100779794
100779794
Copyright © 2018 ICC. ALL RIGHTS RESERVED. Accessed by Tiankai Min (mintiankai@yahoo.com), (-) Order Number #100779794 on Feb 18, 2020 07:28 PM (PST) pursuant to License Agreement with ICC. No
further reproduction, no further reproductions by any third party, or distribution authorized. Single user only, copying and networking prohibited. ANY UNAUTHORIZED REPRODUCTION OR DISTRIBUTION
IS A VIOLATION OF THE FEDERAL COPYRIGHT ACT AND THE LICENSE AGREEMENT, AND SUBJECT TO CIVIL AND CRIMINAL PENALTIES THEREUNDER.
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Seismic and Wind Forces: Structural Design Examples
5th Edition
ISBN: 978-1-60983-844-7
Publications Manager:
Project Editor:
Typesetting:
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COPYRIGHT © 2018
by
INTERNATIONAL CODE COUNCIL, INC.
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First Printing: December 2018
PRINTED IN THE USA
T023374
Copyright © 2018 ICC. ALL RIGHTS RESERVED. Accessed by Tiankai Min (mintiankai@yahoo.com), (-) Order Number #100779794 on Feb 18, 2020 07:28 PM (PST) pursuant to License Agreement with ICC. No
further reproduction, no further reproductions by any third party, or distribution authorized. Single user only, copying and networking prohibited. ANY UNAUTHORIZED REPRODUCTION OR DISTRIBUTION
IS A VIOLATION OF THE FEDERAL COPYRIGHT ACT AND THE LICENSE AGREEMENT, AND SUBJECT TO CIVIL AND CRIMINAL PENALTIES THEREUNDER.
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TABLE OF CONTENTS
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
1
SEISMIC DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 Seismic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Design procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Site classification characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Earthquake response spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Site-specific procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Site coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Adjusted maximum considered earthquake spectral response accelerations . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Fundamental period of vibration of the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 General approximate method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Approximate method for moment-resisting frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Rational analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Design spectral response acceleration parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Risk categories and importance factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Seismic design category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Seismic design category A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.2 Seismic design category B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.3 Seismic design category C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.4 Seismic design category D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.5 Seismic design category E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.6 Seismic design category F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Lateral-force-resisting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Bearing wall systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.2 Building frame system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.3 Moment-resisting frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.4 Dual systems with special moment frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.5 Dual systems with intermediate moment frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.6 Shear wall-frame interactive system with ordinary reinforced concrete moment frames
and ordinary reinforced concrete shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.7 Cantilever column systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.8 Steel systems not specifically detailed for seismic resistance, excluding cantilever
column systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.9 Wind effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Response modification coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Seismic-force-resisting system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12.2 Combinations of seismic-force-resisting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Overstrength factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Deflection amplification factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Effective seismic weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Seismic response coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Seismic base shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Simplified lateral force procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19 Vertical distribution of seismic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.20 Simplified vertical distribution of base shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.21 Vertical seismic load effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.21.1 Overturning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.21.2 Foundation design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.21.3 Optional vertical seismic load effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22 Diaphragm loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
7
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Copyright © 2018 ICC. ALL RIGHTS RESERVED. Accessed by Tiankai Min (mintiankai@yahoo.com), (-) Order Number #100779794 on Feb 18, 2020 07:28 PM (PST) pursuant to License Agreement with ICC. No
further reproduction, no further reproductions by any third party, or distribution authorized. Single user only, copying and networking prohibited. ANY UNAUTHORIZED REPRODUCTION OR DISTRIBUTION
IS A VIOLATION OF THE FEDERAL COPYRIGHT ACT AND THE LICENSE AGREEMENT, AND SUBJECT TO CIVIL AND CRIMINAL PENALTIES THEREUNDER.
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1.23 Story drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.24 Simplified determination of drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.25 P-delta effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.26 Building separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.27 Redundancy factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28 Load combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28.1 Strength design loads and load factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28.2 Special seismic load combinations for the strength design method . . . . . . . . . . . . . . . . . . . . . . . . . 1.28.3 Allowable stress design method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.28.4 Special seismic load combinations for the allowable stress design method . . . . . . . . . . . . . . . . . . 1.29 Structural elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.29.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.29.2 Lateral design force on walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.29.3 Lateral design force on parapets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.30 Anchorage of structural walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.30.1 Anchorage to flexible diaphragms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.30.2 Anchorage to rigid diaphragms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.30.3 Subdiaphragms and continuous ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.31 Architectural, mechanical, and electrical components supported by structures . . . . . . . . . . . . . . . . . . . . . 1.31.1 Design force on mechanical and electrical components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.31.2 Design force on architectural components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.31.3 Wall cladding displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.31.4 Wall cladding seismic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.32 Rigidity and torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.32.1 Shear wall stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.32.2 Rigid diaphragm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33 Modal analysis procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.1 Horizontal structural irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.2 Vertical structural irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.3 Selection of lateral force procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.4 Modal shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.5 Modal participation factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.6 Modal base shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.7 Scaling factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.33.8 Vertical distribution of modal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
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DESIGN FOR WIND LOADS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.1 Wind effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Analysis procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Exposure category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Basic wind speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Velocity pressure exposure coefficients for the whole building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Topographic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Directionality factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Building types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7 Gust effect factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.8 Enclosure classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.9 Ground elevation factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Analytical directional design method for MWFRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Minimum design wind loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Design wind load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Wind velocity pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Internal pressure coefficients and internal pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 External pressure coefficients and external pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
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Copyright © 2018 ICC. ALL RIGHTS RESERVED. Accessed by Tiankai Min (mintiankai@yahoo.com), (-) Order Number #100779794 on Feb 18, 2020 07:28 PM (PST) pursuant to License Agreement with ICC. No
further reproduction, no further reproductions by any third party, or distribution authorized. Single user only, copying and networking prohibited. ANY UNAUTHORIZED REPRODUCTION OR DISTRIBUTION
IS A VIOLATION OF THE FEDERAL COPYRIGHT ACT AND THE LICENSE AGREEMENT, AND SUBJECT TO CIVIL AND CRIMINAL PENALTIES THEREUNDER.
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2.5 Simplified directional design method for MWFRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Wall pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Roof pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Analytical envelope design method for MWFRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Wind velocity pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Internal pressure coefficients and internal pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 External pressure coefficients and external pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Design wind load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Simplified envelope design procedure for MWFRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Adjustment of net pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Simplified method applied to the MWFRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Components and cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Determination of components and cladding loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Analytical envelope design method for components and cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Velocity pressure exposure coefficients and velocity pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3 Internal pressure coefficients and internal pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.4 External pressure coefficients and external pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Simplified envelope design method for components and cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Adjustment of net pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.3 Net wind pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Analytical directional design method for components and cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.2 Velocity pressure exposure coefficients and velocity pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.3 Internal pressure coefficients and internal pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.4 External pressure coefficients and external pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Simplified directional design method for components and cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.2 Adjustment of net pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.3 Net wind pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
202
204
208
209
209
210
210
212
215
215
216
217
221
221
223
224
224
225
227
231
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232
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236
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237
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241
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SEISMIC DESIGN OF STEEL STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1 General design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Material strength and ductility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Capacity design and expected material strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Demand critical welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Protected zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Loads and load combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Concentrically braced frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Ordinary concentrically braced frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Diagonal braces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Beams in chevron configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.4 Diagonal brace connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Special concentrically braced frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Capacity design basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2 Diagonal braces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.3 Diagonal brace connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.4 Beams in chevron configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.5 Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Eccentrically braced frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Basic requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
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269
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Copyright © 2018 ICC. ALL RIGHTS RESERVED. Accessed by Tiankai Min (mintiankai@yahoo.com), (-) Order Number #100779794 on Feb 18, 2020 07:28 PM (PST) pursuant to License Agreement with ICC. No
further reproduction, no further reproductions by any third party, or distribution authorized. Single user only, copying and networking prohibited. ANY UNAUTHORIZED REPRODUCTION OR DISTRIBUTION
IS A VIOLATION OF THE FEDERAL COPYRIGHT ACT AND THE LICENSE AGREEMENT, AND SUBJECT TO CIVIL AND CRIMINAL PENALTIES THEREUNDER.
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3.10.2 Link requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.3 Link shear strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.4 Link length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.5 Link rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.6 Link stiffeners for I-shaped cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.7 Beam requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10.8 Diagonal brace requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.9 Column requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Special moment frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.1 Beam-to-column connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11.2 Design principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.3 Strong column-weak beam concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.4 Beam details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.5 Column details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.6 Panel zone design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.7 Continuity plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Buckling-restrained braced frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.1 Buckling-restrained brace applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.2 Brace requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12.3 Brace connection requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.4 Beam design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.5 Column design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Steel special plate shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.1 Web requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.2 Strip model methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
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320
323
327
327
331
334
340
341
343
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351
353
354
360
361
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365
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SEISMIC DESIGN OF CONCRETE STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1 Special moment frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Design loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Beam details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Beam design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Column details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Column design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Joint design and details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Special structural walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Shear capacity of shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Special boundary elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Nonspecial boundary elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Slender wall design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Required strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Service load deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Anchorage in concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Design requirements for tensile loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Design requirements for shear loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Interaction of tensile and shear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
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381
384
397
401
414
418
419
420
424
433
434
435
437
445
448
452
455
459
SEISMIC DESIGN OF WOOD STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.1 General provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Building classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Design methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Lateral-force-resisting system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Lateral load path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Connection details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
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Seismic and Wind Forces: Structural Design Examples
Copyright © 2018 ICC. ALL RIGHTS RESERVED. Accessed by Tiankai Min (mintiankai@yahoo.com), (-) Order Number #100779794 on Feb 18, 2020 07:28 PM (PST) pursuant to License Agreement with ICC. No
further reproduction, no further reproductions by any third party, or distribution authorized. Single user only, copying and networking prohibited. ANY UNAUTHORIZED REPRODUCTION OR DISTRIBUTION
IS A VIOLATION OF THE FEDERAL COPYRIGHT ACT AND THE LICENSE AGREEMENT, AND SUBJECT TO CIVIL AND CRIMINAL PENALTIES THEREUNDER.
00_SeismicWindForces_5th.indd 6
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5.3 Diaphragms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Diaphragm strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Diaphragm deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Diaphragm flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Subdiaphragm requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.6 Design of collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Shear wall strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Shear wall deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Design using the segmented shear wall method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.5 Design using the perforated shear wall method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Design using the force transfer round openings method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Wood structural panels to resist combined shear and uplift from wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
466
467
477
480
484
487
492
492
499
508
512
517
524
529
529
532
SEISMIC DESIGN OF MASONRY STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . .
535
6.1 Reinforced masonry shear walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Reinforcement requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Design loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Strength reduction factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Shear capacity of a shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Axial load capacity of a shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 Flexural capacity of a shear wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.7 Boundary elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.8 Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.9 Sliding shear and shear friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Walls with out-of-plane loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Strength reduction factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Shear capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Flexural demand on a slender wall with one layer of centered reinforcement . . . . . . . . . . . . . . . . 6.2.4 Flexural capacity of a slender wall with one layer of centered reinforcement . . . . . . . . . . . . . . . . 6.2.5 Deflection under service loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Headed anchor bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Headed anchor bolt installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Seismic design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Headed anchor bolts in tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Headed anchor bolts in shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
540
541
542
543
547
550
563
565
569
571
572
572
575
578
586
587
587
589
589
592
597
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
599
6
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About the Author
Dr. Alan Williams was educated in the United Kingdom where he obtained his B.Sc. and Ph.D. degrees
at Leeds University. He subsequently has had extensive and diverse experience in the practice and
teaching of structural engineering.
Dr. Williams’ practical experience includes bridge design with the Division of Roads in Zimbabwe
and the design of bridges and industrial and commercial structures as a Consulting Engineer in South
Africa and the United States. He has been employed as a Senior Engineer with the State of California
Department of Transportation and as Principal for structural safety with the California Division of the
State Architect.
His academic positions include Associate Professor at the University of Science and Technology in
Ghana, Professor of Structural Analysis at Ahmadu Bello University in Nigeria, External Examiner at
the University of Cape Town, and Lecturer in structural steel design and reinforced concrete design at
the University of California, Irvine.
The author’s published works include textbooks on structural engineering design, structural analysis,
seismic design, and steel and reinforced concrete design. He has authored numerous technical papers
for international journals and conferences.
Dr. Williams is a Fellow and Life Member of the Institution of Civil Engineers, a Chartered Engineer
in the United Kingdom, and a registered Structural Engineer in California.
Seismic and Wind Forces: Structural Design Examples
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Introduction
The purpose of this publication is to provide an understanding of the application of the 2018 International Building Code®1 (IBC®) to current design practice. The IBC is a design standard that is adopted
by jurisdictions throughout the world as the mandated building code.
This text is intended to facilitate the transition of designers, teachers, and students from the previous
code and aid with code compliance. In the text, sections of the code are presented, analyzed, and
explained in a logical and simple manner and are followed by an illustrative example. Each example
concentrates on a specific section of the code and provides a clear and concise interpretation of the
issue.
The text is organized into six chapters that correspond to the primary structural design sections of the
code. These are:
•
earthquake loads
•
wind loads
•
design of steel structures
•
design of concrete structures
•
design of wood structures
•
design of masonry structures
Chapter 16 of the IBC deals with structural design loads. These provisions are derived from ASCE
72 and the NEHRP3 provisions. In this text, seismic design loads are covered in Chapter 1 and wind
design loads are covered in Chapter 2.
Chapter 22 of the IBC deals with the seismic design of steel structures and is based on the AISC4 seismic provisions. These requirements are covered in Chapter 3 of this text.
Seismic design of concrete structures is covered in Chapter 19 of the IBC and Chapter 4 of this text.
These provisions are derived from the ACI5 building code.
Seismic design of wood structures is covered in Chapter 23 of the IBC and Chapter 5 of this text.
These requirements are derived from the NDS6 code.
Seismic design of masonry structures is dealt with in Chapter 21 of the IBC. These provisions are
derived from Masonry Society code TMS 402.7 Chapter 6 of this text covers these requirements.
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About the International Code Council
The International Code Council® (ICC®), a membership association dedicated to building safety, fire
prevention, and energy efficiency, develops the codes and standards used to construct residential and
commercial buildings, including homes and schools. The mission of the ICC is to provide the highest
quality codes, standards, products, and services for all concerned with the safety and performance of
the built environment. Most United States cities, counties, and states choose the International Codes®
(I-Codes®)—building safety codes developed by the International Code Council.
The I-Codes also serve as the basis for construction of federal properties around the world and as a
reference for many nations outside the Untied States. The Code Council is also dedicated to innovation and sustainability. ICC Evaluation Service® (ICC-ES®), a subsidiary of ICC, issues Evaluation
Reports and Listings for innovative building products as well as environmental documents such as
ICC-ES VAR Environmental Reports and ICC-ES Environmental Product Declarations (EPDs).
ICC Headquarters:
500 New Jersey Avenue, NW
6th Floor
Washington, DC 20001
District Offices:
Birmingham, AL • Chicago, IL • Los Angeles, CA
Telephone: 1-888-422-7233 (ICC SAFE)
www.iccsafe.org
References
1. International Code Council. 2018 International Building Code. Washington, DC, 2018.
2. American Society of Civil Engineers. Minimum Design Loads and Associated Criteria for Buildings and Other Structures: ASCE 7-16. Reston, VA, 2016.
3. Building Seismic Safety Council. NEHRP Recommended Seismic Provisions for New Buildings
and Other Structures. Washington, DC, 2009.
4. American Institute of Steel Construction. Seismic Provisions for Structural Steel Buildings. Chicago, IL, 2016.
5. American Concrete Institute. Building Code Requirements and Commentary for Structural Concrete (ACI 318-14). Farmington Hills, MI, 2014.
6. American Wood Council. National Design Specification for Wood Construction (ANSI/AWC
NDS-2018). Leesburg, VA, 2018.
7. The Masonry Society. Building Code Requirements for Masonry Structures (TMS 402-16). Longmont, CO, 2016.
xii
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CHAPTER
1
Seismic Design
Nomenclature
AT
tributary wall area
ft2
Cd
deflection amplification factor from ASCE 7 Table 12.2-1
–
Cs
seismic response coefficient specified in ASCE 7 Section 12.8.1
–
Cu
coefficient for upper limit on calculated period from ASCE 7 Table 12.8-1
–
Cv
vertical coefficient given in ASCE 7 Table 11.9-1
–
D
effect of dead load
lb or kips
E
calculated seismic load on an element of a structure resulting from both
horizontal and vertical earthquake-induced forces as given by ASCE 7
Equation (12.4-1) and Equation (12.4-2)
lb or kips
Eh
calculated horizontal seismic load on an element of a structure as given by
ASCE 7 Equation (12.4-3)
lb or kips
Ev
calculated vertical seismic load on an element of a structure as given by
ASCE 7 Equation (12.4-4a)
lb or kips
fi
design seismic lateral force at level i
lb or kips
Fa
short-period site coefficient
–
Fp
force on diaphragm
lb or kips
Fv
long-period site coefficient
–
Fx
design seismic lateral force at level x as specified in ASCE 7 Section 12.8.3
lb or kips
g
gravitational acceleration
32.2 ft/sec2,
386.4 in/sec2
hi
height above the base to level i
ft
hn
height of the roof above the base, not including the height of penthouses or
parapets
ft
hs
story height
ft
hsx
story height below level x
ft
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2
Seismic Design
hx
height above the base to level x
ft
Ie
seismic importance factor
–
k
distribution exponent given in ASCE 7 Section 12.8.3
–
k
member stiffness
kips/in
L
effect of live load
lb or kips
MCER
risk-targeted maximum considered earthquake
–
MP
primary moment
kip-ft
MS
secondary moment
kip-ft
N
number of stories
–
N
notional load
lb or kips
PI
plasticity index
–
Px
total unfactored vertical design load at and above level x
lb or kips
QE
effect of horizontal seismic forces
lb or kips
R
response modification coefficient for a specific structural system from
ASCE 7 Table 12.2-1
–
su
undrained shear strength
lb/ft2
S
effect of snow load
lb or kips
S1
maximum considered response acceleration for a period of 1.0 second
–
Sa
design spectral response acceleration
–
SaM
spectral response acceleration at any period
–
SaMv
vertical spectral response acceleration at any period
–
SDS
design spectral response acceleration at a period of 0.2 second
–
SD1
design spectral response acceleration at a period of 1.0 second
–
SMS
modified spectral response acceleration at a period of 0.2 second
–
SM1
modified spectral response acceleration at a period of 1.0 second
–
SS
maximum considered response acceleration for a period of 0.2 second
–
T
fundamental period of vibration, defined in ASCE 7 Section 12.8.2
sec
T0
defined in ASCE 7 Section 11.4.6 as 0.2SD1/SDS
–
Ta
approximate fundamental period of vibration determined using ASCE 7
Section 12.8.2.1
sec
TL
long-period transition period
sec
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3
Chapter 1
Tr
fundamental period of vibration determined by the Rayleigh procedure
sec
TS
defined in ASCE 7 Section 11.4.6 as SD1/SDS
–
Tv
vertical period of vibration
sec
V
total seismic base shear
lb or kips
Vx
total shear force at level x
lb or kips
Vx
seismic shear force acting between levels x and (x 2 1)
lb or kips
VY
base shear at formation of the collapse mechanism
lb or kips
wi
seismic weight located at level i
lb or kips
wp
seismic weight tributary to diaphragm
lb or kips
wx
seismic weight located at level x
lb or kips
ww
weight of wall tributary to connection
lb or kips
W
wind load applied to a structural element
lb or kips
W
effective seismic weight defined in ASCE 7 Section 12.7.2
lb or kips
ΣFi
total shear force at level i
lb or kips
Σwi
total seismic weight at level i and above
lb or kips
β
ratio of shear demand to shear capacity for the story between levels x and
x 2 1, defined in ASCE 7 Section 12.8.7
–
δi
elastic horizontal deflection at level i, due to the forces fi
in
δmax
maximum elastic displacement at the critical location, considering torsion
in
δM
maximum inelastic displacement given by ASCE 7 Equation (12.12-1)
in
δMT
required separation between buildings given by ASCE 7 Equation (12.12-2)
in
δx
amplified horizontal deflection at level x, defined in ASCE 7 Section 12.8.6
in
δxe
horizontal deflection at level x, determined by an elastic analysis using
strength seismic forces, defined in ASCE 7 Section 12.8.6
in
Δ
design story drift, occurring simultaneously with the story shear Vx, defined
in ASCE 7 Section 12.8.6, and calculated using the amplification factor Cd
in
Δa
allowable story drift, defined in ASCE 7 Table 12.12-1
in
Ω0
overstrength factor tabulated in ASCE 7 Table 12.2-1
–
ρ
redundancy factor defined in ASCE 7 Section 12.3.4
–
θ
stability coefficient defined in ASCE 7 Section 12.8.7
–
Symbols
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4
Seismic Design
1.1 Seismic loads
Earthquakes are generated by a rupture along a fault zone in the underlying rock. The resultant shaking
in the rock propagates to the earth’s surface and causes vibrations in a structure. As a result of these
vibrations, inertial forces are created in the structure. The inertial forces produced are given by Newton’s Second Law of Motion, which states that the inertial force, F, equals the mass, m, multiplied by
the acceleration, a. Thus,
F 5 ma
At the onset of an earthquake, the ground displacement produces a corresponding displacement of
the foundation of a structure. Because of the inertia of the structure, the roof and upper stories do
not immediately respond. When the roof begins to move in the same direction as the foundation, the
ground displacement has reversed in direction, taking the foundation with it. Thus, the roof and the
foundation are moving in opposite directions and this whiplash effect may cause severe damage unless
the structure is appropriately designed and constructed.
Seismic loads are dynamic in nature and require a complex dynamic analysis for a complete solution.
An alternative method of analysis, the equivalent lateral force (ELF) procedure, provides a simple and
direct approach where a sophisticated dynamic analysis is not warranted. The procedure consists of
applying a single static force at the base of the building, as shown in Figure 1-1. This static force, V,
is termed the seismic base shear and is intended to reproduce in the structure forces similar to those
caused by the earthquake. After the seismic base shear is determined, the forces acting on the individual structural elements in the building may be calculated. The equivalent lateral force procedure is
applicable to regular structures, defined as structures without irregular features that have a reasonably
uniform distribution of stiffness, strength, and mass over the height of the structure. The structure mass
and the equivalent lateral forces are assumed to be concentrated at floor and roof levels as shown in
Figure 1-1.
In accordance with Newton’s Second Law of Motion, the lighter a structure is, the smaller the inertial force generated. Hence, lightweight buildings such as wood-frame houses and light-frame industrial buildings perform better in earthquakes than other types of buildings. The International Building
Code® (IBC®)1 Section 2308 exempts conventional light-frame buildings from seismic design requirements, provided that prescriptive construction limitations are complied with.
Conventional light-frame construction is defined in IBC Section 202 as:
Construction whose primary structural elements are formed by a system of repetitive wood-framing
members.
The IBC adopts by reference most of the seismic provisions of ASCE 7.2 In accordance with IBC
Section 1613.1:
Every structure, and portion thereof, including nonstructural components that are permanently
attached to structures and their supports and attachments, shall be designed and constructed to
resist the effects of earthquake motions in accordance with Chapters 11, 12, 13, 15, 17 and 18 of
ASCE 7, as applicable.
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level 4
F4
level 3
5
m4
F3
level 2
m3
F2
level 1
m2
F1
V
Base shear
m1
V
Equivalent frame
Figure 1-1 Equivalent lateral force procedure
The seismic design criteria of ASCE 7 are primarily based on FEMA P-7503, the 2009 edition of the
NEHRP Recommended Seismic Provisions for New Buildings and Other Structures. FEMA P-750
Section 1.1 states that the intent of the provisions is to avoid structural collapse in a major earthquake
and provide reasonable assurance of seismic performance that will:
•
avoid serious injury and loss of life
•
avoid loss of function in critical facilities
•
minimize structural and nonstructural repair costs where practical
The most severe earthquake ground motion considered by the IBC is the risk-targeted maximum considered earthquake (MCER). This is defined in IBC Section 202 as:
The most severe earthquake effects considered by this code, determined for the orientation that
results in the largest maximum response to horizontal ground motions and with adjustment for
targeted risk.
The design earthquake ground motion is defined in IBC Section 202 as:
The earthquake ground motion that buildings and structures are specifically proportioned to resist.
In accordance with ASCE 7 Section 11.2, the design earthquake ground motion is two-thirds of the
corresponding MCER ground motion. This factor of two-thirds accounts for the margin against collapse inherent in structures designed in accordance with ASCE 7. This is judged to correspond to
two-thirds times the MCER.
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Seismic Design
The consequences of damage to a structure are not the same for all types of structures. The collapse of
an essential building, such as a hospital, has a more severe effect on a community than the collapse of
an agricultural facility. IBC Table 1604.5 subdivides buildings into four risk categories and ASCE 7
Table 1.5-2 assigns a seismic importance factor, Ie , to each risk category. By this means, an essential
structure is designed for a higher seismic load than a less important structure. This results in a reduction in structural damage to an essential structure in a severe earthquake. Risk category is defined in
IBC Section 202 as:
A categorization of buildings and other structures for determination of flood, wind, snow, ice and
earthquake loads based on the risk associated with unacceptable performance.
In IBC Table 1604.5, risk categories are listed as:
•
Risk category I structures are low-hazard structures such as agricultural facilities, minor storage buildings, and temporary facilities. Risk category I structures are allocated an importance
factor of 1.0.
•
Risk category II structures are standard occupancy structures such as residential, commercial,
and office buildings. Risk category II structures are allocated an importance factor of 1.0.
•
Risk category III structures are facilities that represent a substantial hazard to human life in
the event of failure, including buildings with public assembly facilities, educational facilities,
health care facilities, jails, and power-generating stations, and facilities containing quantities of
toxic or explosive materials. Risk category III structures are allocated an importance factor of
1.25. This ensures that a risk category III structure is designed for a seismic load of 1.25 times
greater than a risk category II structure.
•
Risk category IV structures are buildings designated as essential facilities, including hospitals,
fire and police stations, postearthquake-recovery centers, and buildings that house equipment
for these facilities, as well as facilities housing quantities of toxic materials that are of sufficient quantity to pose a threat to public safety if released. Risk category IV structures are allocated an importance factor of 1.5. This ensures that a risk category IV structure is designed for
a seismic load of 1.5 times greater than a risk category II structure.
As shown in ASCE 7 Table 1.3-2, a risk category I and a risk category II structure have a conditional
probability of failure of 10 percent. This is equivalent to an absolute failure probability of 1 percent
in 50 years. The conditional probability of failure for a risk category III structure is 5 percent and 2.5
percent for a risk category IV structure.
In accordance with IBC Section 1613.1, the following buildings are exempt from seismic design
requirements:
•
detached one- and two-family dwellings, assigned to seismic design category A, B, or C, or
located where the mapped short-period spectral response acceleration, SS , is less than 0.4g
•
wood-frame buildings that conform to the provisions of IBC Section 2308
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7
•
agricultural storage structures intended only for incidental human occupancy
•
structures that require special consideration of their response characteristics and environment
that are not addressed by the IBC or ASCE 7 and for which other regulations provide seismic
criteria, such as vehicular bridges, electrical transmission towers, hydraulic structures, buried
utility lines and their appurtenances, and nuclear reactors
IBC Section 2308 includes provisions for buildings of conventional light-frame construction and
buildings complying with the International Residential Code® (IRC®).4
1.2 Design procedures
To determine the seismic response of a structure, several factors must be considered, and these include:
•
site classification characteristics
•
risk-targeted maximum considered earthquake spectral response accelerations
•
site coefficient
•
adjusted maximum considered earthquake spectral response accelerations
•
fundamental period of vibration of the structure
•
design spectral response accelerations
•
importance factors and risk category
•
seismic design category
•
lateral-force-resisting systems
•
response modification coefficient
•
overstrength factor
•
deflection amplification factor
•
effective seismic weight
•
seismic response coefficient
•
seismic base shear
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1.3 Site classification characteristics
The ground motion produced by an earthquake is affected by the soil profile through which the vibrations travel. The amplification of long-period spectral vibrations is significantly larger on soft soil than
on hard soil or rock. To account for this potential amplification, six different soil types are identified
in ASCE 7 Table 20.3-1, ranging from hard rock to soft clay soil and to sites containing peat, highly
plastic clay, or collapsible soil. The classification may be made by determining on-site the average
shear wave velocity in the top 100 feet of material. Alternatively, for site classification types C, D, and
E, the classification may be made by measuring the standard penetration resistance or undrained shear
strength of the material. Soil classification type B is defined as rock and occurs mainly in the western states. Soil classification type A is defined as hard rock and has the effect of reducing the ground
response by 20 percent. Soil classification type A occurs mainly in the eastern states. Soil classification type E is defined as soft soil and has the effect of increasing the long-period ground response by
up to 350 percent. Soil classification type F is defined as peat, highly plastic clay, or collapsible soil
and generally requires a site-specific evaluation to determine the risk-targeted maximum considered
earthquake response parameters. Where soil parameters are unknown, in accordance with ASCE 7
Section 11.4.3, soil classification type D may be assumed unless the building official determines that
soil classification type E or F is likely to be present at the site. The site classifications are defined in
ASCE 7 Table 20.3-1 and an abbreviated listing is provided in Table 1-1.
Table 1-1 Site classification definitions
Site
classification
Soil profile
name
Shear wave velocity,
ft/sec
A
Hard rock
. 5000
B
Rock
2500 to 5000
C
Soft rock
1200 to 2500
D
Stiff soil
600 to 1200
E
Soft clay soil
, 600
F
—
—
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Chapter 1
Example 1-1
The two-story steel-frame building shown in Figure 1-2 is used as an office building. The soil profile
at the site consists of a 100-foot depth of stiff soil with a shear wave velocity of 1300 feet per second.
Determine the applicable site classification.
level 2
N
h=
s = 40 ft
level 1
h=
b = 20 ft
Section
20 ft
20 ft
Plan
Figure 1-2 Details for Example 1-1
Solution
From ASCE 7 Table 20.3-1, the applicable site classification for this soil profile is site classification C.
1.4 Earthquake response spectra
The ground motion parameters SS and S1 are defined in ASCE 7 Section 11.4.2 and are mapped in
ASCE 7 Figures 22-1 through 22-8. Alternatively, the ground motion parameters may be obtained
from the Applied Technology Council website https://hazards.atcouncil.org for locations with known
latitude and longitude or postal address. The two values provided, SS and S1, represent the risk-targeted
maximum considered earthquake (MCER) response accelerations at periods of 0.2 second and 1.0
second for 5-percent damping. Periods of 0.2 second and 1.0 second represent the approximate natural
period of a short and tall building, respectively. The ground motion parameters are adjusted to a reference site condition with an average shear wave velocity of 2500 ft/sec.
To achieve uniformity in structural collapse throughout the United States, the ground motion parameters are risk targeted to provide a uniform risk with a 1-percent probability of building collapse in 50
years.5
For a seismically active region such as coastal California, a probabilistic approach results in much
higher accelerations than that of the characteristic earthquakes in the region. Hence, for this region,
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the values represent the deterministic event defined as the median estimate of the accelerations of the
characteristic earthquakes increased by 50 percent. The characteristic earthquake is defined as the
maximum acceleration capable of occurring in the region but not less than the largest acceleration that
has been recorded in the region.
Two procedures are available for determining the risk-targeted maximum considered earthquake and
the response spectrum. These are the general procedure and the site-specific procedure.
In accordance with ASCE 7 Section 20.3.1, a site response analysis is necessary for structures on site
class F sites with the following exceptions:
•
a structure having a fundamental period of 0.5 second or less situated on liquefiable soil and
where the site class is assigned in accordance with ASCE 7 Table 20.3-1
•
for highly plastic clays with a plasticity index exceeding 75 provided that Fa and Fv are obtained
from Table 1-2 for site class D or E multiplied by a factor that varies linearly from 1.0 at PI 5
75 to 1.3 for PI 5 125 and is equal to 1.3 for PI . 125 and, in addition, the resulting values
of SDS and SD1 do not exceed the upper bound values for seismic design category B given in
Table 1-6
•
for very thick soft/medium stiff clays with a thickness exceeding 120 feet and an undrained
shear strength of less than 1000 psf provided that Fa and Fv are obtained from Table 1-2 for
site class E and, in addition, the resulting values of SDS and SD1 do not exceed the upper bound
values for seismic design category B given in Table 1-6
In accordance with ASCE 7 Section 11.4.8, a ground motion hazard analysis is necessary in the following situations:
•
seismically isolated structures and structures with damping systems at sites with S1 greater than
or equal to 0.6g
•
structures on site class E sites with values of SS greater than or equal to 1.0g
•
structures on site class D or E sites for values of S1 greater than or equal to 0.2g
However, with the exception of seismically isolated structures and structures with damping systems, a
ground motion hazard analysis is not required where the structure is located on:
•
a site class E site with a value of SS greater than or equal to 1.0g provided that the site coefficient Fa is taken as equal to that of site class C (Fa 5 1.2)
•
a site class D site with a value of S1 greater than or equal to 0.2g, provided that the value of the
seismic response coefficient Cs is conservatively calculated using ASCE 7 Equation (12.8-2)
for T ≤ 1.5TS and using 1.5 times the value computed in accordance with either ASCE 7 Equation (12.8-3) for TL ≥ T . 1.5TS or ASCE 7 Equation (12.8-4) for T . TL
•
a site class E site with a value of S1 greater than or equal to 0.2g provided that T is less than or
equal to TS and the ELF procedure is used for design
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1.4.1 General procedure
To apply the general procedure, reference may be made to the risk-targeted maximum considered
earthquake spectral response accelerations mapped in the ASCE 7 provisions. Two sets of maps are
provided to designate the two parameters, SS and S1. SS represents the 5-percent damped, risk-targeted
maximum considered earthquake spectral response acceleration for a period of 0.2 second for structures founded on rock with an average shear wave velocity of 2500 ft/sec and is applicable to short
period structures. S1 represents the 5-percent damped, risk-targeted maximum considered earthquake
spectral response acceleration for a period of 1.0 second for structures founded on rock with an average shear wave velocity of 2500 ft/sec. Figure 1-3 shows the effect produced on the response spectra
by different soil types.
Precise values of the two parameters, SS and S1, are difficult to determine in congested areas of the
maps. To obviate this problem, a software program that calculates the spectral parameters from the
latitude and longitude of a specific location is available on the Applied Technology Council website at
https://hazards.atcouncil.org
The latitude and longitude for a specific location may be obtained from several websites.
Alternatively, the spectral parameters may be determined by the program for a given postal address.
Site classification D
1.0
Acceleration, Sa g
Site classification A
0.5
0
0.5
1.0
1.5
Period, T sec
2.0
Figure 1-3 Representative response spectra
1.4.2 Site-specific procedure
A site-specific study must account for the regional seismicity and geology, the magnitudes, recurrence rates, and locations of earthquakes on known active faults in the region, and the soil profile. The
procedure for determining the site-specific risk-targeted maximum considered earthquake response
spectrum is detailed in ASCE 7 Section 21.2. This consists of comparing the spectra resulting from
a probabilistic and a deterministic risk-targeted maximum considered earthquake with a predefined
deterministic lower limit.
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In accordance with ASCE 7 Section 21.2.1, the probabilistic spectral response acceleration is taken as
the spectral response acceleration in the direction of maximum horizontal ground motions represented
by a 5-percent damped acceleration response spectrum that is expected to achieve a 1-percent probability of collapse within a 50-year period.
In accordance with ASCE 7 Section 21.2.2, the deterministic spectral response acceleration at each
period is calculated as the largest 84th percentile 5-percent damped spectral response acceleration
in the direction of maximum horizontal response computed at that period for characteristic earthquakes on all known active faults within the region. The ordinates of the deterministic ground motions
response spectrum are not taken lower than the corresponding ordinates of the deterministic lower
limit on risk-targeted maximum considered response spectrum shown in ASCE 7 Figure 21.2-1. This
figure is reproduced in Figure 1-4.
For site classes A, B, or C
Fa is the site coefficient at a period of 0.2 second and is obtained from ASCE 7 Table 11.4-1, with SS
taken as 1.5g. Fv is the site coefficient at a period of 1.0 second and is obtained from ASCE 7 Table
11.4-2, with S1 taken as 0.6g.
For site class D
Fa is taken as 1.0 and Fv is taken as 2.5
For site classes E and F
Fa is taken as 1.0 and Fv is taken as 4.0
= 0.6FvTL/T 2
0.08Fv /Fa
0.4Fv /Fa
TL
Figure 1-4 Deterministic lower limit on risk-targeted maximum considered earthquake response spectrum
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In accordance with ASCE 7 Section 21.2.3, the site-specific MCER spectral response acceleration at
any period, SaM , shall be taken as the lesser of the spectral response accelerations from the probabilistic
ground motions of ASCE 7 Section 21.2.1 and the deterministic ground motions of ASCE 7 Section
21.2.2.
1.5 Site coefficient
Site coefficients are amplification factors applied to the maximum considered earthquake response
parameters, obtained by the general procedure at a specific site, to account for the site classification
characteristics and response parameters at the site. Fa is the short period or acceleration-based amplification factor and is defined in ASCE 7 Table 11.4-1. Fv is the long period or velocity-based amplification factor and is defined in ASCE 7 Table 11.4-2. In general, as the soil profile becomes progressively
softer, the value of the site coefficient increases. However, the short period site coefficient for a value
of SS ≥ 1.0 reduces for site classification type D, reflecting the tendency for the ground response to
attenuate as the seismicity increases. ASCE 7 Tables 11.4-1 and 11.4-2 are reproduced in Table 1-2.
Linear interpolation may be used to obtain intermediate values. The site coefficients are presented in
graphical form in Figure 1-5.
Table 1-2 Site coefficients Fa corresponding to Ss , and Fv corresponding to S1
Site
classification  0.25
Response acceleration, SS
Response acceleration, S1
0.50
0.75
1.00
1.25
 1.50
 0.1
0.2
0.3
0.4
0.5
 0.6
A
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
B
0.9
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.8
C
1.3
1.3
1.2
1.2
1.2
1.2
1.5
1.5
1.5
1.5
1.5
1.4
D
1.6
1.4
1.2
1.1
1.0
1.0
2.4
2.2c
2.0c
1.9c
1.8c
1.7c
E
2.4
1.7
1.3
a
a
a
4.2
b
b
b
b
b
F
d
d
d
d
d
d
d
d
d
d
d
d
Notes:
a. A ground motion hazard analysis is required unless the site coefficient Fa is taken as equal to that of site class C (Fa 5 1.2).
b. A ground motion hazard analysis is required unless T is less than or equal to TS and the ELF procedure is used for design.
c. A ground motion hazard analysis is required unless the value of the seismic response coefficient Cs is conservatively calculated
using ASCE 7 Equation (12.8-2) for T ≤ 1.5TS and taken as equal to 1.5 times the value computed in accordance with either
ASCE 7 Equation (12.8-3) for TL ≥ T . 1.5TS or ASCE 7 Equation (12.8-4) for T . TL.
d. A site response analysis is required unless any of the exceptions to ASCE 7 Section 20.3.1 are applicable.
For situations in which site investigations reveal competent rock conditions with moderate fracturing
and weathering consistent with site class B, but site-specific velocity measurements are not made, the
site coefficients Fa and Fv are taken as unity (1.0).
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Seismic Design
Where site class D is selected as the default site class in accordance ASCE 7 Section 11.4.3, the value
of Fa must not be less than 1.2.
E
3
4
E
3
2
D
Fv
Fa
D
C
1 A
2
C
1 A
0
1.0
2.0
0
0.2
SS
0.4
0.6
0.8
S1
Figure 1-5 Site coefficients
Example 1-2
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. The
risk-targeted maximum response accelerations are SS 5 1.26g and S1 5 0.457g. Determine the site
coefficients for this structure.
Solution
From Example 1-1, the site classification at the location of this structure is site classification C. From
the problem statement, the maximum considered earthquake response accelerations are
SS
5 1.260g
S1
5 0.457g
From Table 1-2, the site coefficients are determined as
Fa
5 1.2
Fv
5 1.5
Example 1-3
The two-story steel-frame building shown in Figure 1-2 is located in Miami, Florida. The risk-targeted
maximum considered response accelerations are SS 5 0.05g and S1 5 0.019g. Determine the site coefficients for this structure.
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15
Solution
From Example 1-1, the site classification at the location of this structure is site classification C. From
the problem statement, the maximum considered earthquake response accelerations are
SS
5 0.050g
S1
5 0.019g
From Table 1-2, the site coefficients are determined as
Fa
5 1.3
Fv
5 1.5
1.6 Adjusted maximum considered earthquake spectral response
accelerations
The maximum considered earthquake spectral response accelerations, obtained by the general procedure, must be modified for the site classification effects. ASCE 7 Equations (11.4-1) and (11.4-2)
define the modified spectral response accelerations at short periods and at a period of 1.0 second as
SMS
5 FaSS
SM1
5 FvS1
No adjustment is necessary to the maximum considered earthquake spectral response accelerations,
SaM , derived by the site-specific procedure, as these values already reflect the site classification effects.
Example 1-4
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Determine the adjusted maximum considered earthquake spectral response accelerations for the structure.
Solution
From Example 1-2, the site coefficients are
Fa
5 1.2
Fv
5 1.5
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Seismic Design
From ASCE 7 Equations (11.4-1) and (11.4-2) and Example 1-2, the adjusted spectral response accelerations at short periods and at a period of 1.0 second are
SMS
5 FaSS
5 1.2 3 1.260g
5 1.51g
SM1
5 FvS1
5 1.5 3 0.457g
5 0.69g
Example 1-5
The two-story steel-frame building shown in Figure 1-2 is located in Miami, Florida. Determine the
adjusted maximum considered earthquake spectral response accelerations for the structure.
Solution
From Example 1-3, the site coefficients are
Fa
5 1.3
Fv
5 1.5
From ASCE 7 Equations (11.4-1) and (11.4-2) and Example 1-3, the adjusted spectral response accelerations at short periods and at a period of 1.0 second are
SMS
5 FaSS
5 1.3 3 0.050g
5 0.065g
SM1
5 FvS1
5 1.5 3 0.019g
5 0.029g
1.7 Fundamental period of vibration of the structure
Each structure has a unique natural or fundamental period of vibration that is the time required to
complete one cycle in the first mode of free vibration. The factors determining the fundamental period
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include the stiffness and height of the structure. ASCE 7 Sections 12.8.2 and 12.8.2.1 provide three
methods for determining the fundamental period of a structure. These are the general approximate
method, the approximate method for moment-resisting frames, and the rational analysis method.
1.7.1 General approximate method
The general approximate method utilizes ASCE 7 Equation (12.8-7) and the approximate fundamental
period in seconds is given by
where:
Ta
5 0.028(hn)0.8 . . . for steel moment-resisting frames
Ta
5 0.016(hn)0.9 . . . for reinforced concrete moment-resisting frames
Ta
5 0.030(hn)0.75 . . . for eccentrically braced steel frames and bucklingrestrained braced frames
Ta
5 0.020(hn)0.75 . . . for all other structural systems
hn
5 height in feet of the roof above the base, not including the height of
penthouses or parapets
In order to use these values for moment-resisting frames, the moment-resisting frames must resist 100
percent of the required seismic force.
The calculated base shear for a structure is dependent on the magnitude of the fundamental period,
with a larger value of Ta producing a smaller value of the base shear. The approximate fundamental
period determined by ASCE 7 Equation (12.8-7) underestimates the actual value of the fundamental
period, thus providing a conservative value for the base shear.
Example 1-6
Determine the approximate fundamental period of vibration for the two-story steel-frame building
shown in Figure 1-2.
Solution
The approximate fundamental period is given by ASCE 7 Equation (12.8-7) as
where:
Ta
5 0.028(hn)0.8
hn
5 roof height
5 24 ft
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Then, the fundamental period is
Ta
5 0.028(24)0.8
5 0.36 sec
1.7.2 Approximate method for moment-resisting frames
For moment-resisting frames not exceeding 12 stories in height and with an average story height of at
least 10 feet, the approximate fundamental period may be determined by ASCE 7 Equation (12.8-8),
which is
where:
Ta
5 0.1N
N
5 number of stories
Example 1-7
Determine the approximate fundamental period of vibration for the two-story steel frame building
shown in Figure 1-2.
Solution
The story height is
hs
5 12 ft
. 10 ft . . . satisfactory
The number of stories is
N
52
, 12 . . . satisfactory
Then, for a moment-resisting frame, ASCE 7 Equation (12.8-8) specifies a value for the building
period of
Ta
5 0.1N
5 0.1 3 2
5 0.20 sec
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1.7.3 Rational analysis method
ASCE 7 Section 12.8.2 permits the fundamental period to be determined by a “properly substantiated
analysis.” The Rayleigh procedure is an acceptable method and the fundamental period is given by
Tr
5 2π(Σwi d2i /gΣfi di )1/2
5 (0.32)(Σwi d2i /Σfi di )1/2
where:
di
5 elastic horizontal deflection at level i due to the forces fi
fi
5 lateral force at level i
wi
5 seismic weight located at level i
g
5 acceleration due to gravity
The lateral forces, fi , represent any lateral force distribution increasing approximately uniformly with
height as shown in Figure 1-6. This distribution, in the form of an inverted triangle, corresponds to the
distribution of base shear that is assumed in ASCE 7 and is equivalent to the inertial forces produced
in a frame with uniform mass distribution, equal story heights, and with acceleration increasing uniformly with height. The mathematical model representing the structure must include all significant
elements of the lateral-force-resisting system. If the contribution of the nonstructural elements to the
stiffness of the structure is underestimated, the calculated deflections and natural periods are over­
estimated, giving a value for the base shear that is too low. To reduce the effects of this error, ASCE 7
Section 12.8.2 specifies that the value of the natural period determined by this method may not exceed
the value of
where:
T
5 CuTa . . . when Tr . CuTa
Ta
5 approximate fundamental period determined by ASCE 7 Equation (12.8-7)
Cu
5 coefficient for upper limit on calculated period
level level -1
-1
level -1
level 2
level 1
Frame
-1
2
2
2
1
1
1
Deflections
Lateral force
Story weights
C-104
Figure 1-6 Rayleigh procedure
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Values of Cu are given in ASCE 7 Table 12.8-1 and are shown in Table 1-3. As indicated, the values
of Cu are dependent on SD1, the 5-percent damped, design spectral response acceleration at a period of
1.0 second.
Table 1-3 Coefficient for upper limit on the calculated period
SD1
≥ 0.40
0.30
0.20
0.15
 0.10
Cu
1.4
1.4
1.5
1.6
1.7
Example 1-8
Using Rayleigh’s method, determine the fundamental period of vibration of the two-story steel-frame
building shown in Figure 1-2, which is located in an area with a value for SD1 exceeding 0.40. The
force system shown in Figure 1-7 may be utilized, and the seismic weight at each level and the total
stiffness of each story are indicated.
Figure 1-7 Details for Example 1-8
Solution
Applying the force system indicated, the displacements at each level are given by
δ1
5 ( f2 1 f1)/k1
5 (20 1 10)/30
5 1.00 in
δ2
5 f2 /k2 1 δ1
5 20/30 1 1.00
5 1.67 in
The fundamental period is given by the Rayleigh procedure as
Tr
5 0.32(Σwi d2i /Σfi di )1/2
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The relevant values are given in Table 1-4.
Table 1-4 Rayleigh procedure for Example 1-8
Then:
Tr
Level
wi
fi
di
wi d2i
fi di
2
25.60
20
1.67
71.40
33.40
1
51.20
10
1.00
51.20
10.00
Total
76.80
–
–
122.60
43.40
5 0.32(122.60/43.40)1/2
5 0.538 sec
In a location with a value for the design spectral response acceleration at a period of 1.0 second of
SD1 . 0.4, the value of the coefficient for the upper limit on the calculated period is obtained from
Table 1-3 as
Cu
5 1.4
The approximate fundamental period was determined in Example 1-6 as
Ta
5 0.36 sec
Hence, the fundamental period, in accordance with ASCE 7 Section 12.8.2, is limited to
CuTa
5 1.4Ta
5 1.4 3 0.36
5 0.50 sec
, 0.538 sec
Therefore, use a maximum value of
T
5 0.50 sec
1.8 Design spectral response acceleration parameters
The design objective of ASCE 7 is to provide a uniform risk against collapse for structures throughout all regions of the United States. For ground motions in excess of the design ground motions, the
intention is that there shall be a low likelihood of collapse. The ground motion parameters SS and
S1 represent the risk-targeted maximum considered earthquake ground motions. The design ground
motions SDS and SD1 represent a lower bound estimate of the margin against collapse of a structure.
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This lower bound is equivalent to a factor of 1.5 on the maximum considered earthquake ground
motions. Hence, the design ground motions are calculated as 1⁄1.5 or 2⁄3 times the maximum considered
earthquake motions. Because of the inherent seismic margin built into a structure, it is anticipated that
a structure experiencing a level of ground motion 150 percent of the design ground motion will have
a low likelihood of collapse.
The 5-percent damped, design spectral response accelerations, for a period of 0.2 second and for a
period of 1.0 second, are given by ASCE 7 Equations (11.4-3) and (11.4-4) as
SDS
5 2SMS /3
SD1
5 2SM1/3
The general procedure response spectrum is constructed as indicated in Figure 1-8. The short-period
spectral response value of 0.2 second represents the short-period range of the response spectra. The
long-period spectral response value of 1.0 second represents the long-period range of the response
spectra.
Figure 1-8 Construction of design response spectra
The relevant parameters are defined in ASCE 7 Section 11.4.5 and are given by
SDS
5 5-percent damped, design spectral response acceleration at short periods
SD1
5 5-percent damped, design spectral response acceleration at a period of 1.0
second
T
5 fundamental period of the structure
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TS
5 constant-velocity transition period 5 SD1/SDS
T0
5 constant-acceleration transition period 5 0.2SD1/SDS
TL
5 long-period transition period given on ASCE 7 Figures 22-14 through 22-17
For periods not greater than T0 , the design spectral response acceleration is given by ASCE 7 Equation
(11.4-5) as
Sa
5 SDS(0.4 1 0.6T/T0)
For periods greater than or equal to T0 and less than or equal to TS , the design spectral response acceleration is equal to SDS and this forms the flat-topped, constant-acceleration portion of the spectrum.
For periods greater than TS and less than or equal to TL, the curve forms the descending portion, constant-velocity section of the spectrum. Over this section, the design spectral response acceleration is
given by ASCE 7 Equation (11.4-6) as
Sa
5 SD1/T
For periods greater than TL, the curve forms the constant-displacement section of the spectrum. Values
of TL range from 4 seconds to 16 seconds and are obtained from the contour maps provided in ASCE
7. Over this section, the design spectral response acceleration is given by ASCE 7 Equation (11.4-7) as
Sa
5 SD1TL/T 2
For the site-specific procedure, design spectral response accelerations are given by ASCE 7 Equation
(21.3-1) as
where:
Sa
5 2SaM /3
SaM
5 maximum considered earthquake response accelerations derived by the sitespecific procedure
In accordance with ASCE 7 Section 21.3, the design spectral response accelerations derived from a
site-specific maximum considered earthquake response spectrum may not be taken as less than 80
percent of the values obtained from the corresponding general procedure response spectrum. The
design spectral response accelerations derived from a site-specific maximum considered earthquake
response spectrum for a site classification type F profile may not be taken less than 80 percent of the
values obtained from the corresponding general procedure response spectrum determined for a site
classification type E profile.
Example 1-9
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Determine the design spectral response accelerations and draw the general procedure response spectrum.
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Solution
From Example 1-4, the modified spectral response accelerations at short periods and at a period of 1.0
second are
SMS
5 1.51g
SM1
5 0.69g
The corresponding design spectral response accelerations are
SDS
5 2SMS /3
5 1.01g
SD1
5 2SM1/3
5 0.46g
The response spectrum parameters are given by
TS
5 SD1/SDS
5 0.46 sec
T0
5 0.2SD1/SDS
5 0.092 sec
At T 5 0, the design spectral response acceleration is given by ASCE 7 Equation (11.4-5) as
Sa
5 SDS(0.4 1 0.6T/T0)
5 1.01g(0.4 1 0)
5 0.40g
For a building at this location, the long-period transition period is given on ASCE 7 Figure 22-14 as
TL
5 8 seconds
The design spectral response acceleration at a period of T 5 8 seconds is now obtained from ASCE 7
Equation (11.4-7) as
Sa
5 SD1TL/T 2
5 0.46g 3 8/82
5 0.058g
The response spectrum is shown in Figure 1-9.
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Figure 1-9 Design response spectrum for Example 1-9
Example 1-10
The two-story steel-frame building shown in Figure 1-2 is located in Miami, Florida. Determine the
design spectral response accelerations.
Solution
From Example 1-5, the modified spectral response accelerations at short periods and at a period of 1.0
second are
SMS
5 0.065g
SM1
5 0.029g
The corresponding design spectral response accelerations are
SDS
5 2SMS /3
5 0.043g
SD1
5 2SM1/3
5 0.019g
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1.9 Risk categories and importance factors
Importance factor is tabulated in ASCE 7 Table 1.5-2 and defined in ASCE 7 Section 1.2.1 as:
A factor that accounts for the degree of risk to human life, health, and welfare associated with
damage to property or loss of use or functionality.
The seismic importance factor provides enhanced performance for those facilities that constitute a
substantial public hazard because of high levels of occupancy or because of the storage of toxic or
explosive substances and for those essential facilities that are required to resume operation immediately after a severe earthquake. An increase in the seismic importance factor increases the design base
shear for these buildings, with a consequent reduction in the inelastic behavior and damage caused to
the structure by the design earthquake. Four risk categories are listed in IBC Table 1604.5 and each is
assigned an importance factor in ASCE 7 Table 1.5-2. Details of the risk categories and corresponding
seismic importance factors are given in Table 1.5.
Table 1-5 Risk categories and importance factors
Seismic importance
factor, Ie
Risk category
Nature of occupancy
I and II
Standard occupancy and
low-hazard structures
1.00
III
Assembly structures
1.25
IV
Essential or hazardous
structures
1.50
Example 1-11
For the two-story steel-frame building shown in Figure 1-2, determine the applicable risk category and
importance factor.
Solution
The building is used as an office building, which is a standard occupancy structure. In accordance with
ASCE 7 Tables 1.5-1 and 1.5-2, the applicable risk category designation is II and the seismic importance factor is
Ie
5 1.00
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1.10 Seismic design category
Seismic design category is defined in IBC Section 202 as:
A classification assigned to a structure based on its risk category and the severity of the design
earthquake ground motion at the site.
The seismic design category establishes the allowable height, structural system and irregularity, analy­
sis procedure, and detailing requirements necessary in the structure. In accordance with ASCE 7 Section 11.6 and ASCE 7 Tables 11.6-1 and 11.6-2, six design categories, A through F, are established
based on the design spectral response accelerations in conjunction with the risk category. The seismic
design category is determined twice: first as a function of the design spectral response acceleration at
short periods, using ASCE 7 Table 11.6-1, and then as a function of the design spectral response acceleration at a period of 1.0 second, using ASCE 7 Table 11.6-2. The most severe seismic design category
governs. Table 1-6 combines ASCE 7 Tables 11.6-1 and 11.6-2 and lists the six design categories.
Table 1-6 Seismic design category
Risk category
SDS
SD1
I, II
or III
IV
SDS , 0.167g
SD1 , 0.067g
A
A
0.167g ≤ SDS , 0.33g
0.067g ≤ SD1 , 0.133g
B
C
0.33g ≤ SDS , 0.50g
0.133g ≤ SD1 , 0.20g
C
D
0.50g ≤ SDS
0.20g ≤ SD1
D
D
E
F
MCER acceleration at 1.0 second period,
S1 ≥ 0.75g
MCER 5 maximum considered earthquake
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A brief summary of the design requirements necessary in each seismic design category is given in
Table 1-7.
Table 1-7 Design requirements
Seismic design category
Design requirements
A
Minimal ground movements anticipated. A nominal amount of structural integrity provided in accordance with ASCE 7 Section 11.7.
B
Low seismicity anticipated. Equivalent lateral-force procedure
required.
C
Moderate seismicity anticipated. Some structural systems are
restricted. Some nonstructural components must be designed for
seismic resistance. Detached one- and two-story family dwellings
are exempt from these requirements.
D
High seismicity anticipated. Some structural systems are restricted.
Irregular structures must be designed by dynamic analysis
methods.
E or F
Very high seismicity anticipated. Severe restrictions are placed on the
use of some structural systems, irregular structures, and analysis
methods.
Where S1 is less than 0.75, the seismic design category is permitted to be determined from the shortperiod ground motion of ASCE 7 Table 11.6-1 alone, provided that all of the following apply:
•
in each of the two orthogonal directions, the approximate fundamental period of the structure
is Ta , 0.8Ts
•
in each of two orthogonal directions, the fundamental period of the structure is T , Ts
•
the seismic response coefficient, Cs, is determined from the expression Cs 5 SDS I/R
•
the diaphragms are rigid, or for diaphragms that are flexible, the distance between vertical
elements of the seismic-force-resisting system does not exceed 40 feet
1.10.1 Seismic design category A
Seismic design category A represents structures where SDS , 0.167g and SD1 , 0.067g. No real damage is anticipated in this seismic design category and anticipated ground movements are minor, even
for very long return periods. Also, where S1 is less than or equal to 0.04g and SS is less than or equal to
0.15g, the structure is assigned to seismic design category A. The design requirements for category A
structures are detailed in ASCE 7 Sections 1.4 and 11.7. The objective of the requirements is to provide
a nominal amount of structural integrity that will improve the performance of buildings in the event of
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a possible earthquake. The structure shall be provided with a complete lateral-force-resisting system
designed to resist the minimum notional lateral force, applied simultaneously at each floor level, given
by ASCE 7 Equation (1.4-1) as
where:
Fx
5 0.01wx
wx
5 that portion of the total dead load of the structure that is assigned to level x
Lateral forces may be applied separately in each of two orthogonal directions and orthogonal effects
may be neglected. The application of this provision is illustrated in Figure 1-10.
ASCE 7 Section 1.4.1 requires all smaller elements of a structure to be tied to the remainder of the
structure with a connection capable of resisting a notional horizontal force of
where:
FE
5 0.05wE
wE
5 weight of the smaller element
Figure 1-10 Lateral loads for seismic design category A
In addition, in accordance with ASCE 7 Section 1.4.3, for each beam, girder, or truss, a connection to
a support shall be provided to resist a notional horizontal force acting parallel to the member of
where:
FR
5 0.05wR
wR
5 reaction due to unfactored dead 1 live load
ASCE 7 Section 1.4.4 requires anchoring concrete and masonry walls to elements supplying lateral
support to the wall to provide a minimum out-of-plane notional strength-level resistance of
Fw
5 0.2 ww
≥ AT (5 lb/ft2)
where:
ww
5 weight of wall tributary to the connection
AT
5 tributary wall area
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The notional loads, N, specified in ASCE 7 Sections 1.4.1 through 1.4.4 are combined with other loads
in accordance with ASCE 7 Section 2.3.6 for strength design and ASCE 7 Section 2.4.5 for allowable
stress design. The effect of vertical earthquake forces is disregarded.
The strength design notional load combinations are
1.2D 1 1.0N 1 L 1 0.2S
0.9D 1 1.0N
The allowable stress design notional load combinations are
D 1 0.7N
D 1 0.75(0.7N) 1 0.75L 1 0.75S
0.6D 1 0.7N
where:
D
5 dead load
L
5 live load
S
5 snow load
1.10.2 Seismic design category B
Seismic design category B includes risk category I, II, and III structures in regions of moderate seismicity. Structures in this design category, with the exception of detached one- and two-story family
dwellings, must be designed for the calculated seismic forces. The equivalent lateral force procedure
may be used to analyze the structure. Light, nonstructural damage is anticipated in this seismic design
category.
1.10.3 Seismic design category C
Seismic design category C includes risk category IV structures in regions of moderate seismicity as
well as risk category I, II, and III structures in regions of somewhat more severe seismicity. The use of
some structural systems is restricted in this design category and some nonstructural components must
be designed for seismic resistance. In accordance with IBC Section 1613.1, detached one- and twostory family dwellings in seismic design category A, B or C are exempt from these requirements as are
these structures where located on a site with a spectral response acceleration SS , 0.4g. Hazardous,
nonstructural damage is anticipated in this seismic design category in a risk category II structure.
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1.10.4 Seismic design category D
Seismic design category D includes risk category I, II, III, and IV structures in regions of high seismicity, but not located close to a major active fault, as well as Occupancy Category IV structures in regions
of somewhat less severe seismicity. The use of some structural systems is restricted in this design
category and irregular structures must be designed by dynamic analysis methods. Hazardous damage
to susceptible structures is anticipated in this seismic design category in a risk category II structure.
1.10.5 Seismic design category E
Seismic design category E includes risk category I, II, and III structures located close to a major active
fault that is defined as a region with a maximum considered earthquake spectral response acceleration
at a 1.0-second period of S1 ≥ 0.75g. Severe restrictions are placed on the use of some structural systems, irregular structures, and analysis methods. Hazardous damage to robust structures is anticipated
in this seismic design category in a risk category II structure.
1.10.6 Seismic design category F
Seismic design category F includes risk category IV structures located close to a major active fault
that is defined as a region with a maximum considered earthquake spectral response acceleration at a
1.0-second period of S1 ≥ 0.75g. Severe restrictions are placed on the use of some structural systems,
irregular structures, and analysis methods.
Example 1-12
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Determine the seismic design category.
Solution
From Example 1-9, the design spectral response acceleration at short periods is
SDS
5 1.01g
. 0.50g
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From Example 1-2, the maximum considered earthquake response acceleration is
S1
5 0.457g
, 0.75g
From Example 1-11, the risk category 5 II
From Table 1-6, the seismic design category 5 D
From Example 1-9, the design spectral response acceleration at a period of 1.0 second is
SD1
5 0.46g
. 0.20g
From Example 1-11, the risk category 5 II
From Table 1-6, the seismic design category 5 D . . . governs
Example 1-13
The two-story steel-frame building shown in Figure 1-2 is located in Miami, Florida. Determine the
seismic design category.
Solution
From Example 1-10, the design spectral response acceleration at short periods is
SDS
5 0.043g
, 0.167g
From Example 1-11, the risk category 5 II
From Table 1-6, the seismic design category 5 A
From Example 1-10, the design spectral response acceleration at a period of 1.0 second is
SD1
5 0.019g
, 0.067g
From Example 1-11, the risk category 5 II
From Table 1-6, the seismic design category 5 A . . . governs
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1.11 Lateral-force-resisting systems
From observation of the behavior of structural systems in past earthquakes, it has been noted that
some systems perform better than others. Hence, ASCE 7 groups lateral-force-resisting systems into
different categories.
ASCE 7 Section 12.2.1 and ASCE 7 Table 12.2-1 detail eight major categories of building types characterized by the method used to resist the lateral force. These consist of:
•
bearing wall system. A structural system without a complete vertical load-carrying space
frame. Bearing walls or bracing elements provide support for vertical loads. Seismic lateral
force resistance is provided by the same shear walls or braced frames.
•
building frame system. A structural system with an essentially complete space frame providing support for vertical loads. Seismic lateral force resistance is provided by shear walls or
braced frames.
•
moment-resisting frame system. A structural system with an essentially complete space
frame providing support for vertical loads. Seismic lateral force resistance is provided by the
same moment frames.
•
dual system with special moment frames. A structural system with an essentially complete
space frame providing support for vertical loads. Seismic lateral force resistance is provided by
the moment frame and also by shear walls or braced frames with the moment frame contributing a minimum of 25 percent of the lateral resistance.
•
dual system with intermediate moment frames. A structural system with an essentially complete space frame providing support for vertical loads. Seismic lateral force resistance is provided by a moment frame and also by shear walls or braced frames with the moment frame
contributing a minimum of 25 percent of the lateral resistance. Intermediate moment frames
have less stringent detailing requirements than special moment frames.
•
shear wall-frame interactive system. A structural system that uses combinations of shear
walls and frames designed to resist seismic lateral forces in proportion to their rigidities, considering interaction between shear walls and frames on all levels. Support of vertical loads is
