)*

Table 12.5-7

Maximum downward force (kips) Line

84 / 85

160 / 65

149 / 128

150 / 135

67 / 146

144 / 58

197 / 199

224 / 56

125 / 146

235 / 160

269 / 265

234 / 223

1.0 kip = 4.45 kN

12.5.4.2.5 MCER forces and displacements for isolator units. Simultaneous application of gravity loads and unreduced MCER story forces, as summarized in Table 12.5-5, to the ETABS model of the EOC results in the maximum downward forces on isolator units shown in Guide Table 12.5-8 and the maximum uplift displacements shown in Table 12.5-9. The load orientations and MCER load combinations are described in Guide Section 12.5.2.5. Table 12.5-9 shows that uplift did not occur at most bearing locations and was very small at the few locations where MCER overturning forces exceeded factored gravity loads. Table 12.5-8

Maximum (MCER) Downward Force (kips) on Isolator Units (1.45D + 1.0L + QMCE)* Maximum downward force (kips)

Line

269 / 270

467 / 644

404 / 439

406 / 445

616 / 476

754 / 869

845 / 883

793 / 1,063

430 / 395

804 / 905

785 / 783

732 / 705

1.0 kip = 4.45 kN. *Forces at Column Lines 5, 6 and 7 (not shown) are similar to those at Column Lines 3, 2 and 1, respectively; loads at Column Lines D and E (not shown) are similar to those at Column Lines B and A, respectively.

Table 12.5-9

Maximum (MCER) Uplift Displacement (in.) of Isolator Units (0.8D - QMCE)* Maximum uplift displacement (in.)

Line

No Uplift

0 / 0.007

No Uplift

0 / 0.011

No Uplift

1.0 in. = 25.4 mm. * Displacements at Column Lines 5, 6 and 7 (not shown) are similar to those at Column Lines 3, 2 and 1, respectively; displacements at Column Lines D and E (not shown) are similar to those at Column Lines B and A, respectively.

12-46

Chapter 12: Seismically Isolated Structures

12.5.4.2.6 Limits on dynamic analysis. The displacements and forces determined by the ELF procedure provide a basis for expeditious assessment of size and capacity of isolator units and the required strength of the superstructure. The results of the ELF procedure also establish limits on design parameters when dynamic analysis is used as the basis for design. Specifically, the total design displacement, DTD and the total maximum displacement of the isolation system, DTM, determined by dynamic analysis cannot be less than 90 percent and 80 percent, respectively, of the corresponding ELF procedure values: DTD, dynamic ≥ 0.9DTD, ELF = 0.9(17.6) = 15.8 in. DTM, dynamic ≥ 0.8DTM, ELF = 0.8(34.0) = 27.2 in. Similarly, the lateral force, Vb, required for design of the isolation system, foundation and all structural elements below the isolation system and the lateral force, Vs, required for design of the superstructure determined by dynamic analysis are limited to 90 percent and 80 percent, respectively, of the ELF procedure values: Vb, dynamic ≥ 0.9Vb, ELF = 0.9(1,456) = 1,310 kips (= 0.144W) Vs, dynamic ≥ 0.8Vs, ELF = 0.8(1,092) = 874 kips (= 0.096W) As an exception to the above, lateral forces less than 80 percent of the ELF results are permitted for design of superstructures of regular configuration, if justified by response history analysis (which is seldom, if ever, the case). However, Vs,dynamic is also subject to the limits of Section 17.5.4.3 for response history analysis (and also, by inference, for response spectrum analysis) which govern the value of Vs,ELF in this example and thus govern the minimum value of design base shear (regardless of the analysis method used for design): Vs, dynamic ≥ Vs, ELF = 1.0(1,092) = 1,092 kips (= 0.12W) 12.5.4.3 Preliminary design of the superstructure. The lateral forces, developed in the previous section, in combination with gravity loads, provide a basis for preliminary design of the superstructure, using methods similar to those used for a conventional building. In this example, selection of member sizes was made based on the results of ETABS model calculations. Detailed descriptions of the design calculations are omitted, since the focus of this section is on design aspects unique to isolated structures (i.e., the design of the isolation system, which is described in the next section). Figures 12.5-12 and 12.5-13 are elevation views at Column Lines 2 and B, respectively. Figure 12.5-14 is a plan view of the building that shows the framing at the first floor level.

12-47

FEMA P-751, NEHRP Recommended Provisions: Design Examples

S10

x10 x5/ 8

W24x146

W18x76

/8 0x5 0x1

HS W24x146

W18x76

W18x76 HS S10 x10 x5/ 8 W24x146

W14x68

W14x132

Roof W14x68

HS W18x76 HS S10 x10 x5/ 8

W18x76

Third Floor W14x68

/8 0x5 0x1

Penthouse Roof

Second Floor

W14x68

W14x132

W18x76

W14x132

/8 0x5 0x1

W18x76

W14x132

W24x146

W18x76

W14x132

/8 0x5 0x1

W18x76

W18x76 HS S10 x10 x5/ 8

W14x132

W14x68

W18x76

W14x68

W18x76

W14x132 W14x132

W14x68

W18x76

W14x68

W18x76

W14x132

W14x132 W14x132

First Floor Base

a) Lines 2 and 6

W24x146

W18x76

W24x146

W14x68

Third Floor W14x68

HS W18x76 HS S12 x12 x1/ 2

W18x76

Roof

Second Floor

W14x68

/2 2x1 2x1

W14x68

W18x76

W14x132

W18x76

W14x68

W14x132

W18x76 HS S12 x12 x1/ 2

Penthouse Roof

W14x132

HS W18x76 HS S12 x12 x1/ 2 W18x76 2 / x1 x12 S12 HS W24x146

W14x132

/2 2x1 2x1

W14x132

W14x68

W18x76

W14x132 W14x132

W24x146

W14x132

W18x76

W14x132 W14x132

W18x76

W14x132

W14x68

W18x76

W14x68

W18x76

W14x132

First Floor Base

b) Line 4

Figure 12.5-12 Elevation of framing on Column Lines 2 ,4 and 6

12-48

Chapter 12: Seismically Isolated Structures

0x S1 HS W24x146

W24x146

W18x76 HS S1 0x 10 x5 /8 W24x146

W18x76

/8 x5

0x S1 HS W24x146

W18x76 W14x68

W18x76 HS S1 0x 10 x5

Penthouse Roof

W18x76

W14x68

W18x76

W14x68

W18x76 HS S1 0x 10 x5

W14x68

W18x76 W14x68 W14x68

W18x76

W18x76 HS S1 0x 10 x5 W24x146

W14x132 W14x68

/8 x5

W18x76 5/8 0x x1 0 S1 HS W18x76 HS S1 0x 10 x5 /8 W24x146

W18x76 /8 x5 10 0x 1 S HS W18x76 HS S1 0x 10 x5 /8 W18x76

W14x132 W14x132

W18x76

W18x76 HS S1 0x 10 x5 /8

W14x132 W14x132 W14x132

W14x68

W18x76

W14x132 W14x132 W14x68

W14x132 W14x68

W14x68

W18x76

W14x132 W14x132 W14x132 W14x132 W14x132

W14x68

W18x76

W14x68

W14x132 W14x132

Roof

Third Floor

Second Floor

First Floor Base

Figure 12.5-13 Elevation of framing on Column Lines B and D

W24 x146 W24 x146 W24 x146 W24 x146

W16x26

W24 x146 W16x26

W24 x146

W16x26

W24 x146

W16x26

W24 x146

W24 x146 W16x26

W16x26

W24 x146

W16x26

W24 x146

W16x26

W24 x146 W24 x146

W16x26

W24 x146 W24 x146 W24 x146 W24 x146

W24 x146

W24 x146 W16x26

W24 x146

W16x26

W24 x146

W16x26

W24 x146

W24 x146 W16x26

W16x26

W24 x146

W16x26

W16x26 W16x26

W24 x146

W16x26

W24 x146 W24 x146

W16x26

W24 x146 W24 x146 W24 x146 W24 x146

W16x26

W24 x146

W24 x146 W16x26

W24 x146

W16x26

W24 x146

W16x26

W16x26 W16x26

W16x26

W24 x146

W16x26

W24 x146 W24 x146 W24 x146

W16x26

W24 x146

W16x26

W24 x146

W16x26

W24 x146

W16x26

W24 x146

W16x26

W24 x146

Figure 12.5-14 First floor framing plan

12-49

FEMA P-751, NEHRP Recommended Provisions: Design Examples

As shown in the elevations (Figures 12.5-12 and 12.5-13), relatively large (HSS 10×10×5/8) tubes are conservatively used throughout the structure for diagonal bracing. A quick check of these braces indicates that stresses will be at or below yield for earthquake loads. For example, the four braces at the third story, on Lines 2, 4 and 6 (critical floor and direction of bracing) resist a reduced design earthquake force of approximately 175 kips (628 kips / 4 braces × cos[25°]), on average. The corresponding average brace stress is only approximately 8 ksi (for reduced design earthquake forces) and only approximately 11 ksi and 16 ksi for unreduced DE and MCER earthquake forces, respectively, indicating that the structure is expected to remain elastic even for MCER demand. As shown in Figure 12.5-14, the first-floor framing has relatively heavy, W24×146 girders along lines of bracing (Lines B, D, 2, 4 and 6). These girders resist P-delta moments as well as other forces. A quick check of these girders indicates that only limited yielding is likely, even for the MCER loads (and approximately 3 feet of MCER displacement). Girders on Line 2 that frame into the column at Line B (critical columns and direction of framing) resist a P-delta moment due to the MCER force of approximately 800 kip-ft (1,063 kips / 2 girders × 3 feet / 2). Additional moment in these girders due to MCER shear force in isolators is approximately 500 kip-ft (1,063 kips × 0.24 × 4 feet / 2 girders). Thus, total moment is approximately 1,300 kips due to gravity and MCER loads (and 3 feet of MCER displacement), which is conservatively less than the plastic capacity of these girders (i.e., φbMp = 1,570 kip-ft). 12.5.4.4 Isolator connection detail. The isolation system has a similar detail at each column for the example EOC, as shown in Figure 12.5-15. The column has a large, stiffened base plate that bears directly on the top of the isolator unit. The column base plate is circular, with a diameter comparable to that of the top plate of the isolator unit. Heavy, first-floor girders frame into and are moment connected to the columns (moment connections are required at this floor only). The columns and base plates are strengthened by plates that run in both horizontal directions, from the bottom flange of the girder to the base. Girders are stiffened above the seat plates and at temporary jacking locations. The top plate of the isolator unit is bolted to the column base plate and the bottom plate of the isolator unit is bolted to the foundation. The foundation (and grout) directly below the isolator must have sufficient strength to support concentrated bearing loads from the concave plate and articulated slider above (for all possible lateral displacements of the isolator). Similarly, the column base plate and structure just above the isolator must have sufficient strength to transfer concentrated bearing loads to the concave plate and articulated slider below. The foundation connection accommodates the removal and replacement of isolator units, as required by Standard Section 17.2.4.8. Anchor bolts pass through holes in this plate and connect to threaded couplers that are attached to deeply embedded rods. Figure 12.5-15 illustrates connection details typical of isolated structures; other details could be developed that work just as well and certainly details would be different for structures that have a different configuration or material.

12-50

Chapter 12: Seismically Isolated Structures

W12x120

W24x146

unit

isolator

bolt (milled) heavy plate nut bolt or threaded rod into coupler grout

Figure 12.5-15 Typical detail of the isolation system at columns (for clarity, some elements are not shown) With the exception of the portion of the column above the first-floor slab, each element shown in Figure 12.5-11 is an integral part of the isolation system (or foundation) and is designed for the gravity and unreduced design earthquake loads. In particular, the first-floor girder, the connection of the girder to the column and the connection of the column to the base plate are designed for gravity loads and forces caused by horizontal shear and P-delta effects due to the unreduced design earthquake loads (as shown earlier in Guide Figure 12.4-1). 12.5.5 Design Verification Using Nonlinear Response History Analysis The Standard requires dynamic analysis for design of isolated buildings like the example EOC that are located at sites with 1-second spectral acceleration, S1, greater than or equal to 0.6g (see Guide Table 12.2-1). The Standard also requires that the dynamic analysis requirement be satisfied using response history analysis (RHA) for isolated buildings like the example EOC that have an isolation systems that does not fully comply with the criteria of Item 7 of Standard Section 17.4.1. In practice it is common to use the RHA procedure to verify the design of structures with sliding or high-damping isolation systems. This example uses RHA to determine final design displacements of the isolation system and to confirm that ELF-based forces used for preliminary design of the superstructure and isolation system are valid. While results of RHA could be used to refine the preliminary design of the superstructure (e.g., reduce size of lateral bracing, etc.), this example uses RHA primarily for “design verification” of key global

12-51

FEMA P-751, NEHRP Recommended Provisions: Design Examples response parameters (i.e., confirmation of ELF-based story shear force). If the RHA procedure were used as the primary basis for superstructure design, then response results would be required for design of individual elements, rather than for checking a limited number of global response parameters. 12.5.5.1 Ground motion records. The RHA procedure evaluates DE and MCER response of the example EOC using the set of seven ground motion records previously described in Guide Section 12.5.4.4. Seven records are used for RHA so that the average value of the response parameter of interest can be used for design (or design verification). Each record has two horizontal components, one of which is identified as the “larger” component in terms of spectral acceleration at the effective period of the isolation system (i.e., 3.9 seconds). The ground motion records are grouped (oriented) in terms of the larger component (Step 2) and are applied to the analytical model (Step 4) in accordance with procedure recommended in Guide Sec. 12.4.4. Guide Table 12.5-2 summarizes key source and site properties used as the basis for record selection and Guide Table 12.5-3 summarizes factors used to scale each record to match design earthquake and MCER response spectra, respectively. Guide Figure 12.5-7 compares the average spectra of the seven scaled records (for the SRSS combination of components and the larger component, respectively) with the MCER response spectrum demonstrating compliance with scaling requirements of Standard Section 17.3.2 and the recommendations of Guide Section 12.4.4 (Step 3). 12.5.5.2 ETABS models. The ETABS computer program is used for RHA of analytical models of the isolated structure, as previously discussed in Guide Section 12.5.3. Two basic ETABS models are developed for RHA analysis, both having the same superstructure but representing lower-bound and upper-bound stiffness properties of isolators (FP bearings), respectively. Elements of the superstructure and isolation system (other than the isolators) are modeled as linear elastic elements based on member sizes developed by preliminary design (Guide Section 12.5.4.3). Elements are not expected to yield significantly, even for MCER response and the assumption of linear elastic behavior is quite reasonable. For RHA, ETABS internally calculates modal damping based on the hysteretic properties of the nonlinear elements (isolators) and an additional amount of user-specified modal damping. For the EOC example, additional damping is limited to 2 percent of critical for all modes, since the steel superstructure remains essentially linear elastic. Two types of isolator nonlinearity are modeled explicitly. First, gap elements, just above each isolator, model potential uplift of the isolation system. The gap elements transmit downward (compressive) forces but release and permit upward displacement of the structure, if upward (tension) forces occur. The purpose of the gap elements is to check for possible uplift due to overturning loads exceeding factored gravity loads and to calculate the maximum amount of vertical displacement, should uplift occur. The second type of isolator nonlinearity is that associated with horizontal force-deflection properties of isolators. For this example, nonlinear force-deflection properties of the isolator units are modeled using the ETABS nonlinear “Isolator2” element. This element was originally developed for FP bearing with a single concave plate and requires the user to input the initial stiffness (effective stiffness before the articulated slider displaces on the concave plate), the friction coefficient properties and the radius of the concave plate. For the double concave FP bearing of this example, an “effective” radius, reff, is used to account for simultaneous sliding of the articulated slider on the two concave plates. The effective radius, reff, of the two dishes is 185 inches, based on the radius of each concave plate (rp = 88 inches), the height of the articulated slider (hs = 9 inches), the radius of the articulated slider (rs = 12 inches) and the height of the internal core (hc = 6 inches): 12-52

Chapter 12: Seismically Isolated Structures

reff = 2rp - hs + 2rs - hc = 2 × 88 in. - 9 in. + 2 × 12 in. - 6 in. = 185 inches The Isolator2 element simulates bilinear, hysteretic behavior commonly used to model the nonlinear properties of friction-pendulum bearings, although the double-concave configuration generally requires more sophisticated representation of bearing properties. For example, very complex models of this bearing have been developed and implemented in 3D-BASIS (Constantinou, et al.) and SAP2000 (CSI). Use of these more sophisticated models would be required for double-concave friction pendulum bearings configured to have different friction coefficients for the surfaces of top and bottom concave plates, respectively, or for projects which explicitly model the stiffening effects of articulated slider engagement with the boundary of concave plates. Such modeling is not required for the EOC example and the bilinear curve of the Isolator2 element (available in ETABS) is used for design verification. The Isolator2 element requires the user to input three sliding friction parameters: (1) the “fast” value of the sliding coefficient of friction, (2) the “slow” value of the sliding coefficient of friction and (3)a “rate” parameter that essentially governs transition between slow and fast properties (e.g., when the bearing reaches peak displacement and reverses direction of travel). For the Isolator2 element, “fast” represents velocities in excess of approximately 1 to 2 inches per second. Typically, during strong ground motions, such as those of this example, sliding velocities are well in excess of 2 inches per second and isolation system response is dominated by “fast” sliding friction properties. Lower-bound and upper-bound values of sliding friction coefficient are used to represent lower-bound and upper-bound values FP bearing of “stiffness,” as previously discussed in Guide Section 12.5.4.1. Lower-bound and upper-bound values of sliding friction coefficient, 4 percent and 8 percent respectively, are used to model “fast” sliding friction properties of the FP bearings. One-half of these values, 2 percent and 4 percent respectively, are used to model “slow” sliding friction properties. These values are consistent with observed behavior (“slow” properties are approximately one-half of “fast” properties) and have little influence on peak dynamic response. Figure 12.5-16 illustrates hysteresis loops based on upper-bound and lower-bound stiffness properties of FP bearings, normalized by vertical load (P). The slope of these loops is proportional to the inverse of the effective radius (185 inches) and the thickness (vertical height) of these loops is twice the value of the sliding friction coefficient. The upper-bound loop, plotted to DD = 16 inches, is based on 8 percent sliding friction and the lower-bound loop, plotted to DM = 30.9 inches, is based on 4 percent sliding friction. The hysteresis loops shown in Figure 12.5-16 could be modeled using appropriate values of bilinear element. While the bearings of this example all have identical properties, stiffness varies as a function of vertical load (since friction force is a function of vertical load). Thus, values of the initial stiffness (a required parameter of the Isolator2 element of ETABS) are a function of vertical load. While technically different for each bearing, the same value of initial stiffness is assumed for bearings that have similar amounts of vertical load. Table 12.5-10 provides values of initial FP bearing stiffness for three bearing load groups (regions): Region A: bearings at interior column locations with heavier loads (15 locations), Region B: bearings at perimeter column locations, except corners (16 locations) and Region C: bearings at corner columns with lighter loads (4 locations). Similarly, Table 12.5-11 shows values of effective isolator stiffness, corresponding to 16 inches of peak design earthquake and 30.9 inches of peak MCER response, respectively. Values of effective stiffness shown in Table 12.5-11 are not required for response history analysis but would be required (along with corresponding values of the damping coefficient) for response spectrum analysis using ETABS.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Normalzed bearing force, F/P

0.3

0.2

Maximum effective stiffness

0.1

Minimum effective stiffness

-0.1

-0.2 -DM

-0.3 -40

-D D

-30

-20

DD -10

Bearing displacement (in.)

Figure 12.5-16 Example hysteresis loops used to model EOC bearings (1.0 in. = 25.4 mm) Table 12.5-10 Summary of Initial Stiffness Properties of FP Bearings Used in ETABS Response History Analysis of the Example EOC Region A B C All

12-54

Column Location Interior Perimeter Corner All

Bearing Load Group Number of Average Load Bearings (kips) 15 325 16 230 4 135 35 260

Bearing Initial Stiffness (kips/in.) Lower-Bound Upper-Bound (4% Friction) (8% Friction) 13.5 9.0 9.6 6.4 5.6 3.8 10.8 7.2

Chapter 12: Seismically Isolated Structures Table 12.5-11 Summary of Effective Stiffness Properties of FP Bearings of the Example EOC Building (Appropriate for Response Spectrum Analysis) Bearing Load Group Region A B C All

Bearing Effective Stiffness (kips/in.) Design Earthquake Response MCER Response (at DD = 16.0 in.) (at DM = 30.9 in.) Lower-Bound Upper-Bound Lower-Bound Upper-Bound (4% Friction) (8% Friction) (4% Friction) (8% Friction) 2.57 3.38 2.18 2.60 1.82 2.39 1.54 1.84 1.07 1.40 0.90 1.08 2.06 2.71 1.74 2.08

V/W

For isolators with known properties, parameters defining bilinear stiffness may be based on test data provided by the manufacturer. For example, Figure 12.5-17 compares hysteresis loops based on upperbound and lower-bound stiffness properties (normalized by weight) with actual hysteresis loops of cyclic load tests of a prototype of the double-concave friction pendulum bearing. In this case, design loops are constructed for the same peak displacement (approximately 30 inches) as that of the prototype test. Test results indicate an average effective stiffness (normalized by vertical load) of keff/P = 0.0074 kip/in./kip at 30 inches of peak displacement that is well bounded by maximum and minimum values of effective stiffness kMmin = 0.0067 kip/in./kip (1.74 kips/in. / 260 kips) to kMmax = 0.0080 kip/in./kip (2.08 kips/in. / 260 kips), respectively. Test results also indicate an average effective damping of approximately 18 percent for the three cycles of prototype testing, which supports the use of an effective damping value, βM = 15 percent, for calculating MCER response (ELF procedure). 0.3

0.2

0.1

0 -40

-30

-20

-10

30 40 Horizontal displacement (in.)

-0.1

-0.2

-0.3

Figure 12.5-17 Comparison of modeled and tested hysteresis loops for EOC bearings (1.0 in. = 25.4 mm, 1.0 kip/in. = 0.175 kN/mm) 12.5.5.3 Analysis process. The two primary ETABS models, incorporating lower-bound (4 percent sliding friction) and upper-bound (8 percent sliding friction) properties are analyzed for the set of seven ground motion records applied to the models in each of four orientations of the larger component of

12-55

FEMA P-751, NEHRP Recommended Provisions: Design Examples records with respect to the primary horizontal axes of the model: (1) positive X-axis direction, (2) negative X-axis direction, (3) positive Y-axis direction and (4) negative Y-axis direction, respectively. The average value of the parameter of interest is calculated for each of four record set orientations and the more critical value of positive and negative orientations of the record set is used for design verification in the (x or y) direction of interest (e.g., story shear in the x direction). The more critical value of each of the four record set orientations is used for design verification when the response parameter of interest is not direction dependent (e.g., peak isolation system displacement). This process is repeated for ground motion records scaled to DE and MCER spectral accelerations. Although a large number of data are generated by these analyses, only limited number of response parameters is required for design verification. Response parameters of interest include the following: §

Design Earthquake: 1. Peak isolation system displacement 2. Peak story shear forces (envelope over building height)

§

Maximum Considered Earthquake: 1. Peak isolation system displacement 2. Peak downward load on any isolator unit (1.45D + 1.0L + QMCE) 3. Peak uplift displacement of any isolator unit (0.8D - QMCE).

12.5.5.4 Summary of results of response history analyses. RHA results are generally consistent with and support design values based on ELF formulas. The ETABS model with lower-bound stiffness properties (4 percent sliding friction) governs peak isolation system displacements. The lower-bound stiffness model also governs peak uplift displacement of isolators. The ETABS model with upper-bound stiffness properties (8 percent sliding friction) governs peak story shear forces and peak downward loads on isolators, although the value of the response parameters typically are similar for the two ETABS models. Specific results are summarized and compared to ELF values in the following sections. 12.5.5.4.1 Peak isolation system displacement. Table 12.5-12 summarizes peak isolation system displacements calculated by RHA for DE and MCER ground motions (average displacement of seven records). RHA results show the larger value of peak displacements in the X-axis and Y-axis directions and the peak displacement in the X-Y plane. Total displacement (at corners due to rotation) is based on the minimum 10 percent increase due to accidental mass eccentricity for both ELF and RHA results. Results of the evaluation of uplift are also included in Table 12.5-12.

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Chapter 12: Seismically Isolated Structures Table 12.5-12 Summary of Peak DE and MCER Displacements of the Isolation System Calculated Using RHA (Average Displacement of Seven Records) and Comparison with DE and MCER Displacements Calculated Using ELF Formulas Method of Analysis Response Parameter RHA - Average of Seven Records ELF Formulas Maximum (X, Y) X-Y Plane Design Earthquake - Isolation System Displacement (inches) Design (Center) 16.0 15.0 15.9 Total (Corner) 17.6 16.5 17.5 Uplift NA No uplift (all records) MCER - Isolation System Displacement (inches) MCER (Center) 30.9 28.2 29.6 Total (Corner) 34.0 31.1 32.5 Uplift NA Less than 0.01 in. (2/7 records) The average value of peak DE displacement in the X-Y plane is 15.9 inches, as compared to 16.0 inches calculated using ELF formulas. The average value of peak MCER displacement is 29.6 inches, as compared to 30.8 inches calculated using ELF formulas. The total DE displacement is found to be 17.5 inches and the total MCER displacement is found to be 32.5 inches (including a 10 percent increase in displacement to account for the effects of accidental mass eccentricity not explicitly modeled in the RHA analysis). Peak isolation system displacements calculated using RHA are slightly less than the corresponding values determined by ELF formulas, but within the RHA limits (minimum) based on ELF formulas (see Guide Section 12.5.4.2.6). The total MCER displacement of 32.6 inches is just less than the 33-inch displacement capacity of the FP bearing (displacement at which the articulated slider would engage the restraining edge of concave plates). Uplift did not occur at bearing locations for the DE ground motions. A very small amount of uplift occurred during two of the seven MCER ground motion records, which is less than but consistent with the amount of uplift previously determined by nonlinear static (pushover) analysis (see Guide Table 12.5-8). The Standard does not require the response of individual records to be used for design, or design verification, where seven or more records are used for RHA. Nonetheless, it is prudent to consider the implications of the results of individual record analyses and it is helpful to use ELF formulas to evaluate RHA results (serving as a sanity check on results). Table 12.5-12 summarizes peak displacements calculated by RHA for individual records scaled to DE and MCER ground motions. The results in this table are based on an application of records with the larger components aligned with the X-axis direction of the model. RHA results show values of peak displacement in the X-axis and Y-axis directions and in the X-Y plane. A couple of observations may be made. First, the peak displacement in the direction of the larger component is almost the same, on average, as the peak response in the X-Y plane. The second observation that may be made is that the displacement response varies greatly between records and that this variation is correlated directly with the variation of spectral response at long periods. For example, peak design earthquake displacement of Record Number NF-25 (the Yarmica record from the 1999 Kocaeli earthquake) is 30.7 inches in the X-Y plane (30.5 inches in the X-axis direction), which is less than the unrestrained displacement capacity of the FP bearing, but approximately twice the average 12-57

FEMA P-751, NEHRP Recommended Provisions: Design Examples displacement of the seven records. As shown in Guide Table 12.5-12, this record was scaled by 0.97 for DE analysis of the ETABS model (i.e., the as-recorded ground motions are actually slightly stronger than the scaled record). Large displacements (e.g., in excess of 30 inches) are realistic for long-period systems near active sources. Review of the response spectra of the scaled records shows that the Yarmica record also has the largest values of response spectral acceleration and displacement at long periods. Table 12.5-13 summarizes values of response spectral acceleration and displacement at the effective DE period, TD = 3.5 seconds, for records scaled to the design earthquake and at the effective MCER period, TM = 3.9 seconds, for records scaled to the MCER. Using the same concepts as those of the ELF formulas (e.g., reducing response based on the effective damping of the isolation system), peak isolation displacements are calculated directly from the response spectra of individual records and the results are summarized in Table 12.5-13. The very close agreement between displacements calculated using RHA and ELF methods (applied to response spectra of individual records) illustrates the usefulness of ELF methods and the related concepts of effective period and damping, to provide a “sanity check” of RHA results. Table 12.5-13 Summary of Peak DE and MCER Displacements of the Isolation System Calculated Using RHA and Comparison with DE and MCER Displacements Calculated Using ELF Methods Applied to the Response Spectra Of Individual Records Response Parameter

Seven Scaled Earthquake Ground Motion Records (FEMA P-695 ID No.) NF-8 FF-10 NF-25 FF-3 NF-14 FF-4 NF-28 RHA - Peak Isolation System Displacement - Design Earthquake (in.) 14.9 18.3 30.5 7.5 14.2 5.8 13.6 3.2 4.9 11.3 7.1 9.5 5.6 7.7

Averag e Value

X Direction 15.0 Y Direction 7.1 X-Y 15.0 18.8 30.7 8.9 14.9 7.4 15.8 15.9 Direction ELF Estimate of Peak Isolation System Displacement - Design Earthquake (in.) - TD = 3.5 sec. SaD [TD] (g) 0.182 0.124 0.305 0.106 0.186 0.096 0.133 0.187 SdD [TD] 21.9 14.9 36.6 12.7 22.3 11.5 16.0 22.4 (in.) DD = SdM/BD 14.6 9.9 24.4 8.5 14.9 7.7 10.6 15.0 RHA - Peak Isolation System Displacement - MCER (in.) X Direction 28.6 36.5 58.1 11.4 27.3 13.0 22.7 28.2 Y Direction 4.5 9.7 21.8 10.3 18.8 9.0 13.2 12.5 X-Y 28.7 38.1 58.5 13.6 28.6 13.6 26.0 29.6 Direction ELF Estimate of Peak Isolation System Displacement - MCER (in.) – TM = 3.9 sec. SaM [TM] (g) 0.310 0.295 0.536 0.118 0.225 0.150 0.159 0.256 SdM [TM] 46.2 43.9 79.9 17.6 33.6 22.4 23.8 38.2 (in.) DM = 35.5 33.8 61.5 13.5 25.8 17.2 18.3 29.4 SdM/BD

12-58

Chapter 12: Seismically Isolated Structures Table 12.5-13 shows that the peak MCER displacement for two scaled records (Yarmica and Arcelik) exceeds the 33-inch displacement capacity of the FP bearing—that is, the articulated slider will engage and be restrained by the concave plates. Such engagement would reduce displacement of the isolation system but also increase story shear force and overturning loads and increase the possibility of damage to the isolation system and the superstructure. It should be expected that displacements calculated for one or two records of a set of seven records scaled to match an MCER response spectrum for a site close to an active fault will exceed the average displacement of the set, even by as much as by a factor of two. The Standard requires design for the average and therefore tacitly accepts that displacements calculated for some records will be larger than the average displacement required for design of the isolation system. It would be prudent, however, for designers to consider the consequences of the very unlikely event that the MCER displacement is exceeded in an actual earthquake. In the case of the example EOC, displacements larger than MCER displacement would be expected to cause damage to the bearings, but not catastrophic failure of the isolation system or superstructure. The concave plates (and surrounding moat) would limit displacement of the isolation system (i.e., the isolation system would not displace laterally to failure). As the bearings (and surrounding moat) limit displacement, forces would increase significantly in the seismic force-resisting system of the structure above, comparable to peak response of a conventional (fixed-base) building. Yielding and inelastic response would likely occur, but the Standard requirement that special framing (e.g., SCBF) be used for the superstructure provides the requisite ductility to inelastically resist the response due to impact restraint of isolators (and the surrounding moat). 12.5.5.4.2 Peak story shear force. Table 12.5-14 summarizes story shear force results in the X-axis and Y-axis directions and compares these values with story shear forces calculated by ELF formulas for unreduced design earthquake loads. Figure 12.5-17 shows story shears calculated by ELF formulas and by RHA methods. Table 12.5-14 and Figure 12.5-18 show the inherent (and intentional) conservatism of the ELF formulas, which assume an inverted triangular pattern of lateral loads to account for potential higher-mode effects. The example EOC has minimal contribution from higher modes, as reflected in the lower values of story shear at upper floor and roof elevations. For the example EOC, design story shear forces are based on the limits of Standard Section 17.5.4.3 (Vs = 1.5 × 0.08W = 0.12W, where 0.08W represents the “breakaway” friction level of a sliding system), which apply to all methods of analysis, including RHA. Table 12.5-14 Summary of Peak Design Story Shear Forces and Comparison with Design Earthquake Story Shear Values Calculated Using ELF Methods Response Parameter

Penthouse Third Story Second Story First Story Vb (Isolators)

Method of Analysis Peak RHA Result ELF Formulas X-axis Direction Y-axis Direction Design Earthquake - Story Shear (kips) 261 150 147 837 546 531 1,192 874 855 1,403 1,183 1,173 1,456 1,440 1,449

12-59

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Average results of RHA

Eelevation (feet)

ELF

0 0

200

400

600

800

1,000

1,200

1,400

1,600

Story shear force (kips)

Figure 12.5-18 Comparison of story shear force based on RSA and ELF procedures 12.5.5.4.3 Peak downward loads on isolators. Table 12.5-15 summarizes maximum downward forces on isolator units determined from the MCER response history analyses. This table reports the two values for each isolator location representing both the X-axis and Y-axis orientations of the larger component of shaking of the ground motion record. Table 12.5-15 Maximum Downward Force (kips) on Isolator Units (1.45D + 1.0L + QMCE)* Maximum downward force (kips) Line

247 / 245

512 / 532

376 / 383

378 / 385

534 / 512

639 / 637

753 / 764

796 / 825

380 / 371

718 / 726

707 / 707

620 / 620

1.0 kip = 4.45 kN. * Forces at Column Lines 5, 6 and 7 (not shown) are similar to those at Column Lines 3, 2 and 1, respectively; loads at Column Lines D and E (not shown) are similar to those at Column Lines B and A, respectively.

12-60

Chapter 12: Seismically Isolated Structures Table 12.5-15 indicates a maximum downward force of 825 kips (at Column B4), which is somewhat less than the maximum value of downward force predicted using the ELF (pushover) methods (1,063 kips in Guide Table 12.5-7). The difference between RHA and ELF results reflects differences in overturning loads and patterns of story shear forces shown in the previous section. For this example, RHA results are used to verify ELF (pushover) values, but they could play a more meaningful role in reducing overturning loads for projects with a more challenging configuration of the superstructure. 12.5.6 Design and Testing Criteria for Isolator Units Detailed design of the isolator units typically is the responsibility of the manufacturer subject to the design and testing (performance) criteria included in the construction documents (drawings and/or specifications). Performance criteria typically include a basic description and size(s) of isolator units; design life, durability, environmental loads and fire-resistance criteria; quality assurance and quality control requirements (including QC testing of production units); design criteria (loads, displacements, effective stiffness and damping); and prototype testing requirements. This section summarizes the design criteria and prototype testing requirements for the isolator units (FP bearings) of the example EOC. 12.5.6.1 Bearing design loads. §

Maximum Long-Term Load (from Table 12.5-1, Col. C3): 1.2D + 1.6L = 600 kips

§

Maximum Short-Term Load (from Tables 12.5-8/12.5-15, Col. B4): 1.45D + 1.0L + |E| = 1,000 kips

§

Minimum Short-Term Load (from Table 12.5-9, Col. B4): 0.8D - |E| = 0 kips (less than 0.1 inch of uplift)

§

For Cyclic-Load Testing (Standard Sec. 17.8.2.2): Typical Load (Table 12.5-1, average all bearings): 1.0D + 0.5L = 290 kips Upper-Bound Load (Table 12.5-6, average all bearings): 1.37D + 0.5L + QDE = 500 kips Lower-Bound Load (Table 12.5-7, average all bearings): 0.73D - QDE = 150 kips

Item 2 of Standard Section 17.8.2.2 requires that cyclic tests at different displacement amplitudes be performed for typical vertical load (1.0D + 0.5L) and for upper-bound and lower-bound values of vertical load (for load-bearing isolators) where upper-bound and lower-bound loads are based on the load combinations of Standard Section 17.2.4.6 (vertical load stability). The load combinations of Standard Section 17.2.4 define “worst case” MCER loading conditions for checking isolator stability. In the opinion of the author, these load combinations are, in general, too extreme for cyclic load testing to verify isolator response properties (at lower levels of response). For the EOC example, upper-bound and lowerbound values of vertical load are based (roughly) on average bearing load at peak design earthquake response, as defined by the loads in Tables 12.5-6 and 12.5-7, respectively. The range of vertical load, from 150 kips to 500 kips, addresses the intent of the cyclic testing requirements (i.e., to measure possible variation in effective stiffness and damping properties). 12.5.6.2 Bearing design displacements (Table 12.5-12). §

Design earthquake displacement: DD = 16 in.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

§

Total design earthquake displacement: DTD = 17.5 in.

§

Maximum considered earthquake displacement: DM = 30 in.

§

Total maximum considered earthquake displacement: DTM = 32.5 in.

12.5.6.3 Bearing effective stiffness and damping criteria. §

Minimum effective stiffness normalized by load (typical vertical load): Design earthquake displacement : kDmin ≥ 0.0079 kips/in./kip Maximum considered earthquake displacement: kMmin ≥ 0.0067 kips/in./kip

§

Maximum effective stiffness normalized by weight (typical vertical load): Design earthquake displacement: kDmax ≤ 0.0104 kips/in./kip Maximum considered earthquake displacement: kMmax ≤ 0.0080 kips/in./kip

The kDmin, kDmax, kMmin and kMmax criteria are properties of the isolation system as a whole (calculated from the properties of individual isolator units using Standard Equations 17.8-3 through 17.8-6). The above limits on effective stiffness of the isolation system include changes in isolator properties over the life of the structure (due to contamination, aging, etc.), as well as other effects that may not be addressed by prototype testing (e.g., changes in properties under repeated cycles of dynamic load). Ideally, prototype bearings would have properties near the center of the specified range of effective stiffness. Effective damping (typical vertical load): §

Design earthquake displacement: βD ≥ 20 percent

§

Maximum considered earthquake displacement: βM ≥ 13 percent

12.5.6.4 Prototype bearing testing criteria. Standard Section 17.8 prescribes a series of prototype tests to establish and validate design properties used for design of the isolation system and defines “generic” acceptance criteria (Sec. 17.8.2) with respect to force-deflection properties of test specimens. Table 12.5-16 summarizes the sequence and cycles of prototype testing found in Standard Section 17.8.2, applicable to the FP bearing of the example EOC.

12-62

Chapter 12: Seismically Isolated Structures Table 12.5-16 Prototype Test Requirements No. of Cycles

Standard Criteria Vertical Load

Lateral Load

Example EOC Criteria Vertical Load

Lateral Load

Cyclic Load Tests to Establish Effective Stiffness and Damping (Standard Sec. 17.8.2.2, w/o Item 1) 3 cycles

Typical

0.25DD

290 kips

4 in.

3 cycles

Upper-bound

0.25DD

500 kips

4 in.

3 cycles

Lower-bound

0.25DD

150 kips

4 in.

3 cycles

Typical

0.5DD

290 kips

8 in.

3 cycles

Upper-bound

0.5DD

500 kips

8 in.

3 cycles

Lower-bound

0.5DD

150 kips

8 in.

3 cycles

Typical

1.0DD

290 kips

16 in.

3 cycles

Upper-bound

1.0DD

500 kips

16 in.

3 cycles

Lower-bound

1.0DD

150 kips

16 in.

3 cycles

Typical

1.0DM

290 kips

30 in.

3 cycles

Upper-bound

1.0DM

500 kips

30 in.

3 cycles

Lower-bound

1.0DM

150 kips

30 in.

3 cycles

Typical

1.0DTM

290 kips

32.5 in.

Cyclic Load Tests to Check Durability (Standard Sec. 17.8.2.2) 17 cycles*

Typical

1.0DTD

290 kips

17.5 in.

Static Load Test of Isolator Stability (Standard Sec. 17.8.2.5) N/A

Maximum

1.0DTM

1,000 kips

32.5 in.

N/A

Minimum

1.0DTM

0.1 in. of uplift

32.5 in.

1.0 in. = 25.4 mm, 1.0 kip = 4.45 kN. * 17 cycles = 30SD1/SDSBD (may be performed in 3 sets of tests (6 cycles each), if tests are dynamic.

12-63

13 Nonbuilding Structure Design By J. G. (Greg) Soules, P.E., S.E.

Originally developed by Harold O. Sprague, Jr., P.E.

Contents 13.1

NONBUILDING STRUCTURES VERSUS NONSTRUCTURAL COMPONENTS .............. 4

13.1.1

Nonbuilding Structure ........................................................................................................... 5

13.1.2

Nonstructural Component ...................................................................................................... 6

13.2

PIPE RACK, OXFORD, MISSISSIPPI ...................................................................................... 6

13.2.1

Description ............................................................................................................................. 7

13.2.2

Provisions Parameters ........................................................................................................... 7

13.2.3

Design in the Transverse Direction ....................................................................................... 8

13.2.4

Design in the Longitudinal Direction .................................................................................. 11

13.3

STEEL STORAGE RACK, OXFORD, MISSISSIPPI ............................................................. 13

13.3.1

Description ........................................................................................................................... 13

13.3.2

Provisions Parameters ......................................................................................................... 14

13.3.3

Design of the System ........................................................................................................... 15

13.4

ELECTRIC GENERATING POWER PLANT, MERNA, WYOMING .................................... 17

13.4.1

Description ........................................................................................................................... 17

13.4.2

Provisions Parameters ......................................................................................................... 19

13.4.3

Design in the North-South Direction ................................................................................... 20

13.4.4

Design in the East-West Direction ...................................................................................... 21

13.5

PIER/WHARF DESIGN, LONG BEACH, CALIFORNIA ...................................................... 21

13.5.1

Description ........................................................................................................................... 21

13.5.2

Provisions Parameters ......................................................................................................... 22

13.5.3

Design of the System ........................................................................................................... 23

13.6

TANKS AND VESSELS, EVERETT, WASHINGTON .......................................................... 24

13.6.1

Flat-Bottom Water Storage Tank......................................................................................... 25

13.6.2

Flat-Bottom Gasoline Tank ................................................................................................. 28

FEMA P-751, NEHRP Recommended Provisions: Design Examples 13.7

13-2

VERTICAL VESSEL, ASHPORT, TENNESSEE ................................................................... 31

13.7.1

Description ........................................................................................................................... 31

13.7.2

Provisions Parameters ......................................................................................................... 32

13.7.3

Design of the System ........................................................................................................... 33

Chapter 13: Nonbuilding Structure Design Chapter 15 of the Standard is devoted to nonbuilding structures. Nonbuilding structures comprise a myriad of structures constructed of all types of materials with markedly different dynamic characteristics and a wide range of performance requirements. Nonbuilding structures are a general category of structure distinct from buildings. Key features that differentiate nonbuilding structures from buildings include human occupancy, function, dynamic response and risk to society. Human occupancy, which is incidental in most nonbuilding structures, is the primary purpose of most buildings. The primary purpose and function of nonbuilding structures can be incidental to society, or the purpose and function can be critical for society. In the past, many nonbuilding structures were designed for seismic resistance using building code provisions developed specifically for buildings. These code provisions were inadequate to address the performance requirements and expectations that are unique to nonbuilding structures. For example, consider secondary containment for a vertical vessel containing hazardous materials. Nonlinear performance and collapse prevention, which are performance expectations for buildings, are insufficient for a secondary containment structure, which must not leak. Seismic design requirements specific to nonbuilding structures were first introduced in the 2000 Provisions. Before the introduction of the 2000 Provisions, the seismic design of nonbuilding structures depended on the various trade organizations and standards development organizations that were not connected with the building codes. This chapter develops examples specifically to help clarify Chapter 15 of the Standard. The solutions developed are not intended to be comprehensive but instead focus on correct interpretation of the requirements. Complete solutions to the examples cited are beyond the scope of this chapter. In addition to the Provisions and Commentary, the following publications are referenced in this chapter: API 650

American Petroleum Institute, Welded Steel Tanks for Oil Storage, 10th edition, Addendum 4, 2005.

ASCE

American Society of Civil Engineers, Guidelines for Seismic Evaluation and Design of Petrochemical Facilities, 1997.

ASME BPVC

American Society of Mechanical Engineers, Section VIII, Division 2, Alternate Rules, Rules for Construction of Pressure Vessels, 2007 Edition, 2008 Addenda.

AWWA D100

American Water Works Association, Welded Steel Tanks for Water Storage, 2005.

Bachman and Dowty

Bachman, Robert and Dowty, Susan, “Nonstructural Component or Nonbuilding Structure?”, Building Safety Journal, International Code Council, AprilMay 2008.

Jacobsen

Jacobsen, L.S., “Impulsive Hydrodynamics of Fluid Inside a Cylindrical Tank and of Fluid Surrounding a Cylindrical Pier,” Bulletin of the Seismological Society of America, 39(3), 189-204, 1949.

Morison

Morison, J.R., O’Brien, J.W. and Sohaaf, S.A., “The Forces Exerted by Surface Waves on Piles,” Petroleum Transactions, AIME, Vol. 189; 1950.

13-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples RMI

Rack Manufacturers Institute, Specification for the Design, Testing and Utilization of Industrial Steel Storage Racks, MH16.1, 2008

Soules

Soules, J. G., “The Seismic Provisions of the 2006 IBC – Nonbuilding Structure Criteria,” Proceedings of 8th National Conference on Earthquake Engineering, San Francisco, CA, April 18, 2006.

13.1 NONBUILDING STRUCTURES VERSUS NONSTRUCTURAL COMPONENTS Many industrial structures are classified as either nonbuilding structures or nonstructural components. This distinction is necessary to determine how the practicing engineer designs the structure. The intent of the Standard is to provide a clear and consistent design methodology for engineers to follow regardless of whether the structure is a nonbuilding structure or a nonstructural component. Central to the methodology is how to determine which classification is appropriate. Table 13-1 provides a simple method to determine the appropriate classification. Additional discussion on this topic can be found in Bachman and Dowty (2008). The design methodology contained in Chapter 13 of the Standard focuses on nonstructural component design. As such, the amplification by the supporting structure of the earthquake-induced accelerations is critical to the design of the component and its supports and attachments. The design methodology contained in Chapter 15 of the Standard focuses on the direct effects of earthquake ground motion on the nonbuilding structure.

Table 13-1 Applicability of the Chapters of the Standard Supporting Structure

Supported Item Nonstructural Component

Nonbuilding Structure

Building

Chapter 12 for supporting structure; Chapter 13 for supported item

Chapter 12 for supporting structure; Chapter 15 for supported item

Nonbuilding

Chapter 15 for supporting structure; Chapter 13 for supported item

Chapter 15 for both supporting structure and supported item

The example shown in Figure 13-1 is a combustion turbine, electric-power-generating facility with four bays. Each bay contains a combustion turbine and supports an inlet filter on the roof. The uniform seismic dead load of the supporting roof structure is 30 psf. Each filter weighs 34 kips. The following two examples illustrate the difference between nonbuilding structures that are treated as nonstructural components, using Standard Chapter 13 and those which are designed in accordance with Standard Chapter 15. In many instances, the weight of the supported nonbuilding structure is relatively small compared to the weight of the supporting structure (less than 25 percent of the combined weight) such that the supported nonbuilding structure will have a relatively small effect on the overall nonlinear earthquake response of the primary structure during design-level ground motions. It is permitted to treat such structures as nonstructural components and use the requirements of Standard Chapter 13 for their design. Where the weight of the supported structure is relatively large (greater than or equal to 25 percent of the combined weight) compared to the weight of the supporting structure, the overall response can be affected significantly. In such cases it is intended that seismic design loads and detailing requirements be 13-4

Chapter 13: Nonbuilding Structure Design determined following the procedures of Standard Chapter 15. Where there are multiple large nonbuilding structures, such as vessels supported on a primary nonbuilding structure and the weight of an individual supported nonbuilding structure does not exceed the 25 percent limit but the combined weight of the supported nonbuilding structures does, it is recommended that the combined analysis and design approach of Standard Chapter 15 be used. This difference in design approach is explored in the following example.

Inlet filter

25'

30'

Figure 13-1 Combustion turbine building (1.0 ft = 0.3048 m)

13.1.1 Nonbuilding Structure For the purpose of illustration, assume that the four filter units are connected in a fashion that couples their dynamic response through a rigid diaphragm. Therefore, it is appropriate to combine the masses of the four filter units for both the transverse and longitudinal direction responses. 13.1.1.1 Calculation of seismic weights. All four inlet filters = WIF = 4(34 kips) = 136 kips Support structure = WSS = 4(30 ft)(80 ft)(30 psf) = 288 kips The combined weight of the nonbuilding structure (inlet filters) and the supporting structural system is: Wcombined = 136 kips + 288 kips = 424 kips 13.1.1.2 Selection of design method. The ratio of the supported weight to the total weight is:

WIF

WCombined

136 = 0.321 > 25% 424

13-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Because the weight of the inlet filters is 25 percent or more of the combined weight of the nonbuilding structure and the supporting structure (Standard Sec. 15.3.2), the inlet filters are classified as “nonbuilding structures” and the seismic design forces must be determined from analysis of the combined seismic-resistant structural systems. This would require modeling the filters, the structural components of the filters and the structural components of the combustion turbine supporting structure to determine accurately the seismic forces on the structural elements as opposed to modeling the filters as lumped masses. 13.1.2 Nonstructural Component For the purpose of illustration, assume that the inlet filters are independent structures, although each is supported on the same basic structure. Unlike the previous example where the filter units were connected to each other through a rigid diaphragm, the four filter units are not connected in a fashion that couples their dynamic response. In other words, the four independent structures do not significantly affect the response of the support structure. In this instance, one filter is the nonbuilding structure. The question is whether it is heavy enough to significantly change the response of the combined system. 13.1.2.1 Calculation of seismic weights. One inlet filter = WIF = 34 kips Support structure = WSS = 4(30 ft)(80 ft)(30 psf) = 288 kips The combined weight of the nonbuilding structures (all four inlet filters) and the supporting structural system is: Wcombined = 4(34 kips) + 288 kips = 424 kips 13.1.2.2 Selection of design method. The ratio of the supported weight to the total weight is:

WIF WCombined

34 = 0.08 < 25% 424

Because the weight of an inlet filter is less than 25 percent of the combined weight of the nonbuilding structures and the supporting structure (Standard Sec. 15.3.1), the inlet filters are classified as “nonstructural components” and the seismic design forces must be determined in accordance with Standard Chapter 13. In this example, the filters could be modeled as lumped masses. The filters and the filter supports could then be designed as nonstructural components.

13.2 PIPE RACK, OXFORD, MISSISSIPPI This example illustrates the calculation of design base shears and maximum inelastic displacements for a pipe rack using the equivalent lateral force (ELF) procedure. The pipe rack in this example is supported at grade and is considered a nonbuilding structure.

13-6

Chapter 13: Nonbuilding Structure Design 13.2.1 Description A two-tier, 12-bay pipe rack in a petrochemical facility has concentrically braced frames in the longitudinal direction and ordinary moment frames in the transverse direction. The pipe rack supports four runs of 12-inch-diameter pipe carrying naphtha on the top tier and four runs of 8-inch-diameter pipe carrying water for fire suppression on the bottom tier. The minimum seismic dead load for piping is 35 psf on each tier to allow for future piping loads. The seismic dead load for the steel support structure is 10 psf on each tier. Pipe supports connect the pipe to the structural steel frame and are designed to support the gravity load and resist the seismic and wind forces perpendicular to the pipe. The typical pipe support allows the pipe to move in the longitudinal direction of the pipe to avoid restraining thermal movement. The pipe support near the center of the run is designed to resist longitudinal and transverse pipe movement as well as provide gravity support; such supports are generally referred to as fixed supports. Pipes themselves must be designed to resist gravity, wind, seismic and thermally induced forces, spanning from support to support. If the pipe run is continuous for hundreds of feet, thermal/seismic loops are provided to avoid a cumulative thermal growth effect. The longitudinal runs of pipe in this example are broken up into sections by providing thermal/seismic loops at spaced intervals as shown in Figure 13-2. In Figure 13-2, it is assumed thermal/seismic loops are provided at each end of the pipe run.

20'-0"

5 bays @ 20'-0" = 100'-0"

20'-0"

6 bays @ 20'-0" = 120'-0"

Expansion loop breaks the continuity Horizontal bracing at braced bay only

10'-0"

15'-0"

8'-0"

3'-0"

PLAN

ELEVATION

SECTION

Figure 13-2 Pipe rack (1.0 ft = 0.3048 m)

13.2.2 Provisions Parameters 13.2.2.1 Ground motion. See Section 3.2 for an example illustrating the determination of design ground motion parameters. For this example, the parameters are as follows:

13-7

FEMA P-751, NEHRP Recommended Provisions: Design Examples

SDS = 0.40 SD1 = 0.18 13.2.2.2 Occupancy category and importance factor. The upper piping carries a toxic material (naphtha) (Occupancy Category III – Standard Table 1-1) and the lower piping is required for fire suppression (Occupancy Category IV – Standard Table 1-1). The naphtha piping and the fire water piping are included in Standard Section 1.5.1; therefore, the pipe rack is assigned to Occupancy Category IV based on the more severe category. Standard Section 15.4.1.1 directs the user to use the largest value of I based on the applicable reference document listed in Standard Chapter 23, the largest value selected from Standard Table 11.5-1, or as specified elsewhere in Standard Chapter 15. It is important to be aware of the requirements of Standard Section 15.4.1.1. While the importance factor for most structures will be determined based on Standard Table 11.5-1, there are reference documents that define importance factors greater than those found in Standard Table 11.5-1. Additionally, Standard Section 15.5.3 requires that steel storage racks in structures open to the public be assigned an importance factor of 1.5. This additional requirement for steel storage racks addresses a risk to the public that is not addressed by Standard Table 11.5-1 and Standard Table1-1. For this example, Standard Table 11.5-1 governs the choice of importance factor. According to Standard Section 11.5.1, the importance factor, I, is 1.5 based on Occupancy Category IV. 13.2.2.3 Seismic design category. For this structure assigned to Occupancy Category IV with SDS = 0.40 and SD1 = 0.18, the Seismic Design Category is D according to Standard Section 11.6. 13.2.3 Design in the Transverse Direction 13.2.3.1 Design coefficients. According to Standard Section 15.4-1, either Standard Table 12.2-1 or Standard Table 15.4-1 may be used to determine the seismic parameters, although mixing and matching of values and requirements from the tables is not allowed. In Standard Chapter 15, selected nonbuilding structures similar to buildings are provided an option where both lower R values and less restrictive height limits are specified. This option permits selected types of nonbuilding structures which have performed well in past earthquakes to be constructed with fewer restrictions in Seismic Design Categories D, E and F provided seismic detailing is used and design force levels are considerably higher. The R value-height limit trade-off recognizes that the size of some nonbuilding structures is determined by factors other than traditional loadings and result in structures that are much stronger than required for seismic loadings (Soules, 2006). Therefore, the structure’s ductility demand is generally much lower than a corresponding building. The R value-height trade-off also attempts to obtain the same structural performance at the increased heights. The user will find that the option of reduced R value with less restricted height will prove to be the economical choice in most situations due to the relative cost of materials and construction labor. It must be emphasized that the R value-height limit trade-off of Standard Table 15.4-1 applies only to nonbuilding structures similar to buildings and cannot be applied to building structures. In Standard Table 12.2-1, ordinary steel moment frames are not permitted in Seismic Design Category D (with some exceptions) and cannot be used in this example. There are several options for ordinary steel moment frames found in Standard Table 15.4-1. These options are as follows: 1. Standard Table 15.4-1, Ordinary moment frames of steel, R = 3.5, Ωo = 3, Cd = 3. According to Note c in Standard Table 15.4-1, this system is allowed for pipe racks up to 65 feet high using bolted end plate moment connections and per Note d this system is allowed for pipe racks up to 35 feet without limitations on the connection type. This option requires the use of the AISC 341.

13-8

Chapter 13: Nonbuilding Structure Design

2. Standard Table 15.4-1, Ordinary moment frames of steel with permitted height increase, R = 2.5, Ω0 = 2, Cd = 2.5. This option is intended for pipe racks with height greater than 65 feet and limited to 100 feet. This option is not applicable for this example. 3. Standard Table 15.4-1, Ordinary moment frames of steel with unlimited height, R=1, Ω0 = 1, Cd = 1. This option does not require the use of the AISC 341. For this example, Option 1 above is chosen. Using Standard Table 15.4-1, the parameters for this ordinary steel moment frame are: R = 3.5 Ω0 = 3 Cd = 3 Ordinary steel moment frames are retained for use in nonbuilding structures such as pipe racks because they allow greater flexibility for accommodating process piping and are easier to design and construct than special steel moment frames.

13.2.3.2 Seismic response coefficient. Using Standard Equation 12.8-2:

Cs =

S DS 0.4 = = 0.171 R I 3.5 1.5

From analysis, T = 0.42 second. For nonbuilding structures, the fundamental period is generally approximated for the first iteration and must be verified with final calculations. Standard Section 15.4.4 makes clear that the approximate period equations of Standard Section 12.8.2 do not apply to nonbuilding structures. Using Standard Equation 12.8-3 for T ≤ TL, Cs does not need to exceed

Cs =

S D1 0.18 = = 0.184 T ( R I ) 0.42(3.5 1.5)

Using Standard Equation 12.8-5, Cs must not be less than Cs = 0.044ISDS ≥ 0.01 = 0.044(1.5)(0.4) = 0.0264 Standard Equation 12.8-2 controls; Cs = 0.171.

13-9

FEMA P-751, NEHRP Recommended Provisions: Design Examples

13.2.3.3 Seismic weight. The seismic weight resisted by the moment frame in the transverse direction is shown below based on two levels of piping, a 20 ft bent spacing, a bent width (perpendicular with the piping) of 20 ft, piping dead weight of 35 psf and structure dead weight of 10 psf. W = 2(20 ft)(20 ft)(35 psf + 10 psf) = 36 kips 13.2.3.4 Base shear. Using Standard Equation 12.8-1: V = CsW = 0.171(36 kips) = 6.2 kips 13.2.3.5 Drift. Although not shown here, drift of the pipe rack in the transverse direction was calculated by elastic analysis using the design forces calculated above. The calculated lateral drift, δxe = 0.328 inch. Using Standard Equation 12.8-15:

δx =

Cd δ xe 3( 0.328) = = 0.656 in. 1.5 I

The lateral drift must be checked with regard to acceptable limits. The acceptable limits for nonbuilding structures are not found in codes. Rather, the limits are what is acceptable for the performance of the piping. In general, piping can safely accommodate the amount of lateral drift calculated in this example. P-delta effects must also be considered and checked as required in Standard Section 15.4.5. 13.2.3.6 Redundancy factor. Some nonbuilding structures are designed with parameters from Standard Table 12.2-1 or 15.4-1 if they are termed “nonbuilding structures similar to buildings”. For such structures (assigned to Seismic Design Category D, E, or F) the redundancy factor applies. Pipe racks, being fairly simple moment frames or braced frames, are in the category similar to buildings. Because this structure is assigned to Seismic Design Category D, Standard Section 12.3.4.2 applies. Considering the transverse direction, the seismic force-resisting system is an ordinary moment resisting frame with only two columns in a single frame. The frames repeat in an identical pattern. Loss of moment resistance at the beam-to-column connections at both ends results in a loss of more than 33 percent in story strength. Therefore, Standard Section 12.3.4.2, Condition (a) is not met. The moment frame as described above consists only of a single bay. Therefore, Standard Section 12.3.4.2, Condition b is not met. The value of ρ in the transverse direction is therefore 1.3. 13.2.3.7 Determining E. In Standard Section 12.4.2, E is defined to include the effects of horizontal and vertical ground motions and can be summarized as follows: E = ρQE ± 0.2 SDS D where QE is the effect of the horizontal earthquake ground motions, which is determined primarily by the base shear just computed and D is the effect of dead load. By putting a simple multiplier on the effect of dead load, the last term is an approximation of the effect of vertical ground motion. For the moment frame, the joint moment is influenced by both terms. E with the “+” on the second term where combined with dead and live loads will generally produce the largest negative moment at the joints, while E with the “-” on the second term where combined with the minimum dead load (0.6D) will produce the largest positive joint moments.