provided by the same shear walls and frames.
•
cantilever column system. A structure with a large portion of its mass concentrated at the top;
therefore, having essentially one degree of freedom in horizontal translation. Seismic lateral
force resistance is provided by the columns acting as cantilevers.
•
steel systems not specifically detailed for seismic resistance, excluding cantilever column
systems. A steel structure in seismic design category B or C may be designed with a response
modification coefficient of R 5 3 using the design requirements of AISC 3606 rather than AISC
3417 in conformity with IBC Section 2205.2.1.1.
Six of those categories are illustrated in Figure 1-11. These categories are further subdivided into the
types of construction material used.
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Bearing wall system
Building frame system
Moment-resisting frame
Dual system with
intermediate
moment frame
Dual system with special
moment frames
Cantilever column
system
Figure 1-11 Structural systems
1.11.1 Bearing wall systems
In a bearing wall system, shear walls provide support for all or most of the gravity loads and for resisting all lateral loads. Shear walls are of masonry or concrete, or of wood in wood-frame construction. In
general, deformations in a masonry or concrete bearing wall system are negligible and these systems
provide an excellent method to limit damage to nonstructural components. However, the system has a
poor inelastic response capacity and lacks redundancy because the lateral support members also carry
gravity loads and their failure will result in failure of gravity load-carrying capacity. In addition, shear
walls and braced frames restrict architectural expression by limiting free access in a building. Bearing
wall buildings are typically used in residential construction, warehouses, and low-rise commercial
buildings.
The building shown in Figure 1-12(a) consists of steel joists spanning in the north-south direction supported on concrete bearing walls on the north and south faces. The roof area tributary to the bearing
walls is shown hatched and is given by
Atrib wall 5 3a 3 2a
5 6a2
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The total roof area is
5 4a 3 2a
Aroof
5 8a2
The ratio of tributary area to total area is
Atrib wall /Aroof 5 6a2/8a2
5 0.75
. 0.5
Hence, this is a bearing wall system since the bearing walls support the major portion of the gravity
load.
Bearing wall
a
a
a
a
a
a
a
1.5a
a
2a
a
Joist
Column
3a
3a
(a) Bearing wall system
(b) Building frame system
Figure 1-12 Bearing wall and building frame systems
For structures assigned to seismic design categories D, E, and F, concrete and masonry shear walls
are required to be specially detailed reinforced walls. These walls are generally limited to a maximum
height of 160 feet with the exception of walls in seismic design category F, which are limited to 100
feet. Wood-frame construction with wood structural panels may be utilized in seismic design categories D, E, and F to a maximum height of 65 feet.
Ordinary reinforced concrete shear walls and intermediate reinforced masonry shear walls may be
used in seismic design categories A, B, and C without limitations on their height.
Bearing wall systems may comprise the types summarized in Table 1-8. These are the values given by
ASCE 7 Table 12.2-1.
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Table 1-8 Bearing wall systems
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Special reinforced concrete shear walls
NL
NL
160
160
100
Ordinary reinforced concrete shear walls
NL
NL
NP
NP
NP
Special reinforced masonry shear walls
NL
NL
160
160
100
Intermediate reinforced masonry shear walls
NL
NL
NP
NP
NP
Ordinary reinforced masonry shear walls
NL
160
NP
NP
NP
Light-frame walls with wood structural panels
NL
NL
65
65
65
NL 5 not limited, NP 5 not permitted
1.11.2 Building frame system
A building frame system has separate systems to provide support for lateral forces and gravity loads.
A frame provides support for the major portion of the gravity loads with independent shear walls or
braced frames resisting all lateral forces. The gravity load supporting frame does not require special
ductile detailing, but in seismic design categories D, E, and F, it is required to satisfy the deformation
compatibility requirements of ASCE 7 Section 12.12.5, and this imposes a practical limitation on the
height of a building frame system. Failure of the lateral support members will not result in collapse of
the building because the frame continues to support gravity loads.
The building shown in Figure 1-12(b) consists of steel joists spanning in the north-south direction supported on concrete bearing walls on the north and south faces and on steel pipe columns in the interior
of the building. The roof area tributary to the pipe columns is shown hatched and is given by
Atrib col 5 3a 3 1.5a
5 4.5a2
The total roof area is
Aroof
5 4a 3 2a
5 8a2
The ratio of tributary area to total area is
Atrib col /Aroof 5 4.5a2/8a2
5 0.56
. 0.5
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Hence, this is a building frame system since the pipe columns support the major portion of the gravity
load.
For structures assigned to seismic design categories D, E, and F, specially detailed concrete and
masonry shear walls, as specified for bearing wall systems, may be utilized. These are limited to a
maximum height of 100 feet for category F structures and 160 feet for category D and E structures.
Steel braced frames in seismic design categories D, E, and F may be special concentrically braced
frames, as specified in AISC 341 Section F2, or eccentrically braced frames, as specified in AISC 341
Section F3, with a maximum height of 100 feet for category F structures and 160 feet for category D
and E structures. Ordinary concentrically braced frames, as specified in AISC 341 Section F1, may
also be utilized with a maximum height of 35 feet for category D and E structures and are not permitted for category F structures. Wood-frame construction with wood structural panels may be utilized in
seismic design categories D, E, and F to a maximum height of 65 feet.
Ordinary reinforced concrete shear walls and intermediate reinforced masonry shear walls may be
used in seismic design categories A, B, and C without limitations on their height.
Building frame systems may comprise the types summarized in Table 1-9.
Table 1-9 Building frame systems
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Steel eccentrically braced frames
NL
NL
160
160
100
Steel special concentrically braced frames
NL
NL
160
160
100
Steel ordinary concentrically braced frames
NL
NL
35a
35a
NPa
Special reinforced concrete shear walls
NL
NL
160
160
100
Ordinary reinforced concrete shear walls
NL
NL
NP
NP
NP
Steel and concrete composite special concentrically braced frames
NL
NL
160
160
100
Special reinforced masonry shear walls
NL
NL
160
160
100
Intermediate reinforced masonry shear walls
NL
NL
NP
NP
NP
Light-frame walls with wood structural panels
NL
NL
65
65
65
NL 5 not limited, NP 5 not permitted
a. Permitted in penthouse structures and in single-story buildings up to a height of 60 feet where the dead load of the roof does not
exceed 20 lb/ft2.
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1.11.3 Moment-resisting frames
Moment-resisting frames are specially detailed to provide good ductility and support for both lateral
and gravity loads by flexural action. In seismic design categories D, E, and F, special reinforced concrete and structural steel moment-resisting frames are required to be detailed to satisfy ACI8 Sections
21.5 through 21.7 or AISC 341 Section E3. No restrictions are placed on the height of these systems.
Moment-resisting frames have the advantage of affording unlimited free access in a building. In addition, a high degree of redundancy can be provided and the system has an excellent inelastic response
capacity. Large lateral displacements may be developed while the gravity load-carrying capacity
remains intact. The large displacements, however, may cause damage to nonstructural elements.
Steel intermediate and ordinary moment frames and intermediate reinforced concrete moment frames
may be used in seismic design categories A, B, and C without limitations on their height.
Moment-resisting frame systems may comprise the types summarized in Table 1-10.
Table 1-10 Moment-resisting frame systems
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Steel special moment frames
NL
NL
NL
NL
NL
Steel special truss moment frames
NL
NL
160
100
NP
Steel intermediate moment frames
NL
NL
35a
NPa
NPa
Steel ordinary moment frames
NL
NL
NPa
NPa
NPa
Special reinforced concrete moment frames
NL
NL
NL
NL
NL
Intermediate reinforced concrete moment frames
NL
NL
NP
NP
NP
NL 5 not limited, NP 5 not permitted
a. Permitted in single-story buildings and light-frame construction complying with the requirements of ASCE 7 Sections 12.2.5.6 and
12.2.5.7.
1.11.4 Dual systems with special moment frames
A dual system provides a comparably high level of seismic safety since a secondary redundant lateral
support system is available to assist the primary nonbearing lateral support system. These systems
may be used in regions of high seismic risk. Nonbearing walls or bracing supply the primary lateral
support system with a special moment frame, providing primary support for gravity loads and acting as
a backup lateral force system. The special moment frame must be designed to independently resist at
least 25 percent of the base shear and, in addition, the two systems shall be designed to resist the total
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base shear in proportion to their relative rigidities. The primary lateral-support system may comprise
the types summarized in Table 1-11, and these may be used in all seismic design categories without
limitations on their height.
Table 1-11 Dual systems with special moment frames
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Special reinforced concrete shear walls
NL
NL
NL
NL
NL
Special reinforced masonry shear walls
NL
NL
NL
NL
NL
Steel eccentrically braced frame
NL
NL
NL
NL
NL
Steel special concentrically braced frames
NL
NL
NL
NL
NL
Steel buckling-restrained braced frame
NL
NL
NL
NL
NL
Steel special plate shear walls
NL
NL
NL
NL
NL
NL 5 not limited
1.11.5 Dual systems with intermediate moment frames
These systems may be used in regions of moderate seismic risk and may be used in seismic design
categories A, B, and C without limitations on their height. The primary lateral support system may
comprise the types summarized in Table 1-12.
Table 1-12 Dual systems with intermediate moment frames
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Special reinforced concrete shear walls
NL
NL
160
100
100
Ordinary reinforced concrete shear walls
NL
160
NP
NP
NP
Intermediate reinforced masonry shear walls
NL
NL
NP
NP
NP
Steel special concentrically braced frames
NL
NL
35
NP
NP
NL 5 not limited, NP 5 not permitted
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1.11.6 Shear wall-frame interactive system with ordinary reinforced concrete
moment frames and ordinary reinforced concrete shear walls
This system may be used only in seismic design categories A and B without limitations on their height.
The primary lateral support system is an ordinary reinforced concrete shear wall with an ordinary reinforced concrete space frame providing primary support for gravity loads.
This system is similar to dual systems. In accordance with ASCE 7 Section 12.2.5.8, the shear strength
of the shear walls shall be at least 75 percent of the design story shear at each story. The moment
frames shall be capable of resisting at least 25 percent of the design story shear at each story.
The height limitation requirements are indicated in Table 1-13.
Table 1-13 Shear wall-frame interactive system
with ordinary reinforced concrete moment frames and shear walls
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Ordinary reinforced concrete shear walls and ordinary reinforced concrete moment frames
NL
NP
NP
NP
NP
NL 5 not limited, NP 5 not permitted
1.11.7 Cantilever column systems
A cantilever column structure resists lateral forces by columns cantilevering from the base, and the
columns also provide support of the building’s weight. These structures have limited redundancy and
overstrength and concentrate inelastic response at their bases, producing a side-sway collapse mechanism. Hence, failure of the system due to lateral forces will also cause failure of the gravity load-carrying capacity. Cantilever column structures may comprise the systems indicated in Table 1-14.
Table 1-14 Cantilever column systems
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Steel special cantilever column
35
35
35
35
35
Steel ordinary cantilever column
35
35
NP
NP
NP
Special reinforced concrete moment frames
35
35
35
35
35
NP 5 not permitted
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The required axial strength of a cantilever column, considering only the load combinations that include
seismic load effects, shall not exceed 15 percent of the available axial strength, including slenderness
effects.
Inverted pendulum-type structures are defined in ASCE 7 Section 11.2 as structures in which more
than 50 percent of the structure’s mass is concentrated at the top of a slender, cantilevered structure
and in which stability of the mass at the top of the structure relies on rotational restraint to the top of
the cantilevered element. Supporting columns of inverted pendulum-type structures shall be designed
for the bending moment calculated at the base determined using the equivalent lateral force procedure
and varying uniformly to a moment at the top of the column equal to one-half the calculated bending
moment at the base. Foundations of the cantilever columns must be designed to resist the design seismic load, including the overstrength factor.
1.11.8 Steel systems not specifically detailed for seismic resistance, excluding
cantilever column systems
These systems may be used only in seismic design categories A, B and C without limitations on their
height. In accordance with IBC Section 2205.2.1.1, a steel structure in seismic design category B or
C may be designed for a response modification coefficient of R 5 3 using the design requirements of
AISC 360 rather than AISC 341. This ensures a nominally elastic response to the science loads.
Structures designed in accordance with the requirements of AISC 360 lack the ductility to provide the
inelastic deformation necessary in major seismic events. Hence, these structures are restricted to low
seismic risk areas and are designed for a response modification coefficient of R 5 3 with conventional
structural detailing. The elimination of seismic detailing generally results in a structure that is less
expensive to construct.
The height limitation requirements are indicated in Table 1-15.
Table 1-15 Steel systems not specifically detailed for seismic resistance, excluding cantilever column systems
Building height limitation as determined by
seismic design category, feet
System type
A or B
C
D
E
F
Steel systems not specifically detailed for seismic
resistance, excluding cantilever column systems
NL
NL
NP
NP
NP
NL 5 not limited, NP 5 not permitted
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1.11.9 Wind effects
In locations where wind effects exceed seismic effects, the building elements must still be detailed in
accordance with AISC 341 provisions. These provisions provide the design requirements for the building elements to sustain the large inelastic deformations produced by seismic loads.
1.12 Response modification coefficient
It is uneconomical to design a structure to remain entirely within its elastic range for a major earthquake, and advantage is taken of the nonlinear energy absorbing capacity of the system to allow
limited structural damage without impairing the vertical load-carrying capacity of the system. This
energy dissipation occurs as a result of the hysteresis effect. In ASCE 7 Table 12.2-1, structural systems are assigned a response modification coefficient R corresponding to their perceived ability to
resist a major seismic event. A structure with good hysteretic behavior, sufficiently ductile to sustain
several cycles of inelastic deformation, adequate redundancy, and material overstrength will dissipate
imposed seismic forces without significant loss of strength. Hence, a structure may be designed for
a significantly smaller force than is predicted by a linear-elastic analysis without collapse, provided
that inelastic deformations are accommodated by careful detailing. In addition, as yielding occurs, the
natural period of the structure lengthens and the damping ratio increases, thus reducing the seismic
force developed in the structure. A single seismic-force-resisting system may be used throughout a
building with a single value for the response modification coefficient, or several different systems may
be combined.
1.12.1 Seismic-force-resisting system
The structure response modification coefficient, R, is defined as the ratio of the theoretical seismic
base shear, which would develop in a linear elastic system, to the prescribed design base shear and is a
measure of the ability of the system to absorb energy and sustain cyclic inelastic deformations without
collapse. As shown in Figure 1-13, the modification coefficient is given by
where:
R
5 VE /VS
VE
5 theoretical base shear in an elastic structure
VS
5 design base shear
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0
Figure 1-13 Inelastic force deformation curve4
In addition to compensating for the energy dissipation capability, lateral-force system redundancy,
and increase in natural period and damping ratio, the response modification coefficient allows for the
provision of secondary lateral support systems and the observed performance of specific materials and
structural systems in past earthquakes. The value of R increases as the overall ductility of the structure and its energy dissipation capacity increase and as the degree of redundancy increases. Lightly
damped structures constructed of brittle materials are unable to tolerate appreciable deformation in
excess of initial yield and are assigned low values of R. Highly damped structures constructed of
ductile materials are assigned larger values of R. To justify these larger values of the response modification coefficient, it is necessary to implement the special detailing requirements specified for each
seismic-force-resisting system so as to sustain the cyclic inelastic deformations that occur. Even in
the event that wind forces govern the design, ASCE 7 Section 11.1.1 mandates that the application
of the detailing requirements prescribed for the lateral-force-resisting system are utilized. Values of
the response modification coefficient for various seismic-force-resisting systems are given in ASCE 7
Table 12.2-1 and are summarized in Table 1-16.
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Table 1-16 Design factors for seismic-force-resisting systems
Dual system SMF
Moment frames
Building frames
Bearing walls
Seismic-force-resisting system
R
0
Cd
Special reinforced concrete shear walls
5.0
2.5
5.0
Ordinary reinforced concrete shear walls
4.0
2.5
4.0
Special reinforced masonry shear walls
5.0
2.5
3.5
Intermediate reinforced masonry shear walls
3.5
2.5
2.25
Ordinary reinforced masonry shear walls
2.0
2.5
1.75
Light-frame walls with wood structural panels
6.5
3.0
4.0
Steel eccentrically braced frames
8.0
2.0
4.0
Steel special concentrically braced frames
6.0
2.0
5.0
Steel ordinary concentrically braced frames
3.25
2.0
3.25
Special reinforced concrete shear walls
6.0
2.5
5.0
Ordinary reinforced concrete shear walls
5.0
2.5
4.5
Steel and concrete composite special concentrically braced frames
5.0
2.0
4.5
Special reinforced masonry shear walls
5.5
2.5
4.0
Intermediate reinforced masonry shear walls
4.0
2.5
4.0
Light-frame walls with wood structural panels
7.0
2.5
4.5
Steel special moment frames
8.0
3.0
5.5
Steel special truss moment frames
7.0
3.0
5.5
Steel intermediate moment frames
4.5
3.0
4.0
Steel ordinary moment frames
3.5
3.0
3.0
Special reinforced concrete moment frames
8.0
3.0
5.5
Intermediate reinforced concrete moment frames
5.0
3.0
4.5
Special reinforced concrete shear walls
7.0
2.5
5.5
Special reinforced masonry shear walls
5.5
3.0
5.0
Steel eccentrically braced frames
8.0
2.5
4.0
Steel special concentrically braced frames
7.0
2.5
5.5
Steel buckling-restrained braced frames
8.0
2.5
5.0
Steel special plate shear walls
8.0
2.5
6.5
(continued)
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Table 1-16 Design factors for seismic-force-resisting systems—continued
R53
Cant. col.
Wall-frame
Dual IMF
Seismic-force-resisting system
R
0
Cd
Special reinforced concrete shear walls
6.5
2.5
5.0
Ordinary reinforced concrete shear walls
3.0
3.0
2.5
Intermediate reinforced masonry shear walls
3.5
3.0
3.0
Steel special concentrically braced frames
6.0
2.5
5.0
Ordinary reinforced concrete shear walls and ordinary reinforced
concrete moment frames
4.5
2.5
4.0
Steel special cantilever columns
2.5
1.25
2.5
Steel ordinary cantilever columns
1.25
1.25
1.25
Special reinforced concrete moment frames
2.5
1.25
2.5
Steel systems not specifically detailed for seismic resistance,
excluding cantilever column systems
3.0
3.0
3.0
SMF 5 special moment frame, IMF 5 intermediate moment frame
1.12.2 Combinations of seismic-force-resisting systems
Where combinations of seismic-force-resisting systems are used in a building, ASCE 7 introduces
controls to ensure that an adequate value for the response modification coefficient is adopted. For
the situation where structural components are common to systems with different R-values, ASCE 7
Section 12.2.4 requires design of the component for the higher R-value to ensure that adequate ductile
details are provided, and the component remains functional during inelastic deformations.
In accordance with ASCE 7 Section 12.2.2, where different seismic-force-resisting systems are used
along two orthogonal axes of a structure, the appropriate value of R shall be used for each system. As
illustrated in the building frame system structure shown in Figure 1-14, special reinforced concrete
shear walls provide lateral resistance in the longitudinal direction, and steel ordinary moment frames
provide lateral resistance in the transverse direction. The concrete shear walls have an R-value of 6 and
the steel ordinary moment frames an R-value of 3.5, and these values apply to each system.
In addition, it is possible that one of the systems will restrict the use of the structure in some seismic
design categories to limited heights. In this example in the transverse direction, steel ordinary moment
frames restrict the use of the structure in seismic design categories D, E and F. In accordance with
ASCE 7 Table 12.2-1, this system is not permitted in seismic design categories D, E and F except in
single-story buildings and light-frame construction complying with the requirements of ASCE 7 Sections 12.2.5.6 and 12.2.5.7.
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Seismic Design
ordinary
Figure 1-14 Different systems used along two orthogonal axes
Where different structural systems are used over the height of a building, ASCE 7 Section 12.2.3.1
requires use of the more stringent seismic design parameters (R, W0, and Cd) so as to prevent mixed
systems that could concentrate inelastic behavior in the lower stories.
As shown in Figure 1-15(a), where the upper system has a response modification coefficient higher
than that of the lower system, both systems are designed using their individual seismic design parameters. Forces transferred from the upper system to the lower system are increased by multiplying by
the ratio of the higher response modification coefficient to the lower response modification coefficient.
As shown in Figure 1-15(b), where the upper system has a response modification coefficient lower
than that of the lower system, the seismic design parameters (R, W0, and Cd) for the upper system are
used for both systems.
Detached one- and two-family dwellings constructed of light framing, and supported structural systems with a weight not exceeding 10 percent of the total weight of the structure, are exempted from
this requirement. As shown in Figure 1-15(c), the penthouse with wood structural panel shear walls,
with an R-value of 6.5, is supported by a steel special moment frame with an R-value of 8.0. The
moment frame and the penthouse may be designed independently for response modification coefficients of 8.0 and 6.5, respectively.
A two-stage equivalent lateral force procedure, specified in ASCE 7 Section 12.2.3.2, may be used for
structures having a flexible upper portion above a rigid lower portion, provided that the design of the
structure complies with the following:
•
the stiffness of the lower portion is at least 10 times the stiffness of the upper portion
•
the period of the entire structure is not greater than 1.1 times the period of the upper portion
considered as a separate structure supported at the transition from the upper to the lower portion
•
the upper portion shall be designed as a separate structure using the appropriate values of the
response modification coefficient, R, and redundancy factor, r
•
the lower portion shall be designed as a separate structure using the appropriate values of the
response modification coefficient, R, and redundancy factor, r. The reactions from the upper
portion shall be those determined from the analysis of the upper portion amplified by the ratio
of the R/r of the upper portion over R/r of the lower portion. This ratio shall be not less than
1.0.
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47
the upper portion is analyzed with the equivalent lateral force or modal response spectrum procedure, and the lower portion is analyzed with the equivalent lateral force procedure
As shown in Figure 1-16, the upper portion of the structure, consisting of moment-resisting frames, is
analyzed using the appropriate values of R and r. This gives a shear at the base of the upper portion of
VU. The shear force transferred to the lower portion is
VL
5 VURU rL/RL rU
Special moment-resisting frame
R = 8, Ω0 = 3, Cd = 5.5
6.5
R = 5.0, Ω0 = 2.5, Cd = 5
Design for:
R = 5.0, Ω0 = 2.5, Cd = 5
Special reinforced
concrete walls
R = 5, Ω0 = 2.5, Cd = 5
(a)
R = 8, Ω0 = 3, Cd = 5.5
(b)
(c)
Figure 1-15 Different systems used over the height of a structure
Moment-resisting frames
Ru ρu
VL
Vu
Shear wall
RL ρL
Figure 1-16 Two-stage analysis procedure
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Seismic Design
In accordance with ASCE 7 Section 12.2.3.3, where different seismic-force-resisting systems are used
in horizontal combination, the least value of R for any of the systems shall be used for that direction.
As illustrated in the building in Figure 1-17, special reinforced masonry shear walls that also support
gravity loads and steel special moment frames provide lateral resistance in the transverse direction.
The masonry shear walls have an R-value of 5.0 and the steel moment frames an R-value of 8.0. The
R-value of 5.0 governs for the transverse direction. As specified in ASCE 7 Section 12.2.3.3, resisting
elements may be designed using the least value of R for the different structural systems found on each
independent line of resistance, provided that all of the following apply:
•
the building is classified as Occupancy Category I or II
•
the building is two stories or less in height
•
the building is constructed of light framing or with flexible diaphragms
The value of the response modification coefficient used for design of the diaphragms shall not be
greater than the least value for any of the systems utilized in that same direction.
Figure 1-17 Different systems used in the same direction
Example 1-14
The two-story steel-frame building, shown in Figure 1-2, is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the transverse direction
and steel special concentrically braced frames in the longitudinal direction. Determine the applicable
design factors.
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Chapter 1
Solution
For a steel special moment-resisting frame, the following values are obtained from Table 1-16
R
58
W0
53
Cd
5 5.5
For a steel special concentrically braced frame, the following values are obtained from Table 1-16
R
56
W0
52
Cd
55
A value of R 5 8 may be used in the transverse direction and a value of R 5 6 may be used in the
longitudinal direction.
1.13 Overstrength factor
The overstrength factor is a measure of the reserve capacity of a structure to resist the actual seismic
forces generated by the design ground motions and is given in ASCE 7 Table 12.2-1. As shown in Figure 1-13, the overstrength factor is given by
where:
W0
5 VY /VS
VY
5 base shear at formation of the collapse mechanism
The factors contributing to the overstrength of a structure are:
•
energy dissipation capabilities
•
overstrength of materials
•
application of the resistance factor f to members to ensure adequacy under design loading
•
selection of member sizes in excess of the minimum required by the design
•
the design may be governed by drift limitations rather than strength
From ASCE 7 Table 12.2-1 Note b, the tabulated value of the overstrength factor W0 may be reduced
by subtracting 1⁄2 for structures with flexible diaphragms but shall not be taken as less than 2.0 for any
structure.
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Seismic Design
1.14 Deflection amplification factor
The deflection amplification factor is used to determine the actual displacements produced by the
design ground motions and is given in ASCE 7 Table 12.2-1. The deflection amplification factor is
given by
where:
Cd
5 dx /dxe
dx
5 anticipated horizontal displacement caused by the strength seismic forces
dxe
5 horizontal displacement caused by the strength seismic forces, as
determined by an elastic analysis
Taking into consideration the seismic importance factor, ASCE 7 Equation (12.8-15) gives the value
of the actual displacement as
where:
dx
5 Cd dxe /Ie
Ie
5 importance factor given in Table 1-5
1.15 Effective seismic weight
When a building vibrates in an earthquake, the lateral acceleration of the building mass produces
inertial forces. The summation of these inertial forces produces the seismic base shear and only that
portion of the mass that is physically attached to the building contributes to the inertial force. Hence,
live loads need not be included in the building mass when determining the seismic base shear.
The effective seismic weight, W, as specified in ASCE 7 Section 12.7.2, is the total dead load of the
structure and that part of the service load that may be expected to be attached to the building. This
consists of:
•
25 percent of the reduced floor live load for storage and warehouse occupancies. Live load in
public parking structures and storage loads adding not more than 5-percent effective seismic
weight need not be included.
•
a minimum allowance of 10 pounds per square foot (lb/ft2) for movable partitions or the actual
weight, whichever is greater
•
20 percent of flat roof snow loads exceeding 30 lb/ft2, regardless of the actual roof slope
•
the weight of landscaping and other materials at roof gardens and similar areas
Roof and floor live loads, except as noted in this section, are not included in the value of W as they are
considered negligible by comparison with the dead loads. For movable partitions, an overall average
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value of 10 lb/ft2 is adopted for seismic loads. For permanent walls that are constructed of heavier
materials, the actual weight of the walls shall be used. Freshly fallen snow, not exceeding 30 lb/ft2, has
little effect on the seismic load as it tends to be shaken off the roof in the initial phase of an earthquake.
However, ice and compacted snow, exceeding 30 lb/ft2, may be expected to partially adhere to the roof
and contribute to the seismic load.
Example 1-15
The two-story, steel-frame building shown in Figure 1-2 is used as an office building and has the following component weights, including framing:
Roof diaphragm, wr
20 lb/ft2
Second floor diaphragm, wd
30 lb/ft2
Walls, ww
40 lb/ft2
No allowance is required for permanent equipment or snow loads. Determine the applicable effective
seismic weight for an interior bent, in the north-south direction, at the roof and second floor levels if
the roof and second floor diaphragms may be considered flexible.
Solution
The relevant dead load tributary to the roof diaphragm in the north-south direction is due to the north
and south walls and the roof dead load and is given by
Roof
5 wr 3 s
5 20 3 40
5 800 lb/ft
North wall
5 ww 3 h/2
5 40 3 12/2
5 240 lb/ft
South wall
5 ww 3 h/2
5 40 3 12/2
5 240 lb/ft
The effective seismic weight tributary to an interior bent at roof level is
w2
5 (800 1 240 1 240)b/1000
5 1.28 3 20
5 25.60 kips
The relevant dead load tributary to the second floor diaphragm in the north-south direction is due to
the north and south walls, floor dead load, and partition loads and is given by
Floor
5 wd 3 s
5 30 3 40
5 1200 lb/ft
North wall
5 ww 3 h
5 40 3 12
5 480 lb/ft
South wall
5 ww 3 h
5 40 3 12
5 480 lb/ft
Partitions
5 wp 3 s
5 10 3 40
5 400 lb/ft
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Seismic Design
The effective seismic weight tributary to an interior bent at the second floor level is
w1
5 (1200 1 480 1 480 1 400)b/1000
5 2.56 3 20
5 51.20 kips
1.16 Seismic response coefficient
The seismic response coefficient, Cs , given in ASCE 7 Section 12.8.1.1, represents the code design
spectrum, and determination of the coefficient forms the basis of the equivalent lateral force procedure.
The maximum value of the seismic response coefficient, which is applicable for periods between
T 5 0 and T 5 TS and which defines the flat top or acceleration-related region of the spectrum, is given
by ASCE 7 Equation (12.8-2) as
where:
Cs
5 SDS Ie /R
SDS
5 design spectral response acceleration at short periods
Ie
5 importance factor from Table 1.5
R
5 response modification factor from Table 1.16
This value of Cs need not exceed the value given by ASCE 7 Equation (12.8-3), which defines the longer period, velocity-related region of the spectrum that is valid for periods between T 5 TS and T 5 TL.
ASCE 7 Equation (12.8-3) is given by
where:
Cs
5 SD1Ie /RT
SD1
5 design spectral response acceleration at a period of 1.0 second
T
5 fundamental period of the structure
TS
5 SD1/SDS
TL
5 long-period transition period
For periods exceeding the value T 5 TL , the value of Cs is given by ASCE 7 Equation (12.8-4), which
defines the constant-displacement region of the spectrum. ASCE 7 Equation (12.8-4) is given by
Cs
5 SD1TLIe /RT2
To prevent too low a value being adopted for tall buildings, the minimum allowable value of Cs is given
by ASCE 7 Equation (12.8-5), which is
Cs
5 0.044SDS Ie
≥ 0.01
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In addition, where the maximum considered earthquake spectral response acceleration at a period of
1.0 second is not less than 0.6g, the minimum permitted value is given by ASCE 7 Equation (12.8-6) as
where:
Cs
5 0.5S1Ie /R
S1
5 maximum considered earthquake spectral response acceleration for a period
of 1.0 second
In accordance with ASCE 7 Section 12.8.1.3, for regular structures five stories or less in height with
a period (T) of 0.5 second or less, the seismic response coefficient is permitted to be calculated using
a value of SDS 5 1.0 but not less than 0.7SDS , as determined by ASCE 7 Equation (11.4-3). Provided
that ρ 5 1.0, as determined by ASCE 7 Section 12.3.4.2, the structure is assigned to risk category I or
II and is not located on a site defined as site class E or F.
A graphical presentation of Cs is plotted in Figure 1-18.
Seismic response coefficient, Cs
Constant
acceleration
Constant
velocity
Constant
displacement
SDS Ie /R
SD1 Ie /RT
SD1TL Ie /RT 2
0.44SDS Ie
TL
Ts = SD1 / SDS
Period, T
Figure 1-18 Seismic response coefficient
Example 1-16
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the north-south direction
and steel special concentrically braced frames in the longitudinal direction. Determine the seismic
response coefficient in the north-south direction.
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Seismic Design
Solution
From previous examples, the relevant parameters are given by
SD1
5 0.46g
SDS
5 1.01g
S1
5 0.457g
Ie
5 1.0
R
5 8.0
T
5 0.50 sec
TS
5 0.46 sec
Seismic design category 5 D
The seismic design coefficient is given by ASCE 7 Equation (12.8-3) as
Cs
5 SD1Ie /RT
5 0.46 3 1.0/(8.0 3 0.50)
5 0.115
The maximum value of the seismic design coefficient is given by ASCE 7 Equation (12.8-2) as
Cs
5 SDS Ie /R
51.01 3 1.0/8.0
5 0.126
Since T . TS , the governing value is
Cs
5 0.115
1.17 Seismic base shear
The lateral forces produced in a structure by the ground vibration may be determined by the static or
equivalent lateral force procedure. This utilizes Newton’s Second Law to estimate the horizontal shear
force at the base of the structure. The seismic base shear is prescribed by ASCE 7 Equation (12.8-1),
which is the code representation of Newton’s Second Law, as
V
5 CsW
Seismic and Wind Forces: Structural Design Examples
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This equation is based on the assumption that the structure will undergo several cycles of inelastic
deformation and energy dissipation without collapse. Forces and displacements in the structure are
derived assuming linear elastic behavior. The actual forces and displacements produced in the structure are presumed to be greater than these values as specified for critical elements in ASCE 7 Sections
12.3.3.2, 12.3.3.3, 12.3.3.4, and 12.10.2.1.
Example 1-17
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the north-south direction.
Determine the seismic base shear for an interior bent in the north-south direction.
Solution
The value of the total effective seismic weight was derived in Example 1-15 as
W
5 w2 1 w1
5 25.60 1 51.20
5 76.80 kips
The value of the seismic response coefficient was derived in Example 1-16 as
Cs
5 0.115
Hence, the base shear is given by ASCE 7 Equation (12.8-1) as
V
5 CsW
5 0.115 3 76.80
5 8.83 kips
1.18 Simplified lateral force procedures
For small bearing wall or building frame-type structures classified as risk category I or II and not
exceeding three stories in height, ASCE 7 Section 12.14 permits an alternative design method. The
method is restricted to buildings for which drift is not a controlling factor in design. This method gives
values for the base shear higher than those obtained by using ASCE 7 Equation (12.8-2) that delineates
the flat top of the response spectrum.
Seismic and Wind Forces: Structural Design Examples
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Seismic Design
The simplified design procedure may be used, provided that all of the following conditions apply:
•
the soil profile at the location of the building cannot consist of site class E or F
•
the building must have at least two lines of lateral resistance in each of two major axis directions
•
at least one line of resistance must be provided on each side of the center of mass in each
direction
•
as shown in Figure 1-19, for structures with flexible diaphragms, overhangs beyond the outside
line of shear walls or braced frames must conform to ASCE 7 Equation (12.14-2), which is
where:
$$
a
, d/5
a
5 distance perpendicular to the forces being considered from the extreme
edge of the diaphragm to the line of vertical resistance closest to that edge
d
5 depth of the diaphragm parallel to the forces being considered at the line of
vertical resistance closest to the edge
for cast-in-place concrete diaphragms, the overhang is restricted by ASCE 7 Equation
(12.14‑1) to
a
≤ d/3
Figure 1-19 Flexible diaphragm overhang
•
the distance between the center of rigidity and the center of mass in each story must not exceed
10 percent of the length of the diaphragm parallel to the eccentricity
•
for buildings with a nonflexible diaphragm, forces in the vertical elements are determined as if
the diaphragm is flexible and in addition
$$
for buildings with two lines of resistance in a given direction, the distance between the two
lines is at least 50 percent of the diaphragm length perpendicular to the two lines
Seismic and Wind Forces: Structural Design Examples
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$$
for buildings with more than two lines of resistance in a given direction, the distance
between the two most extreme lines of resistance in that direction is at least 60 percent of
the diaphragm length perpendicular to the lines
$$
for buildings with two or more lines of resistance closer together than one-half the length of
the longer of the walls, as shown in Figure 1-20, the walls may be replaced by a single wall
at the centroid of the group for the initial distribution of forces, and the resultant force to
the group is then distributed to the members of the group based on their relative stiffnesses
l
Centroid of walls
x
Figure 1-20 Closely spaced walls
•
lines of resistance of the lateral-force-resisting system must be oriented at angles of not more
than 15 degrees from alignment with the major orthogonal horizontal axes of the building
•
the simplified design procedure must be used for each major orthogonal horizontal axis direction of the building
•
system irregularities caused by in-plane or out-of-plane offsets of lateral-force-resisting elements are not permitted, except in two-story buildings of light-frame construction provided
that the upper wall is designed for a factor of safety of 2.5 against overturning
The simplified seismic base shear is given by ASCE 7 Equation (12.14-12) as
where:
V
5 (FSDS /R)W
SDS
5 design spectral response acceleration at short periods
5 2FaSS /3
SS
5 5-percent damped, maximum considered earthquake spectral response
acceleration, for a period of 0.2 second for structures founded on rock
≤ 1.5g
Fa
5 short-period site coefficient obtained from ASCE 7 Table 11.4-1, or may be
taken as 1.0 for rock sites or 1.4 for soil sites
F
5 modification factor for building type
5 1.0 for one-story buildings
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Seismic Design
5 1.1 for two-story buildings
5 1.2 for three-story buildings
W
5 effective seismic weight of the structure
R
5 response modification factor from ASCE 7 Table 12.14-1 that is
summarized in Table 1-17
ASCE 7 Table 12.14-1 also indicates the limitations on the use of the various lateral-force-resisting
systems. ASCE 7 Table 12.14-1 is summarized in Table 1-17. Detailing requirements for the different lateral-force-resisting systems are addressed in IBC Chapters 19 through 23. ASCE 7 Section
12.14.8.1 defines a rock site as one in which the height of soil between the rock surface and the bottom
of the building’s foundations does not exceed 10 feet.
Table 1-17 Design factors for seismic-force-resisting systems
Limitations
Seismic design
category
Building frame
Bearing wall
Seismic force-resisting system
Response
modification
coefficient
B
C
D, E
R
Special reinforced concrete shear walls
P
P
P
5
Ordinary reinforced concrete shear walls
P
P
NP
4
Special reinforced masonry shear walls
P
P
P
5
Intermediate reinforced masonry shear walls
P
P
NP
3.5
Ordinary reinforced masonry shear walls
P
NP
NP
2
Light-frame walls with wood structural panels
P
P
P
6.5
Steel eccentrically braced frame
P
P
P
8
Steel special concentrically braced frames
P
P
P
6
Steel ordinary concentrically braced frames
P
P
P
3.25
Special reinforced concrete shear walls
P
P
P
6
Ordinary reinforced concrete shear walls
P
P
NP
5
Steel and concrete composite eccentrically braced frames
P
P
P
8
Special reinforced masonry shear walls
P
P
P
5.5
Intermediate reinforced masonry shear walls
P
P
NP
4
Light-frame walls with wood structural panels
P
P
P
7
Steel buckling-restrained braced frames
P
P
P
8
Steel special plate shear walls
P
P
P
7
P 5 permitted, NP 5 not permitted
Seismic and Wind Forces: Structural Design Examples
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In accordance with ASCE 7 Section 12.14.8.5, when the simplified design method is used, it is not
necessary to calculate the drift of a structure. If a drift value is required, it may be assumed to be 1
percent of building height. In addition, the redundancy factor r may be taken as 1.0 for structures
designed by the simplified method. In accordance with ASCE 7 Section 12.14.3.2.1, the overstrength
factor is given by
W0
5 2.5
Example 1-18
The interior bent of a two-story eccentrically braced steel-frame building is shown in Figure 1-21. The
building is located in Orange County, California, on a site with a soil profile type D and a 5-percent
damped, maximum considered earthquake spectral response acceleration, for a period of 0.2 second of
SS 5 1.239g. The effective seismic weight of the bent is indicated in the figure. Determine the seismic
base shear for the bent using the simplified design procedure.
w2 = 25.6 kips
w1 = 51.2 kips
Figure 1-21 Details for Example 1-18
Solution
The relevant parameters are
Fa
5 short-period site coefficient
5 1.4 for a soil site . . . from ASCE 7 Section 12.14.8.1
SDS
5 design spectral response acceleration at short periods
5 2FaSS /3
5 2 3 1.4 3 1.239/3
5 1.156g
Seismic and Wind Forces: Structural Design Examples
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Seismic Design
R
5 8.0 . . . from Table 1-17
F
5 modification factor for building type
5 1.1 for a two-story building from . . . ASCE 7 Section 12.14.8.1
W
5 25.6 1 51.2
5 76.80 kips
Hence, the simplified base shear is given by ASCE 7 Equation (12.14-12) as
V
5 (FSDS /R)W
5 (1.1 3 1.156/8)76.80
5 0.159 3 76.80
5 12.21 kips
1.19 Vertical distribution of seismic forces
The distribution of base shear over the height of a building results from the superposition of all modes
of vibration of the multiple-degree-of-freedom system. The magnitude of the lateral force at a particular node9, 10 depends on the mass of that node, the distribution of stiffness over the height of the
structure, and the nodal displacements in a given mode and is given by
where:
Fx
5 Vwxfx /Swifi
V
5 modal base shear
wi
5 seismic weight located at level i
fi
5 mode shape component at level i for the given mode
wx
5 seismic weight located at level x
fx
5 mode shape component at level x for the given mode
For a structure with a uniform distribution of mass over its height and assuming a linear mode shape,
as shown in Figure 1-22, this reduces to the equation
where:
Fx
5 Vwxhx /Swihi
hi
5 height above the base to level i
hx
5 height above the base to level x
V
5 base shear
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ф
Figure 1-22 Vertical force distribution
This equation is valid if only a linear first mode shape is considered, and it is applicable to short-period,
regular structures with a fundamental vibration period not exceeding 0.5 second. To allow for higher
mode effects in long-period buildings, ASCE 7 Equations (12.8-11) and (12.8-12) yield the equation
where:
Fx
5 Vwxhxk /Swihik
Fx
5 design lateral force at level x
hi
5 height above the base to any level i
hx
5 height above the base to a specific level x
Swihik
5 summation, over the whole structure, of the product of wi and hik
k
5 distribution exponent
To allow for higher mode effects in long-period buildings with a fundamental vibration period of 2.5
seconds or more, a parabolic mode shape is assumed and the distribution exponent, k, is given by
k
52
Where the building has a fundamental vibration period not exceeding 0.5 second, a linear mode shape
is assumed and the distribution exponent is
k
51
For intermediate values of the fundamental vibration period, a linear variation of k may be assumed or
k may be taken as 2.
Seismic and Wind Forces: Structural Design Examples
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Example 1-19
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the north-south direction.
Determine the vertical force distribution for an interior bent in the north-south direction.
Solution
The fundamental period of vibration was derived in Example 1-8 as
T
5 0.50 sec
The value of the distribution exponent factor is obtained from ASCE 7 Section 12.8.3 as
k
5 1.0
Hence, the expression for Fx reduces to
Fx
5 Vwxhx /Swihi
The seismic dead loads located at levels 1 and 2 are obtained from Example 1-15, and the relevant
values are given in Table 1-18.
From Example 1-17, the base shear is given by
V
5 8.83 kips
The design lateral force at level x is
Fx
5 Vwxhx /Swihi
5 8.83(wxhx)/1228.80
5 0.00719wxhx
The values of Fx are given in Table 1-18.
Table 1-18 Vertical force distribution
Level
wx
hx
wxhx
Fx
2
25.60
24
614.40
4.42
1
51.20
12
614.40
4.42
Total
76.80
–
1228.80
8.84
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Example 1-20
The two-story steel-frame building shown in Figure 1-2 is located in Miami, Florida. Determine the
vertical force distribution for an interior bent in the north-south direction.
Solution
The building was determined in Example 1-13 to be in seismic design category A. Hence, the requirements of ASCE 7 Section 1.4.2 are applicable and the lateral force at each level is given by ASCE 7
Equation (1.4-1) as
where:
Fx
5 0.01wx
wx
5 that portion of the effective seismic weight that is assigned to level x
The effective seismic weights at each level are given in Example 1-15.
The values of the lateral force at each level are given in Table 1-19.
Table 1-19 Vertical force distribution
Level
wx
Fx
2
25.60
0.26
1
51.20
0.51
1.20 Simplified vertical distribution of base shear
In accordance with ASCE 7 Section 12.14.8.2, when the simplified procedure is used to determine the
seismic base shear, the forces at each level may be determined from ASCE 7 Equation (12.14-13) as
where:
Fx
5 wxV/W
wx
5 effective seismic weight located at level x
V
5 seismic base shear determined by ASCE 7 Equation (12.14-12)
W
5 total effective seismic weight
This method provides a rapid and simple determination of the forces at each level based on the effective seismic weight located at that level.
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Seismic Design
Example 1-21
The two-story steel-frame building shown in Figure 1-21 is located in Orange County, California.
Lateral force resistance is provided by eccentrically braced frames. Determine the vertical force distribution for an interior bent using the simplified procedure.
Solution
From Example 1-18, the value of the effective seismic weight is
W
5 76.80 kips
From Example 1-18, the value of the seismic base shear, determined using the simplified lateral force
procedure, is
V
5 12.21 kips
The effective seismic weights at each level are given in Example 1-15.
The relevant values are given in Table 1-20.
The forces at each level are determined from ASCE 7 Equation (12.14-13) as
Fx
5 wxV/W
5 wx 3 12.21/76.80
5 0.159wx
The values of Fx are given in Table 1-20.
Table 1-20 Vertical force distribution
Level
wx
Fx
2
25.60
4.07
1
51.20
8.14
Total
76.80
12.21
1.21 Vertical seismic load effects
Vertical seismic forces are considered in the design of foundations and in the determination of the factor of safety against overturning. The vertical seismic forces may be determined using the appropriate
load combination or by using a vertical response spectrum.
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1.21.1 Overturning
In accordance with ASCE 7 Section 12.8.5, buildings shall be designed to resist the overturning effects
caused by the seismic forces at each level. At any story, the increment in overturning moment is distributed to the vertical-force-resisting elements in the same proportion as the distribution of horizontal
shears to those elements. The determination of the overturning moment at level x is illustrated in Figure 1-23 and is given by
where:
Mx
5 SFi(hi 2 hx)
Fi
5 design lateral force at level i
hi
5 height above the base to level i
hx
5 height above the base to level x
n
5 top level of the building
Figure 1-23 Overturning moment
The factor of safety against overturning is determined at the allowable stress design level. The effect
of vertical seismic forces, as defined in ASCE 7 Section 12.4.2.2, must be considered in the analysis.
Where the effects of gravity loads and vertical seismic loads counteract, the applicable load combination is given by ASCE 7 Section 2.4.5 combination 10, which is
0.6D 2 0.7Ev 1 0.7Eh
where:
D
5 effect of dead load
Eh
5 effect of horizontal seismic load in accordance with ASCE 7 Equation
(12.4-3)
5 rQE
Ev
5 effect of vertical seismic load in accordance with ASCE 7 Equation
(12.4-4a)
5 0.2SDS D
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Seismic Design
QE
5 effect of horizontal seismic forces
5 effect of horizontal seismic forces Fi in Figure 1-23
SDS
5 design spectral response acceleration at a period of 0.2 second
r
5 redundancy factor defined in ASCE 7 Section 12.3.4
ASCE 7 Section 2.4.5 combination 10 may now be rewritten as
(0.6 2 0.14SDS)D 1 rQE
In accordance with ASCE 7 Section 12.4.2.2, Exception 2a, the vertical seismic load effect, Ev , may
be taken as zero for structures assigned to seismic design category B.
The horizontal forces determined using ASCE 7 Equation 12.8-11 do not reflect the actual inertial
forces imparted on a structure at a specific time. Instead, they represent values that envelope the maximum values that can occur. Hence, overturning moments based on these values are conservative. In
accordance with ASCE 7 Section 12.13.4, overturning moments at the soil-foundation interface are
permitted to be reduced by 25 percent for foundations of structures that satisfy both of the following
conditions:
•
the structure is designed by the equivalent lateral force procedure
•
the structure is not a cantilever column structure
The modal response spectrum analysis provides horizontal forces that more accurately reflect the
actual values. Hence, overturning moments at the soil-foundation interface of structures designed by
this method are permitted to be reduced by only 10 percent.
No overturning reduction is allowed in the above-grade portion of the structure.
Example 1-22
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the north-south direction.
Determine the factor of safety against overturning at the base for an interior bent in the north-south
direction. The redundancy factor is 1.0.
Solution
The effective seismic weight determined in Example 1-15 includes an allowance for partitions at the
second floor level and this is used to calculate the seismic base shear. However, for other procedures,
ASCE 7 Section 4.3.2 requires partitions to be considered a live load. The dead load, not including
partitions, tributary to an interior bent at the second floor level is obtained from Example 1-15 as
w1
5 (1200 1 480 1 480)20/1000
5 43.20 kips
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The seismic weight at the roof is obtained from Example 1-15 as
w2
5 25.60 kips
The factor of safety against overturning may be determined using the allowable stress load factors
given in ASCE 7 Section 2.4.1.5 of 0.6 for dead loads and 0.7 for seismic loads. Horizontal seismic
forces acting on the structure, as determined in Example 1-19, are shown in Figure 1-24. The factored
overturning moment is given by
MO
5 0.7(4.42 3 24 1 4.42 3 12)
5 111.38 kip-ft
A reduction of 25 percent is allowed to give
M
5 0.75 3 111.38
5 83.54 kip-ft
The factored restoring moment produced by the seismic weights at level 1 and level 2 is given by
MRD
5 0.6 3 20(43.2 1 25.6)
5 825.60 kip-ft
The design spectral response acceleration at a period of 0.2 second is obtained from Example 1.12 as
SDS
5 1.01g
The factored moment produced by the upward seismic load at levels 1 and 2 is
Mv
5 220 3 0.14SDS(w1 1 w2)
5 220 3 0.14 3 1.01(43.2 1 25.6)
5 2194.57 kip-ft
The net factored restoring moment is
MR
5 MRD 1 Mv
5 825.60 2 194.57
5 631.03 kip-ft
The factor of safety against overturning is
MR /M 5 631.03/83.54
5 7.55
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Seismic Design
Figure 1-24 Details for Example 1-22
1.21.2 Foundation design
The applicable allowable stress design load combinations for foundation design, including seismic
load effects, are given by ASCE 7 Section 2.4.5 combinations 8 and 9, which are
1.0D 1 0.7Ev 1 0.7Eh . . . (combination 8)
1.0D 1 0.525Ev 1 0.525Eh 1 0.75L 1 0.75S . . . (combination 9)
In accordance with ASCE 7 Section 12.4.2.2, Exception 2b, the vertical seismic load effect may be
taken as zero when determining effects at the soil-foundation interface. In accordance with ASCE 7
Section 12.13.4, overturning moments at the soil-foundation interface are permitted to be reduced by
25 percent for foundations of structures that satisfy both of the following conditions:
•
the structure is designed by the equivalent lateral force procedure
•
the structure is not a cantilever column structure
Example 1-23
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the north-south direction. For seismic forces acting in the north-south direction, determine the design vertical force at the
soil-foundation interface at the base of an interior bent. The redundancy factor is 1.0. The ASCE 7
Section 2.4.5 loading combination 8 governs.
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Solution
From Example 1-22, the total dead load is
W
5 w1 1 w2
5 43.2 1 25.6
5 68.8 kips
At the base of one column of the bent, the vertical force produced by the total dead load is
PD
5 W/2
5 68.8/2
534.4 kips
Allowing for the 25-percent reduction given in ASCE 7 Section 12.13.4 and the load factor of 0.7
given in ASCE 7 Section 2.4.5, the factored overturning moment produced by the horizontal seismic
forces at the base of the bent is
M
5 0.75 3 0.7S(Fi hi)
5 83.54 kip-ft . . . (from Example 1-22)
At the base of one column of the bent, the vertical force produced by the horizontal seismic forces is
PE
5 M/s
5 83.54/40
5 2.1 kips
The design vertical force at the base of one column of the bent is
P
5 PD 1 PE
5 34.4 1 2.1
5 36.5 kips
1.21.3 Optional vertical seismic load effect
For structures in seismic design categories C through F, ASCE 7 Section 11.9.1 permits the vertical
seismic load effect, Ev , to be determined using the MCER vertical response spectrum. The vertical
response spectral acceleration, SaMv , is obtained from the horizontal response spectral acceleration
at short periods, SMS , using the vertical coefficient, Cv. The coefficient, Cv , is defined in terms of the
spectral response acceleration parameter at short periods, SS , in ASCE 7 Table 11.9-1. The values of
Cv are given in Table 1-21.
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Table 1-21 Values of vertical coefficient Cv
Mapped MCER spectral
response parameter at
short periods
Site classes A, B
Site class C
Site classes D, E, F
SS ≥ 2.0
0.9
1.3
1.5
SS 5 1.0
0.9
1.1
1.3
SS 5 0.6
0.9
1.0
1.1
SS 5 0.3
0.8
0.8
0.9
SS ≤ 0.2
0.7
0.7
0.7
The vertical response spectrum consists of four segments and is constructed as shown in Figure 1-25:
•
for vertical periods less than or equal to 0.025 second, the vertical response spectral acceleration is given by ASCE 7 Equation (11.9-1) as
SaMv
•
5 0.3Cv SMS
for vertical periods greater than 0.025 second and less than or equal to 0.05 second, the vertical
response spectral acceleration is given by ASCE 7 Equation (11.9-2) as
SaMv 5 20CvSMS(Tv 2 0.025) 1 0.3Cv SMS
•
for vertical periods greater than 0.05 second and less than or equal to 0.15 second, the vertical
response spectral acceleration is given by ASCE 7 Equation (11.9-3) as
SaMv 5 0.8Cv SMS
•
for vertical periods greater than 0.15 second and less than or equal to 2.0 seconds, the vertical
response spectral acceleration is given by ASCE 7 Equation (11.9-4) as
SaMv 5 0.8Cv SMS(0.15/Tv)0.75
where:
SMS
5 horizontal response spectral acceleration parameter at short periods
Tv
5 the vertical period of vibration
The vertical response spectral acceleration, SaMv , derived from ASCE 7 Equations (11.9-1) through
(11.9-4) must be not less than one-half of the corresponding horizontal response spectral acceleration,
SaM .
For vertical periods greater than 2.0 seconds, the vertical response spectral acceleration must be determined from a site-specific procedure. The vertical response spectral acceleration, SaMv , derived from
the site-specific procedure, must be not less than one-half of the corresponding design horizontal
response spectral acceleration, Sa.
For vertical periods less than or equal to 2.0 seconds, the vertical response spectral acceleration is
permitted to be determined from a site-specific procedure. The vertical response spectral acceleration,
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Vertical acceleration
0.8CvSMS
0.8CvSMS (0.15/Tv)0.75
0.3CvSMS
0.5
0.025
0.05
1.0
Vertical period. Tv sec
0.15
Figure 1-25 Vertical response spectrum
SaMv , derived from this site-specific procedure, must be not less than 80 percent of the corresponding
value derived from ASCE 7 Equations (11.9-1) through (11.9-4).
The design vertical response spectral acceleration, Sav , is taken as two-thirds of the value of the vertical
response spectral acceleration, SaMv.
1.22 Diaphragm loads
Precast concrete diaphragms, including chords and collectors, in structures assigned to seismic design
categories C through F must be designed in accordance with ASCE 7 Section 12.10.3. All other diaphragms, chords, and collectors may be designed using the procedure given in ASCE 7 Sections
12.10.1 and 12.10.2. This procedure is as follows:
The load acting on a horizontal diaphragm is given by ASCE 7 Equation (12.10-1) as
Fpx
5 wpx SFi /Swi
≥ 0.2SDS Iewpx . . . ASCE 7 Equation (12.10-2)
≤ 0.4SDS Iewpx . . . ASCE 7 Equation (12.10-3)
where:
Fi
5 lateral force at level i
SFi
5 total shear force at level i
wi
5 seismic dead load located at level i
Swi
5 total seismic weight at level i and above
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wpx
5 seismic weight tributary to the diaphragm at level x, including walls normal
to the direction of the seismic load
For a single-story structure, this reduces to
Fp
5 Vwpx /W
5 Cswpx
Also, in a multistory structure at the second floor level
SFi /Swi
5 V/W
5 Cs
where:
V
5 seismic base shear determined by ASCE 7 Equation (12.8-1)
W
5 effective seismic weight
Cs
5 seismic design coefficient
Example 1-24
The two-story steel-frame building, with wood structural panel diaphragms, shown in Figure 1-2 is
located in Orange County, California. Determine the diaphragm loads for the structure.
Solution
ASCE 7 Equation (12.10-1) is applicable and the diaphragm loads are given by
Fpx
5 wpx SFi /Swi
The effective seismic weight tributary to the diaphragm at level 2 is obtained from Example 1-15 as
Roof
5 0.02 3 40 3 20
5 16.00 kips
Walls 5 2 3 0.04 3 20 3 12/2
5 9.60 kips
wp2
5 16.00 1 9.60
5 25.60 kips
The effective seismic weight tributary to the diaphragm at level 1 is obtained from Example 1-15 as
Second floor 5 0.03 3 40 3 20
5 24.00 kips
Walls
5 2 3 0.04 3 20 3 12
5 19.20 kips
Partitions
5 0.01 3 40 3 20
5 8.00 kips
wp1
5 24.00 1 19.20 1 8.00
5 51.20 kips
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The maximum applicable value for the diaphragm load is given by ASCE 7 Equation (12.10-3) as
Fpx
5 0.4SDS Iewpx . . . SDS 5 1.01 from Example 1-9
5 0.4 3 1.01 3 1.0wpx
5 0.404wpx
The minimum applicable value for the diaphragm load is given by ASCE 7 Equation (12.10-2) as
Fpx
5 0.2SDS Iewpx
5 0.2 3 1.01 3 1.0wpx
5 0.202wpx . . . governs at both levels
The values of the diaphragm loads are given in Table 1-22.
Table 1-22 Diaphragm loads
Level
Swi
SFi
SFi /Swi
max
min
wpx
Fpx
2
25.60
4.42
0.173
0.404
0.202
25.60
5.17
1
76.80
8.84
0.115
0.404
0.202
51.20
10.34
1.23 Story drift
As shown in Figure 1-26, the application of the seismic design force to the top of a single-story bent
causes the inelastic lateral deflection, or drift, D. Excessive drift in a structure may cause damage to
cladding and finishes and to nonstructural walls and partitions. In addition, the secondary stresses, or
P-delta effects, introduced into the columns may produce instability that results in collapse.
∆
P
V = CsW
hs
Figure 1-26 Story drift
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Story drift is defined in ASCE 7 Section 12.8.6 as the difference of the inelastic deflections at the
centers of mass at the top and bottom of the story under consideration. The maximum allowable story
drift, Da, is given in ASCE 7 Table 12.12-1 and shown in Table 1-23.
Table 1-23 Maximum allowable story drift, Da
Risk category
Building type
I or II
III
IV
One-story buildings with fittings designed to accommodate drift
NL
NL
NL
Buildings, other than masonry shear wall buildings, of
four stories or less with fittings designed to accommodate drift
0.025hsx
0.020hsx
0.015hsx
Masonry cantilever shear wall buildings
0.010hsx
0.010hsx
0.010hsx
Other masonry shear wall buildings
0.007hsx
0.007hsx
0.007hsx
All other buildings
0.020hsx
0.015hsx
0.010hsx
hsx 5 story height below level x, NL 5 not limited
Note: For moment-resisting frames in seismic design categories D through F, these values are divided by r.
To allow for inelastic deformations, drift is determined using the deflection amplification factor, Cd ,
given in Table 1-16. Taking into account the importance factor given in Table 1-5, the amplified deflection at level x is defined by ASCE 7 Equation (12.8-15) as
where:
dx
5 Cd dxe /Ie
Cd
5 deflection amplification factor given in Table 1-16
dx
5 design displacement of the structure
5 anticipated inelastic displacement caused by the design ground motion and
defined as the product of dxe and Cd
dxe
5 horizontal displacement caused by the code-prescribed strength level
forces, as determined by an elastic analysis
Ie
5 importance factor given in Table 1-5
In accordance with ASCE 7 Section 12.8.7, P-delta effects need not be included in the calculation of
drift when the stability coefficient, q, does not exceed 0.10.
The stability coefficient is defined by ASCE 7 Equation (12.8-16) as
q
5 Px DIe /Vx hsxCd
5 Px(dxe 2 d(x–1)e)/Vx hsx
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where:
Px
5 total unfactored vertical design load at and above level x
D
5 anticipated inelastic story drift occurring simultaneously with Vx
75
5 Cd(dxe 2 d(x–1)e)/Ie
Vx
5 seismic shear force acting between levels x and (x 2 1)
hsx
5 story height below level x
Cd
5 deflection amplification factor defined in Table 1-16
dxe
5 horizontal displacement caused by the code-prescribed strength level
forces, as determined by an elastic analysis
For the calculation of drift, in accordance with ASCE 7 Section 12.8.6.2, the full value of Tr, the fundamental period determined using the Rayleigh method, may be utilized to determine the seismic base
shear. The upper bound limitation imposed by ASCE 7 Section 12.8.2 is not applicable.
Where allowable stress design methods are used, deflections must be calculated using the codeprescribed design level forces without multiplying by the factor 0.7.
When calculating drift, the redundancy factor, r, is not used.
Example 1-25
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Determine the drift in the top and bottom stories of the frame. The relevant details are shown in Figure 1-27
and Table 1-24, where k1 5 stiffness of story 1, and k2 5 stiffness of story 2.
14
08
21 b/b
Figure 1-27 Details for Example 1-25
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Solution
From previous examples, the relevant parameters are
SDS
5 1.01g
SD1
5 0.46g
TS
5 0.46 sec
Ie
5 1.0
R
58
W
5 76.8 kips
risk category
5 II
From Example 1-8, the fundamental period obtained by using the Rayleigh method is
Tr
5 0.538 sec
In accordance with ASCE 7 Section 12.8.6.2, this value of T may be utilized to determine the seismic
base shear. The seismic response coefficient is given by ASCE 7 Equation (12.8-3) as
Cs
5 SD1Ie /RT
5 0.46 3 1.0/(8.0 3 0.538)
5 0.107
The maximum value of the seismic response coefficient is given by ASCE 7 Equation (12.8-2) as
Cs
5 SDS Ie /R
5 1.01 3 1.0/8.0
5 0.126
Since Tr . TS , the governing value is
Cs
5 0.107
Hence, the seismic base shear is given by ASCE 7 Equation (12.8-1) as
V
5 CsW
5 0.107 3 76.80
5 8.22 kips
The lateral forces, Fx , are calculated at each level using ASCE 7 Equations (12.8-11) and (12.8-12) to
give
Fx
5 Vwxhxk /Swihik
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The fundamental period of vibration is
T
5 0.538 sec
. 0.5 sec
The value of the distribution exponent factor, k, is obtained by linear interpolation from ASCE 7 Section 12.8.3 as
k
5 1 1 (0.538 2 0.5)(2 2 1)/(2.5 2 0.5)
5 1.019
The effective seismic weights located at levels 1 and 2 are obtained from Example 1-15, and the relevant values are given in Table 1-24.
The design lateral force at level x is
Fx
5 Vwxhxk /Swihik
5 8.22(wxhx1.019)/1296.74
5 0.00634wxhx1.019
The values of Fx are shown in Table 1-24.
Table 1-24 Vertical force distribution
Level
wx
hx
wxhx1.019
Fx
2
25.60
24
652.64
4.14
1
51.20
12
644.10
4.08
Total
76.80
—
1296.74
8.22
For a moment-resisting frame, the amplification factor is obtained from Table 1-16 as
Cd
5 5.5
Using the lateral forces given in Table 1-24, the anticipated inelastic drift in the bottom story is
D1
5 Cd dxe /Ie
5 Cd(F2 1 F1)/k1Ie
5 (5.5)(4.14 1 4.08)/(30 3 1.00)
5 1.51 in
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In accordance with Table 1-23, the maximum allowable drift for a two-story structure in risk category
II is
Da
5 0.025hs1
5 0.025 3 12 3 12
5 3.60 in
. 1.51 in . . . satisfactory
Similarly, the drift of the top story is
D2
5 0.76 in . . . satisfactory
1.24 Simplified determination of drift
In accordance with ASCE 7 Section 12.14.8.5, where the simplified procedure is used to determine the
seismic base shear, the design story drift in any story shall be taken as
where:
Dx
5 0.01hsx
Dx
5 drift in story x
hsx
5 height of story x
Example 1-26
The two-story building shown in Figure 1-21 is located in Orange County, California. Lateral force
resistance is provided by eccentrically braced steel frames in the north-south direction, as indicated.
Determine the drift in the bottom story of the frame in the north-south direction using the simplified
procedure.
Solution
The design story drift in the bottom story is given by
D1
5 0.01hs1
5 0.01 3 12 3 12
5 1.44 in
, Da
5 3.60 in . . . satisfactory (as determined in Example 1-25 for a structure in
risk category II)
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1.25 P-delta effects
A compression force in a column introduces secondary stresses into the column due to the additional
moment produced by the column displacements. The P-delta effects are produced by the gravity loads
acting on the sidesway of the structure. As shown in Figure 1-26, the primary moment on the frame is
MP
5 Vhs
The secondary moment on the frame is
MS
5 PD
P-delta moments represent the additional overturning moments produced by the gravity loads acting
on the deflections produced by the lateral loads. The P-delta effects may significantly increase the
displacements and reduce the stability of the structure.
The P-delta effects are calculated using the design level seismic forces and elastic displacements determined in accordance with ASCE 7 Section 12.8.1 with the exception of ASCE 7 Equation (12.8‑5).
P-delta effects in a given story are due to the secondary moments, caused by the eccentricity of the
gravity loads above that story. The secondary moment in a story is defined as the product of the total
dead load, floor live load, and snow load above the story multiplied by the elastic drift of that story.
The primary moment in a story is defined as the seismic shear in the story multiplied by the height
of the story. In determining drift for P-delta effects, the upper bound limitation imposed by ASCE 7
Section 12.8.2 on the calculated period is not applicable.
The ratio of the secondary moment to primary moment is termed the stability coefficient and is an
indication of how sensitive the structure is to P-delta effects. The stability coefficient is given by
where:
q
5 MS /MP
MS
5 product of the total unfactored gravity load above a story and the elastic
drift of that story
Mp
5 total seismic shear in a story multiplied by the height of the story
The stability coefficient is also defined by ASCE 7 Equation (12.8-16) as
q
5 PxDIe /VxhsxCd
5 Px(dxe 2 d(x–1)e)/Vx hsx
where:
Px
5 total unfactored vertical design load at and above level x
D
5 anticipated inelastic story drift
5 dx 2 d(x–1)
5 Cd(dxe 2 d(x–1)e)/Ie
Vx
5 seismic shear force acting between levels x and (x 2 1)
hsx
5 story height below level x
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If:
Cd
5 deflection amplification factor given in Table 1-16
dxe
5 horizontal displacement caused by the code-prescribed strength level
forces, as determined by an elastic analysis
q
. qmax, the structure is unstable and must be redesigned
The stability coefficient in any story shall not exceed the value given by ASCE 7 Equation (12.8-17) as
qmax
5 0.5/bCd
≤ 0.25
The term b is the ratio of the shear demand to the shear capacity in a story and may conservatively be
considered equal to 1.0. If the stability coefficient (q) in any story exceeds 0.1, the effects of the secondary moments shall be included in the determination of story drifts and element forces for the whole
structure. In accordance with ASCE 7 Section 12.8.7, the revised story drift, allowing for P-delta
effects, is obtained as the product of the calculated drift and the factor 1/(1 2 q).
As shown in Figure 1-28, with the designated lateral forces and story drift, and with the combined dead
load plus floor live load indicated by W1 and the combined dead load plus roof snow load indicated by
W2, the primary moment in the second story of the frame is
MP2
5 F2 hs2
The secondary moment in the second story is
MS2
5 P2D2 Ie /Cd
5 W2(d2e 2 d1e)
Figure 1-28 P-delta effects
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The stability coefficient in the second story is
q2
5 MS2 /MP2
The primary moment in the first story of the frame is
MP1
5 (F1 1 F2)hs1
The secondary moment in the first story is
MS1
5 P1D1Ie /Cd
5 (W2 1 W1)d1e
The stability coefficient in the first story is
q1
5 MS1/MP1
Example 1-27
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by special steel moment-resisting frames in the north-south direction.
The structure is used as an office building and snow load on the roof is not applicable. Determine the
stability coefficient for the bottom story of the frame.
Solution
Live load on the second floor is obtained from ASCE 7 Table 4.3-1 as 50 lb/ft2. To this must be added
a 15-lb/ft2 allowance for partitions as specified in ASCE 7 Section 4.3.2. The total live load on the
second floor is
WL
5 (0.05 1 0.015)40 3 20
5 52.00 kips
The dead load on the second floor is obtained from Example 1-24 as
WD1
5 0.03 3 40 3 20 1 2 3 0.04 3 20 3 12
5 43.20 kips
The dead load on the roof is obtained from Example 1-24 as
WD2
5 25.60 kips
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The total dead load on the structure is
WD
5 43.20 1 25.60
5 68.80 kips
The inelastic drift in the bottom story is derived in Example 1-25 as
D1
5 1.51 in
The primary moment in the bottom story is
MP1
5 (F2 1 F1)hs1
5 (4.14 1 4.08) 3 12 3 12
5 1183.68 kip-in
The secondary moment is
MS1
5 (WD 1 WL)D1Ie /Cd
5 (68.80 1 52.00)1.51 3 1.0/5.5
5 33.17 kip-in
The stability coefficient is
q1
5 MS1/MP1
5 33.17/1183.68
5 0.028
, 0.1 . . . satisfactory
1.26 Building separation
ASCE 7 Section 12.12.3 requires that all parts of a building or of adjacent buildings be separated a
sufficient distance to permit independent seismic motion without impact between adjacent parts. The
separation shall allow for the maximum inelastic response displacement (dM ). The dM is determined
at critical locations with consideration for translational and torsional displacements of the structure,
including torsional amplifications. The maximum inelastic displacement is given by ASCE 7 Equation
(12.12-1), which is
where:
dM
5 Cddmax/Ie
dmax
5 maximum elastic displacement at the critical location
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Where a structure adjoins a property line not common to a public way, the structure is set back from
the property line by at least the displacement (dM) of that structure. The required separation between
adjacent buildings determined from ASCE 7 Equation (12.12-2) is
where:
dMT
5 [(dM1)2 1 (dM2)2]½
dM1
5 inelastic displacement of building 1
dM2
5 inelastic displacement of building 2
The displacements of both buildings are determined at the same height.
Example 1-28
Determine the separation required for the two steel moment-resisting frames shown in Figure 1-29.
The drift in each story of building 1 is 0.025hs and the drift in each story of building 2 is 0.020hs.
Solution
The maximum inelastic displacement at the roof level of building 1 is given by
dM1
5 4 3 0.025hs
5 4 3 0.025 3 12 3 12
5 14.40 in
Building 1
Building 2
Figure 1-29 Details for Example 1-28
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The maximum inelastic displacement at level 4 of building 2 is given by
dM2
5 4 3 0.020hs
5 4 3 0.020 3 12 3 12
5 11.52 in
The required separation is
dMT
5 [(dM1)2 1 (dM2)2]½
5 [(14.40)2 1 (11.52)2]½
5 18.44 in
1.27 Redundancy factor
The redundancy factor, r, is based on the extent of redundancy present in a building, and its effect
is to penalize structures with relatively few lateral-load-resisting elements. The value of the factor is
either 1.0 or 1.3 and it is applicable in seismic design categories D, E, and F. The redundancy factor is
an attempt to quantify the effects of redundancy and, in effect, reduce the response modification factor values for less redundant structures. The redundancy factor penalizes less redundant structures by
increasing the design horizontal force by 30 percent and provides an economic incentive for the design
of structures with well-distributed lateral-force-resisting systems.
By providing multiple lateral-load-resisting paths in a structure, a degree of redundancy is provided
to the system. Yield of one element in the system results in redistribution of load to the remaining elements, thus controlling displacements and the deterioration of the structure and delaying the formation
of a collapse mechanism. Thus, to improve the seismic performance of a structure, it is desirable to
provide multiple load paths so as to make the lateral-load-resisting system as redundant as possible.
In accordance with ASCE 7 Section 12.3.4.1, the value of r is permitted to equal 1.0 in the following
situations:
•
structures assigned to seismic design categories B and C
•
drift calculations and P-delta effects
•
design of nonbuilding structures that are not similar to buildings
•
design of nonstructural components
•
design of collector elements, splices and their connections for which the overstrength factor is
required
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•
design of members or connections for which the overstrength factor is required
•
diaphragm loads determined using ASCE 7 Equation (12.10-1)
•
systems with passive energy damping systems
•
design of structural walls for out-of-plane forces, including their anchorages
85
For structures assigned to seismic design category D and having an extreme torsional irregularity, r
shall equal 1.3. For other structures assigned to seismic design category D and for structures assigned
to seismic design categories E or F, r shall equal 1.3 unless one of the conditions given in ASCE 7
Section 12.3.4.2 as condition (a) or condition (b) is met, whereby r is permitted to be taken as 1.0.
As specified in condition (a), the redundancy factor may be taken as 1.0 provided that at each story
resisting more than 35 percent of the base shear, the lateral-load-resisting system satisfies the following redundancy requirements:
•
for a braced frame, removal of an individual brace, or connection thereto, does not result in
more than a 33-percent reduction in story strength, nor create an extreme torsional irregularity
•
for a moment frame, loss of moment resistance at the beam-to-column connections at both
ends of a single beam does not result in more than a 33-percent reduction in story strength, nor
create an extreme torsional irregularity
•
for a shear wall or wall pier system with a height-to-length ratio greater than 1.0, removal of a
wall or pier, or collector connections thereto, does not result in more than a 33-percent reduction in story strength, nor create an extreme torsional irregularity
•
for a cantilever column, loss of moment resistance at the base connections of any single cantilever column does not result in more than a 33-percent reduction in story strength, nor create
an extreme torsional irregularity
•
for all other structural systems, there are no requirements
Alternatively, as specified in condition (b), the value of the redundancy factor may be assumed equal
to 1.0, provided that the building is regular in plan at all levels with not less than two bays of lateralload-resisting perimeter framing on each side of the building in each orthogonal direction at each story,
resisting more than 35 percent of the base shear. The number of bays for a shear wall is defined as the
length of the shear wall divided by the story height. For light-frame construction, the number of bays
for a shear wall is defined as twice the length of the shear wall divided by the story height.
Note that, in accordance with ASCE 7 Section 12.3.3.1, structures that are assigned to seismic design
category E or F and have an extreme torsional irregularity are prohibited.
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Example 1-29
Determine the value of the redundancy factor for the building shown in Figure 1-30. Lateral resistance
is provided in the north-south direction by steel special moment frames, all with the same stiffness.
In the east-west direction, lateral resistance is provided by eccentrically braced frames with identical
stiffness. The building is assigned to seismic design category D.
Figure 1-30 Details for Example 1-29
Solution
The structure is regular in plan and has two bays of special moment-resisting perimeter framing on
each side of the building in the north-south direction. Two bays of eccentrically braced frames are
provided on the perimeter of each side of the building in the east-west direction. Hence, the building
complies with condition (b) of ASCE 7 Section 12.3.4.2 and the redundancy factor is
r
5 1.0
Example 1-30
Determine the value of the redundancy factor for the building shown in Figure 1-31. Lateral resistance
is provided in the north-south direction by steel special moment frames, all with the same stiffness.
In the east-west direction, lateral resistance is provided by eccentrically braced frames with identical
stiffness. The roof diaphragm may be considered flexible. The building is assigned to seismic design
category D.
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Figure 1-31 Details for Example 1-30
Solution
The building does not comply with ASCE 7 Section 12.3.4.2 condition (b) since in the east-west direction, only one eccentrically braced frame is provided on the perimeter of each side of the building.
Removing brace 12 on the north side of the building results in a reduction of the lateral resistance in
the east-west direction of
Vr
5 100(2 2 1)/2
5 50 percent
. 33 percent . . . unsatisfactory
Hence, the redundancy factor is
r
5 1.3
Example 1-31
Determine the value of the redundancy factor for the building shown in Figure 1-32. Lateral resistance
is provided in the north-south direction by steel special moment frames, all with the same stiffness.
In the east-west direction, lateral resistance is provided by eccentrically braced frames with identical stiffness. The stiffness of the braced frames is 10 times the stiffness of the moment frames. The
roof diaphragm is rigid and the center of mass is located at the center of the building. The building is
assigned to seismic design category D.
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Figure 1-32 Details for Example 1-31
Solution
The building does not comply with ASCE 7 Section 12.3.4.2 condition (b) as it is not regular in plan.
For north-south seismic forces, the loss of moment resistance at both ends of beam 12 results in a