13-10

Chapter 13: Nonbuilding Structure Design

The Standard also requires the consideration of an overstrength factor, Ω0, on the effect of horizontal motions in defining Em for components susceptible to brittle failure. Standard Section 12.4.3 defines Em and this definition can be summarized as follows: Em = Ω0 QE ± 0.2 SDS D The moment frame portion of the pipe rack does not have components that require such consideration. 13.2.4 Design in the Longitudinal Direction

13.2.4.1 Design coefficients. In Standard Section 15.4-1, either Standard Table 12.2-1 or Standard Table 15.4-1 may be used to determine the seismic parameters. In Standard Table 12.2-1, ordinary steel concentrically braced frames are not permitted for Seismic Design Category D (with some exceptions) and cannot be used for this example. There are several options for ordinary steel concentrically braced frames found in Standard Table 15.4-1. These options are as follows: 1. Standard Table 15.4-1, Ordinary steel concentrically braced frame, R = 3.25, Ω0 = 2, Cd = 3.25. According to Note b in Standard Table 15.4-1, this system is allowed for pipe racks up to 65 feet high. This option requires the use of AISC 341. 2. Standard Table 15.4-1, Ordinary steel concentrically braced frames with permitted height increase, R = 2.5, Ω0 = 2, Cd = 2.5. This option is intended for pipe racks with height greater than 65 feet and limited to 160 feet. This option is not applicable for this example. 3. Standard Table 15.4-1, Ordinary steel concentrically braced frames with unlimited height, R = 1.5, Ω0 = 1, Cd = 1.5. This option does not require the use of AISC 341. For this example, Option 1 above is chosen. Using Standard Table 15.4-1, the parameters for this ordinary steel concentrically braced frame are: R = 3.25 Ω0 = 2 Cd = 3.25 13.2.4.2 Seismic response coefficient. Using Standard Equation 12.8-2:

Cs =

S DS 0.4 = = 0.185 R I 3.25 1.5

From analysis, T = 0.24 second. The fundamental period for nonbuilding structures is generally approximated for the first iteration and must be verified with final calculations. Using Standard Equation 12.8-3, Cs does not need to exceed:

13-11

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Cs =

S D1 0.18 = = 0.346 T ( R I ) 0.24(3.25 1.5 )

Using Standard Equation 12.8-5, Cs must not be less than: Cs = 0.044ISDS ≥ 0.01 = 0.044(1.5)(0.4) = 0.0264 Standard Equation 12.8-2 controls; Cs = 0.185. 13.2.4.3 Seismic weight. W = 2(240 ft)(20 ft)(35 psf + 10 psf) = 432 kips 13.2.4.4 Base shear. Using Standard Equation 12.8-1: V = CsW = 0.185(432 kips) = 79.9 kips 13.2.4.5 Redundancy factor. The pipe rack in this example does not meet either of the two

redundancy conditions specified in Standard Section 12.3.4.2. Condition a is not met because only one set of bracing is provided on each side, so removal of one brace would result in a reduction of greater than 33 percent in story strength. Condition b is not met because two bays of seismic force-resisting perimeter framing are not provided in each orthogonal direction. Therefore, the redundancy factor, ρ, is 1.3. If two bays of bracing were provided on each side of the pipe rack in the longitudinal direction, the pipe rack would meet Condition (a) and qualify for a redundancy factor, ρ, of 1.0 in that direction. 13.2.4.6 Determine E. In Standard Section 12.4.2, E is defined to include the effects of horizontal and vertical ground motions and can be summarized as follows: E = ρQE ± 0.2 SDS D where QE is the effect of the horizontal earthquake ground motions, which is determined primarily by the base shear just computed and D is the effect of dead load. By putting a simple multiplier on the effect of dead load, the last term is an approximation of the effect of vertical ground motion. The Standard also requires the consideration of an overstrength factor, Ω0, on the effect of horizontal motions in defining Em for components susceptible to brittle failure. Standard Section 12.4.3 defines Em and this definition can be summarized as follows: Em = Ω0 QE ± 0.2 SDS D The ordinary steel concentrically braced frame portion of the pipe rack does have components that require such consideration. The beams connecting each moment frame in the longitudinal direction act as collectors and, as required by Standard Section 12.10.2.1, must be designed for the seismic load effect including overstrength factor. 13.2.4.7 Orthogonal loads. Because the pipe rack in this example is assigned to Seismic Design Category D, Standard Section 12.5.4 requires that the braced sections of the pipe rack be evaluated using the orthogonal combination rule of Standard Section 12.5.3a. Two cases must be checked: 100 percent

13-12

Chapter 13: Nonbuilding Structure Design transverse seismic force plus 30 percent longitudinal seismic force and 100 percent longitudinal seismic force plus 30 percent transverse seismic force. The vertical seismic force represented by 0.2SDSD is only applied once in each load case. Do not include the vertical seismic force in with both horizontal seismic load combinations. In this pipe rack example, due to the bracing configuration, the foundation and column anchorage would be the only components impacted by the orthogonal load combinations.

13.3 STEEL STORAGE RACK, OXFORD, MISSISSIPPI This example uses the ELF procedure to calculate the seismic base shear in the east-west direction for a steel storage rack. 13.3.1 Description A four-tier, five-bay steel storage rack is located in a retail discount warehouse. There are concentrically braced frames in the north-south and east-west directions. The general public has direct access to the aisles and merchandise is stored on the upper racks. The rack is supported on a slab on grade. The design operating load for the rack contents is 125 psf on each tier. The weight of the steel support structure is assumed to be 5 psf on each tier.

13-13

FEMA P-751, NEHRP Recommended Provisions: Design Examples

8'-0"

3'-0"

3'-

N W S

Figure 13-3 Steel storage rack (1.0 ft = 0.3048 m)

13.3.2 Provisions Parameters 13.3.2.1 Ground motion. The spectral response acceleration coefficients at the site are as follows: SDS = 0.40 SD1 = 0.18 13.3.2.2 Occupancy category and importance factor. Use Standard Section 1.5.1. The storage rack is in a retail facility. Therefore, the storage rack is assigned to Occupancy Category II. According to Standard Section 15.5.3 (item 2), I = Ip = 1.5 because the rack is in an area open to the general public. 13.3.2.3 Seismic design category. Use Standard Tables 11.6-1 and 11.6-2. Given Occupancy Category II, SDS = 0.40 and SD1 = 0.18, the Seismic Design Category is C. 13.3.2.4 Design coefficients. According to Standard Table 15.4-1, the design coefficients for this steel storage rack are as follows:

13-14

Chapter 13: Nonbuilding Structure Design

R=4 Ω0 = 2 Cd = 3½ 13.3.3 Design of the System 13.3.3.1 Seismic response coefficient. Standard Section 15.5.3 allows designers some latitude in selecting the seismic design methodology. Designers may use the Rack Manufacturer’s Institute specification (MH16.1-2008) to design steel storage racks. In other words, racks designed using the RMI method of Section 15.5.3 are deemed to comply. As an alternate, designers may use the requirements of Standard Sections 15.5.3.1 through 15.5.3.4. The RMI approach will be used in this example. Using RMI Section 2.6.3, from analysis, T = 0.24 seconds. For this particular example, the short period spectral value controls the design. The period for taller racks, however, may be significant and will be a function of the operating weight. As shown in the calculations that follow, in the RMI method the importance factor appears in the equation for V rather than in the equation for Cs. The seismic response coefficient from RMI is:

Cs =

S D1 0.18 = = 0.188 T ( R ) 0.24( 4 )

But need not be greater than:

Cs =

S DS 0.4 = = 0.10 R 4

Nor less than:

Cs = 0.044SDS = 0.044( 0.4 ) = 0.0176 The governing value of Cs = 0.10. From RMI Section 2.6.2, the seismic base shear is calculated as follows:

V = Cs I p Ws = 0.1(1.5 )Ws = 0.15Ws 13.3.3.2 Condition 1 (each rack loaded). 13.3.3.2.1 Seismic weight. In accordance with RMI Section 2.6.8, Item 1: Ws = 4(5)(8 ft)(3 ft)[0.67(125 psf)+5 psf] = 42.6 kips 13.3.3.2.2 Design forces and moments. Using RMI Section 2.6.2, the design base shear for Condition 1 is calculated as follows: V = CsIpWs = 0.15(42.6 kips) = 6.39 kips

13-15

FEMA P-751, NEHRP Recommended Provisions: Design Examples In order to calculate the design forces, shears and overturning moments at each level, seismic forces must be distributed vertically in accordance with RMI Section 2.6.6. The calculations are shown in Table 13.31. Table 13.3-1 Seismic Forces, Shears and Overturning Moment

wx hxk

0.40

Fx (kips) 2.56

95.85

0.30

1.92

63.90

0.20

1.28

31.95

0.10

0.63

Level x 5

Wx (kips) 10.65

hx (ft) 12

(k = 1) 127.80

10.65

∑ wi hi i =1

Σ 42.6 319.5 1.0 ft = 0.3048 m, 1.0 kip = 4.45 kN, 1.0 ft-kip = 1.36 kN-m.

Vx (kips)

Mx (ft-kips)

2.56

7.68

4.48

21.1

5.76

38.4

6.39

57.6

13.3.3.2.3 Resisting moment at the base. MOT, resisting = Ws (1.5 ft) = 42.6(1.5 ft) = 63.9 ft-kips 13.3.3.3 Condition 2 (only top rack loaded). 13.3.3.3.1 Seismic weight. In accordance with RMI Section 2.6.2, Item 2: Ws = 1(5)(8 ft)(3 ft)(125 psf) + 4(5)(8 ft)(3 ft)(5 psf) = 17.4 kips 12.3.3.3.2 Base shear. Using RMI Section 2.6.2, the design base shear for Condition 2 is calculated as follows: V = CsIpWs = 0.15(17.4 kips) = 2.61 kips 13.3.3.3.3 Overturning moment at the base. Although the forces could be distributed as shown above for Condition 1, a simpler, conservative approach for Condition 2 is to assume that a seismic force equal to the entire base shear is applied at the top level. Using that simplifying assumption, MOT = Vb (12 ft) = 2.61 kip (12 ft) = 31.3 ft-kips 13.3.3.3.4 Resisting moment at the base. MOT, resisting = Ws (1.5 ft) = 17.4(1.5 ft) = 26.1 ft-kips

13-16

Chapter 13: Nonbuilding Structure Design 13.3.3.4 Controlling conditions. Condition 1 controls shear demands at all but the top level. Although the overturning moment is larger under Condition 1, the resisting moment is larger than the overturning moment. Under Condition 2 the resistance to overturning is less than the applied overturning moment. Therefore, the rack anchors must be designed to resist the uplift induced by the base shear for Condition 2. 13.3.3.5 Torsion. It should be noted that the distribution of east-west seismic shear will induce torsion in the rack system because the east-west brace is only on the back of the storage rack. The torsion should be resisted by the north-south braces at each end of the bay where the east-west braces are placed. If the torsion were to be distributed to each end of the storage rack, the engineer would be required to calculate the transfer of torsional forces in diaphragm action in the shelving, which may be impractical. Therefore, north-south braces are provided in each bay.

13.4 ELECTRIC GENERATING POWER PLANT, MERNA, WYOMING This example highlights some of the differences between the design of nonbuilding structures and the design of building structures. The boiler building in this example illustrates a solution using the ELF procedure. Due to mass irregularities, the boiler building would probably also require a modal analysis. For brevity, the modal analysis is not illustrated. 13.4.1 Description Large boilers in coal-fired electric power plants generally are suspended from the supporting steel near the roof level. Additional lateral supports (called buck stays) are provided near the bottom of the boiler. The buck stays resist lateral forces but allow the boiler to move vertically. Lateral seismic forces are resisted at the roof and at the buck stay level. Close coordination with the boiler manufacturer is required in order to determine the proper distribution of seismic forces. In this example, a boiler building for a 950 MW coal-fired electric power generating plant is braced laterally with ordinary concentrically braced frames in both the north-south and east-west directions. The facility is part of a grid and is not for emergency backup of an Occupancy Category IV facility. The dead load of the structure, equipment and piping, WDL, is 16,700 kips. The weight of the boiler in service, WBoiler, is 31,600 kips. The natural period of the structure (determined from analysis) is as follows: North-South, TNS = 1.90 seconds East-West, TEW = 2.60 seconds

13-17

FEMA P-751, NEHRP Recommended Provisions: Design Examples

240'-0"

5'-

230'-0" N

S W

Plate girder to support boiler (typical) Buck stays (typical) BOILER

Section A-A

Figure 13-4 Boiler building (1.0 ft = 0.3048 m)

13-18

Chapter 13: Nonbuilding Structure Design

13.4.2 Provisions Parameters Occupancy Category (Standard Sec. 1.5.1) (for continuous operation, but not for emergency backup of an Occupancy Category IV facility)

III

Occupancy Importance Factor, I (Standard Sec. 11.5.1)

1.25

Short-period Response, SS

0.864

One-second Period Response, S1

0.261

Site Class (Standard Sec. 11.4.2)

D (default)

Short-period Site Coefficient, Fa (Standard Table 11.4-1)

1.155

Long-period Site Coefficient, Fv (Standard Table 11.4-2)

1.877

= =

0.665 0.327

Seismic Design Category (Standard Sec. 11.6)

Seismic Force-resisting System (Standard Table 15.4-1)

Ordinary steel concentrically braced frame with unlimited height

Response Modification Coefficient, R

1.5

System Overstrength Factor, Ω0

Deflection Amplification Factor, Cd

1.5

Height Limit (Standard Table 15.4-1)

Unlimited

Design Spectral Acceleration Response Parameters SDS = (2/3)SMS = (2/3)FaSS = (2/3)(1.155)(0.864) SD1 = (2/3)SM1 = (2/3)FvS1 = (2/3)(1.877)(0.261)

According to Standard Section 15.4-1, either Standard Table 12.2-1 or Standard Table 15.4-1 may be used to determine the seismic parameters, although mixing and matching of values and requirements from the tables is not allowed. If the structure were classified as a “building,” its height would be limited to 35 feet for a Seismic Design Category D ordinary steel concentrically braced frame, according to Standard Table 12.2-1. A review of Standard Table 12.2-1 shows that three steel high ductility braced frame systems (two eccentrically braced systems and the special concentrically braced system) and two special moment frame systems can be used at a height of 240 feet. In most of these cases, the additional requirements of Standard Section 12.2.5.4 must be met to qualify the system at a height of 240 feet. Boiler buildings normally are constructed using ordinary concentrically braced frames.

13-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples As discussed in Section 13.2.3.1 above, Chapter 15 of the Standard presents options to increase height limits for design of some nonbuilding structures similar to buildings where R factors are reduced. For this example, an ordinary steel concentrically braced frame with unlimited height is chosen from Standard Table 15.4-1. By using a significantly reduced R value, the seismic design and detailing requirements of AISC 341 need not be applied. 13.4.3 Design in the North-‐South Direction 13.4.3.1 Seismic response coefficient. Using Standard Equation 12.8-2:

Cs =

S DS 0.665 = = 0.554 R I 1.5 1.25

From analysis, T = 1.90 seconds. Using Standard Equation 12.8-3, CS does not need to exceed S D1 0.327 Cs = = = 0.143 T ( R I ) 1.90(1.5 1.25) but using Standard Equation 12.8-5, Cs must not be less than: Cs = 0.044ISDS ≥ 0.01 = 0.044(1.25)(0.665) = 0.0366 Standard Equation 12.8-3 controls; Cs = 0.143. 13.4.3.2 Seismic weight. Calculate the total seismic weight, W, as follows: W = WDL + WBoiler = 16,700 kips + 31,600 kips = 48,300 kips 13.4.3.3 Base shear. Using Standard Equation 12.8-1: V = CsW = 0.143(48,300 kips) = 6,907 kips 13.4.3.4 Redundancy factor. The structure in this example meets the requirements of Condition b

specified in Standard Section 12.3.4.2, because two bays of seismic force-resisting perimeter framing are provided in each orthogonal direction. Therefore, the redundancy factor, ρ, is 1.0.

It is important to note that each story resists more than 35 percent of the base shear because the boiler is hung from the top of the structure. Therefore, each story must comply with the requirements of Condition b. If a story resisted less than 35 percent of the base shear, the requirements of Standard Section 12.3.4.2 would not apply and that story would not be considered in establishing the redundancy factor. 13.4.3.5 Determining E. E is defined to include the effects of horizontal and vertical ground motions as follows: E = ρQE ± 0.2 SDS D

13-20

Chapter 13: Nonbuilding Structure Design where QE is the effect of the horizontal earthquake ground motions, which is determined primarily by the base shear just computed and D is the effect of dead load. By putting a simple multiplier on the effect of dead load, the last term is an approximation of the effect of vertical ground motion. The Standard also requires the consideration of an overstrength factor, Ω0, on the effect of horizontal motions in defining E for components susceptible to brittle failure. E = Ω0 QE ± 0.2 SDS D The ordinary steel concentrically braced frames have components that require such consideration. The beams transferring shear from one set of braces to another act as collectors and, as required by Standard Section 12.10.2.1, must be designed for the seismic load effect including overstrength factor. 13.4.4 Design in the East-‐West Direction 13.4.4.1 Seismic response coefficient. Using Standard Equation 12.8-2:

Cs =

S DS 0.665 = = 0.554 R I 1.5 1.25

From analysis, T = 2.60 seconds. Using Standard Equation 12.8-3, Cs does not need to exceed: S D1 0.327 Cs = = = 0.105 T ( R I ) 2.60(1.5 1.25) Using Standard Equation 12.8-5, Cs must not be less than: Cs = 0.044ISDS ≥ 0.01 = 0.044(1.25)(0.665) = 0.0366 Standard Equation 12.8-3 controls; Cs = 0.105. 13.4.4.2 Seismic weight. Calculate the total seismic weight, W, as follows: W = WDL + WBoiler = 16,700 kips + 31,600 kips = 48,300 kips 13.4.4.3 Base shear. Using Standard Equation 12.8-1: V = CsW = 0.105(48,300 kips) = 5072 kips

13.5 PIER/WHARF DESIGN, LONG BEACH, CALIFORNIA This example illustrates the calculation of the seismic base shear in the east-west direction for the pier using the ELF procedure. Piers and wharves are covered in Standard Section 15.5.6. 13.5.1 Description A cruise ship company is developing a pier in Long Beach, California, to service ocean liners. The pier contains a large warehouse owned by the cruise ship company. In the north-south direction, the pier is tied directly to an abutment structure supported on grade. In the east-west direction, the pier resists

13-21

FEMA P-751, NEHRP Recommended Provisions: Design Examples seismic forces using moment frames. Calculations for the abutment are not included in this example, but it is assumed to be much stiffer than the moment frames.

PLAN

3'-0"

15'-0"

3'-0"

The design live load for warehouse storage is 1,000 psf.

W 10'-0"

10'-0"

3'-0"

20'-0"

Mean Sea Level

Mud

30'-0"

Dense sand

ELEVATION

Figure 13-5 Pier plan and elevation (1.0 ft = 0.3048 m)

13.5.2 Provisions Parameters Occupancy Category (Standard Sec. 1.5.1) (The pier serves cruise ships that carry no hazardous materials.)

Importance Factor, I (Standard Sec. 11.5.1)

1.0

Short-period Response, SS

1.75

13-22

10'-0"

Chapter 13: Nonbuilding Structure Design

One-second Period Response, S1

0.60

Site Class (Standard Chapter 20)

D (dense sand)

Short-period Site Coefficient, Fa (Standard Table 11.4-1)

1.0

Long-period Site Coefficient, Fv (Standard Table 11.4-2)

1.5

= =

1.167 0.60

Seismic Design Category (Standard Sec. 11.6)

Seismic Force-resisting System (Standard Table 15.4-1)

Intermediate concrete moment frame with permitted height increase

Response Modification Coefficient, R

System Overstrength Factor, Ω0

Deflection Amplification Factor, Cd

2.5

Height Limit (Standard Table 15.4-1)

50 ft

Design Spectral Acceleration Response Parameters SDS = (2/3)SMS = (2/3)FaSS = (2/3)(1.0)(1.75) SD1 = (2/3)SM1 = (2/3)FvS1 = (2/3)(1.5)(0.60)

If the structure were classified as a building, an intermediate reinforced concrete moment frame would not be permitted in Seismic Design Category D. 13.5.3 Design of the System 13.5.3.1 Seismic response coefficient. Using Standard Equation 12.8-2:

Cs =

S DS 1.167 = = 0.389 R I 3 1.0

From analysis, T = 0.596 seconds. Using Standard Equation 12.8-3, Cs does not need to exceed:

Cs =

S D1 0.60 = = 0.336 T ( R I ) 0.596(3 1.0 )

Using Standard Equation 12.8-5, Cs must not be less than: Cs = 0.044ISDS ≥ 0.01 = 0.044(1.0)(1.167) = 0.0513 Standard Equation 12.8-3 controls; Cs = 0.336.

13-23

FEMA P-751, NEHRP Recommended Provisions: Design Examples 13.5.3.2 Seismic weight. In accordance with Standard Section 12.7.2, calculate the dead load due to the deck, beams and support piers, as follows: WDeck = 1.0(43 ft)(21 ft)(0.150 kip/ft3) = 135.5 kips WBeam = 4(2 ft)(2 ft)(21 ft)(0.150 kip/ft3) = 50.4 kips WPier = 8[π(1.25 ft)2][(10 ft - 3 ft) + (20 ft)/2](0.150 kip/ft3) = 100.1 kips WDL = WDeck + WBeams + WPiers = 135.5 + 50.4 + 100.1 = 286.0 kips Calculate 25 percent of the storage live load, as follows: W1/4 LL = 0.25(1,000 psf)(43 ft)(21 ft) = 225.8 kips Standard Section 15.5.6.2 requires that all applicable marine loading combinations be considered (such as those for mooring, berthing, wave and current on piers and wharves). For this example, additional seismic loads from water flowing around the piles will be considered. A “virtual” mass (Jacobsen, 1959) of water equal to a column of water of identical dimensions of the circular pile is to be considered in the effective seismic mass. This additional weight is calculated as follows: WVirtual Mass = 8[π(1.25 ft)2][(20 ft)/2](64 pcf) = 25.1 kips Therefore, the total seismic weight is W = WDL + W1/4LL + WVirtual Mass = 286.0 + 225.8 + 25.1 = 536.9 kips Additional seismic forces from the water due to wave action may also act on the piles. These additional forces are highly dependent on the acceleration and velocity of the waves and are heavily dependent on the geometry of the body of water. These forces can be calculated using the Morison Equation (Morison, 1950). The determination of these forces is beyond the scope of this example. 13.5.3.3 Base shear. Using Standard Equation 12.8-1: V = CsW = 0.336(536.9 kips) = 180.4 kips 13.5.3.4 Redundancy factor. The pier in this example has a sufficient number of moment frames that loss of moment resistance at both ends of a single beam would not result in more than a 33 percent reduction in story strength. However, the direct tie to a much stiffer abutment at the north end likely would cause an extreme torsional irregularity for east-west motion, so that Condition (a) would not be met. Condition (b) is not met because two bays of seismic force-resisting perimeter framing are not provided in each orthogonal direction. Therefore, the redundancy factor, ρ, is taken to be 1.3.

13.6 TANKS AND VESSELS, EVERETT, WASHINGTON The seismic response of tanks and vessels can be significantly different from that of buildings. For a structure composed of interconnected solid elements, it is not difficult to recognize how ground motions accelerate the structure and cause inertial forces within the structure. Tanks and vessels, where empty, respond in a similar manner.

13-24

Chapter 13: Nonbuilding Structure Design Where there is liquid in the tank, the response is much more complicated. As earthquake ground motions accelerate the tank shell, the shell applies lateral forces to the liquid. The response of the liquid to those lateral forces may be amplified significantly if the period content of the earthquake ground motion is similar to the natural sloshing period of the liquid. Earthquake-induced impulsive fluid forces are those calculated assuming that the liquid is a solid mass. Convective fluid forces are those that result from sloshing in the tank. It is important to account for convective forces on columns and appurtenances inside the tank, because they are affected by sloshing in the same way that waves affect a pier in the ocean. Freeboard considerations are critical. Oftentimes, the roof acts as a structural diaphragm. If a tank does not have sufficient freeboard, the sloshing wave can rip the roof from the wall of the tank. This could result in failure of the wall and loss of the liquid within. Seismic design for liquid-containing tanks and vessels is complicated. The fluid mass that is effective for impulsive and convective seismic forces is discussed in AWWA D100 and API 650. 13.6.1 Flat-‐Bottom Water Storage Tank 13.6.1.1 Description. This example illustrates the calculation of the design base shear and the required freeboard using the procedure outlined in AWWA D100 for a steel water storage tank used to store potable water for fire protection within a chemical plant (Figure 13-6). According to Standard Section 15.7.7.1, the governing reference document for this tank is AWWA D100. Standard Chapter 15 makes no modifications to this document for the seismic design of flat-bottom water storage tanks. AWWA D100 is written in terms of allowable stress design (ASD) while the seismic requirements of the Standard are written in terms of strength design. AWWA D100 translates the force equations from the Standard by substituting 1.4R for R. For the purposes of this example, all loads are calculated in terms of strength design. Where appropriate, AWWA D100 equations are referenced for the determination of impulsive and convective (sloshing) masses.

Freeboard

δS

H = 10'-0"

HR = 15'-0"

5'-0"

D = 20'-0"

Figure 13-6 Storage tank section (1.0 ft = 0.3048 m) The weight of the tank shell, roof, bottom and equipment is 15,400 pounds.

13-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples 13.6.1.2 Seismic design parameters. Occupancy Category (Standard Sec. 1.5.1)

Importance Factor, I (Standard Sec. 11.5.1)

1.5

Short-period Response, SS

1.236

One-second Period Response, S1

0.406

Long-period Transition Period, TL

6 seconds

Site Class (Standard Chapter 20)

C (per geotech)

Short-period Site Coefficient, Fa (Standard Table 11.4-1)

1.0

Long-period Site Coefficient, Fv (Standard Table 11.4-2)

1.39

= =

0.824 0.376

Seismic Force-resisting System (Standard Table 15.4-2)

Flat-bottom, groundsupported, mechanically anchored steel tank

Response Modification Coefficient, R

System Overstrength Factor, Ω0

Deflection Amplification Factor, Cd

2.5

Design Spectral Acceleration Response Parameters SDS = (2/3)SMS = (2/3)FaSS = (2/3)(1.0)(1.236) SD1 = (2/3)SM1 = (2/3)FvS1 = (2/3)(1.39)(0.406)

13.6.1.3 Calculations for impulsive response. 13.6.1.3.1 Natural period for the first mode of vibration. AWWA D100 Section 13.5.1 does not require the computation of the natural period for the first mode of vibration. The impulsive acceleration is assumed to be equal to SDS. 13.6.1.3.2 Spectral acceleration. Based on AWWA D100 Section 13.5.1, the impulsive acceleration is set equal to SDS. Sai = SDS = 0.824 13.6.1.3.3 Seismic (impulsive) weight. Wtank = 15.4 kips Wiwater = π(10 ft)2(10 ft)(0.0624 kip/ft3) (Wi/WT)= 196.0 (0.542) kips = 106.2 kips

13-26

Chapter 13: Nonbuilding Structure Design The ratio Wi/WT (= 0.542) was determined from Equation 13-24 (only valid for D/H ≥ 1.333) of AWWA D100 for a diameter-to-liquid height ratio of 2.0 as shown below:

D ⎞ 20 ⎞ ⎛ ⎛ tanh ⎜ 0.866 ⎟ tanh ⎜ 0.866 ⎟ Wi H ⎠ 10 ⎠ ⎝ ⎝ = = = 0.542 20 D WT 0.866 0.866 10 H Wi = Wtank + Wiwater = 15.4 + 106.2 = 121.6 kips 13.6.1.3.4 Base Shear. According to Standard Equation 15.7-5:

Vi =

Sai Wi 0.824(121.6 ) = = 50.1 kips R I 3 1.5

13.6.1.4 Calculations for convective response natural period for the first mode of sloshing. 13.6.1.4.1 Natural period for the first mode of sloshing. Using Standard Equation 15.7-12: = 2

3.68 tanh

3.68

= 2

20 ft ft 3.68 10 ft 3.68 32.174 tanh 20 ft

= 2.65 s

13.6.1.4.2 Spectral acceleration. Using Standard Equation 15.7-10 with Tc < TL = 6 seconds:

S ac =

1.5S D1 1.5( 0.376 ) = = 0.212 Tc 2.65

13.6.1.4.3 Seismic (convective) weight. Wc = Wwater (Wc/WT) = 196 (0.437) = 85.7 kips The ratio Wc/WT (= 0.437) was determined from Equation 13-26 (valid for all D/H) of AWWA D100 for a diameter-to-liquid height ratio of 2.0 as shown below:

Wc 10 ⎞ H ⎞ ⎛ D ⎞ ⎛ ⎛ 20 ⎞ ⎛ = 0.230⎜ ⎟ tanh ⎜ 3.67 ⎟ = 0.230⎜ ⎟ tanh ⎜ 3.67 ⎟ = 0.437 20 ⎠ WT D ⎠ ⎝ H ⎠ ⎝ ⎝ 10 ⎠ ⎝ 13.6.1.4.4 Base shear. According to Standard Equation 15.7-6:

Vc =

0.212(1.5) Sac I Wc = (85.7 ) = 18.2 kips 1.5 1.5

13-27

FEMA P-751, NEHRP Recommended Provisions: Design Examples 13.6.1.5 Design base shear. Item b of Standard Section 15.7.2 indicates that impulsive and convective components may, in general, be combined using the SRSS method. Standard Equation 15.7-4 requires that the direct sum be used for ground-supported storage tanks for liquids. Note b under Standard Section 15.7.6.1 allows the use of the SRSS method in lieu of using Standard Equation 15.7-4. Therefore, the base shear is computed as follows:

V = Vi 2 + Vc2 = 50.12 + 18.22 = 53.3 kips 13.6.1.6 Minimum freeboard. Because the tank is assigned to Occupancy Category IV, the full value of the theoretical wave height must be provided for freeboard. For the case of Occupancy Category IV tanks, the wave height is calculated based on the convective acceleration using the actual value of TL and an importance factor of 1.0. Standard Table 15.7-3 indicates that a minimum freeboard equal to δs is required for this tank. Using Standard Equation 15.7-13 and Note c (sets I = 1.0 for Occupancy Category IV for wave height determination) from Standard Section 15.7.6.1: δs = 0.5DiISac = 0.5(20 ft)(1.0)(0.212) = 2.12 ft The 5 feet of freeboard provided is adequate. 13.6.2 Flat-‐Bottom Gasoline Tank 13.6.2.1 Description. This example illustrates the calculation of the base shear and the required freeboard using the procedure outlined in API 650 for a petrochemical storage tank in a refinery tank farm (Figure 13-7). An impoundment dike is not provided to control liquid spills. According to Standard Section 15.7.8.1, the governing reference document for this tank is API 650. API 650 is written in terms of allowable stress design (ASD) while the seismic requirements of the Standard are written in terms of strength design. API 650 translates the force equations from the Standard by substituting Rw for R, where Rw is equal to 1.4R. For the purposes of this example, all loads are calculated in terms of strength design. Where appropriate, API 650 equations are referenced for the determination of impulsive and convective (sloshing) masses. The tank is a flat-bottom, ground-supported, self-anchored, welded steel tank constructed in accordance with API 650. The weight of the tank shell, roof, bottom and equipment is 490,000 pounds.

13-28

Chapter 13: Nonbuilding Structure Design

3/16 Thk roof PL

Top angle (Toe out) 3x3x3/8 12

Ring No. 5 - 5/16 Thk

Roof rafters 1R (54) W14x26

33' High liquid level

3'-9 5/16" 3/4

Ring No. 4 - 5/16 Thk

Center column 1C 20 Dia Sch 10

Shell Height =40'-0"

Ring No. 3 - 5/16 Thk

Internal aluminum floating roof

Ring No. 2 - 5/16 Thk

Top of bottom PL

Ring No. 1 - 0.401 Thk 3'-0" Low Liquid Level

10'

Low rest position 3'-0" High rest position 6'-0"

1" 6" bottom crown at center

1/4" Thk bottom PL

Ringwall foundation

120'-0" Nominal Diameter

Figure 13-7 Storage tank section (1.0 ft = 0.3048 m) 13.6.2.2 Seismic design parameters. Occupancy Category (Standard Sec. 1.5.1) (The tank is used for storage of toxic or explosive material.)

III

Importance Factor, I (Standard Sec. 11.5.1)

1.25

= =

0.824 0.376

Seismic Force-Resisting System (Standard Table 15.4-2)

Flat-bottom, groundsupported, self- anchored steel tank

Response Modification Coefficient, R

2.5

System Overstrength Factor, Ω0

Deflection Amplification Factor, Cd

Design Spectral Acceleration Response Parameters (Using the same site as in Section 13.6.1) SDS = (2/3)SMS = (2/3)FaSS = (2/3)(1.0)(1.236) SD1 = (2/3)SM1 = (2/3)FvS1 = (2/3)(1.39)(0.406)

13.6.2.3 Calculations for impulsive response. 13.6.2.3.1 Natural period for the first mode of vibration. API 650 Section E.4.8.1 does not require the computation of the natural period for the first mode of vibration. The impulsive acceleration is assumed to be equal to SDS.

13-29

FEMA P-751, NEHRP Recommended Provisions: Design Examples 13.6.2.3.2 Spectral acceleration. Based on API 650 Section 13.5.1, the impulsive acceleration is set equal to SDS. Sai = SDS = 0.824 13.6.2.3.3 Seismic (impulsive) weight. Wtank = 490.0 kips WGas = π(60 ft)2(33 ft)(0.0474 kip/ft3)(Wi/Wp) = 17,691 kips (0.316) = 5,590 kips The ratio Wi/Wp (= 0.316) was determined from Equation E-13 (only valid for D/H ≥ 1.333) of API 650 for a diameter-to-liquid height ratio of 3.636 as shown below:

D ⎞ 120 ⎞ ⎛ ⎛ tanh ⎜ 0.866 ⎟ tanh ⎜ 0.866 Wi H ⎠ 33 ⎟⎠ ⎝ ⎝ = = = 0.316 120 D Wp 0.866 0.866 33 H Wi = Wtank + WGas = 490 + 5590 = 6,080 kips 13.6.2.3.4 Base shear. According to Standard Equation 15.7-5:

Sai Wi 0.824( 6080 ) = = 2,505 kips R I 2.5 1.25 13.6.2.4 Calculations for convective response. Vi =

13.6.2.4.1 Natural period for the first mode of sloshing. Using Standard Equation 15.7-12: = 2

3.68 3.68 tanh

= 2

120 ft ft 3.68 33 ft 3.68 32.174 tanh 120 ft

= 7.22 s

13.6.2.4.2 Spectral acceleration. Using Standard Equation 15.7-11 with Tc > TL = 6 seconds:

Sac =

1.5S D1TL 1.5( 0.376 )( 6 ) = = 0.0649 Tc2 7.222

13.6.2.4.3 Seismic (convective) weight. Wc = WGAS (Wc/Wp) = 17691 (0.640) = 11,322 kips The ratio Wc/Wp (= 0.640) was determined from Equation E-15 (valid for all D/H) of API 650 for a diameter-to-liquid height ratio of 3.636 as shown below:

13-30

Chapter 13: Nonbuilding Structure Design

Wc 33 ⎞ H ⎞ ⎛ D ⎞ ⎛ ⎛ 120 ⎞ ⎛ tanh ⎜ 3.67 = 0.230⎜ ⎟ tanh ⎜ 3.67 ⎟ = 0.230⎜ = 0.640 ⎟ 120 ⎟⎠ WT D ⎠ ⎝ H ⎠ ⎝ ⎝ 33 ⎠ ⎝ 13.6.2.4.4 Base shear. According to Standard Equation 15.7-6:

Vc =

0.0649(1.5 ) Sac I Wc = (11,322 ) = 735 kips 1.5 1.5

13.6.2.5 Design base shear. Item (b) of Standard Section 15.7.2 indicates that impulsive and convective components may, in general, be combined using the SRSS method. Standard Equation 15.7-4 requires that the direct sum be used for ground-supported storage tanks for liquids. Note b under Standard Section 15.7.6.1 allows the use of the SRSS method in lieu of using Standard Equation 15.7-4. Therefore, the base shear is computed as follows:

V = Vi 2 + Vc2 = 25052 + 7352 = 2,611 kips 13.6.2.6 Minimum freeboard. Because the tank is assigned to Occupancy Category III and SDS is greater than 0.50, the freeboard provided must be at least 70 percent of the full value of the theoretical wave height (based on TL = 4 s). For the case of Occupancy Category III tanks, the wave height is calculated based on the convective acceleration using a value of TL equal to 4 seconds and an importance factor of 1.25 according to Standard Section 15.7.6.1, Note (d). Standard Table 15.7-3 indicates that a minimum freeboard equal to 0.7δs is required for this tank. Using Standard Equation 15.7-13 and Note (d) (sets I = 1.25 and TL to 4 seconds for Occupancy Category III for wave height determination) from Standard Section 15.7.6.1: δs = 0.5DiISac = 0.5(120 ft)(1.25)(0.0433) = 3.25 ft

Sac =

1.5S D1TL 1.5( 0.376 )4 = = 0.0433 Tc2 7.222

0.7 δs = 2.27 ft The 7 feet of freeboard provided also includes a 3-foot allowance for an aluminum internal floating roof and the roof framing. The seismic freeboard must be sufficient to avoid forcing the floating roof into the fixed roof framing. The freeboard provided is adequate. The reduced freeboard requirement recognizes that providing seismic freeboard for Occupancy Category I, II, or III tanks is an economic decision (reducing damage) and not a life-safety issue. Because of this, a reduced freeboard is allowed. If secondary containment were provided, no freeboard would be required based on Standard Table 15.7-3, Footnote (b).

13.7 VERTICAL VESSEL, ASHPORT, TENNESSEE 13.7.1 Description This example illustrates the calculation of the base shear using the ELF procedure for a flexible vertical vessel (Figure 13-8). The vertical vessel contains highly toxic material (Occupancy Category IV).

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

H=100'

D=10'

T=3/8"

Figure 13-8 Vertical vessel (1.0 ft = 0.3048 m) The weight of the vertical vessel plus contents is 300,000 pounds. Standard Section 15.4.4 allows the fundamental period of a nonbuilding structure to be determined using a properly substantiated analysis. The period of the vertical vessel is calculated using the equation for a uniform vertical cylindrical steel vessel as found in Appendix 4.A of ASCE (1997). The period of the vessel is calculated as follows:

7.78 ⎛ H ⎞ T = 6 ⎜ ⎟ 10 ⎝ D ⎠ where:

12W D 7.78 ⎛ 100 ⎞ = 6 ⎜ ⎟ t 10 ⎝ 10 ⎠

12(300000 100)10 0.375

= 0.762s

T = period (s) W = weight (lb/ft) H = height (ft) D = diameter (ft) T = shell thickness (in.)

13.7.2 Provisions Parameters 13.7.2.1 Ground motion. The design response spectral accelerations are defined as follows:

13-32

Chapter 13: Nonbuilding Structure Design SDS = 1.86 SD1 = 0.79 13.7.2.2 Importance factor. The vertical vessel contains highly toxic material. Therefore, it is assigned to Occupancy Category IV, as required by Standard Section 1.5.1. Using Standard Table 11.5.1, the importance factor, I, is equal to 1.5. 13.7.2.3 Seismic coefficients. The vertical vessel used in this example is a skirt-supported distributed mass cantilevered structure. There are three possible entries in Standard Table 15.4-2 that describe the vessel in question: 1. Elevated tanks, vessels, bins, or hoppers: Single pedestal or skirt supported – welded steel. 2. Elevated tanks, vessels, bins, or hoppers: Single pedestal or skirt supported – welded steel with special detailing. 3. All other steel and reinforced concrete distributed mass cantilever structures not covered herein including stacks, chimneys, silos and skirt-supported vertical vessels that are not similar to buildings. All three options are keyed to the detailing requirements of Standard Section 15.7.10. Two of the options specifically require that Items (a) and (b) of Standard Section 15.7.10 be met. The intent of Standard Section 15.7.10 and Standard Table 15.4-2 is that skirt-supported vessels be checked for seismic loads based on R/I = 1.0 if the structure is assigned to Occupancy Category IV or if an R factor of 3.0 is used in the design of the vessel. Skirt-supported vessels fail in buckling, which is not a ductile failure mode, so a more conservative design approach is required. The R/I = 1.0 check typically will govern the design of the skirt over using loads determined with an R factor of 3 in a moderate to high area of seismic activity. The only benefit of using an R factor of 3 in this case is in the design of the foundation. The foundation is not required to be designed for the R/I = 1.0 load. For the R/I = 1.0 load, the skirt can be designed based on critical buckling (factor of safety of 1.0). The critical buckling strength of a skirt can be determined using a number of published sources. The two most common methods for determining the critical buckling strength of a skirt are ASME BVPC Section VIII, Division 2, Paragraph 4.4 using a factor of safety of 1.0 and AWWA D100 Section 13.4.3.4. Calculating the critical buckling strength of a skirt is beyond the scope of this example. For this example, the skirt-supported vertical vessel will be treated as “all other steel and reinforced concrete distributed mass cantilever structures not covered herein including stacks, chimneys, silos and skirt-supported vertical vessels that are not similar to buildings” from Standard Table 15.4-2. The seismic design parameters for this structure are as follows: Response Modification Coefficient, R

System Overstrength Factor, Ω0

Deflection Amplification Factor, Cd

2.5

13.7.3 Design of the System 13.7.3.1 Seismic response coefficient. Using Standard Equation 12.8-2:

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Cs =

S DS 1.86 = = 0.930 R I 3 1.5

Using Standard Equation 12.8-3, Cs does not need to exceed:

Cs =

S D1 0.79 = = 0.518 T ( R I ) 0.762(3 1.5)

Using Standard Equation 15.4-1, Cs must not be less than: Cs = 0.044ISDS ≥ 0.03 = 0.044(1.5)(1.86) = 0.123 Standard Equation 12.8-3 controls; Cs = 0.518. 13.7.3.2 Base shear. Using Standard Equation 12.8-1: V = CsW = 0.518(300 kips) = 155.4 kips 13.7.3.3 Vertical distribution of seismic forces. Standard Section 12.8.3 defines the vertical distribution of seismic forces in terms of an exponent, k, related to structural period. If the structural period is less than or equal to 0.5 second, k = 1 and results in an inverted triangular distribution of forces. If the structural period is greater than or equal to 2.5 seconds, k = 2 and results in a parabolic distribution of forces. For periods between 0.5 second and 2.5 seconds, the value of k is determined by linear interpolation between 1 and 2. The significance of the distribution requirements of Standard Section 12.8.3 is that the height of the centroid to the horizontal seismic force increases (thus increasing the overturning moment) as the period increases above 0.5 second (Figure 13-9).

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Chapter 13: Nonbuilding Structure Design

H h

k = 1 (triangle)

k = value between 1 & 2

k = 2 (parabola)

T = 0.5 s

0.5 s < T < 2.5 s

T = 2.5 s

h = 2/3 H

(k+1) (k+2)

h = 3/4 H

Figure 13-9 Vertical distribution of seismic forces Once k is determined, the height to the centroid, h , of the horizontal seismic force is equal to:

( k + 1) H ( k + 2)

where H is the vertical height of the vertical vessel. For the vessel period of T = 0.762 second,

k =1+

0.762 − 0.5 = 1.131 2.5 − 0.5

The value of h is then calculated as:

(1.131 + 1) 100 = 0.681 100 = 68.1ft ( ) ( ) (1.131 + 2)

The overturning moment is then calculated as M = V h = 155.4(68.1) = 10,583 ft-kips.

13-35

14 Design for Nonstructural Components Robert Bachman, S.E., John Gillengerten, S.E. and Susan Dowty, S.E.

Contents 14.1

DEVELOPMENT AND BACKGROUND OF THE REQUIREMENTS FOR NONSTRUCTURAL COMPONENTS ........................................................................................ 3

14.1.1

Approach to Nonstructural Components ............................................................................... 3

14.1.2

Force Equations ..................................................................................................................... 4

14.1.3

Load Combinations and Acceptance Criteria ........................................................................ 5

14.1.4

Component Amplification Factor .......................................................................................... 6

14.1.5

Seismic Coefficient at Grade ................................................................................................. 7

14.1.6

Relative Location Factor........................................................................................................ 7

14.1.7

Component Response Modification Factor ........................................................................... 7

14.1.8

Component Importance Factor .............................................................................................. 7

14.1.9

Accommodation of Seismic Relative Displacements ............................................................ 8

14.1.10

Component Anchorage Factors and Acceptance Criteria .................................................. 9

14.1.11

Construction Documents ................................................................................................... 9

14.2

ARCHITECTURAL CONCRETE WALL PANEL ................................................................... 10

14.2.1

Example Description ........................................................................................................... 10

14.2.2

Design Requirements ........................................................................................................... 12

14.2.3

Spandrel Panel ..................................................................................................................... 12

14.2.4

Column Cover ...................................................................................................................... 19

14.2.5

Additional Design Considerations ....................................................................................... 20

14.3

HVAC FAN UNIT SUPPORT .................................................................................................... 21

14.3.1

Example Description ........................................................................................................... 21

14.3.2

Design Requirements ........................................................................................................... 22

14.3.3

Direct Attachment to Structure ............................................................................................ 23

14.3.4

Support on Vibration Isolation Springs ............................................................................... 26

FEMA P-751, NEHRP Recommended Provisions: Design Examples 14.3.5 14.4

ANALYSIS OF PIPING SYSTEMS........................................................................................... 33

14.4.1

ASME Code Allowable Stress Approach ............................................................................ 33

14.4.2

Allowable Stress Load Combinations ................................................................................. 34

14.4.3

Application of the Standard................................................................................................. 36

14.5

PIPING SYSTEM SEISMIC DESIGN ....................................................................................... 38

14.5.1

Example Description ........................................................................................................... 38

14.5.2

Design Requirements. .......................................................................................................... 43

14.5.3

Piping System Design .......................................................................................................... 45

14.5.4

Pipe Supports and Bracing................................................................................................... 48

14.5.5

Design for Displacements .................................................................................................... 53

14.6

14-2

Additional Considerations for Support on Vibration Isolators ............................................ 31

ELEVATED VESSEL SEISMIC DESIGN ................................................................................ 55

14.6.1

Example Description ........................................................................................................... 55

14.6.2

Design Requirements ........................................................................................................... 58

14.6.3

Load Combinations .............................................................................................................. 60

14.6.4

Forces in Vessel Supports .................................................................................................... 60

14.6.5

Vessel Support and Attachment........................................................................................... 62

14.6.6

Supporting Frame ................................................................................................................ 65

14.6.7

Design Considerations for the Vertical Load-Carrying System .......................................... 69

Chapter 14: Design for Nonstructural Components Chapter 13 of the Standard addresses architectural, mechanical and electrical components of buildings. The examples presented here illustrate many of the requirements and procedures. Design and anchorage are illustrated for exterior precast concrete cladding and for a roof-mounted HVAC unit. The rooftop unit is examined in two common installations: directly attached and isolated with snubbers. This chapter also contains an explanation of the fundamental aspects of the Standard and an explanation of how piping, designed according to the ASME Power Piping code, is checked for the force and displacement requirements of the Standard. Examples are also provided that illustrate how to treat non-ASME piping located within a healthcare facility and a platform-supported vessel located on an upper floor within a building. The variety of materials and industries involved with nonstructural components is large and numerous documents define and describe methods of design, construction, manufacture, installation, attachment, etc. Some of the documents address seismic issues, but many do not. Standard Chapter 23 contains a listing of approved standards for various nonstructural components. In addition to the Standard, the following are referenced in this chapter: §

ACI 318

American Concrete Institute. 2008. Building Code Requirements for Structural Concrete.

§

ASHRAE APP IP

American Society of Heating, Refrigeration and Air-Conditioning Engineers (ASHRAE). 1999. Seismic and Wind Restraint Design, Chapter 53.

§

ASME B31.1

American Society of Mechanical Engineers. 2001. Power Piping Code.

§

IBC

International Code Council. 2006. International Building Code.

The symbols used in this chapter are drawn from Chapter 11 of the Standard or reflect common engineering usage. The examples are presented in U.S. customary units.

14.1 DEVELOPMENT AND BACKGROUND OF THE REQUIREMENTS FOR NONSTRUCTURAL COMPONENTS 14.1.1 Approach to Nonstructural Components The Standard requires that nonstructural components be checked for two fundamentally different demands placed upon them by the response of the structure to earthquake ground motion: resistance to inertial forces and accommodation of imposed displacements. Building codes have long had requirements for resistance to inertial forces. Most such requirements apply to the component mass and acceleration that vary with the basic ground motion parameter and a few broad categories of components. These broad categories are intended to distinguish between components whose dynamic response couples with that of the supporting structure in such a fashion as to cause the component response accelerations to be amplified above the accelerations of the structure and those components that are rigid enough with respect to the structure so that the component response is not amplified over the structural response. In recent years, a coefficient based on the function of the building or of the component has been introduced as another multiplier for components important to life safety or essential facilities. The Standard includes an equation to compute the inertial force that involves two additional concepts: variation of acceleration with relative height within the structure and reduction in design force based upon

14-3

FEMA P-751, NEHRP Recommended Provisions: Design Examples available ductility in the component or its attachment. The Standard also includes a quantitative measure for the deformation imposed upon nonstructural components. The inertial force demands tend to control the seismic design for isolated or heavy components, whereas the imposed deformations are important for the seismic design for elements that are continuous through multiple levels of a structure or across expansion joints between adjacent structures, such as cladding or piping. The remaining portions of this section describe the sequence of steps and decisions prescribed by the Standard to check these two seismic demands on nonstructural components. 14.1.2 Force Equations The following seismic force equations are prescribed for nonstructural components (Standard Eq. 13.3-1 through 13.3-3):

Fp =

0.4a p S DSW p Rp Ip

z ⎞ ⎛ ⎜1 + 2 h ⎟ ⎝ ⎠

Fpmax = 1.6S DS I pWp Fpmin = 0.3S DS I pWp where: Fp = horizontal equivalent static seismic design force centered at the component’s center of gravity and distributed relative to the component’s mass distribution ap = component amplification factor (between 1.0 and 2.5) as tabulated in Standard Table 13.5-1 for architectural components and Standard Table 13.6-1 for mechanical and electrical components SDS = five percent damped spectral response acceleration parameter at short period as defined in Standard Section 11.4.4 Wp = component operating weight Rp = component response modification factor (between 1.0 to 12.0) as tabulated in Standard Table 15.5-1 for architectural components and Standard Table 13.6-1 for mechanical and electrical components Ip =

component importance factor (either 1.0 or 1.5) as indicated in Standard Section 13.1.3

elevation in structure of component point of attachment relative to the base

roof elevation of the structure or elevation of highest point of the seismic force-resisting system of the structure relative to the base

The seismic design force, Fp, is to be applied independently in the longitudinal and transverse directions. Fp should be applied in both the positive and negative directions if higher demands will result. The

14-4

Chapter 14: Design for Nonstructural Components effects of these loads on the component are combined with the effects of static loads. Standard Equations 13.3-2 and 13.3-3 provide maximum and minimum limits for the seismic design force. For each point of attachment, a force, Fp, should be determined based on Standard Equation 13.3-1. The minima and maxima determined from Standard Equations 13.3-2 and 13.3-3 must be considered in determining each Fp. The weight, Wp, used to determine each Fp should be based on the tributary weight of the component associated with the point of attachment. For designing the component, the attachment force, Fp, should be distributed relative to the component’s mass distribution over the area used to establish the tributary weight. With the exception of structural walls, which are covered by Standard Section 12.11.1 and anchorage of concrete or masonry structural walls, which is covered by Standard Section 12.11.2, each anchorage force should be based on simple statics determined by using all the distributed loads applied to the complete component. Cantilever parapets that are part of a continuous element should be checked separately for parapet forces. 14.1.3 Load Combinations and Acceptance Criteria Load combinations for use in determining the overall demand on an item are defined in Standard Section 2.3. Earthquakes cause loads on structures and nonstructural components in both the horizontal and vertical directions. Where these loads are applied to structural and nonstructural systems, the results (forces, stresses, displacements, etc.) are called “effects”. In Standard Section 12.4.2, seismic load effects are defined. The effects resulting from horizontally applied loads are termed horizontal load effects, Eh and the effects resulting from vertically applied loads are termed vertical load effects, Ev. The Ev term is simply a constant 0.2SDS multiplied by the dead load. Because the load combinations defined in Standard Section 2.3 provide a single term, E, to define the earthquake, the horizontal and vertical load effects were sometimes misapplied by casual users of the Standard and the effects Eh and Ev were simply added as if they were applied in the same direction. To eliminate this confusion, when the Standard was reorganized as part of the development of its 2005 edition, a new Section 12.4 was added to separate the horizontal and vertical components of the seismic load and provide a reconstituted version of the load combinations provided in Section 2.3 of the Standard. The seismic load combinations substituted the vertical coefficient term, 0.2SDS (representing Ev) directly into the load combinations. These are not alternate versions of Section 2.3 but instead present expanded versions of the load combinations of Section 2.3. It was intended by the ASCE 7 Seismic Task Committee that unless otherwise noted or excepted, the load combinations provided in Section 12.4 of the Standard be used for the design of all structures and nonstructural components. The 2006 IBC has its own set of load combinations that are very similar to those provided in ASCE 7-05. In general, 2006 IBC load combinations take precedence over those of ASCE 7-05 where they are in conflict. However, for the remainder of the discussion and examples provided herein, the load combinations of Standard Section 12.4 are used. 14.1.3.1 Seismic load effects. From Section 12.4.2, the horizontal seismic load effect Eh and vertical seismic load effect Ev are determined by applying the horizontal component load Fp and the vertical dead load D, respectively, in the structural analysis as indicated below. Eh = ρQE

(Standard Eq. 12.4-3)

Ev = 0.2SDSD

(Standard Eq. 12.4-4)

where:

14-5

FEMA P-751, NEHRP Recommended Provisions: Design Examples

QE = effect of horizontal seismic forces (due to application of Fp for nonstructural components) ρ = redundancy factor = 1.0 for nonstructural components

(Standard Sec. 12.4.2-1) (Standard Sec 13.3.1)

D = dead load effect (due to vertical load application) Where the effects of vertical gravity loads and horizontal earthquake loads are additive, E = ρQE + 0.2SDSD And where the effects of vertical gravity load counteract those of horizontal earthquake loads, E = ρQE - 0.2SDSD where: E = effect of horizontal and vertical earthquake-induced forces 14.1.3.2 Strength load combinations. The Standard provides load combinations that are to be used to determine design member forces, stresses and displacements in Standard Sections 2.3 and 2.4. In Standard Section 2.3, load combinations are provided for Strength Design and in Standard Section 2.4, load combinations are provided for Allowable Stress Design. For purposes of the Chapter 13 examples, only the Strength Load Combinations are used. For Strength Load Combinations involving seismic loads, the terms defined above in Section 14.1.3.1 are substituted for E in the Basic Load Combinations for Strength Design of Standard Section 2.3.2 to determine the design member and connection forces to be used in conjunction with seismic loads. Once the substitutions have been made, the strength load combinations of Section 2.3.2 are presented in Standard Section 12.4.2.3, as follows: (1.2 + 0.2SDS) D + ρQE + L + 0.2S

(Standard Basic Load Combination 5)

(0.9 - 0.2SDS) D + ρQE + 1.6H

(Standard Basic Load Combination 7)

For nonstructural components, the terms L, S and H typically are zero and load combinations with overstrength generally are not applicable. 14.1.4 Component Amplification Factor The component amplification factor, ap, found in Standard Equation 13.3-1 represents the dynamic amplification of the component relative to the maximum acceleration of the component support point(s). Typically, this amplification is a function of the fundamental period of the component, Tp and the fundamental period of the support structure, T. When components are designed or selected, the effective fundamental period of the structure, T, is not always available. Also, for most nonstructural components, the component fundamental period, Tp, can be obtained accurately only by expensive shake-table or pullback tests. As a result, the determination of a component’s fundamental period by dynamic analysis, considering T/Tp ratios, is not always practicable. For this reason, acceptable values of ap are provided in the Standard tables. Therefore, component amplification factors from either these tables or a dynamic

14-6

Chapter 14: Design for Nonstructural Components analysis may be used. Values for ap are tabulated for each component based on the expectation that the component will behave in either a rigid or a flexible manner. For simplicity, a step function increase based on input motion amplifications is provided to help distinguish between rigid and flexible behavior. If the fundamental period of the component is less than 0.06 second, no dynamic amplification is expected and ap may be taken to equal 1.0. If the fundamental period of the component is greater than 0.06 second, dynamic amplification is expected and ap is taken to equal 2.5. In addition, a rational analysis determination of ap is permitted if reasonable values of both T and Tp are available. Acceptable procedures for determining ap are provided in Commentary Chapter 13. 14.1.5 Seismic Coefficient at Grade The short-period design spectral acceleration, SDS, considers the site seismicity and local soil conditions. The site seismicity is obtained from the design value maps (or software) and SDS is determined in accordance with Standard Section 11.4.4. The coefficient SDS is the used to design the structure. The Standard approximates the effective peak ground acceleration as 0.4SDS, which is why 0.4 appears in Standard Equation 13.3-1. 14.1.6 Relative Location Factor

z ⎞ ⎛ The relative location factor, ⎜ 1 + 2 ⎟ , scales the seismic coefficient at grade, resulting in values varying h ⎠ ⎝ linearly from 1.0 at grade to 3.0 at roof level. This factor approximates the dynamic amplification of ground acceleration by the supporting structure. 14.1.7 Component Response Modification Factor The component response modification factor, Rp, represents the energy absorption capability of the component’s construction and attachments. In the absence of applicable research, these factors are based on judgment with respect to the following benchmark values: §

Rp = 1.0 or 1.5: brittle or buckling failure mode is expected

§

Rp = 2.5: some minimal level of energy dissipation capacity

§

Rp = 3.5: ductile materials and detailing

§

Rp = 4.5: non-ASME B31 conforming piping and tubing with threaded joints and/or mechanical couplings

§

Rp = 6.0: ASME 31 conforming piping and tubing with thread joints and/mechanical couplings

§

Rp = 9.0 or 12.0: highly ductile piping and tubing joined with brazing or butt welding

14.1.8 Component Importance Factor The component importance factor, Ip, represents the greater of the life safety importance and the hazard exposure importance of the component. The factor indirectly accounts for the functionality of the component or structure by requiring design for a lesser amount of inelastic behavior (or higher force

14-7

FEMA P-751, NEHRP Recommended Provisions: Design Examples level). It is assumed that a lesser amount of inelastic behavior will result in a component that will have a higher likelihood of functioning after a major earthquake. 14.1.9 Accommodation of Seismic Relative Displacements The Standard requires that seismic relative displacements, Dp, be determined in accordance with several equations. For two connection points on Structure A (or on the same structural system), one at Level x and the other at Level y, Dp is determined from Standard Equation 13.3-5 as follows:

Dp = δ xA − δ yA Because the computed displacements frequently are not available to the designer of nonstructural components, one may use the maximum permissible structural displacements per Standard Equation 13.3-6:

Dp =

– hy hsx

)Δ

For two connection points on Structures A and B (or on two separate structural systems), one at Level x and the other at Level y, DP is determined from Standard Equations 13.3-7 and 13.3-8 as follows:

D p = δ xA + δ yB Dp =

hx Δ aA hy Δ aB + hsx hsx

where: Dp = seismic relative displacement that the component must be designed to accommodate. δxA = deflection of building Level x of Structure A, determined by an elastic analysis as defined in Standard Section 12.8.6 including being multiplied by the Cd factor. δyA = deflection of building Level y of Structure A, determined in the same fashion as δxA. hx = height of upper support attachment at Level x as measured from the base. hy = height of lower support attachment at Level y as measured from the base. ΔaA = allowable story drift for Structure A as defined in Standard Table 12.2-1. hsx = story height used in the definition of the allowable drift, Δa, in Standard Table 12.2-1. δyB = deflection of building Level y of Structure B, determined in the same fashion as δxA. ΔaB = allowable story drift for Structure B as defined in Standard Table 12.2-1. Note that ΔaA/hsx = the drift index.