reduction of the lateral resistance in the north-south direction of
Vr
5 100(4 2 3)/4
5 25 percent
, 33 percent . . . satisfactory
It is now necessary to determine if an extreme torsional irregularity exists.
The lateral-force-resisting arrangement is now as indicated in Figure 1-33. The stiffness of a braced
frame is
where:
kB
5 10kM . . . from problem statement
kM
5 stiffness of a moment frame
The center of rigidity is located a distance from the west wall given by
rw
5 SxkM /SkM
5 (90 3 kM)/3kM
5 30 feet
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Brace
V
Moment
connection
89
N
B1
B2
M3
40 ft
CM
CR
M2
M1
rs = 32 ft
B5
30 ft
B4
30 ft
40 ft
B3
30 ft
L = 90 ft
Figure 1-33 North-south seismic load
The center of rigidity is located a distance from the south wall given by
rs
5 SykB /SkB
5 (80 3 2kB)/5kB
5 32 feet
The distances of the braced frames and the moment frames from the center of rigidity are
rB1
5 rB2
5 48 feet
rB3
5 rB4 5 rB5
5 32 ft
rM1
5 60 feet
rM2
5 rM3
5 30 feet
The polar moment of inertia of the walls is
J
5 Skr2
5 kM(1 3 602 1 2 3 302 1 2 3 10 3 482 1 3 3 10 3 322)
5 82,200kM
5 8220kB
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Seismic Design
For a north-south seismic load, the eccentricity is
e
5 x 2 rw
5 45 2 30
5 15 feet
Accidental eccentricity, in accordance with ASCE 7 Section 12.8.4.2 is
ea
5 60.05 3 L
5 60.05 3 90
5 64.5 feet
An accidental displacement of the center of mass to the east results in the maximum eccentricity of
emax
5 e 1 ea
5 15 1 4.5
5 19.5 feet
The maximum torsional moment acting about the center of rigidity is
T
5 Vemax
5 19.5V
The total shear force produced in a wall is the algebraic sum of the in-plane shear force and the torsional shear force. The in-plane shear force in the west wall is
Fs
5 Vkww /Sk
5 V 3 2kM /3kM
5 0.667V
The torsional shear force in the west wall is
Ft
5 2Trwwkww /J
5 219.5V 3 30 3 2kM /82,200kM
5 20.014V
The total shear force in the west wall is
Fww
5 Fs 1 Ft
5 0.667V 2 0.014V
5 0.653V
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The drift of the west wall is
Dww
5 Fww /kww
5 0.653V/2kM
5 0.327V/kM
The in-plane shear force in the east wall is
Fs
5 Vkew /Sk
5 V 3 kM /3kM
5 0.333V
The torsional shear force in the east wall is
Ft
5 Trewkew /J
5 19.5V 3 60 3 kM /82,200kM
5 0.014V
The total shear force in the east wall is
Few
5 Fs 1 Ft
5 0.333V 1 0.014V
5 0.347V
The drift of the east wall is
Dew
5 Few /kew
5 0.347V/kM
The average of the drifts of the east and west walls is
Dav
5 (Dew 1 Dww)/2
5 (0.347 1 0.327)V/2kM
5 0.337V/kM
The ratio of maximum story drift to average story drift is
Dew /Dav 5 0.347/0.337
5 1.03
, 1.4 . . . satisfactory, in accordance with ASCE 7 Table 12.3-1, extreme
torsional irregularity is not produced
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For east-west seismic forces, the removal of brace 34 from the central bay of the north wall (see Figure
1-32) results in a reduction of the lateral resistance in the east-west direction of
Vr
5 100(5 2 4)/5
5 20 percent
, 33 percent . . . satisfactory
It is now necessary to determine if an extreme torsional irregularity exists.
Brace
Moment
connection
B1
M4
M1
40 ft
CM
L = 80 ft
V
CR
M3
B4
30 ft
rs = 20 ft
M2
40 ft
B2
B3
30 ft
30 ft
Figure 1-34 East-west seismic load
The lateral-force-resisting arrangement is now as indicated in Figure 1-34. The stiffness of a braced
frame is
where:
kB
5 10kM . . . from the problem statement
kM
5 stiffness of a moment frame
From symmetry, the center of rigidity is located a distance from the west wall given by
rw
5 45 ft
The center of rigidity is located a distance from the south wall given by
rs
5 SykB /SkB
5 (80 3 kB)/4kB
5 20 ft
The distances of the braced frames and the moment frames from the center of rigidity are
rB1
5 60 ft
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rB2
93
5 rB3 5 rB4
5 20 ft
rM1
5 rM2 5 rM3 5 rM4
5 45 ft
The polar moment of inertia of the walls is
J
5 Skr2
5 kM(4 3 452 1 10 3 602 1 3 3 10 3 202)
5 56,100kM
5 5610kB
For an east-west seismic load, the eccentricity is
e
5 y 2 rs
5 40 2 20
5 20 feet
Accidental eccentricity, in accordance with ASCE 7 Section 12.8.4.2 is
ea
5 60.05 3 L
5 60.05 3 80
5 64 feet
An accidental displacement of the center of mass to the north results in the maximum eccentricity of
emax
5 e 1 ea
5 20 1 4
5 24 feet
The maximum torsional moment acting about the center of rigidity is
T
5 Vemax
5 24V
The total shear force produced in a wall is the algebraic sum of the in-plane shear force and the torsional shear force. The in-plane shear force in the south wall is
Fs
5 Vksw /Sk
5 V 3 3kB /4kB
5 0.75V
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Seismic Design
The torsional shear force in the south wall is
Ft
5 2Trswksw /J
5 224V 3 20 3 3kB /5610kB
5 20.257V
The total shear force in the south wall is
Fsw
5 Fs 1 Ft
5 0.75V 2 0.257V
5 0.493V
The drift of the south wall is
Dsw
5 Fsw /ksw
5 0.493V/3kB
5 0.164V/kB
The in-plane shear force in the north wall is
Fs
5 Vknw /Sk
5 V 3 kB /4kB
5 0.25V
The torsional shear force in the north wall is
Ft
5 Trnwknw /J
5 24V 3 60 3 kB /5610kB
5 0.257V
The total shear force in the north wall is
Fnw
5 Fs 1 Ft
5 0.25V 1 0.257V
5 0.507V
The drift of the north wall is
Dnw
5 Fnw /knw
5 0.507V/kB
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95
The average of the drifts of the north and south walls is
Dav
5 (Dnw 1 Dsw)/2
5 (0.507 1 0.164)V/2kB
5 0.336V/kB
The ratio of maximum story drift to average story drift is
Dnw /Dav 5 0.507/0.336
5 1.51
. 1.4 . . . unsatisfactory; in accordance with ASCE 7 Table 12.3-1, extreme
torsional irregularity is produced
Hence, the redundancy factor is
r
5 1.3
1.28 Load combinations
Design is permitted in the IBC by either the allowable stress method or the strength design method.
Either method requires the application of prescribed load combinations to determine the most critical
effect on any particular element in a structure.
1.28.1 Strength design loads and load factors
When strength design principles are utilized, the basic requirement is to ensure that the design strength
of a member is not less than the required ultimate strength. The required strength consists of the
service level loads multiplied by appropriate load factors, as defined in ASCE 7 Section 2.3.6. The
ASCE 7 load combinations that include seismic forces are
U
5 1.2D 1 Ev 1 Eh 1 f1L 1 0.2S . . . combination 6
and
U
5 0.9D 2 Ev 1 Eh . . . combination 7
where:
D
5 dead load
L
5 floor live load
S
5 snow load
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Seismic Design
Eh
5 effect of horizontal seismic load in accordance with ASCE 7 Equation
(12.4-3)
5 rQE
Ev
5 effect of vertical seismic load in accordance with ASCE 7 Equation
(12.4-4a)
5 0.2SDS D
QE
5 effect of horizontal seismic forces
5 effect of horizontal seismic forces Fi in Figure 1-22
SDS
5 design spectral response acceleration at a period of 0.2 second
r
5 redundancy factor defined in ASCE 7 Section 12.3.4
f1
5 1.0 for floors in garages and places of public assembly and for floor loads
in excess of 100 lb/ft2
5 0.5 for other live loads
Imposed live load is omitted where this results in a more critical effect in a member subjected to seismic loads. Since seismic load is determined at the strength design level, it has a load factor of 1.0.
Where the effects of gravity and seismic loads are additive, ASCE 7 Section 2.3.6 combination 6 may
be written as
U
5 (1.2 1 0.2SDS)D 1 rQE 1 f1L 1 0.2S . . . combination 6
Where the effects of gravity and seismic loads counteract, ASCE 7 Section 2.3.6 combination 7 may
be written as
U
5 (0.9 2 0.2SDS)D 1 rQE . . . combination 7
Example 1-32
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the north-south direction.
Determine the maximum and minimum strength design axial loads acting on the column footings for
the applied loads shown in Figure 1-35. The redundancy factor is r 5 1.0. The 5-percent damped,
design spectral response acceleration for a period of 0.2 second is SDS 5 0.840g.
Seismic and Wind Forces: Structural Design Examples
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Figure 1-35 Details for Example 1-32
Solution
The force in one column due to the effects of dead load is
D
5 (SwDx)/2
where:
wDx
5 dead load at level x
and
D
5 (43.2 1 25.6)/2
5 34.4 kips
The force in one column due to the effects of superimposed floor load is
L
5 (SwLx)/2
where:
wLx
5 floor live load at level x
and
L
5 56.0/2
5 28.0 kips
The force in one column due to the effects of superimposed roof load is
Lr
5 wr /2
where:
wr
5 roof live load
and
Lr
5 16/2
5 8 kips
The force in one column due to the effects of horizontal seismic forces is
where:
rQE
5 6r(SFx hx)/s
s
5 width of frame 5 40 feet
Fx
5 design lateral force at level x
hx
5 height above base to level x
Seismic and Wind Forces: Structural Design Examples
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Seismic Design
and
rQE
5 61.0(4.06 3 24 1 4.06 3 12)/40
5 63.65 kips
The force in one column due to the effects of vertical seismic forces is
QV
5 60.2SDSD
5 60.2 3 0.840 3 34.4
5 65.78 kips
Applying load combination 6 in ASCE 7 Section 2.3.6 gives the strength design load as
FC(max) 5 (1.2 1 0.2SDS)D 1 rQE 1 f1L 1 0.2S
5 1.2 3 34.4 1 5.78 1 3.65 1 0.5 3 28.0 1 0
5 64.71 kips . . . compression
Applying load combination 2 of ASCE 7 Section 2.3.1 gives the strength design load as
FC(max) 5 1.2D 1 1.6L 1 0.5Lr
5 1.2 3 34.4 1 1.6 3 28.0 1 0.5 3 8
5 90.08 kips . . . compression, governs
Applying load combination 7 of ASCE 7 Section 2.3.6 gives the strength design load as
FC(min) 5 (0.9 2 0.2SDS)D 1 rQE
5 0.9 3 34.4 2 5.78 2 3.65
5 21.53 kips . . . compression, no uplift
1.28.2 Special seismic load combinations for the strength design method
The special load combinations incorporate the overstrength factor, W0, of ASCE 7 Section 12.4.3
to protect critical elements in a building. These critical elements must be designed with sufficient
strength to protect against their failure and a subsequent building collapse. Seismic loads multiplied by
the overstrength factor are an approximation of the maximum load an element will experience. These
situations where W0 is included in the design force are covered in the following code sections:
•
ASCE 7 Section 12.10.2.1: In seismic design categories C through F, for collector elements
and their connections, including connections to vertical elements where the seismic force is
determined by ASCE 7 Section 12.8, Section 12.9, or ASCE 7 Equation (12.10-1). The overstrength factor is not applied where the seismic force is determined using the minimum value of
Fpx from ASCE 7 Equation (12.10-2). An exception is permitted to the use of the overstrength
factor for structures braced entirely with light-frame shear walls. An example of collectors is
shown in Figure 1-36.
Seismic and Wind Forces: Structural Design Examples
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•
ASCE 7 Section 12.3.3.3: For elements supporting discontinuous walls or frames of structures
having in-plane or out-of-plane discontinuities type 4 of ASCE 7 Table 12.3-1 or type 4 of
ASCE 7 Table 12.3-2. An example of in-plane discontinuity is shown in Figure 1-37.
•
ASCE 7 Section 12.13.8.4: For batter piles and their connections as shown in Figure 1-38.
•
ASCE 7 Section 12.3.3.2: For buildings in seismic design categories B and C exceeding two
stories or 30 feet in height and having an extreme weak story type 5b in ASCE 7 Table 12.3-2,
the weak story shall be designed using the overstrength factor. In accordance with ASCE 7 Section 12.3.3.1, buildings with extreme weak stories are prohibited in seismic design categories
D through F. An example of an extreme weak story is shown in Figure 1-39.
Figure 1-36 Collector elements
Shear wall
Offset
Shear wall
Columns designed
for Ω0
Figure 1-37 In-plane discontinuity
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Seismic Design
Batter piles
Figure 1-38 Batter piles
Shear wall
Columns designed
for 0
Figure 1-39 Extreme weak story
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101
Where the seismic load effect including overstrength is combined with the effects of other loads,
ASCE 7 Section 2.3.6 requires the use of combinations 6 and 7, which are
U
5 1.2D 1 Ev 1 Emh 1 f1L 1 0.2S . . . combination 6
and
U
5 0.9D 2 Ev 1 Emh . . . combination 7
where:
D
5 dead load
L
5 floor live load
S
5 snow load
Emh
5 maximum effect of horizontal earthquake forces that can be developed in an
element, as defined in ASCE 7 Equation (12.4-7)
5 W0QE
Ev
5 effect of vertical seismic load in accordance with ASCE 7 Equation
(12.4-4a)
W0
5 structure overstrength factor given in ASCE 7 Table 12.2-1 and tabulated
for an abbreviated number of structures in Table 1-16
5 amplification factor to account for the overstrength of the structure in the
inelastic range
QE
5 effect of horizontal seismic forces Fi in Figure 1-22
SDS
5 design spectral response acceleration at a period of 0.2 second
r
5 redundancy factor defined in ASCE 7 Section 12.3.4
f1
5 1.0 for floors in garages and places of public assembly and for floor loads
in excess of 100 lb/ft2
5 0.5 for other live loads
Imposed live load is omitted where this results in a more critical effect in a member subjected to seismic loads. Since seismic load is determined at the strength design level, it has a load factor of 1.0.
Where the effects of gravity and seismic loads are additive, ASCE 7 Section 2.3.6 combination 6 is
defined by
U
5 (1.2 1 0.2SDS)D 1 W0QE 1 f1L 1 0.2S . . . combination 6
Where the effects of gravity and seismic loads counteract, ASCE 7 Section 2.3.6 combination 7 is
defined by
U
5 (0.9 2 0.2SDS)D 1 W0QE . . . combination 7
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Seismic Design
Example 1-33
The special reinforced masonry shear wall, supported on columns as shown in Figure 1-39, forms part
of a bearing wall lateral-force-resisting system. The building is assigned to seismic design category
C, and the bottom story is classified as an extreme weak story. The 5-percent damped, design spectral
response acceleration for a period of 0.2 second is SDS 5 0.40g. The applied axial loads on each column are dead load (D 5 80 kips), floor live load (L 5 20 kips), and the effect of horizontal seismic
force (QE 5 630 kips). Determine the maximum and minimum strength design axial loads acting on
the column footings. The building is an office structure.
Solution
Because the building exceeds two stories in height and the bottom story is classified as an extreme
weak story, the special seismic load combinations are applicable, and the structure overstrength factor
for special reinforced masonry shear walls, given in Table 1-16, is
W0
5 2.5
Applying load combination 6 of ASCE 7 Section 2.3.6 gives the strength design load as
FC(max) 5 (1.2 1 0.2SDS)D 1 W0QE 1 f1L 1 0.2S
5 (1.2 1 0.2 3 0.40)80 1 2.5 3 30 1 0.5 3 20 1 0
5 187 kips . . . compression
Applying load combination 7 of ASCE 7 Section 2.3.6 gives the strength design load as
FC(min) 5 (0.9 2 0.2SDS)D 1 W0QE
5 (0.9 2 0.2 3 0.40)80 2 2.5 3 30
5 29.40 kips . . . tension
1.28.3 Allowable stress design method
When allowable stress design principles are utilized, the basic requirement is to ensure that, under the
action of service level loads, the stress in an element does not exceed permissible limits. The design
level load combinations that include seismic forces are defined by combinations 8, 9, and 10 of ASCE
7 Section 2.4.5 and, after substituting the values of Eh and Ev from ASCE 7 Equations (12.4-3) and
(12.4-4a), are given by
(1.0 1 0.14SDS)D 1 0.7rQE . . . combination 8
(1.0 1 0.105SDS)D 1 0.525rQE 1 0.75L 1 0.75S . . . combination 9
(0.6 2 0.14SDS)D 1 0.7rQE . . . combination 10
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where:
103
D
5 dead load
L
5 floor live load
S
5 snow load
E
5 seismic load
QE
5 effect of horizontal seismic forces
SDS
5 5-percent damped, design spectral response acceleration, for a period of 0.2
second
r
5 redundancy factor
No increase in allowable stress is permitted with these load combinations, with the exception of the
duration of load increase specified for wood members in NDS Section 2.3.2.11
Example 1-34
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Lateral force resistance is provided by steel special moment-resisting frames in the north-south direction.
The 5-percent damped, design spectral response acceleration for a period of 0.2 second is SDS 5 0.840g
and the redundancy coefficient is r 5 1.0. Determine the maximum and minimum loads acting on the
column footings using the allowable stress design method.
Solution
The loads acting on the structure are shown in Figure 1-35, and the forces acting on the columns are
determined in Example 1-32.
The force in one column due to the effects of dead load is
D
5 34.4 kips
The force in one column due to the effects of superimposed floor load is
L
5 28.0 kips
The strength level force in one column due to the effects of horizontal seismic forces is
rQE
5 63.65 kips
The strength level force in one column due to the effects of vertical seismic forces is
QV
5 60.2SDSD
5 65.78 kips
Seismic and Wind Forces: Structural Design Examples
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Seismic Design
Applying load combination 2 of ASCE 7 Section 2.4.1 gives the allowable stress design load as
FC(max) 5 D 1 L
5 34.4 1 28.0
5 62.40 kips . . . compression
Load combination 4 of ASCE 7 Section 2.4.1 is less critical than load combination 2.
Applying load combination 8 of ASCE 7 Section 2.4.5 gives the allowable stress design load as
FC(max) 5 (1.0 1 0.14SDS)D 1 0.7rQE
5 1.0 3 34.4 1 0.7 3 5.78 1 0.7 3 3.65
5 41.00 kips . . . compression
Applying load combination 9 of ASCE 7 Section 2.4.5 gives the allowable stress design load as
FC(max) 5 (1.0 1 0.105SDS)D 1 0.525rQE 1 0.75L 1 0.75S
5 1.0 3 34.4 1 0.525 3 5.78 1 0.525 3 3.65 1 0.75 3 28 1 0
5 60.35 kips . . . compression, governs
Applying load combination 10 of ASCE 7 Section 2.4.5 gives the allowable stress design load as
FC(min) 5 (0.6 2 0.14SDS)D 1 0.7rQE
5 0.6 3 34.4 2 0.7 3 5.78 2 0.7 3 3.65
5 14.04 kips . . . compression, no uplift
1.28.4 Special seismic load combinations for the allowable stress design method
The maximum force that can be delivered to the system, when the effects of gravity and seismic loads
are additive, is determined by load combinations 8 and 9 given by ASCE 7 Section 2.4.5, which are
FC(max) 5 (1.0 1 0.14SDS)D 1 0.7W0QE
FC(max) 5 (1.0 1 0.105SDS)D 1 0.525W0QE 1 0.75L 1 0.75S
where:
D
5 dead load
L
5 floor live load
QE
5 strength level effect of horizontal seismic forces
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105
SDS
5 5-percent damped, design spectral response acceleration, for a period of 0.2
second
W0
5 structure overstrength factor given in ASCE 7 Table 12.2-1 and tabulated
for an abbreviated number of structures in Table 1.16
5 amplification factor to account for the overstrength of the structure in the
inelastic range
The maximum force that can be delivered to the system, when the effects of gravity and seismic loads
counteract, is determined by load combination 10 given by ASCE 7 Section 2.4.5, which is
FC(min) 5 (0.6 2 0.14SDS)D 1 0.7W0QE
In applying these load combinations, the allowable stress in a member may be increased by a factor
of 1.2. No additional stress increases are permitted with the exception of the duration of load increase
specified for wood members in NDS Section 2.3.2.
1.29 Structural elements
Structural elements and their attachments are designed to resist the design seismic forces detailed in
ASCE 7 Section 12.1. These requirements are applicable to structures in seismic design category B
and higher to ensure the structural integrity of a building in the event of an earthquake. All elements in
a building must be connected so as to act together as a single unit.
1.29.1 Connections
ASCE 7 Section 12.1.3 requires all smaller elements of a structure to be tied to the remainder of the
structure with a connection capable of resisting a horizontal force given by
Fp
5 0.133SDSwp
with a minimum value given by
where:
Fp
5 0.05wp
wp
5 weight of the smaller element
SDS
5 5-percent damped, design spectral response acceleration for a period of 0.2
second
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Seismic Design
In addition, as specified in ASCE 7 Section 12.1.4, for each beam, girder, or truss, a connection shall
be provided to resist a horizontal force given by
where:
FR
5 0.05wR
wR
5 reaction due to dead 1 live loads
Example 1-35
The two-span glued-laminated girder shown in Figure 1-40 supports a dead load, including its own
weight, of 450 pounds per foot and a live load of 500 pounds per foot. Sliding bearings are provided at
supports 3 and 4 with a hinge at support 1. The 5-percent damped, design spectral response acceleration for a period of 0.2 second is SDS 5 0.826g. Determine the required tie force at the hinge connector
and the horizontal force at support 1.
Figure 1-40 Details for Example 1-35
Solution
The tie force required at the hinge connector is given by ASCE 7 Section 12.1.3
where:
Fp
5 0.05wp
wp
5 dead load of beam 12
5 0.45 3 40
5 18 kips
then
Fp
5 0.05 3 18
5 0.90 kip
Alternatively the tie force required at the hinge connector is given by
Fp
5 0.133SDSwp
5 0.133 3 0.826 3 18
5 1.98 kips . . . governs
Seismic and Wind Forces: Structural Design Examples
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107
At support 1, the required horizontal force is given by
where:
FR
5 0.05wR
wR
5 reaction due to dead 1 live loads
5 (0.45 1 0.5)40/2
5 19 kips
then
FR
5 0.05 3 19
5 0.95 kip
1.29.2 Lateral design force on walls
The out-of-plane seismic force on structural walls is specified in ASCE 7 Section 12.11.1 and is given
by
Fp
5 0.40IeSDSWc
≥ 0.1Wc
where:
Ie
5 occupancy importance factor given in Table 1-5
SDS
5 5-percent damped, design spectral response acceleration for a period of 0.2
second
Wc
5 weight of the wall
Example 1-36
The 7-inch concrete wall shown in Figure 1-41 forms part of a building assigned to seismic design
category B with an importance factor of 1.0. The factored roof load is 300 pounds per foot and it acts at
an eccentricity of 7 inches with respect to the center of the wall. The 5-percent damped, design spectral
response acceleration for a period of 0.2 second is SDS 5 0.30g. Assuming that seismic loads govern
the design, determine the strength level design moment in the wall.
Solution
Weight of the wall is
Wc
5 150 3 7/12
5 87.50 lb/ft2
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Seismic Design
128 lb-ft
Figure 1-41 Details for Example 1-36
The seismic lateral force on the wall is given by ASCE 7 Section 12.11.1 as
Fp
5 0.40IeSDSWc
5 0.40 3 1.0 3 0.30 3 87.50
5 10.50 lb/ft2
The horizontal force per linear foot of wall at roof level is obtained by taking moments about the
hinged base
HA
5 (10.5 3 232/2 2 300 3 7/12)/20
5 130.11 lb/ft
The horizontal force per linear foot of wall at the base of the wall is
HB
5 (10.5 3 23 3 8.5 1 300 3 7/12)/20
5 111.39 lb/ft
The maximum moment in the wall occurs at a height y above the base given by
M
5 yHB 2 Fp y2/2
5 111.39y 2 10.5y2/2
Differentiating with respect to y and equating dM/dy to zero gives
y
5 10.61 ft
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109
The maximum moment is given by
Mw
5 111.39 3 10.61 2 5.25 3 (10.61)2
5 591 lb-ft/ft
1.29.3 Lateral design force on parapets
To compensate for the poor seismic performance and lack of redundancy of parapets, which may
create a safety hazard to the public, parapets are designed for a higher design load than walls. For
the design of parapets, ASCE 7 Section 13.5.2 requires the application of ASCE 7 Equation (13.3-1).
However, in determining the design moment in a wall with a parapet or the design force in an anchorage, ASCE 7 Equation (13.3-1) is not applied to the parapet and ASCE 7 Section 12.11.1 is applied
to the entire wall, including the parapet.12 ASCE 7 Equation (13.3-1) is applied to parapets in seismic
design category B and above and is given by
where:
Fp
5 (0.4apSDS Ip /Rp)(1 1 2z/h)Wp
Ip
5 component importance factor given in ASCE 7 Section 13.1.3
SDS
5 5-percent damped, design spectral response acceleration for a period of 0.2
second
Wp
5 weight of parapet
ap
5 component amplification factor from ASCE 7 Table 13.5-1
5 2.5 . . . for unbraced parapets
h
5 height of roof above the base
z
5 height of parapet at point of attachment
5h
Rp
5 component response modification factor from ASCE 7 Table 13.5-1
5 2.5 . . . for unbraced parapets
In accordance with ASCE 7 Equation (13.3-2), Fp need not be taken greater than
Fp
5 1.6SDS IpWp
In accordance with ASCE 7 Equation (13.3-3), Fp shall not be taken less than
Fp
5 0.3SDS IpWp
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Example 1-37
The 7-inch concrete parapet shown in Figure 1-41 forms part of a building with a component importance factor of 1.0. The 5-percent damped, design spectral response acceleration for a period of 0.2
second is SDS 5 0.30g. Determine the strength level seismic design moment in the parapet.
Solution
Weight of the parapet per linear foot is
Wp
5 150 3 3 3 7/12
5 262.50 lb/ft
The seismic lateral force acting on the parapet is given by ASCE 7 Equation (13.3-1) as
where:
Fp
5 (0.4apSDS Ip /Rp)(1 1 2z/h)Wp
Ip
5 component importance factor
5 1.0
SDS
5 5-percent damped, design spectral response acceleration for a period of 0.2
second
5 0.30g
Wp
5 weight of parapet
5 262.50 lb/ft
ap
5 component amplification factor from ASCE 7 Table 13.5-1
5 2.5
h
5 height of roof above the base
5 20 ft
z
5 height of parapet at point of attachment
5 20 ft
Rp
5 component response modification factor from ASCE 7 Table 13.5-1
5 2.5
and
Fp
5 (0.4 3 2.5 3 0.30 3 1.0/2.5)(1 1 2 3 20/20)Wp
5 0.36Wp
5 94.5 lb/ft
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111
Neither ASCE 7 Equation (13.3-2) nor (13.3-3) govern, and the bending moment at the base of the
parapet is
Mp
5 1.5Fp
5 142 lb-ft/ft
1.30 Anchorage of structural walls
During past earthquakes, a major cause of failure has been the separation of flexible diaphragms from
concrete and masonry supporting walls. This is due to diaphragm flexibility amplifying out-of-plane
accelerations. To prevent separation occurring, anchorage ties must be provided as shown in Figure
1-41 to tie the diaphragms and walls together. These forces apply only to the design of the ties and not
to the overall wall design. Where the wall anchor spacing exceeds 4 feet, in accordance with ASCE
7 Section 12.11.2.1, the wall must be designed to span between anchors. In accordance with ASCE 7
Section 12.11.2.2.2, steel elements in the anchorage system are required to resist 1.4 times the calculated force in structures assigned to seismic design categories C through F. This provides a factor of
safety of approximately two against tensile rupture.
For buildings in seismic design categories B through F, ASCE 7 Section 12.11.2.1 requires anchors to
be designed for the force.
Fp
5 0.4SDS kaIeWp
≥ 0.2kaIeWp
where:
Ie
5 importance factor
SDS
5 design response acceleration, for a period of 0.2 second
Wp
5 weight of the wall tributary to the anchor
ka
5 amplification factor for diaphragm flexibility
5 1.0 1 Lf /100
≤2
≤ 1 . . . for a diaphragm that is not flexible
Lf
5 span in feet of a flexible diaphragm measured between vertical elements
that provide lateral support to the diaphragm in the direction considered
5 0 . . . for rigid diaphragms
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1.30.1 Anchorage to flexible diaphragms
A diaphragm is considered flexible, in accordance with ASCE 7 Section 12.3.1.3, where the maximum
displacement of the diaphragm under lateral load exceeds twice the average displacement of the end
supports.
This is shown in Figure 1-42 and the diaphragm is flexible if:
where:
dM
. 2dA
dM
5 maximum displacement of the diaphragm
dA
5 average story drift
In accordance with ASCE 7 Section 12.3.1.1, the following types of diaphragms may be considered
flexible:
•
untopped steel decking or wood structural panels supported by vertical elements of steel or
composite braced frames, or concrete, masonry, steel, or composite shear walls
•
untopped steel decking or wood structural panels in one- and two-family residential buildings
In addition, diaphragms of untopped steel decking or wood structural panels are considered flexible
provided all of the following conditions are met:
•
in structures of light-frame construction, toppings of concrete or similar materials are not
placed over wood structural panel diaphragms except for nonstructural toppings not greater
than 1.5 inches thick
•
each line of the lateral-force-resisting system complies with the allowable story drift of ASCE
7 Table 12.12-1
The inertial forces developed in a building by an earthquake must be transferred by a suitable seismic-force-resisting system to the foundation. This system consists of two parts: horizontal diaphragms
that transfer the seismic forces at each floor to the vertical seismic-force-resisting elements and the
vertical elements that transfer the lateral forces to the foundation. A flexible diaphragm is assumed to
act as a simply supported beam between vertical seismic-force-resisting elements. Hence, lateral force
is distributed to the vertical elements based on tributary mass, without producing any torsional effects.
The anchorage of structural walls to supporting construction must be capable of resisting the lateral
seismic force given by ASCE 7 Equation (12.11-1) as
Fp
5 0.4SDS kaIeWp
≥ 0.2kaIeWp
where:
Wp
5 weight of the wall tributary to the anchor
ka
5 amplification factor for diaphragm flexibility
5 1.0 1 Lf /100
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Lf
113
5 span in feet of the flexible diaphragm
5 0 for rigid diaphragms
Figure 1-42 Diaphragm flexibility
Example 1-38
The 7-inch concrete wall shown in Figure 1-41 forms part of a building assigned to seismic design category C with an importance factor of 1.0. The 5-percent damped, design spectral response acceleration
for a period of 0.2 second is SDS 5 0.40g. If the roof diaphragm may be considered flexible and has a
span of 30 feet, determine the strength level seismic design force in each anchor.
Solution
Weight of the wall is
w
5 150 3 7/12
5 87.50 lb/ft2
The equivalent area of wall tributary to each anchor is obtained by taking moments about the hinged
base
Aw
5 4 3 232/(2 3 20)
5 52.90 ft2
The weight of the wall tributary to each anchor is
Wp
5 wAw
5 87.50 3 52.90/1000
5 4.63 kips
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Seismic Design
The span in feet of the flexible diaphragm is
Lf
5 30 feet
The amplification factor for diaphragm flexibility is
ka
5 1.0 1 Lf /100
5 1.0 1 30/100
5 1.3
For seismic design category C, the seismic lateral force on an anchor is given by ASCE 7 Equation
(12.11-1) as
Fp
5 0.4SDS ka IeWp
5 0.4 3 0.4 3 1.3 3 1.0 3 4.63
5 0.96 kip
The minimum permissible force on one anchor is
Fp
5 0.2ka IeWp
5 0.2 3 1.3 3 1.0 3 4.63
5 1.20 kips . . . governs
The required seismic design force for the anchors is
Fp
5 1.2 kips
1.30.2 Anchorage to rigid diaphragms
A diaphragm is considered rigid in accordance with ASCE 7 Section 12.3.1.2 where it consists of concrete slabs or concrete-filled metal decks with span-to-depth ratios of three or less in structures that
have no horizontal irregularities. In accordance with ASCE 7 Section 12.11.2.1, the anchorages for a
rigid diaphragm, with the exception of roof diaphragms, shall resist the horizontal forces determined
from
where:
Fp
5 (0.4SDS Ie)(1 1 2z/h)Wp /3
Fp
5 seismic design force on the anchor
Ie
5 importance factor
SDS
5 5-percent damped, design spectral response acceleration for a period of
0.2 second
Wp
5 weight of the wall tributary to the anchor
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h
5 height of roof above the base
z
5 height of anchor above the base
115
In accordance with ASCE 7 Section 12.11.2.1, Fp shall not be taken less than
Fp
5 0.2IeWp
Anchorage forces for rigid roof diaphragms are determined using ASCE 7 Equation (12.11.1) with
ka 5 1.0, which gives
Fp
5 0.4SDS IeWp
≥ 0.2IeWp
Example 1-39
The 7-inch concrete wall shown in Figure 1-43 forms part of a building assigned to seismic design category C with an importance factor of 1.0. The 5-percent damped, design spectral response acceleration
for a period of 0.2 second is SDS 5 0.40g. If the roof diaphragm and the second-floor diaphragm may
be considered rigid, determine the anchorage force in the roof diaphragm.
7-in concrete wall
10 ft
10 ft
Figure 1-43 Details for Example 1-39
Solution
Weight of the wall is
w
5 150 3 7/12
5 87.50 lb/ft2
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Seismic Design
The area of the wall tributary to the roof diaphragm is
Aw
5 1 3 10/2
5 5 ft2/ft
The weight of the wall tributary to the roof diaphragm is
Wp
5 wAw
5 87.50 3 5
5 438 lb/ft
For a rigid roof diaphragm, ASCE 7 Section 12.11.2.1 stipulates that
Lf
50
ka
5 1.0
and ASCE 7 Equation (12.11-1) reduces to
Fp
5 0.4SDS IeWp
5 0.4 3 0.4 3 1.0 3 438
5 70 lb/ft
The minimum permissible anchor force is
Fp
5 0.2IeWp
5 0.2 3 1.0 3 438
5 88 lb/ft . . . governs
The required anchor force is
Fp
5 88 lb/ft
Example 1-40
The 7-inch concrete wall shown in Figure 1-43 forms part of a building assigned to seismic design
category C. The 5-percent damped, design spectral response acceleration for a period of 0.2 second is
SDS 5 0.4g. If the roof diaphragm and the second-floor diaphragm may be considered rigid, determine
the anchorage force in the second-floor diaphragm.
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Solution
From Figure 1-43
h
5 height of roof above the base
5 20 ft
z
5 height of the second-floor diaphragm above the base
5 10 ft
The weight of the wall is
w
5 150 3 7/12
5 87.50 lb/ft2
The area of the wall tributary to the second-floor diaphragm is
Aw
5 1 3 hs
5 1 3 10
5 10 ft2/ft
The weight of the wall tributary to the second-floor diaphragm is
Wp
5 wAw
5 87.50 3 10
5 875 lb/ft
For seismic design category C, the anchor force on a rigid diaphragm is given by ASCE 7 Section
12.11.2.1 as
Fp
5 (0.4SDS Ie)(1 1 2z/h)Wp /3
5 (0.4 3 0.4 3 1.0)(1 1 2 3 10/20)875/3
5 93 lb
The minimum permissible force on the diaphragm is
Fp
5 0.2IeWp
5 0.2 3 1.0 3 875
5 175 lb . . . governs
The required seismic design force for the diaphragm is
Fp
5 175 lb
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1.30.3 Subdiaphragms and continuous ties
In seismic design categories C through F, to transfer anchorage forces across the complete depth of
the diaphragm and to prevent the walls and diaphragm from separating, ASCE 7 Section 12.11.2.2.1
requires the provision of continuous ties across the complete depth of the diaphragm. To reduce the
number of full depth ties required, subdiaphragms and added chords are used to transmit the anchorage
forces to the main continuous crossties. The maximum permitted length-to-width ratio of the subdiaphragm is 2.5 to 1.
In accordance with ASCE 7 Sections 12.11.2.2.3 and 12.11.2.2.4, neither plywood sheathing nor metal
deck may be considered effective as providing the ties. In addition, anchorage may not be accomplished by use of toenails or nails subject to withdrawal, nor may wood ledgers be used in cross-gain
bending. Connections must extend into the diaphragm a sufficient distance to develop the force transferred into the diaphragm.
Example 1-41
For the north-south direction, determine a suitable subdiaphragm arrangement for the plywood roof
diaphragm of the building shown in Figure 1-44. The plywood diaphragm may be considered flexible
with joists spaced at 40-foot centers and purlins spaced at 8-foot centers. The pull-out force on the
north and south walls in the north-south direction is p 5 300 lb/ft.
40 ft
40 ft
40 ft
N
24 ft
beams at 24 ft o.c.
joists at 40 ft o.c.
24 ft
24 ft
purlins at 8 ft o.c.
typical
24 ft
p = 300 lb/ft
Figure 1-44 Details for Example 1-41
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Solution
Provide three subdiaphragms, with dimensions b 5 40 feet and d 5 24 feet along the north and south
walls with crossties at 40-foot centers, as shown in Figure 1-45.
40 ft x 24 ft subdiaphragm
typical
continuous crosstie
d = 24 ft
b = 40 ft
Figure 1-45 Details of subdiaphragms
The aspect ratio of each subdiaphragm is
b/d
5 40/24
5 1.67 . . . complies with ASCE 7 Section 12.11.2.2.1
, 2.5
The purlins at 8-foot centers provide the subdiaphragm ties and the force in each is
Fp
5 300 3 8
5 2400 lb
The unit shear stress along the ends of the subdiaphragm is
q
5 pb/2d
5 300 3 40/(2 3 24)
5 250 lb/ft
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Seismic Design
The subdiaphragm chord force is
Pc
5 pb2/8d
5 300 3 402/(8 3 24)
5 2500 lb
The force in the crossties is
Pt
5 pb
5 300 3 40
5 12,000 lb
1.31 Architectural, mechanical, and electrical components supported
by structures
Nonstructural architectural components include nonbearing walls, wall elements, cantilevered parapet
walls, signs, ornamentation, chimneys, and penthouses. Mechanical and electrical components include
boilers, tanks, machinery, and nonbuilding structures supported by other structures. Design levels are
specified for components and their anchorage to ensure that life safety is not endangered and, in the
case of safety-related equipment, to ensure the continued function of essential facilities. The design
force for components must account for the dynamic response of the component to the motion of the
structure, the amplified response of equipment relative to the fundamental period of the structure, the
lack of redundancy and ductility in the component itself, and the weight of the component. These
effects are covered in ASCE 7 Chapter 13, and this section of the code is primarily concerned with the
design of attachments and supports that connect components to the structure.
In accordance with ASCE 7 Section 13.1.2, components are considered to have the same seismic
design category as the building in which they are located and based on ASCE 7 Section 13.1.3, are
allocated a component importance factor, Ip , as indicated in Table 1-25.
Table 1-25 Component importance factor
Component type
Ip
Life safety, required to function after an earthquake
1.5
Contains hazardous material
1.5
Required for continued operation of an Occupancy Category IV facility
1.5
All other
1.0
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Several exemptions are made from the requirements of ASCE 7 Chapter 13, Section 13.1.4, and these
are listed in Table 1-26.
Table 1-26 Exemptions to requirements
Seismic
design
category
Ip
Weight
Height
above floor
All
A
Any
Any
Any
Architectural (other than parapets)
B
1.0
Any
Any
Mechanical and electrical
B
Any
Any
Any
Mechanical and electrical
C
1.0
Any
Any
Mechanical and electrical*
D, E, F
1.0
≤ 400 lb
≤ 4 ft
Mechanical and electrical*
D, E, F
1.0
≤ 20 lb
Any
Component
* Flexible connections are provided between the components and associated ductwork, piping, and conduit and the
component is positively attached to the structure.
The design seismic force is given by ASCE 7 Equation (13.3-1), which is
where:
Fp
5 (0.4ap SDS Ip /Rp)(1 1 2z/h)Wp
Fp
5 seismic design force centered at the component’s center of gravity and
distributed relative to the component’s mass distribution
Ip
5 component importance factor given in ASCE 7 Section 13.1.3 and Table 1-25
SDS
5 5-percent damped, design spectral response acceleration for a period of
0.2 second
Wp
5 component operating weight
ap
5 component amplification factor from ASCE 7 Table 13.5-1 or 13.6-1
h
5 height of roof above the base
z
5 height of component at point of attachment
Rp
5 component response modification factor from ASCE 7 Table 13.5-1 or 13.6-1
In accordance with ASCE 7 Equation (13.3-2), Fp need not be taken greater than
Fp
5 1.6SDS IpWp
In accordance with ASCE 7 Equation (13.3-3), Fp shall not be taken less than
Fp
5 0.3SDS IpWp
As specified in ASCE 7 Section 13.3.1.2, the component shall be designed for a concurrent vertical
force of
Fpv
5 0.2SDSWp
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The component amplification factor represents the dynamic amplification of the component relative to
the fundamental period of the structure. The values assigned in ASCE 7 Table 13.5-1 or 13.6-1 to the
component amplification factor are dependant on the relative rigidity of the component.
The values assigned in ASCE 7 Table 13.5-1 or 13.6-1 to the component response modification factor
reflect the method of attachment of the component to the structure and its energy absorption capacity.
In accordance with ASCE 7 Section 15.4.9.2, anchors in masonry are designed in accordance with
TMS 402.13 As specified in ASCE 7 Section 15.4.9.1, anchors in concrete are designed in accordance
with ACI 318 Chapter 17.
Post-installed anchors in concrete are prequalified for seismic applications in accordance with ACI
355.214 or other approved qualification procedures. Post-installed anchors in masonry are prequalified
for seismic applications in accordance with approved qualification procedures.
In determining the design seismic force on a component, the value of the reliability factor, r, shall be
taken as unity, in accordance with ASCE 7 Section 13.3.1, and the overstrength factor of ASCE 7 Table
12.2-1 does not apply.
1.31.1 Design force on mechanical and electrical components
The design factors for mechanical and electrical components are tabulated in ASCE 7 Table 13.6-1,
and an abbreviated listing is given in Table 1-27.
Example 1-42
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California and
has a 5-percent damped, design spectral response acceleration for a period of 0.2 second of SDS 5
0.840g. The building is assigned to seismic design category D. An electrical transformer weighing 2
kips is mounted on the concrete roof of the building. A component importance factor of Ip 5 1.0 may
be assumed. Determine the design seismic force on the equipment.
Solution
The design seismic force is given by ASCE 7 Equation (13.3-1), which is
where:
Fp
5 (0.4apSDS Ip /Rp)(1 1 2z/h)Wp
Ip
5 component importance factor
5 1.0
SDS
5 5-percent damped, design spectral response acceleration for a period of
0.2 second
Seismic and Wind Forces: Structural Design Examples
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5 0.840g
Wp
5 component operating weight
5 2 kips
ap
5 component amplification factor from Table 1-27
5 1.0
h
5 height of roof above the base
5 24 ft
z
5 height of component at point of attachment
5 24 ft
Rp
5 component response modification factor from Table 1-27
5 2.5
and
Fp
5 (0.4 3 1.0 3 0.840 3 1.0/2.5)(1 1 2 3 24/24)Wp
5 0.40Wp
5 0.80 kip
Neither ASCE 7 Equation (13.3-2) nor (13.3-3) governs.
Table 1-27 Coefficients for mechanical and electrical components
Component
ap
Rp
Generators, motors, transformers
1.0
2.5
Communication equipment
1.0
2.5
Motor control centers, panel boards, switch gear
2.5
6.0
Air-conditioning units, fans, cabinet heaters
2.5
6.0
Boilers and furnaces
1.0
2.5
Skirt-supported pressure vessels
2.5
2.5
Elevator and escalator components
1.0
2.5
Roof-mounted stacks, etc., braced below the center of mass
2.5
3.0
Roof-mounted stacks, etc., braced above the center of mass
1.0
2.5
Electrical conduit and cable trays
2.5
6.0
Lighting fixtures
1.0
1.5
Seismic and Wind Forces: Structural Design Examples
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1.31.2 Design force on architectural components
The design factors for architectural components are tabulated in ASCE 7 Table 13.5-1, and an abbreviated listing is given in Table 1-28.
Table 1-28 Coefficients for architectural components
Component
ap
Rp
Nonstructural interior unreinforced masonry walls
1.0
1.5
Other nonstructural interior walls and partitions
1.0
2.5
Unbraced parapets
2.5
2.5
Nonstructural exterior wall element
1.0
2.5
Body of wall panel connecting system
1.0
2.5
Fasteners of the wall panel connecting system
1.25
1.0
Chimneys braced below center of mass
2.5
2.5
Chimneys braced above center of mass
1.0
2.5
Penthouses except where framed by the building frame
2.5
3.5
Signs and billboards
2.5
3.0
Flexible high deformability elements and attachments
2.5
3.5
Flexible limited deformability elements and attachments
2.5
2.5
Flexible low deformability elements and attachments
2.5
1.5
Rigid high deformability elements and attachments
1.0
3.5
Rigid limited deformability elements and attachments
1.0
2.5
Rigid low deformability elements and attachments
1.0
1.5
Ceilings
1.0
2.5
Example 1-43
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California and has
a 5-percent damped, design spectral response acceleration for a period of 0.2 second of SDS 5 0.840g.
The building is assigned to seismic design category D. A penthouse weighing 10 kips is located on
the roof of the building. A component importance factor of Ip 5 1.0 may be assumed. Determine the
design seismic force on the penthouse.
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Solution
The design seismic force is given by ASCE 7 Equation (13.3-1), which is
where:
Fp
5 (0.4apSDS Ip /Rp)(1 1 2z/h)Wp
Ip
5 component importance factor
5 1.0
SDS
5 5-percent damped, design spectral response acceleration for a period of
0.2 second
5 0.840g
Wp
5 component operating weight
5 10 kips
ap
5 component amplification factor from Table 1-28
5 2.50
h
5 height of roof above the base
5 24 ft
z
5 height of component at point of attachment
5 24 ft
Rp
5 component response modification factor from Table 1-28
5 3.5
and
Fp
5 (0.4 3 2.5 3 0.840 3 1.0/3.5)(1 1 2 3 24/24)Wp
5 0.72Wp
5 7.20 kips
Neither ASCE 7 Equation (13.3-2) nor (13.3-3) govern.
1.31.3 Wall cladding displacements
External wall cladding panels and their connections must be designed, in accordance with ASCE 7
Section 13.5.3, to accommodate the maximum inelastic seismic relative displacement, Dp , specified
in ASCE 7 Section 13.3.2, with a minimum value of 0.5 inch. As shown in Figure 1-46, the relative
seismic displacement is determined from ASCE 7 Equation (13.3-7) as
Dp
5 dxA 2 dyA
≤ (hx 2 hy)DaA /hsx . . . ASCE 7 Equation (13.3-8)
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where:
dxA
5 inelastic seismic displacement of the structure at level x, as determined
by ASCE 7 Equation (12.8-15)
dyA
5 inelastic seismic displacement of the structure at level y, as determined
by ASCE 7 Equation (12.8-15)
DaA
5 allowable story drift for the structure, as defined in Table 1-23
hsx
5 story height below level x
hx
5 height of the upper support attachment at level x, as measured from the base
hy
5 height of the lower support attachment at level y, as measured from the base
p
Figure 1-46 Seismic relative displacement, Dp
Example 1-44
Wall panels weighing 40 lb/ft2 are externally mounted on the two-story steel-frame building shown in
Figure 1-2 that is located in Orange County, California. The building has a 5-percent damped, design
spectral response acceleration for a period of 0.2 second of SDS 5 0.840g and is assigned to seismic
design category D. The panels project 3 feet above the roof and 3 feet below the second floor, as shown
in Figure 1-47. Determine the allowance required to accommodate seismic movements.
Solution
The relative displacement need not exceed the value given by ASCE 7 Equation (13.3-8), which is
Dp
5 (hx 2 hy)DaA /hsx
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where:
hx
127
5 height of the upper support attachment at level x, as measured from the base
5 24 ft
hy
5 height of the lower support attachment at level y, as measured from the base
5 12 ft
DaA
5 allowable story drift for the structure, as determined in Example 1-25
5 3.6 in
hsx
5 story height below level x
5 12 ft
and
Dp
≤ (hx 2 hy)DaA /hsx
5 (24 2 12)3.6/12
5 3.6 in
The value for the relative displacement given by ASCE 7 Equation (13.3-7) is
Dp
5 dxA 2 dyA
≤ (hx 2 hy)DaA /hsx
where:
dxA
5 inelastic seismic displacement of the structure at level x as determined in
Example 1-25
5 1.51 1 0.76
5 2.27 in
dyA
5 inelastic seismic displacement of the structure at level y as determined in
Example 1-25
5 1.51 in
and
Dp
5 2.27 2 1.51
5 0.76 in . . . governs
1.31.4 Wall cladding seismic forces
Wall panels, the connecting system, and fasteners in the connecting system must be designed for the
force, Fp , determined by ASCE 7 Equation (13.3-1), applied at the center of mass of the panel. The
panel and the body of the connecting system are designed for the force, Fp , determined by ASCE
7 Equation (13.3-1), using values of Rp 5 2.5 and ap 5 1.0. Fasteners in the connecting system are
designed for the force, Fp , determined by ASCE 7 Equation (13.3-1), using values of Rp 5 1.0 and
ap 5 1.25.
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Figure 1-47 Details for Examples 1-44 and 1-45
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Example 1-45
Wall panels weighing 40 lb/ft2 are externally mounted on the two-story steel-frame building shown in
Figure 1-2 that is located in Orange County, California. The building has a 5-percent damped, design
spectral response acceleration for a period of 0.2 second of SDS 5 0.840g and is assigned to seismic
design category D. The panels project 3 feet above the roof and 3 feet below the second floor, as shown
in Figure 1-47. Determine the out-of-plane design seismic force on (a) the wall panel, (b) the connecting system, and (c) the fasteners.
Solution
(a) The basic design seismic force on the wall panel is determined as the average of the forces calculated for the top and bottom connectors, as given by ASCE 7 Equation (13.3-1). The force at the
level of the top connectors is
where:
Fp
5 (0.4apSDS Ip /Rp)(1 1 2z/h)Wp
Ip
5 component importance factor
5 1.0
SDS
5 5-percent damped, design spectral response acceleration for a period of
0.2 second
5 0.840g
Wp
5 weight of panel
5 40 3 18 5 720 lb/ft
ap
5 component amplification factor from Table 1-28
5 1.0
h
5 height of roof above the base
5 24 ft
z
5 height of the top connectors
5 24 ft
Rp
5 component response modification factor from Table 1-28
5 2.5
and
Fp
5 (0.4 3 1.0 3 0.840 3 1.0/2.5)(1 1 2 3 24/24)Wp
5 0.403Wp
5 290 lb/ft
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At the level of the bottom connectors, z 5 12 feet and the force at the level of the bottom connectors is
Fp
5 (0.4 3 1.0 3 0.840 3 1.0/2.5)(1 1 2 3 12/24)Wp
5 0.269Wp
5 194 lb/ft
The minimum permissible force is given by ASCE 7 Equation (13.3-3) as
Fp
5 0.3SDS IpWp
5 0.3 3 0.840 3 1.0Wp
5 0.252Wp . . . does not govern
, 0.269Wp
Hence, the out-of-plane design seismic force acting at the centroid of the wall panel is
Fp
5 (290 1 194)/2
5 242 lb/ft
(b) For the connecting system, the values of the component amplification factor and component
response modification factor are the same as for the wall panel. Hence, the out-of-plane design
seismic force on both the top and bottom connecting systems is
Fp
5 242/2
5 121 lb/ft
(c) For the fasteners, the component amplification factor is ap 5 1.25 and the component response
modification factor is Rp 5 1.0. Hence, the out-of-plane design seismic force on both the top and
bottom fasteners is
Fp
5 121 3 2.5 3 1.25
5 378 lb/ft
1.32 Rigidity and torsion
In ASCE 7 Section 12.3.1.2, a rigid diaphragm is defined as a concrete slab or a concrete-filled metal
deck with a span-to-depth ratio of three or less in structures that have no horizontal irregularities. In
a building with rigid diaphragms, lateral force is distributed to the shear walls based on the relative
stiffness of the walls and the torsional displacements produced by the rigid-body rotation of the diaphragm and walls.
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1.32.1 Shear wall stiffness
The rigidity, or stiffness, of a concrete or masonry shear wall is the force required to produce unit displacement at the top of the wall. This is most readily obtained as the reciprocal of the deflection of the
wall due to unit load applied at the top. The deflection of a wall due to unit load, as shown in Figure
1-48, is the sum of the flexural and shear deflections and is given by
where
d
5 dF 1 dS
dF
5 deflection due to flexure
5 4(H/L)3/Et for a cantilever wall
5 (H/L)3/Et for a wall fixed at top and bottom
H
5 height of wall
L
5 length of wall
E
5 modulus of elasticity of wall
t
5 thickness of wall
dS
5 deflection due to shear
5 1.2H/GA
5 3(H/L)/Et
G
5 rigidity modulus of wall
5 0.4E
A
5 cross-sectional area of wall
5 tL
δ
1.0
H
L
Figure 1-48 Shear wall stiffness
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The stiffness of the wall is given by
s
5 1/d
An opening in a wall reduces its stiffness and the stiffness may be determined by the following technique. The deflection of the wall is first obtained as though it is a solid wall. From this is subtracted
the deflection of that portion of the wall that contains the opening. The deflection of each wall, formed
by the openings, is now added back.
Example 1-46
Determine the stiffness of the concrete wall shown in Figure 1-49. The wall is 8 inches thick, with a
modulus of elasticity of Ec 5 3000 kips/in2, and is fixed at the top and bottom.
3
3 ft
1
4
2
4 ft
3 ft
4 ft
8 ft
Figure 1-49 Details for Example 1-46
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Solution
The relevant details are shown in Table 1-29.
Table 1-29 Wall stiffness
Wall
H
L
Type
(H/L)3
5 EtdF
3H/L
5 Etds
Et(dF 1 ds )
5 Etd
s/Et
S Wall
11
11
Fixed
1
3
4.000
—
11214
8
11
Fixed
20.385
22.182
22.566
—
1
8
4
Fixed
8
6
—
0.071
2
8
4
Fixed
8
6
—
0.071
112
—
—
—
—
—
7.042
← 0.142
Total
—
—
—
—
—
8.476 →
0.118
The actual stiffness of the wall is
s
5 0.118Et
5 0.118 3 3000 3 8
5 2832 kips/in
1.32.2 Rigid diaphragm
Figure 1-50 shows a single-story building with a rigid roof diaphragm supported on four shear walls.
The center of mass of the building is shown as point CM, and this is the point through which the seismic base shear, V, acts. The center of mass is obtained by taking statical moments of the wall and roof
weights about a convenient origin. From Figure 1-50, the center of mass is located a distance from
wall 1 given by
x
5 SWx/SW
5 (WR 3 B/2 1 W1 3 0 1 W2 3 B/2 1 W3 3 B 1 W4 3 B/2)/(WR 1 W1 1
W2 1 W3 1 W4)
where:
WR
5 weight of roof
Wi
5 weight of wall i
In this instance, from the symmetry of the walls and the roof, the center of mass lies midway between
walls 1 and 3 and
x
5 B/2
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Similarly, the center of mass is located a distance from wall 4, given by
y
5 SWy/SW
5 (WR 3 L/2 1 W1 3 L/2 1 W2 3 L 1 W3 3 L/2 1 W4 3 0)/(WR 1 W1 1
W2 1 W3 1 W4)
r3
B
r1
FT2
FS2
2
N
r2
3
V
+ CM
+ CR
=
1
r4
4
+
+
FT3
V +
FT1
L
+ T
FT4
FS4
(a) Layout
+
(b) Translation
(c) Rotation
Figure 1-50 Torsional effects
The center of rigidity is shown as point CR, and this is the point about which the structure rotates
when subjected to a torsional moment. The location of the center of rigidity is obtained by taking
statical moments of the wall rigidities about a convenient origin. For seismic loads in the north-south
direction, walls 2 and 4, which have no stiffness in this direction, are omitted, and only walls 1 and 3
are considered. From Figure 1-50, by taking moments of the wall stiffness about wall 1, the center of
rigidity is located a distance from wall 1 given by
r1
5 Ssyx/Ssy
5 (s3 3 B 1 s1 3 0)/(s1 1 s3)
5 s3B/(s1 1 s3)
Similarly, the center of rigidity is located a distance from wall 4 given by
r4
5 Ssxy/Ssx
5 (s2 3 L 1 s4 3 0)/(s2 1 s4)
5 s2L/(s2 1 s4)
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The polar moment of inertia of the walls is given by
J
5 Sr2i si
5 r12s1 1 r22s2 1 r32s3 1 r42s4
where:
si
5 stiffness of wall i
sx
5 stiffness of a wall in the x-direction (east-west)
sy
5 stiffness of a wall in the y-direction (north-south)
ri
5 distance of the centroid of wall i from the center of rigidity
The torsional moment acting on the building is
where:
T
5 Ve
e
5 eccentricity of the center of mass with respect to the center of rigidity
For east-west seismic force
The displacement of the building consists of an east-west translation and a clockwise rotation about
the center of rigidity. As shown in Figure 1-50(b), the translation produces in-plane forces in shear
walls 2 and 4 proportional to their relative translational stiffness. These forces are given by
and
FSi
5 Vsi /Ssy
FS2
5 Vs2 /(s2 1 s4)
FS4
5 Vs4 /(s2 1 s4)
No forces are produced in shear walls 1 and 3 by this translation.
The clockwise rotation produces forces in all four walls, proportional to their torsional stiffness and
distance from the center of rigidity, as shown in Figure 1-50(c). These forces are given by
and
FTi
5 Tri si /J
FT1
5 Tr1s1/J
FT2
5 Tr2s2 /J
FT3
5 Tr3s3 /J
FT4
5 Tr4s4 /J
The total force in a wall is
F
5 FS 1 FT
For the direction of V shown in Figure 1-50, FS and FT are additive in wall 2 and are of opposite sense
in wall 4.
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In a perfectly symmetrical building, the centers of mass and rigidity coincide and torsion is not produced. However, the exact locations of the centers of mass and rigidity are uncertain. The calculated
location of the center of mass may not be exact due to the distribution of structure weight being
imprecisely known. Similarly, inaccuracies in calculating the rigidity of shear walls and the neglect
of nonstructural components, such as partitions and stairs, lead to the inexact location of the center
of rigidity. Hence, accidental eccentricity may in fact exist even in a nominally symmetric structure.
Torsion resulting from this accidental eccentricity is referred to as accidental torsion. To account for
accidental torsion, ASCE 7 Section 12.8.4.2 specifies that the center of mass is assumed displaced
each way from its actual location by a distance equal to 5 percent of the building dimension perpendicular to the direction of the applied force.
When a building assigned to seismic design categories C through F has a torsional irregularity, as
defined in ASCE 7 Table 12.3-1 (horizontal structural irregularity type 1a or 1b), the accidental torsion
is amplified as specified in ASCE 7 Section 12.8.4.3. This is to account for the possibility of unsymmetrical yielding of the perimeter vertical seismic-force-resisting elements resulting in a large increase
in torsional effects. The amplification factor is given by ASCE 7 Section 12.8.4.3 as
Ax
5 (dmax /1.2davg)2
≤ 3.0
≥ 1.0
where:
dmax
5 maximum displacement at level x computed assuming Ax 5 1
davg
5 average of displacements at extreme points of the structure at level x
computed assuming Ax 5 1
Accidental torsion is applied to all structures to determine if a horizontal structural irregularity exists,
as defined in ASCE 7 Table 12.3-1. For a structure assigned to seismic design category C through F
with type 1a horizontal structural irregularity or a structure assigned to seismic design category B
through D with type 1b horizontal structural irregularity, accidental torsion is included in the determination of seismic forces and story drift. In accordance with ASCE 7 Section 12.3.3.1, structures
assigned to seismic design category E or F that have horizontal structural irregularity type 1b are not
permitted.
Example 1-47
Determine the force acting on shear wall 2 of the building shown in Figure 1-50 for a base shear of
V 5 40 kips. The building dimensions and the relative shear wall stiffness are
L
5 42 ft
B
5 20 ft
s4
52
s1 5 s2 5 s3 5 1
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The diaphragm is rigid and the building mass is symmetrically disposed about the centerlines of the
building. The building is assigned to seismic design category D.
Solution
From the symmetry of the structure, for an east-west seismic load, the center of mass is located midway between walls 2 and 4 and its distance from wall 4 is
y
5 42/2 5 21 ft
In locating the center of rigidity for an east-west seismic load, walls 1 and 3, which have no stiffness in
the east-west direction, are omitted. Taking moments about wall 4, the distance of the center of rigidity
from wall 4 is given by
r4
5 Ssxy/Ssx
5 (1 3 42 1 2 3 0)/(1 1 2)
5 14 ft
The distance of the center of rigidity from wall 2 is
r2
5 42 2 14
5 28 ft
In locating the center of rigidity for a north-south seismic load, walls 2 and 4, which have no stiffness
in the north-south direction, are omitted. Due to the symmetry of walls 1 and 3, the center of rigidity
is located midway between walls 1 and 3 and
r1
5 r3
5 10 ft
The polar moment of inertia of the walls is
J
5 Sr2i si
5 r12 3 s1 1 r22 3 s2 1 r32 3 s3 1 r42 3 s4
5 102 3 1 1 282 3 1 1 102 3 1 1 142 3 2
5 1376 ft2
For a seismic load in the east-west direction, the eccentricity is
ey
5 y 2 r4
5 21 2 14
5 7 ft
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Accidental eccentricity, in accordance with ASCE 7 Section 12.8.4.2, is
ea
5 0.05 3 L
5 0.05 3 42
5 2.1 ft
An accidental displacement of the center of mass to the north gives the maximum eccentricity of
e
5 ey 1 ea
5 7 1 2.1
5 9.1 ft
The maximum eccentricity governs for the force in wall 2 since the torsional force and the in-plane
force are of the same sense and are additive.
The maximum torsional moment acting about the center of rigidity is
T
5 Ve
5 40 3 9.1
5 364 kip-ft
The force produced in a wall by the base shear acting in the east-west direction is the algebraic sum of
the in-plane shear force and the torsional shear force.
The sum of the wall rigidities for a seismic load in the east-west direction is
Ssx
5 s2 1 s4
5112
53
The in-plane shear force is
FSi
5 Vsx /Ssx
5 40sx /3
5 13.33sx
The maximum torsional shear force is
FTi
5 Tri si /J
5 364ri si /1376
5 0.265ri si
The total force in a wall is
F
5 FS 1 FT
with a negative value for FT indicating that the torsional force is opposite in sense to the in-plane force.
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The total forces produced in walls 2 and 4 by the maximum torsional moment are
F2
5 13.33 3 s2 1 0.265 3 r2 3 s2
5 13.33 3 1 1 0.265 3 28 3 1
5 20.75 kips
F4
5 13.33 3 s4 2 0.265 3 r4 3 s4
5 13.33 3 2 2 0.265 3 14 3 2
5 19.24 kips
Check for torsional irregularity
To determine if amplification of the torsional moment is necessary, the displacements of walls 1 and
4 must be determined.
The relative displacement of a wall is given by
d
5 F/s
The relative displacements of walls 2 and 4 are
d2
5 20.75/1
5 20.75
d4
5 19.24/2
5 9.62
The ratio of the maximum displacement of wall 2 to the average displacement of walls 2 and 4 is
m
5 2d2 /(d2 1 d4)
5 2 3 20.75/30.37
5 1.37
, 1.40
. 1.20
This constitutes a torsional irregularity type 1a, as defined in ASCE 7 Table 12.3-1, and for a structure
assigned to seismic design category D, the accidental eccentricity must be amplified, as specified in
ASCE 7 Section 12.8.4.3, by the factor
Ax
5 (m/1.2)2
5 (1.37/1.2)2
5 1.30
, 3.00 . . . satisfactory
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The revised accidental eccentricity is
e9
5 6Ax ea
5 61.30 3 2.1
5 62.73 ft
The revised maximum eccentricity for a displacement of the center of mass to the north is
e0
5 ey 1 e9
5 7 1 2.73
5 9.73 ft
The amplified torsional moment is
T9
5 Ve0
5 40 3 9.73
5 389 kip-ft
The revised total force produced in wall 2 by the maximum amplified torsional moment is
F2A
5 FS 1 FTA
5 Vs2 /Ssx 1 T9r2s2 /J
5 40 3 1/3 1 389 3 28 3 1/1376
5 21.25 kips
1.33 Modal analysis procedure
The equivalent lateral force procedure is applicable to structures that are of essentially regular construction, possess a uniform distribution of mass and stiffness, and are without irregular features.
When these conditions are satisfied, the equivalent lateral force procedure provides a reasonable envelope of the forces and deformations due to the actual dynamic response. Structures that posses plan
or vertical irregularities may require a modal analysis or dynamic analysis procedure to determine an
accurate distribution of seismic forces. ASCE 7 Tables 12.3-1 and 12.3-2 define six possible horizontal
and seven possible vertical structural irregularities.
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1.33.1 Horizontal structural irregularities
The six types of horizontal structural irregularities illustrated in Figure 1-51 are:
1a. torsional irregularity, which exists when the maximum story drift at one end of a rigid diaphragm, including accidental torsion with Ax 5 1.0, exceeds 1.2 times the average story drift
1b. extreme torsional irregularity, which exists when the maximum story drift at one end of a
rigid diaphragm, including accidental torsion with Ax 5 1.0, exceeds 1.4 times the average
story drift
2.
re-entrant corners, where both projections of the structure beyond a re-entrant corner exceed
15 percent of the plan dimension of the structure in the given direction
3.
diaphragm discontinuity, where the area of an opening exceeds 50 percent of the area of the
diaphragm or where the diaphragm stiffness from one story to the next changes by more than
50 percent
4.
out-of-plane offsets, where there is a discontinuity in the lateral-force-resistance path such as
an out-of-plane offset of at least one of the vertical elements
5.
nonparallel systems, where the vertical lateral-force-resisting elements are not parallel to the
major orthogonal axes of the lateral-force-resisting system
Additional design requirements are imposed on structures with horizontal irregularities, depending on
their seismic design category, and these are given in Table 1-30.
Extreme torsional
Diaphragm discontinuity
Out-of-plane offset
Re-entrant corners
Nonparallel systems
Figure 1-51 Horizontal structural irregularities
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Table 1-30 Additional design requirements for horizontal irregularities
Seismic
design
category
Irregularity
1a. Torsional (Applies only
to structures with rigid or
semirigid diaphragms)
Additional design requirements
B, C, D, E, F
Design required using a three-dimensional representation (12.7.3) (16.3.4).
C, D, E, F
Amplification of torsion required (12.8.4.3).
Story drift is determined from the largest difference in deflection along the top and bottom
edges of the story (12.8.6).
1b. Extreme torsional
(Applies only to structures
with rigid or semirigid
diaphragms)
2. Re-entrant corners
D, E, F
Equivalent lateral force procedure not permitted
(12.6).
D, E, F
An increase of 25% is required in the calculated
design forces for connections of diaphragms to
collectors and vertical elements, and for connections of collectors to vertical elements (12.3.3.4).
B, C, D
Design required using a three-dimensional representation (12.7.3) (16.3.4).
C, D
Amplification of torsion required (12.8.4.3).
C, D, E, F
Story drift is determined from the largest difference in deflection along the top and bottom
edges of a story (12.8.6).
D
Equivalent lateral force procedure not permitted
(12.6) and redundancy factor 5 1.3 (12.3.5.2).
D, E, F
An increase of 25% is required in the calculated
design forces for connections of diaphragms to
collectors and vertical elements, and for connections of collectors to vertical elements (12.3.3.4).
E, F
Not permitted (12.3.3.1).
D, E, F
An increase of 25% is required in the calculated
design forces for connections of diaphragms to
collectors and vertical elements, and for connections of collectors to vertical elements (12.3.3.4).
Equivalent lateral force method not permitted
(12.6).
(continued)
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Table 1-30 Additional design requirements for horizontal irregularities—continued
Seismic
design
category
Irregularity
3. Diaphragm discontinuity
D, E, F
Additional design requirements
An increase of 25% is required in the calculated
design forces for connections of diaphragms to
collectors and vertical elements, and for connections of collectors to vertical elements (12.3.3.4).
Equivalent lateral force method not permitted
(12.6).
4. Out-of-plane offsets
5. Nonparallel system
B, C, D, E, F
Columns, beams, trusses, or slabs supporting
discontinuous walls or frames shall be designed
for the special seismic load combinations of
ASCE 7 Section 12.4.3 (12.3.3.3).
B, C, D, E, F
Design required using a three-dimensional representation (12.7.3) (16.3.4) except for structures
with flexible diaphragms.
D, E, F
An increase of 25% is required in the calculated
design forces for connections of diaphragms to
collectors and vertical elements, and for connections of collectors to vertical elements (12.3.3.4).
B, C, D, E, F
Design required using a three-dimensional representation (12.7.3) (16.3.4).
C, D, E, F
Design required for 100% of forces for one
direction plus 30% of the forces for the perpendicular direction (12.5.3).
Example 1-48
A three-story office building with special moment-resisting frames in the north-south direction and
eccentrically braced frames in the east-west direction is located in Orange County, California, and
is assigned to seismic design category D. At each level, the dead load (W), stiffness (k), and shear
strength (v) in each frame in the north-south direction are indicated in Figure 1-52. The fundamental
period of the building is T , 3.5TS. Identify the horizontal irregularities for the building and indicate
additional code requirements and procedures required for each.
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Figure 1-52 Details for Examples 1-48 and 1-49
Solution
The lateral-force-resisting system in the east-west direction above the first story consists of braced
frames located on the tower section walls. In the first story, under the tower, no bracing is provided and
this out-of-plane offset of the vertical elements constitutes a horizontal irregularity type 4. ASCE
7 Section 12.3.3.4 specifies that, for this irregularity, connection of diaphragm and collectors to the
vertical elements shall be designed for an increase of 25 percent in the calculated design forces. In
addition, ASCE 7 Section 12.3.3.3 requires that the first-story columns under the tower section, which
support the discontinuous braced frames in story two and above, shall be especially designed and
detailed for the load combinations given in ASCE 7 Sections 2.3.6 and 12.4.3, which are
U
5 1.2D 1 f1L 1 1.0Em 1 0.2S
5 (1.2 1 0.2SDS)D 1 W0QE 1 f1L 1 0.2S
and
U
5 0.9D 1 1.0Em
5 (0.9 2 0.2SDS)D 1 W0QE
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where:
145
D
5 dead load
L
5 floor live load
f1
5 1.0 for floors in garages and places of public assembly and for floor loads
in excess of 100 lb/ft2
5 0.5 for other live loads
Em
5 maximum effect of horizontal and vertical earthquake forces that can be
developed in an element as defined in ASCE 7 Equations (12.4-5) and
(12.4-7)
5 W0QE 6 0.2SDS D
W0
5 structure overstrength factor given in ASCE 7 Table 12.2-1 and tabulated
for an abbreviated number of structures in Table 1-16
5 amplification factor to account for the overstrength of the structure in the
inelastic range
QE
5 effect of horizontal seismic forces
SDS
5 5-percent damped, design spectral response acceleration for a period of 0.2
second
At the third floor of the tower, the gross area of the diaphragm is
Ag
5 25 3 25
5 625 ft2
The area of the opening in the third floor diaphragm is
Ao
5 18 3 18
5 324 ft2
. 0.5 3 Ag
This constitutes a horizontal irregularity type 3 and ASCE 7 Section 12.3.3.4 requires an increase
of 25 percent in the calculated design forces for connections of diaphragms to collectors and vertical
elements, and for connections of collectors to vertical elements.
In the east-west direction, the projection of the structure beyond the re-entrant corner is 50 percent of
the plan dimension of the structure in the east-west direction. In the north-south direction, the projection of the structure beyond the re-entrant corner is 50 percent of the plan dimension of the structure
in the east-west direction. Since both of these values exceed 15 percent, this constitutes a horizontal
irregularity type 2. ASCE 7 Section 12.3.3.4 specifies that this irregularity requires an increase of 25
percent in the calculated design forces for connections of diaphragms to collectors and vertical elements, and for connections of collectors to vertical elements.
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1.33.2 Vertical structural irregularities
The seven types of vertical structural irregularities illustrated in Figure 1-53 are:
1a. stiffness—soft story, which exists when the stiffness of one story is less than 70 percent of the
stiffness of the story above or less than 80 percent of the average stiffness of the three stories
above
1b. stiffness—extreme soft story, which exists when the stiffness of one story is less than 60
percent of the stiffness of the story above or less than 70 percent of the average stiffness of the
three stories above
2.
weight (mass) irregularity, which exists when the mass of any story is more than 150 percent of the mass of an adjacent story. A roof that is lighter than the floor below need not be
considered
3.
vertical geometric irregularity, where the horizontal dimension of the lateral-force-resisting
system is more than 130 percent of that in an adjacent story
4.
in-plane discontinuity, where an in-plane offset of the lateral-force-resisting elements is
greater than the length of those elements
5a. discontinuity in lateral strength—weak story, where the lateral strength of a story is less
than 80 percent of that in the story above
5b. discontinuity in lateral strength—extreme weak story, where the lateral stength of a story
is less than 65 percent of that in the story above
ASCE 7 Section 12.3.2.2 exempts one-story buildings in any seismic design classification and twostory buildings in seismic design categories A through D from the consideration of vertical irregularity
types 1a, 1b, and 2. These irregularities may also be ignored when no story drift ratio is greater than
130 percent of the story drift ratio of the next story above.
In accordance with ASCE 7 Section 12.3.3.2, buildings in seismic design category B or C with vertical
irregularity type 5b shall not be over two stories or 30 feet in height. However, where the weak story
can resist a seismic force not less than W0 times the calculated design force, the height limitation does
not apply. Buildings in seismic design category D, E, or F with vertical irregularity type 5b are not
permitted.
Additional design requirements are imposed on structures with vertical irregularities, depending on
their seismic design category, and these are given in Table 1-31.
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Stiffness—soft story
Stiffness—extreme soft story
Vertical geometric irregularity
Discontinuity in lateral strength—weak story
147
Weight (mass) irregularity
In-plane discontinuity
Discontinuity in lateral strength—extreme weak story
Figure 1-53 Vertical structural irregularities
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Table 1-31 Additional design requirements for vertical irregularities
Seismic
design
category
Irregularity
Additional design requirements
1a. Soft story
D, E, F
Equivalent lateral force procedure not permitted
(12.6).
1b. Extreme-soft story
D
Equivalent lateral force procedure not permitted
(12.6).
E, F
Not permitted (12.3.3.1).
2. Mass
D, E, F
Equivalent lateral force procedure not permitted
(12.6).
3. Geometric
D, E, F
Equivalent lateral force procedure not permitted
(12.6).
4. Discontinuity
B, C, D, E, F
Columns, beams, trusses, or slabs supporting
discontinuous walls or frames shall be designed
for the special seismic load combinations of
ASCE 7 Section 12.4.3.2 (12.3.3.3).
D, E, F
An increase of 25% is required in the calculated
design forces for connections of diaphragms to
collectors and vertical elements, and for connections of collectors to vertical elements (12.3.3.4).
Equivalent lateral force procedure not permitted
for structures with T ≥ 3.5TS (12.6).
5a. Weak story
5b. Extreme weak story
D
Equivalent lateral force procedure not permitted
for structures with T ≥ 3.5TS (12.6).
E, F
Not permitted (12.3.3.1).
D, E, F
Not permitted (12.3.3.1).
B, C
Maximum height two stories unless designed for
W0 forces (12.3.3.2).
Example 1-49
A three-story office building with special moment-resisting frames in the north-south direction and
eccentrically braced frames in the east-west direction is located in Orange County, California, and is
assigned to seismic design category D. At each level, the dead load (W), the stiffness (k), and the shear
strength (v) in each frame in the north-south direction are indicated in Figure 1-52. The fundamental
period of the building is T , 3.5TS. Identify the vertical irregularities for the building and indicate
additional code requirements and procedures required for each.
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Solution
The total stiffness of the second story in the north-south direction is
k2
5 2 3 200
5 400 kips/in
The total stiffness of the first story in the north-south direction is
k1
5 60 1 2 3 100
5 260 kips/in
5 65% 3 k2
, 70% 3 k2
. 60% 3 k2
Hence, the first story constitutes a stiffness–soft story and is considered a vertical irregularity type
1a in ASCE 7 Table 12.3-2. ASCE 7 Section 12.6 requires the structure to be designed using the modal
analysis procedure.
The effective mass of the second story is
W2
5 300 kips
The effective mass of the first story is
W1
5 600 kips
. 150% 3 W2
Hence, this constitutes a vertical irregularity type 2 and the additional code requirements are identical to those given for the vertical irregularity type 1a.
In the east-west direction, the horizontal dimension of the lateral-force-resisting system in the second
story is
L2
5 25 feet
The horizontal dimension of the lateral-force-resisting system in the first story is
L1
5 50 feet
. 130% 3 L2
Hence, this constitutes a vertical irregularity type 3 and the additional code requirements are identical to those given for the vertical irregularity type 1a.
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The total shear strength of the second story in the north-south direction is
v2
5 2 3 400
5 800 kips
The total shear strength of the first story in the north-south direction is
v1
5 150 1 2 3 200
5 550 kips
5 69% 3 v2
, 80% 3 v2
. 65% 3 v2
Hence, the first story constitutes a discontinuity in lateral strength–weak story and is considered a
vertical irregularity type 5a.
1.33.3 Selection of lateral force procedure
Structural irregularities produce seismic loads that may differ significantly from the loads that are predicted by the elastic lateral force procedure. Inelastic demand can concentrate in the area of the irregularity, resulting in the failure of structural elements in these regions. The elastic lateral force procedure
is unable to predict the stress concentrations produced in an irregular structure.
The modal analysis procedure is suitable for calculating the response of complex multipledegrees-of-freedom structures to earthquake motion. The structural response is modeled as the maximum response of a number of single-degree-of-freedom oscillators, each representing a specific mode
of vibration of the actual structure. Combining the responses of the individual modes produces the equivalent external forces and base and story shears, which may then be used in the same manner as in the
equivalent lateral force procedure. The modal analysis procedure has the advantage of determining the
actual distribution of lateral forces, from the actual mass and stiffness distribution over the height of an
irregular structure, which may differ appreciably from the simplified linear distribution assumed in the
equivalent lateral force method. In addition, it accounts for the effects of the higher modes of response
of a structure, some of which may contribute significantly to the overall response of the structure.
As specified in ASCE 7 Table 12.6-1, the equivalent lateral force method may be used in the design of
a structure under the following conditions:
•
the building is assigned to seismic design category B or C
•
the building is assigned to seismic design category D, E, or F and is of light-frame construction
•
the building is assigned to seismic design category D, E, or F with a risk category of I or II and
is of any construction not exceeding two stories in height
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•
the building is assigned to seismic design category D, E, or F with a height not exceeding 160
feet and is a regular building
•
the building is assigned to seismic design category D, E, or F with a height not exceeding 160
feet and has neither horizontal irregularities 1a (torsional) or 1b (extreme torsional) nor vertical
irregularities 1a (soft story), 1b (extreme-soft story), 2 (mass), or 3 (geometric)
•
the building is assigned to seismic design category D, E, or F with a height exceeding 160 feet,
a fundamental period T , 3.5TS , and with no structural irregularities
As specified in ASCE 7 Table 12.6-1, a modal analysis is necessary for all other structures assigned to
seismic design categories D, E, and F.
The determination of the necessary lateral force analysis procedure is illustrated in Figure 1-54.
Exempt
A
Select seismic
design
category
B, C
Use equivalent
lateral force
procedure
D, E, F
Seismic design
Structures of light-frame
category
construction or risk
Yes
No E or F with horizontal Yes
category I or II buildings
irregularity 1b or
of 2 stories or less
vertical irregularity 1b, 5
No
Not permitted
Seismic design
Yes
category D with
vertical irregularity 5b
No
No Horizontal irregularity
1, vertical irregularity
1, 2, 3
No
Yes
Use modal
analysis procedure
Height > 160 ft
Yes
No
Regular building
T < 3.5Ts
Yes
Use equivalent
lateral force
procedure
Figure 1-54 Selection of analysis procedure
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1.33.4 Modal shapes
The multistory structure shown in Figure 1-55 may be idealized as a multistory shear building by
assuming that the mass is lumped at the floor and roof diaphragms, the diaphragms are infinitely rigid,
and the columns are axially inextensible but laterally flexible. The dynamic response of the system is
represented by the lateral displacements of the lumped masses with the number of degrees of dynamic
freedom, or modes of vibration, n, being equal to the number of masses. The resultant response of the
system is given by the superposition of the responses of each lumped mass. Each individual mode of
vibration has its own frequency and may be represented by a single-degree-of-freedom system of the
same period, and each mode shape, or eigenvector, remains of constant relative shape, regardless of
the amplitude of the displacement. The actual amplitudes must be obtained from the initial conditions.
Figure 1-55 shows the four modes of the four-story building. The mode of vibration with the longest
period (lowest frequency) is termed the first fundamental mode. Modes with shorter periods (higher
frequencies) are termed higher modes or harmonics.
Figure 1-55 Modal shapes
A modal analysis procedure may be utilized to determine the dynamic response of a multipledegrees-of-freedom structure.9 Since each degree of dynamic freedom provides one equation of
dynamic equilibrium, the resultant vibration of the system consists of n such equations and may be
expressed in matrix form, for undamped free vibrations, as
{0}
5 [M]{ẍ} 1 [K]{x}
For simple harmonic motion, this reduces to
{0}
5 ([K] 2 w2[M]){x}
This expression is a representation of the eigenvalue equation with
[K]
5 stiffness matrix of the system
[M]
5 mass matrix, the diagonal matrix of lumped masses
{x}
5 eigenvector or mode shape associated with the eigenvalue w
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The eigenvalue equation has a nontrivial solution only if the determinant of the coefficient matrix is
zero. Thus, the frequency determinant is
[K] 2 w2[M] 5 0
Expansion of this determinant yields the characteristic polynomial of degree n in (w2), the roots
of which provide the eigenvalues, and from the eigenvalues, the corresponding natural periods are
obtained. Back substituting the eigenvalues in the eigenvalue equation yields the eigenvectors for each
mode.
Example 1-50
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Determine the natural periods of vibration and the eigenvectors for each of the two modes of vibration. The
relevant details are shown in Figure 1-56.
Figure 1-56 Details for Examples 1-50 and 1-51
Solution
Unit shear displacement is imposed on each node in turn, and the coefficient kij of the stiffness matrix
is obtained as the force produced at node i by a unit displacement at node j. The stiffness matrix is then
[K]
 k
k12 
11


5
 k21 k22 


 (30 + 30) −30 

5
30 
−30

 60 −30 
5

 −30 30 
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The mass at each node is given by
m1
5 w1/g
5 51.2/386.4
5 0.133 kip-sec2/in
m2
5 w2 /g
5 25.6/386.4
5 0.066 kip-sec2/in
The diagonal mass matrix is
[M]
 m
0.0 

 0.0 m2 


5
1
 0.133
5

0.0
0.0 

0.066 
The eigenvalue equation is
{0}
5 ([K] 2 w2[M]){f}
  60 −30 
 0.133 0.0    x1 
2

 
−ω 
 0.0 0.066    x 2 
  −30 30 
5 
The frequency determinant is
T
5
(60 − 0.133ω 2 )
−30
−30
(30 − 0.066ω 2 )
5 0.0088w4 2 7.95w2 1 900
Equating this polynomial in w to zero provides the circular natural frequencies for the two modes of
vibration and these are
w1
5 11.52 radians/sec
w2
5 27.76 radians/sec
The corresponding natural periods are
T1
5 2p/w1
5 0.545 sec
T2
5 2p/w2
5 0.226 sec
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Substituting the value for w1 in the eigenvalue equation gives
and
 42.35
0
5
f11
5 0.708f21
 −30


−30   φ11 

21.24   φ21 


Substituting the value for w2 in the eigenvalue equation gives
and
 −42.79


−30   φ12 

21.01   φ22 


0
5
f12
5 20.701f22

−30
The eigenvectors or matrix of relative modal shapes is
[Φ]
 φ
5
21
 φ11

φ22 

φ12 

 1.00
1.00 

 0.708 −0.701 
5
1.33.5 Modal participation factor
Numerical methods15 may be used to facilitate the modal analysis procedure. For a given mode of
vibration, the participation factor is defined by
where:
Pm
5 Swi fim /Swi f2im
Pm
5 participation factor associated with the specific mode m
wi
5 seismic dead load at floor level i
fim
5 mode shape component for node point i for the given mode m and the
summation extends over all the nodes in the structure
For a specific system, the participation factors have the property
SPmf1m
where:
f1m
5 1.0
5 mode shape component, for the first node of the system, of the eigenvector
associated with the specific mode
The effective modal gravity load, associated with the specific mode m, is defined by
Wm
5 (Swi fim)2/Swi f2im
5 PmSwi fim
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The higher modes do not contribute significantly to the total response of the structure and only the significant modes need be included to obtain an acceptable degree of accuracy in the modal analysis. As
specified in ASCE 7 Section 12.9.1, this may be achieved by including a sufficient number of modes
to ensure that 90 percent of the participating mass of the structure, in each orthogonal horizontal direction, is included in the calculation.
The total structure weight is given by
W
5 Swi
The relationship between effective modal gravity load and total structure weight is given by9, 10, 15
where:
SWm
5W
SWm
5 sum of the effective modal gravity load for all modes
This provides a method of satisfying the requirement that sufficient modes are included in the analysis
to ensure that 90 percent of the structural mass participates in the derivation of the response parameters. Thus, sufficient modes may be defined to ensure that the sum of their effective weights is
SWm
≥ 0.9W
By this means, a minimum of 90 percent of the structural mass participates in the determination of the
response parameters.
Example 1-51
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California.
For the north-south direction, determine the number of modes of vibration that must be included in
the analysis to satisfy the requirements of ASCE 7 Section 12.9.1. The relevant details are shown in
Figure 1-56.
Solution
The relevant values for the first mode are shown in Table 1-32.
Table 1-32 Details for the first mode
wi
fi1
wi fi1
wi f2i1
Roof
25.60
1.000
25.60
25.60
2nd Floor
51.20
0.708
36.25
25.67
Total
76.80
–
61.85
51.27
Level
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From Table 1-32, the participation factor for the first mode is given by
P1
5 Swi fi1/Swif2i1
5 61.85/51.27
5 1.21
The effective modal gravity load for the first mode is given by
W1
5 P1Swi fi1
5 1.21 3 61.85
5 74.84 kips
As a percentage of the structural weight, the effective modal gravity load for the first mode is
100W1/W 5 100 3 74.84/76.80
5 97.45%
. 90%
Hence, consideration of the first mode is adequate to satisfy ASCE 7 Section 12.9.1.
The relevant values for the second mode are shown in Table 1-33.
Table 1-33 Details for the second mode
wi
fi2
wi fi2
wi f2i2
Roof
25.60
1.000
25.60
25.60
2nd Floor
51.20
20.701
235.89
25.16
Total
76.80
–
210.29
50.76
Level
From Table 1-33, the participation factor for the second mode is given by
P2
5 Swi fi2 /Swif2i2
5 210.29/50.76
5 20.203
The effective modal gravity load for the second mode is given by
W2
5 P2Swifi2
5 20.203 3 210.29
5 2.09 kips
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As a percentage of the structural weight, the effective modal gravity load for the second mode is
100W2 /W 5 100 3 2.09/76.80
5 2.72%
Summing the effective modal gravity loads for both modes gives
SWm
5 W1 1 W2
5 74.84 1 2.09
5 76.93 kips
 W . . . satisfactory
Also:
SPmf1m 5 P1f11 1 P2f12
5 1.21 3 0.708 1 (20.203) 3 (20.701)
5 0.999
 1.0 . . . satisfactory
1.33.6 Modal base shear
The stages necessary in the modal analysis procedure consist of selecting the appropriate ground
motion response spectrum, applying a dynamic analysis technique to a mathematical model of the
structure, combining the response of a sufficient number of modes to ensure a 90-percent participation
of the mass of the structure, and scaling the results to ensure consistency with the static lateral force
procedure.
Three methods of dynamic analysis are referred to in ASCE 7 Table 12.6-1. The modal response spectrum analysis technique uses an appropriate response spectrum to calculate the peak modal response
of all significant modes. Alternatively, two seismic response history techniques may be utilized, using
either linear or nonlinear analysis. Seismic response history analysis determines the structural response
through numerical integration over short time increments for a site-specific, time-dependent, seismic
input motion that is representative of actual earthquake motions.
The design response spectra presented in ASCE 7 Section 11.4 and shown in Figure 1-8 may be used
after applying the appropriate scaling factors to provide the requisite response spectrum. Alternatively,
site-specific design spectra, as specified in ASCE 7 Section 11.4.8, may be utilized to obtain the input
spectrum.
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The modal seismic response coefficient, Csm, is determined for each mode of vibration of the structure
using its associated period of vibration and, in accordance with ASCE 7 Section 12.9.1.3, is given by
where:
Csm
5 SamIe /R
Sam
5 modal design spectral response acceleration at a period, Tm, as determined
from either the general response spectrum or a site-specific response
spectrum
Ie
5 occupancy importance factor from Table 1-5
R
5 response modification factor from Table 1-16
Tm
5 modal period of vibration (in seconds) of mode, m, of the structure
That portion of the base shear contributed by mode, m, is given by
Vm
5 CsmWm
Because the modal maximums do not all occur simultaneously or act in the same direction, a statistical
combination of these values is necessary. As indicated in ASCE 7 Section 12.9.1.3, the square-rootof-the-sum-of-the-squares method10, 15 is acceptable unless the periods of adjacent modes are closely
spaced. In this case, the complete-quadratic-combination (CQC) technique shall be used.
Example 1-52
Determine the modal base shears for the two-story building analyzed in Example 1-51.
The relevant parameters are
SDS
5 0.840g
SD1
5 0.469g
TS
5 0.558 second
T0
5 0.112 second
R
58
Ie
5 1.0
T1
5 0.545 second
T2
5 0.226 second
W1
5 74.84 kips
W2
5 2.09 kips
site classification
5D
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Solution
For the first mode of vibration, the natural period is
T1
5 0.545 sec
, TS
and
. T0
hence
5 SDS
Sa1
5 0.840g
5 Sa2
The modal seismic response coefficient is given by ASCE 7 Section 12.9.1.2 as
Cs1
5 Sa1Ie /R
5 0.840 3 1.0/8
5 0.105g
The portion of the base shear contributed by the first mode is given by
V1
5 Cs1W1
5 0.105 3 74.84
5 7.86 kips
For the second mode of vibration, the natural period is
T2
5 0.226 sec
The modal seismic response coefficient is given by ASCE 7 Section 12.9.1.2 as
Cs2
5 Sa2 I/R
5 0.840 3 1.0/8
5 0.105g
The portion of the base shear contributed by the second mode is given by
V2
5 Cs2W2
5 0.105 3 2.09
5 0.22 kip
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The ratio of the period of the second mode of vibration to the fundamental mode is
T2 /T1 5 0.226/0.545
5 0.42
, 0.75
Hence, the square-root-of-the-sum-of-the-squares method is an acceptable method10, 15 of combining
the modal values of the base shears, and the design value of the modal base shear is
Vt
5 [(V1)2 1 (V2)2]0.5
5 [(7.86)2 1 (0.22)2]0.5
5 7.86 kips
1.33.7 Scaling factors
To ensure consistency with the basic design principles adopted in the equivalent lateral force procedure,
a minimum value is stipulated in ASCE 7 Section 12.9.1.4 for the base shear derived by a dynamic
analysis. Some reduction is allowed, in comparison with the base shear derived from an equivalent
lateral force analysis, with a limit imposed to account for the underestimation of the stiffness of the
mathematical model assumed. The limit is imposed by comparison with an equivalent lateral force
analysis with a maximum value for the fundamental period T 5 CuTa.
In determining the base shear by the equivalent lateral force procedure, the fundamental period assumed
is given by ASCE 7 Section 12.8.2 as
where:
T
5 CuTa
Cu
5 coefficient for upper limit on the calculated period given in Table 1-3
Ta
5 approximate fundamental period of vibration as determined by ASCE 7
Equation (12.8-7)
The comparative base shear is then obtained from ASCE 7 Equation (12.8-1) as
where:
V
5 CsW
W
5 seismic dead load
Cs
5 seismic response coefficient
5 SD1I/RT . . . from ASCE 7 Equation (12.8-3) when T . TS ≤ TL
or
SD1
5 SDS I/R . . . from ASCE 7 Equation (12.8-2) when T , TS
5 design spectral response acceleration at a period of 1.0 second
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SDS
5 5-percent damped, design spectral response acceleration for a period of
0.2 second
Ie
5 occupancy importance factor from Table 1-5
R
5 response modification factor from Table 1-16
Where the modal base shear, Vt , is less than 100 percent of the base shear, V, determined by the equivalent lateral force procedure, all modal response forces must be multiplied by the scaling factor given
by ASCE 7 Section 12.9.1.4 as
Cm
5 V/Vt
Example 1-53
The two-story steel-frame building shown in Figure 1-2 is located in Orange County, California. Determine the modified modal base shear for the two-story building analyzed in Example 1-52.
The relevant parameters are
SDS
5 0.840g
SD1
5 0.469g
Ta
5 0.36 second
TS
5 0.558 second
T0
5 0.112 second
R
58
Ie
5 1.0
T1
5 0.545 second
Vt
5 7.86 kips
W
5 76.8 kips
site classification
5D
Solution
In a location with a value for the design spectral response acceleration at a period of 1.0 second of
SD1 . 0.4g, the value of the coefficient for the upper limit on the calculated period is obtained from
Table 1-3 as
Cu
5 1.4
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Hence, the natural period, in accordance with ASCE 7 Section 12.8.2, is limited to
T
5 CuTa
5 1.4 3 0.36
5 0.50 sec
, T1
Hence, use a maximum value of
T
5 0.50 sec
, Ts . . . ASCE 7 Equation (12.8-2) is applicable
Hence, the seismic response coefficient is given by
Cs
5 SDS I/R
5 0.840 3 1.0/8
5 0.105
The comparative base shear is then obtained from ASCE 7 Equation (12.8-1) as
V
5 CsW
5 0.105 3 76.8
5 8.06 kips
. Vt
Hence, the scaling factor is given by ASCE 7 Section 12.9.4 as
Cm
5 V/Vt
5 8.06/7.86
5 1.025
and the scaled modal base shear is
V
5 Cm 3 Vt
5 1.025 3 7.86
5 8.06 kips
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1.33.8 Vertical distribution of modal forces
The modal forces at each node may be determined by using the seismic dead load at each node, the
mode shape, and the modal base shear. The vertical distribution factor is given by
where:
Cvxm
5 wxfxm /Swi fim
Cvxm
5 vertical distribution factor for the given mode, m
wx
5 seismic dead load at a specific floor level x
wi
5 seismic dead load at floor level i
fxm
5 mode shape component for a specific node point, x, for the given mode, m
fim
5 mode shape component for node point i for the given mode, m, and the
summation extends over all the nodes in the structure
The modal force at each node for mode, m, is given by
where:
Fxm
5 CvxmVm
Vm
5 that portion of the base shear contributed by mode, m
Example 1-54
Determine the distribution of modal forces for each mode of the two-story building analyzed in Example 1-53. The relevant details are shown in Figure 1-57.
3.253
0.538
3.297
3.379
4.607
0.754
4.668
4.785
Figure 1-57 Details for Example 1-54
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Solution
The relevant values for the first mode are shown in Table 1-34.
Table 1-34 Details for the first mode
Level
wi
fi1
wi fi1
Fi1
Roof
25.60
1.000
25.60
3.253
2nd Floor
51.20
0.708
36.25
4.607
Total
76.80
–
61.85
7.860
From Table 1-34, the modal force at each node for the first mode is given by
Fi1
5 Cvi1V1
5 V1(wi fi1/Swi fi1)
where:
V1
5 that portion of the base shear contributed by the first mode
5 7.86 kips . . . from Example 1-52
and
Fi1
5 7.86(wi fi1)/61.85
5 0.127(wi fi1)
The modal force at each node is shown in Table 1-34 and Figure 1-57.
The relevant values for the second mode are shown in Table 1-35.
Table 1-35 Details for the second mode
Level
wi
fi2
wi fi2
Fi2
Roof
25.60
1.000
25.60
20.538
2nd Floor
51.20
20.701
235.89
0.754
Total
76.80
–
210.29
0.216
From Table 1-35, the modal force at each node for the second mode is given by
Fi2
5 Cvi2V2
5 V2(wi fi2 /Swi fi2)
where:
V2
5 that portion of the base shear contributed by the second mode
5 0.22 kip . . . from Example 1-52
and
Fi2
5 0.22(wi fi2)/(210.29)
5 20.021(wi fi2)
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The modal force at each node is shown in Table 1-35 and Figure 1-57.
The square-root-of-the-sum-of-the-squares method may be used to combine the modal values of the
forces at each node, and the design values of the modal forces are
Fit
5 [(Fi1)2 1 (Fi2)2]0.5
The scaled values of the modal forces are
Fi
5 CmFit
5 1.025 3 Fit
Values of Fit and Fi are given in Table 1-36 and Figure 1-57.
Table 1-36 Combined vertical force distribution
Level
Fi1
Fi2
Fit
Fi
Roof
3.253
20.538
3.297
3.379
2nd Floor
4.607
0.754
4.668
4.785
References
1. International Code Council. 2018 International Building Code. Washington, DC, 2018.
2. American Society of Civil Engineers. Minimum Design Loads and Associated Criteria for Buildings and Other Structures: ASCE 7-16. Reston, VA, 2016.
3. Federal Emergency Management Agency. NEHRP Recommended Seismic Provisions for New
Buildings and Other Structures. FEMA P-1050. Washington, DC, 2015.
4. International Code Council. 2018 International Residential Code. Washington, DC, 2018.
5. Luco, N. et al. “Risk-Targeted versus Current Seismic Design Maps for the Conterminous United
States.” Proceedings of the 2007 SEAOC Annual Convention. SEAOC. Sacramento, CA, 2007.
6. American Institute of Steel Construction. Specification for Structural Steel Buildings. AISC 360.
Chicago, IL, 2016.
7. American Institute of Steel Construction. Seismic Provisions for Structural Steel Buildings. AISC
341-16. Chicago, IL, 2016.
8. American Concrete Institute. Building Code Requirements and Commentary for Structural Concrete (ACI 318-14). Farmington Hills, MI, 2014.
9. Paz, M. Structural Dynamics. Kluwer Academic Publishers. New York, NY, 2003.
10. Chopra, A. K. Dynamics of Structures. Prentice Hall. New York, NY, 2000.
11. American Wood Council. National Design Specification for Wood Construction, (ANSI/AWS
NDS-2018). Washington, DC, 2018.
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12. Sheedy, P. “Anchorage of Concrete and Masonry Walls.” Building Standards. October 1983 and
April 1984. International Conference of Building Officials.
13. The Masonry Society. Building Code for Masonry Structures (TMS 402-16). Longmont, CO,
2016.
14. American Concrete Institute. Qualification of Post-Installed Mechanical Anchors in Concrete
(ACI 355.2-07). Farmington Hills, MI, 2007.
15. Structural Engineering Association of California. Recommended Lateral Force Requirements and
Commentary. Sacramento, CA, 1999.
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2
Design for Wind Loads
Nomenclature
a
width of pressure coefficient zone
ft
A
effective wind area
ft2
Ag
gross area of wall in which Ao is identified
ft2
Agi
sum of gross surface areas of building envelope (walls and roof), excluding Ag
ft2
Ao
total area of openings in a wall receiving positive external pressure
ft2
Aoi
sum of areas of all openings in building envelope (walls and roof), excluding Ao ft2
be
effective joist spacing
ft
B
horizontal dimension of building measured normal to wind direction
ft
Cp
external pressure coefficient
–
Cpi
internal pressure coefficient
–
G
gust effect factor
–
(GCp)
product of gust effect factor and external pressure coefficient
–
(GCpf)
product of gust effect factor and equivalent external pressure coefficient for
determining wind loads in MWFRS of low-rise buildings
–
(GCpi)
product of internal pressure coefficient and gust effect factor
–
h
mean roof or eave height
ft
h
eave height for roof angle, θ, less than or equal to 10°
ft
K1, K2, K3
multipliers from ASCE 7 Figure 26.8-1 used to obtain Kzt
–
Kd
wind directionality factor given in ASCE 7 Table 26.6-1
–
Ke
ground elevation factor given in ASCE 7 Table 26.9-1
–
Kh
velocity pressure exposure coefficient evaluated at height z 5 h
–
Kz
velocity pressure exposure coefficient evaluated at height z
–
Kzt
topographic factor as defined in ASCE 7 Section 26.8.1
–
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l
span of joist
ft
L
horizontal dimension of building measured parallel to wind direction
ft
MWFRS
main windforce-resisting system
–
p
design pressure for determining wind loads
lb/ft2
pe
external wind pressure given by ASCE 7 Equation (27.3-1)
lb/ft2
pi
internal wind pressure given by ASCE 7 Equation (27.3-1)
lb/ft2
q
velocity pressure given by ASCE 7 Equation (26.10-1)
lb/ft2
qh
velocity pressure evaluated at height z 5 h
lb/ft2
qs
wind stagnation pressure
lb/ft2
qz
velocity pressure evaluated at height z above ground
lb/ft2
s
joist spacing
ft
V
basic wind speed
mph
w
distributed load
lb/ft
z
height above ground
ft
zg
gradient height
ft
γ
exposure adjustment factor
–
θ
angle of plane of roof from horizontal
degree
λ
adjustment factor for building height and exposure
–
Symbols
2.1 Wind effects
On striking an enclosed building, wind flows around the sides and over the roof and produces either
a pressure or a suction on the external surfaces of the building. As shown in Figure 2-1, the windward
wall that is perpendicular to the wind direction experiences an inward, positive pressure. As wind
flows around the corners of the windward wall, the turbulence produced separates the airflow from
the walls and causes an outward, negative pressure or suction on the sidewalls and the leeward wall.
As wind flows over a high-sloping gable roof, a positive pressure is produced on the windward side of
the ridge and a suction on the leeward side of the ridge. However, for gable roofs with shallow slopes,
suction also develops on the windward side of the ridge and for flat roofs, suction develops over the
whole roof.
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Suction
Pressure
Suction
Suction
Suction
Pressure
Plan
Elevation
Figure 2-1 Wind pressure effects
Procedures are available for determining pressures on the main windforce-resisting system (MWFRS)
and on components and cladding. The main windforce-resisting system is defined in the International Building Code® (IBC®)1 Section 202 as the structural elements assigned to provide support and
stability for the overall structure. Components and cladding are defined in ASCE 72 Section 26.2 as
elements of the building envelope that do not qualify as part of the main windforce-resisting system.
The cladding of a building receives wind loading directly. Examples of cladding include wall and
roof sheathing, windows, and doors. Components receive wind loading from the cladding and transfer
the load to the main windforce-resisting system. Components include purlins, studs, girts, fasteners,
and roof trusses. Some elements, such as roof trusses and sheathing, may also form part of the main
windforce-resisting system and must be designed for both conditions. Because of local turbulence,
which may occur over small areas at ridges and corners of buildings, components and cladding are
designed for higher wind pressures than the main windforce-resisting system.
The design procedures consist of two basic approaches:
•
the directional procedure determines the wind loads on buildings for specific wind directions,
in which the external pressure coefficients are based on wind tunnel testing of prototypical
building models for the corresponding direction of wind
•
the envelope procedure determines the wind load cases on buildings, in which pseudo external pressure coefficients are derived from wind tunnel testing of prototypical building models
successively rotated through 360 degrees, such that the pseudo pressure cases produce key
structural actions (uplift, horizontal shear, bending moments, and so on) that envelope their
maximum values among all possible wind directions
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2.2 Analysis procedures
Several analysis procedures are permitted by IBC Section 1609.1.1 for determining the wind loads
on buildings. For determining wind loads on the main windforce-resisting system of buildings, the
permitted procedures are:
•
the analytical directional design method of ASCE 7 Chapter 27 Part 1 Section 27.3. This is
applicable to enclosed, partially enclosed, and open buildings of all heights and roof geometry.
Wind pressure is calculated using specific wind pressure equations applicable to each building
surface. The method uses the directional procedure to separate applied wind loads onto the
windward walls, leeward walls, and sidewalls of the building to correctly assess the forces in
the members.
•
the simplified directional method of ASCE 7 Chapter 27 Part 2 Section 27.5. This is based on
the analytical method of ASCE 7 Chapter 27 Part 1, and wind pressures are obtained directly
from a table. The method is applicable to enclosed, simple diaphragm buildings of any roof
geometry complying with the requirements of either Class 1 or Class 2 buildings. For a Class 1
building, the dimensions must be such that
h ≤ 60 ft
0.2 ≤ L/B ≤ 5.0
where:
h
5 mean roof height
L
5 horizontal dimension of building parallel to the wind direction
B
5 horizontal dimension of building normal to the wind direction
For a Class 2 building, the dimensions must be such that
60 ft , h ≤ 160 ft
0.5 ≤ L/B ≤ 2.0
In addition, the fundamental natural frequency of the building shall be not less than 75/h where
h is in feet.
•
the analytical envelope design method of ASCE 7 Chapter 28 Part 1 Section 28.3. This is
applicable to enclosed, partially enclosed, and open low-rise buildings that have a flat, gable,
or hip roof with a height not exceeding 60 feet and not exceeding the least horizontal dimension. Wind pressure is calculated using specific wind pressure equations applicable to each
building surface. The method uses the envelope procedure to separate applied wind loads onto
the windward walls, leeward walls, and sidewalls of the building to correctly assess the forces
in the members.
•
the simplified envelope method of ASCE 7 Chapter 28 Part 2 Section 28.5. This is based on
the envelope procedure of ASCE 7 Chapter 28 Part 1 and is applicable to enclosed, simple
diaphragm low-rise buildings that have a flat, gable, or hip roof with a height not exceeding 60
feet. Wind pressures are obtained directly from a table and applied to vertical and horizontal
projected surfaces of the building.
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•
the wind tunnel procedure of ASCE 7 Chapter 31, which may be used for any structure. This
is a procedure for determining wind loads, using a model of the building or other structure and
its surroundings, in which pressures, forces, and moments may be determined for each wind
direction considered. The wind tunnel procedure must be used when the limiting conditions of
the previous methods are not satisfied. This method is considered to produce the most accurate
wind pressure values.
•
the prescriptive provisions of ICC 600: Standard for Residential Construction in High-Wind
Regions3 is permitted for applicable Group R-2 and R-3 buildings, subject to the limitations of
IBC Section 1609.1.1.1.
•
the prescriptive provisions of AWC Wood Frame Construction Manual for One- and Two-Family
Dwellings4, subject to the limitations of IBC Section 1609.1.1.1.
•
the prescriptive provisions of AISI S230 Standard for Cold-Formed Steel Framing—
Prescriptive Method for One- and Two-Family Dwellings5, subject to the limitations of IBC
Section 1609.1.1.1.
2.3 General requirements
To calculate the wind loads on a building, it is necessary to determine a number of prerequisites,
including:
•
the exposure category of the site
•
wind speed at the location of the structure
•
the velocity pressure exposure coefficient
•
the topography at the location of the building
•
the probable direction of the wind
•
the building type
2.3.1 Exposure category
Exposure category accounts for the effect of terrain roughness on wind speed and is defined in ASCE
7 Section C26.7. The height and density of topographic features and buildings for a selected upwind
fetch distance are considered. Three surface roughness categories are specified and listed in Table 2-1
and are illustrated in Figure 2-2.
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Surface roughness category B
Surface roughness category D
Surface roughness category C
Figure 2-2 Surface roughness categories
The three exposure categories are listed in Table 2-1, and exposure categories B and D are illustrated
in Figure 2-3.
Wind direction
Surface roughness B
h Exposure category B
< d3 or d4
h Exposure category D
Surface roughness D
< d1
Surface roughness B or C
h Exposure category D
Surface roughness D
< d1
≤ d2
Figure 2-3 Exposure categories (See Table 2-1 for notation)
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Table 2.1 Surface roughness and exposure categories
Exposure category
Applicable ground surface roughness
B
Applicable to urban, suburban and mixed wooded areas with numerous
closely spaced obstructions the size of single-family dwellings or larger.
The minimum specified upwind fetch distance is the greater of d3 or d4.
C
Applicable to open terrain with scattered obstructions having heights
generally less than 30 feet. This category includes flat open country,
grasslands, and direct coastal exposure in hurricane-prone regions.
Exposure C shall apply for all cases where exposure B or D does not
apply.
D
Applicable to flat, unobstructed areas and to wind flowing over open
water for a minimum distance of 5000 feet or 20h. Exposure D also
applies where surface roughness categories B or C extend upwind for a
distance d2 followed by surface roughness category D for a distance d1.
Note: h 5 building height, d1 5 greater of 5000 feet or 20h, d2 5 greater of 600 feet or 20h,
d3 5 greater of 2600 feet or 20h . . . h . 30 feet, or d4 5 1500 feet . . . h ≤ 30 feet
2.3.2 Basic wind speed
Wind speed, V, is determined from the wind speed maps ASCE 7 Figures 26.5-1 and 26.5-2. The values given are based on the 3-second gust wind speed, in miles per hour, adjusted to a reference height
of 33 feet and for exposure category C. As shown in Figure 2-4, drag effects retard wind flow close
to the ground, and wind speed increases with height above ground level until the gradient height is
reached and the speed becomes constant. The gradient heights, zg, for different exposure conditions are
given in ASCE 7 Table 26.11-1 together with the 3-second gust speed power law exponent, α. These
are reproduced in Table 2-2. The wind speed at height, z, is obtained from the power law as
where:
Vz
5 V33(z/33)1/α
V33
5 wind speed at height 33 feet above ground
z
5 height above ground
The wind speed is given at the strength level or ultimate design value. This is similar to the approach
used for seismic design and gives a load factor of 1.0 for wind loads in the strength design load
combinations.
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76
90 98
127
Figure 2-4 Gradient height
Table 2-2 Gradient heights and power law exponents
Exposure
B
C
D
zg ft
1200
900
700
α
7.0
9.5
11.5
In the ASCE 7 standard, an importance factor is not used to provide enhanced performance for those
facilities assigned to a high-risk category. ASCE 7-16 achieves the same objective by using a probabilistic approach with four wind speed maps provided for buildings with different risk categories. An
increased return period provides enhanced performance for those facilities that constitute a substantial
public hazard because of high levels of occupancy or because of the essential nature of their function. The design wind speed return period for each map is based on the risk category assigned to the
building and the importance factor is eliminated. This ensures that high-risk facilities are designed for
higher loads so as to reduce possible structural damage. Four risk categories are listed in IBC Table
1604.5, as follows:
•
risk category IV buildings are essential facilities such as hospitals with emergency treatment
facilities, fire and police stations, emergency centers, hurricane or other emergency shelters,
and buildings housing equipment required to maintain the functionality of these installations.
Also included in risk category IV are structures housing toxic materials that will endanger the
safety of the public if released.
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•
risk category III buildings are facilities with a high occupant load, such as buildings where
more than 300 people congregate, schools with a capacity exceeding 250, colleges with a
capacity exceeding 500, health care facilities with a capacity of 50 or more or those that do not
have emergency treatment facilities, jails, and power stations.
•
risk category II buildings are standard occupancy structures that consist of all other types of
facilities.
•
risk category I buildings are low-hazard structures such as agricultural facilities, minor storage
facilities, and temporary facilities.
The four wind speed maps provided for the contiguous United States, Alaska, and Puerto Rico are:
•
ASCE 7 Figure 26.5-1A, which gives basic wind speeds for risk category I buildings and provides a return period of 300 years.
•
ASCE 7 Figure 26.5-1B, which gives basic wind speeds for risk category II buildings and provides a return period of 700 years.
•
ASCE 7 Figure 26.5-1C, which gives basic wind speeds for risk category III buildings and
provides a return period of 1700 years.
•
ASCE 7 Figure 26.5-1D, which gives basic wind speeds for risk category IV buildings and
provides a return period of 3000 years.
Details of the different occupancy categories and corresponding risk categories and return periods are
given in Table 2.3.
Similarly, ASCE 7 Figures 26.5-2A through 26.5-2D give basic wind speeds for risk category I through
IV buildings for Hawaii.
Precise values of the basic wind speeds are difficult to determine in congested areas of the maps. To
obviate this problem, the basic wind speed at a specific location with a known latitude and longitude
or mailing address may be obtained from the website
https://hazards.atcouncil.org
Table 2.3 Risk category and return period
Risk category
Nature of occupancy
Return period
Wind speed map
I
Low-hazard structures
300
26.5-1A
II
Standard occupancy structures
700
26.5-1B
III
Assembly structures
1700
26.5-1C
IV
Essential or hazardous structures
3000
26.5-1D
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2.3.3 Velocity pressure exposure coefficients for the whole building
The velocity pressure exposure coefficient, Kz , reflects the change in wind speed with height and
exposure category. The velocity pressure exposure coefficient is defined by ASCE 7 Table 26.10-1 as
5 2.01(z/zg)2/α . . . for 15 ft ≤ z ≤ zg
Kz
5 2.01(15/zg)2/α . . . for z , 15 ft
where:
z
5 height above ground level
zg
5 gradient height
The velocity pressure exposure coefficients, Kz , for different exposure conditions are given in
ASCE 7 Table 26.10-1. These are tabulated in Table 2-4, for a limited number of heights, for main
windforce-​resisting systems for the purpose of determining overall wind loads on the building. The
main windforce-​resisting system is defined in ASCE 7 Section 26.2 as an assemblage of structural elements assigned to provide support and stability for the overall structure. The system generally receives
wind loading from more than one surface.
Table 2-4 Velocity pressure exposure coefficients
for main windforce-resisting systems
Height above ground level, ft
Exposure
0–15
20
25
30
40
50
Ba
0.57
0.62
0.66
0.70
0.76
0.81
Bb
0.70
0.70
0.70
0.70
0.76
0.81
C
0.85
0.90
0.94
0.98
1.04
1.09
D
1.03
1.08
1.12
1.16
1.22
1.27
Note: a 5 directional procedure, b 5 envelope procedure
2.3.4 Topographic effects
The topographic factor, Kzt , accounts for the higher wind speeds experienced by buildings sited on or
adjacent to an abrupt change in topography such as an isolated hill, ridge, or escarpment. To adjust for
this effect, the velocity pressure exposure coefficient is multiplied by the topographic factor. As shown
in Figure 2-5, the wind velocity near the ground surface is most affected and the topographic factor is
given by ASCE 7 Equation (26.8-1) as
Kzt
5 (1 1 K1K2K3)2
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where:
K1
5 factor that accounts for the gradient of the slope
K2
5 factor that accounts for the distance of the building from the crest
K3
5 factor that accounts for the height above toe of slope
179
Figure 2-5 Wind speed-up at topographic feature
These three factors are determined from ASCE 7 Figure 26.8-1 using the notation shown in Figure 2-6.
In accordance with ASCE 7 Section 26.8-1, the topographic factor is applicable, provided that all of
the following conditions apply:
•
the hill, ridge, or escarpment is unobstructed upwind by similar features for a distance given by
the lesser of 100 times the height of the topographic feature or 2 miles
•
the topographic feature protrudes above the height of the upwind terrain, within a radius of
2 miles, by a factor of not less than 2
•
the building is located on the upper one-half of a hill or ridge or near the crest of an escarpment
•
H/Lh ≥ 0.2
•
the height of the topographic feature, H, is not less than 15 feet for exposures C and D and 60
feet for exposure B
Where no topographic effect is to be considered, the topographic factor is given by
Kzt
5 1.0
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Figure 2-6 Topographic factor parameters
2.3.5 Directionality factor
The directionality factor, Kd , is obtained from ASCE 7 Table 26.6-1 and for buildings is given as 0.85.
The directionality factor accounts for the reduced probability of:
•
extreme winds occurring in any specific direction
•
the peak pressure coefficient occurring for a specific wind direction
2.3.6 Building types
Several types of building construction are defined in ASCE 7 Section 26.2, including:
Low-rise building: an enclosed or partially enclosed building that satisfies both of the following
conditions:
•
mean roof height, h, is less than or equal to 60 feet
•
mean roof height, h, does not exceed least horizontal dimension
Applying the analytical method to low-rise buildings requires the use of specific velocity pressure
exposure coefficients.
Regular building: a building having no unusual geometrical irregularity in spatial form.
Diaphragm: Roof, floor, or other membrane or bracing system acting to transfer lateral forces to the
vertical main windforce-resisting system. Diaphragms constructed of wood structural panels are considered flexible diaphragms. Diaphragms constructed of untopped metal decks, concrete-filled metal
decks, and concrete slabs, each having a span-to-depth ratio of two or less, are considered rigid.
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Simple diaphragm building: a building in which both windward and leeward wind loads are transmitted by roof and vertically spanning wall elements through continuous roof and floor diaphragms
to the main windforce-resisting system. As shown in Figure 2-7, the wind loads consist of pressure on
the vertically spanning walls normal to the wind direction. These loads are collected by the diaphragm
and transferred to the shear walls parallel to the wind direction. Diaphragms must be continuous and
without expansion joints.
Diaphragm
She
ar W
all
Wind direction
Figure 2-7 Simple diaphragm building
Building envelope: cladding, roofing, exterior walls, glazing, door assemblies, window assemblies,
skylight assemblies, and other components enclosing the building.
Rigid building: a building with a fundamental natural frequency of n1 ≥ 1 Hz. A general guidance,
given in ASCE 7 Section C26.2, is that most buildings with a height-to-minimum-width ratio less than
4 may be considered rigid. Where necessary, the fundamental frequency may be determined using the
procedures given in ASCE 7 Section 26.11.3. In accordance with ASCE 7 Section 26.11.2, a low-rise
building is permitted to be considered rigid. A structure with a fundamental frequency less than 1 Hz
is considered flexible. A flexible structure exhibits a significant dynamic resonant response to wind
gusts.
Seismic and Wind Forces: Structural Design Examples
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Class 1 building: defined in ASCE 7 Section 27.4.2 as an enclosed, simple diaphragm building with
the following dimensions:
h ≤ 60 ft
0.2 ≤ L/B ≤ 5.0
where:
h
5 mean roof height
L
5 horizontal dimension of building parallel to the wind direction
B
5 horizontal dimension of building normal to the wind direction
Class 2 building: defined in ASCE 7 Section 27.4.2 as an enclosed, simple diaphragm building with
the following dimensions:
60 ft , h ≤ 160 ft
0.5 ≤ L/B ≤ 2.0
2.3.7 Gust effect factor
The gust effect factor, G, accounts for along-wind loading effects caused by dynamic amplification in
flexible structures and for wind turbulence-structure interaction. For a rigid structure, in accordance
with ASCE 7 Section 26.11.1, the gust effect factor may be taken as 0.85. Alternatively, the gust effect
factor for a rigid structure may be calculated using the procedures given in ASCE 7 Section 26.11.4.
For a flexible or dynamically sensitive structure, the gust effect factor is determined using the procedures given in ASCE 7 Section 26.11.5.
2.3.8 Enclosure classifications
The internal pressure produced in a structure by wind depends on the size and location of openings in the
external walls of the structure. As shown in Figure 2-8, an opening in the windward wall of a structure
produces an internal pressure. An opening in the leeward wall of a structure produces an internal suction.
Figure 2-8 Effect of openings on internal pressure
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Glazing that is breached by missiles must be treated as openings, as this may result in the development
of high internal pressures. In accordance with ASCE 7 Sec. 26.12.3.1, in a wind-borne debris region,
glazing in the lower 60 feet of structures shall be assumed to be openings unless such glazing is impact
resistant or protected with an impact-resistant covering. The same requirement applies to glazing that
is less than 30 feet above aggregate-surfaced roofs, including roofs with gravel or stone ballast, located
within 1500 feet of the structure. In accordance with ASCE 7 Section 26.12.3.1, glazed openings in
risk categories II, III, and IV buildings shall be protected in the following locations:
•
within 1 mile of the coastal mean high water line where the basic wind speed is equal to or
greater than 130 miles per hour
•
within a region where the basic wind speed is not less than 140 miles per hour
An open building is defined in ASCE 7 Section 26.2 as a building having each wall at least 80 percent
open. This is given for each wall by the expression
where:
Ao
≥ 0.8Ag
Ao
5 total area of openings in a wall that receives positive external pressure
Ag
5 the gross area of the wall in which Ao is identified
A partially enclosed building is defined as satisfying both of the following requirements:
•
the total area of openings in a wall that receives positive external pressure exceeds the sum of
the areas of openings in the balance of the building envelope (walls and roof) by more than 10
percent
•
the total area of openings in a wall that receives positive external pressure exceeds the smaller
of 4 ft2 or 1 percent of the area of the wall, and the percentage of openings in the balance of the
building envelope does not exceed 20 percent
These requirements are given by the following expressions
Ao
. 1.10Aoi
Aoi/Agi ≤ 0.20
and the smaller of
Ao
. 0.01Ag
or
. 4 ft2
where:
Aoi
5 sum of the areas of openings in the building envelope (walls and
roof) not including Ao
Agi
5 sum of the gross surface area of the building envelope (walls and
roof) not including Ag
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An enclosed building is defined as a building having the total area of openings in a wall that receives
positive external pressure less than or equal to 4 ft2 or 1 percent of the area of the wall, whichever is
smaller. These requirements are given by the following expressions:
but not more than
Ao
, 0.01Ag
Ao
5 4 ft2
A partially open building is defined as a building that does not comply with the requirements for
open, partially enclosed, or enclosed buildings.
2.3.9 Ground elevation factor
Air density and air pressure decrease with increasing altitude. The ground elevation factor, Ke, accounts
for this and is determined from ASCE 7 Table 26.9-1. It is permitted to take Ke 5 1.0 for all elevations.
2.4 Analytical directional design method for MWFRS
This method, detailed in ASCE 7 Section 27.3, is applicable to enclosed, partially enclosed, and open
buildings of all heights and roof geometry. Wind pressure is calculated using specific wind pressure
equations applicable to each building surface. The method uses the directional procedure to separate
applied wind loads onto the windward walls, leeward walls, sidewalls, and roof of the building so as
to correctly assess the forces in the members. The procedure consists of the determination of the following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Figures 26.5-1 and 26.5-2)
•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
•
velocity pressure exposure coefficients, Kz , for the applicable exposure category . . . (Table 2-4)
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
ground elevation factor, Ke . . . (ASCE 7 Table 26.9-1)
•
directionality factor, Kd . . . (ASCE 7 Table 26.6-1)
•
gust effect factor, G . . . (ASCE 7 Section 26.11)
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•
enclosure classification . . . (ASCE 7 Section 26.12)
•
internal pressure coefficient, (GCpi) . . . (ASCE 7 Table 26.13-1)
•
wind velocity pressure, q . . . [ASCE 7 Equation (26.10-1)]
•
external pressure coefficient, Cp or CN . . . (ASCE 7 Figures 27.3-1 through 27.3-7)
•
internal wind pressure, pi . . . [ASCE 7 Equation (27.3-1)]
•
external wind pressure, pe . . . [ASCE 7 Equation (27.3-1)]
•
combined internal and external wind pressures, p . . . [ASCE 7 Equation (27.3-1)]
•
check minimum design wind loads . . . (ASCE 7 Section 27.1.5)
•
apply design wind load cases . . . (ASCE 7 Figure 27.3-8)
185
2.4.1 Minimum design wind loads
The minimum design wind loads for an enclosed or partially enclosed building are given in ASCE 7
Section 27.1.5. As shown in Figure 2-9, these consist of an external pressure of 16 lb/ft2 on wall areas
and 8 lb/ft2 on roof areas projected onto a vertical plane normal to the wind direction. The minimum
loads are to be applied as a separate load case in addition to the normal load cases specified.
2
ft
b/
8l
t2
b/f
l
16
16
lb/f 2
t
Figure 2-9 Minimum design wind load
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2.4.2 Design wind load cases
Torsional effects are caused by nonuniform pressure on the different faces of the building, interference
effects of nearby buildings and terrain, and by dynamic effects on flexible structures. Hence, buildings
must be designed for the four load cases given in ASCE 7 Section 27.3.5 and ASCE 7 Figure 27.3-8
and shown in Figure 2-10. Design load cases 2 and 4 are the torsion load cases.
0.75PWY
0.75PWX
PWX
0.75PLX
0.75PLY
PLX
Case 1
Case 3
BY
BX
0.563PWY
+
MT
+
MT
0.563PLX
0.563PLY
0.563PWX
0.75PWX
0.75PLX
eX = ± 0.15BX
eX = ± 0.15BX
Case 2
eY = ± 0.15BY
Case 4
Figure 2-10 Design wind load cases
The four load cases consist of:
Load case 1: full design wind pressure acting along each principal axis of the structure, considered
separately along each principal axis
Load case 2: 75 percent of full design wind pressure acting along each principal axis of the structure
in conjunction with a torsional moment, considered separately along each principal axis with
MT
5 0.75(PWX 1 PLX)BX eX . . . wind acting in the x direction
MT
5 0.75(PWY 1 PLY)BY eY . . . wind acting in the y direction
eX
5 0.15BX . . . wind acting in the x direction
eY
5 0.15BY . . . wind acting in the y direction
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Load case 3: 75 percent of full design wind pressure acting simultaneously along both principal axes
of the structure
Load case 4: 56.3 percent of full design wind pressure acting simultaneously along both principal
axes of the structure in conjunction with a torsional moment of
where:
MT
5 0.563(PWX 1 PLX)BX eX 1 0.563(PWY 1 PLY)BY eY
PWX
5 windward face design pressure acting in the x direction
PWY
5 windward face design pressure acting in the y direction
PLX
5 leeward face design pressure acting in the x direction
PLY
5 leeward face design pressure acting in the y direction
eX
5 eccentricity for wind acting in the x direction measured from the geometric
center of the building face perpendicular to the direction of the wind
eY
5 eccentricity for wind acting in the y direction measured from the geometric
center of the building face perpendicular to the direction of the wind
MT
5 torsional moment per unit height acting about a vertical axis of the building
In accordance with ASCE 7 Appendix D, the following buildings need not be designed for torsion load
cases 2 and 4 and need only be designed for no torsion load cases 1 and 3:
•
one-story buildings with h ≤ 30 ft
•
one- or two-story buildings of light-frame construction
•
one- or two-story buildings with flexible diaphragms
•
buildings meeting the spatial distribution and stiffness requirements of ASCE 7 Appendix D
Sections D.3 through D.6
As indicated in Appendix CD, the design objective is to minimize the inherent torsion from wind on
the building. This is achieved by placing and proportioning the vertical elements of the main windforce-resisting system in each direction so that the center of pressure from wind forces is located near
the center of rigidity of the main windforce-resisting system. A torsional eccentricity exceeding 5
percent of the building width may produce large shear forces and torsional story drift, causing damage
to cladding and interior partitions.
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2.4.3 Wind velocity pressure
Taking the ground elevation factor as Ke 5 1, as permitted by ASCE 7 Section 26.9, the velocity pressure q at any height above the ground is given by ASCE 7 Equation (26.10-1) as
q
5 0.00256Kz Kzt KdV2
Example 2-1
The two-story steel-frame office building with flexible diaphragms shown in Figure 2-11 is located in
a flat suburban area of Wyoming. The basic wind speed is obtained from ASCE 7 Figure 26.5-1B as
110 miles per hour. Determine the wind velocity pressure at roof height for the main windforce-resisting system. Use the analytical directional design method of ASCE 7 Section 27.3.
h
B
L
Figure 2-11 Details for Example 2-1
Solution
For a suburban area, the exposure is category B. The relevant parameters are obtained as
Kz
5 velocity pressure exposure coefficient
5 0.65 . . . from Table 2-4 for a height of 24 ft for the main windforceresisting system and exposure category B, using the analytical directional
design method
Kzt
5 topographic factor
5 1.0 . . . from ASCE 7 Figure 26.8-1
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Kd
189
5 wind directionality factor
5 0.85 . . . from ASCE 7 Table 26.6-1 for a building
The velocity pressure, q, at a height of 24 feet above the ground is given by ASCE 7 Equation
(26.10-1) as
q
5 0.00256Kz Kzt KdV2
5 0.00256 3 0.65 3 1.0 3 0.85 3 1102
5 17.11 lb/ft2
Example 2-2
The two-story steel-frame office building with flexible diaphragms shown in Figure 2-11 is located
adjacent to the shoreline in Miami, Florida. The basic wind speed is obtained from ASCE 7 Figure
26.5-1B as 170 miles per hour. Determine the wind velocity pressure at roof height, and at a height
of 15 feet, for the main windforce-resisting system. Use the analytical directional design method of
ASCE 7 Section 27.3.
Solution
For a location on the shoreline, the exposure is category D. The relevant parameters are obtained as
Kz
5 velocity pressure exposure coefficient
5 1.11 . . . from Table 2-4 for a height of 24 ft for the main windforceresisting system and exposure category D, using the analytical directional
design method
5 1.03 . . . from Table 2-4 for a height of 15 ft for the main windforceresisting system and exposure category D
Kzt
5 topographic factor
5 1.0 . . . from ASCE 7 Figure 26.8-1
Kd
5 wind directionality factor
5 0.85 . . . from ASCE 7 Table 26.6-1 for a building
The velocity pressure, qh , at a roof height of 24 feet above the ground is given by AISC Equation
(26.10‑1) as
qh
5 0.00256Kz Kzt KdV2
5 0.00256 3 1.11 3 1.0 3 0.85 3 1702
5 69.80 lb/ft2 . . . at roof height
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Design for Wind Loads
The velocity pressure, q15, at a height of 15 feet above the ground is given by ASCE 7 Equation
(26.10-1) as
q15
5 0.00256Kz Kzt KdV2
5 0.00256 3 1.03 3 1.0 3 0.85 3 1702
5 64.77 lb/ft2 . . . at a height of 15 ft
Example 2-3
The two-story steel-frame office building shown in Figure 2-12 is located adjacent to the shoreline in
Miami, Florida, at the top of an escarpment. The basic wind speed is 170 miles per hour. Determine
the wind velocity pressure at roof height for the main windforce-resisting system. Use the analytical
directional design method of ASCE 7 Section 27.3.
Figure 2-12 Details for Example 2-3
Solution
For a location on the shoreline, the exposure is category D. The relevant parameters are obtained as
Kz
5 velocity pressure exposure coefficient
5 1.11 . . . from Table 2-4 for a height of 24 ft for the main windforceresisting system and exposure category D
H
5 20 ft
. 15 ft . . . satisfies ASCE 7 Section 26.8.1 for exposure D
H/Lh
5 20/50
5 0.4
. 0.2 . . . satisfies ASCE 7 Section 26.8.1
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For exposure D
K1
5 0.95H/Lh . . . ASCE 7 Figure 26.8.1
5 0.95 3 0.4
5 0.38
x/Lh
5 25/50
5 0.5
m
5 4 . . . downwind of crest
K2
5 1 2 x/mLh . . . from ASCE 7 Figure 26.8-1
5 1 2 0.5/4
5 0.875
z
5 24 ft . . . roof height
z/Lh
5 24/50
5 0.48
γ
5 2.5 . . . for an escarpment
K3
5 e2γz /Lh . . . ASCE 7 Figure 26.8-1
5 e22.5 3 0.48
5 0.301
Kzt
5 topographic factor given by ASCE 7 Figure 26.8-1
5 (1 1 K1K2K3)2
5 (1 1 0.38 3 0.875 3 0.301)2
5 1.21
Kd
5 wind directionality factor
5 0.85 . . . from ASCE 7 Table 26.6-1
The velocity pressure, q, at a roof height of 24 feet above the ground is given by ASCE 7 Equation
(26.10‑1) as
q
5 0.00256Kz Kzt KdV2
5 0.00256 3 1.11 3 1.21 3 0.85 3 1702
5 84.46 lb/ft2 . . . at roof height
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2.4.4 Internal pressure coefficients and internal pressure
Internal pressure coefficients are used in conjunction with velocity pressure values to determine internal pressure in buildings. The product of the gust effect factor and the internal pressure coefficient is
denoted in ASCE 7 Section 26.13 as (GCpi). Values of (GCpi) are tabulated in ASCE 7 Table 26.13-1
for the four different building enclosure classifications. Pressures act normal to wall and roof surfaces
and are positive when acting toward the surface and negative when acting away from the surface. The
conditions that produce internal suction and internal pressure are shown in Figure 2-8. Both cases must
be considered for any building and added algebraically to external pressures to determine the most
critical loading conditions. Values of (GCpi) are given in Table 2-5.
Table 2-5 Values of internal pressure coefficients
Enclosure classification
(GCpi)
Open buildings
0.00
Partially open buildings
0.18
Partially enclosed buildings
0.55
Enclosed buildings
0.18
For a rigid building, the pressure acting on internal surfaces is obtained from the second term of ASCE
7 Equation (27.3-1) as
where:
pi
5 qi (GCpi)
qi
5 qh . . . for all surfaces of enclosed buildings and for negative internal
pressure evaluation in partially enclosed buildings
5 wind velocity pressure at roof height
or
qi
5 qz . . . for positive internal pressure evaluation in partially enclosed buildings
5 wind velocity pressure at level of the highest opening that can affect the
positive internal pressure in partially enclosed buildings
and
z
5 height of the highest opening that can affect the positive internal pressure
(GCpi)
5 product of the gust effect factor and the internal pressure coefficient from
Table 2-5
For a conservative evaluation of positive internal pressure, qi 5 qh .
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Example 2-4
The two-story steel-frame office building with flexible diaphragms analyzed in Example 2-2 is located
adjacent to the shoreline in Miami, Florida. All glazing in the building is impact resistant. Determine
the internal pressure acting on the building. Use the analytical directional design method.
Solution
Since the glazing is impact resistant, the building may be considered an enclosed building and the
product of the internal pressure coefficient and the gust effect factor is obtained from Table 2-5 as
(GCpi)
5 0.18
The wind velocity pressure at roof height is obtained from Example 2-2 as
qi
5 qh
5 69.80 lb/ft2
The pressure acting on all internal surfaces is obtained from the second term of ASCE 7 Equation
(27.3-1) as
pi
5 qi(GCpi)
5 qh(GCpi) . . . for an enclosed building
5 69.80 3 0.18
5 12.56 lb/ft2
2.4.5 External pressure coefficients and external pressures
For a rigid building, the pressure acting on external surfaces is obtained from the first term of ASCE
7 Equation (27.3-1) as
where:
pe
5 qGCp
q
5 qh . . . for leeward walls, sidewalls, and roof, evaluated at a height, h
5 wind velocity pressure at roof height
or
q
5 qz . . . for windward walls evaluated at a height, z, above the ground
5 wind velocity pressure at a specific height, z, above the ground
and
z
5 any specific height above the ground
G
5 gust effect factor given in ASCE 7 Section 26.11-1
Cp
5 external pressure coefficient from ASCE 7 Figures 27.3-1 and 27.3-2
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For the windward wall, the external pressure coefficient increases with height and is independent of
the wall dimensions. For the leeward wall, the external pressure coefficient is constant over the height
of the wall and is utilized with the wind velocity pressure at roof height, and is a function of the wall
dimensions. For the sidewalls, the external pressure coefficient is constant over the height of the wall
and is utilized with the wind velocity pressure at roof height and is independent of the wall dimensions. Values of the external pressure coefficient for walls are given in ASCE 7 Figure 27.3-1 and are
tabulated in Table 2-6. The distribution of external pressure on walls is shown in Figure 2-13.
Table 2-6 Wall external pressure coefficients
Surface
L/B
Cp
Use with
Windward wall
All values
0.8
qz
0–1
20.5
qh
2
20.3
≥4
20.2
All values
20.7
Leeward wall
Sidewall
qh
Note: L
5 horizontal dimension of building measured parallel to wind direction
B 5 horizontal dimension of building measured normal to wind direction
qhGCp
qhGCp
Wind direction
qhGCp
A
A
qhGCp
qZGCp
qhGCp
Elevation
Section A-A
Figure 2-13 External wind pressure distribution
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For a flat roof, or gable roof with the ridge parallel to wind direction, or a gable roof with a pitch of
less than 10 degrees and with the ridge normal to the wind direction, the external pressure coefficient
depends on the horizontal distance from the windward edge, the applicable area of the roof, and the
h/L ratio. Values of the external pressure coefficient for these conditions are given in ASCE 7 Figure
27.3-1 and are tabulated in Table 2-7.
Table 2-7 Roof external pressure coefficients
Windward direction
h/L
Horizontal
distance from
windward edge
Applicable
area (ft2)
0 to h/2
≤ 0.5
Normal to ridge for
θ , 10° and parallel
to ridge for all θ and
for flat roofs
20.9, 20.18
h/2 to h
All
h to 2h
. 2h
≥ 1.0
Pressure
coefficient Cp
20.9, 20.18
20.5, 20.18
20.3, 20.18
≤ 100
21.3, 20.18
250
21.17, 20.18
≥ 1000
21.04, 20.18
All
20.7, 20.18
0 to h/2
. h/2
Note: θ
5 angle of plane of roof from horizontal, in degrees
h 5 mean roof height except that eave height shall be used for θ ≤ 10°
For a gable roof, with the ridge normal to wind direction and with a pitch of not less than 10 degrees,
the external pressure coefficient depends on the location windward or leeward of the ridge, the pitch
of the roof, the applicable area of the roof, and the h/L ratio. The leeward slope of the roof is subject
to pressure acting away from the surface for all pitch angles. For a pitch angle of 10 degrees, the
windward slope of the roof is also subjected to pressure acting away from the surface but, as the pitch
increases, may be subjected to pressure acting toward the surface. Both cases must be considered to
determine the most critical loading conditions. Values of the external pressure coefficient for these
conditions are given in ASCE 7 Figure 27.3-1 and, for a limited number of cases, are tabulated in Table
2-8. The distribution of external pressure on roofs is shown in Figure 2-13.
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Design for Wind Loads
Table 2-8 Gable roof external pressure coefficients
Windward
direction
Normal to ridge
for θ ≥ 10°
0.5
≥ 1.0
Leeward slope
Roof pitch 
Roof pitch 
45°
≥ 60°
0.3
0.4
b
20.4
20.2
0.0
20.18
0.0
0.2
0.4
a
20.7
20.3
0.0
20.18
20.18
0.2
0.3
h/L
≤ 0.25
Windward slope
10°
20°
30°
20.7
20.3
20.2
20.18
0.2
20.9
b
b
10°
15°
≥ 20°
20.3
20.5
20.6
20.5
20.5
20.6
20.7
20.6
20.6
Note: a. Cp 5 21.3 for A ≤ 100 ft2, Cp 5 21.17 for A 5 250 ft2, Cp 5 21.04 for A ≥ 1000 ft2
b. Cp 5 0.01 × θ for 60° ≤ θ ≤ 80°, Cp 5 0.8 for θ . 80°
Example 2-5
The two-story steel-frame office building with flexible diaphragms analyzed in Example 2-2 is located
adjacent to the shoreline in Miami, Florida. All glazing in the building is impact resistant. Determine
the design wind pressure acting on the whole building for wind flowing normal to the 100-foot-long
side and the resultant base shear. Use the analytical directional design method.
Solution
The relevant parameters are
L
5 horizontal dimension of building measured parallel to wind direction
5 40 ft
B
5 horizontal dimension of building measured normal to wind direction
5 100 ft
L/B
5 length/width ratio
5 40/100
5 0.4
, 1.0
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197
5 roof height
5 24 ft
, 60 ft . . . low-rise building in accordance with ASCE 7 Section 26.2
h/L
5 roof height/length ratio
5 24/40
5 0.6
, 4.0
Hence, in accordance with ASCE 7 Sections C26.2 and 26.11.2, the building may be considered a rigid
structure and the gust effect factor is given by ASCE 7 Section 26.11.1 as
G
5 0.85
The roof area, measured from the windward edge for a distance of h/2 is given by
A
5 Bh/2
5 100 3 24/2
5 1200 ft2
. 1000 ft2
qh
5 wind velocity pressure at roof height
5 69.80 lb/ft2 . . . from Example 2-2
q15
5 wind velocity pressure at a height of 15 ft
5 64.77 lb/ft2 . . . from Example 2-2
For the windward wall, the external pressure is independent of the wall dimensions and is obtained
from Table 2-6 as
Cp
5 0.8
For the windward wall, the external pressure acting toward the wall surface is proportional to the wind
velocity pressure. The external pressure at roof height is obtained from ASCE 7 Equation (27.3-1) as
ph
5 qhGCp
5 69.80 3 0.85 3 0.8
5 47.46 lb/ft2
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Design for Wind Loads
The external pressure at a height of 15 feet is obtained from ASCE 7 Equation (27.3-1) as
p15
5 q15GCp
5 64.77 3 0.85 3 0.8
5 44.04 lb/ft2
For the leeward wall, the external pressure coefficient is dependent on the length/width ratio of the
building and, for a value of L/B , 1.0, is obtained from Table 2-6 as
Cp
5 20.5
For the leeward wall, the external pressure coefficient is utilized with the wind velocity pressure at
roof height to give a uniform pressure acting away from the wall surface. The external pressure acting
on the full height of the wall is obtained from ASCE 7 Equation (27.3-1) as
pe
5 qhGCp
5 69.80 3 0.85 3 (20.5)
5 229.67 lb/ft2
For the sidewalls, the external pressure coefficient is independent of the wall dimensions and is
obtained from Table 2-6 as
Cp
5 20.7
For the sidewalls, the external pressure coefficient is utilized with the wind velocity pressure at roof
height to give a uniform pressure acting away from the wall surface. The external pressure acting on
the full height of the wall is obtained from ASCE 7 Equation (27.3-1) as
pe
5 qhGCp
5 69.80 3 0.85 3 (20.7)
5 241.53 lb/ft2
For the roof, the external pressure coefficient is dependent on the roof height/length ratio and varies
over the length of the roof and with the applicable area over which the pressure acts. For h/L 5 0.5
and a distance of 0 to h/2 from the windward edge, the external pressure coefficient is obtained from
Table 2-7 as
Cp
5 20.9 or 20.18
For h/L 5 1.0 and a distance of 0 to h/2 from the windward edge, and for an applicable area A . 1000
ft2, the external pressure coefficient is obtained from Table 2-7 as
Cp
5 21.04 or 20.18
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By interpolation for h/L 5 0.6, the external pressure coefficient is obtained as
Cp
5 20.9 2 (1.04 2 0.9) 3 0.1/0.5 or 20.18
5 20.93 or 20.18
For the roof, the external pressure coefficient is utilized with the wind velocity pressure at roof height
to give a uniform pressure acting away from the roof surface. The external pressure acting on the segment a distance of 0 to h/2 from the windward edge is obtained from ASCE 7 Equation (27.3-1) as
pe
5 qhGCp
5 69.80 3 0.85 3 (20.93) or 69.80 3 0.85 3 (20.18)
5 255.18 lb/ft2 or 210.68 lb/ft2
For the roof, for h/L 5 0.5 and a distance of h/2 to h from the windward edge, the external pressure
coefficient is obtained from Table 2-7 as
Cp
5 20.9 or 20.18
For h/L 5 1.0 and a distance greater than h/2 from the windward edge, the external pressure coefficient
is obtained from Table 2-7 as
Cp
5 20.7 or 20.18
By interpolation for h/L 5 0.6, the external pressure coefficient is obtained as
Cp
5 20.7 2 (0.9 2 0.7) 3 0.4/0.5
5 20.86 or 20.18
For the roof, the external pressure acting on the segment a distance of h/2 to h from the windward edge
is obtained from ASCE 7 Equation (27.3-1) as
pe
5 qhGCp
5 69.80 3 0.85 3 (20.86) or 69.80 3 0.85 3 (20.18)
5 251.02 lb/ft2 or 210.68 lb/ft2
For the roof, the external pressure coefficient for h/L 5 0.5 and a distance of h to 2h from the windward edge, the external pressure coefficient is obtained from Table 2-7 as
Cp
5 20.5 or 20.18
For h/L 5 1.0 and a distance greater than h/2 from the windward edge, the external pressure coefficient
is obtained from Table 2-7 as
Cp
5 20.7 or 20.18
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By interpolation for h/L 5 0.6, the external pressure coefficient is obtained as
Cp
5 20.5 2 (0.7 2 0.5) 3 0.1/0.5
5 20.54 or 20.18
For the roof, the external pressure acting on the segment a distance greater than h from the windward
edge is obtained from ASCE 7 Equation (27.3-1) as
pe
5 qhGCp
5 69.80 3 0.85 3 (20.54) or 69.80 3 0.85 3 (20.18)
5 232.04 lb/ft2 or 210.68 lb/ft2
The design wind pressures for the main windforce-resisting system are shown in Figure 2-14. The values of the combined internal and external pressures are given in Table 2-9. Minimum values of design
wind pressure
do not govern. Since the building is of two stories and has flexible diaphragms, torsional
wind load cases 2 and 4 need not be considered.
12 ft
–55.18 lb/ft2
12 ft
16 ft
–51.02 lb/ft2
Wind direction
–32.04 lb/ft2
47.46 lb/ft2
9 ft
Add internal pressure
±12.56 lb/ft2
–29.67 lb/ft2
15 ft
44.04 lb/ft2
Figure 2-14 Design wind pressures for Example 2-5
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Table 2-9 Design wind pressures for Example 2-5
Location
Internal pressure
Internal suction
0–15 ft
31.48 lb/ft2
56.60 lb/ft2
15–24 ft
34.90 lb/ft2
60.02 lb/ft2
Leeward wall
242.23 lb/ft2
217.11 lb/ft2
Sidewall
254.12 lb/ft2
229.00 lb/ft2
267.74 lb/ft2
242.62 lb/ft2
or
or
223.24 lb/ft2
1.88 lb/ft2
263.58 lb/ft2
238.46 lb/ft2
or
or
223.24 lb/ft2
1.88 lb/ft2
244.60 lb/ft2
219.48 lb/ft2
or
or
Windward wall
Roof
0–12 ft
12–24 ft
24–40 ft
2
223.24 lb/ft
1.88 lb/ft2
Using the values shown in Figure 2-14, the strength level base shear acting normal to the 100-foot
side is determined from the product of the external pressure and the area over which it occurs and is
given by
Fw
5 100(44.04 3 15 1 47.46 3 9 1 29.67 3 24)/1000
5 180 kips
2.5 Simplified directional design method for MWFRS
The simplified method of ASCE 7 Chapter 27 Part 2 Section 27.5 is based on the analytical method of
ASCE 7 Chapter 27 Part 1. Wind pressures are obtained directly from a table. The method is applicable to enclosed, simple diaphragm buildings of any roof geometry complying with the requirements
of either Class 1 buildings or Class 2 buildings. The procedure consists of the determination of the
following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Figures 26.5-1 and 26.5-2)
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•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
•
velocity pressure exposure coefficients, Kz , for the applicable exposure category . . . (Table 2-4)
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
ground elevation factor, Ke . . . (ASCE 7 Table 26.9-1)
•
directionality factor, Kd . . . (ASCE 7 Table 26.6-1)
•
gust effect factor, G . . . (ASCE 7 Section 26.11)
•
enclosure classification . . . (ASCE 7 Section 26.12)
•
net pressures at top of walls, ph , and bottom of walls, po . . . (ASCE 7 Table 27.5-1)
•
net roof pressures, ph . . . (ASCE 7 Table 27.5-2)
•
if topographic factor Kzt . 1.0, apply factor to wall and roof pressures . . . (ASCE 7 Section
26.8)
•
apply wall pressures simultaneously with roof pressures . . . (ASCE 7 Section 27.5.1)
•
check minimum design wind loads . . . (ASCE 7 Section 27.1.5)
•
apply design wind load cases . . . (ASCE 7 Figure 27.3-8)
2.5.1 Wall pressure
In a simple diaphragm building where the wind loads are collected by diaphragms and transferred to
the main windforce-resisting system, windward and leeward wind loads may be combined into a single
load and applied to the windward wall. Similarly, internal pressures cancel out and need not be considered for the design of the main windforce-resisting system. ASCE 7 Table 27.5-1 tabulates net alongwind pressure for walls. For a specific exposure category, wind speed, building height, and building
aspect ratio, two values of wind pressure are obtained from the table. As shown in Figure 2-15, these
are ph, the pressure at the top of the building, and p0 , the pressure at the bottom of the building. These
values do not include the effect of internal pressures since the internal pressures cancel out when considering the net pressures on simple diaphragm buildings.
The wind pressure values, ph and p0 , are the sum of the wind pressures acting simultaneously on the
windward and leeward walls. As indicated in Figure 2-15, these combined wind pressures are shown
acting on the windward wall of the building and this is an adequate representation when designing the
MWFRS.
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For Class 1 buildings, ASCE 7 Section 27.5.1 specifies that for L/B ratios less than 0.5, the wind pressure values tabulated for L/B 5 0.5 are applicable. For L/B ratios greater than 2.0, the wind pressure
values tabulated for L/B 5 2.0 are applicable.
ph
p0
Elevation
Figure 2-15 Net pressure distribution on windward wall
As shown in Figure 2-16, sidewall pressures act outward, are constant over the full height of the building, and are determined from the applicable value of ph. The sidewall pressures are given by ASCE 7
Table 27.5-1 Note 2 as
psidewall
5 0.54ph . . . for 0.2 ≤ L/B ≤ 1.0
5 0.64ph . . . for 2.0 ≤ L/B ≤ 5.0
interpolate for 1.0 , L/B , 2.0
Wind direction
psidewall
psidewall
Plan
Figure 2-16 Net pressure distribution on sidewalls
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Design for Wind Loads
For the design of drag struts and wall elements, it is necessary to determine the individual pressures on
the windward and leeward walls. The leeward wall pressure acts outward, is constant for the full height
of the building, and is determined from the applicable value of ph. The leeward wall pressure is given
by ASCE 7 Table 27.5-1 Note 4 as
pleeward wall
5 0.38ph . . . for 0.2 ≤ L/B ≤ 1.0
5 0.27ph . . . for 2.0 ≤ L/B ≤ 5.0
interpolate for 1.0 , L/B , 2.0
The windward wall pressure is calculated as the difference between the total net pressure from ASCE
7 Table 27.5-1 using the ph and p0 values and the constant leeward wall pressure. The effect of internal
pressure must also be included in order to obtain the total wall forces.
2.5.2 Roof pressure
ASCE 7 Table 27.5-2 tabulates net wind pressures for roofs of various configurations for exposure
category C. The net pressures are combined external pressures and internal pressures appropriate to
an enclosed building condition. For a specific roof configuration, wind speed, and building height,
values of wind pressure are obtained from the table for designated areas of the roof. Where two values
are given in the table, both values must be investigated. An adjustment for exposure category B or D
is made to the pressures by multiplying by the applicable factor from ASCE 7 Table 27.5-2, which
depends on the height of the structure. Adjustment factors for a limited number of heights are shown
in Table 2-10.
Table 2-10 Exposure adjustment factor
h (ft)
15
20
30
40
50
60
70
80
90
100
Exposure B