14-8

Chapter 14: Design for Nonstructural Components The effects of seismic relative displacements must be considered in combination with displacements caused by other loads as appropriate. Specific methods for evaluating seismic relative displacement effects of components and associated acceptance criteria are not specified in the Standard. However, the intention is to satisfy the purpose of the Standard. Therefore, for nonessential facilities, nonstructural components can experience serious damage during the design-level earthquake provided they do not constitute a serious life-safety hazard. For essential facilities, nonstructural components can experience some damage or inelastic deformation during the design-level earthquake provided they do not significantly impair the function of the facility. 14.1.10

Component Anchorage Factors and Acceptance Criteria

Design seismic forces in the connected parts, Fp, are prescribed in Standard Section 13.4. Anchors embedded in concrete or masonry are proportioned to carry the least of the following: §

1.3 times the prescribed seismic design force, or

§

The maximum force that can be transferred to the anchor by the component or its support.

The value of Rp used in Section 13.3.1 to determine the forces in the connected part (i.e., the anchor) shall not exceed 1.5 unless at least one of the following conditions is satisfied: §

The component anchorage is designed to be governed by the strength of a ductile steel element.

§

The anchorage design of post-installed anchors is tested for seismic application in accordance with the procedures of ACI 355.2 and has a design capacity determined in accordance with ACI 318 Appendix D.

§

The anchor is designed in accordance with Standard Section 14.2.2.14.

Determination of design seismic forces in anchors must consider installation eccentricities, prying effects, multiple anchor effects and the stiffness of the connected system. Use of power actuated fasteners is not permitted for seismic design tension forces in Seismic Design Categories D, E and F unless approved for such loading. It should be noted that the term used in previous editions of the Standard was “powder” actuated instead of “power” actuated. The term was changed to cover a broader range of fastener types than is implied by “powder-driven”. Per Standard Sections 14.2.2.17 and 14.2.2.18, the design strength of anchors in concrete is to be determined in accordance with ACI 318 Appendix D as modified by these Standard Sections. 14.1.11

Construction Documents

Construction documents must be prepared by a registered design professional and must include sufficient detail for use by the owner, building officials, contractors and special inspectors; Standard Section 13.2.7 includes specific requirements.

14-9

FEMA P-751, NEHRP Recommended Provisions: Design Examples

14.2 ARCHITECTURAL CONCRETE WALL PANEL 14.2.1 Example Description In this example, the architectural components are a 4.5-inch-thick precast normal-weight concrete spandrel panel and a column cover supported by the structural steel frame of a five-story building, as shown in Figures 14.2-1 and 14.2-2.

Spandrel panel under consideration

Typical window frame system

Structural steel frame

5 at 13'-6" = 67'-6"

Column cover under consideration

24'-0"

Figure 14.2-1 Five-story building elevation showing panel location (1.0 ft = 0.3048 m)

14-10

Chapter 14: Design for Nonstructural Components

24'-0"

3'-0"

Typical window frame system

13'-6" story height

7'-0" column cover

T.O.S

Column cover under consideration

6'-6" panel

T.O.S

~ Spandrel panel under consideration

Figure 14.2-2 Detailed building elevation (1.0 ft = 0.3048 m) The columns at the third level of the five-story office building support the spandrel panel under consideration. The columns between the third and fourth levels of the building support the column cover under consideration. The building, located near a significant active fault in Los Angeles, California, is assigned to Occupancy Category II. Wind pressures normal to the building are 17 psf, determined in accordance with the Standard. The spandrel panel supports glass windows weighing 10 psf. This example develops prescribed seismic forces for the selected spandrel panel and prescribed seismic displacements for the selected column cover. It should be noted that details of precast connections vary according to the preferences and local practices of the precast panel supplier. In addition, some connections may involve patented designs. As a result, this example will concentrate on quantifying the prescribed seismic forces and displacements. After the prescribed seismic forces and displacements are determined, the connections can be detailed and designed according to the appropriate AISC and ACI codes and the recommendations of the Precast/Prestressed Concrete Institute (PCI).

14-11

FEMA P-751, NEHRP Recommended Provisions: Design Examples 14.2.2 Design Requirements 14.2.2.1 Provisions parameters and coefficients ap = 1.0 for wall panels

(Standard Table 13.5-1)

ap = 1.25 for fasteners of the connecting system

(Standard Table 13.5-1)

SDS = 1.487 (for the selected location and site class) Seismic Design Category = D

(given) (Standard Table 11.6-1)

Spandrel panel Wp = (150 lb/ft3)(24 ft)(6.5 ft)(0.375 ft) = 8,775 lb Glass Wp = (10 lb/ft2)(21 ft)(7 ft) = 1,470 lb

(supported by spandrel panel)

Column cover Wp = (150 lb/ft3)(3 ft)(7 ft)(0.375 ft) = 1,181 lb Rp = 2.5 for wall panels

(Standard Table 13.5-1)

Rp = 1.0 for fasteners of the connecting system

(Standard Table 13.5-1)

Ip = 1.0

z 40.5 ft = = 0.6 h 67.5 ft

(Standard Sec. 13.1.3) (at third floor)

According to Standard Section 13.3.1 (and repeated in Section 12.3.4.1 Item 3), the redundancy factor, ρ, does not apply to the design of nonstructural components and therefore may be taken as 1.0 in load combinations where it appears. 14.2.2.2 Performance criteria. Component failure must not cause failure of an essential architectural, mechanical, or electrical component (Standard Sec. 13.2.3). Component seismic attachments must be bolted, welded, or otherwise positively fastened without considering the frictional resistance produced by the effects of gravity (Standard Sec. 13.4). The effects of seismic relative displacements must be considered in combination with displacements caused by other loads as appropriate (Standard Sec. 13.3.2). Exterior nonstructural wall panels that are attached to or enclose the structure must be designed to resist the forces in accordance with Standard Section 13.3.1 and must be able to accommodate movements of the structure resulting from response to the design basis ground motion, Dp, or temperature changes (Standard Sec. 13.5.3). 14.2.3 Spandrel Panel 14.2.3.1 Connection details. Figure 14.2-3 shows the types and locations of connections that support one spandrel panel.

14-12

Chapter 14: Design for Nonstructural Components

7'-0"

column cover/ window frame

24'-0"

6'-6"

panel

3rd Story beam A

Panel C.G.

Figure 14.2-3 Spandrel panel connection layout from interior (1.0 ft = 0.3048 m) The connection system must resist the weight of the panel and supported construction including the eccentricity between that load and the supports as well as inertial forces generated by response to the seismic motions in all three dimensions. Furthermore, the connection system must not create undue interaction between the structural frame and the panel, such as restraint of the natural shrinkage of the panel or the transfer of floor live load from the beam to the panel. The panels are usually very stiff compared to the frame and this requires careful release of potential constraints at connections. PCI’s Architectural Precast Concrete (Third Edition, 1989) provides an extended discussion of important design concepts for such panels. For this example, the basic gravity load and vertical accelerations are resisted at the two points identified as A, which provide the recommended simple and statically determinant system for the main gravity weight. The eccentricity of vertical loads is resisted by a force couple at the two pairs of A1 and A connections. Horizontal loads parallel to the panel are resisted by the A connections. Horizontal loads perpendicular to the panel are resisted by two pairs of A and A1 connections and the pair of midspan B connections. The A connections, therefore, restrain movement in three dimensions while the A1 and B connections restrain movement in only one dimension, perpendicular to the panel. Connection components can be designed to resolve some eccentricities by bending of the element; for example, the eccentricity of the horizontal in-plane force with the structural frame can be resisted by bending the A connection. The practice of resisting the horizontal in-plane force at two points varies with seismic demand and local industry practice. The option is to resist all of the in-plane horizontal force at one connection in order to avoid restraint of panel shrinkage. The choice made here depends on local experience indicating that

14-13

FEMA P-751, NEHRP Recommended Provisions: Design Examples precast panels of this length have been restrained at the two ends without undue shrinkage restraint problems. The A and A1 connections are often designed to take the loads directly to the columns, particularly on steel moment frames where attachments to the flexural hinging regions of beams are difficult to accomplish. The lower B connection often requires an intersecting beam to provide sufficient stiffness and strength to resist the loads. The column cover is supported both vertically and horizontally by the column, transfers no loads to the spandrel panel and provides no support for the window frame. The window frame is supported both vertically and horizontally along the length of the spandrel panel and transfers no loads to the column covers. 14.2.3.2 Prescribed seismic forces. Lateral forces on the wall panels and connection fasteners include seismic loads and wind loads. Design for wind forces is not illustrated here. 14.2.3.2.1 Panels. D = Wp = 8,775 lb + 1,470 lb = 10,245 lb

Fp =

0.4 (1.0 )(1.487 )(10,245 lb ) (1 + 2 (0.6)) = 5,362 lb 2.5 1.0

(

)

(vertical gravity effect) (Standard Eq. 13.3-1)

Fpmax = 1.6(1.487)(1.0)(10,245 lb) = 24,375 lb

(Standard Eq. 13.3-2)

Fpmin = 0.3(1.487)(1.0)(10,245 lb) = 4,570 lb

(Standard Eq. 13.3-3)

Eh = ρQE

(Standard Eq. 12.4-3)

Ev = 0.2SDSD

(Standard Eq. 12.4-4)

where: QE (due to horizontal application of Fp) = 5,362 lb ρ = 1.0 (because panels are nonstructural components)

(Standard Sec. 12.4.2-1) (Standard Sec 13.3.1)

D = dead load effect (due to vertical load application) Substituting, the following is obtained: Eh = ρQE = (1.0)(5,362 lb) = 5,362 lb Ev = 0.2SDSD = 0.2SDSD = (0.2)(1.487)(10,245 lb) = 3,047 lb

(horizontal earthquake effect) (vertical earthquake effect)

The above terms are then substituted into the following Basic Load Combinations for Strength Design from Section 12.4.2.3 to determine the design member and connection forces to be used in conjunction with seismic loads. 14-14

Chapter 14: Design for Nonstructural Components

(1.2 + 0.2SDS) D + ρQE + L + 0.2S

(Standard Basic Load Combination 5)

(0.9 - 0.2SDS) D + ρQE + 1.6H

(Standard Basic Load Combination 7)

For nonstructural components, the terms L, S and H typically are zero and load combinations with overstrength generally are not applicable. 14.2.3.2.2 Connection fasteners. The Standard specifies a reduced Rp and an increased ap for “fasteners” with the intention of preventing premature failure in those elements of connections that are inherently brittle, such as embedments that depend on concrete breakout strength, or are simply too small to adequately dissipate energy inelastically, such as welds or bolts. The net effect more than triples the design seismic force.

Fp =

0.4 (1.25)(1.487 )(10,245 lb ) (1 + 2 (0.6)) = 16,757 lb 1.0 1.0

(

)

(Standard Eq. 13.3-1)

Fpmax = 1.6(1.487)(1.0)(10,245 lb) = 24,375 lb

(Standard Eq. 13.3-2)

Fpmin = 0.3(1.487)(1.0)(10,245 lb) = 4,570 lb

(Standard Eq. 13.3-3)

Eh = ρQE

(Standard Eq. 12.4-3)

Ev = 0.2SDSD

(Standard Eq. 12.4-4)

where: QE (due to horizontal application of Fp) = 16,757 lb ρ = 1.0 (because panels are nonstructural components)

(Standard Sec. 12.4.2-1) (Standard Sec 13.3.1)

D = dead load effect (due to vertical load application) Substituting, the following is obtained: Eh = ρQE = (1.0)(16,757 lb) = 16,757 lb Ev = 0.2SDSD = 0.2SDSD = (0.2)(1.487)(10,245 lb) = 3,047 lb

(horizontal earthquake effect) (vertical earthquake effect)

(Standard Basic Load Combination 5)

(0.9 - 0.2SDS) D + ρQE + 1.6H

(Standard Basic Load Combination 7)

For precast panels, the terms L, S and H typically are zero. Load combinations with overstrength generally are not applicable to nonstructural components. 14-15

FEMA P-751, NEHRP Recommended Provisions: Design Examples

14.2.3.3 Proportioning and design. 14.2.3.3.1 Panels. The wall panels should be designed for the following loads in accordance with ACI 318. The design of the reinforced concrete panel is standard and is not illustrated in this example. Spandrel panel moments are shown in Figure 14.2-4. Reaction shears (Vu), forces (Hu) and moments (Mu) are calculated for applicable strength load combinations. For this example, the values of F, L, S and H are assumed to be zero.

V u = Dead and/or earthquake load Strong axis

L = Column spacing M ux

Vu L 8

H u = Wind or earthquake load

Weak axis

M uy

Hu L 32

Figure 14.2-4 Spandrel panel moments Standard Basic Load Combination 1: 1.4(D + F) Vu = 1.4(10,245 lb) = 14,343 lb

M ux =

14-16

(14,343 lb )( 24 ft ) = 43,029 ft-lb 8

(from Standard Sec. 2.3.2) (vertical load downward) (strong axis moment)

Chapter 14: Design for Nonstructural Components Standard Basic Load Combination 5: (1.2 + 0.2SDS) D + ρQE + L + 0.2S Vumax = [1.2 + 0.2(1.487)] (10,245 lb) = 15,341 lb

(vertical load downward)

⇔ H u = 1.0(5,362 lbs) = 5,362 lb

(horizontal load parallel to panel)

⊥ H u = 1.0(5,362 lbs) = 5,362 lb

(horizontal load perpendicular to panel)

M uxmax = M uy =

(15,341 lb )( 24 ft ) = 46,023 ft-lb

(strong axis moment)

(5,362 lb )( 24 ft ) = 4,022 ft-lb

(weak axis moment)

Standard Basic Load Combination 7: (0.9 - 0.2SDS) D + ρQE + 1.6H Vumin = [1.2 – 0.2(1.487)] (10,245 lb) = 6,174 lb

⇔ H u = 1.0(5,362 lb) = 5,362 lb ⊥ H u = 1.0(5,362 lb) = 5,362 lb M uxmin = M uy =

( 6,174 lb )( 24 ft ) = 18,522 ft-lb 8

(5,362 lb )( 24 ft ) = 4,022 ft-lb 32

(vertical load downward) (horizontal load parallel to panel) (horizontal load perpendicular to panel) (strong axis moment)

(weak axis moment)

14.2.3.3.2 Connection fasteners. The connection fasteners should be designed for the following loads in accordance with ACI 318 (Appendix D) and the AISC specification. There are special reduction factors for anchorage in high seismic demand locations and those reduction factors would apply to this example. The design of the connection fasteners is not illustrated in this example. Spandrel panel connection forces are shown in Figure 14.2-5. Reaction shears (Vu), forces (Hu) and moments (Mu) are calculated for applicable strength load combinations.

14-17

FEMA P-751, NEHRP Recommended Provisions: Design Examples RA D = Dead E = Earthquake W = Wind R = Reaction

1'-6"

1'-4"

RA1

E, W Panel C.G. D E

Figure 14.2-5 Spandrel panel connection forces Standard Basic Load Combination 1: 1.4(D + F)

VuA =

1.4 (10, 245 lb ) 2

= 7,172 lb

(from Standard Section 2.3.2) (vertical load downward at Points A and A1)

MuA = (7,172 lb)(1.5 ft) = 10,758 ft-lb

(moment resisted by paired Points A and A1)

Horizontal couple from moment at A and A1 = 10758 / 1.33 = 8071 lb Standard Basic Load Combination 5: (1.2 + 0.2SDS) D + ρQE + L + 0.2S

⎡1.2 + 0.2 (1.487 )⎤⎦ (10,245 lb ) = 7,671 lb VuA max = ⎣ 2 ⊥ H uA = 1.0(16,757 lb)

3 = 3,142 lb 16

(vertical load downward at Point A)

(horizontal load perpendicular to panel at Points A and A1)

HAin = (7,671 lb)(1.5 ft) / (1.33 ft) + (3142 lb)(2.0 ft) / (1.33 ft) = 13366 lb

(inward force at Point A)

HA1out = (7671 lb)(1.5 ft) / (1.33 ft) + (3,142 lb)(0.67) / (1.33 ft) = 10222 lb(outward force at Point A1)

⇔ H uA =

1.0(16,757 lb) = 8,378 lb 2

Mu2A = (8,378 lb)(1.5 ft) = 12,568 ft-lb

14-18

(horizontal load parallel to panel at Point A) (flexural moment at Point A)

Chapter 14: Design for Nonstructural Components

5 ⊥ H uB = 1.0(16,757 lb) = 10,473 lb 8

(horizontal load perpendicular to panel at Points B and B1)

HB = (10,743 lb)(2.0 ft) / (1.33 ft) = 15,714 lb

(inward or outward force at Point B)

HB1 = (10,473 lb)(0.67 ft) / (1.33 ft) = 5,237 lb

(inward or outward force at Point B1)

Standard Basic Load Combination 7: (0.9 - 0.2SDS) D + ρQE + 1.6H

⎡1.2 - 0.2 (1.487 )⎤⎦ (10,245 lb ) = 3,086 lb VuA min = ⎣ 2

(vertical load downward at Point A)

Horizontal forces are the same as combination 1.2D + 1.0E. No uplift occurs; the net reaction at Point A is downward. Maximum forces are controlled by prior combination. It is important to realize that inward and outward acting horizontal forces generate different demands where the connections are eccentric to the center of mass, as in this example. Only the maximum reactions are computed above. 14.2.3.4 Prescribed seismic displacements. Prescribed seismic displacements are not applicable to the building panel because all connections are at essentially the same elevation. 14.2.4 Column Cover 14.2.4.1 Connection details. Figure 14.2-6 shows the key to the types of forces resisted at each column cover connection.

24'-0"

6'-0"

Cover C.G.

6'-6" panel

1'-3"

7'-0" column cover window frame

1'-6"

Figure 14.2-6 Column cover connection layout (1.0 ft = 0.3048 m)

14-19

FEMA P-751, NEHRP Recommended Provisions: Design Examples Vertical loads, horizontal loads parallel to the panel and horizontal loads perpendicular to the panel are resisted at Point C. The eccentricity of vertical loads is resisted by a force couple at Points C and D. The horizontal load parallel to the panel eccentricity between the panel and the support is resisted in flexure of the connection. The connection is designed to take the loads directly to the columns. Horizontal loads parallel to the panel are all resisted at Point D. The vertical load eccentricity between the panel and the support is resisted by a force couple of Points C and D. The eccentricity of horizontal loads parallel to the panel is resisted by flexure at the connection. The connection is designed to not restrict vertical movement of the panel due to thermal effects or seismic input. The connection is designed to take the loads directly to the columns. Horizontal loads perpendicular to the panel are resisted equally at Points C and D and the two points identified as E. The connections are designed to take the loads directly to the columns. There is no load eccentricity associated with the horizontal loads perpendicular to the panel. In this example, all connections are made to the sides of the column because usually there is not enough room between the outside face of the column and the inside face of the cover to allow a feasible loadcarrying connection. 14.2.4.2 Prescribed seismic forces. Calculation of prescribed seismic forces for the column cover is not shown in this example. They should be determined in the same manner as illustrated for the spandrel panels. 14.2.4.3 Prescribed seismic displacements. The results of an elastic analysis of the building structure usually are not available in time for use in the design of the precast cladding system. As a result, prescribed seismic displacements usually are calculated based on allowable story drift requirements: hsx = story height = 13’-6” hx = height of upper support attachment = 47’-9” hy = height of lower support attachment = 41’-9” ΔaA = 0.020hsx Dpmax =

(Standard Table 12.12-1)

)

– hy Δ aA hsx

( 72 in.) 0.020hsx hsx

= 1.44 in.

(Standard Eq. 13.3-6)

The joints at the top and bottom of the column cover must be designed to accommodate an in-plane relative displacement of 1.44 inches. The column cover will rotate somewhat as these displacements occur, depending on the nature of the connections to the column. If the supports at one level are “fixed” to the columns while the other level is designed to “float,” then the rotation will be that of the column at the point of attachment. 14.2.5 Additional Design Considerations 14.2.5.1 Window frame system. The window frame system is supported by the spandrel panels above and below. Assuming that the spandrel panels move rigidly in-plane with each floor level, the window frame system must accommodate a prescribed seismic displacement based on the full story height. 14-20

Chapter 14: Design for Nonstructural Components

Dpmax =

)

– hy Δ aA hsx

(162 in.) 0.020hsx hsx

= 3.24 in.

(Standard Eq. 13.3-6)

The window frame system must be designed to accommodate an in-plane relative displacement of 3.24 inches between the supports. Normally this is accommodated by a clearance between the glass and the frame. Standard Section 13.5.9.1 prescribes a method of checking such a clearance. It requires that the clearance be large enough so that the glass panel will not fall out of the frame unless the relative seismic displacement at the top and bottom of the panel exceeds 125 percent of the predicted value amplified by the building importance factor. If hp and bp are the respective height and width of individual panes and if the horizontal and vertical clearances are designated c1 and c2, respectively, then the following expression applies:

⎛ hp c2 ⎞ Dclear = 2c1 ⎜1 + ⎟ ≥ 1.25Dp ⎜ bp c1 ⎟ ⎝ ⎠

(Standard Sec. 13.5.9.1)

For hp = 7 feet, bp = 5 feet and Dp = 3.24 inches and setting c1 = c2, the required clearance is 0.84 inch. 14.2.5.2 Building corners. Some thought needs to be given to seismic behavior at external building corners. The preferred approach is to detail the corners with two separate panel pieces, mitered at a 45 degree angle, with high grade sealant between the sections. An alternative choice of detailing Lshaped corner pieces would introduce more seismic mass and load eccentricity into connections on both sides of the corner column. 14.2.5.3 Dimensional coordination. It is important to coordinate dimensions with the architect and structural engineer. Precast concrete panels must be located a sufficient distance from the building structural frame to allow room for the design of efficient load transfer connection pieces. However, distances must not be so large as to increase unnecessarily the load eccentricities between the panels and the frame.

14.3 HVAC FAN UNIT SUPPORT 14.3.1 Example Description In this example, the mechanical component is a 4-foot-high, 5-foot-wide, 8-foot-long, 3,000-pound HVAC fan unit that is supported on the two long sides near each corner (Figure 14.3-1). The component is located at the roof level of a five-story office building, near a significant active fault in Los Angeles, California. The building is assigned to Occupancy Category II. Two methods of attaching the component to the 4,000 psi, normal-weight roof slab are considered, as follows: §

Direct attachment to the structure with 36 ksi, carbon steel, cast-in-place anchors.

§

Support on vibration isolation springs that are attached to the slab with 36 ksi, carbon steel, postinstalled expansion anchors. The nominal gap between the vibration spring seismic restraints and the base frame of the fan unit is presumed to be greater than 0.25 in.

14-21

5'-0"

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Attachment location (typical). Direct attachment shown

HVAC fan unit W = 3,000 lbs

8'-0" Center-of-mass

2'-0"

4'-0"

Plan

Concrete

a = 7'-0"

b = 5'-6"

Elevation

Figure 14.3-1 HVAC fan unit (1.0 ft = 0.3048 m, 1.0 lb = 4.45 N)

14.3.2 Design Requirements 14.3.2.1 Seismic design parameters and coefficients. ap = 2.5 for both direct attachment and spring isolated

(Standard Table 13.6-1)

ap = 2.5

(Standard Table 13.6-1)

SDS = 1.487 (for the selected location and site class) Seismic Design Category = D Wp = 3,000 lb

(given) (Standard Table 11.6-1) (given)

Rp = 6.0 for HVAC fans, directly attached (not vibration isolated)

(Standard Table 13.6-1)

Rp = 2.0 for spring isolated components with restraints

(Standard Table 13.6-1)

Rp = 1.5 for anchors in concrete or masonry unless criteria of Standard Section 13.4.2 are satisfied Ip = 1.0 z/h = 1.0

14-22

(Standard Sec. 13.1.3) (for roof-mounted equipment)

Chapter 14: Design for Nonstructural Components 14.3.2.2 Performance criteria. Component failure should not cause failure of an essential architectural, mechanical, or electrical component (Standard Sec. 13.2.3). Component seismic attachments must be bolted, welded, or otherwise positively fastened without consideration of frictional resistance produced by the effects of gravity (Standard Sec. 13.4). Anchors embedded in concrete or masonry must be proportioned to carry the lesser of (a) 1.3 times the force in the component and its supports due to the prescribed forces and (b) the maximum force that can be transferred to the anchor by the component and its supports (Standard Sec. 13.4.2). Attachments and supports transferring seismic loads must be constructed of materials suitable for the application and must be designed and constructed in accordance with a nationally recognized structural standard (Standard Sec. 13.6.5). Components mounted on vibration isolation systems must have a bumper restraint or snubber in each horizontal direction. Vertical restraints must be provided where required to resist overturning. Isolator housings and restraints must also be constructed of ductile materials. A viscoelastic pad, or similar material of appropriate thickness, must be used between the bumper and equipment item to limit the impact load (Standard Table 13.6-1, footnote b). Such components also must resist doubled seismic design forces if the nominal clearance (air gap) between the equipment support frame and restraints is greater than 0.25 in. (Standard Table 13.6-1, footnote b). 14.3.3 Direct Attachment to Structure This section illustrates design for cast-in-place concrete anchors that satisfies the requirements of ACI 318, Appendix D, where the component anchorage embedment strength is greater than the strength capacity of the ductile steel anchorage element. Therefore, the Rp of the component (Rp = 6) can be used for the component anchorage design. 14.3.3.1 Prescribed seismic forces. See Figure 14.3-2 for a free-body diagram for seismic force analysis.