0.667
0.692
0.713
0.729
0.741
0.751
0.760
0.768
0.775
0.781
Exposure D
1.214
1.201
1.183
1.171
1.161
1.154
1.147
1.141
1.137
1.132
The distribution of wind pressures for a flat roof is shown in Figure 2-17.
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h/2
205
h/2
Wind direction
h
Figure 2-17 Net pressure distribution on a flat roof
Example 2-6
The two-story steel-frame office building with flexible diaphragms, shown in Figure 2-11, is located
adjacent to the shoreline in Miami, Florida. All glazing in the building is impact resistant. Determine
the design net wind pressure acting on the whole building for wind flowing normal to the 100-footlong side and the resultant base shear. Use the simplified directional method.
Solution
The relevant parameters are
V
Exposure category
L
5 170 mph . . . from Example 2-2
5 D . . . from Example 2-2
5 horizontal dimension of building measured parallel to wind direction
5 40 ft
B
5 horizontal dimension of building measured normal to wind direction
5 100 ft
L/B
5 length/width ratio
5 40/100
5 0.4
, 5.0
. 0.2
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Design for Wind Loads
5 roof height
h
5 24 ft
, 60 ft
Hence, in accordance with ASCE 7 Section 27.4.2, the building may be considered a Class 1 building.
From the problem statement, the building is an enclosed simple diaphragm structure. Hence, the net
wind pressures given in ASCE 7 Tables 27.5-1 and 27.5-2 are applicable. Also, in accordance with
ASCE 7 Section 27.5.1, use wind pressures tabulated for L/B 5 0.5. From ASCE 7 Table 27.5-1 for
exposure D, the windward wall net pressures are obtained by interpolation for L/B 5 0.5 as shown in
Table 2-11 to give
ph
5 80.9 lb/ft2
p0
5 78.8 lb/ft2
Table 2-11 Determination of ph and p0
ph (lb/ft2)
p0 (lb/ft2)
h (ft)
V 5 160
mph
V 5 170
mph
V 5 180
mph
V 5 160
mph
V 5 170
mph
V 5 180
mph
30
74.8
85.2
95.5
71.1
81.0
90.9
24
80.9
20
68.7
78.1
78.8
87.5
68.0
77.3
86.5
For a value of L/B 5 0.4, the sidewall pressures are given by ASCE 7 Table 27.5-1 Note 2 as
psidewall
5 0.54ph
5 0.54 3 80.9
5 43.7 lb/ft2
From ASCE 7 Table 27.5-2, three wind zones are applicable. These are zones 3, 4, and 5, as shown in
Figure 2-18. For the roof, the adjustment factor for exposure category D and a height (h) of 24 feet is
obtained by interpolation from Table 2-10 as
γ
5 1.183 1 6(1.201 2 1.183)/10
5 1.194
For the roof, for exposure category D at h 5 24 ft and V 5 170 mph, the net external pressure is
obtained by interpolating values from ASCE 7 Table 27.5-2, as shown in Table 2-12, and multiplying
by γ 5 1.194 to give
proof
5 1.194 3 262.5 . . . zone 3
5 274.63 lb/ft2
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proof
207
5 1.194 3 255.7 . . . zone 4
5 266.51 lb/ft2
proof
5 1.194 3 245.7 . . . zone 5
5 254.57 lb/ft2
Table 2-12 Determination of proof
Zone
h (ft)
3
4
5
30
265.7
258.6
248.1
24
262.5
255.7
245.7
20
260.4
253.8
244.1
The net design wind pressures for the main windforce-resisting system are shown in Figure 2-18. Minimum values of design wind pressure do not govern. Since the building is of two stories and has flexible diaphragms, torsional wind load cases 2 and 4 of ASCE 7 Figure 27.3-8 need not be considered.
Zone 3
Zone 4
Zone 5
12 ft
12 ft
16 ft
–74.63 lb/ft2
– 66.51 lb/ft2
Wind direction
–54.57 lb/ft2
80.9 lb/ft2
78.8 lb/ft2
Figure 2-18 Net wind pressures for Example 2-6
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Design for Wind Loads
The base shear acting normal to the 100-foot side is determined from the product of the external pressure and the area over which it occurs and is given by
Fw
5 100(80.9 1 78.8) 3 24/2000
5 191.6 kips
2.6 Analytical envelope design method for MWFRS
The envelope design method of ASCE 7 Chapter 28 Part 1 Section 28.3 is applicable to enclosed, partially enclosed, and open low-rise regular buildings that have a flat, gable, or hip roof with a height not
exceeding 60 feet and not exceeding the least horizontal dimension. In addition, the structure must not
have response characteristics making it subject to across wind loading, vortex shedding, or instability
due to galloping or flutter. Also, the structure must not be located at a site subject to channeling effects
or buffeting in the wake of upwind obstructions.
Wind pressure is calculated using the specified wind pressure equation as applicable to each building
surface. The method uses the envelope procedure to separate applied wind loads onto the windward
walls, leeward walls, and sidewalls of the building so as to correctly assess the forces in the members.
The procedure consists of the determination of the following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Figures 26.5-1 and 26.5-2)
•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
•
velocity pressure exposure coefficients, Kz, for the applicable exposure category . . . (Table 2-4)
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
ground elevation factor, Ke . . . (ASCE 7 Table 26.9-1)
•
directionality factor, Kd . . . (ASCE 7 Table 26.6-1)
•
enclosure classification . . . (ASCE 7 Section 26.12)
•
internal pressure coefficient, (GCpi) . . . (ASCE 7 Table 26.13-1)
•
wind velocity pressure, qh . . . [ASCE 7 Equation (26.10-1)]
•
external pressure coefficient, (GCpf) . . . (ASCE 7 Figure 28.3-1)
•
internal wind pressure, pi 5 qh(GCpi) . . . [ASCE 7 Equation (28.3-1)]
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•
external wind pressure, pe 5 qh(GCpf) . . . [ASCE 7 Equation (28.3-1)]
•
combined internal and external wind pressures, p . . . [ASCE 7 Equation (28.3-1)]
•
check minimum design wind loads . . . (ASCE 7 Section 28.3.4)
•
apply loading patterns to each building corner . . . (ASCE 7 Figure 28.3-1)
209
2.6.1 Design parameters
The following design parameters are determined as for the directional procedure:
•
risk category
•
basic wind speed, V
•
exposure category
•
ground elevation factor, Ke
•
topographic factor, Kzt
•
directionality factor, Kd
•
enclosure classification
•
internal pressure coefficient, (GCpi)
The velocity pressure exposure coefficients, Kz , are obtained from ASCE 7 Table 26.10-1, and for
exposure category B, differ from the values used in the directional procedure for a mean roof height
of 0 feet to 25 feet.
2.6.2 Wind velocity pressure
Taking a value for the ground elevation factor of Ke 5 1.0, the velocity pressure, q, at any height above
the ground is given by ASCE 7 Equation (26.10-1) as
q
5 0.00256Kz Kzt KdV2
The velocity pressure varies with the height above ground level since the value of the velocity pressure
exposure coefficient also varies with the height above ground level.
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Design for Wind Loads
2.6.3 Internal pressure coefficients and internal pressure
For the envelope procedure of ASCE 7 Chapter 28 Part 1, the gust effect factor is combined with the
internal pressure coefficients and denoted by (GCpi). Values of (GCpi) are tabulated in ASCE 7 Table
26.13-1 for the three different building enclosure classifications and are shown in Table 2-5, which is
repeated here.
Table 2-5 Values of internal pressure coefficients
Enclosure classification
Open buildings
(GCpi)
0.00
Partially open buildings
0.18
Partially enclosed buildings
0.55
Enclosed buildings
0.18
The pressure acting on internal surfaces is obtained from the second term of ASCE 7 Equation (28.3‑1)
as
where:
pi
5 6qh(GCpi)
qh
5 wind velocity pressure at mean roof height
(GCpi)
5 product of the gust effect factor and the internal pressure coefficient
Pressures act normal to wall and roof surfaces and are positive when acting toward the surface and
negative when acting away from the surface. The conditions that produce internal suction and internal
pressure are shown in Figure 2-8. Both cases must be considered for any building and added algebraically to external pressures to determine the most critical loading conditions.
2.6.4 External pressure coefficients and external pressure
For the envelope procedure of ASCE 7 Chapter 28 Part 1, the gust effect factor is combined with the
external pressure coefficients and denoted by (GCpf). Values of (GCpf) are tabulated in ASCE 7 Figure
28.3-1 and are shown in Tables 2-13 and 2-14. Values are given for two separate loading conditions.
These are load case A for wind acting transversely and load case B for wind acting longitudinally to
the building.
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Chapter 2
Table 2-13 External pressure coefficients load case A
Load case A
Roof
angle 
(degrees)
1
2
3
4
1E
2E
3E
4E
0–5
0.40
20.69
20.37
20.29
0.61
21.07
20.53
20.43
20
0.53
20.69
20.48
20.43
0.80
21.07
20.69
20.64
30–45
0.56
0.21
20.43
20.37
0.69
0.27
20.53
20.48
90
0.56
0.56
20.37
20.37
0.69
0.69
20.48
20.48
Building surface
Table 2-14 External pressure coefficients load case B
Load case B
Roof
angle 
(degrees)
0–90
Building surface
1
2
3
4
5
6
1E
2E
3E
4E
5E
6E
20.45 20.69 20.37 20.45 0.40 20.29 20.48 21.07 20.53 20.48 0.61 20.43
The pressure acting on external surfaces is obtained from the first term of ASCE 7 Equation (28.3-1) as
where:
pe
5 qh(GCpf)
qh
5 wind velocity pressure at mean roof height
(GCpf)
5 product of the gust effect factor and the external pressure coefficient
External pressure coefficients are given for two zones on each wall and roof surface: an end zone and
an interior zone. The end zone width is given by ASCE 7 Figure 28.3-1 as either a or 2a where a is
the lesser of
or
a
5 0.1 3 (least horizontal dimension)
a
5 0.4h
but not less than
either
a
5 0.04 3 (least horizontal dimension)
or
a
5 3 ft
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Design for Wind Loads
Pressures act normal to wall and roof surfaces and are positive when acting toward the surface and negative when acting away from the surface. The design wind pressure on the main windforce-resisting
system is given by ASCE 7 Equation (28.3-1) as
5 qh[(GCpf) 2 (GCpi)]
p
5 pe 2 pi
2.6.5 Design wind load cases
The envelope procedure utilizes pseudo external pressure coefficients derived from wind tunnel tests
on building models successively rotated through 360 degrees. The pseudo pressure cases envelope the
desired structural actions (bending moment, shear, and thrust) independent of the wind direction. To
ensure that all possible conditions are considered, ASCE 7 Figure 28.3.1 indicates that both load case
A and load case B must be applied in turn to all four corners of the building, giving eight loading cases.
Two of these loading cases are shown in Figure 2-19.
6
3
4
4
3
2
6E
3E
4E
θ
2
3E
2E
1
4E
5
θ
2E
1
1E
2a
5E
Windward
Corner
Load Case A
a
1E
2a
Load Case B
Wind
Direction
Windward
Corner
Wind
Direction
Figure 2-19 Loading cases A and B
For each of the eight cases, both positive and negative internal pressure must be considered, resulting
in a total of 16 loading cases. Where the building is symmetrical about one axis, only two corners need
to be investigated. If the building is doubly symmetrical, only one corner needs to be investigated.
Where torsion must be considered, each of these load cases is also modified, as indicated in ASCE 7
Figure 28.3-1 Note 5.
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Example 2-7
The two-story steel-frame office building with flexible diaphragms, shown in Figure 2-11, is located
adjacent to the shoreline in Miami, Florida. All glazing in the building is impact resistant. Determine
the external wind pressure acting on the whole building for wind flowing normal to the 100-foot-long
side and the resultant base shear. Use the analytical envelope method.
Solution
The relevant parameters are
V
Exposure category
qh
5 170 mph . . . from Example 2-2
5 D . . . from Example 2-2
5 wind velocity pressure at roof height
5 69.80 lb/ft2 . . . from Example 2-2
L
5 horizontal dimension of building measured parallel to the wind direction
5 40 ft
B
5 horizontal dimension of building normal to wind direction
5 100 ft
h
5 roof height
5 24 ft
, 60 ft
h/L
5 roof height/least horizontal dimension
5 24/40
5 0.6
, 1.0
Hence, in accordance with ASCE 7 Section 26.2, the building may be considered a low-rise structure
and the envelope method of ASCE 7 Section 28.3 is applicable. Values of (GCpf) may be obtained from
Table 2-13.
For a two-story building with flexible diaphragms, ASCE 7 Figure 28.3-1 Note 5 specifies that torsional load cases may be neglected.
To determine the base shear of the building, the external pressures on surfaces 1 and 1E on the windward face and 4 and 4E on the leeward face must be determined. Roof pressures and internal pressures
are not required.
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Design for Wind Loads
The pressure acting on external surfaces is obtained from the first term of ASCE 7 Equation (28.3-1) as
pe
5 qh(GCpf)
5 69.80(GCpf)
Values of pe are obtained from Table 2-13 and are tabulated in Table 2-15.
Table 2-15 External pressures for Example 2-7
Surface
1
1E
4
4E
(GCpf)
0.40
0.61
20.29
20.43
pe lb/ft2
27.92
42.58
220.24
230.01
The width of end zones 1E and 4E are given by ASCE 7 Figure 28.3-1 as the lesser of
2a
5 0.2 3 L
5 0.2 3 40
5 8 ft . . . governs
2a
≤ 0.8h
5 0.8 3 24
5 19.2 ft
but not less than
either
2a
5 0.08 3 L
5 0.08 3 40
5 3.2 ft
or
2a
5 6 ft
The base shear acting normal to the 100-foot side is determined from the product of the external pressure and the area over which it occurs and is given by
Fw
5 h[2a(pe1E 2 pe4E) 1 (B 2 2a)(pe1 2 pe4)]
5 24[8(42.58 1 30.01) 1 (100 2 8)(27.92 1 20.24)]/1000
5 120 kips
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2.7 Simplified envelope design procedure for MWFRS
The simplified method of ASCE 7 Chapter 28 Part 2 Section 28.5 is based on the envelope method
of ASCE 7 Chapter 28 Part 1. The combined windward wall and leeward wall wind pressures are
obtained directly from a table and applied to the windward vertical projected surface of the building.
Internal pressures are neglected as they cancel out. Roof pressures consist of combined external pressures and internal pressures appropriate to an enclosed building and are applied to the horizontal projection of the roof. In accordance with ASCE 7 Section 28.6.2, the simplified method may be applied
to the design of buildings complying with all the following conditions:
•
building is classified as an enclosed building as defined in ASCE 7 Section 26.2
•
building is a low-rise building as defined in ASCE 7 Section 26.2
•
building is a regular-shaped building as defined in ASCE 7 Section 26.2
•
building has an approximately symmetrical cross section with either a flat roof, or a gable or
hip roof with q ≤ 45°
•
building is a simple diaphragm building as defined in ASCE 7 Section 26.2
•
building is not classified as flexible as defined in ASCE 7 Section 26.2
•
building is not sensitive to dynamic effects and is not subject to channeling effects or buffeting
•
torsional loads do not govern or the building complies with the requirements of ASCE 7 Appendix D and is exempt from consideration of torsional load cases
2.7.1 Design procedure
The procedure consists of the determination of the following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Figures 26.5-1 and 26.5-2)
•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
net pressures at walls and roofs, ps30 . . . (ASCE 7 Figure 28.5-1)
•
if topographic factor Kzt . 1.0, apply factor to wall and roof pressures . . . (ASCE 7 Figure
26.8-1)
•
adjustment for building height and exposure category, l . . . (ASCE 7 Figure 28.5-1)
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Design for Wind Loads
•
check minimum design wind loads . . . (ASCE 7 Section 28.3.4)
•
apply design wind loads to each corner of the building in turn as the windward corner . . .
(ASCE 7 Figure 28.5-1 Note 2)
2.7.2 Adjustment of net pressures
The simplified method uses tabulated values of net design pressures for various wind speeds, which
are based on exposure classification B at a height of h 5 30 feet. For other exposure classifications
and mean roof heights, an adjustment factor, l, is applied to the tabulated values. These height and
exposure factors are provided in ASCE 7 Figure 28.5-1 and, for a limited number of conditions, are
reproduced in Table 2-16. The net adjusted pressures are given by ASCE 7 Equation (28.5-1) as
where:
ps
5 lKzt ps30
l
5 adjustment factor from Table 2-16
ps30
5 simplified design wind pressure for exposure B, at h 5 30 feet,
from ASCE 7 Figure 28.5-1
Kzt
5 topographic factor
The mean roof height is defined in ASCE 7 Section 26.2 as the average of the roof eave height to the
highest point on the roof surface, except that eave height is used for roof angles not exceeding 10
degrees.
Table 2-16 Height and exposure adjustment factors, l
Exposure
Mean roof
height, ft
B
C
D
15
1.00
1.21
1.47
20
1.00
1.29
1.55
25
1.00
1.35
1.61
30
1.00
1.40
1.66
40
1.09
1.49
1.74
50
1.16
1.56
1.81
60
1.22
1.62
1.87
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For the purpose of designating the variation of wind pressure over the building, the building surface is
divided into interior zones, edge strips, end zones, and corner zones. Edge strips and end zones have a
width of 2a, corner zones have a width of a. The dimension a is defined as the lesser of
or
a
5 0.1 3 (least horizontal dimension)
a
5 0.4h
but not less than
either
a
5 0.04 3 (least horizontal dimension)
or
a
5 3 ft
2.7.3 Simplified method applied to the MWFRS
As shown in Figure 2.20, the design wind pressures are assumed to act normal to the projected wall
and roof areas. The values tabulated in ASCE 7 Figure 28.5-1 for the roof are composite pressures,
which include the internal pressures appropriate to an enclosed building condition. Values given for the
walls represent the sum of the positive pressure on the windward face of the building and the negative
(or suction) pressure on the leeward face and are applied to the windward projection of the building as
shown. Internal pressures for the walls are not included since they cancel.
2a
Figure 2-20 Application of wind loads
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Design for Wind Loads
As specified in ASCE 7 Figure 28.5-1 Note 2, the longitudinal and the transverse loading patterns are
applied to each corner of the building in turn as the windward corner. However, where the building is
symmetrical about one axis, only two corners need to be investigated. If the building is doubly symmetrical, only one corner needs to be investigated.
The locations of end zones and interior zones for the main windforce-resisting system are shown in
Figure 2-21. For buildings that have flat roofs, a ridge line is assumed along the longitudinal axis of
the building. Values for the design wind pressure are given in ASCE 7 Figure 28.5-1 for wind flowing
parallel to the longitudinal axis of the building and for wind flowing transversely. Wind pressures are
positive when acting toward the projected surface and negative when acting away from the projected
surface. Values are provided for wind speeds of 85 to 200 miles per hour, roof slopes up to 45 degrees,
and overhangs.
Figure 2-21 Application of design wind pressures
Example 2-8
The regular two-story simple diaphragm steel-frame office building with flexible diaphragms shown
in Figure 2-11 and analyzed in Example 2-2 is located adjacent to the shoreline in Miami, Florida. All
glazing in the building is impact resistant. The building is not sensitive to dynamic effects and is not
subjected to buffeting or channeling effects. Using the simplified envelope method, determine the design
wind pressures acting on the whole building for the transverse wind direction and the resultant base shear.
Seismic and Wind Forces: Structural Design Examples
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Solution
From the problem statement and Figure 2-11:
•
glazing is impact resistant . . . the building is enclosed
h
5 24 ft
, 60 ft
h/L
5 24/40
5 0.6
, 1.0 . . . the building is low rise
•
the building is a regular-shaped building
•
the building has a symmetrical cross section with a flat roof
•
the building is a simple diaphragm building
h/L
5 0.6
, 4.0 . . . the building is not flexible
•
the building is not subjected to channeling effects or buffeting
•
the building is of two stories with flexible diaphragms and is exempt from consideration of
torsional load cases
Hence, the simplified envelope procedure is applicable.
The relevant parameters are
V
5 wind speed
5 170 mph
h
5 mean roof height
5 24 ft
The dimension a is given by ASCE 7 Figure 28.5-1 as the lesser of
a
5 0.1 3 L
5 0.1 3 40
5 4 ft . . . governs
or
a
5 0.4h
5 0.4 3 24
5 9.6 ft
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Design for Wind Loads
but not less than
either
a
5 0.04 3 L
5 0.04 3 40
5 1.6 ft
or
a
5 3 ft
2a
5 width of end zone
5 8 ft
Kzt
5 1.0
The exposure classification is D and the combined height and exposure adjustment factor is obtained
from Table 2-16 as
l
5 1.60
Design wind pressures interpolated from ASCE 7 Figure 28.5-1 are multiplied by 1.60 and are shown
in Table 2-17.
Table 2-17 Design wind pressures for Example 2-8
Zone
Wind
pressure,
lb/ft2
Zone
width,
ft
Zone
height,
ft
Horizontal
force,
kips
A
73.3
8
24
14.1
C
48.6
92
24
107.3
E
288.2
8
–
–
F
250.1
8
–
–
G
261.3
92
–
–
H
238.7
92
–
–
Total base shear
121.4
The horizontal force on each zone is obtained as the product of the zone pressure and the zone area and
is shown in Table 2-17 together with the total base shear.
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2.8 Components and cladding
Components and cladding are defined in ASCE 7 Section 26.2 as elements of the building envelope, which do not qualify as part of the main windforce-resisting system. The cladding of a building receives wind loading directly. Examples of cladding include wall and roof sheathing, windows,
and doors. Components receive wind loading from the cladding and transfer the load to the main
windforce-​resisting system. Components include purlins, studs, fasteners, and roof trusses. Some elements, such as roof trusses and sheathing, may also form part of the main windforce-resisting system
and must be designed for both conditions.
Because of local turbulence, which may occur over small areas and at ridges and corners of buildings,
components and cladding are designed for higher wind pressures than the main windforce-resisting
system.
The effective wind area is used to determine external pressure coefficients. This is defined in ASCE 7
Section 26.2 as
where:
A
5 bel
be
5 effective tributary width
l
5 element span length
≥ l/3
For cladding fasteners, the effective wind area shall be not greater than the area that is tributary to an
individual fastener.
In accordance with ASCE 7 Section 30.2.3, component and cladding elements with tributary areas
greater than 700 ft2 may be designed using provisions for the main windforce-resisting system.
In accordance with ASCE 7 Section 30.2.2, the design wind pressure for components and cladding
of buildings shall be not less than a net pressure of 16 lb/ft2 acting in either direction normal to the
surface.
2.8.1 Determination of components and cladding loads
Several procedures are available for determining the loads on components and cladding of buildings.
All procedures require compliance with the following conditions:
•
the structure is a regular-shaped building without irregularities such as projections or indentations
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Design for Wind Loads
•
the structure does not have response characteristics making it subject to across wind loading,
vortex shedding, or instability due to galloping or flutter
•
the structure is not located at a site subject to channeling effects or buffeting in the wake of
upwind obstructions
The available procedures include:
1. The analytical envelope design method of ASCE 7 Chapter 30 Part 1 Section 30.3. Wind pressures are calculated using equations specific to each building surface. This method is applicable
to buildings with all of the following characteristics:
•
enclosed and partially enclosed buildings
•
low-rise buildings as defined in ASCE 7 Section 26.2
•
buildings with h ≤ 60 ft
•
buildings that have flat roofs, gable roofs, multispan gable roofs, hip roofs, monoslope
roofs, stepped roofs, or sawtooth roofs
2. The simplified envelope design method of ASCE 7 Chapter 30 Part 2 Section 30.4. This method
is based on the analytical procedure of Part 1. Wind pressures are determined from a table and
adjusted for height and exposure. This method is applicable to buildings with all of the following characteristics:
•
enclosed buildings
•
low-rise buildings as defined in ASCE 7 Section 26.2
•
buildings with h ≤ 60 ft
•
buildings that have flat roofs, gable roofs, or hip roofs
3. The analytical directional design method of ASCE 7 Chapter 30 Part 3 Section 30.5. Wind pressures are calculated using equations specific to each building surface. This method is applicable
to buildings with all of the following characteristics:
•
enclosed and partially enclosed buildings
•
buildings with h . 60 ft
•
buildings that have flat roofs, gable roofs, pitched roofs, hip roofs, mansard roofs, or domed
roofs
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4. The simplified directional design method of ASCE 7 Chapter 30 Part 4 Section 30.6. This
method is based on the analytical procedure of Part 3. Wind pressures are determined from a
table and adjusted for height and exposure. This method is applicable to buildings with all of
the following characteristics:
•
enclosed buildings
•
buildings with 60 ft , h ≤ 160 ft
•
buildings that have flat roofs, gable roofs, monoslope roofs, and mansard roofs or hip roofs
2.9 Analytical envelope design method for components and cladding
The envelope design method of ASCE 7 Chapter 30 Part 1 Section 30.3 is applicable to enclosed and
partially enclosed low-rise regular buildings with a height not exceeding 60 feet and having a flat,
gable, multispan gable, hip, monoslope, stepped, or sawtooth roof. Wind pressure is calculated using
the specified wind pressure equation as applicable to each building surface. The procedure consists of
the determination of the following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Section 26.5)
•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
•
velocity pressure exposure coefficients, Kh, for the applicable exposure category . . . (ASCE 7
Table 26.10-1)
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
ground elevation factor, Ke . . . (ASCE 7 Table 26.9-1)
•
directionality factor, Kd . . . (ASCE 7 Table 26.6-1)
•
enclosure classification . . . (ASCE 7 Section 26.12)
•
internal pressure coefficient, (GCpi) . . . (ASCE 7 Table 26.13-1)
•
wind velocity pressure, qh . . . [ASCE 7 Equation (26.10-1)]
•
external pressure coefficient, (GCpf) . . . (ASCE 7 Figures 30.3-1 through 30.3-7)
•
internal wind pressure, pi 5 qh(GCpi) . . . [ASCE 7 Equation (30.3-1)]
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Design for Wind Loads
•
external wind pressure, pe 5 qh(GCpf) . . . [ASCE 7 Equation (30.3-1)]
•
combined internal and external wind pressures, p . . . [ASCE 7 Equation (30.3-1)]
•
check minimum design wind loads . . . (ASCE 7 Section 30.2.2)
2.9.1 Design parameters
The following design parameters are determined as for the main windforce-resisting system procedure:
•
risk category
•
basic wind speed, V
•
exposure category
•
topographic factor, Kzt
•
directionality factor, Kd
•
enclosure classification
•
internal pressure coefficient, (GCpi)
The velocity pressure exposure coefficients, Kh , are obtained from ASCE 7 Table 26.10-1.
2.9.2 Velocity pressure exposure coefficients and velocity pressure
The velocity pressure exposure coefficients for components and cladding are given in ASCE 7 Table
26.10-1.
Kz
5 2.01(15/zg)2/a . . . for z , 15 ft
5 2.01(z/zg)2/a . . . for 15 ft ≤ z ≤ zg
Values are tabulated in Table 2-18 for a limited number of heights.
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Chapter 2
Table 2-18 Velocity pressure exposure coefficients for components and cladding
Height above ground level, ft
Exposure
0–15
20
25
30
40
50
B
0.70
0.70
0.70
0.70
0.76
0.81
C
0.85
0.90
0.94
0.98
1.04
1.09
D
1.03
1.08
1.12
1.16
1.22
1.27
Taking the ground elevation factor as Ke 5 1, the velocity pressure, qh , at mean roof height is given by
ASCE 7 Equation (26.10-1) as
qh
5 0.00256Kh Kzt KdV2
2.9.3 Internal pressure coefficients and internal pressure
For the envelope procedure of ASCE 7 Chapter 30 Part 1, the gust effect factor is combined with the
internal pressure coefficients and denoted by (GCpi). Values of (GCpi) are tabulated in ASCE 7 Table
26.13-1 and the values for enclosed and partially enclosed buildings are shown in Table 2-19.
Table 2-19 Values of internal pressure coefficients
Enclosure classification
(GCpi)
Partially enclosed buildings
60.55
Enclosed buildings
60.18
The pressure acting on internal surfaces is obtained from the second term of ASCE 7 Equation (30.3‑1)
as
where:
pi
5 6qh(GCpi)
qh
5 wind velocity pressure at mean roof height
(GCpi)
5 product of the gust effect factor and the internal pressure coefficient
Pressures act normal to wall and roof surfaces and are positive when acting toward the surface and negative when acting away from the surface. Both cases must be considered for any building and added
algebraically to external pressures to determine the most critical loading conditions.
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Design for Wind Loads
Example 2-9
The two-story simple diaphragm steel-frame office building with flexible diaphragms, shown in Figure
2-11, is located adjacent to the shoreline in Miami, Florida. The structure is not sensitive to dynamic
effects and is not located on a site at which channeling effects or buffeting occurs. Determine the wind
velocity pressure at roof height for cladding and components. If all glazing in the building is impact
resistant, determine the internal pressure acting on cladding and components. Use the analytical envelope design method.
Solution
The analytical envelope design method of ASCE 7 Chapter 30 Part 1 Section 30.3 is applicable. For
a location on the shoreline, the exposure is category D and the wind speed, V, is obtained from ASCE
7 Figure 26.5-1B as 170 miles per hour. The relevant parameters are obtained from Example 2-2 as
Kh
5 velocity pressure exposure coefficient
5 1.11 . . . from Table 2-18 for a height of 24 ft for cladding and components
and exposure category D
Kzt
5 topographic factor
5 1.0 . . . from ASCE 7 Figure 26.8-1
Kd
5 wind directionality factor
5 0.85 . . . from ASCE 7 Table 26.6-1
Taking Ke 5 1.0, the velocity pressure, qh, at a roof height of 24 feet above the ground is given by
ASCE 7 Equation (26.10-1) as
qh
5 0.00256Kh Kzt Kd KeV2
5 0.00256 3 1.11 3 1.0 3 0.85 3 1.0 3 1702
5 69.80 lb/ft2 . . . at roof height
The product of the internal pressure coefficient and the gust effect factor for an enclosed building is
obtained from Table 2-19 as
(GCpi)
5 60.18
The internal pressure acting on all internal surfaces is given by the second term of ASCE 7 Equation
(30.3-1) as
pi
5 6qh(GCpi)
5 669.80 3 0.18
5 612.56 lb/ft2
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2.9.4 External pressure coefficients and external pressure
Because of local turbulence at corners and at roof eaves, an increase in pressure is produced in these
areas. Hence, as shown in Figure 2-22, walls are divided into two zones and roofs are divided into four
zones, with a different wind pressure coefficient assigned to each.
1.2h
0.6h
3
4
2
4
2
4
2h
1
4
1
4
0.
2
4
0.6h
1.
2h
4
1′
3
4
4
1′
h
3
4
5
2
4
1
4
5
a
0.2
2
4
a
2
4
4
4
3
4
5
a
5
a
Figure 2-22 Components and cladding loading diagram for roof slope  ≤ 7°
For the walls
The end zone width is given by ASCE 7 Figure 30.3-1 as a, where a is the lesser of
or
a
5 0.1 3 (least horizontal dimension)
a
5 0.4h
but not less than
either
a
5 0.04 3 (least horizontal dimension)
or
a
5 3 ft
Where the least horizontal dimension of the building is greater than 300 feet and the roof angle, θ, is
less than or equal to 7 degrees, the end zone width is given by the lesser of
or
a
5 0.1 × (least horizontal dimension)
a
5 0.8h
Seismic and Wind Forces: Structural Design Examples
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Design for Wind Loads
For the roof
The zone width is given by ASCE 7 Figure 30.3-2A as
a
5 0.6h
The values of (GCp) are a function of the effective area attributed to the element considered and are
given in ASCE 7 Figure 30.3-1 for walls and ASCE 7 Figure 30.3-2A for roofs with a slope of q ≤
7 degrees. The values may also be derived from logarithmic expressions given in Tables C30.3-1 and
C30.3-2.
The pressure acting on external surfaces is obtained from the first term of ASCE 7 Equation (30.3-1) as
where:
pe
5 qh(GCp)
qh
5 wind velocity pressure at mean roof height
(GCp)
5 product of the gust effect factor and the external pressure coefficient
Pressures act normal to wall and roof surfaces and are positive when acting toward the surface and
negative when acting away from the surface. The design wind pressure on components and cladding is
given by ASCE 7 Equation (30.3-1) as
p
5 qh[(GCp) 2 (GCpi)]
5 pe 2 pi
For cladding fasteners, the effective wind area shall be not greater than the area that is tributary to
an individual fastener. In accordance with ASCE 7 Figure 30.3-1 Note 5, the values of (GCp) may be
reduced by 10 percent for the walls of buildings with a roof slope of 10 degrees or less.
Example 2-10
The two-story steel-frame office building, shown in Figure 2-11, is located adjacent to the shoreline in
Miami, Florida. The roof framing consists of open web joists at s 5 5-foot centers spanning 40 feet,
and all glazing in the building is impact resistant. Determine the design wind pressure acting on an
interior roof joist.
Solution
The relevant parameters are
The velocity pressure, qh , at a roof height of 24 feet above the ground is given by Example 2-9 as
qh
5 69.80 lb/ft2 . . . at roof height
The interior pressure acting on all internal surfaces is obtained from Example 2-9 as
pi
5 612.56 lb/ft2
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The width of zone 1 and zone 2 is given by ASCE 7 Figure 30.3-2A as
a
5 0.6 3 h
5 0.6 3 24
5 14.4 ft . . . governs
The combined width of zones 1 and 2 is
Σa
5 4 3 14.4 ft
5 57.6 ft
This exceeds the span of the joist, which is
l
5 40 ft
Hence, only zones 1 and 2 are applicable to the joist and zone 1 does not develop.
The effective tributary width of a roof joist is defined in ASCE 7 Section 26.2 as the larger of
be
5 joist spacing
5 5 ft
or
be
≥ l/3
5 40/3
5 13.33 ft . . . governs
The effective wind area attributed to the roof joist is then
A
5 bel
5 13.33 3 40
5 533.2 ft2
The negative external pressure coefficient for zone 1 is obtained from ASCE 7 Figure 30.3‑2A as
(GCp)
5 21.0
The negative design wind pressure on a roof joist for zone 1 is obtained from ASCE 7 Equation
(30.3‑1) as
p
5 qh(GCp) 2 qh(GCpi)
5 pe2 pi
5 69.80 3 (21.0) 2 12.56
5 282.36 lb/ft2
Seismic and Wind Forces: Structural Design Examples
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Design for Wind Loads
The upward load on the roof joist over zone 1 is
w
5 ps
5 282.36 3 5
5 2412 lb/ft
The positive external pressure coefficient for zone 1 is obtained from ASCE 7 Figure 30.3-2A as
(GCp)
5 0.2
The positive design wind pressure on a roof joist for zone 1 is obtained from ASCE 7 Equation (30.3‑1)
as
p
5 qh(GCp) 1 qh(GCpi)
5 pe 1 pi
5 69.80 3 0.2 1 12.56
5 26.52 lb/ft2
The downward load on the roof joist over zone 1 is
w
5 ps
5 26.52 3 5
5 133 lb/ft
The negative external pressure coefficient for eave zone 2 is obtained from ASCE 7 Figure 30.3-2A as
(GCp)
5 21.4
The negative design wind pressure on a roof joist for eave zone 2 is obtained from ASCE 7 Equation
(30.3-1) as
p
5 qh(GCp) 2 qh(GCpi)
5 69.80 3 (21.4) 2 12.56
5 2110 lb/ft2
The upward load on the roof joist over eave zone 2 is
w
5 ps
5 2110 3 5
5 2550 lb/ft
The positive external pressure coefficient for eave zone 2 is obtained from ASCE 7 Figure 30.3-2A as
(GCp)
5 0.2
Seismic and Wind Forces: Structural Design Examples
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The positive design wind pressure on a roof joist for eave zone 2 is obtained from ASCE 7 Equation
(30.3-1) as
5 qh(GCp) 1 qh(GCpi)
p
5 69.80 3 0.2 1 12.56
5 26.52 lb/ft2
The downward load on the roof joist over eave zone 2 is
w
5 ps
5 26.52 3 5
5 133 lb/ft
The wind loading acting on the roof joist is shown in Figure 2-23.
–412 lb/ft
–550 lb/ft
–550 lb/ft
133 lb/ft
14.4 ft
11.2 ft
14.4 ft
Upward load
40 ft
Downward load
Figure 2-23 Wind loading on roof joist
2.10 Simplified envelope design method for components and cladding
The simplified envelope design method of ASCE 7 Chapter 30 Part 2 Section 30.4 is applicable to
enclosed low-rise regular buildings with a height not exceeding 60 feet and having flat, gable, or hip
roof shapes. Wind pressures are determined from a table and adjusted for height and exposure. The
procedure consists of the determination of the following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Section 26.5)
•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
Seismic and Wind Forces: Structural Design Examples
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Design for Wind Loads
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
net pressures at walls and roofs, pnet30 . . . (ASCE 7 Figure 30.4-1)
•
adjustment for building height and exposure category, l . . . (ASCE 7 Figure 30.4-1)
•
check minimum design wind pressure . . . (ASCE 7 Section 30.2.2)
2.10.1 Design parameters
The following design parameters are determined as for the main windforce-resisting system procedure:
•
risk category
•
basic wind speed, V
•
exposure category
•
topographic factor, Kzt
The net pressure values, pnet30, are obtained from ASCE 7 Figure 30.4-1, and the adjustment factors for
building height and exposure category, l, are obtained from ASCE 7 Figure 30.4-1.
2.10.2 Adjustment of net pressures
The simplified method uses tabulated values of net design pressures for various wind speeds, which
are based on exposure classification B at a height of h 5 30 feet. For other exposure classifications and
mean roof heights, an adjustment factor, l, is applied to the tabulated values. These height and exposure adjustment coefficients are provided in ASCE 7 Figure 30.4-1. The net pressures are also based on
the effective wind area of the element under consideration, which is defined in ASCE 7 Section 26.2 as
where:
A
5 bel
be
5 effective tributary width
l
5 element span length
≥ l/3
A positive value for the pressure indicates that the pressure is acting toward the surface. A negative
value for the pressure indicates that the pressure is acting away from the surface. The mean roof height
is defined in ASCE 7 Section 26.2 as the average of the roof eave height to the highest point on the roof
surface, except that eave height is used for roof angles not exceeding 10 degrees.
Seismic and Wind Forces: Structural Design Examples
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2.10.3 Net wind pressure
Because of local turbulence at corners and at roof eaves, an increase in pressure is produced in these
areas. Hence, as shown in ASCE 7 Figure 30.4-1, for roofs with a slope of θ ≤ 7 degrees, the building
surface is divided into interior zones, edge strips, and corner zones. Walls are divided into two zones
and roofs are divided into four zones, with a different net wind pressure assigned to each. The net
pressure values tabulated in ASCE 7 Figure 30.4-1 are composite pressures that include the interior
pressures appropriate to an enclosed building condition.
The applicable zone widths are given by ASCE 7 Figure 30.4-1.
In each zone, values are provided for wind speeds of 95 to 200 miles per hour, for gable roof slopes
up to 45 degrees, and for overhangs. The net design wind pressure is given by ASCE 7 Equation
(30.4-1) as
where:
pnet
5 lKzt pnet30
l
5 adjustment factor from ASCE 7 Figure 30.4-1
Kzt
5 topographic factor from ASCE 7 Figure 26.8-1
pnet30
5 net wind pressure for exposure B, at h 5 30 ft, from ASCE 7 Figure 30.4-1
Example 2-11
The regular two-story simple diaphragm steel-frame office building with flexible diaphragms shown in
Figure 2-11 is located adjacent to the shoreline in Miami, Florida. The roof framing consists of open
web joists at 5-foot centers spanning 40 feet, and all glazing in the building is impact resistant. The
building is not sensitive to dynamic effects and is not subject to buffeting or channeling effects. Using
the simplified envelope design method, determine the design wind pressure acting on an interior roof
joist.
Solution
The relevant parameters are
V
5 wind speed
5 170 mph
h
5 mean roof height
5 24 ft
Kzt
5 topographic factor
5 1.0
Seismic and Wind Forces: Structural Design Examples
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Design for Wind Loads
a
5 width of zone 1 and zone 2 from ASCE 7 Figure 30.4-1
5 0.6 3 h
5 0.6 3 24
5 14.4 ft
The combined width of zones 1 and 2 is
Σa
5 4 3 14.4 ft
5 57.6 ft
This exceeds the span of the joist, which is
l
5 40 ft
Hence, only zones 1 and 2 are applicable to the joist and zone 1 does not develop.
The effective tributary width of a roof joist is defined in ASCE 7 Section 26.2 as the larger of
be
5 joist spacing
5 5 ft
or
be
≥ l/3
5 40/3
5 13.33 ft . . . governs
The effective wind area used to determine the external pressure coefficients is then
A
5 bel
5 13.33 3 40
5 533.2 ft2
The exposure classification is D and the combined height and exposure adjustment factor is obtained
from ASCE 7 Figure 30.4-1 as
l
5 1.59
Hence, design wind pressures interpolated from ASCE 7 Figure 30.4-1 are multiplied by 1.59.
The negative design wind pressure on a roof joist for zone 1 is obtained from ASCE 7 Equation
(30.4‑1) as
pnet
5 lKzt pnet30
5 1.59 3 1.0 3 (264.7)
5 2102.87 lb/ft2
Seismic and Wind Forces: Structural Design Examples
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The upward load on the roof joist over interior zone 1 is
w
5 pnet30s
5 2102.87 3 5
5 2514 lb/ft
The positive design wind pressure on a roof joist for zone 1 is obtained from ASCE 7 Figure 30.4-1 as
pnet
5 lKzt pnet30
5 1.59 3 1.0 3 16.7
5 26.55 lb/ft2
The downward load on the roof joist over interior zone 1 is
w
5 pnet30s
5 26.55 3 5
5 133 lb/ft
The negative design wind pressure on a roof joist for eave zone 2 is obtained from ASCE 7 Figure
30.4-1 as
pnet
5 lkzt pnet30
5 1.59 3 1.0 3 (285.9)
5 2136.58 lb/ft2
The upward load on the roof joist over eave zone 2 is
w
5 pnet30s
5 2136.58 3 5
5 2683 lb/ft
The positive design wind pressure on a roof joist for eave zone 2 is obtained from ASCE 7 Figure
30.4-1 as
pnet
5 lKzt pnet30
5 1.59 3 1.0 3 16.7
5 26.55 lb/ft2
The downward load on the roof joist over eave zone 2 is
w
5 pnet30s
5 26.55 3 5
5 133 lb/ft
Seismic and Wind Forces: Structural Design Examples
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Design for Wind Loads
The wind loading acting on the roof joist is shown in Figure 2-24.
–514 lb/ft
–683 lb/ft
–683 lb/ft
133 lb/ft
14.4 ft
11.2 ft
14.4 ft
40 ft
Downward load
Upward load
Figure 2-24 Wind loading on roof joist
2.11 Analytical directional design method for components and cladding
The analytical directional design method of ASCE 7 Chapter 30 Part 3 Section 30.5 is applicable to
enclosed and partially enclosed regular buildings with a height exceeding 60 feet and having a flat,
pitched, gable, hip, mansard, arched, or domed roof. Wind pressure is calculated using the specified
wind pressure equation as applicable to each building surface. The procedure consists of the determination of the following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Section 26.5)
•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
•
velocity pressure exposure coefficients, Kz or Kh, for the applicable exposure category . . .
(ASCE 7 Table 26.10-1)
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
ground elevation factor, Ke . . . (ASCE 7 Table 26.9-1)
•
directionality factor, Kd . . . (ASCE 7 Table 26.6-1)
•
enclosure classification . . . (ASCE 7 Section 26.12)
•
internal pressure coefficient, (GCpi) . . . (ASCE 7 Table 26.13-1)
•
wind velocity pressure, qz . . . [ASCE 7 Equation (26.10-1)]
Seismic and Wind Forces: Structural Design Examples
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•
external pressure coefficient, (GCp) . . . (ASCE 7 Figure 30.5-1)
•
internal wind pressure, pi 5 qz(GCpi) . . . [ASCE 7 Equation (30.5-1)]
•
external wind pressure, pe 5 qz(GCp) . . . [ASCE 7 Equation (30.5-1)]
•
combined internal and external wind pressures, p . . . [ASCE 7 Equation (30.5-1)]
•
check minimum design wind loads . . . (ASCE 7 Section 30.2.2)
237
2.11.1 Design parameters
The following design parameters are determined as for the main windforce-resisting system procedure:
•
risk category
•
basic wind speed, V
•
exposure category
•
topographic factor, Kzt
•
directionality factor, Kd
•
enclosure classification
•
internal pressure coefficient, (GCpi)
The velocity pressure exposure coefficients, Kz or Kh , are obtained from ASCE 7 Table 26.10-1.
2.11.2 Velocity pressure exposure coefficients and velocity pressure
The velocity pressure exposure coefficients for components and cladding are given in ASCE 7 Table
26.10-1.
Kz
5 2.01(z/zg)2/a . . . for 15 ft ≤ z ≤ zg
Taking Ke 5 1, the velocity pressure, qh , at mean roof height is given by ASCE 7 Equation (26.10-1) as
qh
5 0.00256Kh Kzt KdV2
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Design for Wind Loads
2.11.3 Internal pressure coefficients and internal pressure
For the directional procedure of ASCE 7 Chapter 30 Part 3, the gust effect factor is combined with the
internal pressure coefficients and denoted by (GCpi). Values of (GCpi) are tabulated in ASCE 7 Table
26.13-1, and the values for enclosed and partially enclosed buildings are shown in Table 2-20.
Table 2-20 Values of internal pressure coefficients
Enclosure classification
(GCpi)
Partially enclosed buildings
60.55
Enclosed buildings
60.18
The pressure acting on internal surfaces is obtained from the second term of ASCE 7 Equation
(30.5‑1) as
where:
pi
5 6qi (GCpi)
qi
5 qh . . . for all surfaces of enclosed buildings and for negative internal
pressure evaluation in partially enclosed buildings
5 wind velocity pressure at roof height
or
qi
5 qz . . . for positive internal pressure evaluation in partially enclosed
buildings
5 wind velocity pressure at level of the highest opening that can affect the
positive internal pressure in partially enclosed buildings
and
z
5 height of the highest opening that can affect the positive internal pressure
(GCpi)
5 product of the gust effect factor and the internal pressure coefficient from
Table 2-5
For a conservative evaluation of positive internal pressure
qi
5 qh
Pressures act normal to wall and roof surfaces and are positive when acting toward the surface and negative when acting away from the surface. Both cases must be considered for any building and added
algebraically to external pressures to determine the most critical loading conditions.
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2.11.4 External pressure coefficients and external pressure
Because of local turbulence at corners and at roof eaves, an increase in pressure is produced in these
areas. Hence, as shown in Figure 2-25, walls are divided into two zones and roofs are divided into three
zones, with a different wind pressure coefficient assigned to each.
Figure 2-25 Components and cladding loading diagram for roof slope  ≤ 7°
The end zone width is given by ASCE 7 Figure 30.5-1 as a, where a is given as
a
5 0.1 3 (least horizontal dimension)
but not less than
a
5 3 ft
The values of (GCp) are a function of the effective area attributed to the element considered and are
given in ASCE 7 Figure 30.5-1 for walls and for roofs with a slope of q ≤ 10 degrees. The values may
also be derived6 from logarithmic expressions, and these are tabulated in Table 2-21 in terms of log to
the base 10 of the effective area.
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Design for Wind Loads
Table 2-21 Values of external pressure coefficients (GCp) for roof slope  ≤ 10°
Zone
1. Roof interior
2. Roof eaves
3. Roof corner
4. Wall interior
5. Wall corner
Effective area A, ft2
(GCp)
A ≤ 10 ft2
21.4
10 , A ≤ 500 ft2
21.6943 1 0.2943 log A
A . 500 ft2
20.9
A ≤ 10 ft2
22.3
10 , A ≤ 500 ft2
22.7120 1 0.4120 log A
A . 500 ft2
21.6
A ≤ 10 ft2
23.2
10 , A ≤ 500 ft2
23.7297 1 0.5297 log A
A . 500 ft2
22.3
A ≤ 20 ft2
20.9 or 0.9
20 , A ≤ 500 ft2(2ve)
21.0861 1 0.1431 log A
20 , A ≤ 500 ft2(1ve)
1.1792 2 0.2146 log A
2
A . 500 ft
20.7 or 0.6
A ≤ 20 ft2
21.8 or 0.9
20 , A ≤ 500 ft2 (2ve)
22.5445 1 0.5723 log A
20 , A ≤ 500 ft2 (1ve)
1.1792 2 0.2146 log A
A . 500 ft2
21.0 or 0.6
The pressure acting on external surfaces is obtained from the first term of ASCE 7 Equation (30.5-1) as
where:
pe
5 q(GCp)
q
5 qh . . . use with negative values of (GCp) for leeward walls, sidewalls, and
roof, evaluated at a height, h
5 wind velocity pressure at roof height
or
q
5 qz . . . use with positive values of (GCp) for windward walls evaluated at
height z above the ground
5 wind velocity pressure at a specific height, z, above the ground
and
z
(GCp)
5 any specific height above the ground
5 product of the gust effect factor and the external pressure coefficient from
ASCE 7 Figure 30.5-1
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For the windward wall, the external pressure coefficient increases with height. For the leeward wall
and the sidewalls, the external pressure coefficient is constant over the height of the wall. The design
wind pressure on components and cladding is given by ASCE 7 Equation (30.5-1) as
p
5 q(GCp) 2 qi(GCpi)
5 pe2 pi
2.12 Simplified directional design method for components and cladding
The simplified directional design method of ASCE 7 Chapter 30 Part 4 Section 30.6 is applicable to
enclosed regular buildings with a height greater than 60 feet and not exceeding 160 feet and having
a flat, gable, hip, mansard, or monoslope roof. Wind pressure is selected directly from a table and
adjusted for exposure category and effective wind area. The procedure consists of the determination
of the following items:
•
risk category I, II, III, or IV . . . (ASCE 7 Table 1.5-1)
•
basic wind speed, V, for the applicable risk category . . . (ASCE 7 Section 26.5)
•
exposure category B, C, or D . . . (ASCE 7 Section 26.7)
•
topographic factor, Kzt . . . (ASCE 7 Figure 26.8-1)
•
net pressures at walls and roofs, ptable . . . (ASCE 7 Table 30.6-2)
•
adjustment for effective area, RF, and exposure category, EAF . . . [ASCE 7 Equation (30.6-1)]
•
check minimum design wind pressure . . . (ASCE 7 Section 30.2.2)
2.12.1 Design parameters
The following design parameters are determined as for the main windforce-resisting system procedure:
•
risk category
•
basic wind speed, V
•
exposure category
•
topographic factor, Kzt
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Design for Wind Loads
The net pressure values, ptable , are obtained from ASCE 7 Table 30.6-2. The adjustment factors for
building height and exposure category, EAF, are obtained from ASCE 7 Table 30.6-2. The adjustment
factors for effective area, RF, are obtained from ASCE 7 Table 30.6-2.
2.12.2 Adjustment of net pressures
The simplified directional method uses tabulated values of net design pressures for various wind
speeds, which are based on exposure classification C at a height of h 5 30 feet. For other exposure
classifications and mean roof heights, an exposure adjustment factor, EAF, is applied to the tabulated
values. The height and exposure adjustment factors are provided in ASCE 7 Table 30.6-2. The net pressures are also based on an effective wind area of 10 ft2 and must be adjusted for larger areas, depending
on the type of roof and the location on the surface. The area reduction factors, RF, are provided in
ASCE 7 Table 30.6-2. The effective area of an element is defined in ASCE 7 Section 26.2 as
where:
A
5 bel
be
5 effective tributary width
l
5 element span length
≥ l/3
A positive value for the pressure indicates that the pressure is acting toward the surface. A negative
value for the pressure indicates that the pressure is acting away from the surface. The mean roof height
is defined in ASCE 7 Section 26.2 as the average of the roof eave height to the highest point on the roof
surface, except that eave height is used for roof angles not exceeding 10 degrees.
2.12.3 Net wind pressure
Because of local turbulence at corners and at roof eaves, an increase in pressure is produced in these
areas. Hence, as shown in ASCE 7 Table 30.6-2 and Figure 2-26, the building surface is divided into
interior zones, edge strips, and corner zones. Walls are divided into two zones and roofs are divided
into three zones, with a different net wind pressure assigned to each. The net pressure values tabulated
in ASCE 7 Table 30.6-2 are composite pressures that include the interior pressures appropriate to an
enclosed building condition.
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2a
a
3
2
2
2
a
1
3
1
2a
3
5
2
5
2
a
2
a
4
4
3
5
a
5
a
Figure 2-26 Components and cladding loading diagram for roof slope  ≤ 7° and h . 60 feet
For a building height exceeding 60 feet
The end zone width is given by ASCE 7 Table 30.6-2 as a, where a is given as
a
5 0.1 3 (least horizontal dimension)
but not less than
a
5 3 ft
In each zone, values are provided for wind speeds of 110 to 200 miles per hour. Interpolation between
h values is permitted. The required pressure, pr , for a wind speed, Vr , that is not provided in the table
is given by
where:
pr
5 ptable(Vr /Vtable)2
ptable
5 tabulated pressure from ASCE 7 Table 30.6-2 for wind speed, Vtable
Vr
5 wind speed at which pressure is required
Vtable
5 tabulated wind speed
For the design of parapets and overhangs, modifications are made to roof pressure and wall pressure
values as indicated in ASCE 7 Figures 30.6-1 and 30.6-2.
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Design for Wind Loads
The net design wind pressure is given by ASCE 7 Equation (30.6-1) as
where:
p
5 ptable(EAF)(RF)Kzt
RF
5 effective area reduction factor from ASCE 7 Table 30.6-2
EAF
5 exposure adjustment factor from ASCE 7 Table 30.6-2
Kzt
5 topographic factor as defined in ASCE 7 Figure 26.8-1
For a building height not exceeding 60 feet
For flat, hip, gable, monoslope, and mansard roofs with h ≤ 60 feet, roof pressures are obtained from
ASCE 7 Chapter 30 Part 2 and ASCE 7 Figure 30.4-1. Wind loads are determined using the procedure
of ASCE 7 Chapter 30 Part 2 Section 30.4.
For a building height of h ≤ 60 feet and a roof slope of θ ≤ 7 degrees, the zone widths are given by
ASCE 7 Table 30.6-2. Walls are divided into two zones and a roof is divided into four zones. The zone
width is
a
5 0.6h
Example 2-12
The regular six-story simple diaphragm steel-frame office building with flexible diaphragms, shown in
Figure 2-27, is located adjacent to the shoreline in Miami, Florida. The roof framing consists of open
web joists at 5-foot centers spanning 40 feet, and all glazing in the building is impact resistant. The
structure is not sensitive to dynamic effects and is not located at a site subject to channeling effects or
buffeting in the wake of upwind obstructions. Determine the design wind pressure acting on an interior
roof joist. Use the simplified directional design method.
Wind
Direction
h = 72 ft
B = 100 ft
L = 40 ft
Figure 2-27 Details for Example 2-12
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Solution
The height of 72 feet exceeds 60 feet and simplified directional design method of ASCE 7 Chapter 30
Part 4 Section 30.6 is applicable. The relevant parameters from previous examples are
V
5 wind speed
5 170 mph
exposure category
h
5D
5 mean roof height
5 72 ft
. 60 ft . . . ASCE 7 Section 30.6 is applicable
Kzt
5 topographic factor
5 1.0
The width of zone 2 is given by ASCE 7 Figure 30.6-1 as
a
5 0.1 3 L
5 0.1 3 40
5 4 ft . . . governs
but not less than
a
5 3 ft
The effective tributary width of a roof joist is defined in ASCE 7 Section 26.2 as the larger of
be
5 joist spacing
5 5 ft
or
be
≥ l/3
5 40/3
5 13.33 ft . . . governs
The effective wind area attributed to the roof joist is then
A
5 bel
5 13.33 3 40
5 533.2 ft2
The exposure adjustment factor is obtained from ASCE 7 Table 30.6-2 as
EAF
5 1.146
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Design for Wind Loads
The reduction factor for effective wind area is obtained from ASCE 7 Table 30.6-2 as
RF
5 NA . . . for roof zones 1 and 2, use a conservative value of 1.0
Hence, wind pressures from ASCE 7 Table 30.6-2 must be multiplied by the factor
(EAF)(RF)Kzt
5 1.146 3 1.0 3 1.0
5 1.146
To obtain wind pressures for a wind speed of 170 miles per hour, tabulated wind pressure values for a
wind speed of 160 miles per hour are multiplied by the factor
(170/160)2
5 1.13
Interpolation for h 5 72 feet is shown in Table 2-22.
Table 2-22 Determination of net wind pressures
Zone 1
Zone 2
h (ft)
V 5 160 mph
V 5 170 mph
V 5 160 mph
V 5 170 mph
80
2106.3
2120.1
2166.8
2188.5
72
2117.4
70
2103.3
2165.0
2116.7
2162.2
2159.1
The design wind pressure on a roof joist for interior zone 1 is
p
5 ptable(EAF)(RF)Kzt
5 2117.4 3 1.146
5 2134.5 lb/ft2
The upward load on the roof joist over interior zone 1 is
w
5 ps
5 2134.5 3 5
5 2673 lb/ft
The design wind pressure on a roof joist for eave zone 2 is
p
5 ptable(EAF)(RF)Kzt
5 2165.0 3 1.146
5 2189.1 lb/ft2
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The upward load on the roof joist over interior zone 2 is
w
5 ps
5 2189.1 3 5
5 2946 lb/ft
The wind loading acting on the roof joist is shown in Figure 2-28.
–946 lb/ft
–946 lb/ft
–673 lb/ft
4 ft
32 ft
4 ft
Figure 2-28 ​Wind loading on roof joist for Example 2-12
References
1. International Code Council. 2018 International Building Code. Washington, DC, 2018.
2. American Society of Civil Engineers. Minimum Design Loads and Associated Criteria for Buildings and Other Structures: ASCE 7-16. Reston, VA, 2016.
3. International Code Council. Standard for Residential Construction in High-Wind Regions. ICC
600-14. Washington, DC, 2014.
4. American Wood Council. Wood Frame Construction Manual for One- and Two-Family Dwellings: SBC High Wind Edition. WFCM-18. Washington, DC, 2018.
5. American Iron and Steel Institute. Standard for Cold-Formed Steel Framing—Prescriptive Method
for One- and Two-Family Dwellings. AISI S230-15. Washington, DC, 2015.
6. Mehta, K. C. and Perry, D. C. Guide to the Use of the Wind Load Provisions of ASCE 7-98. ASCE
Press. Reston, VA, 2002.
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Seismic Design of Steel Structures
Nomenclature
Ab
cross-sectional area of a horizontal boundary element
in2
Ac
cross-sectional area of a vertical boundary element
in2
Ag
gross area
in2
Alw
web area of link (excluding flanges)
in2
Asc
cross-sectional area of the yielding segment of steel core
in2
Ast
horizontal cross-sectional area of the link stiffener
in2
Ca
ratio of required strength to available axial yield strength
–
Cd
coefficient relating relative brace stiffness and curvature
–
D
dead load
kips
D
outside diameter of round HSS
in
E
seismic load effect
kips
E
modulus of elasticity of steel 5 29,000
ksi
Emh
horizontal seismic load effect, including the overstrength factor
kips, kip-in
Fe
elastic critical buckling stress
ksi
Fcr
critical stress
ksi
Fcre
critical stress calculated using expected yield stress
ksi
Fy
specified minimum yield stress
ksi
Fyb
specified minimum yield stress of beam
ksi
Fyc
specified minimum yield stress of column
ksi
Fu
specified minimum tensile strength
ksi
I
moment of inertia
in4
Ib
moment of inertia of a horizontal boundary element
in4
Ic
moment of inertia of a vertical boundary element
in4
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K
effective length factor
–
L
live load due to occupancy and moveable equipment
kips
L
length of column
in
L
length of brace
in
L
distance between vertical boundary element centerlines
in
Lb
unbraced length
in
Lcf
clear distance between column flanges
in
Lh
distance between beam plastic hinge locations
in
Ma
required flexural strength, using ASD load combinations
kip-in
Mf
maximum probable moment at the column face
kip-in
Mp
plastic bending moment
kip-in
Mp
plastic bending moment of a link
kip-in
Mp,exp expected flexural strength
kip-in
Mpr
maximum probable moment at the location of the plastic hinge
kip-in
Mr
required flexural strength
kip-in
Mu
required flexural strength, using LRFD load combinations
kip-in
My
yield moment corresponding to yielding of the member in flexure
kip-in
Pa
required axial strength using ASD load combinations
kips
Pc
available axial strength
kips
Pn
nominal axial compressive strength
kips
Pr
required axial compressive strength
kips
Pu
required axial strength using LRFD load combinations
kips
Py
axial yield strength
kips
Pysc
axial yield strength of steel core
kips
QE
effect of horizontal seismic forces
kips, kip-in
R
seismic response modification coefficient
–
Ra
required tensile strength using ASD load combinations
kips
Rn
nominal strength
kips
Rt
ratio of the expected tensile strength to the specified minimum tensile strength
–
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Ry
ratio of the expected yield stress to the specified minimum yield stress
–
S
snow load
kips
SDS
design spectral response acceleration at a period of 0.2 second
ft/sec2
Sh
hinge location distance from face of column
in
Va
required shear strength using ASD load combinations
kips
Vn
nominal shear strength of link
kips
Vp
plastic shear strength of a link
kips
Vr
required shear strength using LRFD or ASD load combinations
kips
Vu
required shear strength using LRFD load combinations
kips
Vy
shear yield strength
kips
Z
plastic section modulus about the axis of bending
in3
Zc
plastic section modulus of column about the axis of bending
in3
bbf
width of beam flange
in
bf
width of flange
in
d
overall depth of beam
in
d
overall depth of link
in
d*
distance between centroids of beam flanges
in
e
length of link
in
h
clear distance between flanges less the fillet for rolled shapes
in
h
distance between horizontal boundary element centerlines
in
ho
distance between flange centroids
in
r
governing radius of gyration
in
sh
hinge location distance from center of column
in
t
thickness of column web or individual doubler plate
in
tbf
thickness of beam flange
in
tf
thickness of flange
in
tw
thickness of web
in
tw
web-plate thickness
in
wz
width of panel zone between column flanges
in
251
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Symbols
αs
force level adjustment factor 5 1.0 for LRFD and 1.5 for ASD
–
β
compressive strength adjustment factor
–
γtotal
total link rotation angle
rad
Δ
design story drift
in
Δb
total brace axial deformation for the brace test specimen
in
θ
story drift angle
rad
λhd
slenderness parameter for highly ductile compression elements
–
λmd
slenderness parameter for moderately ductile compression elements
–
f
resistance factor
–
fc
resistance factor for compression
–
fv
resistance factor for shear
–
w
strain hardening adjustment factor
–
Ωc
safety factor
–
Ωc
safety factor for compression
–
Ω0
system overstrength factor
–
3.1 General design requirements
In accordance with IBC1 Section 2205.2, steel building structures assigned to seismic design category
D, E, or F must be designed and detailed as specified by AISC 341.2 In accordance with IBC Section
2205.2.2, steel building structures assigned to seismic design category B or C may also be designed
and detailed as specified by AISC 341. In this case, the seismic loads are computed using the response
modification coefficient, R, given in ASCE 7 Table 12.2-1.3 However, in accordance with IBC Section
2205.2.1.1, steel building structures assigned to seismic design category B or C, with the exception of
cantilever column systems, may be designed and detailed as specified by AISC 360.4 In this case, the
seismic loads are computed using a response modification coefficient of R 5 3, and this alternative
may often result in a more economical structure. For seismic design category A, special detailing is not
required and steel building structures may be designed and detailed as specified by AISC 360.
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3.2 Material strength and ductility
Structural steels used in seismic applications must exhibit the following characteristics:
•
a pronounced stress-strain plateau at the yield stress
•
a large inelastic strain capability
•
good weldability
Elements of the structural system that undergo extremely large plastic rotations in excess of 0.04
radians under the design earthquake are designated as highly ductile members. These members have
severe restrictions placed on their width-to-thickness ratios to prevent local buckling as plastic hinges
develop. An example of this is the link in an eccentrically braced frame. As shown in Figure 3-1,
inelastic action occurs primarily in the link, and the remaining members in the system remain essentially elastic.
Plastic
hinge
Link
Figure 3-1 Eccentrically braced frame
Elements of the structural system that undergo moderate plastic rotations not exceeding 0.02 radians
under the design earthquake are designated as moderately ductile members. These members have less
restrictive limits placed on their width-to-thickness ratios. An example of this is the diagonal brace in
an eccentrically braced frame. As shown in Figure 3-1, the link serves as a fuse to limit the load transferred to the diagonal braces, which are designed to remain essentially elastic without the possibility
of buckling and are designed, as specified in AISC 341 Section F3.5a, as moderately ductile members.
As specified in AISC 341 Section F3.5b(1), the link is designed as a highly ductile member.
The diagonal braces in a special concentrically braced frame with chevron configuration, as shown
in Figure 3-2, act as the fuses in the system. Inelastic action occurs primarily in the braces and, as
specified in AISC 341 Section F2.5a, these are designed as highly ductile members. Beams remain
essentially elastic and are designed as moderately ductile members.
In order to prevent local buckling in elements that undergo large plastic deformations, stringent widthto-thickness ratio limits are specified for highly ductile elements. Values of limiting width-to-thickness
ratios for moderately ductile compression members, lmd , and highly ductile compression members, lhd ,
are tabulated in AISC 341 Table D1.1 and are given in Table 3-1 for the more commonly used sections.
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•
Plastic
hinges
•
•
Figure 3-2 Buckled compression brace
Table 3-1 Limiting width-to-thickness ratios
Width-tothickness
ratio
Moderately ductile, lmd
Highly ductile, lhd
Round HSS used as
diagonal bracesa
D/t
0.062E/Ry Fy
0.053E/Ry Fy
Rectangular HSS used
as diagonal braces
b/t
0.76(E/Ry Fy)0.5
0.65(E/Ry Fy)0.5
Rectangular HSS used
in beams or columns
b/t
1.18(E/Ry Fy)0.5
0.65(E/Ry Fy)0.5
Angles
b/t
0.40(E/RyFy)0.5
0.32(E/Ry Fy)0.5
Flanges of I-shaped
members and channels
b/t
0.40(E/Ry Fy)0.5
0.32(E/Ry Fy)0.5
3.96(1 2 3.04Ca)(E/Ry Fy)0.5
. . . for Ca ≤ 0.114
2.57(1 2 1.04Ca)(E/Ry Fy)0.5
. . . for Ca ≤ 0.114
Element
Limiting width-to-thickness ratio
Webs of I-shaped sections used as beams or
columnsb
h/tw
1.29(2.12 2 Ca)(E/Ry Fy)0.5
≥ 1.57(E/Ry Fy)0.5
. . . for Ca . 0.114
0.88(2.68 2 Ca)(E/Ry Fy)0.5
≥ 1.57(E/Ry Fy)0.5
. . . for Ca . 0.114
Webs of I-shaped sections used as diagonal
braces
h/tw
1.57(E/Ry Fy)0.5
1.57(E/Ry Fy)0.5
Ca 5 Pu /fc Py . . . LRFD, Ca 5 Wc Pa /Py . . . ASD, SMF 5 special moment frames, IMF 5 intermediate moment frames
a. The limiting diameter-to-thickness ratio of round HSS members used as beams or columns shall not exceed 0.077E/Ry Fy.
b. For I-shaped beams in SMF systems, where Ca is less than or equal to 0.114, the limiting ratio h/tw shall not exceed 2.57(E/Ry Fy)0.5.
For I-shaped beams in IMF systems, where Ca is less than or equal to 0.114, the limiting width-to-thickness ratio shall not exceed
3.96(E/Ry Fy)0.5.
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3.3 Capacity design and expected material strength
For the design of some elements, a capacity design, or capacity-limited, approach is adopted. One element of the system is designated as the yielding element, or structural fuse. The remaining elements in
the system are designed to remain elastic for the anticipated force developed in the yielding element.
An example of this is the link in an eccentrically braced frame. As shown in Figure 3-1, yielding
occurs at the ends of the link and a mechanism is formed. Forces in the remaining elements of the
system are obtained by removing the link and applying the gravity loads and link-induced loads to the
remaining structure. The remaining beams and columns are designed to resist the force produced in
the link so as to remain essentially elastic.
ASCE 7 Section C12.4.3.2 describes the basis of the capacity design method as the expected strength
of one or more elements in a structure being used to generate the required strength for other elements,
because the yielding of the former limits the forces delivered to the latter.
Steel sections invariably have a yield stress and a tensile strength greater than the specified minimum
values. An accurate estimate of the link strength at yield is required and this requires an accurate estimate of the expected yield stress and tensile strength. Then,
expected yield stress
5 Ry Fy
expected tensile strength 5 Rt Fu
where:
Ry
5 ratio of the expected yield stress to the specified minimum yield strength
Rt
5 ratio of the expected tensile strength to the specified minimum tensile
strength
Fy
5 specified minimum yield stress of the type of steel used
Fu
5 specified minimum tensile strength of the type of steel used
Values of Ry and Rt are tabulated in AISC 341 Table A3.1 and are given in Table 3-2 for the more commonly used steels.
Table 3-2 Values of Ry and Rt
Application
Grade
Fy
Fu
Ry
Rt
Hot-rolled structural shapes
and bars
A36
A992
A572 Grade 50
36
50
50
58
65
65
1.5
1.1
1.1
1.2
1.1
1.1
Hollow structural sections
A500 Grade C, rectangular
A500 Grade C, round
50
46
62
62
1.3
1.3
1.2
1.2
Pipes
A53 Grade B
35
60
1.6
1.2
Plates, strips, and sheets
A36
A572 Grade 50
36
50
58
65
1.3
1.1
1.2
1.2
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The factors Ry and Rt are applied only in the determination of the force developed in the designated
member at yield and not in the determination of the required capacity of other members in the system.
The required strength of the designated member is determined by elastic analysis methods for the prescribed load combinations. In determining the required capacity of other elements in the system, neither the resistance factor, used in the load and resistance factor design (LRFD) method, nor the safety
factor, used in the allowable stress design (ASD) method, is applied to the strength of the designated
yielding member.
Table 3-3 lists the ductility requirements for a number of bracing systems.
Table 3-3 Ductility requirements
System
Highly ductile
Moderately ductile
Ordinary concentrically braced frames
Diagonal braces

Special concentrically braced frames
Diagonal braces

Beams

Columns

Eccentrically braced frames
Diagonal braces

Columns

Link beams

Beams outside the link

Special moment frames
Beams

Columns

Buckling-restrained braced frames
Beams

Columns

3.4 Demand critical welds
Welds that are located in a joint subjected to high stress demands, the failure of which would result in
the severe degradation of the structure, are designated as demand critical. These welds must be made
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with filler metals that have twice the Charpy V-notch (CVN) toughness values of filler metals commonly used. An example of demand critical welds is the groove welds at column splices in a special
concentrically braced frame. The actual stresses that occur at a column splice during a severe earthquake are not known with any certitude since the location of points of inflection in the column cannot
be reliably predicted. Failure of a column splice may lead to catastrophic failure of the system so a
conservative design approach is justified.
3.5 Protected zones
Discontinuities introduced into plastic hinge zones produce stress concentrations that may lead to fracture. For this reason, discontinuities are prohibited in some areas of the seismic-force-resisting system.
The discontinuities that are prohibited are listed in AISC 341 Section I2.1 and consist of:
•
holes, tack welds, erection aids, air-arc gouging, and unspecified thermal cutting
•
steel-headed stud anchors and decking attachments that penetrate the beam flange; arc spot
welds as required to secure decking are permitted
•
welded, bolted, screwed or shot-in attachments for perimeter edge angles, exterior facades,
partitions, duct work, piping, or other construction
The protected zones for a special concentrically braced frame are defined in AISC 341 Section F2.5c
and are shown in Figure 3-3.
d
4
L
L/
d
Figure 3-3 Protected zones for a special concentrically braced frame
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3.6 Loads and load combinations
For LRFD, the prescribed load combinations given in ASCE 7 Section 2.3.6 are
U
5 1.2D 1 Ev 1 Eh 1 f1L 1 0.2S . . . combination 6
and
U
5 0.9D 2 Ev 1 Eh . . . combination 7
where:
D
5 dead load
L
5 floor live load
S
5 snow load
Eh
5 effect of horizontal seismic load in accordance with ASCE 7 Equation
(12.4-3)
5 rQE
Ev
5 effect of vertical seismic load in accordance with ASCE 7 Equation
(12.4-4a)
5 0.2SDSD
QE
5 effect of horizontal seismic forces
SDS
5 design spectral response acceleration at a period of 0.2 second
r
5 redundancy factor defined in ASCE 7 Section 12.3.4
f1
5 1.0 for floors in garages and places of public assembly and for floor loads
in excess of 100 lb/ft2
5 0.5 for other live loads
Imposed live load is omitted where this results in a more critical effect in a member subjected to seismic loads. Since seismic load is determined at the strength design level, it has a load factor of 1.0.
Where the effects of gravity and seismic loads are additive, ASCE 7 Section 2.3.6 combination 6 may
be written as
(1.2 1 0.2SDS)D 1 rQE 1 0.5L 1 0.2S . . . for live load ≤ 100 lb/ft2
Where the effects of gravity and seismic loads counteract, load combination 7 is applicable, which is
(0.9 2 0.2SDS)D 1 rQE
The amplified seismic loads are given by load combinations 6 and 7 of ASCE 7 Section 2.3.6, which
are
(1.2 1 0.2SDS)D 1 W0QE 1 0.5L 1 0.2S . . . for live load ≤ 100 lb/ft2
(0.9 2 0.2SDS)D 1 W0QE
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where:
W0
259
5 overstrength factor given in ASCE 7 Table 12.2-1
W0QE 5 Emh
5 effect of horizontal seismic forces, including overstrength
For some structural systems, the value of Emh is defined in AISC 341, and this is substituted in the
previous equations after multiplying by 1.0 for LRFD load combinations.
For ASD, the prescribed load combinations are given in ASCE 7 Section 2.4.5. Where the effects of
gravity and seismic loads are additive, load combinations 8 and 9 are applicable, which are
(1.0 1 0.14SDS)D 1 0.7rQE . . . combination 8
(1.0 1 0.105SDS)D 1 0.525rQE 1 0.75L 1 0.75S . . . combination 9
Where the effects of gravity and seismic loads counteract, load combination 10 is applicable, which is
(0.6 2 0.14SDS)D 1 0.7rQE . . . combination 10
The amplified seismic loads are given by load combinations 8, 9, and 10 of ASCE 7 Section 2.4.5,
which are
(1.0 1 0.14SDS)D 1 0.7 W0QE
(1.0 1 0.105SDS)D 1 0.525 W0QE 1 0.75L 1 0.75S
(0.6 2 0.14SDS)D 1 0.7W0QE
where:
W0
5 overstrength factor given in ASCE 7 Table 12.2-1
W0QE 5 Emh
5 effect of horizontal seismic forces, including overstrength
For some structural systems, the value of Emh is defined in AISC 341, and this is substituted in the
above equations.
3.7 Concentrically braced frames
Chevron bracing, X bracing, K bracing, diagonal bracing, two-story X bracing, and zipper column
bracing are classified as concentrically braced frames5, 6, 7 and are shown in Figure 3-4. The bracing
members of a concentrically braced frame act as a truss system to resist lateral forces and are subjected
primarily to axial stress in the elastic range. During a severe earthquake, the bracing members and their
connections may undergo significant inelastic deformations into the post-buckling range and are subject to cyclic tension and compression. Cyclic rotations occur at plastic hinges, and bracing members
and their connections are specially detailed to avoid premature failure.
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Figure 3-4 Concentrically braced frames
Concentrically braced frames are subdivided into two categories: ordinary concentrically braced
frames and special concentrically braced frames. Special concentrically braced frames are used where
significant ductility is required. Ordinary concentrically braced frames are designed for a relatively
higher load, using a lower value of the response modification factor, to obviate the need for significant
ductility in the system.
Inelastic deformation and buckling of K bracing members may produce lateral deflection of the connected columns, causing instability and collapse. For concentrically braced frames, K bracing is not
allowed.
As shown in Figure 3-2, chevron or V bracing that is loaded in the inelastic range may cause large
unbalanced forces in the horizontal floor beam as the compressive strength of a bracing member deteriorates rapidly with reversing load cycles. For this reason, AISC 341 Section F1.4a requires that the
intersecting beam, in an ordinary concentrically braced frame, be designed for the unbalanced vertical
force produced. The two-story X bracing configuration and the zipper column configuration eliminate
the unbalanced force.
In accordance with AISC 341 Commentary Section F2.4d, tension-only bracing is allowed for ordinary concentrically braced frames but not for special concentrically braced frames.
3.8 Ordinary concentrically braced frames
Ordinary concentrically braced frames, as specified in AISC 341 Section F1, may be utilized in building frame systems in all seismic design categories, with the exception of F, using a value of 3.25 for the
response modification coefficient and a value of 2.0 for the overstrength factor. As specified in ASCE
7 Table 12.2-1, no limitation is imposed on the building height in seismic design categories A, B, and
C. The maximum height permitted in seismic design categories D and E is 35 feet. An exception is
permitted in seismic design categories D, E, and F for single-story buildings up to a height of 60 feet
where the dead load of the roof does not exceed 20 pounds per square foot.
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Ordinary concentrically braced frames may not be utilized in dual systems with moment frames.
In accordance with AISC 341 Commentary Section F1.3, ordinary concentrically braced frames are
designed with a low R factor so as to remain essentially elastic under a seismic event and preclude
the need for significant ductility of the system. Diagonal braces are required to be moderately ductile
members with no ductility conditions imposed on the other members in the system. No protected
zones are specified.
To reduce the likelihood of buckling causing large unbalanced forces in the floor beam, AISC 341
Section F1.5b requires bracing members in a chevron configuration to be designed with a slenderness
ratio not exceeding
where:
KL/r
5 4(E/Fy)0.5
K
5 effective length factor
L
5 length of the bracing member
r
5 governing radius of gyration
The required values of KL/r are given in Table 3-4.
Table 3-4 Limiting KL/r values
Fy kips/in2
KL/r
36
114
42
105
46
100
50
96
3.8.1 Diagonal braces
To determine the design force in a diagonal brace, the design seismic loads and the factored gravity
loads are applied to the frame as shown in Figure 3-5. The frame is analyzed as a vertical pin-jointed
truss and, ignoring gravity loads, the force in a brace in the bottom story is obtained as
FB
5 (F1 1 F2)/2cos q
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F2
θ
(1.2 + 0.2SDS)D + 0.5L
F1
FB
FB
Figure 3-5 Design of brace
The brace undergoes only moderate inelastic demands and AISC 341 Section F1.5a designates the
brace as a moderately ductile member. Moderately ductile members have less restrictive limits placed
on their width-to-thickness ratios. In tension-only frames, braces with slenderness ratios exceeding
200 need not comply with this requirement.
3.8.2 Beams in chevron configuration
As shown in Figure 3-2, the post-elastic behavior of the braces in a chevron configuration produces
a large, unbalanced force on the beam. Because of this, AISC 341 Section F1.4b requires beams to
be continuous between columns. In addition, both flanges of the beam must be provided with lateral
braces at the point of intersection of the braces to ensure the stability of the beam.
In designing the beam, a capacity approach is adopted and two load distributions are checked. It is
assumed in AISC 341 Section F1.4a that the diagonal braces provide no support to the beam and that
the beam supports all gravity loads and the unbalanced brace forces shown in Figure 3-2. It is also
assumed that the compression brace has buckled and has a residual strength of
0.3Pn 5 0.3Fcr Ag
where:
Ag
5 gross area of member
Fcr
5 critical stress given by AISC 360 Section E3 using the specified yield
stress Fy
Pn
5 nominal axial strength
The force in the tension brace is assumed to be the least of the following:
(i) The force developed in the brace when the amplified seismic load is applied to the system, which
is given by
RW
5 W0QE
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where:
W0
5 overstrength factor given in ASCE 7 Table 12.2-1
QE
5 effect of horizontal seismic forces
263
This produces the load distribution shown in Figure 3-6(a), and the beam is designed for the loading
condition shown in Figure 3-6(b).
Ω0F2
θ
WG = (1.2 + 0.2SDS)D + 0.5L
WG
Ω0F1
RΩ
RΩ
0.3Pn
0.3Pn
(a)
(b)
Figure 3-6 Design of beam for amplified seismic force
(ii) The maximum force that can be developed by the system. As shown in Figure 3-7(a), when foundation uplift occurs in the system, the force in the tension brace is R and the beam is designed for
the loading condition shown in Figure 3-7(b).
θ
WG
WG = (1.2 + 0.2SDS)D + 0.5L
R
R
0.3Pn
0.3Pn
(a)
(b)
Figure 3-7 Design of beam for maximum force that can be developed
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3.8.3 Columns
To determine the design force in a column, the design seismic loads and the factored gravity loads are
applied to the pion-jointed truss to give the post-elastic forces shown in Figure 3-8. It is assumed that
the compression brace has buckled and has a residual strength of
0.3Pn 5 0.3Fcr Ag
The force in the tension brace is assumed to be the expected yield strength, which is given by
Ru
5 Ry Fy Ag
Then, the required column axial compressive strength is
Pr
5 (SRu 1 S0.3Pn)sin q 1 S[(1.2 1 0.2SDS)D 1 0.5L]/2
where: [(1.2 1 0.2SDS)D 1 0.5L] 5 total gravity load on the beam
F2
Ru
θ
WG
F1
0.3Pn
Ru
0.3Pn
Figure 3-8 Design of column for post-elastic forces
3.8.4 Diagonal brace connections
Bracing connections are designed for forces of sufficient magnitude to ensure that brace yielding or
buckling will occur prior to a connection failure.
As shown in Figure 3-9, the required strength of diagonal brace connections is determined from the
load effect based on the amplified seismic load.
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Ω0F2
265
RΩ
θ
WG
Ω0F1
RΩ
RΩ
RΩ
Figure 3-9 Design of brace connections
Required tensile strength
In accordance with AISC 341 Section F1.6a, the required strength of the connection in tension need
not exceed the value
Ru
5 Ry Fy Ag . . . for LRFD
Ra
5 Ry Fy Ag /1.5 . . . for ASD
Required compressive strength
The required strength of the connection in compression need not exceed the lesser value of
Pu
5 Ry Fy Ag . . . for LRFD
≤ 1.10Fcre Ag
Pa
5 Ry Fy Ag /1.5 . . . for ASD
≤ 1.10Fcre Ag /1.5
Fcre
5 critical stress determined using the expected yield stress Ry Fy
Where KL/r ≤ 4.71(E/Ry Fy)0.5 or Ry Fy /Fe ≤ 2.25, AISC 360 Equation (E3-2) governs and the critical
stress is
where:
Fcre
5 (0.658k)Ry Fy
k
5 RyFy /Fe
Fe
5 elastic critical buckling stress
5 p2E/(KL/r)2 . . . from AISC 360 Equation (E3-4)
≥ Ry Fy /2.25
Where KL/r . 4.71(E/Ry Fy)0.5 or Ry Fy /Fe . 2.25, AISC 360 Equation (E3-3) governs and the critical
stress is
Fcre
5 0.877Fe
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The brace length used for the determination of Fcre must not exceed the distance from brace end to
brace end.
When oversized holes are used, the available slip resistance of the connection need not exceed the
force in the connection determined from the design seismic loads, not including the amplified seismic
load. Bolt slip does not constitute connection failure and the associated energy dissipation can serve
to reduce seismic response.
Example 3-1
The two-story ordinary concentrically braced steel frame shown in Figure 3-10 forms part of the
building frame system of a structure in seismic design category D with a redundancy factor of 1.0 and
a design response acceleration of SDS 5 1.0g. The loads acting on the brace in the bottom story are
dead load, D 5 20 kips
live load, L 5 10 kips
design seismic force, QE 5 90 kips
Determine a suitable steel A53 pipe section for the brace in the bottom story.
Solution
Factored loads
The factored design load on the brace is given by load combination 7 of ASCE 7 Section 2.3.6 as
Put
5 (0.9 2 0.2 SDS)D 1 rQE
5 (0.9 2 0.2 3 1.0)20 2 1.0 3 90
5 276 kips, tension
Figure 3-10 Details for Example 3-1
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The factored design load on the brace is given by load combination 6 of ASCE 7 Section 2.3.6 as
Puc
5 (1.2 1 0.2SDS)D 1 0.5L 1 rQE 1 0.2S
5 (1.2 1 0.2 3 1.0)20 1 0.5 3 10 1 1.0 3 90 1 0.2 3 0
5 123 kips, compression . . . governs
Select section
The diagonal length of the brace, between workpoints, is
l
5 H/sin q
5 14/sin 45°
5 19.80 ft
Allowing for a connection length at each end of 2 feet, the actual brace length is
L
5 19.8 2 4.0
5 15.8 ft
The effective length factor for the brace, assuming hinged ends, is given by AISC 360 Table C–A-7.1,
item (d), as
K
5 1.0
The effective length of the brace is
KL
5 15.8 ft
The design strength in axial compression is defined in AISC 360 Section E3 as fcPn and is given by
AISC 360 Equation (E3-1) as
where:
fc Pn
5 fc Ag Fcr
fc
5 resistance factor for compression
5 0.90
Ag
5 gross area of member
Fcr
5 critical stress
Pn
5 nominal axial compressive strength
From AISC Manual8 Table 4-6, select a steel pipe 6 XS, which has a design strength in axial compression, for an effective length of 15.8 feet, of
fc Pn
5 169 kips
. Puc . . . satisfactory
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Member properties
The section properties of a steel pipe 6 XS are given in AISC Manual Table 4-6 and Table 1-14 as
A
5 7.83 in2
r
5 2.20 in
t
5 0.403 in
D
5 6.63 in
Fy
5 35 ksi
Fu
5 60 ksi
The ratio of expected yield stress to specified minimum yield strength is
Ry
5 1.6 . . . from Table 3-2
Local buckling
The diameter-to-thickness ratio of a moderately ductile pipe section is limited by AISC 341 Table
D1.1, as shown in Table 3-1, to a maximum value of
D/t
5 lmd
5 0.062E/Ry Fy
5 0.062 3 29,000/(1.6 3 35) . . . from Table 3-2
5 32.1
The actual diameter-to-thickness ratio is
D/t
5 6.63/0.403
5 16.5
, lmd . . . satisfactory
Slenderness ratio
AISC 341 Section F1.5b requires bracing members in a chevron configuration to be designed with a
slenderness ratio not exceeding
KL/r
5 4.0(E/Fy)0.5
5 4.0(29,000/35)0.5
5 115
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The actual slenderness ratio is
KL/r
5 15.8 3 12/2.20
5 86
, 115 . . . satisfactory
Hence, the steel pipe 6 XS brace satisfies all requirements.
3.9 Special concentrically braced frames
Special concentrically braced frames, as specified in AISC 341 Section F2, may be utilized in building
frame systems in all seismic design categories using a value of 6 for the response modification coefficient and a value of 2.0 for the overstrength factor. As specified in ASCE 7 Table 12.2-1, no limitation
is imposed on the building height in seismic design categories A, B, and C. The maximum height permitted in seismic design categories D and E is 160 feet, and in seismic design category F, it is 100 feet.
Special concentrically braced frames may be utilized in dual systems with special moment frames in
all seismic design categories. In accordance with ASCE 7 Table 12.2-1, no limitation is imposed on
the building height, and a value of 7 is used for the response modification coefficient and a value of
2.5 for the overstrength factor.
Special concentrically braced frames may be utilized in dual systems with intermediate moment
frames, in all seismic design categories with the exceptions of E and F, using a value of 6 for the
response modification coefficient and a value of 2.5 for the overstrength factor. As specified in ASCE
7 Table 12.2-1, no limitation is imposed on the building height in seismic design categories A, B, and
C. The maximum height permitted in seismic design category D is 35 feet.
Special concentrically braced frames provide inelastic deformation capacity by brace buckling and
yielding of the brace in tension. In the elastic range, the frame behaves essentially as a vertical truss
with members subjected to axial load. In a severe earthquake, the diagonal bracing members undergo
significant inelastic deformation in the post-elastic range and provide a stable and ductile response.
Diagonal braces can sustain large inelastic cyclic deformations, as long as brittle failure due to local
buckling is prevented by limiting width-to-thickness ratios and connection failures are prevented.
Figure 3-11 shows a diagonal bracing configuration in which the story shear at every story is resisted
by braces oriented in a single direction. Since the post-buckling strength of a brace is considerably
less than its tensile strength, there is an accumulation of inelastic drift in the direction corresponding
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θ
Figure 3-11 Bracing aligned in one direction
to compression in the braces (to the left in Figure 3-11). After several cycles of inelastic deformation,
this produces excessive lateral deflection and possible instability. To prevent this, AISC 341 Section
F2.4a requires that the sum of neither the horizontal components of the compressive member forces
nor the horizontal components of the tensile member forces, along any line of bracing, shall exceed
70 percent of the total horizontal force along that line. This requirement is relaxed provided that the
braces are designed to resist the forces produced by the amplified seismic force applied to the frame.
3.9.1 Capacity design basis
In accordance with AISC 341 Section F2.3, the required strength of columns, beams, and connections is determined from the amplified seismic load where Emh , the effect of horizontal seismic forces
including overstrength, is replaced by the greater force determined from:
•
an analysis in which all braces are assumed to have reached their maximum forces corresponding to their expected strength in compression or in tension (i.e., all braces have reached their
maximum forces)
•
an analysis in which all braces in tension are assumed to have reached their maximum forces
corresponding to their expected strength and all braces in compression are assumed to have
reached their post-buckling strength (i.e., tension braces have reached their maximum force
and compression braces their post-buckling force)
The expected brace strength in tension is
where:
Ru
5 Ry Fy Ag
Ry
5 ratio of the expected yield stress to the specified minimum yield strength
Fy
5 specified minimum yield stress of the type of steel used
Ag
5 gross area of section
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The expected brace strength in compression is
Pu
5 1.14Fcre Ag
≤ Ry Fy Ag
where:
Fcre
5 critical stress given by AISC 360 Section E3 using the expected yield stress
Ry Fy in lieu of Fy
Where KL/r ≤ 4.71(E/Ry Fy)0.5 or Ry Fy /Fe ≤ 2.25, AISC 360 Equation (E3-2) governs and the critical
stress is
where:
Fcre
5 (0.658k)Ry Fy
k
5 Ry Fy /Fe
Fe
5 elastic critical buckling stress
5 p2E/(KL/r)2 . . . from AISC 360 Equation (E3-4)
≥ Ry Fy /2.25
Where KL/r . 4.71(E/Ry Fy)0.5 or Ry Fy /Fe . 2.25, AISC 360 Equation (E3-3) governs and the critical
stress is
Fcre
5 0.877Fe
The expected post-buckling strength of the compression brace is
0.3Pn 5 0.3Fcr Ag
where:
Fcr
5 critical stress given by AISC 360 Section E3 using the normal value of Fy
A conservative value of
0.3Fcr 5 2.1 kips/in2
may be assumed as this is the value for a brace with the maximum permitted slenderness ratio of 200.
The brace length used for the determination of Fcre must not exceed the distance from brace end to
brace end. Braces shall be determined to be in compression or tension, neglecting the effects of gravity
loads.
3.9.2 Diagonal braces
To determine the design force in a diagonal brace, the design seismic loads and the factored gravity
loads are applied to the frame, as shown in Figure 3-5.
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The brace undergoes large inelastic cyclic deformation demands with plastic hinges forming at the
center and ends of the compression brace. Hence, to prevent local buckling, AISC 341 Section F2.5a
designates the brace as a highly ductile member with severe restrictions on the width-to-thickness
ratio, as indicated in AISC 341 Table D1.1 or in Table 3-1 for the more commonly used sections.
The brace slenderness ratio is limited to a maximum value of
KL/r
5 200
To ensure a ductile response and prevent net section rupture of the brace at the connection, the brace
effective net area must not be less than the brace gross area. This typically requires the connection
of reinforcement at the ends of the brace. In accordance with AISC 341 Section F2.5b(c), the specified minimum yield strength of the reinforcement may not be less than the specified minimum yield
strength of the brace. In addition, the connection of the reinforcement to the brace shall have sufficient
strength to develop the expected reinforcement strength on each side of the reduced section.
As specified in AISC 341 Section F2.5c and shown in Figure 3-3, the protected zones for braces extend
over the center one-quarter of the brace length and adjacent to each connection for a length equal to
the brace depth in the plane of buckling.
K-braced frames are not permitted in special concentrically braced frames. In accordance with AISC
341 Section 2.4d, tension-only frames are not permitted in special concentrically braced frames.
In order to prevent individual buckling of elements in built-up bracing members, stitch spacing closer
than normal is required.
As specified in AISC 341 Section F2.5b(b), a minimum of two stitches is required and bolted stitches
may not be located within the central quarter of the clear brace length. The total design shear strength
of the stitches shall be at least equal to the design tensile strength of each element. The slenderness
ratio of the individual members between stitches may not exceed 40 percent of the governing slenderness ratio of the built-up member.
Example 3-2
The three-story special concentrically braced steel frame shown in Figure 3-12 forms part of the building frame system of a structure in seismic design category D with a redundancy factor of 1.0 and a
design response acceleration of SDS 5 1.0. The loads acting on the brace in the bottom story are
dead load, D 5 20 kips
live load, L 5 10 kips
design seismic force, QE 5 90 kips
Determine a suitable steel hollow structural section for the brace in the bottom story.
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Figure 3-12 Details for Examples 3-2 through 3-5
Solution
Factored loads
The factored design loads on the brace are given by
Puc
5 (1.2 1 0.2SDS)D 1 rQE 1 0.5L 1 0.2S
5 (1.2 1 0.2 3 1.0)20 1 1.0 3 90 1 0.5 3 10 1 0.2 3 0
5 123 kips, compression
and
Put
5 (0.9 2 0.2SDS)D 1 rQE
5 (0.9 2 0.2 3 1.0)20 2 1.0 3 90
5 276 kips, tension
Select section
The diagonal length of the brace, between workpoints, is
l
5 H/sin q
5 14/sin 45°
5 19.80 ft
Allowing for a connection length at each end of 2 feet, the actual brace length is
L
5 19.8 2 4.0
5 15.8 ft
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The effective length factor for the brace, assuming hinged ends, is given by AISC 360 Table C–A-7.1,
item (d), as
K
5 1.0
The effective length of the brace is
KL
5 15.8 ft
Second order effects are negligible and may be neglected.
From AISC Manual Table 4-5, select an A500 Grade C HSS 6.625 3 0.280, which has a design
strength in axial compression, for an effective length of 15.8 feet, and a yield stress of Fy 5 46 ksi, of
fc Pn
5 134 kips
. Puc . . . satisfactory
Member properties
The section properties of an A500 Grade C HSS 6.625 3 0.280 are given in AISC Manual Table 4-5
and Table 1-13 as
Ag
5 5.20 in2
r
5 2.25 in
t
5 0.26 in
5 design wall thickness
D/t
5 25.5 in
Fy
5 46 ksi
Fu
5 62 ksi
Ry
5 1.3
Rt
5 1.2
Local buckling
The diameter-to-thickness ratio of a highly ductile, round hollow section is limited by Table 3-1 to a
maximum value of
D/t
5 0.053E/Ry Fy
5 0.053 3 29,000/(1.3 3 46)
5 25.7
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The actual diameter-to-thickness ratio is
D/t
5 25.5
, 25.7 . . . satisfactory
Slenderness ratio
AISC 341 Section F2.5b(a) requires bracing members in a chevron configuration to be designed with
a slenderness ratio not exceeding
KL/r
5 200
The actual slenderness ratio is
KL/r
5 15.8 3 12/2.25
5 84.3
, 200 . . . satisfactory
Hence, the HSS 6.625 3 0.280 brace satisfies all requirements.
3.9.3 Diagonal brace connections
Required tensile strength
In accordance with AISC 341 Section F2.6c.1(a), the required strength of the connection in tension is
not less than the expected yield strength of the brace, which is
where:
Ru
5 Ry Fy Ag . . . for LRFD
Ra
5 Ry Fy Ag /1.5 . . . for ASD
Ag
5 area of the brace
However, the required strength in tension need not exceed the maximum force that can be transferred
to the brace by the system.
Required compressive strength
The required strength of the connection in compression is given by AISC 341 Section F2.6c.2 as
Pu
5 Ry Fy Ag . . . for LRFD
≤ 1.14Fcre Ag
Pa
5 Ry Fy Ag /1.5 . . . for ASD
≤ 1.14Fcre Ag /1.5
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Where KL/r ≤ 4.71(E/Ry Fy)0.5 or Ry Fy /Fe ≤ 2.25, AISC 360 Equation (E3-2) governs and the critical
stress is
where:
Fcre
5 (0.658k)Ry Fy
k
5 Ry Fy /Fe
Fe
5 elastic critical buckling stress
5 p2E/(KL/r)2 . . . from AISC 360 Equation (E3-4)
≥ Ry Fy /2.25
Where KL/r . 4.71(E/Ry Fy)0.5 or Ry Fy /Fe . 2.25, AISC 360 Equation (E3-3) governs and the critical
stress is
Fcre
5 0.877Fe
The brace length used for the determination of Fcre must not exceed the distance from brace end to
brace end.
When oversized holes are used, the available slip resistance of the connection need not exceed the
force in the connection determined from the design seismic loads, including the amplified seismic
load. Bolt slip does not constitute connection failure and the associated energy dissipation can serve
to reduce seismic response.
Accommodation of brace buckling
Brace connections are subject to severe stress reversals due to the cyclic buckling of the diagonal
braces. To prevent fracture of the connection resulting from brace rotations, bracing connections must
have either sufficient strength to confine inelastic rotation to the bracing member or sufficient ductility
to accommodate brace end rotations. For brace buckling in the plane of the gusset plates, the end connections should be designed to resist the expected flexural strength of the brace, which is
M
5 Ry Fy Zb . . . for LRFD
5 Ry Fy Zb /1.5 . . . for ASD
where:
Zb
5 plastic section modulus of the brace
For brace buckling out of the plane of single plate gussets, weak-axis bending in the gusset is invoked
by restraint-free member end rotation. The brace is terminated on the gusset plate a minimum of twice
the gusset plate thickness from a line about which the gusset plate can bend unrestrained by the column or beam. This is shown in Figure 3-13.
The interface forces at a gusset plate may be determined by the uniform force method.8, 9 Using this
method, equilibrium is achieved at a bracing connection by means of linear forces at the interface and
without any moments. The design of the gusset plate is facilitated by using the Whitmore construction,8, 10 which determines the effective section resisting the applied forces from the brace. As shown
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Figure 3-13 Gusset plate requirements
in Figure 3-14, two lines inclined at 30 degrees to the direction of the tensile force are drawn from the
first connectors in the bolt group to a line drawn through the last line of connectors to establish the
length of the Whitmore section. The design capacity of the gusset plate in tension yielding is determined using the area of the Whitmore section.
Figure 3-14 Whitmore section
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Example 3-3
The three-story special concentrically braced steel frame shown in Figure 3-12 forms part of the
building frame system of a structure in seismic design category D with a redundancy factor of 1.0. The
braces in all stories are identical.
Determine a suitable thickness for the grade A36 gusset plate on the second floor beam shown in Figure 3-15 and the size of fillet weld required.
8
60
Figure 3-15 Details for Example 3-3
Solution
Brace expected yield stress
The minimum required tensile strength of the gusset plate, in accordance with AISC 341 Section
F2.6c.1(a), must not be less than the expected yield strength of the A500 Grade C HSS 6.625 3 0.280
brace, determined as
Ru
5 Ry Fy Ag
5 1.3 3 46 3 5.20
5 311 kips
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Fillet weld design
For design purposes, it is convenient to determine the design strength of a 1⁄16-inch fillet weld per inch
run of E70XX grade electrodes, which is given by
qu
5 fFw te
5 0.75 3 0.6 3 70 3 0.707 3 1⁄16
5 1.39 kips per inch per 1⁄16 inch
If D denotes the number of 1⁄16 inch in weld size, the design capacity of a weld is
Qu
5 Dqu kips/in
5 1.39D kips/in
The total length of weld provided on the brace is obtained from Figure 3-15 as
L
5 4 3 17
5 68 in
The required fillet weld size per 1⁄16 inch is
Ru
5 max strength of brace
5 max force on connection
5 Pu
D
5 Pu /Lqu
5 311/(68 3 1.39)
5 3.3 sixteenths
The required weld size is
w
5 1⁄4 in . . . to the nearest 1⁄16 in
For this weld size, the maximum thickness of the gusset plate is limited by AISC 360 Table J2.4 to
tg
5 3⁄4 in
. 1⁄2 in . . . satisfactory
Select a 1⁄2-inch-thick gusset plate.
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Base metal thickness
The design shear rupture capacity of a plate is given by AISC 360 Equation (J4-4) as
Qb
5 fFBM t
5 0.75 3 0.60Fu t
5 0.45Fu t kips/in
5 1.39D kips/in
Hence, to develop the full capacity of the weld, for welds on one side only of a plate, the minimum
required plate thickness is
t
5 1.39D/0.45Fu
5 3.09D/Fu
To develop the full capacity of the weld, for welds on both sides of a plate, the minimum required plate
thickness is
t
5 6.18D/Fu
Hence, using A36 steel with a tensile strength of 58 kips per square inch (ksi), the minimum thickness
of gusset plate to develop the full strength of the 1⁄4-inch fillet welds opposite to each other on both
sides of the plate is
tg
5 6.18 3 4/58
5 0.43 in
Hence, a 1⁄2-inch-thick gusset plate is adequate.
Using an A500 Grade C HSS brace with a tensile strength of 62 kips per square inch, the minimum
wall thickness of the brace to develop the full strength of the 1⁄4-inch fillet welds on one side only of
the brace is
t
5 3.09 3 4/62
5 0.20 in
Hence, the wall thickness of the brace of 0.26 inch is adequate.
Brace tension rupture limit state
The brace selected in Example 3-2 is an HSS 6.625 3 0.280. As shown in Figure 3-15, a slot is cut in
the end of the brace that is fitted over the gusset plate and welded in place. To allow clearance for the
gusset plate, the slot is cut 1⁄8-inch oversize. Therefore, the net section of the brace at the slot is
As
5 A 2 2(tg 1 0.125)t
5 5.20 2 2(0.5 1 0.125) 3 0.26
5 4.88 in2
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In order to reinforce the end of the brace, two sections of an HSS 7.000 3 0.188 are welded to the
brace as shown in Figure 3-16. The inside diameter of the HSS 7.000 3 0.188 is 6.624 inches and this
matches the outside diameter of the HSS 6.625 3 0.280 brace. From the dimensions shown in Figure
3-16, the chord length of the reinforcement is
c
5 3.5 in
HSS 7.000 × 0.188 reinforcement
c = 3.5 in
HSS 6.625 × 0.280 brace
Figure 3-16 Reinforcement for brace
The design wall thickness of the HSS 7.000 3 0.188 reinforcement is
tr
5 0.174 in
The radius of curvature at the center line of the reinforcement is
Rr
5(D 1 tr)/2
5 (6.625 1 0.174)/2
5 3.40 in
The angle subtended by the chord c at the center of the brace is
q
5 2sin21(c/2Rr)
5 2sin21(3.5/6.80)
5 62°
The arc length of the reinforcement is
s
5 Rrq
5 2pRrq/360
5 2 3 3.14 3 3.40 3 62/360
5 3.68 in
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The area of the reinforcement is
Ar
5 2str
5 2 3 3.68 3 0.174
5 1.28 in2
The total net area of the brace plus reinforcement at the gusset plate is
An
5 Ar 1 As
5 1.28 1 4.88
5 6.16 in2
The length of weld at the slot is l 5 17 inches, and the outside diameter of the brace plus reinforcement
is Dr 5 7 inches. Hence,
l/Dr
5 17/7
5 2.4
. 1.3
Hence, from AISC Table D3.1, the shear lag coefficient is given by
U
5 1.0
The total effective area of the brace plus reinforcement at the gusset plate is
Ae
5 UAn
5 6.16 in2
. 5.20 . . . satisfies AISC 341 Section F2.5b(c)
The factor Rt given in Table 3-2 is used to determine the expected increase in the tensile strength of the
brace. Hence, the design capacity for the tensile rupture condition of the brace is given by AISC 341
Section A3.2 as
ft Rn
5 0.75Rt Fu Ae
5 0.75 3 1.2 3 62 3 6.16
5 344 kips
. Ru . . . satisfactory
The expected yield strength of each section of the reinforcement is given by AISC 341 Section A3.2 as
Ru
5 Ry Fy Ar /2
5 1.3 3 46 3 1.28/2
5 38 kips
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To develop the full capacity of the reinforcement requires a length of 1⁄8-inch fillet weld of
L
5 Ru /1.39D
5 38/(1.39 3 2)
5 14 inches
This length is distributed on both sides of the reinforcement, and the total length of reinforcement
required is 14 inches, as shown on Figure 3-15.
Gusset plate block shear
From Figure 3-15, the gross shear area of the gusset, which equals the net shear area, is
Agv
5 2ltg
5 2 3 17 3 0.50
5 17 in2
From Figure 3-15, the net tension area of the gusset, which equals the gross tension area, is
Agt
5 Dtg
5 6.625 3 0.5
5 3.31 in2
For uniform tensile stress, the reduction coefficient is given by AISC 360 Section J4.3, as
Ubs
5 1.0
Hence, the rupture strength in tension is given by
Ubs Fu Agt 5 1.0 3 58 3 3.31
5 192 kips
The strength in shear is given by
0.6Fy Agv 5 0.6 3 36 3 17
5 367 kips
The block shear design strength is given by AISC 360 Equation (J4-5) as
fRn
5 f(0.6Fy Agv 1 Ubs Fu Agt)
5 0.75(367 1 192)
5 419 kips
. Ru . . . satisfactory
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Gusset plate tension yielding limit state
The length of the Whitmore section is given by
lw
5 D 1 2l tan 30°
5 6.625 1 2 3 17tan 30°
5 26 in
The design capacity of the gusset plate in tension yielding, which governs, is given by AISC 360 Section D2 as
ftPn
5 0.9Fy lw t
5 0.9 3 36 3 26 3 0.5
5 421 kips
. Ru . . . satisfactory
Gusset plate compressive strength
The radius of gyration of the gusset plate is
rg
5 tg /(12)0.5
5 0.50/3.46
5 0.145 in
The length of the gusset plate from the end of the pipe to the junction of the beam and column is
lg
5 18 in
The effective length factor for a gusset plate welded on two edges is given by11
K
5 0.5
The slenderness ratio is
Klg /rg 5 0.5 3 18/0.145
5 62
From AISC Manual Table 4-14, the design axial compressive stress for the gusset plate is
fc Fcr 5 26.5 ksi
The design compressive strength of the gusset plate is
fc Pgn 5 fc Fcr lw tg
5 26.5 3 26 3 0.5
5 345 kips
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Gusset plate design loads
From Example 3-2, the slenderness ratio of the brace is
KL/r
4.71(E/Ry Fy)0.5
5 84.3
5 4.71 3 [29,000/(1.3 3 46)]0.5
5 103.7
. KL/r . . . AISC 360 Equation (E3-2) governs
Fe
5 elastic critical buckling stress
5 p2E/(KL/r)2 . . . from AISC 360 Equation (E3-4)
5 3.142 3 29,000/84.32
5 40.23 kips/in2
k
5 Ry Fy /Fe
5 1.3 3 46/40.23
5 1.49
Fcre
5 critical stress
5 (0.658k)Ry Fy
5 0.6581.49 3 1.3 3 46
5 32.05 kips/in2
The required compressive strength of the connection in compression is given by AISC 341 Section
F2.6c.2 as the lesser of
Pu
5 Ry Fy Ag
5 1.3 3 46 3 5.20
5 311 kips
or
Pu
5 1.14Fcre Ag
5 1.14 3 32.05 3 5.20
5 190 kips . . . governs
, fc Pgn . . . satisfactory
Hence, the A36 1⁄2-inch-thick gusset plate satisfies all requirements.
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3.9.4 Beams in chevron configuration
As shown in Figure 3-2, the post-elastic behavior of the braces in a chevron configuration produces
a large, unbalanced force on the beam. Because of this, AISC 341 Section F2.4b requires beams to
be continuous between columns. In addition, both flanges of the beam must be provided with lateral
braces at the point of intersection of the braces to ensure the stability of the beam. The beam is classified as a moderately ductile member and requires lateral bracing, as specified by AISC 341 Equation
(D1-2), at a maximum spacing of
Lb
5 0.19Ry E/Ry Fy
The required strength of the lateral bracing is given by AISC 360 Equation (A-6-7) as
where:
Prb
5 0.02MrCd /ho
Mr
5 Ry Fy Z . . . for LRFD
Cd
5 1.0 . . . for single curvature
ho
5 distance between flange centroids
As indicated in Section 3.9.1, the beam is designed for the more critical of the two loading distributions shown in Figure 3-17(a) or (b).
The expected brace strength in tension is
Ru
5 Ry Fy Ag . . . for LRFD
The expected brace strength in compression is
Pu
5 1.14Fcre Ag . . . for LRFD
≤ Ry Fy Ag
The expected post-buckling strength of the compression brace is
0.3Pn 5 0.3Fcr Ag . . . for LRFD
WG
WG
(a)
(b)
Ru
0.3Pn
Ru
Pu
Figure 3-17 Beam loading
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The beam-to-column connection is specified in AISC 341 Section F2.6b as either a simple connection
with a rotation capacity of 0.025 radian or a fully restrained moment connection. If a moment connection is used, the connection must be designed to resist a moment equal to the lesser of the following:
(i)
M
5 1.1Ry Fy Zb . . . for LRFD
5 1.1Ry Fy Zb /1.5 . . . for ASD
(ii)
M
5 S1.1Ry Fy Zc . . . for LRFD
5 S1.1Ry Fy Zc /1.5 . . . for ASD
where:
Zb
5 plastic section modulus of the beam
Zc
5 plastic section modulus of a column
Example 3-4
The three-story special concentrically braced steel frame shown in Figure 3-12 forms part of the building frame system of a structure in seismic design category D with a redundancy factor of 1.0. The loads
acting on the beam at the second floor are
dead load, D 5 1 kip/ft
live load, L 5 0.5 kip/ft
The braces in all stories are identical. The design response acceleration is SDS 5 1.0g.
Select a suitable W27 section, with a yield stress of 50 kips per square inch, for the beam.
Solution
Unbalanced vertical force
The unbalanced vertical force on the beam, in accordance with AISC 341 Section F2.3, is given by the
greater force determined from
(a)
Qb
5 (Ry Fy Ag 2 0.3Pn)sin 45°
(b)
Qb
(Ry Fy Ag 2 Pu)sin 45°
where:
Fy
5 specified minimum yield stress of the type of steel used in the brace
or
5 46 ksi . . . from Example 3-2
Ag
5 gross area of the brace
5 5.20 in2 . . . from Example 3-2
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Pn
5 nominal strength of the 15.8-ft long brace in axial compression from AISC
Manual Table 4-5
5 fc Pn /fc
5 134/0.90
5 149 kips
Pu
5 expected brace strength in compression . . . from Example 3-3
5 1.14Fcre Ag
5 1.14 3 32.05 3 5.20
5 190 kips
Ry Fy Ag 5 expected yield strength of the brace
5 311 kips . . . from Example 3-3
For load case (a)
Qb
5 (311 2 0.3 3 149)0.707
5 188 kips
The factored moment on the beam is
Mu
5 (1.2D 1 0.2D 1 0.5L)l2/8 1 Qbl/4
5 (1.2 3 1.0 1 0.2 3 1.0 1 0.5 3 0.5) 3 282/8 1 188 3 28/4
5 1478 kip-ft
As shown in Figure 3-18, the factored axial load on the beam is
Pr
5 (Ry Fy Ag 1 0.3Pn)cos 45°/2
5 (311 1 0.3 3 149)0.707/2
5 126 kips
1.65 kips/ft
126 kips
126 kips
252 kips
188 kips
Figure 3-18 Post-buckling forces for Example 3-4 load case (a)
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For load case (b)
The unbalanced vertical force on the beam is
Qb
5 (Ry Fy Ag 2 Pu)sin 45°
5 (311 2 190)0.707
5 86 kips
The factored moment on the beam is
Mu
5 (1.4D 1 0.5L)l2/8 1 Qbl/4
5 (1.4 3 1.0 1 0.5 3 0.5) 3 282/8 1 86 3 28/4
5 764 kip-ft
As shown in Figure 3-19, the factored axial load on the beam is
Pr
5 (Ry Fy Ag 1 Pu)cos 45°/2
5 (311 1 190)0.707/2
5 177 kips
Load case (a) governs.
1.65 kips/ft
177 kips
177 kips
354 kips
86 kips
Figure 3-19 Forces due to expected brace compression for Example 3-4 load case (b)
Combined compression and flexure
As specified in AISC 341 Section F2.4b, the top and bottom flanges of the beam at the point of intersection of chevron braces are laterally supported. Ignoring the column depth, the unbraced segment
lengths about the x- and y-axes are
Lbx
5 28 ft
Lby
5 14 ft
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A W27 3 146 trial section is selected and will be analyzed using AISC 360 Equations (H1-1a) or (H11b) as applicable. The relevant properties of the W27 3 146 are
Ag
5 43.2 in2
I
5 5660 in4
bf /2tf 5 7.16
h/tw
5 39.4
rx
5 11.5 in
ry
5 3.20 in
fb Mp 5 1740 kip-ft
fBF
5 29.5 kips
Lp
5 11.3
KL/rx 5 1.0 3 28 3 12/11.5
5 29.2
KL/ry 5 1.0 3 14 3 12/3.20
5 52.5 . . . governs
Fb 5 50 ksi
Local buckling
The flange width-to-thickness ratio for the moderately ductile beam is limited by Table 3-1 to a maximum value of
bf /2tf 5 0.40(E/Ry Fy)0.5
5 0.40[29,000/(1.1 3 50)]0.5
5 9.18
The actual flange width-to-thickness ratio is
bf /2tf 5 7.16
, 9.18 . . . satisfactory
The ratio of required strength to available strength is
Ca
5 Pr /fb Py
5 126/(0.9 3 43.2 3 50)
5 0.065
, 0.114
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Hence, the web height-to-thickness ratio is limited by Table 3-1 to a maximum value of
h/tw
5 3.96(1 2 3.04Ca)(E/Ry Fy)0.5
5 3.96(1 2 3.04 3 0.065)[29,000/(1.1 3 50)]0.5
5 73
The actual web height-to-thickness ratio is
h/tw
5 39.4
, 73 . . . satisfactory
Design compressive strength
From AISC Manual Table 4-14, for KL/ry 5 52.5, the design axial compressive stress for the beam is
fc Fcr 5 36.8 ksi
The design compressive strength of the beam is
fc Pn
5 fc Fcr Ag
5 36.8 3 43.2
5 1590 kips
Pr /fcPn
5 126/1590
5 0.079
, 0.2
Hence, AISC 360 Equation (H1-1b) applies and, for bending about the x-axis only, is given by
Pr /2fc Pn 1 Mux /fb Mnx ≤ 1.0
Second-order analysis
The Euler buckling load for a braced frame is given by AISC 360 Equation (A-8-5) as
Pe1
5 p2EI/(KL)2
5 p229,000 3 5660/(28 3 12)2
5 14,335 kips
The reduction factor for a member in a braced frame, with pinned ends, subjected to transverse loading
is given by AISC 360 Appendix 8.2.1 as
Cm
5 1.0
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From AISC 360 Equation (A-8-3), the multiplier to account for P-d effects is given by
B1
5 Cm /(1 2 Pr /Pe1)
5 1.0/(1 2 126/14,335)
5 1.01
For a braced frame, the required flexural strength is given by AISC 360 Equation (A-8-1) as
Mr
5 B1Mu
5 1.01 3 1478
5 1493 kip-ft
Design flexural strength
For a moment distribution on the beam that is approximately triangular, the bending coefficient dependent on the moment gradient is
Cb
5 1.67
From AISC Manual Table 3-2, a W27 3 146 with an unbraced length of 14 feet has a design flexural
strength about the strong axis of
fbMnx 5 Cb[fb Mp 2 (fBF)(Lb 2 Lp)]
5 1.67[1740 2 29.5(14 2 11.3)]
5 2773 kip-ft
The maximum permitted value is
fb Mp 5 1740 kip-ft
Interaction equation
The left side of AISC 360 Equation (H1-1b) is
Pr /2fc Pn 1 Mr /fb Mnx
5 0.079/2 1 1493/1740
5 0.90
, 1.0 . . . satisfactory
Hence, the W27 3 146 beam is adequate.
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3.9.5 Columns
Columns are designated in AISC 341 Section F2.5a as highly ductile members.
As indicated in AISC 341 Section F2.6a, welds at a column splice are subjected to high stress demands
and are designated as demand critical, as are welds at column-to-base plate connections and beam-tocolumn connections. In accordance with AISC 341 Section F2.6d, welds at a splice must be complete
joint-penetration groove welds, and the splice is designed to develop at least 50 percent of the lesser
available flexural strength of the connected columns. The required shear strength of the joint is
Vr
5 SFy Zc /Hc . . . for LRFD
5 SFy Zc /1.5Hc . . . for ASD
where:
Zc
5 plastic section modulus of a column
SFy Zc 5 sum of the nominal plastic flexural strengths of the columns above and
below the splice
Hc
5 clear height of the column between beam connections
As specified in AISC 341 Section D2.5a, column splices must be located at least 4 feet from the beamto-column connection.
As indicated in AISC 341 Section F2.3, the column is designed for the more critical of the two loading
distributions shown in Figure 3-20(a) or (b) for an X-braced frame.
The expected brace strength in tension is
Ru
5 Ry Fy Ag . . . for LRFD
Ra
5 Ry Fy Ag /1.5 . . . for ASD
The expected brace strength in compression is
Pu
5 1.14Fcre Ag . . . for LRFD
≤ Ry Fy Ag
Pa
5 1.14Fcre Ag /1.5 . . . for ASD
≤ Ry Fy Ag /1.5
The expected post-buckling strength of the compression brace is
0.3Pn 5 0.3Fcr Ag . . . for LRFD
5 0.3Fcr Ag /1.5 . . . for ASD
The brace length used for the determination of Fcre must not exceed the distance from brace end to
brace end.
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Then, the required compressive strength of the column is the more critical of the following two
expressions
Pr
5 (SRu 1 SPu)sin q 1 S(1.2 1 0.2SDS)D 1 0.5L
Pr
5 (SRu 1 S0.3Pn)sin q 1 S(1.2 1 0.2SDS)D 1 0.5L
where the gravity loads acting on the column are D 5 dead load and L 5 live load.
In accordance with AISC 341 Section F2.3, flexural forces on the columns due to story drift may be
neglected.
θ
Ru
Ru
Pu
0.3Pn
Ru
Ru
0.3Pn
Pu
(a)
(b)
Figure 3-20 Column loading
In accordance with AISC 341 Section F2.3, the required strength of columns need not exceed the least
of the following:
•
the forces corresponding to the resistance of the foundation to overturning uplift
•
the forces determined from nonlinear analysis
Example 3-5
The three-story special concentrically braced steel frame shown in Figure 3-12 forms part of the building frame system of a structure in seismic design category D with a redundancy factor of 1.0. The loads
acting on the column in the bottom story are
dead load, D 5 80 kips
live load, L 5 30 kips
The braces in all stories are identical. Uplift of the foundation does not govern.
Select a suitable W14 section, with a yield stress of 50 kips per square inch, for the column.
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Solution
The two applicable loading conditions are shown in Figure 3-21(a) and (b). From Examples 3-3 and
3-4
Ru
5 expected yield strength of the A500 Grade C HSS 6.625 3 0.280 brace
5 Ry Fy Ag
5 311 kips
Pn
5 nominal strength of the 15.8-ft long brace in axial compression
5 fc Pn /fc
5 149 kips
Pu
5 expected brace strength in compression
5 1.14Fcre Ag
5 190 kips
Pu
0.3Pn
Ru
Ru
0.3Pn
Pu
Pu
Ru
Ru
0.3Pn
Pu
0.3Pn
Pu
Ru
Ru
0.3Pn
(a)
0.3Pn
(b)
Pu
Figure 3-21 Column loading cases for Example 3-5
For load case (a)
The unbalanced vertical force on the midpoint of the beam is
Qb
5 (Ry Fy Ag 2 0.3Pn)sin 45°
5 (311 2 0.3 3 149)0.707
5 188 kips
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The required column compressive strength is
Pr
5 3Qb /2 1 2 3 0.3Pnsin 45° 1 (1.2 1 0.2S)D 1 0.5L
5 3 3 188/2 1 2 3 0.3 3 149 3 0.707 1 1.4 3 80 1 0.5 3 30
5 472 kips
For load case (b)
The unbalanced vertical force on the midpoint of the beam is
Qb
5 (Ry Fy Ag 2 Pu)sin 45°
5 (311 2 190)0.707
5 86 kips
The required column compressive strength is
Pr
5 3Qb /2 1 2 3 Pu sin 45° 1 (1.2 1 0.2S)D 1 0.5L
5 3 3 86/2 1 2 3 190 3 0.707 1 1.4 3 80 1 0.5 3 30
5 525 kips . . . governs
Select section
The unbraced length of the column, using centerline dimensions, is
L
5 14 ft
The effective length factor for the column, assuming hinged ends, is given by AISC 360 Table C–A-71,
item (d), as
K
5 1.0
The effective length of the column is
KL
5 14 ft
From AISC Manual Table 4-1a, select a W14 3 68, which has a design strength in axial compression,
for an effective length of 14 feet, of
fc Pn
5 640 kips
. Pr . . . satisfactory
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Member properties
The section properties of a W14 3 68 are
A
5 20.0 in2
ry
5 2.46 in
bf /2tf 5 6.97
h/tw
5 27.5
Fy
5 50 ksi
Fu
5 65 ksi
Local buckling
The flange width-to-thickness ratio for a highly ductile member is limited by Table 3-1 to a maximum
value of
bf /2tf 5 0.32(E/Ry Fy)0.5
5 0.32[29,000/(1.1 3 50)]0.5
5 7.35
The actual flange width-to-thickness ratio is
bf /2tf 5 6.97
, 7.35 . . . satisfactory
The ratio of required strength to available strength is
Ca
5 Pr /fb Py
5 525/(0.9 3 20.0 3 50)
5 0.583
. 0.114
Hence, the web height-to-thickness ratio is limited by Table 3-1 to a maximum value of
h/tw
5 0.88(2.68 2 Ca)(E/Ry Fy)0.5
5 0.88(2.68 2 0.583)[29,000/(1.1 3 50)]0.5
5 42.4
The actual web height-to-thickness ratio is
h/tw
5 27.5
, 42.4 . . . satisfactory
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Slenderness ratio
The actual slenderness ratio is
KL/ry 5 14.0 3 12/2.46
5 68.3 . . . satisfactory
, 200
Hence, the W14 3 68 column satisfies all requirements.
3.10 Eccentrically braced frames
Eccentrically braced frames,12, 13, 14 as illustrated in Figure 3-22, may provide a high degree of stiffness
in the elastic range that is comparable to that of concentrically braced frames. The bracing member in
an eccentrically braced frame is connected to the beam so as to form a short link between the brace
and the column or between two opposing braces. The link acts as a fuse to prevent other elements in
the frame from being overstressed. The shorter the link, the stiffer the frame becomes and the smaller
the drift produced. During a major earthquake, the link is designed to deform inelastically and provide
a nonlinear energy-absorbing capacity similar to a special moment-resisting frame. The other framing
elements are designed to remain elastic and to be sufficiently strong to cause the link to yield. Shear
or flexural yielding of the link provides a ductile response comparable to that obtained in special
moment-resisting frames.
Figure 3-22 Eccentrically braced frames
Eccentrically braced frames, as specified in ASCE 7 Table 12.2-1, may be utilized in building frame
systems in all seismic design categories using a value of 2 for the overstrength factor and a value of
8 for the response modification coefficient. As specified in ASCE 7 Table 12.2-1, no limitation is
imposed on the building height in seismic design categories A, B, and C. The maximum height permitted in seismic design categories D and E is 160 feet, and in seismic design category F, it is 100 feet.
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Eccentrically braced frames may be utilized in dual systems with special moment frames, in all seismic design categories, using a value of 2.5 for the overstrength factor. In accordance with ASCE 7
Table 12.2-1, no limitation is imposed on the building height. A value of 8 is specified for the response
modification coefficient.
3.10.1 Basic requirements
The ductility demands in an eccentrically braced frame are concentrated in the links. Braces, columns
and beams outside the link are designed to be stronger than the link and to remain essentially elastic.
The link acts as a fuse to limit the loads transferred to other members in the frame and prevents buckling of the braces. Usually, the beam and the link are a single continuous wide flange member and any
increase in yield strength present in the link will also be present in the beam segment outside of the
link. Hence, the available strength of the beam can be increased by Ry.
To ensure that stable inelastic deformations can occur in the link, it is designated a highly ductile member. Similarly, columns are designated highly ductile members. Braces and the beam outside the link,
if a different section from the link, are designated moderately ductile members.
Links are designated as protected zones.
3.10.2 Link requirements
AISC 341 Section F3.5b(1) specifies the following design requirements:
•
to ensure stability of the highly ductile link during inelastic deformations, compact sections
shall be used complying with the flange width-to-thickness ratios in Table 3-1 of
bf /2tf ≤ 0.32(E/Ry Fy)0.5
where:
•
bf
5 flange width
tf
5 flange thickness
if e ≤ 1.6Mp /Vp , the link may satisfy the requirements for a moderately ductile member with
bf /2tf ≤ 0.40(E/Ry Fy)0.5
•
doubler plates on the web of the link are not allowed because they are ineffective during inelastic deformation
•
links are a protection zone and holes are not allowed in the web of the link because these affect
the inelastic deformation of the link web
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3.10.3 Link shear strength
Depending on the length of the link, either shear yielding or flexural yielding may occur at the ends of
the link. A balanced shear condition exists when flexural and shear hinges occur simultaneously. This
occurs with a link length of
ey
5 2Mp /Vp
For lengths less than ey , a shear mode predominates, and for lengths greater than ey , a flexural mode
predominates.
For flexural yielding
As shown in Figure 3-23, if flexural plastic hinges form at the ends of the link, a point of inflection
occurs at the center of the link. When as Pr /Pc ≤ 0.15, the effect of axial force on the link moment
capacity need not be considered, and the nominal plastic flexural strength is given by AISC 341 Equation (F3-8) as
where:
Mp
5 FyZ
Z
5 link plastic section modulus
Fy
5 link specified minimum yield stress
as
5 1.0 for LRFD and 1.5 for ASD
Figure 3-23 Forces on link
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When as Pr /Pc . 0.15, the reduced nominal plastic flexural strength is given by AISC 341 Equation
(F3-9) as
where:
Mp
5 Fy Z(1 2 as Pr /Pc)/0.85
Pr
5 required axial strength
5 Pu . . . for LRFD
5 Pa . . . for ASD
Pc
5 available axial strength
5 Py . . . for LRFD
5 Py /1.5 . . . for ASD
Py
5 nominal axial yield strength given by AISC 341 Equation (F3-6)
5 Fy Ag
where:
Ag
5 gross area of link
From Figure 3-23, the shear produced at the ends of the link is given by AISC 341 Equation (F3-7) as
Vn
5 2Mp /e
≤ Vp
where:
e
5 length of link
Vp
5 plastic shear strength
For shear yielding
When shear yielding occurs at the ends of a link and as Pr /Pc ≤ 0.15, the effect of axial force on the
shear yielding capacity need not be considered, and the nominal shear yielding strength is given by
AISC 341 Equation (F3-2) as
Vn
5 Vp
5 0.60Fy Alw
where:
Alw
5 web area
5 (db 2 2tf)tw . . . for I-shaped link sections
db
5 depth of link
tf
5 flange thickness
tw
5 web thickness
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The moment produced at the ends of the link is
Mr
5 Vne/2
When as Pr /Pc . 0.15, the reduced nominal shear yielding capacity is given by AISC 341 Equation
(F3-3) as
Vp
5 0.6Fy Alw[1 2 (as Pr /Pc)2]0.5
3.10.4 Link length
Links with high axial forces subject to flexural yielding may not be able to develop adequate rotations
and may exhibit unstable inelastic behavior. Hence, where high axial forces can develop in the link, its
length is limited to ensure that shear yielding, rather than flexural yielding, governs. Where as Pr /Pc .
0.15 and r9 ≤ 0.5, the length of the link is limited by AISC 341 Equation (F3-10) to
e
≤ 1.6Mp /Vp
Where as Pr /Pc . 0.15 and r9 . 0.5, the length of the link is limited by AISC 341 Equation (F3-11) to
where:
e
≤ [1.15 2 0.3r9]1.6Mp /Vp
r9
5 (Pr /Py)/(Vr /Vy)
Vr
5 required shear strength
5 Vu . . . for LRFD
5 Va . . . for ASD
Vy
5 nominal shear yield strength given by AISC 341 Equation (F3-13) as
5 0.6Fy Alw
Where as Pr /Pc ≤ 0.15, there is no upper limitation on link length.
3.10.5 Link rotation
For the maximum inelastic story drift, the elements of the frame may be considered rigid and the link
rotation angle, gp , is derived as shown in Figure 3-24, which is given by
gp
5 LD/he
5 Lqp /e
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where:
L
5 beam length between column centers
D
5 maximum inelastic story drift
h
5 story height
e
5 length of link
qp
5 story drift angle
gp
5 link rotation angle
θ
303
θ
Figure 3-24 Link rotation
To limit the inelastic deformation of the frame, the link rotation angle is limited by AISC 341 Section
F3.4a to the following values
gp
≤ 0.080 radian . . . for short links of length e ≤ 1.6Mp /Vp
gp
≤ 0.020 radian . . . for long links of length e ≥ 2.6Mp /Vp
These limits are illustrated in Figure 3-25 and linear interpolation may be used for intermediate link
lengths.
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Figure 3-25 Link rotation angle
3.10.6 Link stiffeners for I-shaped cross sections
To ensure stable behavior of the link under cyclic loading, AISC 341 Section F3.5b.4 specifies the
following detailing requirements:
•
To prevent web instability under cyclic loading, full-depth web stiffeners shall be provided on
both sides of the link web at the brace end of the link. As shown in Figure 3-26, the stiffeners
shall have a combined width of
2bst
≥ bf 2 2tw
and a thickness of
tst
where:
•
5 0.75tw
≥ 3⁄8 in
bf
5 link flange width
tw
5 link web thickness
The weld between the stiffener and the web is required to develop the full strength of the stiffener, as shown in Figure 3-26. The weld must be adequate to resist the force as given by
Pw
where:
Ast
≥ Ast Fy . . . for LRFD
≥ Ast Fy /1.5 . . . for ASD
5 area of stiffener
5 bst tst
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Figure 3-26 Stiffener details
The weld between the stiffener and the flange is necessary to develop the rigidity of the stiffener and
restrain flange buckling. The weld force is given by
Pw
•
≥ Ast Fy /4 . . . for LRFD
≥ Ast Fy /4(1.5) . . . for ASD
For a shear link with e ≤ 1.6Mp /Vp, intermediate stiffeners are required, as shown in Figure
3-27, at a spacing of
s
≤ 30tw 2 d/5 . . . for gp 5 0.08 radian
s
≤ 52tw 2 d/5 . . . for gp ≤ 0.02 radian
Linear interpolation may be used for intermediate link rotations.
For
2.6Mp /Vp ≤ e , 5Mp /Vp
intermediate stiffeners are required at a distance of 1.5bf from each end of the link.
For
1.6Mp /Vp ≤ e , 2.6Mp /Vp
intermediate stiffeners are required to satisfy both the above requirements.
For
e . 5Mp /Vp
intermediate stiffeners are not required.
•
Single-sided, full-depth web intermediate stiffeners are permitted, provided the link depth is
less than 25 inches. The required width is given by
bst
≥ bf /2 2 tw
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Figure 3-27 Link details
and the thickness by
tst
5 tw
≥ 3⁄8 in
Where the link depth is 25 inches or greater, similar intermediate stiffeners are required on both sides
of the web.
•
As specified in AISC 341 Section F3.4b, lateral bracing to the top and bottom flanges is necessary at each end of the link to prevent instability and restrain the link from twisting out of
plane. Lateral support must be provided to both flanges at the ends of the link. The required
strength of the lateral bracing is given by AISC 341 Equation (D1-4) as
where:
Pbl
5 0.06Ry Fy Z/ho as
Ry
5 ratio of the expected yield stress to the specified minimum yield
strength as given in Table 3-2
ho
5 distance between flange centroids
Z
5 plastic section modulus of the beam
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Example 3-6
Figure 3-28 shows the bottom story of a five-story eccentrically braced steel frame with a redundancy
factor of 1.0. The total design lateral force acting at the level of the second floor is indicated. If the
effects of gravity loads may be neglected, select a suitable W10 section, with a yield stress of 50 kips
per square inch, for the link.
Solution
Select a link length.
e
5 4 ft
The strength design forces acting on the link are
Vr 5 Vu 5 shear force on the link, from Figure 3-28
5 Vh/Lbm
5 90 3 14/12
5 105 kips
Pr 5 Pu 5 axial force on the link, from Figure 3-28
5 0 kip . . . effect of axial force on the link design need not be considered
Pr /Pc , 0.15 . . . no upper limit on link length
Figure 3-28 Details for Example 3-6
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Select section
From AISC Manual Table 1-1, select a W10 3 68 that has the following member properties
d
5 10.4 in
bf
5 10.1 in
tw
5 0.47 in
tf
5 0.77 in
ho
5 9.63 in
bf /2tf 5 6.58
h/tw
5 16.7
Zx
5 85.3 in3
Fy
5 50 ksi
Mp
5 nominal plastic flexural strength
5 Zx Fy
5 85.3 3 50/12
5 355 kip-ft
Aw
5 web area
5 (d 2 2tf)tw
5 (10.4 2 2 3 0.77)0.47
5 4.16 in2
Vn 5 Vy 5 Vp 5 nominal shear strength of link for Pr /Pc , 0.15, from AISC 341 Equation
(F3-2)
5 0.60Fy Aw
5 0.60 3 50 3 4.16
5 125 kips
f
5 resistance factor given by AISC 360 Section G2.1(a)
5 1.0
fVp
5 design shear strength
5 1.0 3 125
5125 kips
. Vu . . . satisfactory
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1.6Mp /Vp 5 1.6 3 355/125
5 4.54 ft
. e . . . link behavior is governed by shear yielding
Design requirements
The nominal required moment capacity of the link for shear yielding is
Mr
5 required moment at the end of the link
5 Vn e/2 . . . taking a conservative value
5 125 3 4/2
5 250 kip-ft
fMp
5 design flexural strength
5 0.9 3 355
5 320 kip-ft
. Mr . . . satisfactory
Local buckling
The flange width-to-thickness ratio for a highly ductile member is limited by Table 3-1 to a maximum
value of
bf /2tf 5 0.32(E/Ry Fy)0.5
5 0.32[29,000/(1.1 3 50)]0.5
5 7.35
The actual flange width-to-thickness ratio is
bf /2tf 5 6.58
, 7.35 . . . satisfactory
For a value of Pu 5 0 and Ca 5 0, the web height-to-thickness ratio is limited by Table 3-1 to a maximum value of
h/tw
5 2.57(E/Ry Fy)0.5
5 2.57[29,000/(1.1 3 50)]0.5
5 59.0
The actual web height-to-thickness ratio is
h/tw
5 16.7
, 59.0 . . . satisfactory
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Link rotation
The displacements in the bottom story may be determined using the virtual work method.15 To determine the elastic drift in the bottom story, a unit virtual load is applied to the frame as shown in Figure
3-29a, and the forces, u, are obtained. The design loads are applied to the frame as shown in Figure
3-29b, and the forces, P, are obtained. Neglecting the upper stories and the effects of bending moments
in the link and in the beams outside the link, the elastic drift in the bottom story is given by
where:
De
5 SPuL/AE
P
5 axial force in a member due to the applied design loads
u
5 axial force in a member due to the virtual unit load
L
5 length of a member
A
5 cross-sectional area of a member
E
5 modulus of elasticity of a member
(a)
(b)
Figure 3-29 Determination of drift
The area of the W10 3 68 beam is 19.9 square inches and the area of the diagonal brace may be
assumed to be 11.6 square inches. Details of the calculation are shown in Table 3-5.
Table 3-5 Details for Example 3-6
Member
P kips
u kips
L in
A in2
PuL/A
Beam
90
1.00
144
19.9
651
Brace
2138
20.77
221
11.6
2024
Brace
138
0.77
221
11.6
2024
Total
4699
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The elastic drift in the bottom story is given by
De
5 SPuL/AE
5 4699/29,000
5 0.162 in
The total inelastic drift in the bottom story is given by
where:
D
5 Cd De
Cd
5 deflection amplification factor
5 4.0 . . . from ASCE 7 Table 12.2-1
D
and
5 0.65 in
The link rotation angle is
gp
5 LD/he
5 28 3 0.65/(14 3 48)
5 0.027 radian
1.6Mp /Vp
5 1.6 3 355/125
5 4.54 ft
.e
Hence, from AISC 341 Section F3.4a, the link rotation capacity is limited to
ga
5 0.080 radian
. gp . . . satisfactory
Stiffener details
In accordance with AISC 341 Section F3.5b.4, full-depth web stiffeners are required on both sides of
the link web at the diagonal brace ends of the link.
Single-sided, full-depth web intermediate stiffeners are permitted because the link depth is less than
25 inches. The minimum required width is given by
bst
5 bf /2 2 tw
5 10.1/2 2 0.47
5 4.58 in . . . use 4.75 in
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The required thickness is given by the greater of
or
tst
5 3⁄8 in
tst
5 tw
5 1⁄2 in . . . governs
The weld between the grade A36 stiffener and the web is required to develop the force given by
Pw
5 Ast Fy
5 4.58 3 0.5 3 36
5 82.44 kips
The total length of fillet weld provided for welds on both sides of the stiffener, allowing for 0.75-inch
corner snips, is
l
5 2[(d 2 2tf) 2 1.5]
5 2[(10.4 2 2 3 0.77) 2 1.5]
5 14.72 in
The design fillet weld strength per 1⁄16 inch of E70XX electrodes is
qu
5 1.39 kips/in
The required weld size per 1⁄16 inch is
D
5 Pw /lqu
5 82.44/(14.72 3 1.39)
5 4.0 sixteenths
Hence, the required weld size is
w
5 4.0/16
5 1⁄4 in
The minimum size of weld permitted for the 1⁄2-inch-thick stiffener is given by AISC 360 Table J2.4 as
wmin
5 3⁄16 in
, w . . . satisfactory, use 1⁄4-in fillet weld
The minimum thickness of stiffener required to match the shear rupture strength of the welds on opposite sides of the plate is given by
tmin
5 6.18D/Fust
5 6.18 3 4/58
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5 0.43
, tst . . . satisfactory
The weld between the stiffener and the flange is required to develop the force given by
Pw
5 Ast Fy /4
5 4.58 3 0.5 3 36/4
5 20.6 kips
The total length of weld provided for welds on both sides of the stiffener, allowing for 0.75-inch corner
snips, is
l
5 2(bst 2 0.75)
5 2(4.75 2 0.75)
5 8.0 in
Using E70XX fillet welds, the required weld size per 1⁄16 inch is
D
5 Pw /lqu
5 20.6/(8.0 3 1.39)
5 1.9 sixteenths
The required weld size is
w
5 3⁄16 . . . minimum
The link rotation angle is
gp
5 0.027 radian
Hence, by interpolation the required intermediate stiffener spacing is
s
5 49.4tw 2 d/5 . . . for gp ≤ 0.027 radian
5 49.4 3 0.47 2 10.4/5
5 21 in
Provide two intermediate stiffeners to give a spacing of
sp
5 48/3
5 16 in
, s . . . satisfactory
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3.10.7 Beam requirements
AISC 341 Section F3.3 and Commentary Section F3.3 specify the following design requirements for
the beam outside the link:
•
The nominal required axial and flexural capacity of the beam shall be determined from load
combination 6 of ASCE 7 Section 2.3.6 with the amplified seismic force, Emh, replaced by
(1.25 3 0.88) 5 1.1 times the nominal shear capacity of the link to give
where:
Emh
5 1.1RyVn
Ry
5 ratio of the expected yield stress to the specified minimum yield
strength of the link
Vn
5 nominal shear capacity of the link
In accordance with AISC 341 Section F3.5a, the design capacity of the beam determined using
the procedures from AISC 360 Sections C, E, F, and H may be multiplied by Ry . For Grade 50
steel, Ry 5 1.1 and the enhanced design capacity becomes
RyfRn 5 1.1 3 fRn
•
5 0.99Rn . . . for f 5 0.9
Where required, the beam shall be provided with lateral support at both the top and bottom
flanges. Each support shall have a design capacity given by AISC 360 Equation (A-6-7) as
where:
Prb
5 0.02MrCd /ho
Cd
5 1.0 for single curvature
ho
5 distance between flange centroids
Mr
5 Ry Fy Z . . . for LRFD
5 Ry Fy Z/1.5 . . . for ASD
Example 3-7
Figure 3-28 shows the bottom story of a five-story eccentrically braced steel frame with a redundancy
factor of 1.0. The total design lateral force acting at the level of the second floor is indicated. The
effects of gravity loads may be neglected. Determine if the W10 3 68 section selected for the link is
adequate for the beam outside the link.
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Solution
The design forces acting on the beam and link are shown in Figure 3-30. In accordance with AISC 341
Section F3.3, the beam is designed for the maximum forces that can be generated by the link using an
overstrength factor of
W0
5 1.1Ry
5 1.1 3 1.1
5 1.21
Mr
Figure 3-30 Details for Examples 3-7 through 3-9
Before applying the overstrength factor, the forces acting on the beam are
Vbm
5 shear force on the beam, from Figure 3-30
5 Mr /Lbm
5 250/12
5 21 kips
Vn
5 nominal shear strength of the link
5 125 kips . . . from Example 3-6
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Rbr
5 Vn 1 Vbm . . . from Figure 3-30
5 125 1 21
5 146 kips
where:
Mr
5 required moment capacity of the link, from Example 3-6
5 250 kip-ft
Pbm
5 axial force on the beam, from Figure 3-28
5 Vn Lbm /h
5 125 3 12/14
5 107 kips
Design shear strength
Allowing for the overstrength factor, 1.1Ry , the nominal required shear capacity of the beam, in accordance with AISC 341 Section F3.3, is given by
Vu
5 1.1RyVbm
5 1.21 3 21
5 25 kips
The enhanced design shear capacity, in accordance with AISC 341 Section F3.5a, is given by
RyfVn 5 1.1 3 0.9 3 125 kips . . . from Example 3-6
5 124 kips
. Vu . . . satisfactory
Combined compression and flexure
Allowing for the overstrength factor, 1.1Ry , as specified in AISC 341 Section F3.3, the applied moment
on the beam is
Mu
5 1.1Ry Mr
5 1.21 3 250
5 303 kip-ft
Allowing for the overstrength factor, 1.1Ry , as specified in AISC 341 Section F3.3, the applied axial
load on the beam is
Pu
5 1.1Ry Pbm
5 1.21 3 107
5 129 kips
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As specified in AISC 341 Section F3.4b, lateral bracing is required to the top and bottom flanges at
each end of the link with a design strength of
Pb1
5 0.06Ry Fy Zx /ho
5 0.06 3 1.1 3 50 3 85.3/9.63
5 29 kips
In addition, lateral bracing is provided at the center of the beam to the top and bottom flanges with a
design strength of
Pb
5 0.02MrCd /ho
5 0.02Ry Fy ZxCd /ho
5 0.02 3 1.1 3 50 3 85.3 3 1.0/9.63
5 10 kips
The unbraced segment lengths about the x- and y-axes are thus
Lbx
5 12 ft
Lby
5 6 ft