ρ QE D

2'-0"

0.2S DS D

Tu Ru 5'-6"

Figure 14.3-2 Free-body diagram for seismic force analysis (1.0 ft = 0.348 m)

Fp =

0.4(2.5)(1.487)(3,000 lb) ⎡⎣1 + 2 (1)⎤⎦ = 2,231 lb (6.0 / 1.0)

(Standard Eq. 13.3-1)

14-23

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Fpmax = 1.6(1.487)(1.0)(3,000 lb) = 7,138 lb

(Standard Eq. 13.3-2)

Fpmin = 0.3(1.487)(1.0)(3,000 lb) = 1,338 lb

(Standard Eq. 13.3-3)

Since Fp is greater the Fpmin and less than Fpmax, the value determined from Equation 13.3-1 applies. Eh = ρQE

(Standard Eq. 12.4-3)

Ev = 0.2SDSD

(Standard Eq. 12.4-4)

where: QE (due to horizontal application of Fp) = 2,231 lb ρ = 1.0 (HVAC units are nonstructural components)

(Standard Sec. 12.4.2-1) (Standard Sec. 13.3.1)

D = dead load effect (due to vertical load application) Substituting, one obtains: Eh = ρQE = (1.0)(2,231 lb) = 2,231 lb Ev = 0.2SDSD = 0.2SDSD = (0.2)(1.487)(3,000 lb) = 892 lb

(horizontal earthquake effect) (vertical earthquake effect)

The above terms are then substituted into the following Basic Load Combinations for Strength Design of Section 12.4.2.3 to determine the design member and connection forces to be used in conjunction with seismic loads. (1.2 + 0.2SDS) D + ρQE + L + 0.2S

(Standard Basic Load Combination 5)

(0.9 - 0.2SDS) D + ρQE + 1.6H

(Standard Basic Load Combination 7)

Based on the free-body diagram, the seismic load effects can be used to determine bolt shear, Vu and tension, Tu (where a negative value indicates tension). In the calculations below, the signs of SDS and Fp have been selected to result in the largest value of Tu. U = (1.2 + 0.2SDS) D + ρQE Vu =

1.0 (2,231 lb) = 558 lb/bolt 4 bolts

⎡1.2 - 0.2 (1.487 )⎤⎦ ( 3,000 lb )( 2.75 ft ) - 1.0 ( 2,231) ( 2 ft ) = 299 lb/bolt (no tension) Tu = ⎣ (5.5 ft)(2 bolts) U = (0.9 - 0.2SDS)D + ρQE

14-24

Chapter 14: Design for Nonstructural Components

Vu =

1.0 (2,231 lb) = 558 lb/bolt 4 bolts

⎡0.9 - 0.2 (1.487 )⎤⎦ (3,000 lb )( 2.75 ft ) - 1.0 ( 2,231 lb ) ( 2 ft ) = 46 lb/bolt (no tension) Tu = ⎣ (5.5 ft)(2 bolts) 14.3.3.2 Proportioning and design. See Figure 14.3-3 for anchor for direct attachment to structure. Check one 1/4-inch-diameter cast-in-place anchor embedded 2 inches into the concrete slab with no transverse reinforcing engaging the anchor and extending through the failure surface. Although there is no required tension strength on these anchors, design strengths and tension/shear interaction acceptance relationships are calculated to demonstrate the use of the Standard equations. Assume the following material properties have been specified for the concrete slab and anchors: fc’ = 4,000 psi, normal weight concrete, λ = 1.0, with no supplementary anchor reinforcing provided Ase,N = Ase,V = 0.032 in2 fy = 36,000 psi (ductile steel) futa = 58,000 psi

Tu Component base

Figure 14.3-3 Anchor for direct attachment to structure 14.3.3.2.1 Design tension strength on isolated anchor in slab, away from edge, loaded concentrically. Tension capacity of steel, φ = 0.75 (ductile steel element):

φNsa = φ Ase,NFuta = 0.75(0.032 in2)(58,000 psi) = 1,392 lb

(ACI 318-08 Eq. D-1, D-3)

Tension capacity of concrete, φ = 0.70 and apply an additional 0.75 factor to determine seismic concrete capacities as required by Sections D.3.3 and D.3.3.3 of ACI 318-08. Also assume configuration is such that there is no eccentricity, no pull-through and no edge or group effect:

0.75φ Nb = 0.75φ kc λ fcʹ′hef1.5 = 0.75(0.70)24(1.0) 4,000(21.5 ) = 2,254 lb

(ACI 318-08 Eq. D-1, D-7)

14-25

FEMA P-751, NEHRP Recommended Provisions: Design Examples

The steel capacity controls in tension, so Section D.3.3.4 of ACI 318-08 is satisfied, but since there is no tension demand, the point is moot. 14.3.3.2.2 Design shear strength on isolated anchor, away from edge. Shear capacity of steel, φ = 0.65 (ductile steel element):

φVsa = φ 0.6Ase,VFuta = 0.65(0.6)(0.032 in2)(58,000 psi) = 724 lb

(ACI 318-08 Eq. D-1, D-19)

Shear capacity of concrete, far from edge, limited to pry-out, no supplementary reinforcing, φ = 0.70:

φVcp = φ kcpNcb = φ kcpNsa = φ kcpAse,Nfuta

(ACI 318-08 Eq. D-1, D-4, D-30)

φVcp = 0.70(1.0)(0.032 in2)(58,000 psi) = 1,299 lb The steel capacity controls in shear, with φVN = φVsa = 724 lb. Per Standard Section 13.4.2, anchors embedded in concrete or masonry must be proportioned to carry at least 1.3 times the force in the connected part due to the prescribed forces. Thus, Vu = 1.3(558) = 725 pounds, so the anchor is inadequate (mathematically), but would be deemed acceptable given the practical precision of engineering design. 14.3.3.2.3 Combined tension and shear. ACI 318-08 provides an equation (Eq. D-29) for the interaction of tension and shear on an anchor or a group of anchors:

Nu V + u ≤ 1.2 , which applies where either term exceeds 0.2 φ N N φVN Since the tension demand, Nu, is zero, by inspection, this equation is also satisfied. 14.3.3.2.3 Summary. At each corner of the component, provide one 1/4-inch-diameter cast-in-place anchor embedded 2 inches into the concrete slab. Transverse reinforcement engaging the anchor and extending through the failure surface is not necessary. 14.3.4 Support on Vibration Isolation Springs 14.3.4.1 Prescribed seismic forces. Design forces for vibration isolation springs with the seismic stop gap clearance presumed to be greater than 1/4 inch are determined by an analysis of earthquake forces applied in a diagonal horizontal direction as shown in Figure 14.3-4. Terminology and concept are taken from ASHRAE APP IP. In the equations below, Fpv = Ev = 0.2SDSWp: Angle of diagonal loading:

⎛ b ⎞ θ = tan -1 ⎜ ⎟ ⎝ a ⎠ Tension per isolator:

14-26

(ASHRAE APP IP Eq. 17)

Chapter 14: Design for Nonstructural Components

Tu =

Wp − Fpv

−

Fp h ⎛ cosθ sin θ ⎞ + ⎜ a ⎟⎠ 2 ⎝ b

(ASHRAE APP IP Eq. 18)

Compression per isolator:

Cu =

Wp + Fpv 4

Fp h ⎛ cosθ sin θ ⎞ + ⎜ ⎟ a ⎠ 2 ⎝ b

(ASHRAE APP IP Eq. 19)

Shear per isolator:

Vu =

(ASHRAE APP IP Eq. 20)

X 3

4 Fp F p cos (θ )

θ Y

Y F p sin (θ )

W P = Operating weight of equipment F P = Seismic horizontal force FPV = Seismic vertical force a = Distance between vibration isolators along Y-Y b = Distance between vibration isolators along X-X h = Height of center of gravity = Vibration isolator location

2 X

Plan view

F PV

F p cos (θ )

F p sin (θ )

Figure 14.3-4 ASHRAE diagonal seismic force analysis for vibration isolation springs Select the worst-case assumption. Design for post-installed expansion anchors, requiring the use of Rp = 2.0 and nominal clearance between equipment and restraint greater than 1/4 inch. Fp =

0.4(2.5)(1.487) (3,000 lb ) (2.0/1.0)

(1 + 2(1) ) = 6,692 lb

(Standard Eq. 13.3-1)

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Fpmax = 1.6(1.487)(1.0)(3,000 lb) = 7,138 lb

(Standard Eq.13.3-2)

Fpmin = 0.3(1.487)(1.0)(3,000 lb) = 1,338 lb

(Standard Eq. 13.3-3)

Components mounted on vibration isolation systems must have a bumper restraint or snubber in each horizontal direction. Per Standard Table 13.6-1, footnote b, the design force must be taken as 2Fp if nominal clearance (air gap) between equipment and seismic restraint is greater than 0.25 inch. QE = 2Fp = 2(6,692 lb) = 13,384 lb

(Standard Sec. 6.1.3)

ρ = 1.0 (HVAC units are nonstructural components)

(Standard Sec. 13.3.1)

ρQE = (1.0)(13,384 lb) = 13,384 lb

(horizontal earthquake effect)

Fpv(ASHRAE) = 0.2SDSD = (0.2)(1.487)(3,000 lb) = 892 lb

(vertical earthquake effect)

D = Wp = 3,000 lb

(vertical gravity effect)

The above terms are then substituted into the Basic Load Combinations for Strength Design of Section 12.4.2.3 to determine the design member and connection forces to be used in conjunction with seismic loads. (1.2 + 0.2SDS) D + ρQE + L + 0.2S

(Standard Basic Load Combination 5)

(0.9 - 0.2SDS) D + ρQE + 1.6H

(Standard Basic Load Combination 7)

These seismic load effects can be used to determine bolt shear, Vu and tension, Tu (where a negative value indicates tension). In the calculations below, the signs of SDS and Fp have been selected to result in the largest value of Tu. U = (1.2 + 0.2SDS)D + ρQE Tu =

Cu =

Vu =

1.2 (3,000 lb ) – (892 lb ) (13,384 lb )( 2 ft ) ⎡ cos (51.8 deg ) sin (51.8 deg ) ⎤ + ⎢ ⎥ = -2,405 lb 4 2 7 ft 5.5 ft ⎣ ⎦ 1.2 (3,000 lb ) + (892 lb ) 4

13,384 lb = 3,346 lb 4

U = (0.9 - 0.2SDS)D + ρQE

⎛ 7 ft ⎞ = 51.8o ⎟ ⎝ 5.5 ft ⎠

θ = tan −1 ⎜

14-28

(13,384 lb )( 2 ft ) ⎡ cos (51.8 deg ) 2

⎢ ⎣

7 ft

sin (51.8 deg ) ⎤ ⎥ = 4,204 lb 5.5 ft ⎦

Chapter 14: Design for Nonstructural Components

Tu =

Cu =

Vu =

0.9 (3,000 lb ) – (892 lb ) (13,384 lb )( 2 ft ) ⎡ cos (51.8 deg ) sin (51.8 deg ) ⎤ + ⎢ ⎥ = 2,630 lb 4 2 7 ft 5.5 ft ⎣ ⎦ 0.9 (3,000 lb ) + (892 lb ) 4

(13,384 lb )( 2 ft ) ⎡ cos (51.8 deg ) 2

⎢ ⎣

7 ft

sin (51.8 deg ) ⎤ ⎥ = -3,980 lb 5.5 ft ⎦

13,384 lb = 3,346 lb 4

Note that the above values need to be multiplied by 1.3 per Section 13.4.2 to obtain the design values to be compared with design capacities. Therefore, the following values will be used for checking against design capacities of the post-installed anchors. Tension = 1.3(2,630 lb) = 3,419 lb Shear = 1.3(3,980 lb) = 5,174 lb 14.3.4.2 Proportioning and details. Anchor and snubber loads for support on vibration isolation springs are shown in Figure 14.3-5. Check the vibration isolation system within a housing anchored with one 5/8-inch-diameter post-installed wedge expansion anchors embedded 9 inches into the 10-inch-thick, 4,000 psi concrete housekeeping pad. The expansion anchors selected are specifically designed for seismic forces and have been tested for use in cracked concrete, following the procedures of ACI 355.2. Unless at least one of three conditions is satisfied, Standard Section 13.4.2 requires that the anchorage design forces be determined using Rp = 1.5. One of these conditions is that post-installed anchors be tested to verify their seismic capacity in accordance with the procedures of ACI 355.2. Since anchors selected for this example have been tested per ACI 355.2, this condition has been satisfied and anchorage design forces do not need to be modified and are those provided in Section 14.3.4.1 of this example. For the purpose of this example it is assumed that there are no edge effects or groups effects. Also, for this example the local prying effects of attachment members are assumed to be negligible. In most real cases these effects will exist and would either increase the design anchor force or reduce the capacity of the anchor. The Standard refers to ACI 318 to determine the capacity of anchors, including the design strength of post-installed expansion anchors. ACI 318 refers to the testing procedures of ACI 355.2-01 for verifying the seismic capacities of post-installed anchors. The ICC Evaluation Service provides testing procedures and acceptance criteria for post-installed anchors for both non-seismic and seismic applications. For nonstructural components, non-seismic post-installed anchors are permitted but are severely penalized. This example uses seismic anchors, which have been tested for use in cracked concrete and have ICC-ES reports approving their used for seismic applications consistent with the requirements of ACI 318 Appendix D and ACI 355.2. Assume the following material properties have been specified for the concrete slab and anchors: fc’ = 4,000 psi, normal weight concrete, λ = 1.0, with no supplementary anchor reinforcing provided For post-installed anchors, assume sleeves extend through shear plane Ase,N = Ase,V = 0.17 in2 14-29

FEMA P-751, NEHRP Recommended Provisions: Design Examples

fy = 84,800 psi (ductile steel) futa = 106,000 psi

Tu Equipment frame Vu

H = 5"

Vibration isolator w/ seismic housing

"O" Vb

Tb W = 4"

Figure 14.3-5 Anchor and snubber loads for support on vibration isolation springs (1.0 in. = 25.4 mm) 14.3.4.2.1 Tension strength of an isolated anchor in slab, away from edge, with no group effects. Tension capacity of steel, φ = 0.75 (ductile steel element):

φNsa = φ Ase,NFuta = 0.75(0.17 in2)(106,000 psi) = 13,515 lb

(ACI 318-08 Eq. D-1, D-3)

Tension capacity of concrete, φ = 0.65 and apply an additional 0.75 factor to determine seismic concrete capacities as required by Sections D.3.3 and D.3.3.3 of ACI 318-08. Also assume configuration is such that there is no eccentricity, no pull-through and no edge or group effect:

0.75φ Nb = 0.75φ kc λ fcʹ′hef1.5 = 0.75(0.65)17(1.0) 4,000(91.5 ) = 14,151 lb (ACI 318-08 Eq. D-1, D-7) The steel capacity controls in tension, so Section D.3.3.4 of ACI 318-08 is satisfied. Therefore, φNn = φNsa = 13,515 lb, so tension alone is okay since capacity is greater than demand. 14.3.4.2.2 Design shear strength on isolated anchor in slab, away from edge. Allowable stress shear values are obtained from ICC Evaluation Service ESR-2251, ITW Trubolt Wedge anchors. Similar certified allowable values are expected with anchors from other manufacturers. Anchor diameter = 5/8 in. Anchor depth = 9 in. 14-30

Chapter 14: Design for Nonstructural Components

fc' = 4,000 psi Shear capacity of steel, φ = 0.65:

φVsa = φ 0.60Ase,V futa = 0.65(0.60)(0.17 in2)(106,000 psi) = 7,030 lb

(ACI 318-08 Eq. D-1, D-19)

Shear capacity of concrete, far from edge, limited to pry-out, no supplementary reinforcing, φ = 0.70:

φ Vcp = φ kcpNcb = φ kcpNsa = φ kcp Ase,N futa

(ACI 318-08 Eq. D-1, D-4, D-30)

φ Vcp = 0.70(1.0)(0.17 in2)(106,000 psi) = 12,614 lb The steel capacity controls in shear, with φVN = φVsa = 7,030 lb 14.3.4.2.3 Combined tension and shear. ACI 318-08 provides an equation (Eq. D-32) for the interaction of tension and shear on an anchor or a group of anchors: Nu V + u ≤ 1.2 , which applies where either term exceeds 0.2 φ N N φVN

Substituting the demand and capacities computed above, one obtains: (3,419/13,515) + (5,174/7,030) = 0.99 which is less than 1.2 Therefore, the preliminary anchor design is okay. 14.3.4.2.4 Summary. At each corner of the HVAC fan unit, a proposed design provides a vibration isolation system within a housing anchored with one 5/8-inch-diameter post-installed expansion anchor embedded 9 inches into the concrete slab. When checked against the design loadings, these anchors work, but may not be practical in normal design situations. Other anchor configurations should be investigated, such as two anchors or four anchors per corner, which would reduce both the size of anchor and depth of embedment and required concrete pad thickness. Also, consideration should be given to requiring a limit of gap clearance to ¼ inch (which would require special inspection during construction to be sure it happened) to reduce design seismic forces on seismic restraints and associated anchorage. 14.3.5 Additional Considerations for Support on Vibration Isolators Vibration isolation springs are provided for equipment to prevent vibration from being transmitted to the building structure. However, they provide virtually no resistance to horizontal seismic forces. In such cases, some type of restraint is required to resist the seismic forces. Figure 14.3-6 illustrates one concept where a bolt attached to the equipment base is allowed to slide a controlled distance (gap) in either direction along its longitudinal axis before it contacts resilient impact material.

14-31

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Equipment frame

Gap

Impact material Steel bushing bolted or welded to equipment frame

Figure 14.3-6 Lateral restraint required to resist seismic forces Design of restraints for vibration-isolated equipment varies for different applications and for different manufacturers. In most cases, restraint design incorporates all directional capability with an air gap, a soft impact material and a ductile restraint or housing. Restraints should have all-directional restraint capability to resist both horizontal and vertical motion. Vibration isolators have little or no resistance to overturning forces. Therefore, if there is a difference in height between the equipment's center of gravity and the support points of the springs, rocking is inevitable and vertical restraint is required. An air gap between the restraint device and the equipment prevents vibration from transmitting to the structure during normal operation of the equipment. Air gaps generally are no greater than 1/4 inch. Dynamic tests indicate a significant increase in acceleration for air gaps larger than 1/4 inch. A soft impact material, often an elastomer such as bridge bearing neoprene, reduces accelerations and impact loads by preventing steel-to-steel contact. The thickness of the elastomer can significantly reduce accelerations to both the equipment and the restraint device and should be addressed specifically for lifesafety applications. A ductile restraint or housing is critical to prevent catastrophic failure. Unfortunately, housed isolators made of brittle materials such as cast iron often are assumed to be capable of resisting seismic loads and continue to be installed in seismic zones. Overturning calculations for vibration-isolated equipment must consider a worst-case scenario as illustrated in Section 14.3.4.1. However, important variations in calculation procedures merit further discussion. For equipment that is usually directly attached to the structure or mounted on housed vibration isolators, the weight can be used as a restoring force since the equipment will not transfer a tension load to the anchors until the entire equipment weight is overcome at any corner. For equipment installed on any other vibration-isolated system (such as the separate spring and snubber arrangement shown in Figure 13.3-5), the weight cannot be used as a restoring force in the overturning calculations.

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Chapter 14: Design for Nonstructural Components

As the foregoing illustrates, design of restraints for resiliently mounted equipment is a specialized topic. The Standard sets out only a few of the governing criteria. Some suppliers of vibration isolators in the highest seismic zones are familiar with the appropriate criteria and procedures. Consultation with these suppliers may be beneficial.

14.4 ANALYSIS OF PIPING SYSTEMS 14.4.1 ASME Code Allowable Stress Approach Piping systems typically are designed to satisfy national standards such as ASME B31.1. Piping required to be designed to other ASME piping codes uses similar approaches with similar definition of terms. 14.4.1.1 Earthquake design requirements. ASME B31.1 Section 101.5.3 requires that the effects of earthquakes, where applicable, be considered in the design of piping, piping supports and restraints using data for the site as a guide in assessing the forces involved. However, earthquakes need not be considered as acting concurrently with wind. 14.4.1.2 Stresses due to sustained loads. The effects of pressure, weight and other sustained loads must meet the requirements of ASME B31.1 Equation 11A:

SL =

PDo 0.75iM A + ≤ 1.0 S h Z 4tn

where: SL = sum of the longitudinal stresses due to pressure, weight and other sustained loads P = internal design pressure, psig Do = outside diameter of pipe, in. tn = nominal pipe wall thickness, in. i = stress intensification factor from ASME Piping Code Appendix D, unitless = 1.0 for straight pipe ≥ 1.0 for fittings and connections MA = resultant moment loading on cross section due to weight and other sustained loads, in-lb Z = section modulus, in3 Sh = basic material allowable stress at maximum (hot) temperature from ASME Piping Code Appendix A For example, ASTM A53 seamless pipe and tube, Grade B: Sh = 15.0 ksi for -20 to 650 degrees Fahrenheit. 14.4.1.3 Stresses due to occasional loads. The effects of pressure, weight and other sustained loads and occasional loads including earthquake, must meet the requirements of ASME B31.1 Equation 12A: 14-33

FEMA P-751, NEHRP Recommended Provisions: Design Examples

PDo 0.75iM A 0.75iM B + + ≤ kS h Z Z 4tn where: MB = resultant moment loading on cross-section due to occasional loads, such as from thrust loads, pressure and flow transients and earthquake. Use one-half the earthquake moment range. Effects of earthquake anchor displacements may be excluded if they are considered in Equation 13A, in-lb. k= = = =

duration factor, unitless. 1.15 for occasional loads acting less than 10 percent of any 24-hour operating period. 1.20 for occasional loads acting less than 1 percent of any 24-hour operating period. 2.00 for rarely occurring earthquake loads resulting from both inertial forces and anchor movements (per ASME interpretation).

14.4.1.4 Thermal expansion stress range. The effects of thermal expansion must meet the requirements of ASME B31.1 Equation 13A:

SE =

iM C ≤ S A + f ( Sh − S L ) Z

where: SE = sum of the longitudinal stresses due to thermal expansion, ksi. MC = range of resultant moments due to thermal expansion. Also includes the effects of earthquake anchor displacements if not considered in Equation 12A, in-lb. SA = allowable stress range, ksi (per ASME B31.1 Eq. 1, S A = f (1.25Sc + 0.25Sh ) ). f=

stress range reduction factor for cyclic conditions from the ASME Piping Code Table 102.3.2.

Sc = basic material allowable stress at minimum (cold) temperature from the ASME Piping Code Appendix A. 14.4.1.5 Summary. In the ASME B31.1 allowable stress approach, earthquake effects appear only in the MB and MC terms. MB represents earthquake inertial effects and MC represents earthquake displacement effects. 14.4.2 Allowable Stress Load Combinations ASME B31.1 uses an allowable stress approach; therefore, allowable stress force levels and allowable stress load combinations should be used. While the Standard is based on strength design, Section 13.1.7 of the Standard permits the use of allowable stress design for nonstructural components where the reference document that is used for earthquake design of the component provides its acceptance criteria in terms of allowable stresses rather than strengths. For such cases, the allowable stress load combinations must consider dead, live, operating and earthquake loads in addition to those specified in the reference

14-34

Chapter 14: Design for Nonstructural Components document. The earthquake loads determined in accordance with Standard Section 13.3.1 are multiplied by a factor of 0.7. Also, the allowable stress load combinations of Sections 2.4 and 12.4 of the Standard need not be used and the allowable stress increases for load combination which include seismic loads are permitted. Where earthquake effects are not considered, load combinations should be taken from the appropriate piping system design code. Section 13.1.7 also points out that the component and system must accommodate the relative displacements specified in Standard Section 13.3.2. 14.4.2.1 Standard allowable stress load combinations. The following allowable stress load combinations, specified in Standard Section 12.4.3, are permitted to be used in the design of nonstructural components. The load combinations have been adjusted to apply specifically to piping by deleting the terms F and H and substituting with Fp for QE. Other operational loads should be added using a load factor of 1.0. No increases in allowable stress are permitted for these load combinations: (1.0 + 0.14SDS)D + 0.7Fp (0.6 - 0.14SDS)D + 0.7Fp 14.4.2.2 Traditional allowable stress load combinations. The following allowable stress load combinations traditionally have been used in the design of many nonstructural components, including piping. Increases in allowable stress (typically 1/3) generally have been permitted for these combinations: (1.0 + 0.14SDS)D +0.7Fp (0.9 - 0.14SDS)D + 0.7Fp 14.4.2.3 Modified traditional allowable stress load combinations. It is convenient to define separate earthquake load terms to represent the separate inertial and displacement effects: EI = Earthquake horizontal inertial effects (MB term) EΔ = Earthquake displacement effects (MC term) It is also convenient to use the Traditional Allowable Stress Load Combinations modified to use ASME Piping Code terminology, deleting roof load effects (Lr or S or R) and multiplying by 0.75 to account for the 1.33 allowable stress increase where W or E is included. Only modified traditional allowable stress load combinations are considered in the discussion that follows. 0.75[(1.0 + 0.14SDS)D + L + S + 0.7(EI + EΔ)] 0.75[(0.9 + 0.14SDS)D + 0.7 (EI + EΔ)] By replacing EI with Fp, the following is obtained: 0.75[(1.0 + 0.14SDS)D + L + S + 0.7(Fp + EΔ)] 0.75[(0.9 + 0.14SDS)D + 0.7(Fp + EΔ)]

14-35

FEMA P-751, NEHRP Recommended Provisions: Design Examples 14.4.3 Application of the Standard 14.4.3.1 Overview. Piping systems are considered mechanical components. Mechanical component are exempt from the seismic requirements of Standard Chapter 13 under certain conditions, which are listed in Section 13.1.4. Furthermore, under certain conditions listed in Section 13.6.8, pipe supports are not required to be designed to the seismic requirements of Standard Chapter 13. There are many different types of piping systems. Standard Section 13.6.8.1 states that pressure piping designed in accordance ASME B31 is deemed to comply with the force, relative displacement and other requirements of the Standard and that elevator piping must be designed to ASME 17.1. Section 13.6.8.1 requires that the force and relative displacement requirements of Section 13.3.1 be used in lieu of specific force and displacement requirements of ASME B31. Fire sprinkler systems designed and installed in accordance with NPFA 13-2007 are deemed to comply with the force and displacement requirements of the Standard (see Provisions Section 13.6.8.2). Other piping is required to satisfy Standard Section 13.6.11, which refers to Sections 13.4, 13.6.5.3 and 13.6.5. Section 13.6.5 requires piping supports to be designed for the forces and displacements of Standard Sections 13.3.1 and 13.3.2. Section 13.4 provides special requirements for the design and detailing of anchorage that supports or anchors nonstructural components including piping. Standard Section 13.6.3 requires that mechanical components, including piping, that are assigned an Ip greater than 1.0 themselves be designed for the forces and displacements of Sections 13.3.1 and 13.3.2. Also, where Ip is greater than 1.0, the following additional requirements are imposed. §

Provision must be made to eliminate seismic impact for components vulnerable to impact, for components constructed of nonductile materials and in cases where material ductility will be reduced (e.g., low temperature applications).

§

The possibility of loads imposed on components by attached utility or service lines due to differential movement of support points on separate structures must be evaluated.

§

Where piping or HVAC ductwork components are attached to structures than could displace relative to one another and for isolated structures where the components cross the isolation interface, the components must be designed to accommodate the seismic relative displacements defined in Standard Section 13.3.2.

14.4.3.2 Forces. Standard Section 13.3.1 and Table 13.6-1 provide specific guidance regarding the equivalent static forces that must be considered in the design of piping systems. Note that the piping itself need only be designed for seismic forces and displacements if required by the reference standard (i.e., ASME 17.1, ASME B31.1, or NPFA-13) or if the value of Ip assigned to the piping is greater than 1.0. In computing the earthquake forces for piping systems, the inertial portion of the forces (noted as EI in this example) are computed using Standard Equations 13.3-1, 13.3-2 and 13.3-3. Depending on the type of piping system, the value specified for Rp ranges between 3 and 12 while the value of ap equals 2.5 for all values. For anchor points with different elevations, the average value of the Fp may be used, with minimum and maximum observed. In addition, the vertical effects should be considered, as illustrated in Section 14.4.2.

⎛ 0.2S DS ⎞ by the variable β. It is convenient to designate the term ⎜1 ± 1.4 ⎟⎠ ⎝

14-36

Chapter 14: Design for Nonstructural Components The vertical component of EI can now be defined as βMa and applied to all load combinations that include E I. MB can now be defined as the resultant moment induced by the design force 0.7Fp, where Fp is as defined by Standard Equations 13.3-1, 13.3-2 and 13.3-3. 14.4.3.3 Displacements. Standard Section 13.3.2 provides specific guidance regarding the relative displacements that must be considered. Typically piping systems are designed considering forces and displacements using elastic analysis and allowable stresses for code prescribed wind and seismic equivalent static forces in combination with operational loads. However, no specific guidance is provided in the Standard except to say that the relative displacements should be “designed for”. The intent of this wording was not to require that a piping system remain elastic. Indeed, many types of piping systems typically are very ductile and can accommodate large amounts of inelastic strain while still functioning quite satisfactorily. What was intended was that the relative displacements between anchor and constraining points that displace significantly relative to one another be demonstrated to be accommodated by some rational means. This accommodation can be made by demonstrating that the pipe has enough flexibility or inelastic strain capacity to accommodate the displacement by providing loops in the pipeline to permit the displacement or by adding flex lines or articulating couplings which provide free movement to accommodate the displacement. Sufficient flexibility may not exist where branch lines may be forced to move with a ceiling or where other structural systems are connected to main lines. Often this “accommodation” is done by using engineering judgment, without calculations. However, if relative displacement calculations were required for a piping system, a flexibility analysis would be required. A flexibility analysis is one in which a pipe is modeled as a finite element system with commercial pipe stress analysis programs (such as Autopipe or CAESAR II) and the points of attachment are displaced by the prescribed relative displacements. The allowable stress for such a condition may be significantly greater than the normal allowable stress for the pipe. The internal moments resulting from support displacement may be computed by means of elastic analysis programs using the maximum computed relative displacements as described earlier and then adjusted. Adjustments should be in accordance with reference standards. 14.4.3.4 Load combinations. Combining ASME B31.1 Equations 12A and 13A with modified traditional allowable stress load combinations of Section 14.4.2.3 yields the following: For the modified traditional equation:

PDo ⎛ 0.75iM A ⎞ 0.75iM B + β ⎜ ≤ kSh ⎟ ± 4tn Z Z ⎝ ⎠ iM C ≤ S A + f ( Sh − S L ) Z and for the modified IBC Equation 16-18:

PDo ⎛ 0.75iM A ⎞ 0.75iM B + 0.9β ⎜ ≤ kSh ⎟ − 4tn Z Z ⎝ ⎠

14-37

FEMA P-751, NEHRP Recommended Provisions: Design Examples

iM C ≤ S A + f ( Sh − S L ) Z where: β = (1.0 + 0.105SDS) in the first equation and (0.9 - 0.105SDS) in the second equation. MA = the resultant moment due to weight (including pipe contents). MB = the resultant moment induced by the design force 0.7Fp where Fp is as defined by Standard Equations 13.3-1, 13.3-2 and 13.3-3. MC = the resultant moment induced by the design relative seismic displacement Dp where Dp is as defined in Standard Sections 13.3.3 and 13.6.3. SDS and Wp, are as defined in the Standard. P, Do, tn, I, Z, k, Sh, SA, f and SL are as defined in ASME B31.1.

14.5 PIPING SYSTEM SEISMIC DESIGN 14.5.1 Example Description This example illustrates seismic design for a portion of a piping system in an acute care hospital. It illustrates determination of the seismic demands on the system, consideration of anchorage and bracing of the system and design for system displacements within the structure and between structures. The example focuses on the determination of force and displacement demands on the different components of the system. The sizing of the various elements (braces, anchor bolts, etc.) are not covered in detail. The Standard provides requirements for three types of piping systems: ASME B31 pressure piping systems (Sec. 13.6.8.1), fire protection piping systems in accordance with NFPA 13 (Sec. 13.6.8.2 and 13.6.8.3) and other piping systems (Sec. 13.6.8.4). This example considers three piping runs of a chilled water piping system supported from the roof of a two-story structure. The system is not intended to meet the ASME 31 requirements and, therefore, is designed to the “other piping system” requirements of the Standard. The piping system is illustrated in Figures 14.5-1 and 14.5-2 and a typical trapeze-type support assembly is shown in Figures 14.5-3 and 14.5-4. One run of the piping system crosses a seismic separation joint to enter an adjacent structure. The building, located in the San Francisco Bay area of California, is assigned to Occupancy Category IV.

14-38

Chapter 14: Design for Nonstructural Components

Structural Separation Joint 1

2 4'-0"

40'-0"

Transverse Brace Lateral Brace

Pipe Run "A" - 4 inch dia. Pipe Run "B" - 6 inch dia. Pipe Run "C" - 4 inch dia.

2'-6"

Support 1

Support 2 Longitudingal Braces

Support 3 Longitudingal Braces

A -

40'-0"

B -

Transverse Brace

Plan

Support 5

Figure 14.5-1 Piping system

2 Structural Separation Joint

5'-0"

2'-6"

Roof

Support 1 Pipe "A"

Support 2

Support 3

Pipes "A", "B", and "C"

Support M

Pipe "C"

7'-6"

Mechanical Unit

Level 2

Section

Support 4

Figure 14.5-2 Piping system

14-39

FEMA P-751, NEHRP Recommended Provisions: Design Examples

2'-6"

"A"

"B"

"C" 1

A -

Section Support 1

2'-6"

Figure 14.5-3 Typical trapeze-type support assembly

1 1

B -

Section Support 1

Figure 14.5-4 Typical trapeze-type support assembly

14-40

Chapter 14: Design for Nonstructural Components 14.5.1.1 Earthquake design requirements. Earthquake design requirements for piping systems in the Standard depends on the system importance factor (Ip), the pipe diameter and the installation geometry. The importance factor is determined in Standard Section 13.1.3. Given that the structure is assigned to Occupancy Category IV, the components are assigned Ip = 1.5, unless it can be shown that the component is not needed for continued operation of the facility and failure of the component would not impair operations. Since failure of the piping system will result in flooding of the hospital, Ip = 1.5. Some piping is exempt from some or all of the seismic requirements, provided it meets the criteria in Sections 13.1.4 or 13.6.8. The user should refer to important errata in Standard Section 13.1.4, which addresses exemptions. The exemptions in Section 13.1.4 apply only to components with Ip = 1.0 and therefore are not applicable to this example. Section 13.6.8 waives seismic support requirements for piping supported by rods less than 12 inches long and for small-diameter high-deformability piping. Our example piping meets neither condition, so seismic supports will be required. "Other" piping systems must meet the following requirements of the Standard: §

Section 13.4: Nonstructural Component Anchorage

§

Section 13.6.3: Mechanical Components

§

Section 13.6.5: Component Supports

It is important to note that per Standard Section 13.6.3, where Ip is greater than 1.0, the component anchorage, bracing and the component itself (in this example, the pipe) must be designed to resist seismic forces. 14.5.1.2 System configuration. The portion of the piping system under consideration consists of three piping runs: §

Piping Run “A”, a 4-inch-diameter pipe, which connects to a large mechanical unit at Line 1 supported at the second level. It crosses a seismic separation between adjacent structures at Line 3.

§

Piping Run "B", a 6-inch-diameter pipe, which has a vertical riser to the second level at Line 3.

§

Piping Run "C", a 4-inch-diameter pipe, which turns 90 degrees to parallel Line 3 at Column Line 3-A.

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FEMA P-751, NEHRP Recommended Provisions: Design Examples

Y1A Z1A

X1A ZMA M

XMA 3

load to transverse brace at support 1 Y Z

Seismic Separation Joint X

Pipe Run "A" - 4-inch dia.

Figure 14.5-5 Piping Run A

Y1B

Z1B X1B 2

Y Z

load to transverse brace at support 1

Pipe Run "B" - 6-inch dia.

Figure 14.5-6 Piping Run B

14-42

Chapter 14: Design for Nonstructural Components

Y1C

Z1C X1C 2

Y Z

load to transverse brace at support 1

Pipe Run "C" - 4-inch dia. 5

Figure 14.5-7 Piping Run C The system consists of non-ASME B31 piping fabricated from steel Schedule 40 pipe with threaded connections. This example covers determination of the seismic forces acting on the system, a check of the seismically induced stresses in the pipes using simplifying assumptions, determination of bracing and anchorage forces and a check of the system for seismic relative displacements. It should be noted that details of pipe bracing systems vary according to the local preferences and practices of plumbing contractors. In addition, the use of proprietary pipe hanging and bracing systems is relatively common. As a result, this example concentrates on quantifying the prescribed seismic forces and displacements and on simplified stress checks of the piping system itself. After the seismic forces and displacements are determined, the bracing and anchorage connections can be designed and detailed according to the appropriate AISC and ACI codes. 14.5.2 Design Requirements. 14.5.2.1 Seismic design parameters and coefficients. ap = 2.5 for piping systems

(Standard Table 13.6-1)

Rp = 4.5 for piping not in accordance with ASME B31, constructed of high or limited deformability materials, with joints made by threading

(Standard Table 13.6-1)

SDS = 1.0 (for the selected location and site class) Seismic Design Category = D

(given) (Standard Table 11.6-1)

h = 30 feet (roof height)

(given)

hsx = 15 feet (story height)

(given) 14-43

FEMA P-751, NEHRP Recommended Provisions: Design Examples

z = 30 feet (system is braced at the roof level)

(given)

z 30.0 ft = = 1.0 h 30.0 ft

(at roof)

Ip = 1.5

(Standard Sec. 13.1.3)

Gravity (non-seismic) supports provided every 10'-0"

(given)

System working pressure (P) = 200 psi

(given)

ASTM A53 Pipe Fy = 35 ksi, threaded connections

(given)

D = Dead Load = Wp = 16.4 plf (4-inch-diameter water-filled pipe)

(given)

= 31.7 plf (6-inch-diameter water-filled pipe) Longitudinal brace spacing = 80 feet

(given)

According to Standard Section 13.3.1 (and repeated in Sec. 12.3.4.1), the redundancy factor does not apply to the design of nonstructural components. 14.5.2.2 Seismic design forces.

Fp =

0.4(2.5)(1.0)Wp (1 + 2(1)) = 1.00Wp 4.5 1.5

(Standard Eq. 13.3-1)

Fp = 1.6(1.0)(1.5)Wp = 2.40Wp

(Standard Eq. 13.3-2)

Fp = 0.3(1.0)(1.5)Wp = 0.45Wp

(Standard Eq. 13.3-3)

Ev = 0.2(1.0) D = 0.2D = 0.2Wp

(Standard Eq. 12.4-4)

14.5.2.3 Performance criteria. System failure must not cause failure of an essential architectural, mechanical, or electrical component (Standard Sec. 13.2.3). Component seismic attachments must be bolted, welded, or otherwise positively fastened without considering the frictional resistance produced by the effects of gravity (Standard Sec. 13.4). The effects of seismic relative displacements must be considered in combination with displacements caused by other loads as appropriate (Standard Sec. 13.3.2). The piping system must be designed to resist the forces in accordance with Standard Section 13.3.1 and must be able to accommodate movements of the structure resulting from response to the design basis ground motion, Dp.

14-44

Chapter 14: Design for Nonstructural Components 14.5.3 Piping System Design The requirements for the design of the piping system are summarized in Table 13.2-1 of the Standard. The supports and attachments of all mechanical and electrical components must meet the requirements listed in Table 13.2-1. Where Ip > 1.0, the component itself must also meet these requirements. 14.5.3.1 Check of pipe stresses. The spacing of seismic supports is often determined by the need to limit stresses in the pipe. Therefore, the piping stress check is often performed first in order confirm the assumptions on brace spacing. For non-ASME B31 piping that is not subject to high operating temperatures or pressures, the stress check assumptions may be simplified. The pipes can be idealized as continuous beams spanning between lateral braces, while longitudinal forces can be determined using the length of pipe tributary to the longitudinal brace. The permissible stresses in the pipe are given in Standard Section 13.6.11, Item 2. For piping with threaded connections, the permissible stresses are limited to 70 percent of the minimum specified yield strength. The section properties of the Schedule 40 pipes are as follows: 4-inch diameter: Inner diameter, d1 = 4.026 in. Outer diameter, d = 4.5 in. Wall thickness, t = 0.237 in. Plastic modulus, Z =

d 3 d13 (4.5)3 (4.026)3 = 4.31 in3 − = − 6 6 6 6

Moment of inertia, I = 0.049807(d 4 − d14 ) = 0.049087((4.5)4 − (4.026)4 ) = 7.23 in4 6-inch diameter: Inner diameter, d1 = 6.065 in. Outer diameter, d = 6.625 in. Wall thickness, t = 0.28 in. Plastic modulus, Z =

(6.625)3 (6.065)3 = 11.28 in3 − 6 6

Moment of inertia, I = 0.049087((6.625)4 − (6.065)4 ) = 28.14 in4

14-45

FEMA P-751, NEHRP Recommended Provisions: Design Examples 14.5.3.1.1 Gravity and pressure loads. The longitudinal stresses due to pressure and weight may be estimated using the following equation:

SL =

Pd M g + Z 4t

where: SL = sum of the longitudinal stresses due to pressure and weight P = internal design pressure, psig d = outside diameter of pipe, in. t = pipe wall thickness, in. Mg = resultant moment loading on cross section due to weight and other sustained loads, in-lb Z = section modulus, in3 Vertical supports are spaced at 10-foot centers, so the moment due to gravity, Mg, may be conservatively estimated as follows:

Mg =

Dl 2 D(10)2 = = 12.5D 8 8

For a 4-inch-diameter pipe, where D = 16.4 plf:

M g = 12.5D = 12.5(16.4) = 205 ft-lb = 2,460 in-lb SL-DeadLoad =

2, 460 = 571 psi 4.31

S L − Pr essure =

200(4.5) = 949 psi 4(0.237)

For a 6-inch-diameter pipe, where D = 31.7 plf:

M g = 12.5(31.7) = 396 ft-lb = 4,752 in-lb S L − DeadLoad =

4,752 = 421 psi 11.28

SL-Pressure = 1,183 psi 14.5.3.1.2 Seismic loads on Piping Runs A and C. By idealizing the piping runs as continuous beams, the maximum bending moments and reactions can be estimated readily.

14-46

Chapter 14: Design for Nonstructural Components Piping Runs A and C are 4-inch-diameter pipes, shown schematically in Figures 14.5-5 and 14.5-7. They are idealized as a two-span continuous beam. The design lateral load, Fp, is taken as follows:

Fp = 1.00Wp = 1.00(16.4) = 16.4 plf = w The maximum moment due to horizontal seismic load may be approximated as follows:

ME =

wl 2 16.4(40)2 = 3,280 ft-lb = 39,360 in-lb = 8 8

The flexural stress associated with this moment is:

fbh =

M E 39,360 = 9,132 psi = Z 4.31

The moment due to vertical seismic load, Ev = 0.2 Wp, may be approximated as follows:

Mv =

Ev l 2 0.2(16.4)(10)2 = 41 ft-lb = 492 in-lb = 8 8

The flexural stress associated with this moment is:

fbv =

= 492

4.31

= 114 psi

Note that for vertical seismic effects, the span of the pipe is taken as the distance between vertical supports, not the distance between lateral bracing. The basic strength load combination including earthquake effects from Standard Section 12.4.2.3 (based upon Standard Sec. 2.3.2) that will govern is Load Combination 5: U = (1.2 + 0.2SDS)D + 1.0ρQE + 0.5L + 0.2S For nonstructural components, ρ = 1.0 and QE = the forces (or stresses) resulting from applying Fp. In this example, live load, L and snow load, S, are equal to zero. The dead load, D, includes bending stress due to dead load. The load factor for internal pressure is the same as that for dead load. The design stress in the pipe is therefore: U = [1.2 + 0.2(1.0)](421 psi) + 1.2(1,183 psi) + 1.0(1.0)(9,132 psi) = 11,141 psi The permissible stress from Section 13.6.11, Item 2, of the Standard is 0.7Fy = 0.7(35,000) = 24,500 psi. Comparing the demand to capacity: U =11,141 psi < 0.7(35,000 psi) = 24,500 psi

Note that a number of conservative assumptions were made for the sake of simplicity. A more precise analysis can be performed, where the piping is modeled to achieve more accurate bending moments and the effects of biaxial bending in the pipe are considered separately. Also note that at any point in the pipe

14-47

FEMA P-751, NEHRP Recommended Provisions: Design Examples wall, the stresses caused by dead (and vertical seismic) load and by horizontal seismic load occur in different physical locations in the pipe. The peak stresses due to vertically applied load occurs at the top and bottom of the pipe, while the peak stress for horizontally applied load occurs at mid-height of the pipe. So assuming that they are both occurring in the same location and are algebraically additive is quite conservative. 14.5.3.1.3 Seismic loads on Piping Run B. Piping Run B, a 6-inch-diameter pipe, is shown schematically in Figure 14.5-6. It is idealized as a two-span continuous beam. Note that the effects of the 15-foot-high riser between Level 2 and the roof are considered separately. The design lateral load, Fp, is taken as follows:

Fp = 1.00Wp = 1.00(31.7) = 31.7 plf = w The maximum moment due to horizontal seismic load is approximated as follows:

ME =

wl 2 31.7(40)2 = = 6,340 ft-lb = 76,080 in-lb 8 8

The flexural stress associated with this moment is:

fbh =

M E 76,080 = 6,745 psi = Z 11.28

The moment due to the vertical seismic load, Ev = 0.2 Wp, may be approximated as follows:

Mv =

Ev l 2 0.2(31.7)(10)2 = 79.25 ft-lb = 951 in-lb = 8 8

The flexural stress associated with this moment is:

fbv =

Mv 951 = 84 psi = Z 11.28

The design stress in the pipe is: U = [1.2 + 0.2(1.0)](571 psi) + 1.2(941 psi) + 1.0(1.0)(6,745 psi) = 8,674 psi < 24,500 psi

14.5.4 Pipe Supports and Bracing As with the design of the pipe itself, design of the vertical and lateral supports of piping systems can be simplified by making conservative assumptions. In this example, design demands on the support assembly at Support 1 (Figure 14.5-8) are determined.

14-48

Chapter 14: Design for Nonstructural Components

e b

Y X Z

L1 7"

L2 8" 30"

L2 8"

C g

L4 7"

Figure 14.5-8 Design demands on piping support assembly 14.5.4.1 Vertical loads. Vertical pipe supports are often considered separately from lateral bracing. Configuration and spacing of vertical supports may be governed by plumbing codes or other standards and guidelines. Given that the vertical component of seismic force, Ev, is often low relative to other vertical loads, vertical supports proportioned for gravity and operational loads generally are adequate to resist the vertical seismic forces. However, where a support resists the vertical component of a lateral or longitudinal brace force, it should be designed explicitly to resist all applied forces. This example focuses on vertical supports associated with the lateral bracing system. Due to the repetitious nature of the pipe gravity support system, the vertical load at the brace assembly due to gravity or vertical seismic load can be estimated based on the tributary length of pipe. Given a 10-foot spacing of vertical supports, the vertical loads due to a 4-inch-diameter pipe are as follows: Dead load, Pv4 = (10 ft)(16.4 plf) = 164 lb Vertical seismic load, PEv4 = 0.2(10 ft)(16.4 plf) = 33 lb For a 6-inch diameter pipe: Dead load, Pv6 = (10 ft)(31.7 plf) = 317 lb Vertical seismic load, PEv6 = 0.2(10 ft)(31.7 plf) = 63 lb

14-49

FEMA P-751, NEHRP Recommended Provisions: Design Examples 14.5.4.2 Longitudinal lateral loads. Spacing of longitudinal bracing may be dictated by system geometry, thermal demands on the pipe, anchorage and brace capacities, or prescriptive limitations in standards and guidelines. In this example, we assume longitudinal braces are provided every 80 feet, which is twice the transverse brace spacing. For Piping Run A, the total length of pipe tributary to Support 1 is approximately 40 feet (half the distance between longitudinal braces at Supports 1 and 3) plus 9 feet (length of pipe from Support 1 to Support M, the mechanical unit), or 49 feet. The longitudinal seismic load, PX1A, for the 4-inch-diameter Piping Run A is: PX1A = (49 ft)(Fp) = (49 ft)(16.4 plf) = 804 lb For Piping Runs B and C, the total length of pipe tributary to Support 1 is approximately 80 feet. The longitudinal seismic load, PX1B, for the 6-inch-diameter Piping Run B is: PX1B = (80 ft)(Fp) = (80 ft)(31.7 plf) = 2,536 lb The longitudinal seismic load, PX1C, for the 4-inch-diameter Piping Run C is: PX1C = (80 ft)(Fp) = (80 ft)(16.4 plf) = 1,312 lb 14.5.4.3 Transverse lateral loads. To determine the transverse loads at support points, the pipes are idealized as continuous beams spanning between transverse braces. Assuming continuity over a minimum of two spans, the maximum reaction can be approximated conservatively as follows:

5 (2) wl 8 where w is the distributed lateral load and l is the spacing between transverse braces. For Piping Run A, we assume that 5/8 of the total length of pipe between the mechanical unit and the transverse brace at Support 2 is laterally supported at Support 1 (see Figure 14.5-3). The maximum transverse reaction due to Piping Run A at Support 1 may be approximated as follows:

5 5 PZ 1 A = wl = (16.4 plf)(40 ft + 9 ft) = 502 lb 8 8 For Piping Runs B and C, we assume that 5/8 of the total length of pipe on each side of Support 1 is laterally braced at Support 1 (see Figures 14.5-4 and 14.5-5). The maximum transverse reaction due to Piping Run B at Support 1 is then approximated as follows:

5 5 PZ 1B = (2) wl = (2) (31.7 plf)(40 ft) = 1,585 lb 8 8

14-50

Chapter 14: Design for Nonstructural Components The maximum transverse reaction due to Piping Run C at Support 1 is approximated as follows:

5 5 PZ 1C = (2) wl = (2) (16.4 plf)(40 ft) = 820 lb 8 8 14.5.4.4 Support design. The bracing system at Support 1 is shown in Figure 14.5-8. The analysis must consider design of the following bracing elements: Beam f-g, Hangers f-b and g-d, Transverse Brace a-f and Longitudinal Braces f-c and g-e. The connections at a, b, c, d and e must also be designed and are subject to special requirements. 14.5.4.4.1 Beam f-g. Beam f-g is subject to biaxial bending under vertical (Y-direction) and longitudinal (X-direction) forces. The maximum moment, which occurs at the center, is equal to:

PA L1 PB L PC L4 + + 2 4 2

The factored vertical loads for the piping runs are as follows: Piping Run A: PA = 1.2(164 lb) + 1.0(33 lb) = 230 lb Piping Run B: PA = 1.2(317 lb) + 1.0(63 lb) = 443 lb Piping Run C: PA = 1.2(164 lb) + 1.0(33 lb) = 230 lb The maximum moment about the x-axis of the beam due to vertical loads is:

Mx =

230(7) 443(30) 230(7) = 4,933 in-lb + + 2 4 2

The vertical reactions at f and g are equal to (230 + 443 +230)/2 = 452 pounds. The factored lateral loads in the longitudinal direction determined in Section 14.5.4.2 are as follows: Piping Run A: PA = 804 lb Piping Run B: PA = 2,356 lb Piping Run C: PA = 1,312 lb The maximum moment about the y-axis of the beam due to lateral loads is:

My =

804(7) 2,356(30) 1,312(7) = 25,076 in-lb + + 2 4 2

The horizontal reactions at f and g are:

Rf =

PA ( L2 + L3 + L4 ) PB PC L4 804(8 + 8 + 7) 2,356 1,312(7) = 2,101 lb + + = + + L 2 L 30 2 30

14-51

FEMA P-751, NEHRP Recommended Provisions: Design Examples

Rg =

PA L1 PB PC ( L1 + L2 + L3 ) 804(7) 2,356 1,312(7 + 8 + 8) = 2,371 lb + + = + + L 2 L 30 2 30

Beam f-g would be designed for moments Mx and My acting simultaneously. 14.5.4.4.2 Brace design. By inspection, Brace g-e will govern the longitudinal brace design, since the horizontal reaction at g (2,371 lb) is larger than that at f (2,101 lb). The horizontal load that must be resisted by the Transverse Brace a-f is the sum of the loads from the three pipes determined in Section 14.5.4.3: Rz = 502 lb + 1,585 lb + 820 lb = 2,907 lb Assuming the same member will be used for all braces, Brace a-f governs the design. Since the brace is installed at a 1:1 slope (45 degrees), the maximum tension or compression in the brace would be:

Tmax = Cmax + Rz 2 = (2,907)(1.414) = 4,110 lb The brace selected must be capable of carrying Cmax with an unbraced length of ( 30 in.) 2 = 42 inches. Bracing elements subject to compression should meet the slenderness ratio requirements of the appropriate material design standards. 14.5.4.4.3 Hangers. By inspection, Hanger f-b will govern the vertical element design, since the brace force in f-a governs the brace design. Since the brace is installed at a 1:1 slope (45 degrees), the maximum tension or compression due to seismic forces in the hanger is the same as the horizontal force resisted by the brace: 2,907 pounds. The vertical component of the brace force must be combined with gravity loads and the vertical seismic component. The maximum tension force in the hanger is determined using the basic strength Load Combination 5 from Standard Section 12.4.3.2 (based upon Standard Sec. 2.3.2): U = (1.2 + 0.2SDS)D + 1.0ρQE + 0.5L + 0.2S For nonstructural components, ρ = 1.0 and QE = the forces (or stresses) resulting from applying Fp. In this example, live load, L and snow load, S, are equal to zero. The unfactored reactions at f due to the weight of the water-filled pipes is 323 pounds. U = [1.2 + 0.2(1.0)](323 lb) + 1.0(1.0)(2,907 lb) = 3,359 lb The maximum compression force in the hanger is determined using the basic strength Load Combination 7 from Standard Section 12.4.2.3 (based upon Standard Sec. 2.3.2): U = (0.9 - 0.2SDS)D + 1.0 ρQE + 1.6H Substituting the values from above and noting that the lateral earth pressure load, H, is not applicable: U =[0.9 - 0.2(1.0)](323 lb) - 1.0(1.0)(2,907 lb) = -2,681 lb (tension)

14-52

Chapter 14: Design for Nonstructural Components Fp should be applied in the direction which creates the largest value for the item being checked. A negative sign indicates compression. The hanger selected must be capable of carrying the maximum compression with an unbraced length of 30 inches. Again, bracing elements subject to compression should meet the slenderness ratio requirements of the appropriate material design standards. It is also important to note that the length of pipe that contributes dead load to counteract the vertical component of brace force is based on the spacing of the vertical hangers, not the spacing between lateral braces. 14.5.4.5 Anchorage design. Standard Section 13.4 covers the attachment of the hangers and braces to the structure. Component forces are those determined in Sections 13.3.1 and 13.3.2, with important exceptions. Anchors in concrete and masonry are proportioned to carry the least of the following: §

1.3 times the prescribed seismic design force, or

§

The maximum force that can be transferred to the connected part by the component structural system.

In addition, the value of Rp may not exceed 1.5 unless the anchors are prequalified for seismic loading or the component anchorage is governed by yielding of a ductile steel element. To illustrate the effects of these provisions, consider the design of the attachment to the structure at Point a in Figure 14.5-8. The horizontal and vertical components of the seismic brace force at Point a are 2,907 pounds each. Assuming the brace capacity does not limit the force to the anchor, the minimum design forces for the anchors is 1.3 times 2,907 pounds, which equals 3,779 pounds in tension acting currently with 3,779 pounds in shear. The maximum design force for the anchors, assuming that a ductile element does not govern the anchorage capacity and the anchor is not prequalified for seismic load, would be determined using an Rp = 1.5. Substituting Rp = 1.5 into the seismic force Equation 13.3-1, the following is obtained:

Fp =

0.4(2.5)(1.0)Wp (1 + 2(1)) = 3.00Wp 1.5 1.5

(Standard Eq. 13.3-1)

In this case, the seismic maximum force of Equation 13.3-2, Fp = 2.40Wp, governs. This design force compares to the force of Fp = 1.00Wp obtained previously. The amplified design forces for the anchors would be 2.4/1.0 times 2,907 pounds, which equals 6,977 pounds in tension acting currently with 6,977 pounds in shear. 14.5.5 Design for Displacements In addition to design for seismic forces, the piping system must accommodate seismic relative displacements. For the purposes of this example, we assume that the building has a 15-foot story height and has been designed for a maximum allowable story drift:

Δ a = 0.015hsx = 0.015(15)(12) = 2.7 in. per floor 14.5.5.1 Design for displacements within structures. Piping Run A, a 4-inch-diameter pipe, connects to a large mechanical unit at Line 1 supported at the second level. Because the mechanical unit can be assumed to behave as a rigid body and the piping system is rigidly braced to the roof structure, the entire story drift must be accommodated in the 5'-0" piping drop (see Figure 14.5-2). There are several

14-53

FEMA P-751, NEHRP Recommended Provisions: Design Examples approaches to accommodate the drift. The first is to provide a flexible coupling (articulated connections or braided couplings, for example). A second approach is to accommodate the drift through bending in the pipe. Loops are often used to make the pipe more flexible for thermal expansion and contraction and this approach also works for seismic loads. In this example, a straight length of a pipe is assumed. For a 4-inch-diameter Schedule 40 pipe, the moment of inertia, I, is equal to 7.23 in4. Assuming the pipe is fixed against rotation at both ends, the shear and moments required to deflect the pipe 2.7 inches are as follows:

V = 12EI Δ a / l 3 = 12(29,000,000)(7.23)(2.7) / ((5.0)(12))3 = 31,451 lb M = (31,451 lb)(60 in.) = 1,887,060 in-lb

fb =

M 1,887,060 = 437,833 psi = Z 4.31

These demands far exceed the capacity of the pipe and would overload the nozzle on the mechanical unit as well. Therefore, either a flexible coupling or a loop piping layout is required to accommodate the story drift. Piping Run B, a 6-inch-diameter pipe, drops from the roof level to the second level at Line 3. Again, the drift demand is 2.7 inches, but in this case, it may be accommodated over the full story height of 15 feet. A simplified analysis assumes that the pipe is fixed at the roof and second level. This assumption is conservative, since in reality the horizontal runs of the pipe at the roof and Level 2 provide restraint but not fixity. For a 6-inch-diameter Schedule 40 pipe, the moment of inertia, I, is equal to 28.14 in4. The shear and moments required to deflect the pipe 2.7 inches are as follows:

V = 12EI Δ a / l 3 = 12(29,000,000)(28.14)(2.7) / ((15.0)(12))3 = 4,534 lb M = Vl = (4,534)((15.0)(12))/2 = 408,150 in-lb 2 The stress in the pipe displaced ∆ would be:

fb =

M 408,150 = 36,184 psi = 11.28 Z

This exceeds the permissible stress in the pipe, but not by a wide margin. In ASME B31.3, piping allowables are increased by a factor of 3 for stresses associated with seismic anchor movement relative displacements, but no such increases are currently provided for other piping. It is expected that threaded joints would be more highly stressed and a factor of 3 may not be conservative. Other options include refining the analysis to more accurately consider the effects of the rotational restraint provided by threaded couplings in the horizontal piping runs (which will tend to reduce the rigidity of the pipe and therefore reduce the bending stress), providing loops in the piping layout, or providing flexible couplings. 14.5.5.2 Design for displacements between structures. At the roof level, Piping Run A crosses a seismic separation between adjacent two-story structures at Line 3. Assuming story heights of 15 feet and design for a maximum allowable story drift for both buildings, the deflections of the buildings are:

δ xA = δ xB = (2)0.015hsx = (2)0.015(15)(12) = 5.4 in.

14-54

Chapter 14: Design for Nonstructural Components

The displacement demand, DP is determined from Standard Equation 13.3-7 as follows:

Dpmax = δ xA + δ yB = 5.4 + 5.4 = 10.8 in. In addition to motions perpendicular to the pipe, the seismic isolation joint must accommodate movement parallel to the pipe. Assuming a 12-inch seismic separation joint is provided, during an earthquake the joint could vary from 1.2 inches (if the structures move towards each other) to 22.8 inches (if the structures move away from each other). The flexible coupling, which could include articulated connections, braided couplings, or pipe loops, must be capable of accommodating this range of movements.

14.6 ELEVATED VESSEL SEISMIC DESIGN 14.6.1 Example Description This example illustrates seismic design for a small platform-supported vessel in the upper floor of a structure. It includes determination of the seismic design forces, anchorage of the vessel to the supporting platform, design and anchorage of the platform and demands on the floor slab of the supporting structure. The example focuses on the determination of force and displacement demands on the different components of the support system. The sizing of the various elements (beams, columns, braces, connections, anchor bolts, etc.) are not covered in detail. This example considers a vessel supported by a platform on the second floor of a three-story structure. The contents of the vessel, a compressed non-flammable gas, are not hazardous. The structure, located in the Los Angeles area of California, is assigned to Occupancy Category II. The design approach for nonstructural components depends on the type, size and location of the component. The Standard provides requirements for nonstructural components in Chapter 13 and nonbuilding structures in Chapter 15. Vessels are classified as nonbuilding structures, but in this example the provisions of Standard Chapter 13 apply, due to the size and location of the vessel. 14.6.1.1 Earthquake design requirements. Earthquake design requirements for vessels in the Standard depend on the system importance factor (Ip), the mass of the vessel relative to that of the supporting structure and the installation geometry. There are special analytical requirements triggered by Section 13.1.5 where the weight of the nonstructural component exceeds 25 percent of the effective seismic weight of the structure. The example does not cover this condition. If that were the case, the component would be classified as a nonbuilding structure and the requirements of Section 15.3.2 would apply. The importance factor is determined in accordance with Standard Section 13.1.3. Given that the structure is assigned to Occupancy Category II, components are assigned Ip = 1.0, unless they must function for life-safety protection or contain hazardous materials. Neither of these conditions apply, so Ip = 1.0. 14.6.1.2 System configuration. The vessel is of steel construction and supported on four legs, which are bolted to a steel frame. A plan of the second level showing the location of the vessel is shown in Figure 14.6-1. A section through the structure showing the location of the vessel is presented in Figure 14.6-2. The supporting frame system consists of ordinary braced frames (tension-only bracing). An elevation of the vessel and supporting frame is shown in Figure 14.6-3.