The W10 3 68 section will be analyzed using AISC 360 Equations (H1-1a) or (H1-1b). The relevant
properties of the W10 3 68 are
A
5 19.9 in2
I
5 394 in4
rx
5 4.44 in
ry
5 2.59 in
Mp
5 355 kip-ft
fBF
5 3.85
Lp
5 9.15
KLby /ry
5 1.0 3 6 3 12/2.59
5 27.8
KLbx /rx
5 1.0 3 12 3 12/4.44
5 32.4 . . . governs
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Local buckling
The flange width-to-thickness ratio for a moderately ductile member is limited by Table 3-1 to a maximum value of
bf /2tf 5 0.40(E/Ry Fy)0.5
5 0.40[29,000/(1.1 3 50)]0.5
5 9.18
The actual flange width-to-thickness ratio is
bf /2tf 5 6.58
, 9.18 . . . satisfactory
The ratio of required strength to available strength is
Ca
5 Pu /fb Py
5 129/(0.9 3 19.9 3 50)
5 0.144
. 0.114
Hence, the web height-to-thickness ratio is limited by Table 3-1 to a maximum value of
h/tw
5 1.29(2.12 2 Ca)(E/Ry Fy)0.5
5 1.29(2.12 2 0.145)[29,000/(1.1 3 50)]0.5
5 58.5
The actual web height-to-thickness ratio is
h/tw
5 16.7
, 58.5 . . . satisfactory
Design compressive strength
From AISC Manual Table 4-14, the design axial compressive stress for the beam for KL/r 5 32.4 is
fc Fcr 5 41.7 ksi
As specified in AISC 341 Section F3.5a, the enhanced design axial compressive stress for the beam is
Ryfc Fcr 5 1.1 3 41.7
5 45.9 ksi
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The enhanced compressive strength of the beam is
Ryfc Pn 5 Ryfc Fcr Ag
5 45.9 3 19.9
5 913 kips
then:
Pu /Ryfc Pn 5 129/913
5 0.14
, 0.20
Hence, AISC 360 Equation (H1-1b) applies and, after modifying in accordance with AISC 341 Section
F3.5a, this is given by
Pu /2Ryfc Pn 1 Mux /Ryfb Mnx ≤ 1.0
where:
Mux
5 required flexural strength about the strong axis, including second-order
effects
5 B1Mnt
5 B1Mu
5 303B1
Mnx
5 nominal flexural strength about the strong axis in the absence of axial load
5 Mp
5 355 kip-ft
Second-order analysis
The Euler buckling load for the beam is given by AISC 360 Equation (A-8-5) as
Pe1
5 p2EI/(KLx)2
5 p2 3 29,000 3 394/(12 3 12)2
5 5433 kips
The reduction factor for a member in a braced frame, with one end pinned and not subjected to transverse loading, is given by AISC 360 Appendix 8.2.1 as
Cm
5 0.6
From AISC 360 Equation (A-8-3), the multiplier to account for P-d effects is given by
B1
5 Cm /(1 2 Pu /Pe1)
5 0.6/(1 2 129/5433)
5 0.61
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The minimum permitted value is
B1
5 1.0
The required flexural strength is given by AISC 360 Equation (A-8-1) as
Mux
5 B1Mnt
5 1.0 3 303
5 303 kip-ft
Design flexural strength
The beam unbraced length is
Lbm
5 6 ft
, Lp
Hence, the enhanced flexural capacity is
Ryfb Mnx 5 1.1 3 0.9 3 355
5 351 kip-ft
Interaction equation
The left side of AISC 360 Equation (H1-1b) is
Pu/2Ryfc Pn 1 Mux /Ryfb Mnx
5 0.14/2 1 303/351
5 0.93
, 1.0 . . . satisfactory
Hence, the W10 3 68 beam is adequate.
3.10.8 Diagonal brace requirements
AISC 341 Section F3.3 specifies the following design requirements for the diagonal brace:
•
To allow for strain hardening in the link, the nominal required axial and flexural capacity of the
brace are determined from load combination 6 of ASCE 7 Section 2.3.6 with the earthquake
force, Emh, replaced by the amplified nominal shear capacity of the link defined as
1.25RyVn
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where:
321
Ry
5 ratio of the expected yield stress to the specified minimum yield
strength of the link
Vn
5 nominal shear capacity of the link
•
As shown in Figure 3-27, the intersection of the brace and beam centerlines are at the end of
the link or within the link. In accordance with AISC 341 Commentary Section F3.5b.1, when
the intersection of the brace and beam centerlines are located outside the link, the eccentricity
produced creates additional moment in the beam and this must be considered in the design.
•
The required strength of the brace-to-beam connection must be sufficient to resist forces corresponding to link yielding and strain hardening. If the brace resists a portion of the link end
moment, the additional end moment must be considered in the design.
Example 3-8
Figure 3-28 shows the bottom story of a five-story eccentrically braced steel frame with a redundancy
factor of 1.0. The total design lateral force acting at the level of the second floor is indicated. If the
effects of gravity loads may be neglected, select a suitable rectangular hollow structural section, with
a yield stress of 50 kips per square inch, for the diagonal brace.
Solution
The unbraced length of the brace, using conservative centerline dimensions, is given by
Lbr
5 [(Lbm)2 1 h2]0.5
5 (122 1 142)0.5
5 18.4 ft
Factored loads
Before applying the overstrength factor, the forces acting on the beam and link are shown in Figure
3-30. The vertical component of the axial force in the brace is
Rbr
5 Vn 1 Vbm
5 125 1 21
5 146 kips
The axial force in the brace is
Pbr
5 Rbr Lbr /h
5 146 3 18.4/14
5 192 kips
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In accordance with AISC 341 Section F3.3, the brace shall be designed for the maximum forces that
can be generated by the link using an overstrength factor of
W0
5 1.25Ry
5 1.25 3 1.1
5 1.38
Allowing for the overstrength factor, the applied axial force on the brace is
Pu
5 W0Pbr
5 1.38 3 192
5 265 kips
Select section
The effective length of the brace is
KLbr
5 1.0 3 18.4
5 18.4 ft
From AISC Manual Table 4-4, select an HSS 7 3 7 3 1⁄2 that has a design strength in axial compression, for an effective length of 18.4 feet, of
fc Pn
5 312 kips
. Pu . . . satisfactory
Local buckling
The width-to-thickness ratio of a moderately ductile rectangular hollow section is limited by Table 3-1
to a maximum value of
b/t
5 0.76(E/Ry Fy)0.5
5 0.76[29,000/(1.3 3 50)]0.5
5 16.1
The actual width-to-thickness ratio is
b/t
5 7/0.465
5 15.1
, 16.1 . . . satisfactory
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3.10.9 Column requirements
Columns are designed using capacity design principles based on the forces generated by the fully
yielded and strain hardened link so as to prevent failure of a column or formation of a soft story. The
required forces in the column are determined from load combination 6 of ASCE 7 Section 2.3.6 with
the earthquake force, Emh , replaced by the amplified forces developed in the links. In accordance with
AISC 341 Section F3.3, the overstrength factor is
W0
5 1.25Ry
5 1.25 3 1.1
5 1.38
Since all links above the level of the column under consideration are unlikely to reach their maximum
shear strength simultaneously, a relaxation is permitted for multistory frames. For this situation, AISC
341 Commentary Section 3.3 gives a value for the overstrength factor of
W0
5 0.88 3 1.25Ry
5 0.88 3 1.25 3 1.1
5 1.21
In addition, the required strength of columns need not exceed the lesser of the following two conditions:
•
forces corresponding to the resistance of the foundation to overturning uplift
•
forces determined from nonlinear analysis
In designing a column, flexural forces resulting from seismic drift are neglected.
Welds at a column splice are subjected to high stress demands and inelastic strains and are designated
as demand critical in AISC 341 Section F3.6a as are welds at column-to-base plate connections and
beam-to-column connections. In accordance with AISC 341 Section F3.6d, welds at a splice must be
complete joint-penetration groove welds, and the splice is designed to develop at least 50 percent of the
lesser available flexural strength of the connected columns. The required shear strength of the joint is
Vr
5 SFy Zc /Hc . . . for LRFD
5 SFy Zc /1.5Hc . . . for ASD
where:
Zc
5 plastic section modulus of a column
SFy Zc 5 sum of the nominal plastic flexural strengths of the columns above and
below the splice
Hc
5 clear height of the column between beam connections
Columns are designated in AISC 341 Section F3.5a as highly ductile members.
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Example 3-9
Figure 3-28 shows the bottom story of a five-story eccentrically braced steel frame with a redundancy
factor of 1.0. The total design lateral force acting at the level of the second floor is indicated. The gravity loads acting on the column in the bottom story are
dead load, D 5 130 kips
live load, L 5 50 kips
The design response acceleration is SDS 5 1.0g.
Select a suitable W12 section, with a yield stress of 50 kips per square inch, for the column and determine the design uplift on the column.
Solution
The seismic forces, produced by the nominal strength of the link, acting on the link and the beam outside the link are as shown in Figure 3-30 and determined in Example 3-7. These forces are transferred
to the column, as shown in Figure 3-31. Assuming that the beams at all floors and at the roof are W10
3 68, the total compressive force acting on the column at the second floor is
Pcol
5 4Rbr 2 5Vbm
5 4 3 146 2 5 3 21
5 479 kips
The total seismic tensile force acting at the base of the column is
Tcol
5 5Rbr 2 5Vbm
5 5 3 146 2 5 3 21
5 625 kips
Factored loads
For maximum compression load in the column for a frame of five stories, AISC 341 Section F3.3
specifies the following loading for the design of the column
Pu
where:
5 (1.2 1 0.2SDS)D 1 0.5L 1 0.2S 1 W0QE
W0QE 5 forces generated by 1.1RyVn
5 1.1 3 1.1Pcol
5 1.1 3 1.1 3 479
5 580 kips
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Figure 3-31 Details for Example 3-9
and
Pu
5 (1.2 1 0.2)130 1 0.5 3 50 1 580
5 787 kips
Select section
The unbraced length of the column, using centerline dimensions, is
L
5 14 ft
The effective length of the column is
KL
5 1.0 3 14
5 14 ft
From AISC Manual Table 4-1, select a W12 3 96 that has a design strength in axial compression, for
an effective length of 14 feet, of
fc Pn
5 1020 kips
. Pu . . . satisfactory
Seismic and Wind Forces: Structural Design Examples
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Member properties
The section properties of a W12 3 96 are
A
5 28.2
bf /2tf 5 6.76
h/tw
5 17.7
Local buckling
The flange width-to-thickness ratio for a highly ductile member is limited by Table 3-1 to a maximum
value of
bf /2tf 5 0.32(E/Ry Fy)0.5
5 0.32[29,000/(1.1 3 50)]0.5
5 7.35
The actual flange width-to-thickness ratio is
bf /2tf 5 6.76
, 7.23 . . . satisfactory
The ratio of required strength to available strength is
Ca
5 Pu /fb Py
5 787/(0.9 3 28.2 3 50)
5 0.62
. 0.114
Hence, the web height-to-thickness ratio is limited by Table 3-1 to a maximum value of
h/tw
5 0.88(2.68 2 Ca)(E/Ry Fy)0.5
5 0.88(2.68 2 0.62)[29,000/(1.1 3 50)]0.5
5 41.6
The actual web height-to-thickness ratio is
h/tw
5 17.7
, 41.6 . . . satisfactory
Hence, the W12 3 96 column satisfies all requirements.
Seismic and Wind Forces: Structural Design Examples
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Design uplift
For maximum tensile load in the column for a frame of five stories, AISC 341 Section F3.3 specifies
the following loading
Tu
5 (0.9 2 0.2SDS)D 2 W0QE
5 (0.9 2 0.2)130 2 1.21 3 625
5 2665 kips
3.11 Special moment frames
Special moment frames resist seismic forces by means of the large inelastic deformations that occur in
the ductile frame. Inelastic rotations may occur at plastic hinges in the beam-column joints and shear
deformations may occur in the joint panel zone.
Special moment frames, as specified in AISC 341 Section E3, may be utilized in all seismic design
categories using a value of 3 for the overstrength factor and a value of 5.5 for the deflection amplification factor. As specified in ASCE 7 Table 12.2-1, no limitation is imposed on the building height in
any seismic design category.
Special moment frames may be utilized in dual systems with braced frames or shear walls. The system
limitations for the different types are specified in ASCE 7 Table 12.2-1.
3.11.1 Beam-to-column connections
Failures of steel special moment frame systems in the 1994 Northridge earthquake in California
occurred at the beam-column joint. The typical prescriptive joint specified at that time was unable to
sustain the large inelastic deformations that occurred at the joint. The factors contributing to the joint
failures were:
•
stress concentrations at the beam bottom flange weld
•
unsuitable weld metal used at the beam-column connection
•
variations of member strength from prescribed values
•
the use of larger beams than had previously been tested
•
fatigue failure at the joint
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Beam-column connections are now required by AISC 341 Section E3.6b to be capable of developing
an interstory drift angle of at least 0.04 radian with a residual moment capacity of 80 percent of the
nominal plastic moment capacity of the beam. As a result of an extensive research program,16, 17, 18 a
number of joint assemblies were initially determined to meet these criteria and were designated prequalified connections for special moment frames. These prequalified connections may be categorized
into the six main types, which are illustrated in Figure 3-32.
Subsequently, several proprietary joint assemblies have been developed and designated prequalified.
These include the slotted web connection,19 the SidePlate connection,20 the Kaiser bolted bracket,21
the Simpson strong frame moment connection,22 and the Conxtech Conxl moment connection.22
The slotted web connection is shown in Figure 3-33. In this system a slot is introduced into the top
and bottom of the beam web at the beam-column interface. In addition to a welded shear plate, the
beam web is welded to the column flange. The separation of the beam flanges and beam web allows
the flanges and web to buckle independently. This eliminates the lateral-torsional mode of beam buckling and the associated torsional flange/weld stresses that are characteristic of nonslotted beams. In
addition, this provides a uniform distribution of flexural stress and strain at the flange/weld connection
and eliminates vertical shear in the welds at the beam flanges. The beam web resists the entire vertical
shear and its share of the beam moment and the flanges resist the residual moment. The fatigue life of
the connection is more than triple the fatigue life of a nonslotted connection because of the elimination
of vertical shear at the flange/welds.
The SidePlate connection is shown in Figure 3-34 and consists of a column tree with shop-welded
side plates. In the field, the column trees are erected and full-length beams are hoisted into position
between two parallel column side plates. The beams are attached to the side plates with bolts and fillet welds. The parallel full-depth side plates reinforce the connection and force the plastic hinging to
occur in the beam at one-third the depth of the beam from the end of the side plates. The connection
exhibits high resistance to blast and progressive collapse.
The Kaiser bolted bracket moment connection is shown in Figure 3-35 and consists of a cast highstrength steel bracket fastened to each beam flange and bolted to the column flange. The bracket is
either shop welded or bolted to the beam flange and is field bolted to the column flange. Field welding
is entirely eliminated and frame erection is facilitated. The bracket develops the maximum moment
capacity of the connected beam, and plastic hinge formation occurs outside the connection in the beam
at the end of the bracket. Inelastic rotation is intended to occur in the beam in the region near the end
of the brackets.
Seismic and Wind Forces: Structural Design Examples
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Sh
a
b
bbf
d
Reduced beam section
Unreinforced flange-welded web
Lp
Free flange
Welded or bolted flange plate
Lst
Ltee
tbf
tpl
Bolted stiffened or unstiffened end plate
Double split tee
Figure 3-32 Prequalified connections
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Beam slot
Shear plate
Figure 3-33 Slotted web connection
Figure 3-34 SidePlate connection
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Figure 3-35 Kaiser bolted bracket connection
3.11.2 Design principles
The formation of plastic hinges at the beam-column interface during a seismic event results in large
inelastic strain demands at the connection, leading to brittle failure. To prevent this occurrence, the
prequalified connections are designed to produce the plastic hinges within the beam span, as shown
in Figure 3-36. This condition may be achieved by reducing the section of the beam22, 23 at the desired
location of the plastic hinge or by reinforcing the beam at the connection so as to prevent the formation
θp drift angle
Plastic hinge
h
Plastic hinge
Lh
L
Figure 3-36 Formation of plastic hinges
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of a hinge in this region. By this means, the connection at the beam-column interface remains nominally elastic and the inelastic deformation occurs away from the connection. The hinge location distances are provided in AISC 358.
The design principles are formulated on an expected strength basis using the probable strengths of the
materials.
The probable beam plastic moment, allowing for overstrength of the steel; the difference in yield
strengths of the beam flanges and web materials; and the estimated strain hardening is given by AISC
358 Equation (2.4-1) as
where:
Mpr
5 Cpr Ry Ze Fy
Ry
5 overstrength coefficient given in Table 3-2
5 ratio of the expected yield stress to the specified minimum yield strength of
the material
Fy
5 specified minimum yield stress of the beam
Ze
5 effective plastic section modulus of the beam at the zone of plastic hinging
Cpr
5 peak connection strength coefficient defined by AISC 358 Equation (2.4-2)
5 (Fy 1 Fu)/2Fy
≤ 1.2
5 1.15 . . . for Fy 5 50 ksi and Fu 5 65 ksi
5 1.4 . . . for unreinforced flange-welded web connections, from AISC 358
Section 8.7
Fu
5 specified minimum tensile strength of the beam
From Figure 3-37 and AISC 358 Equation (5.8-9), the shear force at the plastic hinge on the left end
of the beam is given by
where:
Vh
5 2Mpr /Lh 1 wuLh /2
Vh9
5 22Mpr /Lh 1 wu Lh /2
wu
5 factored gravity load on the beam, from AISC 358 Section 5.8
5 1.2D 1 0.5L 1 0.2S . . . for L ≤ 100 lb/ft2
Lh
5 distance between plastic hinges
Neglecting the gravity load on the length, Sh , the resulting bending moment at the face of the column
is given by AISC 358 Equation (5.8-6) as
Mf
5 Mpr 1 VhSh . . . for symmetrical loading on the beam
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For reduced beam section connections, in accordance with AISC 358 Equation (5.8-8), the bending
moment at the face of the column is limited to
where:
Mf
≤ fd Ry ZbFy
Zb
5 plastic section modulus of the beam at the column face
fd
5 1.0 . . . from AISC 358 Section 2.4.1
For the portion of the single bay frame indicated, the resulting bending moment at the center of the
column is given by AISC 358 Section 5.4(2) as
*
Mpb
5 Mpr 1 Vhsh
sh
Lh
Sh
Sh
′
h
h
L
h
h
Sh
Figure 3-37 Shear at plastic hinge
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3.11.3 Strong column-weak beam concept
Under normal circumstances, a strong column-weak beam concept is adopted to ensure that inelastic drift is uniformly distributed over the height of the building and inelastic deformations are concentrated at the ends of the beams. As shown in Figure 3-38(a), this prevents frame instability due
to P-delta effects. The formation of plastic hinges in the columns of a story may cause a soft story
condition, as shown in Figure 3-38(b). The large inelastic displacements produced in the lower story
columns increase the P-delta effect and may lead to column failure.
∆
∆
Plastic hinges
Soft story
(a)
(b)
Figure 3-38 Collapse mechanisms
The strong column-weak beam concept may be achieved in accordance with AISC 341 Equation
(E3‑1) by ensuring that
SMpc
* /SMpb
* . 1.0
where:
SMpc
* 5 the sum of the projections of the nominal flexural strengths of the columns
above and below the joint to the beam centerline with a reduction for the
axial force in the column as given by AISC 341 Equation (E3-2)
5 SZc(Fyc 2 Puc /Ag) . . . for LRFD
SMpb
* 5 the sum of the projections of the expected flexural strengths of the beams at
the plastic hinge locations to the column centerline
5 Mpr 1 Vh Sh . . . for one-sided (exterior column) connections
In accordance with AISC 341 Equation (F3-3), SMpb
* is calculated in accordance with AISC 358 Equation (5.8-5) as
* 5 S(Cpr Ry Fyb Zb 1 Vh sh)
SMpb
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where:
Puc
5 required axial compressive strength in the column using LRFD load
combinations, including the amplified seismic load
Zc
5 plastic section modulus of the column
Fyc
5 specified minimum yield stress of the column
Ag
5 gross area of the column
335
AISC 341 Section E3.4a(a) relaxes the strong column-weak beam requirement for columns with Puc
, 0.3Pc and the column is either:
(i) located in a one-story building or in the top story of a multistory building
or
(ii) located in a column line in which the available shear strength of all exempted columns is less
than 33 percent of the total available shear strength of the column line, and the available shear
strength of all exempted columns in the story is less than 20 percent of the total available shear
strength of the story
where:
Pc
5 nominal axial compressive strength of the column
5 Fy Ag
AISC 341 Section E3.4a(b) also provides an exemption for a column located in a story with a ratio of
available shear strength to required shear strength 50 percent greater than that of the story above.
Example 3-10
Figure 3-39 shows the beam-column connection of a reduced beam section steel special moment frame
with a redundancy factor of 1.0. The span between column centers is L 5 25 feet, the factored gravity
load on the beam is wu 5 2 kips per foot, and the factored axial compressive force on the column is
Puc 5 200 kips. Determine if the strong column-weak beam requirement is satisfied.
Solution
The relevant section properties of the W14 3 132 column are
Plastic modulus, Zc
5 234 in3
Yield stress, Fyc
5 50 ksi
Tensile strength, Fuc 5 65 ksi
Depth, dc
5 14.7 in
Flange thickness, tcf
5 1.03 in
Flange width, bcf
5 14.7 in
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Web thickness, tcw
5 0.645 in
hc /tcw
5 17.7
bcf /2tcf
5 7.15
k1
5 1.56 in
kdes
5 1.63 in
kdet
5 2.31 in
Area, Ag
5 38.8 in2
The relevant section properties of the W21 3 122 beam are
Plastic modulus, Zb
5 307 in3
Yield stress, Fyb
5 50 ksi
Depth, db
5 21.7 in
Flange thickness, tbf
5 0.96 in
Flange width, bbf
5 12.4 in
Web thickness, tbw
5 0.60 in
hb /tbw
5 31.3
bbf /2tbf
5 6.45
ho
5 20.7 in
Figure 3-39 Details for Examples 3-10 through 3-14
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Reduced beam section
AISC 358 Section 5.8 specifies the dimensions of the reduced section shown in Figure 3-40 as
a
5 (0.5 to 0.75)bf
b
5 (0.65 to 0.85)d
c
5 (0.1 to 0.25)bf
r
5 (4c2 1 b2)/8c . . . from AISC 358 Figure 5.1
Figure 3-40 Reduced beam section details
Select the following values
a
5 7 in
5 0.56bf . . . satisfactory
b
5 14 in
5 0.65d . . . satisfactory
c
5 3 in
5 0.24bf . . . satisfactory
r
5 (4 3 32 1 142)/(8 3 3)
5 9.7 in
Shear force at plastic hinge
The effective plastic section modulus of the beam at the zone of plastic hinging at the center of the
reduced section is given by AISC 358 Equation (5.8-4) as
Zbe
5 Zb 2 2ctbf (db 2 tbf)
5 307 2 2 3 3 3 0.96(21.7 2 0.96)
5 188 in3
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The probable beam plastic moment is given by AISC 358 Equation (2.4-1) as
Mpr
5 Cpr Ry Zbe Fyb
5 1.15 3 1.1 3 188 3 50
5 11,891 kip-in
The hinge location distance from the center of the column is
sh
5 dc /2 1 a 1 b/2
5 14.7/2 1 7 1 14/2
5 21.35 in
The distance between plastic hinges is
Lh
5 L 2 2sh
5 25 3 12 2 2 3 21.35
5 257.3 in
The shear force at the plastic hinge is given by AISC 358 Equation (5.8-9) as
Vh
5 2Mpr /Lh 1 wu Lh /2
5 2 3 11,891/257.3 1 (2/12)257.3/2
5 114 kips
Bending moment at the column
Neglecting the gravity load on the length Sh , the resulting bending moment at the face of the column is
Mf
5 Mpr 1 Vh Sh
5 Mpr 1 Vh (a 1 b/2)
5 11,891 1 114(7 1 14/2)
5 13,487 kip-in
For reduced beam section connections, in accordance with AISC 358 Equation (5.8-8), the bending
moment at the face of the column is limited to
where
Mpe
≤ fd Ry Zb Fyb
fd
5 ductile resistance factor from AISC 358 Section 2.4.1
5 1.0
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The right side of the expression is
fd Ry Zb Fyb 5 1.0 3 1.1 3 307 3 50
5 16,885 kip-in
. Mf . . . satisfactory
The resulting bending moment at the center of the column is
Mpb
*
5 Mpr 1 Vh sh
5 11,891 1 114 3 21.35
5 14,325 kip-in
Strong column-weak beam
The sum of the nominal flexural strengths of the column above and below the joint at the beam centerline, with a reduction for the factored axial force in the column, is given by AISC 341 Equation (E3-2)
as
* 5 SZc(Fyc 2 Puc /Ag)
SMpc
where:
Puc
5 required axial compressive strength in the column
5 200 kips
Zc
5 plastic section modulus of the column
5 234 in3
Fyc
5 specified minimum yield stress of the column
5 50 ksi
Ag
5 gross area of the column
5 38.8 in2
SMpc
* 5 2 3 234(50 2 200/38.8)
and
5 20,988 kip-in
The ratio of column moments to beam moment is
SMpc
* /SMpb
*
5 20,988/14,325
5 1.5
. 1.0 . . . satisfactory
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3.11.4 Beam details
To limit local flange buckling, AISC 341 Section E3.5a specifies the use of sections with a maximum
flange width-to-thickness ratio, for a highly ductile member, of
bbf /2tbf 5 0.32(E/Ry Fy)0.5
In accordance with AISC 358 Section 5.3.1(6), this ratio may be determined, in reduced beam section
connections, at the ends of the center two-thirds of the reduced section of the beam, unless gravity
loading moves the hinge point significantly from the center of the reduced section.
To prevent stress concentrations resulting in a brittle mode of failure, abrupt changes of flange area are
not permitted in the plastic hinge regions. The hinging area is defined in AISC 341 Section E3.5c as
the distance from the face of the column to one-half the beam depth beyond the theoretical hinge point.
Connections, shear studs, or other attachments shall not be permitted in the hinging area.
To provide adequate web stability for a highly ductile member with Ca ≃ 0, Table 3-1 gives a maximum height-to-thickness ratio of
hb /tbw 5 2.57(E/Ry Fy)0.5
Lateral bracing is necessary, as specified in AISC 341 Section D1.2, to the top and bottom flanges of
the beam to prevent instability. Bracing is required near all concentrated loads, at changes in cross
section, where a hinge may form, and at a maximum spacing of
lcr
5 0.095ry E/Ry Fy
Where the beam supports a concrete slab along its whole length, lateral bracing is not required to the
top flange.
In accordance with AISC 358 Section 5.3.1(8), the protected zone for a reduced beam design extends
from the column face to the end of the reduced section.
The flanges and web of the beam are connected to the column flange with complete joint penetration
groove welds and are designated demand critical welds by AISC 341 Section E3.6a(c).
Example 3-11
Figure 3-39 shows the beam-column connection of a steel special moment frame with a redundancy
factor of 1.0. The beam supports a concrete slab over its full length. Determine if the beam satisfies
local buckling requirements.
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Solution
The flange width-to-thickness ratio is limited by Table 3-1 to a maximum value of
bbf /2tbf 5 0.32(E/Ry Fy)0.5
5 0.32[29,000/(1.1 3 50)]0.5
5 7.35
For a reduced beam section, the flange width may be taken at the ends of the center two-thirds of the
reduced section. However, using the full width, the actual flange width-to-thickness ratio is
bbf /2tbf 5 6.45
, 7.35 . . . satisfactory
For a value of Ca ≃ 0, the height-to-thickness ratio of the web is limited by Table 3-1 to a maximum
value of
hb /tbw 5 2.57(E/RyFy)0.5
5 2.57[29,000/(1.1 3 50)]0.5
5 59.0
The actual web height-to-thickness ratio is
hb /tbw 5 31.3
, 59.0 . . . satisfactory
The beam supports a concrete slab and lateral bracing is not required to the top flange.
3.11.5 Column details
In accordance with AISC 341 Table D1.1, columns shall comply with the slenderness requirements of
a highly ductile member given in Table 3-1.
In accordance with AISC 341 Section E3.4c, where the ratio of column moments to beam moments is
* /SMpb
*
SMpc
, 2.0
lateral bracing of column flanges at beam-column connections is provided at the levels of both the top
and bottom beam flanges. Where a concrete slab is supported, this may be considered to provide the
necessary bracing.
* /SMpb
* ≥ 2.0, bracing is required only at the level of the top flanges of the beams.
Where SMpc
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Example 3-12
Figure 3-39 shows the beam-column connection of a steel special moment-resisting frame with a
redundancy factor of 1.0. The beam supports a concrete slab over its full length. Determine if the column satisfies local buckling requirements.
Solution
The ratio of column moments to beam moment is obtained from Example 3-10 as
* /SMpb
* 5 1.5
SMpc
, 2.0
Hence, the flange width-to-thickness ratio of the column is limited by Table 3-1 to a maximum value
of
bcf /2tcf 5 0.32(E/Ry Fy)0.5
5 0.32[29,000/(1.1 3 50)]0.5
5 7.35
The actual flange width-to-thickness ratio is
bcf /2tcf 5 7.15
, 7.35 . . . satisfactory
The ratio of required strength to available strength is
Ca
5 Puc /fc Pyc
5 200/(0.9 3 38.8 3 50)
5 0.115
. 0.114
Hence, the web height-to-thickness ratio is limited by Table 3-1 to a maximum value of
hc /tcw 5 0.88(2.68 2 Ca)(E/Ry Fy)0.5
5 0.88(2.68 2 0.115)[29,000/(1.1 3 50)]0.5
5 51.8
The actual web height-to-thickness ratio is
hc/tcw 5 17.7
, 51.8 . . . satisfactory
Lateral bracing is provided by the concrete slab.
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3.11.6 Panel zone design
To prevent shear buckling during cyclic loading, the individual thicknesses of column webs and doubler plates shall not be less than the value given by AISC 341 Equation (E3-7) as
where:
t
5 (dz 1 wz)/90
dz
5 db 2 2tf of deeper beam at connection
wz
5 panel zone width between column flanges
The thickness of any doubler plate may be included in t, provided it is connected to the column web
with a minimum of four plug welds adequate to prevent local buckling of the plate, as shown in Figure 3-41. In addition, as specified in AISC 341 Section E3.6e3, where the doubler plate is placed
against the column web, it must be welded to the column flanges to develop the available shear yielding strength of the doubler plate. The doubler plate must be either complete-joint-penetration groove
welded or fillet welded to the column flanges. Where continuity plates are not used, the doubler plate
must be fillet welded top and bottom to develop the proportion of the total force that is transmitted to
the doubler plate. For this situation, the doubler plates must extend a minimum of 6 inches above and
below the top and bottom of the deeper beam. Where continuity plates are used, the doubler plate must
be welded to the continuity plates to develop 75 percent of the available shear strength of the doubler
plate. Where the doubler plates are placed away from the column web, they must be placed symmetrically in pairs and complete-joint-penetration groove welded to continuity plates to develop the pro rata
share of the total force transmitted to the doubler plate.
A
A
k1 + (1/2 in max)
ho
Plug welds
to doubler
1
kdet + 11/2 in
Typ
/4 in
Lnet
kdet + (11/2 in min)
Section A-A
Figure 3-41 Continuity plates
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From AISC 341 Section 3E.6e(1), the required shear strength of the panel zone is determined from
the summation of the moments at the column faces by projecting the expected moments at the plastic
hinges to the column faces. The required shear strength is
where:
Ru
5 Mf /ho
ho
5 distance between beam flange centroids
Mf
5 bending moment at the face of the column
5 Mpr 1 VhSh
Vh
5 shear force at plastic hinge
5 2Mpr /Lh 1 wuLh /2
Lh
5 length between plastic hinges
wu
5 factored gravity load on the beam
5 (1.2 1 0.2SDS)D 1 0.5L 1 0.2S
Sh
5 distance between plastic hinge and face of column
The design shear strength of the column panel zone is given by AISC 360 Equation (J10-11) as
where:
Ru
5 0.60fFy dc tp(1 1 3bcf t 2cf /db dc tp)
f
5 1.0 . . . from AISC 341 Section E3.6e
tp
5 tcw 1 ddbl
ddbl
5 doubler plate thickness
Example 3-13
Figure 3-39 shows the beam-column connection of a steel special moment frame with a redundancy
factor of 1.0. Determine if doubler plates are required.
Solution
Shear buckling
The minimum column web thickness to prevent shear buckling is given by AISC 341 Equation
(E3-7) as
t
5 (dz 1 wz)/90
5 (db 2 2tf 1 dc 2 2tcf)/90
5 (21.7 2 2 3 0.96 1 14.7 2 2 3 1.03)/90
5 0.360 in
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The actual column web thickness is
tcw
5 0.645
. t . . . satisfactory
Required shear strength
From AISC 341 Section 3E.6e(1), the required shear strength of the panel zone is
Ru
5 Mf /ho
5 13,487/20.7
5 652 kips
Available shear strength
The available shear strength of the column panel zone is given by AISC 360 Equation (J10-11) as
fRn
5 0.60fFy dc tcw[1 1 (3bcf t 2cf )/(db dc tcw)]
5 0.60 3 1.0 3 50 3 14.7 3 0.645[1 1 (3 3 14.7 3 1.032)/(21.7 3 14.7 3 0.645)]
5 349 kips
, Ru . . . unsatisfactory
A doubler plate is required with a thickness of 0.75 inch to give a total thickness of
tp
5 tcw 1 ddbl
5 0.645 1 0.75
5 1.395 in
This provides an available shear strength of
fRn
5 680 kips
. Ru . . . satisfactory
3.11.7 Continuity plates
Requirements for the provision and design of continuity plates, also known as stiffener plates, are provided in AISC 358 Section 2.4, AISC 341 Section E3.6f, and AISC 360 Section J10.
As specified in AISC 358 Section 2.4, continuity plates must be provided in accordance with the
details of the particular prequalified connection where these are required. Where continuity plates are
not specifically detailed for a prequalified connection, they must be provided in accordance with AISC
341.
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Where the beam flange is welded to the flange of a W-shape column, AISC 341 Equation (E3-8)
requires the provision of a continuity plate when
tcf
, bbf /6
The beam flange force applied to the column flange is specified by AISC 341 Section E3.6f.1. Where
the beam web is welded to the column flange, it is assumed that the web participates in transferring the
beam moment, Mf , to the column and the beam flange force is
Pf
5 0.85Mf /as d*
Where the beam web is bolted to the column flange, it is assumed that only the beam flanges transfer
the beam moment to the column and the beam flange force is
where:
Pf
5 Mf /as d*
Mf
5 maximum probable moment at face of column
Pf
5 required strength at the column face
ho
5 distance between centroids of beam flanges
as
5 LRFD-ASD force level adjustment factor
Where the required strength at the column face exceeds the available column strength, continuity
plates in accordance with AISC 360 Section J10.8 and AISC 341 Section E3.6f.2 must be provided.
The minimum continuity plate thickness specified by AISC 341 Section E3.6f is
and
tst
5 0.75tbf . . . for two-sided (interior column) connections
tst
5 tbf /2 . . . for one-sided (exterior column) connections
The minimum width of a continuity plate is required to match the beam flange.
In accordance with AISC 360 Section J10.8, where continuity plates are required, they shall be designed
as axially loaded columns to support the beam flange force. The effective length is taken as
where:
Le
5 0.75h
h
5 clear distance between flanges, less the corner radii
5 dc 2 2kdes
dc
5 depth of column
kdes
5 distance from outer face of column flange to web toe of fillet design value
The cross section of the column may be considered to consist of the stiffener and a strip of column web
having a width of 25tw.
Continuity plates are welded to the column flange using complete-joint-penetration groove welds,
as shown in Figure 3-41. Continuity plates are clipped as detailed in AISC 358 Section 3.6 and as
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shown in Figure 3-41 to avoid the column k-area. The plates are welded to column webs using complete-joint-penetration groove welds or fillet welds. The required strength of the sum of the welded
joints of the continuity plates to the column web is the lesser of the following:
(i)
the sum of the design strengths in tension of the contact areas of the continuity plates to the column flanges, which is
Ru
5 2fFyst tst(bst 2 k1 2 0.5 in)
(ii) the design strength in shear of the contact area of the continuity plates with the column web,
which is
where:
Ru
5 2 3 0.60fFyst tst Lnet
Lnet
5 net length of continuity plate
5 dc 2 2(kdet 1 1.5 in) . . . from AISC 358 Section 3.6
kdet
5 distance from outer face of column flange to web toe of fillet detailing value
(iii) the design strength in shear of the column panel zone, which is given by AISC 360 Equation
(J10‑11) as
where:
Ru
5 0.60fFydctp[1 1 (3bcf t 2cf )/(db dc tp)]
f
5 1.0 . . . from AISC 341 Section E3.6c
tp
5 tcw 1 ddbl
ddbl
5 doubler plate thickness
(iv) the sum of the beam flange forces produced by the maximum probable moment at the column
face, which is
where:
Ru
5 Mpr /ho
ho
5 distance between beam flange centroids
Example 3-14
Figure 3-39 shows the beam-column connection of a steel special moment frame with a redundancy
factor of 1.0. Design the continuity plates using A36 steel.
Solution
bbf /6
5 12.4/6
5 2.07 in
. tcf . . . continuity plates are required
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Continuity plate thickness
The minimum continuity plate thickness specified by AISC 341 Section E3.6f.2 is
tst
5 tbf /2 . . . for one-sided (exterior column) connections
5 0.96/2
5 0.48
Use
tst
5 0.75 in
The required continuity plate width is
bst
5 (bbf 2 tcw)/2
5 (12.4 2 0.645)/2
5 5.88 in
Use
bst
5 7 in
The beam flange design force for a welded beam web is given by
Puc
5 0.85Mf /ho
5 0.85 3 13,487/20.7
5 554 kips
The effective column resisting the flange force consists of the two stiffener plates plus a strip of web
having a width of 25tcw . The effective column has a moment of inertia of
I
 tst(2bst 1 tcw)3/12
5 0.75 3 (14.645)3/12
5 196 in4
The area of the effective column is
A
5 2bst tst 1 25(tcw)2
5 2 3 7 3 0.75 1 25 3 0.6452
5 20.9 in2
The radius of gyration of the effective column is
r
5 (I/A)0.5
5 (196/20.9)0.5
5 3.06 in
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The clear distance between flanges less the corner radii is
h
5 dc 2 2kdes
5 14.7 2 2 3 1.63
5 11.44 in
For an effective length factor of K 5 0.75, the slenderness ratio of the effective column is
KL/r
5 0.75h/r
5 0.75 3 11.44/3.06
5 2.80
From AISC Manual Table 4-14, the available axial compressive stress for the stiffener is
Fcr
5 32.4 ksi
The available axial strength of the stiftener
Pc
5 Fcr A
5 32.4 3 20.9
5 677 kips
. Puc . . . satisfactory
Continuity plate welding
The required strength of the sum of the welded joints of the continuity plates to the column web is the
lesser of the following:
(i)
the sum of the design strengths in tension of the contact areas of both continuity plates to the
column flanges, which is
Ru
5 2fFyst tst(bst 2 k1 2 0.5 in)
5 2 3 0.9 3 36 3 0.75 3 (7 2 1.56 2 0.5)
5 240 kips
(ii) the design strength in shear of the contact area of both continuity plates with the column web,
which is
Ru
5 2 3 0.60fFyst tst Lnet
5 2 3 0.6 3 0.9 3 36 3 0.75[14.7 2 2(2.31 1 1.5)]
5 206 kips . . . governs
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(iii) the design strength in shear of the column panel zone, which is
Ru
5 0.60fFy dc tp[1 1 (3bcf t 2cf )/(dbdctp)]
5 680 kips . . . from Example 3-13
(iv) the sum of the expected yield strengths of the beam flanges transmitting force to the continuity
plates, which is
Ru
5 Mpr /ho
5 11,891/20.7
5 574 kips
where:
ho
5 distance between beam flange centroids
The total length of fillet weld provided for welds on both sides of the continuity plates is
l
5 2 3 2(Lnet 2 0.5) . . . from Figure 3-41
5 2 3 2(7.08 2 0.5)
5 26.3 in
The design fillet weld strength per 1⁄16 inch of E70XX electrodes is given by AISC 360 Table J2.5 as
qu
5 0.60fFEXXAw
5 0.60 3 0.75 3 70 3 0.707/16
5 1.39 kips/in
The required weld size per 1⁄16 inch is
D
5 Ru /lqu
5 206/(26.3 3 1.39)
5 5.6 sixteenths
Hence, the required weld size is
w
5 5.6/16
5 3⁄8 in . . . to the nearest 1⁄16 in
The thickness of the column web is 5⁄8 inch and the minimum allowable fillet weld size connecting the
stiffeners to the web is given by AISC 360 Table J2.4 as
wmin
5 1⁄4 in
, w . . . satisfactory
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The minimum thickness of web required to match the shear rupture strength of the welds on opposite
sides of the web is given by AISC Manual Part 9 as
tmin
5 6.19D/Fuc
5 6.19 3 5.6/65
5 0.53 in
, tcw . . . satisfactory
3.12 Buckling-restrained braced frames
A buckling-restrained brace consists of a brace in which buckling is inhibited, thereby permitting compression yielding of the brace to occur, as well as tensile yielding.24, 25, 26 As shown in Figure 3-42, the
brace element consists of a steel core encased in a steel tube filled with mortar that acts as a restraining
element. The steel core is debonded over its length from the mortar fill, thus allowing relative deformation between the two elements. Axial loads are resisted by the core only, whereas the casing and mortar
fill prevents Euler buckling and local buckling of the core. As shown in Figure 3-42, several alternative
configurations are possible for the core and for the end attachments.
In a conventional special concentric braced frame with chevron configuration, the strength of the structure is limited by the Euler buckling load of the brace in compression. A typical hysteretic response for
an unrestrained brace is shown in Figure 3-43. The buckling of the brace under compression loading
produces a significant loss in strength, a reduction in the area under the hysteretic curve, and a decrease
in the amount of energy dissipation. Because of this, the single-diagonal braced frame is not permitted.
The strength of the beam is governed by the unbalanced forces in the tension and compression braces.
The braces are designed to resist buckling, resulting in large forces in the connections and the frame.
In a buckling-restrained braced frame with chevron configuration, ductile yielding in both tension and
compression occurs in the brace. This produces a symmetrical hysteretic curve, as shown in Figure
3-43, with a consequent increase in energy dissipation and the ability to resist numerous cycles of
alternating loads without degradation. Because the tension and compressive strengths of a buckling-restrained brace are almost identical, the single-diagonal braced frame is permitted. The buckling-restrained braced frame provides an elastic stiffness equivalent to that of an eccentrically braced frame,
and ductility and energy dissipation equivalent to that of a special moment frame.
The design of a buckling-restrained braced frame is based on the utilization of buckling-restrained
braces qualified by testing. As specified in AISC 341 Section K3, cyclic testing is required on a buckling-restrained brace test specimen and on a subassemblage test specimen. The performance of the
brace and the subassemblage must be satisfactory up to a deformation corresponding to twice the
design story drift of the prototype frame with a minimum value for the story drift of 1 percent of the
story height. Prequalified braces are available from several manufacturers.27, 28
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Figure 3-42 Buckling-restrained braced frames
Buckling-restrained braced frames, as specified in ASCE 7 Table 12.2-1, may be used in building
frame systems in all seismic design categories. A value of 8 is specified for the response modification
coefficient, a value of 2.5 for the overstrength factor, and a value of 5 for the deflection amplification
factor. As specified in ASCE 7 Table 12.2-1, no limitation is imposed on the building height in seismic
design categories A, B, and C. The maximum height permitted in seismic design categories D and E is
160 feet, and in seismic design category F, it is 100 feet.
Inelastic deformations under the design earthquake occur primarily as brace yielding in tension and
compression, while other members of the frame remain nominally elastic.
Buckling-restrained braced frames may be used in dual systems with special moment frames in all
seismic design categories, using a value of 8 for the response modification coefficient, a value of 2.5
for the overstrength factor, and a value of 5 for the deflection amplification factor. In accordance with
ASCE 7 Table 12.2-1, no limitation is imposed on the building height.
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Figure 3-43 Hysteretic response
3.12.1 Buckling-restrained brace applications
Buckling-restrained braces are typically used in concentrically braced frames. In a special concentrically braced frame, buckling of the compression brace is the controlling factor in the strength of the
frame. Compression buckling results in severe loss of brace capacity and ductility, and the formation
of plastic hinges in the brace leads to eventual fracture. Since buckling strength governs the size of
the braces, the brace is unnecessarily strong in tension. Hence, the design of other members of the
braced frame, using capacity design principles, results in member sizes that are larger than necessary.
For chevron bracing configurations, the beam intersected by the braces has to be designed for the large
unbalanced brace force.
Buckling-restrained brace frames have superior ductile performance compared with special concentrically braced frames. The buckling restrained brace has almost identical strengths in tension and compression and is not subject to compression buckling and degradation in hysteretic response. Hence,
the brace forces are lower than in a special concentrically braced frame and the other members of the
frame are correspondingly smaller. In addition, the beam intersected by the braces in a chevron configuration does not have to be designed for a large unbalanced brace force.
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ASCE 7 Table 12.8-2 gives values for the building period parameter, Ct, of 0.03 for a bucklingrestrained braced frame and 0.02 for a special concentrically braced frame. Hence, the building fundamental period, T, given by ASCE 7 Equation (12.8-7), is 50 percent higher for the special buckling-​
restrained frame. This results in a lower value of the seismic response coefficient, Cs, given by ASCE 7
Equation (12.8-3) and a lower value of the seismic base shear, V, given by ASCE 7 Equation (12.8-1).
ASCE 7 Table 12.2-1 gives values for the response modification coefficient, R, of 8 for a bucklingrestrained braced frame and 6 for a special concentrically braced frame. This results in a lower value
of the seismic response coefficient, Cs, given by ASCE 7 Equation (12.8-3) and a lower value of the
seismic base shear, V, given by ASCE 7 Equation (12.8-1).
The reduction in design forces on the buckling-restrained braced frame results in a decrease in material quantities and reduced foundation costs and makes this a cost-effective alternative to the special
concentrically braced frame.29
3.12.2 Brace requirements
The steel core is designed to resist the design axial force in the brace. The design axial strength of the
brace is given by AISC 341 Equation (F4-1) as
where:
Pysc
5 Fysc Asc
Fysc
5 specified minimum yield stress of the steel core, or actual yield stress as
determined from a coupon test
Asc
5 net area of steel core
The required area of the steel core is determined from
where:
Asc
5 Pu /fFysc
Pu
5 calculated load on the brace from AISC 341 Section F4.3, based on the
seismic base shear and neglecting the effects of gravity loads
f
5 resistance factor
5 0.9
W
5 safety factor
5 1.67
In accordance with AISC 341 Section F4.2, the buckling-restraining system shall prevent Euler buckling and local buckling of the steel core at a deformation corresponding to the larger of twice the design
story drift or 2 percent of the story height. The design drift is given by ASCE 7 Equation (12.8-15) as
Dx
5 DxeCd /Ie
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where:
355
Dxe
5 theoretical drift, caused by the code-prescribed design level forces, as
determined by an elastic analysis
Cd
5 deflection amplification factor for inelastic deformation, given in ASCE 7
Table 12.2-1
Ie
5 seismic importance factor given in ASCE 7 Table 1.5-2
As shown in Figure 3-44, the elongation of the brace, Db, for a story drift of Dx is
Db
5 Dx cos q
Vertical
displacement
Lateral
displacement
Rotation
Figure 3-44 Brace deformation
Because of the brace overstrength in compression, a vertical upward deflection is produced in a beam
intersected by braces in a chevron configuration. In accordance with AISC 341 Section C-F4.4a, the
maximum elongation of the brace must be increased to allow for this vertical deflection. As shown in
Figure 3-44, the elongation of the brace, Db , for a vertical deflection of Dy is
Db
5 Dy sin q
Similarly, the buckling-restrained brace must perform satisfactorily with end connection rotational
demands corresponding to twice the design story drift. As shown in Figure 3-44, the end rotation associated with a design story drift of Dx is
l
 Dx /h
The design of a buckling-restrained brace is based on the testing of a similarly sized specimen and on
a brace subassemblage that includes rotational demands. The uniaxial test is required to demonstrate
adequate brace hysteretic performance and to determine the overstrength factors for the design of the
other prototype members. The subassemblage test is required to demonstrate that deformations and
rotations of the prototype structure will not cause failure of any of the component parts. For some
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subassemblage arrangements, a single test may qualify as both a subassemblage and a brace test. Test
requirements are detailed in AISC 341 Section K3 and are designed to confirm that the brace can
function as intended. Prequalified braces may be selected from catalogs,27, 28 provided that they have
adequate strength and are similar in scale to the prototype.
To be acceptable, the prequalified brace must satisfy the following requirements of AISC 341 Section
K3.3c, which are:
•
the cross-sectional shape and orientation of the steel core shall be the same as that of the
prototype
•
the axial yield stress of the steel core shall not be less than 30 percent nor more than 120 percent of the prototype
•
the method of separation between the steel core and the buckling restraining mechanism, and
the material used, shall be the same as in the prototype
Hence, the axial strength of the steel core of the prototype brace may vary from a maximum of 3.33Pysc
to a minimum of 0.83Pysc , where Pysc is the design axial strength of the steel core of the prequalified
brace.
Testing also provides the designer with the magnitude of the adjusted, or maximum, brace strength
that can be developed in the prototype. The maximum brace force may be significantly greater than the
design strength because of compression overstrength, strain hardening, and the use of a resistance factor. The adjusted brace strength is used in the design of the brace connections and the other prototype
members using capacity-design principles. As shown in Figure 3-45 and defined in AISC 341 Section
F4.2a, the adjusted brace strength in tension is given by
where:
Tmax
5 wRy Pysc
Tmax
5 tensile force in the brace at a brace deformation of Dbm
Dbm
5 deformation of the brace corresponding to the design story drift of the
prototype
Pysc
5 Fysc Asc
w
5 strain hardening adjustment factor
Ry
5 ratio of expected yield stress to specified minimum yield stress
5 1.0 . . . when Fysc is determined from a coupon test of the steel core
The adjusted brace strength in compression is given by
where:
Pmax
5 bwRy Pysc
Pmax
5 compressive force in the brace at a brace deformation of Dm
b
5 compression overstrength adjustment factor
5 Pmax/Tmax
≤ 1.5 . . . from AISC 341 Section K3.8
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Figure 3-45 Brace force/displacement diagram
Braces with values of b and w less than unity are not permitted. In addition, the ratio of maximum
compressive force to maximum tension force shall not exceed 1.5 during the brace tests for cycles with
a deformation greater than Dby
Dby
where:
5 deformation of brace corresponding to yield
The required test loading protocol is specified in AISC 341 Section K3.4c and is illustrated in Figure
3-46. Two cycles of loading are required at brace deformations and rotations corresponding to Dby ,
0.5Dbm , Dbm , 1.5Dbm , and 2Dbm . In addition, for the brace test specimen, additional cycles are required
at a deformation of 1.5Dbm to produce a cumulative inelastic axial deformation of 200 times the yield
deformation.
The requirements for the steel core of the brace test specimen are specified in AISC 341 Section K3.3e
and are:
•
the specified minimum yield stress shall be identical with that of the prototype
•
the measured yield stress shall be not less than 90 percent of that of the prototype, as measured
by coupon tests
•
the specified minimum ultimate stress and strain shall not exceed those of the prototype
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Figure 3-46 Testing protocol
Example 3-15
Figure 3-47 shows the bottom story of a multistory buckling-restrained braced frame with a redundancy factor of r 5 1.0. The structure is located on a site with a design response acceleration of
SDS 5 1.0g. The specified minimum yield stress of the grade A36 steel core is Fysc 5 40 kips/in2. The
design story drift produced by the design loads is estimated as 1 percent of the story height, and the
vertical deflection of the beam at the second floor as 1/500 times the beam span. The load acting on a
brace based on the seismic base shear and neglecting gravity loads is
design seismic force 5 230 kips
Figure 3-47 Details for Example 3-15
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Determine the area required for the steel core of the prototype and the requirements of a suitable prequalified brace.
Solution
The seismic load acting on the prototype brace is given in the problem statement as
Pu
5 230 kips
The required design axial strength of the steel core is
Pysc
5 Pu /f
5 230/0.9
5 256 kips
The required area of the steel core is given by AISC 341 Equation (F4-1) as
Asc
5 Pysc /Fysc
5 256/40
5 6.40 in2
The minimum permissible axial strength of a prequalified brace is given by AISC 341 Section K3.3c as
Pymin
5 0.3 3 Pysc
5 0.3 3 256
5 77 kips
The maximum permissible axial strength of a prequalified brace is given by AISC 341 Section K3.3c as
Pymax 5 1.2 3 Pysc
5 1.2 3 256
5 307 kips
The brace deformation caused by the design story drift is
Dbm
5 Dx cos q
Dbm
5 0.01 3 hcos q
5 0.01 3 14 3 12cos 45°
5 1.19 in
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The brace deformation caused by the vertical deflection of the beam is
Db
5 Dy sin q
5 (l/500)sin q
5 (28 3 12/500)sin 45°
5 0.48 in
The minimum deformation required from the prequalified brace in accordance with AISC 341 Section
F4.4a is
Dbm 1 Db 5 1.19 1 0.48
5 1.67 in
3.12.3 Brace connection requirements
In accordance with AISC 341 Section F4.6c, the required strength of bracing connections is
where:
Pu
5 Pmax
Pmax
5 adjusted brace strength in compression
5 bwRy Pysc
b
5 compression overstrength adjustment factor
w
5 strain hardening adjustment factor
Pysc
5 design axial strength of the brace
Ry
5 ratio of expected yield stress to specified minimum yield stress
5 1.0 . . . when Fysc is determined from a coupon test of the steel core
Example 3-16
Figure 3-47 shows the bottom story of a multistory buckling-restrained braced frame with a design
axial strength of the steel core of Pysc 5 256 kips, which is established using the yield stress determined from a coupon test. The following factors are obtained from the prequalification tests
b
5 compression overstrength adjustment factor
5 1.15
w
5 strain hardening adjustment factor
5 1.35
Determine the required strength of bracing connections.
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Solution
From AISC 341 Section F4.6c, the required strength of bracing connections is
Pu
5 bwRy Pysc
5 1.15 3 1.35 3 1.0 3 256
5 397 kips
3.12.4 Beam design requirements
In a buckling-restrained braced frame utilizing chevron bracing, AISC 341 Section F4.4a requires the
beam to be continuous between columns and to be designed to carry all tributary gravity loads, without
support from the bracing, using the load combinations of ASCE 7 Section 2.3.6. In addition, for load
combinations that include seismic effects, the earthquake load, Emh , is replaced by the unbalanced
force, Qb , produced by the adjusted brace strengths in tension and compression. The beam is then
designed for the load combinations
(1.2 1 0.2SDS)D 1 0.5L 1 0.2S 1 Qb
(0.9 2 0.2SDS)D 2 Qb
As specified in AISC 341 Section F4.4a(b), the top and bottom flanges of the beam must be laterally
supported, as a minimum, at the point of intersection of the chevron braces. This may be achieved by
designing the lateral brace for the force given by AISC 360 Appendix 6, Equation (A-6-7) as
where:
Pbr
5 0.02MrCd /ho
Mr
5 beam required flexural strength
5 Ry ZFy
Ry
5 ratio of expected yield stress to specified minimum yield stress
Cd
5 curvature factor
5 1.0 for bending in single curvature
ho
5 distance between beam flange centroids
The required stiffness of the lateral brace is given by AISC 360 Appendix 6, Equation (A-6-8) as
where:
bbr
5 10Mr Cd /fLb ho
f
5 resistance factor
5 0.75
Lb
5 laterally unbraced length
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Additional bracing is provided, as necessary for a moderately ductile member, to satisfy AISC 341
Equation (D1-2).
To reduce the possibility of local buckling, AISC 341 Section F4.5a requires beams to be compact
sections, as tabulated in Table 3-1 for moderately ductile members.
Example 3-17
Figure 3-47 shows the bottom story of a multistory buckling-restrained braced frame with a design
axial strength of the steel core of Pysc 5 256 kips, which is established using the yield stress determined from a coupon test. The following factors are obtained from the prequalification tests
b
5 compression overstrength adjustment factor
5 1.15
w
5 strain hardening adjustment factor
5 1.35
Determine the unbalanced vertical force, Qb , produced by the adjusted brace strengths in tension and
compression.
Solution
The adjusted brace strength in compression is defined in AISC 341 Section F4.2a as
where:
Pmax
5 bwRy Pysc
b
5 compression overstrength adjustment factor
5 1.15
w
5 strain hardening adjustment factor
5 1.35
Pysc
5 design axial strength of the brace
5 256 kips
Ry
5 ratio of expected yield stress to specified minimum yield stress
5 1.0 . . . when Fysc is determined from a coupon test of the steel core
Hence:
Pmax
5 1.15 3 1.35 3 1.0 3 256
5 397 kips
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The adjusted brace strength in tension is defined in AISC 341 Section F4.2a as
Tmax
5 wRy Pysc
5 1.35 3 1.0 3 256
5 346 kips
The unbalanced vertical force is
Qb
5 (Pmax 2 Tmax)sin q
5 (397 2 346)sin 45°
5 36 kips
3.12.5 Column design requirements
In a buckling-restrained braced frame for load combinations that include seismic effects, AISC 341
Section F4.3 requires the earthquake load, Emh , to be determined from the adjusted brace strengths in
tension and compression. AISC 341 Section F4.5a requires columns to be compact sections, as tabulated in Table 3-1 for moderately ductile members.
In accordance with AISC 341 Section F4.6a, welds at column splices and column-to-base plate connections are designated demand critical.
Example 3-18
The three-story buckling-restrained braced frame shown in Figure 3-48(a) has braces in all stories with
a design axial strength of the steel core of Pysc 5 256 kips, which is established using the yield stress
determined from a coupon test. The following factors are obtained from the prequalification tests
b
5 compression overstrength adjustment factor
5 1.15
w
5 strain hardening adjustment factor
5 1.35
The building is located on a site with a design response acceleration parameter of SDS 5 1.0g. The
gravity loads acting on the columns in the bottom story are
dead load, D 5 80 kips
live load, L 5 30 kips
Determine the required axial strength of the columns in the bottom story.
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14 ft
14 ft
Pmax
Tmax
14 ft
Pmax
θ
Pmax
Tmax
14 ft
Pmax
Pmax
Tmax
14 ft
Pmax
(a)
(b)
Figure 3-48 Details for Example 3-18
Solution
The seismic loading condition is shown in Figure 3-48(b). From Example 3-17
Tmax
5 adjusted brace strength in tension
5 wRy Pysc
5 1.35 3 1.0 3 256
5 346 kips
Pmax
5 bwRy Pysc
5 1.15 3 1.35 3 1.0 3 256
5 397 kips
The unbalanced, vertical upward force on the midpoint of the beam is
Qb
5 (Pmax 2 Tmax)sin q
5 (397 2 346)sin 45°
5 36 kips
Allowing for the gravity loads, and applying load combination 6 of ASCE 7 Section 2.3.6, the required
column compressive strength is
Pr
5 23Qb/2 1 2 3 Pmax sin 45° 1 (1.2 1 0.2SDS)D 1 0.5L
5 23 3 36/2 1 2 3 397 3 sin 45° 1 1.4 3 80 1 0.5 3 30
5 634 kips
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3.13 Steel special plate shear walls
In a steel special plate shear wall, an unstiffened steel plate is connected to the surrounding beams
and columns in a building frame and resists earthquake forces by ductile, hysteretic behavior.30, 31 As
shown in Figure 3-49, the steel plates are installed in one or more bays of the building frame over
the full height of the structure. The horizontal boundary elements, consisting of the beams in the
framework, are connected to the vertical boundary elements, consisting of the columns, with moment-​
resisting connections. The building frame is designed to support all gravity loads without assistance
from the steel plates. Lateral loads are resisted by the buckling of the plate, utilizing diagonal tension
field action, while the boundary elements remain elastic, with the exception of plastic hinges that
occur at the ends of the beams. For intermediate horizontal boundary elements, with a web member
of equal thickness above and below, the net force applied to the element by the vertical components of
the tension fields is zero. At the top panel, the top horizontal boundary element must possess sufficient
strength to resist the vertical component of the tension field. Similarly at the bottom panel, the bottom
horizontal boundary element must resist the tension field in the plate, and this may be achieved by
anchoring the element to the foundation.
Figure 3-49 Steel special plate shear wall
The advantages of the steel special plate shear wall are:
•
the system allows wall thicknesses less than for other systems
•
the system can be constructed more quickly than other systems
•
there is a reduction in weight compared with concrete shear walls
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The disadvantages of the steel special plate shear wall are:
•
the system is more flexible than other systems and may require additional stiffening elements
•
to prevent compression stress being introduced into the plates, the plates must be installed after
dead load deformation has occurred in the building frame
Steel special plate shear walls, as specified in ASCE 7 Table 12.2-1, may be utilized in building frame
systems in all seismic design categories using a value of 2 for the overstrength factor and a value of
6 for the deflection amplification factor. A value of 7 is specified for the response modification coefficient. No limitation is imposed on the building height in seismic design categories A, B, and C. The
maximum height permitted in seismic design categories D and E is 160 feet, and in seismic design
category F, it is 100 feet.
Steel special plate shear walls may be utilized in dual systems, with special moment frames, in all
seismic design categories using a value of 2.5 for the overstrength factor and 6.5 for the deflection
amplification factor. A value of 8 is specified for the response modification coefficient. In accordance
with ASCE 7 Table 12.2-1, no limitation is imposed on the building height.
3.13.1 Web requirements
The minimum aspect ratio to ensure ductility is typically
L/h
5 0.6
The maximum aspect ratio is limited by the strength of the horizontal boundary element.
The L/tw ratio usually ranges from a minimum of 300 to a maximum of 800.
The angle of inclination of the tension field to the vertical is given by AISC 341 Equation (F5-2) as
tan4a 5 (1 1 twL/2Ac)/(1 1 tw h/Ab 1 tw h4/360Ic L)
where:
tw
5 thickness of the web
L
5 distance between vertical boundary element centerlines
Ac
5 cross-sectional area of a vertical boundary element
h
5 distance between horizontal boundary element centerlines
Ab
5 cross-sectional area of a horizontal boundary element
Ic
5 moment of inertia of a vertical boundary element
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367
The nominal shear strength of the plate web is given by AISC 341 Equation (F5-1) as
where:
Vn
5 0.42Fy twLcf sin 2a
Fy
5 specified minimum yield stress of the plate
Lcf
5 clear distance between vertical boundary element flanges
The resistance factor is
f
5 0.90
Example 3-19
Figure 3-50 shows one panel of a steel special plate shear wall with moment-resisting beam-column
connections. The plate is 1⁄4 inch thick with a yield stress of Fy 5 36 kips/in2. Determine the nominal
shear strength of the panel.
W12 × 106
1
W12 × 72
/4-in plate
hcf = 10.98 ft
h = 12 ft
Lcf = 10.93 ft
L = 12 ft
Figure 3-50 ​Details for Example 3-19
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Solution
From Figure 3-50:
tw
5 0.25 in
Ac
5 31.2 in2 . . . for a W12 3 106
L
5 12 3 12 5 144 in
h
5 12 3 12 5 144 in
Ic
5 933 in4
Ab
5 21.1 in2 . . . for a W12 3 72
The panel aspect ratio is
L/h
5 12/12
5 1.0
. 0.6 . . . satisfactory
The width/thickness ratio is
L/tw
5 144/0.25
5 576
. 300 . . . satisfactory
, 800 . . . satisfactory
The angle of inclination of the tension field to the vertical is given by AISC 341 Equation (F5-2) as
tan4a 5 (1 1 tw L/2Ac)/(1 1 tw h/Ab 1 tw h4/360Ic L)
5 (1 1 36/62.4)/[1 1 36/21.1 1 746,496/(360 3 933)]
5 0.32
tan a 5 0.752
a
5 36.95°
The nominal shear strength of the plate web is given by AISC 341 Equation (F5-1) as
Vn
5 0.42Fy tw Lcf sin 2a
5 0.42 3 36 3 0.25 3 10.93 3 12 3 sin 73.90°
5 476 kips
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369
3.13.2 ​Strip model methodology
The boundary elements of a panel must resist the forces developed by the tension field action of the
fully yielding web. These forces are determined from a plane frame analysis with the web represented
by a number of pin-ended strips inclined at an angle, a, to the vertical, as shown in Figure 3-51. A
minimum of 10 equally spaced strips are required to give accurate results, and the expected tensile
strength of a strip is given by AISC 341 Commentary Section F5.6c as
where:
Ts exp
5 RyFyAs
Ry
5 ratio of expected yield stress to specified minimum yield stress of the plate
Fy
5 specified minimum yield stress of the plate
As
5 area of a strip
5 tw(Lcf cos a 1 hcf sin a)/n
hcf
5 clear distance between horizontal boundary element flanges
n
5 number of strips per panel
≥ 10
hcf
Lcf
Figure 3-51 ​Strip model
Example 3-20
Figure 3-50 shows one panel of a steel special plate shear wall with moment-resisting beam-column
connections. The plate is 1⁄4 inch thick with a yield stress of Fy 5 36 kips/in2. The web is divided into
10 equally spaced strips aligned in the direction of the tension field. Determine the expected tensile
strength of a strip.
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Solution
The angle of inclination of the tension field to the vertical is obtained in Example 3-19 as
a
5 36.95°
The area of a strip is given by AISC 341 Commentary Section F5.6c as
As
5 tw(Lcf cos a 1 hcf sin a)/n
5 0.25(10.93 3 12cos 36.95° 1 10.98 3 12sin 36.95°)/10
5 4.60 in2
The ratio of expected yield stress to specified minimum yield stress of the plate is
Ry
5 1.3 . . . from Table 3-2 for A36 steel
The specified minimum yield stress of the plate is
Fy
5 36 kips/in2 . . . from Table 3-2 for A36 steel
The expected tensile strength of a strip is given by AISC Commentary Section F5.6c as
Ts exp
5 Ry Fy As
5 1.3 3 36 3 4.60
5 215 kips
References
1. International Code Council. 2018 International Building Code. Washington, DC, 2018.
2. American Institute of Steel Construction. Seismic Provisions for Structural Steel Buildings. AISC
341-16. Chicago, IL, 2016.
3. American Society of Civil Engineers. Minimum Design Loads and Associated Criteria for Buildings and Other Structures: ASCE 7-16. Reston, VA, 2016.
4. American Institute of Steel Construction. Specification for Structural Steel Buildings. AISC 36016. Chicago, IL, 2016.
5. Tremblay, R. “Seismic Behavior and Design of Concentrically Braced Frames.” Engineering
Journal, 38, No. 3. American Institute of Steel Construction. Chicago, IL, 2001.
6. Engelhardt, M. D. Concentrically Braced Frames. AISC. Chicago, IL, 2007.
7. Sabelli, R. “Mechanism Analysis in the 2010 Seismic Provisions.” NASCC Steel Conference Proceedings. AISC. Chicago, IL, 2011.
8. American Institute of Steel Construction. Manual of Steel Construction, Fifteenth Edition. Chicago, IL, 2017.
Seismic and Wind Forces: Structural Design Examples
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371
9. Thornton, W. A. “Designing for Cost Efficient Fabrication.” Modern Steel Construction. 32, No.
2. AISC. Chicago, IL, February 1992.
10. Astaneh-Asl, A. Steel Tips: Seismic Behavior and Design of Gusset Plates. Structural Steel Educational Council. Moraga, CA, 1998.
11. Gross, J. L. “Experimental Study of Gusseted Connections.” Engineering Journal. 27, No. 3.
AISC. Chicago, IL, 1990.
12. Lindsey, S. D. Eccentric Braced Steel Frames for Wind and Low-to-Moderate Seismic Loads.
American Institute of Steel Construction. Chicago, IL, 2003.
13. Becker, R. and Ishler, M. Seismic Design Practice for Eccentrically Braced Frames. Structural
Steel Educational Council. Moraga, CA, 1996.
14. Structural Engineering Association of California. Seismic Design Manual, Volume 3: Building
Design Examples: Steel, Concrete, and Cladding. International Conference of Building Officials.
Whittier, CA, 2000.
15. Williams, A. Structural Analysis: In Theory and Practice. Butterworth-Heinemann. Burlington,
MA/ICC. Washington, DC, 2009.
16. Federal Emergency Management Agency. FEMA 350 Recommended Seismic Design Criteria for
New Steel Moment-Frame Buildings. SAC Joint Venture. Sacramento, CA, June 2000.
17. American Institute of Steel Construction. Prequalified Connections for Special and Intermediate
Steel Moment Frames for Seismic Applications. AISC 358-16. Chicago, IL, 2016.
18. Hamburger, R. O., Krawinkler, H., Malley, J. O., and Adan, S. M. “Seismic Design of Steel Special Moment Frames: A Guide for Practicing Engineers.” NEHRP Seismic Design Technical Brief