14-55

FEMA P-751, NEHRP Recommended Provisions: Design Examples

30'-0" 6'-0"

12'-0"

6'-0"

4'-6"

12'-0"

6" R/C Slab

Figure 14.6-1 Elevated vessel - second-level plan

14-56

30'-0"

15'-0"

4'-6"

Chapter 14: Design for Nonstructural Components

18'-0"

Roof

46'-0"

14'-0"

Level 2

14'-0"

Level 1

Level 0

Figure 14.6-2 Elevated vessel - section

14-57

FEMA P-751, NEHRP Recommended Provisions: Design Examples

c.g. 3'-0" Vessel

5'-6"

3'-0"

HSS6x2x1/4 (typ)

5'-0"

HSS 2x2x1/4 (typ)

Supporting Frame

6'-0"

Figure 14.6-3 Elevated vessel - supporting frame system The example covers the determination of the seismic forces in the supporting steel frame and its connection to the concrete slab at the second level and concentrates on quantifying the prescribed seismic forces for the different elements in the vessel support system. After the seismic demands are determined, the bracing and anchorage connections can be designed and detailed according to the appropriate AISC and ACI codes. Finally, methods of considering the seismic demands in the floor system due to the vessel are discussed. 14.6.2 Design Requirements 14.6.2.1 Seismic design parameters and coefficients. ap = 1.0 for vessels not supported on skirts and not subject to Chapter 15

(Standard Table 13.6-1)

Rp = 2.5 for vessels not supported on skirts and not subject to Chapter 15

(Standard Table 13.6-1)

SDS = 1.2 (for the selected location and site class) Seismic Design Category = D

(given) (Standard Table 11.6-1)

h = 46 feet (roof height)

(given)

z = 28 feet (system is supported at the second level)

(given)

14-58

Chapter 14: Design for Nonstructural Components

z 28.0 ft = = 0.609 h 46.0 ft Ip = 1.0

(Standard Sec. 13.1.3)

ASTM A500 Grade B steel HSS sections, Fy = 46 ksi, Fu = 58 ksi

(given)

ASTM A36 steel bars and plates, Fy = 36 ksi, Fu = 58 ksi

(given)

ASTM A53 Grade B pipe, Fy = 35 ksi, Fu = 60 ksi

(given)

ASTM A 307 bolts and threaded rods

(given)

D = Dead Load = Wp = 5,000 lb (vessel and legs)

(given)

= 1,000 lb (allowance, supporting frame)

(given)

According to Standard Section 13.3.1 (and repeated in Sec. 12.3.4.1), the redundancy factor does not apply to the design of nonstructural components. 14.6.2.2 Seismic design forces.

Fp =

0.4(1.0)(1.2)Wp (1 + 2(0.609)) = 0.426Wp 2.5 1.0

(Standard Eq. 13.3-1)

Maximum Fp = 1.6(1.2)(1.0)Wp = 1.92Wp

(Standard Eq. 13.3-2)

Minimum Fp = 0.3(1.2)(1.0)Wp = 0.36Wp

(Standard Eq. 13.3-3)

Ev = 0.2(1.2) D = 0.24D = 0.24Wp

(Standard Eq. 12.4-4)

14.6.2.2.1 Vessel. The seismic forces acting on the vessel are as follows:

Fp = 0.426Wp = 0.426(5,000) = 2,129 lb Ev = 0.24Wp = 0.24(5,000) = 1,200 lb 14.6.2.2.2 Supporting frame. The seismic loads due to the supporting frame self-weight are as follows:

Fp = 0.426Wp = 0.426(1,000) = 426 lb Ev = 0.24Wp = 0.24(1,000) = 240 lb 14.6.2.3 Performance criteria. Component failure must not cause failure of an essential architectural, mechanical, or electrical component (Standard Sec. 13.2.3).

14-59

FEMA P-751, NEHRP Recommended Provisions: Design Examples Component seismic attachments must be bolted, welded, or otherwise positively fastened without considering the frictional resistance produced by the effects of gravity (Standard Sec. 13.4). The effects of seismic relative displacements must be considered in combination with displacements caused by other loads as appropriate (Standard Sec. 13.3.2). The component must be designed to resist the forces in accordance with Standard Section 13.3.1 and must be able to accommodate movements of the structure resulting from response to the design basis ground motion, Dp. Local elements of the structure, including connections, must be designed and constructed for the component forces where they control the design. 14.6.3 Load Combinations The basic strength load combinations including earthquake effects from Standard Section 12.4.2.3 (based upon Standard Sec. 2.3.2) that will govern design of the vessel legs, attachments and the supporting frame are the following: §

Load Combination 5: U = (1.2 + 0.2SDS)D + 1.0ρQE + 0.5L + 0.2S

§

Load Combination 7: U = (0.9 - 0.2SDS)D + 1.0 ρQE + 1.6H

For nonstructural components, ρ = 1.0 and QE = the forces resulting from applying Fp. In this example, live load (L), snow load (S) and the lateral earth pressure load (H) are equal to zero. 14.6.4 Forces in Vessel Supports Supports and attachments for the vessel must meet the requirements listed in Standard Table 13.2-1. Seismic design of the vessel itself is not required, since Ip = 1.0. While the vessel itself need not be checked for seismic loading, the component supports listed in Standard Section 13.6.5 must be designed to resist the prescribed seismic forces. The affected components include the following: §

The legs supporting the vessel

§

Connection between the legs and the vessel shell

§

Base plates and the welds attaching them to the legs

§

Bolts connecting the base plates to the supporting frame

Standard Section 13.4.1 states that the lateral force, Fp, must be applied independently in at least two orthogonal directions. For vertically cantilevered systems, the lateral force also must be assumed to act in any horizontal direction. In this example, layout of the vessel legs is symmetric and there are two horizontal directions of interest, separated by 45 degrees. These two load cases are illustrated in Figure 14.6-4.

14-60

Chapter 14: Design for Nonstructural Components

Frame Leg "1" Load Case 2

Vessel Leg "C"

Vessel Leg "A"

x Load Case 1

Frame Leg "2"

Figure 14.6-4 Elevated vessel support load cases 14.6.4.1 Case 1 - moments about the y-y axis. The height of the vessel’s center-of-gravity above the bottom of the leg base plates is 5.5 feet. The moments about the bottom of these base plates are taken as follows:

M = (5.5) Fp = (5.5)(2,129) = 11,710 ft-lb Assuming the vessel acts as a rigid body, in Case 1 the overturning moments is resisted by the two legs along the x-x axis. The vessel is assumed to rotate about the legs on the y-y axis. The maximum tension and compression loads in the legs may be estimated as follows: T =C =M /d

where the distance between legs A and C, d, is 6.0 feet. Therefore:

T = C = 11,710 / 6.0 = 1,952 lb The vertical load in each leg due to gravity is Wp/4 = 5,000/4 = 1,250 pounds. The shear in each leg due to Fp is V= Fp/4 = 2,129/4 = 532 pounds. 14.6.4.2 Case 2 - moments about the x'-x' axis. In Case 2 the overturning moments are resisted by all four legs, two in compression and two in tension. The loads in the legs due to gravity are the same as in Case 1, as is the shear in the legs due to Fp. Under seismic load, the vessel is assumed to rotate about the x'-x axis. The maximum tension and compression loads in the legs may be estimated as follows:

T =C =

M 2(0.707 d ) 14-61

FEMA P-751, NEHRP Recommended Provisions: Design Examples

where the distance between legs A and C, 0.707d , is 4.24 feet. Therefore:

T = C = 11,710 / 2(4.24) = 1,380 lb 14.6.5 Vessel Support and Attachment The axial loads in the vessel legs due to seismic overturning about the y-y axis (Case 1, in Section 14.6.4.1) are substantially larger than those obtained for overturning about the x'-x' axis (Case 2, in Section 14.6.4.2) . Therefore, by inspection Case 1 governs the design of the legs. The design compression loads on the vessel legs is governed by Load Combination 5: U = [1.2 + 0.2(1.2)](1,250 lb) + 1.0(1.0)( 1,952 lb) = 3,752 lb = CU The design tension load on the vessel legs is governed by Load Combination 7: U = [0.9 – 0.2(1.2)](1,250 lb) - 1.0(1.0)(1,952 lb) = -1,127 lb = TU (A negative sign denotes compression.) The design shear in each leg is U = 1.0(1.0)(532 lb) = 532 lb = VU. 14.6.5.1 Vessel leg design. The check of the leg involves a check of the connection between the vessel and the leg and a stress check of the leg itself. The length of the leg, L, is 18 inches and the legs are fabricated from 2-inch-diameter standard pipe. The section properties of the leg are as follows: A = 1.00 in2 Z = 0.713 in3 Assuming the leg is pinned at the connection to the supporting frame and fixed at the connection to the vessel, the moment and bending stress in the leg are as follows: M = VUL = 532(18) = 9,576 in-lb fb = M/Z = 9,756/0.713 = 13,683 psi The maximum axial compressive stress in the leg is: Cu/A = 3,752/1.00 = 3,752 psi The capacities of the leg and the connection to the vessel are determined using the structural steel specifications (AISC 360). The permissible strengths are as follows: Fa = 31,500 psi Fbw = 31,500 psi

14-62

Chapter 14: Design for Nonstructural Components For combined loading:

f a fbw + ≤ 1.0 Fa Fbw

3,752 13,683 + = 0.553 ≤ 1.0 31,500 31,500 14.6.5.2 Connections of the vessel leg. The connection between the vessel leg and the supporting frame is shown in Figure 14.6-5. The design of this connection involves checks of the weld between the pipe leg and the base plate, design of the base plate and design of the bolts to the supporting frame. Load Combination 7, which results in tension in the pipe leg, will govern the design of the base plates and the bolts to the supporting frame. The design of the base plate and bolts should consider the effects of prying on the tension demand in the bolts.

Vessel leg 1-1/2" 5/8"-diameter bolts

1-1/2"

HSS6x2 beam

Figure 14.6-5 Elevated vessel leg connection Each vessel leg is connected to the supporting frame by a pair of 5/8-inch-diameter bolts. The load path for this connection consists of the following elements: the weld of the leg to the connecting plate, the connecting plate acting in bending considering the effects of prying as appropriate, the bolts, the connection plate welded to the supporting frame beam and the welding of the connection plate to the supporting frame beam. Again by inspection, Case 1 (Section 14.6.4.1) governs. The factored loads in the connection are determined using Load Combinations 5 and 7 of Standard Section 12.4.2.3 (based upon Standard Sec. 2.3.2). As previously determined, the maximum compression in the connection (Load Combination 5) is: CU = 3,752 lb The maximum tension in the connection (Load Combination 7) is: U = (0.9 - 0.2SDS)D + 1.0ρQE + 1.6H

14-63

FEMA P-751, NEHRP Recommended Provisions: Design Examples U = [0.9 – 0.2(1.2)](1,250 lb) - 1.0(1.0)(1,952 lb) = -1,127 lb (tension) = TU The maximum shear per bolt is: VU = 532 lb/2 = 266 lb The designs of the vessel leg base plate and of the connection plate at the supporting frame beams are identical. The maximum tension in each bolt is: TU = 1,127 lb/2 = 534 lb The available shear and tensile strengths of the bolt are as follows: vrn = 5,520 lb (shear) rn = 10,400 lb (tension) Therefore, the bolts are adequate. The connection plates are 1/4 inch thick and 3 inches wide.

bd 2 3(0.25)2 = 0.0469 in3 = 4 4

The maximum moment in the plate is: MU = 534 lb (1.5 in.) = 801 in-lb The bending stress is: fb = MU /Z = 801/0.0469 = 17,088 psi

Prying action can have the effect of increasing the tensile forces in the bolts. AISC 360 permits prying action to be neglected if the plate meets minimum thickness requirements, given by:

tmin =

4.44Tb ' pFu

where p = 3 inches is the tributary length per pair of bolts b' = (b-db/2) = (1.5-0.625/2) = 1.1875 in.

tmin =

4.44(534)(1.1875) = 0.13 in. 3(58,000)

This is less than the 0.25-inch thickness provided, so prying need not be considered further. The welds of the vessel leg to the vessel body and of the leg to the upper connection plate are proportioned in a similar manner. The calculation can be simplified by assuming the weld is of unit 14-64

Chapter 14: Design for Nonstructural Components thickness. This yields a demand per inch of weld and an appropriate weld thickness can then be selected. The vessel leg has an outer diameter, d, of 2.38 inches. The weld properties for a weld of unit thickness are as follows:

d 3 2.383 = 2.25 in3 = 6 6

A = πd =3.14(2.38) = 7.45 in. The shear in the weld is: v = 532 lb/7.45 in. = 714 lb/in. The tension in the weld due to axial load is: T = 1,127 lb/7.45 in. = 151 lb/in. The tension to the weld due to bending (at the connection to the vessel) is: T = M/Z = 9,756/7.45 = 1,310 lb/in. For E70 electrodes, the capacity of a fillet weld is given by: Rn = 1.392 Dl (kips/in.) where D is the size of the weld in sixteenths of an inch and l is the weld length. For a unit length, a 3/16-inch fillet weld has a capacity of : Rn = 1.392(3)(1) = 4.18 kips/in. which will be adequate. The same size weld is used for the vessel leg-to-vessel body joint and for the leg-to-upper connection plate joint. A similar design approach is used to proportion the weld of the lower connection plate to the HSS 6x2 beam. 14.6.6 Supporting Frame The design of the supporting frame can be performed separately from that of the vessel. The reactions from the vessel are applied to the frame and combined with the inertial loads resulting from the supporting frame itself. The configuration of the supporting frame is shown in Figure 14.6-6.

14-65

FEMA P-751, NEHRP Recommended Provisions: Design Examples 3'-0"

5'-0"

3'-0"

R1H

R2H R1V

R2V

Figure 14.6-6 Elevated vessel supporting frame The supporting frame uses Steel Ordinary Braced Frames (OBF). While the supporting frame is designed for seismic forces determined in Section 13.3, the design process for the frame itself is similar to that used for building frames or nonbuilding structures similar to buildings. In this example, seismic loads are developed for the following elements: §

Beams supporting the vessel legs

§

Braces

§

Columns supporting the platform and vessel

§

Base plates and anchor bolts

To simplify the analysis, the self weight of the supporting frame is lumped at the vessel leg connection locations. 14.6.6.1 Support frame beams. The beams transfer vertical and horizontal loads from the vessel to the brace frames. The beams, fabricated from HSS6x2x1/4 members, are idealized as simply supported with a span of 6 feet. The reactions from the vessel legs are idealized as point loads applied at mid-span. The vertical loads applied to the beam are as follows: P = [5,000 lb (vessel) + 10,000 lb (frame)]/4 supports = 1,500 lb The lateral load per beam of the combined vessel and supporting frames is: V = 0.426[5,000 lb (vessel) + 10,000 lb (frame)]/4 supports = 639 lb The maximum load on a leg due to overturning of the vessel was computed as P = 1,952 pounds. The maximum factored vertical load, which will generate strong axis bending in the beam, is determined using Load Combination 5:

14-66

Chapter 14: Design for Nonstructural Components

PU = [1.2 + 0.2(1.2)](1,500 lb) + 1.0(1.0)( 1,952 lb) = 4,112 lb = Pv Acting with the horizontal load, VU = 639 pounds. The moment in the beams then is: Mx-x = PUl/4 = (4,112 lb)(6.0 ft)/4 = 6,168 ft-lb My-y = VUl/4 = (639 lb)(6.0 ft)/4 = 959 ft-lb For an HSS6x2x1/4: Zx-x = 5.84 in3 Zy-y = 2.61 in3

fbx fby + ≤ 1.0 Fb Fb Fb = Fy = 0.9(46,000 psi) = 41,400 psi

M y− y Z y− y fbx fby M Z 6,168(12) / 5.84 959(12) / 2.61 + = x− x x− x + = + = 0.41 ≤ 1.0 41, 400 41, 400 Fb Fb Fb Fb

14.6.6.2 Support frame braces. The maximum brace force occurs where loads are applied in the X- or Y-direction and the loads are resisted by two frames. The horizontal force is: V = 0.426[5,000 lb (vessel) + 10,000 lb (frame)]/2 braces = 1,278 lb The length of the brace is:

(5) 2 + (6) 2 = 7.81 ft The force in the brace then is:

Tu =

7.81 1, 278 = 1,664 lb (tension) 6

The braces consist of 5/8-inch-diameter ASTM A307 threaded rod. The nominal tensile capacity is: rn = 10,400 lb (tension) > 1,664 lb

It is good practice to design the supporting frame connections to the same level as a nonbuilding structure subject to Chapter 15. In this example, the supporting frames would be treated as an ordinary braced frame. For this system, AISC 341 requires the strength of the bracing connection to be the lesser of the expected yield strength of the brace in tension, the maximum force that can be developed by the system, or the load effect based on the amplified load. 14-67

FEMA P-751, NEHRP Recommended Provisions: Design Examples

14.6.6.3 Support frame columns. The columns support the vertical loads from the vessel and frame, including the vertical component of the supporting frame brace forces. The columns are fabricated from HSS2x2x1/4 members and are idealized as pinned top and bottom with a length of 5 feet. The case where the vessel rotates about the x'-x' axis governs the design of the supporting frame columns. The overturning moment is:

M = (10.5) Fp −vessel + (5.0) Fp − frame = (10.5)(2,129) + 5.0(426) = 24, 485 ft-lb Assuming the vessel acts as a rigid body, in Case 1 the overturning moment is resisted by the two legs along the y'-y' axis. The vessel is assumed to rotate about the legs on the x'-x' axis. The maximum tension and compression loads in the columns due to overturning may be estimated as follows: T =C =M /d

where the distance between Frame Legs 1 and 2, d = (6.0) 2 = 8.48 feet. Therefore:

T = C = 24, 485 / 8.48 = 2,886 = 2,886 lb The vertical load in each leg due to gravity is Wp/4 = (5,000+1,000)/4 = 1,500 pounds. The design compression load on the supporting frame columns is governed by Load Combination 5: U = [1.2 + 0.2(1.2)](1, 500 lb) + 1.0(1.0)(2,886 lb) = 5,046 lb = CU The design tension load on the supporting frame columns is governed by Load Combination 7: U = [0.9 - 0.2(1.2)](1,500 lb) - 1.0(1.0)(2,886 lb) = -1,896 lb = TU The capacity of the HSS2x2x1/4 column is 38,300 pounds and is therefore adequate. 14.6.6.4 Support frame connection to the floor slab. The connection of the support frame columns to the floor slab includes the following elements: §

Weld of the column and brace connection to the base plate

§

Base plate

§

Anchor bolts

The design of the connection of the base plate and of the base plate itself follows the typical procedures used for other structures. There are special considerations for the design of the anchor bolts to the concrete slab that are unique to nonstructural components. Anchors in concrete and masonry must be proportioned to carry the least of the following: §

1.3 times the prescribed seismic design force, or

§

The maximum force that can be transferred to the connected part by the component structural system.

14-68

Chapter 14: Design for Nonstructural Components The value of Rp may not exceed 1.5 unless the anchors are prequalified for seismic loading or the component anchorage is governed by yielding of a ductile steel element. The horizontal and vertical reactions of the supporting frame columns are as follows: CU = 5,046 lb TU = -1,896 lb VU = 1,278 lb Assuming that no other element of the supporting frame limits the force to the anchor, the minimum design forces for the anchors would be 1.3 times 1,896 pounds, which is 2,465 pounds in tension, acting concurrently with 1.3 times 1,278 pounds, which is 1,661 pounds in shear. The maximum design force for the anchors, assuming a ductile element does not govern the anchorage capacity and the anchor is not prequalified for seismic load, would be determined using Rp = 1.5. Substituting Rp = 1.5 into the seismic force Equation 13.3-1 results in the following:

Fp =

0.4(1.0)(1.2)Wp (1 + 2(0.609)) = 0.71Wp 1.5 1.0

(Standard Eq. 13.3-1)

This design forces compares to Fp = 0.426Wp obtained previously. The amplified overturning moment for the design of the anchors would be:

M = (10.5) Fp −vessel + (5.0) Fp − frame = (10.5)(0.71)(5,000) + 5.0(0.71)(1,000) = 40,825 ft-lb The corresponding tension load becomes:

T = 40,825 / 8.48 = 4,814 lb The factored design tension is: U = [0.9 - 0.2(1.2)](1,500 lb) - 1.0(1.0)(4,814 lb) = -3,824 lb This is represents a 55 percent increase in the design tension. The amplified shear forces for the anchors would be (2.50/1.50)(1,278 lb), which is 2,130 pounds. 14.6.7 Design Considerations for the Vertical Load-‐Carrying System This portion of the example illustrates design considerations for the floor slab supporting the nonstructural component. The floor system at Level 2 consists of a 6-inch-thick reinforced concrete flatslab spanning between steel beams. To illustrate the effects of the vessel, the contribution of the vessel load to the overall slab demand is examined.

14-69

FEMA P-751, NEHRP Recommended Provisions: Design Examples 14.6.7.1 Slab design assumptions. Dead load = 100 psf Live load = 100 psf (non-reducible) 14.6.7.2 Effect of vessel loading. During design, the slab moments and shear are checked at different points along each span. In order to simply illustrate the potential effects of the vessel, this investigation will be limited to the change in the negative moments about the x-x axis over the center support. In an actual design, a complete analysis of the slab for the loads imposed by the vessel would be required. At the center support, the moments due to dead load and live load are as follows: Maximum dead load moment, MDL = wl2/8 = (100)(152)/8 = 2,813 ft-lb/ft Maximum live load moment, MLL = wl2/8 = (100)(152)/8 = 2,813 ft-lb/ft The support frame columns are 6 feet apart. Assuming an additional 3 feet of slab on each side of the frame to resist loads generated by the vessel, the design moments for the strip of slab supporting the vessel are as follows: MDL = 2,813 ft-lb/ft (12 ft) = 33,756 ft-lb MLL = 2,813 ft-lb/ft (12 ft) = 33,756 ft-lb The moments at the center support due to a point load, P, in one of the spans is:

Pab (l + a ) 4l 2

where: a = distance from the end support to the point load b = distance from the point load to the center support l = span between supports = 15 ft The point loads due to the vessel and support frame self weight is: P = 2(1,500 lb) = 3,000 lb The moment in the slab due to the vessel and support frame is as follows:

M VD =

(3,000)(4.5)(10.5) (3,000)(10.5)(4.5) (15 + 10.5) = 7,088 ft-lb (15 + 4.5) + 2 4(15) 4(15) 2

The point loads due to the vessel and support frame caused by seismic in the Y-direction are as follows:

T = C = 24, 485 / 6.00 = 4,081 lb

14-70

Chapter 14: Design for Nonstructural Components The moment in the slab due to the overturning of the vessel and support frame for seismic forces in the Y-direction is:

M VD =

(4,081)(4.5)(10.5) ( −4,081)(10.5)(4.5) (15 + 10.5) = -1,286 ft-lb (15 + 4.5) + 2 4(15) 4(15) 2

M VD =

(−4,081)(4.5)(10.5) (4,081)(10.5)(4.5) (15 + 10.5) = 1,286 ft-lb (15 + 4.5) + 2 4(15) 4(15) 2

The factored moments for the slab without the vessel are: D + L: U = 1.2 D + 1.6 L = 1.2 (33,756 ft-lb) + 1.6(33,756 ft-lb) = 94,517 ft-lb The factored moments for the slab including the vessel are as follows: D + L: U = 1.2 D + 1.6 L = 1.2 (33,756 ft-lb+ 7,088 ft-lb) + 1.6(33,756 ft-lb) = 103,022 ft-lb The factored moments including seismic are as follows: Load Combination 5: U = (1.2 + 0.2SDS)D + 1.0ρQE + 0.5L + 0.2S = (1.2 + 0.2(1.2))(33,756+7,088) + 1.0(1.0)(1,286) + 0.5(33,756) + 0.2(0) = 76,979 ft-lb Load Combination 7: U = (0.9 - 0.2SDS)D + 1.0 ρQE + 1.6H = (0.9 - 0.2(1.2)) (33,756+7,088) + 1.0(1.0)(-1,286) + 1.6(0) = 25,671 ft-lb In this case, the loads from the vessel do not control the design of the slab over the center support.

14-71

The purpose of the Building Seismic Safety Council is to enhance the public's safety by providing a national forum to foster improved seismic safety provisions for use by the building com m unity. For the purposes of the Council, the building community is taken to include all those involved in the planning, design, construction, regulation, and utilization of buildings. To achieve its purposes, the Council shall conduct activities and provide the leadership needed to: 1. 2. 3. 4. 5.

6. 7.

Promote development of seismic safety provisions suitable for use throughout the United States; Recommend, encourage, and promote adoption of appropriate seismic safety provisions in voluntary standards and model codes; Assess implementation progress by federal, state, and local regulatory and construction agencies; Identify opportunities for the improvement of seismic regulations and practices and encourage public and private organizations to effect such improvements; Promote the development of training and educational courses and materials for use by design professionals, builders, building regulatory officials, elected officials, industry representatives, other members of the building community and the public. Provide advice to governmental bodies on their program s of research, development, and implementation; and Periodically review and evaluate research findings, practice, and experience and make recommendations for incorporation into seismic design practices.

The scope of the Council's activities encompasses seismic safety of structures with explicit consideration and assessment of the social, technical, administrative, political, legal, and economic implications of its deliberations and recommendations. Achievement of the Council's purpose is important to all in the public and private sectors. Council activities will provide an opportunity for participation by those with interest, including local, State, and Federal Government, voluntary organizations, business, industry, the design professions, the construction industry, the research com m unity and the public. Regional and local differences in the nature and magnitude of potentially hazardous earthquake events require a flexible approach adaptable to the relative risk, resources and capabilities of each com m unity. The Council recognizes that appropriate earthquake hazard reduction measures and initiatives should be adopted by existing organizations and institutions and incorporated into their legislation, regulations, practices, rules, codes, relief procedures and loan requirements, whenever possible, so that these measures and initiatives become part of established activities rather than being superposed as separate and additional. The Council is established as a voluntary advisory, facilitative council of the National Institute of Building Sciences, a nonprofit corporation incorporated in the District of Columbia, under the authority given the Institute by the Housing and Community Development Act of 1974, (Public Law 93-383); Title V III, in furtherance of the objectives of the Earthquake Hazards Reduction Act of 1977 (Public Law 95-124); and in support of the President's National Earthquake Hazards Reduction Program, June 22, 1978.

Appendix A: Building Seismic Safety Council

Chair Jim. W. Sealy, FAIA, Architect/Consultant, Representing: National Institute of Building Sciences Members Remington B. Brown, P.E., Senior Engineering Manager, Insurance Institute for Building and Home Safety, Representing: Insurance Institute for Building and Home Safety James R. Cagley, P.E., S.E. Chairman of the Board, Cagley & Associates, Representing: Applied Technology Council Charles J. Carter, S.E., P.E., PhD., Vice President of Engineering and Research, American Institute of Steel Construction, Representing: American Institute of Steel Construction Bradford K. Douglas, P.E., Vice President, Engineering, American Wood Council, Representing: American Wood Council Jennifer Goupil, P.E., Director, Structural Engineering Institute, American Society of Civil Engineers, Representing: American Society of Civil Engineers Melvyn Green, P.E., S.E., Structural Engineer, Melvyn Green & Associates, Representing: Earthquake Engineering Research Institute John R. Hayes, Jr. ("Jack"), P.E., PhD., NEHRP Director, National Institute of Standards and Technology (NIST), Representing: National Institute of Standards and Technology Jay W. Larson, P.E., F.ASCE, Managing Director, Construction Technical, American Iron and Steel Institute, Representing: American Iron and Steel Institute Stephen S. Szoke. P.E., FACI, Director, Codes and Standards, Portland Cement Association, Representing: Portland Cement Association Jason J. Thompson, Vice President of Engineering, National Concrete Masonry Association, Representing: National Concrete Masonry Association BSSC Staff Dana K. (Deke) Smith, FAIA, Executive Director, Building Seismic Safety Council Drew N. Rowland, PMP, Program Director Roger J. Grant, CSI, CDT, Program Director

Appendix A: Building Seismic Safety Council

Voting Members American Concrete Institute American Institute of Architects American Institute of Steel Construction American Society of Mechanical Engineers APA-The Engineered Wood Association Applied Technology Council Brick Industry Association Building Owners & Managers Association International California Seismic Safety Commission Concrete Masonry Association of California and Nevada Institute for Business and Home Safety International Code Council, Inc. Masonry Institute of America National Association of Home Builders National Fire Sprinkler Association National Institute of Building Sciences National Institute of Standards and Technology PLANiT Measuring Portland Cement Association Rack Manufacturers Institute Steel Deck Institute Structural Engineers Association of San Diego Structural Engineers Association of Washington

FEMA P-751, NEHRP Recommended Provisions: Design Examples Affiliate Members Architectural Testing, Inc. Baltimore Aircoil Company Building Technology Inc. Collins Engineers, Inc.

Appendix A: Building Seismic Safety Council

For a complete list of all BSSC publications and to download copies free of charge, visit the BSSC website at www.nibs.org/bssc/publications. BSSC Publications are also available free of charge from the Federal Emergency Management Agency at 1-800-4802520 (by FEMA Publication Number). For detailed information about the BSSC and its projects, visit the BSSC website at www.nibs.org/bssc or contact the Council directly at: BSSC 1090 Vermont A venue, N .W., Suite 700, Washington, D .C . 20005; Phone: 202-289-7800; Fax 202-289-1092 e-mail dsmith@ nibs.org

FEMA P-751 Catalog No. 12253-1

FEMA P-751 - 2009 NEHRP Recommended Seismic Provisions Design Examples.pdf - PDFCOFFEE.COM (2024)