No. 2. National Institute of Standards and Technology. Gaithersburg, MD, 2009.
19. International Code Council. Slotted Web Beam-to-Column Steel Moment Frame Connection.
ICC-ES Evaluation Report, ESR-1093. Washington, DC, 2011.
20. SidePlate Systems, Inc. Engineer’s Design Guide. Laguna Hills, CA, 2012.
21. Adan, S. M. and Gibb, W. “Experimental Evaluation of Kaiser Bolted Bracket Steel Moment-​
Resisting Connections.” Engineering Journal, 46, No. 3. American Institute of Steel Construction,
2009.
22. Engelhardt, M. D. and Matthew, M. A. “What’s new with prequalified connections?” Modern
Steel Construction, AISC. Chicago, IL, November 2016.
23. Iwankiw, N. R. “Ultimate strength considerations for seismic design of the reduced beam section
(internal plastic hinge).” Engineering Journal, 34, No. 1. American Institute of Steel Construction. Chicago, IL, 1997.
24. Lopez, W. A. and Sabelli, R. “Seismic Design of Buckling-Restrained Braced Frames.” NASCC
Steel Conference Proceedings. AISC. Chicago, IL, 2008.
25. Structural Engineering Association of California. “Buckling-Restrained Braced Frames.” SEAOC
Blue Book: Seismic Design Recommendations. SEAOC. Sacramento, CA, 2009.
26. Robinson, K. and Black, C. “Specifying Buckling-Restrained Braced Frames: How to Get What
You Want.” NASCC Steel Conference Proceedings. AISC. Chicago, IL, 2011.
Seismic and Wind Forces: Structural Design Examples
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27. CoreBrace. Product Information. West Jordan, UT, 2006.
28. Star Seismic. Product Information. Park City, UT, 2006.
29. Dasse Design Inc. Cost Advantages of Buckling-Restrained Braced Frame Buildings. Dasse. San
Francisco, CA, 2009.
30. Seillie, I. F. and Hooper, J. D. “Steel Plate Shear Walls: Practical Design and Construction.” Modern Steel Construction, 45, No. 4. AISC. Chicago, IL, April 2005.
31. Sabelli, R. Steel Plate Shear Walls. Design Guide No. 20. AISC. Chicago, IL, 2006.
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4
Seismic Design of Concrete Structures
Nomenclature
a
depth of equivalent rectangular stress block
in
Abrg
net bearing area of the head of stud, anchor bolt, or headed deformed bar
in2
Ach
cross-sectional area of a member measured to the outside edges of transverse
reinforcement
in2
Acv
gross area of concrete section bounded by web thickness and length of section in the
direction of shear force considered in the case of walls
in2
Ag
gross area of concrete section
in2
Aj
effective cross-sectional area within a joint in a plane parallel to plane of beam
reinforcement
in2
ANc
projected concrete failure area of a single anchor or group of anchors, for calculation
of strength in tension
in2
ANco
projected concrete failure area of a single anchor, for calculation of strength in
tension if not limited by edge distance or spacing
in2
As
area of nonprestressed longitudinal tension reinforcement
in2
Ase,N
effective cross-sectional area of anchor in tension
in2
Ase,V
effective cross-sectional area of anchor in shear
in2
Ase,w
effective cross-sectional area of longitudinal reinforcement in tension
in2
Ast
total area of nonprestressed longitudinal reinforcement
in2
Atr
total cross-sectional area of all transverse reinforcement that crosses the potential
plane of splitting
in2
Av
area of shear reinforcement within spacing s
in2
b
width of compression face of member
in
bc
cross-sectional dimension of member core measured to the outside edges of the
transverse reinforcement
in
bf
effective flange width of T section
in
bw
web width
in
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c
distance from extreme compression fiber to neutral axis
in
ca1
distance from the center of an anchor shaft to the edge of concrete in one direction
in
ca1
minimum edge distance
in
ca2
distance from center of an anchor shaft to the edge of concrete in the direction
perpendicular to ca1
in
cb
lesser of: (a) the distance from center of a bar to nearest concrete surface and
(b) one-half the center-to-center spacing of bars being developed
in
cc
clear cover of reinforcement
in
c1
dimension of column measured in the direction of the span for which moments are
being determined
in
c2
dimension of column measured in the direction perpendicular to c1
in
Cd
deflection amplification factor
–
d
effective depth of section
in
da
outside diameter of anchor or shaft diameter of headed stud, headed bolt, or hooked
bolt
in
db
nominal diameter of bar
in
D
effect of service dead load
–
eh
distance from the inner surface of the shaft of a J- or L-bolt to the outer tip of the Jor L-bolt
in
E
effect of horizontal and vertical earthquake-induced forces
–
Ec
modulus of elasticity of concrete
psi
Es
modulus of elasticity of reinforcement steel
29,000 ksi
fc9
specified compressive strength of concrete
psi
fr
modulus of rupture
psi
fs
tensile stress in reinforcement at service loads
psi
futa
specified tensile strength of anchor steel
psi
fy
specified yield strength for nonprestressed reinforcement
psi
fya
specified yield strength of anchor steel
psi
fyt
specified yield strength of transverse reinforcement
psi
h
overall thickness, height, or depth of member
in
hef
effective embedment depth of anchor
in
hu
laterally unsupported height at extreme compression fiber of wall
in
hw
height of entire wall from base to top
in
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Chapter 4
hx
maximum center-to-center spacing of longitudinal bars laterally supported
by corners of crossties or hoop legs around the perimeter of the column
in
Hn
column clear height
in
I
moment of inertia of section about centroidal axis
in4
Icr
moment of inertia of cracked section
in4
Ig
moment of inertia of gross concrete section
in4
kcp
coefficient for pryout strength
–
Ktr
transverse reinforcement index
in
l
span length of beam
in
lc
vertical distance between supports
in
lc
length of compression member, measured center-to-center of the joints
in
ld
development length in tension of deformed bar
in
ldc
development length in compression of deformed bar
in
ldh
development length in tension of hooked bar
in
lo
length over which special transverse reinforcement must be provided
in
lsc
compression lap splice length
in
lst
tension lap splice length
in
lw
length of entire wall in direction of shear force
in
L
effect of service live load
–
Ln
beam clear span
in
Lr
effect of service roof live load
–
Ma
maximum moment in member due to service loads including P-delta effects
lb-in
Mcr
cracking moment
lb-in
Mn
nominal flexural strength at section
lb-in
Mnb
nominal flexural strength of beam including slab where in tension
lb-in
Mnc
nominal flexural strength of column framing into joint, calculated for factored axial
force
lb-in
Mpr
probable flexural strength of member assuming a tensile stress in the longitudinal
bars of 1.25fy and a strength reduction factor f of 1.0
lb-in
Msa
maximum moment in wall due to service loads, excluding P-delta effects
lb-in
Mu
factored moment at section
lb-in
n
number of items
–
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n
Es /Ec
–
Nb
basic concrete breakout strength in tension of a single anchor in cracked concrete
lb
Ncb
nominal concrete breakout strength in tension of a single anchor
lb
Ncbg
nominal concrete breakout strength in tension of a group of anchors
lb
Ncp
basic concrete pryout strength of a single anchor
lb
Ncpg
basic concrete pryout strength of a group of anchors
lb
Nn
nominal strength in tension
lb
Np
pullout strength in tension of a single anchor in cracked concrete
lb
Npn
nominal pullout strength in tension of a single anchor
lb
Nsa
nominal strength of a single anchor or individual anchor in a group of anchors in
tension as governed by the steel strength
lb
Nsb
side-face blowout strength of a single anchor
lb
Ps
unfactored axial load at midheight section including effects of self-weight
lb
Pu
factored axial force
lb
PD
secondary moment due to lateral deflection
lb-in
QE
effect of horizontal seismic forces
–
R
effect of rain load
–
R
response modification factor
–
s
center-to-center spacing of hoops or stirrups
in
so
hoop spacing calculated by ACI 318 Equation (18.7.5.3)
in
so
center-to-center spacing of transverse reinforcement within the length lo
in
S
effect of snow load
–
SDS
response acceleration for a period of 0.2 second
–
U
strength of a member to resist factored loads in such combinations as stipulated in
the code
–
V
shear force
lb
Vb
basic concrete breakout strength in shear of a single anchor in cracked concrete
lb
Vc
nominal shear strength provided by concrete
lb
Vcb
nominal concrete breakout strength in shear of a single anchor
lb
Vcbg
nominal concrete breakout strength in shear of a group of anchors
lb
Vcp
nominal concrete pryout strength of a single anchor
lb
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Vcpg
nominal concrete pryout strength of a group of anchors
lb
Ve
shear force due to probable beam moments
lb
Vs
nominal shear strength of shear reinforcement
lb
w
distributed load
kips/ft
yt
distance from the centroidal axis of the gross section to the extreme fiber in tension
in
Symbols
b1
factor relating depth of equivalent rectangular compressive stress block to depth of
neutral axis
–
Dcr
calculated out-of-plane deflection at midheight of wall corresponding to cracking
moment Mcr
in
Dn
calculated out-of-plane deflection at midheight of wall corresponding to nominal
flexural strength Mn
in
Ds
out-of-plane deflection due to service loads
in
Du
calculated out-of-plane deflection at midheight of wall due to factored loads
in
ec
assumed maximum compressive strain in concrete
–
λ
modification factor for lightweight concrete
–
λa
modification factor for lightweight concrete used in concrete anchorage applications
–
r
ratio of As to bd
–
r
redundancy factor
–
rl
ratio of area of distributed longitudinal reinforcement to gross concrete area
perpendicular to that reinforcement
rt
ratio of area of distributed transverse reinforcement to gross concrete area
perpendicular to that reinforcement
–
f
strength reduction factor
–
yc,N
factor used to modify tensile strength of anchors based on presence or absence of
cracks in concrete
–
yc,P
factor used to modify pullout strength of anchors based on presence or absence of
cracks in concrete
–
yc,V
factor used to modify shear strength of anchors based on presence or absence of
cracks in concrete and presence or absence of supplementary reinforcement
–
ye
factor used to modify development length based on reinforcement coating
–
yec,N
factor used to modify tensile strength of anchors based on eccentricity of applied
loads
–
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yec,V
factor used to modify shear strength of anchors based on eccentricity of applied loads –
yed,N
factor used to modify tensile strength of anchors based on proximity to edges of
concrete member
–
yed,V
factor used to modify shear strength of anchors based on proximity to edges of
concrete member
–
ys
factor used to modify development length based on reinforcement size
–
yt
factor used to modify development length for casting location in tension
–
W0
amplification factor to account for overstrength of the structure in the inelastic range
–
4.1 Special moment frames
Special moment frames are the only moment frames permitted in seismic design categories D, E, and
F. No limitation is placed on building height and the following parameters are specified in ASCE 71
Table 12.2-1 as
R
5 response modification factor
58
Cd
5 deflection amplification factor
5 5.5
W0
5 structure overstrength factor
5 amplification factor to account for the overstrength of the structure in the
inelastic range
53
Special moment frames may also be utilized in dual building systems with special reinforced concrete
shear walls. No limitation is placed on building height and the following parameters are specified in
ASCE 7 Table 12.2-1 as
R
57
Cd
5 5.5
W0
5 2.5
Special moment frames are detailed to ensure that absorption of seismic forces can occur at large
inelastic displacements without impairment of the structural integrity. The design principles are formulated on an expected strength basis using the probable strength of the materials. Members are
designed to resist the design seismic forces and gravity loads and, in addition, are required to resist
forces generated by the probable flexural strength of a member after strain hardening effects occur in
the reinforcement.
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4.1.1 Design loads
The 2018 IBC2 adopts by reference the American Concrete Institute’s Building Code and Commentary3 with some exceptions that are given in IBC Sections 1901.2, 1903.1, and 1905. For earthquake
loads, the load combinations given by ACI Equations (5.3.1a) and (5.3.1g) must be utilized to obtain
the required strength U, and these are
U
5 1.2D 1 1.0E 1 1.0L 1 0.2S . . . ACI Equation (5.3.1e)
and
U
5 0.9D 1 1.0E . . . ACI Equation (5.3.1g)
where:
D
5 dead load
L
5 floor live load
S
5 snow load
E
5 strength level seismic load
In Equation (5.3.1e), replace 1.0L with 0.5L except for floors in garages and places of public assembly
and for floor loads in excess of 100 psf.
The seismic load is a function of both horizontal and vertical earthquake-induced forces, and when
the effects of gravity and seismic loads are additive, is given by ASCE 7 Equations (12.4-1), (12.4-3),
and (12.4-4a) as
where:
E
5 rQE 1 0.2SDSD
QE
5 effect of horizontal seismic forces
SDS
5 5-percent damped, design spectral response acceleration for a period of
0.2 second
D
5 effect of dead load
r
5 redundancy factor
The required load combination may now be defined by ASCE 7 Section 2.3.6 load combination 6 as
U
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
In Equation (5.3.1e), replace 1.0L with 0.5L except for floors in garages and places of public assembly
and for floor loads in excess of 100 psf.
Where the effects of gravity and seismic loads counteract, the seismic load is given by ASCE Equations (12.4-2), (12.4-3), and (12.4-4a) as
E
5 rQE 2 0.2SDS D
The required load combination may now be defined by ASCE 7 Section 2.3.6 load combination 7 as
U
5 (0.9 2 0.2SDS)D 1 rQE
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To determine the required strength to resist the effects of gravity loads due to dead load, floor live load,
and roof live load only, the load combination given by ACI Equation (5.3.1b) is applicable, and this is
where:
U
5 1.2D 1 1.6L 1 0.5(Lr or S or R)
Lr
5 roof live load
R
5 rain load
To determine the design strength of a member, the appropriate strength reduction factor, f, is applied
to the nominal strength of the member. The values of the reduction factor for reinforced concrete specified in ACI Sections 2.2, 21.2.1, and 21.2.2 are
f
5 0.9 for flexure of tension-controlled sections
5 1.0 for probable flexural strength in special moment frames
5 0.75 for shear and torsion
5 0.6 for shear in walls and special moment frame members with a nominal
shear strength less than the shear corresponding to their nominal flexural
strengths
5 0.75 for compression-controlled members with spiral reinforcement
5 0.65 for compression-controlled members with lateral ties
5 0.65 for bearing
5 0.85 for shear in joints of special moment frames
The nominal strength of a member is determined in accordance with the principles defined in ACI Section 21.2.2.3. These principles are employed in several design aids.4, 5, 6 The nominal flexural capacity
of a tension-controlled member, with only tensile reinforcement, may also be determined from the
expression6
where:
Mn
5 As fy d(1 2 0.59rfy /fc9)
As
5 area of tensile reinforcement
fy
5 yield strength of the reinforcement
d
5 effective depth of section
r
5 reinforcement ratio
5 As /bd
fc9
5 compressive strength of the concrete
b
5 width of compression face of section
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4.1.2 Beam details
The reinforcement detailing provisions of ACI Chapter 18 are intended to produce a ductile structure
capable of withstanding the large inelastic deformations that occur during a severe earthquake.
Flexural members are defined in ACI Section 18.6.2.1 as elements having a clear span not less than
four times the effective depth.
To provide a compact cross section with good stability during nonlinear displacements, geometrical
constraints are imposed in ACI Section 18.6.2.1. As shown in Figure 4-1, these are
bw/h
≥ 0.3
bw
≥ 10 inches
≤ c2 1 0.75c1 on each side of a column
≤ c2 1 c2 on each side of a column
where:
bw
5 web width
h
5 beam depth
c1
5 column width in direction of span
c2
5 column width perpendicular to c1
c2
h
c2
c2
c1
Figure 4-1 Beam details
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ACI Sections 9.6.1.2 and 18.6.3 stipulate limitations on the amount of longitudinal reinforcement to
prevent steel congestion, ensure nonbrittle ductile behavior, and provide a minimum reinforcement
capacity greater than the tensile strength of the concrete. As shown in Figure 4-1, these limitations are
rmin
≥ 3( fc9)0.5/fy
≥ 200/fy
rmax
≤ 0.025
In addition, to allow for the possibility of moment reversals:
•
at least two continuous reinforcing bars shall be provided at the top and bottom of the beam
•
at the ends of the member, positive moment strength is required at least equal to one-half of the
negative moment strength
•
at any section along the beam, neither the positive nor the negative moment strength shall be
less than one-fourth of the moment strength at the ends of the beam
Reinforcement splices are not permitted in regions of plastic hinging as splices are unreliable under
inelastic cyclic loading conditions. Hence, ACI Section 18.6.3.3 specifies that splices shall not be used:
•
within a beam-to-column joint
•
within a distance of twice the beam depth from the face of the joint
•
within a distance of twice the beam depth from locations of flexural yielding
To prevent the spalling of concrete cover at splice locations, hoop reinforcement shall be provided over
the lap length with a maximum spacing of d/4 or 4 inches.
To account for the reinforcement stress exceeding the yield stress for bars of sizes 3 through 11, ACI
Section 18.8.5.1 specifies that the development length for a hooked bar in normalweight concrete shall
not be less than the larger of
ldh
5 fy db /65l( fc9)0.5
or
5 8db
or
5 6 inches
where:
db
5 bar diameter
fy
5 specified yield strength of reinforcement
l
5 0.75 for lightweight concrete
l
5 1.0 for normalweight concrete
The hook shall be located within the confined core of a column or boundary element. For straight bars
of sizes 3 through 11 embedded in confined concrete, the development length is given as
ld
5 2.5ldh
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and where the depth of concrete cast in one lift beneath the bar exceeds 12 inches
ld
5 3.25ldh
For straight bars not entirely embedded in confined concrete, the development length is given by
where:
ldm
5 1.6ld 2 0.6ldc
ldc
5 length of bar in confined concrete
Transverse reinforcement is required to provide shear resistance, to provide confinement to the concrete at locations of plastic hinging, and to control lateral buckling of longitudinal bars after the concrete cover has spalled. Closed hoops, as shown in Figure 4-2, are required to provide confinement and
may also provide shear resistance. Seismic stirrups or links with 135-degree seismic hooks provide
only shear resistance. Either single-piece or two-piece closed hoops may be provided. The two-piece
hoop consists of a seismic stirrup and a seismic crosstie with one 135-degree seismic hook and one
90-degree hook. Adjacent crossties must have the seismic hooks on opposite sides of the member
unless confined by a slab on only one side. For this situation, the 90-degree hook is placed on the side
with the slab. Hoops are required in accordance with ACI Section 18.6.4.1:
•
over a distance of 2h from face of a column
•
over a distance of 2h on both sides of a section subjected to plastic hinging
The first hoop must be located not more than 2 inches from the face of the column. The hoop spacing
shall not exceed the lesser of
smax
5 d/4
or
5 6db
or
5 6 inches
where:
d
5 beam effective depth
db
5 diameter of smallest longitudinal bar
For diameter, see
ACI Table 25.3.2
Figure 4-2 Seismic hoops and stirrups
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Where hoops are not required, stirrups with seismic hooks at both ends must be provided throughout
the length of the member at a maximum spacing of d/2. Details of hoop and stirrup requirements are
shown in Figure 4-3.
6
6
Figure 4-3 Hoop and stirrup locations
4.1.3 Beam design
To ensure ductile flexural failure of a beam and prevent brittle shear failure, ACI Section 21.5.4 requires
the design shear force to be determined from the probable flexural strength that can be developed at
the ends of the beam plus the factored tributary gravity loads. The probable flexural strength is calculated7, 8, 9 by assuming that strain hardening increases the effective tensile strength of the reinforcement
by 25 percent and by using a strength reduction factor f of 1.0, as specified in ACI Section 2.2. The
probable flexural strength is given by
Mpr
5 As(1.25fy)d[1 2 0.59r(1.25fy)/fc9]
5 As fy d(1.25 2 0.92rfy /fc9)
As shown in Figure 4-4, moments of opposite sign act at the ends of a beam bent in double curvature
and the sense of the moments reverses as the seismic loading reverses. The sign convention adopted
in the figure is that bending moments at the ends of a member are shown acting from the joint to the
member; in other words, the support reactions are considered. The arrowheads point toward the face
of the member that is in tension.
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Figure 4-4 Beam shear due to probable flexural strength
Both the positive and negative probable flexural strengths must be calculated at both ends of the beam
in order to determine the critical shear value. The design shear force at the left end of the beam for
seismic load acting to the left is
where:
Ve
5 (Mpr1 1 Mpr2)/Ln 1 VgL
Ln
5 beam clear span
VgL
5 shear at the left end of the beam due to the factored tributary gravity loads
The design shear force at the right end of the beam for seismic load acting to the right is
where:
Ve
5 (Mpr3 1 Mpr4)/Ln 1 VgR
VgR
5 shear at the right end of the beam due to the factored tributary gravity loads
The design shear capacity of the beam is given by ACI Equation (22.5.1.1) as
where:
fVn
5 fVc 1 fVs
fVc
5 design shear capacity of the concrete from ACI Equation (22.5.5.1)
5 2fbw d l( fc9)0.5
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fVs
5 design shear capacity of the shear reinforcement from ACI Equation
(22.5.10.5.3)
5 fAv fy d/s
l
5 modification factor for lightweight concrete
Av
5 area of shear reinforcement
s
5 spacing of shear reinforcement
In accordance with ACI Section 18.6.5.2, the shear resistance of the concrete shall not be included in
the shear capacity of the beam when both of the following conditions occur:
•
the seismic induced shear force represents one-half or more of the total applied shear
•
the factored axial compressive force is less than Ag fc9/20
Example 4-1
The beam of an interior bay of a special moment-resisting frame is shown in Figure 4-5. The building
is an office building and floor live load is less than 100 psf. The structure has a redundancy factor of
r 5 1.0 and the 5-percent damped, design spectral response acceleration for a period of 0.2 second is
SDS 5 0.826g. The service level gravity loads and bending moments are indicated in the figure together
with the proposed beam and column sections and the moments due to the design level seismic forces.
The bending moments are shown acting at the face of the joint. Bending moments and axial forces due
to roof live load are negligible. Reinforcement consists of Grade 60 bars, the normalweight concrete
cylinder strength is 4000 psi, and 1.5-inch clear cover is provided to stirrups and hoops. Determine the
required reinforcement details.
Solution
The longitudinal reinforcement required to resist the factored loads is determined first.
Load combinations
For dead load and live load, the applicable load combination is given by ACI Equation (5.3.1b), which is
U
5 1.2D 1 1.6L 1 0.5(Lr or S or R)
The factored beam support moment for dead and live load is then
Mu
5 1.2 3 66 1 1.6 3 37
5 138 kip-ft . . . tension on the top face of the beam
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Figure 4-5 Details for Example 4-1
Where the effects of dead load and seismic load are additive, the applicable loading case is given by
ASCE 7 Section 2.3.6 as
U
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
The corresponding factored beam support moment is
Mu
5 (1.2 1 0.2 3 0.826)66 1 1.0 3 200 1 0.5 3 37 1 0 . . . floor live load
, 100 psf
5 309 kip-ft . . . tension on the top face of the beam
Where the effects of dead load and seismic load counteract, the applicable loading case is given by
ASCE 7 Section 2.3.6 as
U
5 (0.9 2 0.2SDS)D 1 rQE
The corresponding factored beam support moment is
Mu
5 (0.9 2 0.2 3 0.826)66 2 1.0 3 200
5 2152 kip-ft . . . tension on the bottom face of the beam
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The factored beam span moment for dead and live load is
Mu
5 1.2 3 45 1 1.6 3 26
5 96 kip-ft . . . tension on the bottom face of the beam
The required design moments, in accordance with ACI Section 18.6.3, are:
•
negative moment at beam support
Mu
•
5 309 kip-ft
positive moment at beam support
Mu
5 152 kip-ft
≥ 309/2
5 155 kip-ft . . . governs
•
positive moment in beam span
Mu
5 96 kip-ft . . . governs
≥ 309/4
5 77 kip-ft
•
negative moment in beam span
Mu
5 309/4
5 77 kip-ft
Longitudinal reinforcement
For the top reinforcement in the beam, at the face of the column, provide two #8 and three #7 bars to
give a reinforcement area of
and
As
5 3.38 in2
r
5 As /bd
5 3.38/(21 3 21.5)
5 0.0075
, 0.025 . . . satisfies ACI Section 18.6.3
3( fc9)0.5/fy 5 3(4000)0.5/60,000
5 0.0032
, r . . . satisfies ACI Section 9.6.1.2
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200/fy 5 200/60,000
5 0.0033
, r . . . satisfies ACI Section 9.6.1.2
The limiting reinforcement ratio for a tension-controlled section is
rt
5 0.319b1 fc9/fy . . . from ACI Section 21.2.2
5 0.319 3 0.85 3 4/60
5 0.0181
. r . . . section is tension controlled
Hence, the design flexural strength provided is given by
Mu
5 fAs fy d(1 2 0.59rfy /fc9)
5 0.9 3 3.38 3 60 3 21.5(1 2 0.59 3 0.0075 3 60/4)/12
5 305 kip-ft
 309 kip-ft . . . satisfactory
For the bottom reinforcement in the beam, at the face of the column, provide three #7 bars to give a
reinforcement area of
and
As
5 1.80 in2
r
5 As /bd
5 1.80/(21 3 21.5) . . . neglecting the flange concrete
5 0.0040
The design flexural strength provided is given by
Mu
5 fAs fy d(1 2 0.59rfy /fc9)
5 0.9 3 1.80 3 60 3 21.5(1 2 0.59 3 0.0040 3 60/4)/12
5 168 kip-ft
. 155 kip-ft . . . satisfactory
For the bottom reinforcement in the beam span, provide three #7 bars to give a reinforcement area of
and
As
5 1.80 in2
r
5 0.0040
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The design flexural strength provided is given by
Mu
5 fAs fy d(1 2 0.59rfy /fc9)
5 168 kip-ft
. 96 kip-ft . . . satisfactory
For the top reinforcement in the beam span, provide two #7 bars to give a reinforcement area of
and
As
5 1.20 in2
r
5 0.0027
The design flexural strength provided is given by
Mu
5 fAs fy d(1 2 0.59rfy /fc9)
5 0.9 3 1.20 3 60 3 21.5(1 2 0.59 3 0.0027 3 60/4)/12
5 113 kip-ft
. 77 kip-ft . . . satisfactory
1.33 3 77 5 102 kip-ft
, 113 kip-ft
Hence, the reinforcement ratio of r 5 0.0027, which is less than 0.0033, conforms to ACI Section
9.6.1.3 and is satisfactory.
Crack control
To limit cracking, ACI Section 24.3.2 limits the center-to-center spacing of the tension reinforcement
to a maximum of
where:
s
5 600/fs 2 2.5cc . . . with fs in ksi
fs
5 (2/3)fy
5 (2/3) 3 60
5 40 ksi
cc
5 clear cover to the tension reinforcement
5 1.5 1 0.38 . . . using #3 seismic stirrups
5 1.88 in
or
s
5 12 3 40/fs . . . with fs in ksi
5 12 in
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For the bottom reinforcement in the beam span, the actual spacing of the three #7 bars is
sa
5 (bw 2 2cc 2 db)/2
5 (21.0 2 2 3 1.88 2 0.88)/2
5 8.2 in
, s . . . satisfactory
Transverse reinforcement
Figure 4-6 shows the reinforcement areas that are effective for seismic load acting to the left. The
probable flexural strength at the left end of the beam is given by
Mpr1
5 As fy d(1.25 2 0.92rfy /fc9)
5 3.38 3 60 3 21.5(1.25 2 0.92 3 0.0075 3 60/4)/12
5 417 kip-ft
Bending moment, kip-ft
Figure 4-6 Reinforcement areas and probable moments
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The probable flexural strength at the right end of the beam is given by
Mpr2
5 As fy d(1.25 2 0.92rfy /fc9)
5 1.80 3 60 3 21.5(1.25 2 0.92 3 0.0040 3 60/4)/12
5 231 kip-ft
The factored gravity load on the beam is
wu
5 (1.2 1 0.2SDS)D 1 0.5L
5 (1.2 1 0.2 3 0.826)1.5 1 0.5 3 0.85
5 2.47 kips/ft
The corresponding shear force at the left end of the beam is
VgL
5 wu Ln /2
5 2.47 3 22/2
5 27 kips
The design shear force at the left end of the beam for seismic load acting to the left is
Ve
5 (Mpr1 1 Mpr2)/Ln 1 VgL
5 (417 1 231)/22 1 27
5 30 1 27
5 57 kips
0.5
8f( fc9) bw d 5 8 3 0.75 3 21 3 21.5(4000)0.5/1000
5 171 kips
. Ve . . . ACI Section 22.5.1.2 is satisfied
The seismic-induced shear force represents more than half of the total shear and the axial compressive
force is less than Ag fc9/20. Hence, in accordance with ACI Section 18.6.5.2, the shear capacity of the
concrete may not be included in the shear capacity of the beam. In addition, the nominal shear strength
of the beam is not less than the shear corresponding to the development of the nominal flexural strength
of the beam, with f 5 0.75 for shear in accordance with ACI Section 21.2.2. The maximum spacing of
hoop reinforcement is given by ACI Section 18.6.4.7 as the lesser of
smax
5 6db
5 6 3 0.88
5 5.3 in . . . governs
or
5 6 in
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393
5 d/4
5 21.5/4
5 5.4 in
The design capacity of shear reinforcement is given by ACI Equation (22.5.10.5.3) as
fVs
5 fAv fy d/s
Providing #4 hoops at 4-inch spacing gives a design shear capacity of
fVs
5 0.75 3 0.40 3 60 3 21.5/4
5 97 kips
. Ve . . . satisfactory
Provide hoops at 4-inch spacing over a length of 2h 5 4 feet from the face of each column. At 4 feet
from the face of the column, the shear force acting on the beam is given by
Vu
5 Ve 2 4wu
5 57 2 4 3 2.47
5 47 kips
4f( fc9)0.5bw d 5 86 kips
. Vu . . . ACI Section 9.7.6.2.2 allows the lesser of s 5 d/2 or 24 in
Using #3 seismic stirrups, the required spacing is
s
5 fAv fy d/Vu
5 0.75 3 0.22 3 60 3 21.5/47
5 4.5 in
To comply with bar curtailment requirements, provide #3 seismic stirrups at 31⁄2-inch spacing over the
remainder of the span with the exception of the location of splices. This gives a design capacity of
fVs
5 fAv fy d/s
5 0.75 3 0.22 3 60 3 21.5/3.5
5 61 kips
. Vu . . . satisfactory
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Curtailment of longitudinal reinforcement
From the top reinforcement at the face of the column, two #8 bars and one #7 bar will be curtailed. The
remaining two #7 bars provide a design flexural strength of
Mu
5 113 kip-ft
The applied moment equals 113 kip-ft at a distance, x, from the face of the column given by
2113 5 Vex 2 Mpr1 2 wu x2/2
2113 5 57x 2 417 2 2.47x2/2
x
5 6.2 ft
The physical cut-off point of the #8 bars is located an additional distance beyond this point, as specified by ACI Section 9.7.3.3, given by the larger of
12db
5 12 3 1.0
5 12 in
or
d
5 21.5 in . . . governs
Hence, the physical cut-off point is a distance from the face of the column given by
Lc
5x1d
5 6.2 1 1.8
5 8 ft
In addition, the physical cut-off point may not be less than a development length from the face of the
column, and since the depth of concrete beneath the bars exceeds 12 inches, this is given by ACI Sections 18.8.5.3 and 18.8.5.4 as
where:
ldm
5 1.6ld 2 0.6ldc
ldc
5 length of bar in confined concrete
5 length over which hoops are provided
5 4 ft
ld
5 3.25ldh
5 3.25 3 fy db/65l( fc9)0.5
5 3.25 3 14.6/12
5 3.95 ft
and
ldm
5 1.6 3 3.95 2 0.6 3 4
5 3.9 ft
, Lc
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Hence, the cut-off point for the two #8 and one #7 top bars is 8 feet from the face of the column.
The point of inflection is located a distance, y, from the face of the column given by
0
5 Ve y 2 Mpr1 2 wu y2/2
5 57y 2 417 2 2.47y2/2
y
5 9.1 ft
. Lc
Hence, the bars are terminated in a tension zone and must satisfy either ACI Section 9.7.3.5(a),
9.7.3.5(b), or 9.7.3.5(c). At 8 feet from the face of the column, the design shear strength provided is
fVn
5 61 kips
At 8 feet from the face of the column, the shear force acting on the beam is given by
Vu
5 Ve 2 8wu
5 57 2 8 3 2.47
5 37 kips
, 2fVn /3 . . . (5 41 kips)
Hence, ACI Section 9.7.3.5(a) is satisfied and the bars may be curtailed at a distance of 8 feet from the
face of the column.
Splicing of longitudinal reinforcement
The three #7 bars in the bottom of the beam will be spliced within the span. Hoop reinforcement consisting of #3 bars at the required spacing of 4 inches will be provided over the length of the splice. The
development length of the bars may be determined using ACI Section 25.4.2.3. From ACI Equation
(25.4.2.3a), the development length is given by
where:
ld /db
5 0.075fyytyeys /[l( fc9)0.5(cb 1 Ktr)/db]
cb
5 minimum of one-half the center-to-center spacing of the bars being
developed or center of bar to nearest concrete surface
5 s/2
5 8.2/2
5 4.1 in
or
5 1.5 1 0.38 1 0.88/2
5 2.32 in . . . governs
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The transverse reinforcement index is
where:
Ktr
5 40Atr /sn
Atr
5 cross-sectional area of transverse reinforcement crossing the plane of
splitting
5 2 3 0.11 in2
s
5 spacing of the transverse reinforcement
5 4 in
n
5 number of bars being developed along the potential plane of splitting
53
Hence,
Ktr
5 40 3 2 3 0.11/(4 3 3)
5 0.73 in
and
(cb 1 Ktr)/db 5 (2.32 1 0.73)/0.88
5 3.5
use 2.5 . . . max
yt
5 reinforcement location factor
5 1.0 . . . for bottom bars
ye
5 reinforcement coating factor
5 1.0 . . . for uncoated reinforcement
ys
5 reinforcement size factor
5 1.0 . . . for #7 bars
l
5 lightweight aggregate factor
5 1.0 . . . for normalweight concrete
and
ld /db
5 0.075 3 60,000 3 1.0 3 1.0 3 1.0/[1.0 3 2.5(4000)0.5]
5 28.5
ld
5 28.5 3 0.88
5 25 in
All bars are spliced at midspan and a class B splice is required. The class B splice length is
ldB
5 1.3ld
5 1.3 3 25
5 33 in
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The beam reinforcement details are shown in Figure 4-7.
31/2
31/2
Figure 4-7 Beam reinforcement details
4.1.4 Column details
Geometrical constraints are imposed on columns based on established design practice and these are
given in ACI Section 18.7.2.1 as
hmin
hmin/hperp
where:
≥ 12 in
≥ 0.4
hmin
5 minimum cross-sectional dimension
hperp
5 dimension perpendicular to minimum dimension
Longitudinal reinforcement limits are imposed by ACI Section 18.7.4.1 in order to control creep,
reduce steel congestion, and provide a flexural capacity in excess of the cracking moment.
These limitations are
rg
≥ 0.01
≤ 0.06
where:
rg
5 ratio of reinforcement area to cross-sectional area
Spalling of the concrete cover typically occurs at the ends of columns, which makes these areas undesirable for the location of lap splices. Lap splices, proportioned as tension lap splices, are restricted
by ACI Section 18.7.4.3 to the center half of the column where moment reversals are less likely. Lap
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splices shall be enclosed with confinement reinforcement, conforming to ACI Sections 18.7.5.2 and
18.7.5.3, over the full length of the splice. In accordance with ACI Section 18.2.7.2, Type 1 mechanical
splices, which develop 125 percent of the specified yield strength of the bar, may not be used within a
distance equal to twice the column depth from the joint face. Type 2 mechanical splices, which develop
the specified tensile strength of the bar, may be used at any location. In accordance with ACI Section
18.2.8.1, welded splices may not be used within a distance equal to twice the column depth from the
joint face.
Transverse reinforcement, consisting of closed hoops and crossties, shall be provided throughout
the height of the column to furnish shear resistance and confinement. As specified in ACI Section
18.7.5.2(e) and shown in Figure 4-8, longitudinal bars supported by the corner of a crosstie or hoop
leg must be spaced a maximum distance of 14 inches on center.
Figure 4-8 Column transverse reinforcement
At the ends of the column, over the length, lo , specified by ACI Section 18.7.5.1, the area of the rectilinear hoop reinforcement required is given by the greater value obtained from ACI Table 18.7.5.4
Equations (a) and (b), which are
Ash
5 0.3sbc(Ag/Ach 2 1)fc9/fyt . . . for Pu ≤ 0.3Ag fc9 and fc9 ≤10,000 psi
and
Ash
5 0.09sbc fc9/fyt . . . for Pu ≤ 0.3Ag fc9 and fc9 ≤10,000 psi
where:
s
5 spacing of hoop reinforcement
Ag
5 gross area of column section
Ach
5 cross-sectional area measured out-to-out of hoop reinforcement
bc
5 dimension of core measured out-to-out of hoop reinforcement
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In accordance with ACI Section 18.7.5.1, confinement reinforcement is required over a distance of lo
from each joint face given by the maximum of
lo
5h
or
lo
5 Hn /6
or
lo
5 18 inches
where:
h
5 depth of column
Hn
5 column clear height
The spacing of the confinement reinforcement is limited by ACI Section 18.7.5.3 to the smaller value
given by
so
5 hmin /4
or
so
5 6db
or
so
5 6 in
or
so
5 4 1 (14 2 hx)/3 . . . ACI Equation (18.7.5.3)
where:
hmin
5 minimum column dimension
hx
5 maximum center-to-center spacing of longitudinal bars, supported by the
corner of a crosstie or hoop leg, on all faces of the column
5 14 inches maximum
db
5 diameter of the smallest longitudinal bar
The spacing need not be taken less than 4 inches.
Where confinement reinforcement is not required, the hoop spacing, in accordance with ACI Section
18.7.5.5, shall not exceed the smaller value given by
smax
5 6db
or
5 6 in
where:
5 diameters of smallest longitudinal bar
db
Details of column reinforcement are shown in Figure 4-9.
If the thickness of the concrete cover outside the hoops exceeds 4 inches, additional transverse reinforcement shall be provided, as required by ACI Section 18.7.5.7, at a maximum spacing of 12 inches.
Concrete cover on the additional transverse reinforcement shall not exceed 4 inches.
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,
min
Figure 4-9 Column reinforcement details
Columns supporting discontinued walls are required by ACI Section 18.7.5.6 to be supplied with
confinement reinforcement over their full height when the axial force due to seismic effects is Pu .
Ag fc9/10. The confinement reinforcement shall extend into the wall for the development length of the
largest longitudinal bar.
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4.1.5 Column design
The formation of plastic hinges at both ends of a story’s columns due to seismic loads may produce
a sidesway mechanism that causes the story to collapse. To prevent this, a strong column-weak beam
design is required by ACI Section 18.7.3.2. A column forming part of the lateral-force-resisting system
must be designed to satisfy ACI Equation (18.7.3.2), which is
SMnc ≥ 1.2SMnb
where:
SMnc 5 sum of the nominal flexural strengths of columns at the face of a joint
calculated for the applicable factored axial force, resulting in the lowest
flexural strength
SMnb 5 sum of the nominal flexural strengths of beams at the face of the joint
and in the same plane as the columns. In T-beam construction, slab
reinforcement within an effective width of the flange is assumed to
contribute to the negative flexural strength. The effective flange width is
defined in ACI Section 6.3.2 as the lesser of
bf
5 l/4
or
5 16hf 1 bw
or
5 sw 1 bw
where:
l
5 beam span
hf
5 flange thickness
bw
5 width of web
sw
5 clear distance between webs
As shown in Figure 4-10, the strong column-weak beam relationship applies to seismic loading from
either direction. The sign convention adopted in the figure is that bending moments at the ends of a
member are shown acting from the joint to the member; in other words, the support reactions are considered. The arrowheads point toward the face of the member that is in tension.
Where the strong column-weak beam concept of ACI Equation (18.7.3.2) cannot be satisfied at a
joint, ACI Section 18.7.3.3 requires the column to be designed as a member, not designated as part
of the seismic-force-resisting system. In addition, such columns are to be ignored in calculating the
lateral strength and stiffness of the structure. However, since the columns contribute to the stiffness of
the structure prior to developing plastic hinges, their influence should be included in determining the
design base shear and torsional effects.
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Figure 4-10 Strong column-weak beam design
In accordance with ACI Section 18.7.6.1.1, the design shear force for columns shall be calculated
using the probable moment strengths at the top and bottom of the column associated with the factored
axial load, Pu, acting on the column. The probable flexural strength is calculated by assuming that
strain hardening increases the effective tensile strength of the reinforcement by 25 percent and by
using a strength reduction factor f of 1.0, as specified in ACI Section 2.2. As shown in Figure 4-11,
the design shear force at the top and bottom of the column is
where:
Ve
5 (Mpr1 1 Mpr2)/Hn
Hn
5 column clear height
u
n
u
Figure 4-11 Column shear due to probable flexural strength
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However, the column design shear need not exceed the value determined from the probable moment
strengths of the beams framing into the top and bottom of the column. As shown in Figure 4-12, the
design shear force for this condition, provided that the column stiffness is the same in all stories, is
given by
Ve
5 (Mpr1 1 Mpr2 1 Mpr3 1 Mpr4)/2Hn
In addition, the transverse reinforcement must also be adequate to resist the factored shear calculated
by analysis of the structure.
n
Figure 4-12 Column shear due to beam probable flexural strength
The cyclical nonlinear effects produced by seismic loading necessitate additional shear requirements
to ensure a ductile flexural failure. Where the factored compressive force in a member is less than
Ag fc9/20 and the seismic-induced shear represents one-half or more of the total design shear, the shear
resistance of the concrete, Vc , shall be neglected over the length, lo. Shear reinforcement shall then be
provided to resist the total design shear as required by ACI Section 18.7.6.2.1.
Example 4-2
The columns of an interior bay of the special moment-resisting frame shown in Figure 4-5 have a clear
height of 10 feet. The structure has a redundancy factor of r 5 1.0 and the 5-percent damped, design
spectral response acceleration for a period of 0.2 second is SDS 5 0.826g. The service level gravity
loads and bending moments are indicated in the figure together with the proposed beam and column
sections and the moments due to the design level seismic forces. The bending moments are shown
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acting at the face of the joint. Second-order effects may be neglected and the axial force due to seismic loads and the bending moments due to dead and live load are negligible. Neglect roof live load.
Reinforcement consists of Grade 60 bars and the normalweight concrete cylinder strength is 4000 psi.
Columns are at 24 3 12 feet on center. Determine the required reinforcement details for the column
above the third floor.
Solution
The longitudinal reinforcement to resist the factored loads is determined first.
Load combinations
For dead load and live load, the applicable load combination for the column above the third floor is
given by ACI Equation (5.3.1b), which is
U
5 1.2D 1 1.6L 1 0.5(Lr or S or R)
The factored column axial load for dead and live load is then
Pu3, D1L
5 1.2 3 400 1 1.6 3 100
5 640 kips
Where the effects of dead load and seismic load are additive, the applicable loading cases are given by
ASCE 7 Section 2.3.6 as
U
5 1.2D 1 1.0E 1 1.0L 1 0.2S
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
The corresponding factored column axial load is
Pu3
5 1.2D 1 1.0E 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826) 3 400 1 1.0 3 0 1 0.5 3 100 1 0
5 596 kips
The corresponding factored column moment is
Mu3
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826) 3 0 1 1.0 3 176 1 0.5 3 0 1 0
5 176 kip-ft
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For the column below the third floor, the applicable factored loads for the dead load plus seismic load
combination are
Pu2
5 1.2D 1 1.0E 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826) 3 470 1 1.0 3 0 1 0.5 3 120 1 0
5 702 kips
The corresponding factored column moment is
Mu2
5 (1.2 1 0.2SDS)D 1 rQE 1 0.5L 1 0.2S
5 (1.2 1 0.2 3 0.826) 3 0 1 1.0 3 224 1 0.5 3 0 1 0
5 224 kip-ft
Column reinforcement
Providing eight #8 bars gives a reinforcement area of
As
5 6.32 in2
Gross area of the column is
Ag
5 23 3 23
5 529 in2
and
rg
5 As /Ag
5 6.32/529
5 0.012
0.01Ag
5 5.29
, As . . . satisfies ACI Section 18.7.4.1
0.06Ag
5 31.74
. As . . . satisfies ACI Section 18.7.4.1
Column axial and flexural capacity
For zero applied moment on the column above the third floor, the design capacity in axial compression
is obtained from the appropriate interaction diagram obtained using the computer program spColumn,5
which is shown in Figure 4-13 as
fPn3
5 1121 kips
. Pu3, D1L . . . satisfactory
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For a factored applied load of 596 kips on the column above the third floor, the design flexural capac­
ity is
fMn3 5 399 kip-ft
. Mu3 . . . satisfactory
For a factored applied load of 702 kips on the column below the third floor, the design flexural capacity is
fMn2 5 379 kip-ft
. Mu2 . . . satisfactory
P (kip)
2500
2.5″
23″
9″
Y
23″
9″
Nominal diagram, fy=75 ksi
2.5″
X
1500
Design diagram, fy=60 ksi
1121
(379 kip-ft, 702 kips)
(640 kip-ft, 596 kips)
(399 kip-ft, 596 kips)
500
200
400
600
Mx (kip-ft)
-500
Figure 4-13 Interaction diagram5 for Example 4-2
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Strong column-weak beam
In determining the negative moment strength of the beam framing into the column, the reinforcement
within an effective width of the flange is assumed to contribute to the negative flexural strength. The
effective flange width is defined in ACI Section 6.3.2 as the lesser of
be
5 16hf 1 bw
5 16 3 0.5 1 21/12
5 9.75 ft
or
5 sw 1 bw
5 12 ft
or
5 l/4
5 24/4
5 6 ft . . . governs
where:
l
5 beam span
hf
5 flange thickness
bw
5 width of stem
sw
5 clear distance between stems
The distribution reinforcement in this width of flange is
Asf
5 0.0018 3 6 3 72
5 0.78 in2
Hence, the total area of reinforcement in the top of the beam at the face of the column is
As
5 3.38 1 0.78
5 4.16 in2
r
5 As /bw d
5 4.16/(21 3 21.5)
5 0.0092
The nominal negative moment strength of the beam framing into the right-hand face of the joint is
determined from the expression6
MnR
5 As fy d(1 2 0.59rfy /fc9)
5 4.16 3 60 3 21.5(1 2 0.59 3 0.0092 3 60/4)/12
5 411 kip-ft
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The nominal positive moment strength of the beam framing into the left-hand face of the joint is
obtained from Example 4-1 as
MnL
5 168/f
5 168/0.9
5 187 kip-ft
Hence,
1.2SMnb 5 1.2(411 1 187)
5 718 kip-ft
The sum of the nominal flexural capacities of the columns framing into the joint at the third floor is
SMnc 5 (fMn3 1 fMn2)/f
5 (399 1 379)/0.65
5 1197 kip-ft
. 1.2SMg . . . satisfies ACI Equation (18.7.3.2)
Column shear
The maximum factored column shear is obtained from Figure 4-5 as
Vu
5 1.2D 1 1.0E 1 1.0L 1 0.2S
5 1.2 3 0 1 1.0 3 35 1 0 1 0
5 35 kips
In accordance with ACI Sections 2.2 and 18.7.6.1.1, the design shear force for the column above the
third floor may be calculated from the probable moment strengths at the top and bottom of the column. The column probable moment strength is determined by assuming a strength reduction factor of
zero and a tensile reinforcement stress of 1.25fy . The maximum probable moment, at both the top and
bottom of the column, occurs at an axial load of Pu 5 596 kips and is obtained from the appropriate
interaction diagram5 using f 5 1.0 and fy 5 75 ksi, as shown in Figure 4-13, to give
Mpr
5 640 kip-ft
The clear height of the column is
Hn
5 10 ft
The design shear force is then
Ve
5 2Mpr /Hn
5 2 3 640/10
5 128 kips
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However, in accordance with ACI Section 18.7.6.1.1, the maximum design shear force in the column
need not exceed that determined from the probable flexural strengths of the beams that frame into
either side of the joint. The probable beam strengths, assuming a strength reduction factor of unity and
a tensile reinforcement stress of 1.25fy , were derived in Example 4-1 as
Mpr1
5 Mpr3
5 417 kip-ft
Mpr2
5 Mpr4
5 231 kip-ft
As shown in Figure 4-12, the design shear force in the column for this condition, since the column
stiffness is the same in all stories, is given by
Ve
5 (Mpr1 1 Mpr2 1 Mpr3 1 Mpr4)/2Hn
5 2(417 1 231)/(2 3 10)
5 65 kips
, 128 kips
. Vu
then:
Ve
5 65 kips . . . governs and ACI Section 18.7.6.1.1 is satisfied
The compressive force value given by
Ag fc9/20 5 529 3 4/20
5 106 kips
, Pu3
Hence, in accordance with ACI Section 18.7.6.2.1, the design shear strength provided by the concrete
may be utilized, and neglecting axial compression, this is given by ACI Equation (22.5.5.1) as
fVc
5 0.75 3 2lbw d( fc9)0.5
5 0.75 3 2 3 1.0 3 23 3 (23 2 1.5 2 0.5 2 0.5)(4000)0.5/1000
5 45 kips
, Ve
The design shear strength required from shear reinforcement is given by ACI Equation (11-2) as
fVs
5 Ve 2 fVc
5 65 2 45
5 20 kips
, 4 3 fVc . . . satisfies ACI Section 22.5.1.2
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The maximum hoop spacing, in accordance with ACI Section 18.7.5.5, may not exceed the lesser
value given by
s
5 6db
5 6 3 1.0
5 6 in
or
s
5 6 in
Hence, a spacing of 6 inches is appropriate, with the exception of confinement reinforcement at the
ends of the column and at lap splices.
The area of shear reinforcement, at a spacing of 6 inches, that is required to provide a shear strength
of fVs is specified by ACI Equation (22.5.10.5.3) as
Av
5 fVs s/fdfy
5 20 3 6/(0.75 3 20.5 3 60)
5 0.13 in2
The minimum size of crosstie required for a #8 longitudinal bar is specified by ACI Section 25.7.2.2(a)
as a #3 bar, and at least one crosstie is required to satisfy the lateral support requirements of ACI Section 18.7.5.2(e).
0.3Ag fc9 5 0.3 3 529 3 4
5 635 kips
. Pu3 5 596 kips
Hence, ACI Section 18.7.5.2(f) does not apply.
The minimum area of shear reinforcement, which may be provided at a spacing of 6 inches, is given
by ACI Section 10.6.2.2 as
Av(min) 5 50bw s/fyt . . . governs for fc9 , 4444 lb/in2
5 50 3 23 3 6/60,000
5 0.12 in2 . . . does not govern
, 0.13 in2
Providing a #4 hoop and one #4 crosstie gives an area of
Av
5 3 3 0.2
5 0.6 in2
. 0.13 in2 . . . satisfactory
From Figure 4-13, the center-to-center spacing between longitudinal bars is
hx
5 9 in
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Confinement reinforcement
The point of inflection of the column lies within the center half of the column clear height. Hence, in
accordance with ACI Section 18.7.5.1, confinement reinforcement is required for a distance from each
joint face given by the greater of
lo
5 Hn /6
5 10 3 12/6
5 20 in
or
lo
5 18 in
or
lo
5h
5 23 in . . . governs
The spacing of the confinement reinforcement is limited by ACI Section 18.7.5.3 to the minimum
value given by
so
5 hmin /4
5 23/4
5 5.75 in
or
so
5 6db
5 6 3 1.0
5 6 in
or
so
5 6 in
or
so
5 4 1 (14 2 hx)/3 . . . ACI Equation (18.7.5.3)
5 4 1 (14 2 9)/3
5 5.7 in . . . governs
Using #4 hoop reinforcement bars at 4 inches on center, and providing 11⁄2 inches clear cover to the
bars, gives a core dimension, measured out-to-out of the hoop reinforcement, of
bc
5 23 2 3
5 20 in
The area, calculated out-to-out of the confining bars, is
Ach
5 bc2
5 202
5 400 in2
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The required area of confinement reinforcement is given by the greater value obtained from ACI Table
18.7.5.4 Equations (a) and (b), which are
Ash
5 0.3sbc(Ag /Ach 2 1) fc9/fy
5 0.3 3 4 3 20(529/400 2 1)4/60
5 0.52 in2 . . . governs
or
Ash
5 0.09sbc fc9/fy
5 0.09 3 4 3 20 3 4/60
5 0.48 in2
A #4 hoop with one #4 crosstie provides an area of confinement reinforcing of
Ash
5 0.60 in2
. Ash . . . satisfactory
To conform with ACI Section 18.7.4.3, a tension splice is required within the center half of the clear
column height. Hoop reinforcement, at a spacing of 4 inches, is provided over the splice length in
accordance with ACI Section 18.7.5.3. The lap length required for a Class B splice is specified by ACI
Section 25.5.2.1 as being equal to 1.3 times the tensile development length. The development length is
given by ACI Equation (25.4.2.3a) as
ld
5 0.075db fy /[( fc9)0.5(cb 1 Ktr)/db] . . . where l 5 yt 5 ye 5 ys 5 1.0
For the reinforcement layout indicated, (cb 1 Ktr)/db equals its maximum permissible value of 2.5 and
1.3ld
5 1.3 3 0.075 3 1.0 3 60,000/[2.5(4000)0.5]
5 37 in
Details of the column reinforcement are shown in Figure 4-14.
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Figure 4-14 Column reinforcement for Example 4-2
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4.1.6 Joint design and details
Joints are designed on an expected strength basis using the probable strength of the materials. At a joint
in a frame, the horizontal design shear force is determined as required by ACI Section 18.8.2.1 and as
shown in Figure 4-15. The shear force produced in the column by the probable moment strengths of
the beams at the joint is
where:
V
5 (Mpr1 1 Mpr2)/Hc
Hc
5 floor-to-floor height
The probable tensile force in the tensile reinforcement in the beam framing into the right-hand face of
the joint is
where:
T1
5 1.25As1 fy
As1
5 area of tensile (top) reinforcement of right-hand beam
Figure 4-15 Forces acting at a joint
The probable compressive force in the concrete in the beam framing into the left-hand face of the joint
is
C2
5 T2
5 1.25As2 fy
where:
As2
5 area of tensile (bottom) reinforcement of left-hand beam
The net shear acting on the joint is given by
Ve
5 T1 1 T2 2 V
5 1.25fy(As1 1 As2) 2 (Mpr1 1 Mpr2)/Hc
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In accordance with ACI Section 18.8.4, the nominal shear capacity of the joint depends on the concrete
strength and effective area of the joint, and the contribution of hoops to the shear strength is neglected.
The nominal shear strength of the joint is given by
Vn
5 20lAj ( fc9)0.5 for joints confined on four faces
5 15lAj ( fc9)0.5 for joints confined on opposite faces or on three faces
5 12lAj ( fc9)0.5 for other conditions
where:
Aj
5 effective cross-sectional area within the joint
l
5 0.75 . . . lightweight concrete
l
5 1.0 . . . normalweight concrete
As shown in Figure 4-16, the effective joint depth equals the overall depth of the column. Where a
beam frames into a column of larger width, the effective joint width is given by
be
5b1h
≤ b 1 2x
where:
b
5 beam width
h
5 column depth
x
5 smaller distance from edge of beam to edge of column
Figure 4-16 Effective area of a joint
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As specified in ACI Section 18.8.3.1, hoop reinforcement shall be provided through the joint as
required at the ends of the column. Where beams frame into three or fewer sides of the joint, hoop
reinforcement, Ash , as specified over the length of the column, lo , shall be provided throughout the
height of the joint. Where beams frame into all four faces of the joint and provide confinement, ACI
Section 18.8.3.2 requires hoop reinforcement with an area of Ash /2 at a maximum spacing of 6 inches.
A joint is considered confined, in accordance with ACI Section 18.8.3.2, where the beam width is at
least three-fourths of the column width.
As required by ACI Section 18.8.2.2, beam reinforcement terminating in a column shall extend to the
far face of the confined concrete core and be provided with an anchorage length as specified in ACI
Section 18.8.5. Typical joint details are shown in Figure 4-17.
Figure 4-17 Typical joint details
Example 4-3
The columns of an interior bay of the special moment-resisting frame shown in Figure 4-5 have a
clear height of 10 feet. Beams frame into the opposite faces of the column as indicated. Reinforcement
consists of Grade 60 bars and the normalweight concrete cylinder strength is 4000 psi. Determine the
required reinforcement details for an interior joint at the third floor.
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Solution
The longitudinal reinforcement in the beams on either side of the joint is shown in Figure 4-6. The
reinforcement areas are
As1
5 3.38 in2
As2
5 1.80 in2
The probable tensile forces in the reinforcement are
T1
5 1.25 3 As1 fy
5 1.25 3 3.38 3 60
5 254 kips
T2
5 1.25 3 As2 fy
5 1.25 3 1.80 3 60
5 135 kips
The probable moment strengths are obtained from Example 4-1 as
Mpr1
5 417 kip-ft
Mpr2
5 231 kip-ft
The shear force produced in the column by the probable moment strengths of the beams is
V
5 (Mpr1 1 Mpr2)/Hc
5 (417 1 231)/12
5 54 kips
The net shear acting on the joint is given by
Ve
5 T1 1 T2 2 V
5 254 1 135 2 54
5 335 kips
The effective joint width is given by the lesser of
be
5b1h
5 21 1 23
5 44 in
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or
Seismic Design of Concrete Structures
be
5 b 1 2x
5 21 1 2 3 1
5 23 in . . . governs
The effective cross-sectional area of the joint is
Aj
5 hbe
5 23 3 23
5 529 in2
The design shear strength of the joint, which is confined on opposite faces, is given by ACI Section
18.8.4 with f 5 0.85 as specified by ACI Section 21.2.4.3 as
fnVn
5 f 3 15Aj ( fc9)0.5 . . . for joints confined on opposite faces or on three faces
5 0.85 3 15 3 529(4000)0.5/1000
5 427 kips
. Ve . . . satisfactory
Provide #4 hoops and crossties at 4 inches on center through the joint.
4.2 Special structural walls
Special structural walls10 are defined in ACI Section 2.3 as walls designed in accordance with ACI
Sections 18.2.3 through 18.2.8. For bearing wall systems, special structural walls are the only shear
walls permitted in seismic design categories D through F. No limitation is placed on building height
in seismic design categories A, B, and C and a limiting height of 160 feet applies in seismic design
categories D and E. In seismic design category F, the height is restricted to 100 feet. The following
parameters are specified in ASCE 7 Table 12.2-1 as
R
5 response modification factor
5 5.0
Cd
5 deflection amplification factor
55
W0
5 structure overstrength factor
5 amplification factor to account for the overstrength of the structure in the
inelastic range
5 2.5
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Similarly, for building frame systems, special reinforced concrete structural walls are the only shear
walls permitted in seismic design categories D through F. No limitation is placed on building height
in seismic design categories A, B, and C, and a limiting height of 160 feet applies in seismic design
categories D and E. In seismic design category F, the height is restricted to 100 feet. The following
parameters are specified in ASCE 7 Table 12.2-1 as
R
56
Cd
55
W0
5 2.5
Special reinforced concrete shear walls may also be utilized in dual building systems with special
moment frames. No limitation is placed on building height in any seismic design category and the
following parameters are specified in ASCE 7 Table 12.2-1 as
R
57
Cd
5 5.5
W0
5 2.5
4.2.1 Shear capacity of shear walls
In accordance with ACI Section 18.10.3, the design shear force, Vu , shall be obtained from a lateral
load analysis of the structure with the appropriate factored load combinations.
The nominal shear strength of a shear wall may be determined as specified in ACI Section 18.10.4 and
given in the ACI Equation (18.10.4.1) as
where:
Vn
5 Acv[ac l( fc9)0.5 1 rt fy]
Acv
5 gross area of concrete section bounded by the web thickness and length of
the section in the direction of the shear force
rt
5 reinforcement ratio of horizontal shear reinforcement
ac
5 3 . . . for hw /lw ≤ 1.5
5 2 . . . for hw /lw ≥ 2.0
l
5 lightweight concrete modification factor
Linear interpolation may be used in the determination of ac for values of hw /lw between 1.5 and 2
hw
5 height of wall
lw
5 length of wall in direction of shear force
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As specified by ACI Section 18.10.4.4, the maximum allowable nominal shear strength for all vertical
wall segments resisting a common lateral force is
where:
Vn
5 8Acv ( fc9)0.5
Acv
5 gross combined area of all wall segments
For any individual wall segment, the maximum allowable nominal shear strength is
where:
Vn
5 10Acv ( fc9)0.5
Acv
5 area of segment considered
In accordance with ACI Section 18.10.2.1, where the design shear force, Vu , exceeds Acv l( fc9)0.5, the
minimum distributed web reinforcement ratios for the horizontal and vertical reinforcement shall be
where:
rt
5 0.0025
rl
5 0.0025
rl
5 reinforcement ratio of vertical shear reinforcement
Where the design shear force, Vu , does not exceed Acv l( fc9)0.5, the minimum reinforcement ratios for
the horizontal and vertical reinforcement may be as specified in ACI Table 11.6.1. For this situation,
the minimum required reinforcement ratios are
rl
5 0.0012 . . . for #5 deformed bars or smaller with fy ≥ 60 ksi
5 0.0015 . . . for other deformed bars
rt
5 0.0020 . . . for #5 deformed bars or smaller with fy ≥ 60 ksi
5 0.0025 . . . for other deformed bars
In addition, the spacing of shear reinforcement shall not exceed 18 inches each way. In order to control cracking and inhibit fragmentation of the wall due to cyclical loading in the inelastic range, ACI
Section 18.10.2.2 specifies the provision of two curtains of reinforcement where the design shear force
exceeds the value
Vu
5 2Acv l( fc9)0.5 or hw /lw ≥ 2.0
4.2.2 Special boundary elements
For shear walls subjected to combined flexural and axial load, ACI Section 18.10.5.1 requires the
wall to be designed in accordance with ACI Section 22.2.2. The strain distribution across the section
is assumed linear with a maximum concrete compressive strain of 0.003. The assumptions used are
shown in Figure 4-18.
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The effective width of flanged sections contributing to the section is specified in ACI Section 18.10.5.2
as half the distance between adjacent walls but not more than 25 percent of the wall height, as shown
in Figure 4-19.
In accordance with ACI Section 18.10.6.2, special boundary elements are required where the distance
from the extreme compression fiber to the neutral axis is not less than the value given by ACI Equation
(18.10.6.2)
where:
c
5 lw /600(1.5du /hw)
lw
5 length of wall
hw
5 height of wall
du
5 design displacement of the wall
5 Cd dxe /Ie
5 actual anticipated inelastic displacement caused by the design ground
motion and defined in ASCE 7 Section 12.8.6
and
Cd
5 deflection amplification factor defined in ASCE 7 Table 12.2-1
dxe
5 theoretical displacement caused by the code-prescribed design level forces,
as determined by an elastic analysis
Ie
5 importance factor
du /hw ≥ 0.005
0.85
Figure 4-18 Assumptions used in shear wall design
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Figure 4-19 Effective flange widths
The depth, c, is calculated for the factored axial force and nominal moment strength, consistent with
the design displacement, du , resulting in the largest neutral axis depth.
Special boundary element confinement reinforcement shall be provided in the zones specified in ACI
Sections 18.10.6.2 (b) and 18.10.6.4 as indicated in Figure 4-20. The area of rectilinear confinement
reinforcement required is given by ACI Table 18.10.6.4(f) as the greater of
where:
Ash
5 0.09sbc fc9/fyt
Ash
5 0.3sbc (Ag /Ach 2 12) fc9/fyt
s
5 spacing of transverse reinforcement
bc
5 dimension of confined core of boundary element measured out-to-out of
transverse reinforcement
fyt
5 yield strength of transverse reinforcement
Ag
5 gross area of concrete section
Ach
5 cross-sectional area measured out-to-out of transverse reinforcement
The spacing of the confinement reinforcement is limited by ACI Section 18.7.5.3 to the minimum
value given by
s
5 hmin /4
or
s
5 6db
or
so
5 4 1 (14 2 hx)/3 . . . ACI Equation (18.7.5.3)
≥ 4 in
≤ 6 in
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b
423
hu/16
Figure 4-20 Special boundary element dimensions
where:
hmin
5 minimum boundary element dimension
db
5 diameter of the smallest longitudinal bar
hx
5 maximum horizontal spacing of hoop or crosstie legs on all faces of the
boundary element
5 14 in maximum
≤ 2b/3
Details of wall reinforcement are shown in Figure 4-21.
Special boundary element confinement reinforcement shall extend into the support at least the development length of the largest longitudinal bar or at least 12 inches into a footing or mat.
Horizontal reinforcement in the wall web shall extend to within 6 inches of the wall end and shall be
anchored within the confined core of the boundary element to develop the full tensile strength of the
reinforcement.
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b
2b/3
2b/3
Figure 4-21 Special boundary element reinforcement
4.2.3 Nonspecial boundary elements
Where special boundary elements are not necessary and the vertical reinforcement ratio at the wall
boundary exceeds 400/fy, ACI Section 18.10.6.5 requires confinement reinforcement extending horizontally from the extreme compression fiber a distance not less than the larger of
xt
or
5 c 2 lw /10
5 c/2
The maximum spacing of the confinement reinforcement shall not exceed the lesser of 8 inches or 8db .
Over a height equal to the greater of lw or Mu /4Vu , the maximum spacing is reduced to the lesser of 6
inches or 6db where yielding of the longitudinal reinforcement is likely to occur.
Except where the factored shear force in the wall is less than Acv l( fc9)0.5, horizontal reinforcement in
the wall web shall be anchored at the end of the wall with a standard hook engaging the edge reinforcement. Alternatively, the edge reinforcement may be enclosed in U-stirrups spliced to the horizontal
reinforcement.
Example 4-4
The shear wall of a bearing wall system is shown in Figure 4-22. The structure has a redundancy factor
of r 5 1.0 and the 5-percent damped, design spectral response acceleration for a period of 0.2 second
is SDS 5 0.826g. The service level gravity loads are indicated in the figure together with the proposed
wall section and the moment and shear due to design level seismic forces. Roof live load is negligible.
Reinforcement consists of Grade 60 bars and the normalweight concrete cylinder strength is 4000 psi.
The importance factor, Ie , is 1.0. Determine the required reinforcement details.
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Solution
The required shear reinforcement is determined first.
Load combinations
For dead load and live load, the applicable load combination is given by ACI Equation (5.3.1b), which is
5 1.2D 1 1.6L 1 0.5(Lr or S or R)
U
The factored axial load for dead and live load is then
Pu,D1L 5 1.2 3 300 1 1.6 3 50
5 440 kips
I
11/2 in
11/2 in
Figure 4-22 Details for Example 4-4
Where the effects of dead load and seismic load are additive, the applicable loading case is given by
ASCE 7 Section 2.3.6, which is
U
5 1.2D 1 1.0E 1 1.0L 1 0.2S
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
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Seismic Design of Concrete Structures
The factored wall axial load caused by dead load and live load is
Pu
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826)300 1 1.0 3 0 1 0.5 3 50 1 0
5 435 kips
The corresponding factored wall moment is
Mu
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826) 3 0 1 1.0 3 8400 1 0 1 0
5 8400 kip-ft
The corresponding factored shear force is
Vu
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826) 3 0 1 1.0 3 250 1 0 1 0
5 250 kips
Where the effects of gravity and seismic loads counteract, the applicable loading case is given by
ASCE 7 Section 2.3.6 as
U
5 (0.9 2 0.2SDS)D 1 rQE
The factored wall axial load is then
Pu, -E
5 (0.9 2 0.2SDS)D 1 rQE
5 (0.9 2 0.2 3 0.826)300 1 1.0 3 0
5 220 kips
The corresponding bending moment and shear force are
Mu
5 8400 kip-ft
Vu
5 250 kips
Shear reinforcement required
The wall parameters are
Acv
5 gross area of concrete section bounded by the web thickness and length of
section
5 12 3 12 3 12
5 1728 in2
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hw/lw
427
5 48/12
54
.2
ac
and
5 2.0 . . . from ACI Section 18.10.4.1
Acv l( fc9)0.5 5 1728 3 1.0 3 (4000)0.5/1000
5 109 kips
, Vu
, Vu /2
Hence, in accordance with ACI Section 18.10.2.2, two curtains of reinforcement are necessary and the
required minimum reinforcement ratios, specified by ACI Section 18.10.2.1, along both the longitudinal and transverse axes are
rv
5 rn 5 0.0025
Horizontal reinforcement consisting of #5 bars in each face at a spacing of 18 inches provides a reinforcement ratio of
rt
5 2 3 0.31/(12 3 18)
5 0.00287
. 0.0025 . . . satisfactory
The proposed spacing does not exceed the maximum permissible value of 18 inches, given by ACI
Section 18.10.2.1, and is satisfactory.
Hence, from ACI Section 18.10.4.1, the design shear force is given by
fVn
5 fAcv[acl( fc9)0.5 1 rtfy]
5 0.75Acv[2( fc9)0.5 1 rtfy]
5 0.75 3 1728[2(4000)0.5 1 0.00287 3 60,000]/1000
5 387 kips
. Vu . . . satisfactory
Vertical reinforcement consisting of #5 bars in each face at a spacing of 18 inches also satisfies all
criteria. Hence, #5 bars in each face at a spacing of 18 inches is satisfactory.
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Design for load combinations
The total steel area in the wall consists of 16 #5 bars and 16 #8 bars, giving a total area of
Ast
5 16 3 0.31 1 16 3 0.79
5 17.60 in2
Ag
5 Acv 1 2 3 15 3 3
5 1818 in2
The design axial load strength of the wall, in the absence of bending moment, is given by ACI Equation (22.4.2.2) as
fPn
5 0.8f[0.85fc9(Ag 2 Ast) 1 fy Ast]
5 0.8 3 0.65[0.85 3 4(1818 2 17.6) 1 60 3 17.6]
5 3732 kips
. Pu,D1L . . . satisfactory
Under combined flexure and axial load, the assumed maximum compressive strain in the concrete is
specified in ACI Section 22.2.2.1 as
ec
5 0.003
In accordance with ACI Section 22.2.1.2, strain in reinforcement and concrete is assumed directly
proportional to the distance from the neutral axis, and assuming the depth to the neutral axis is given
by c 5 24 inches, the strain produced in a reinforcing bar is
es
5 eec /c
5 e 3 0.003/24
5 0.000125e
where:
e
5 distance of a reinforcing bar from the neutral axis
The force produced in a reinforcing bar is given by
F
5 es As Es
5 0.000125 3 29,000eAs
5 3.625eAs
where:
As
5 area of the reinforcing bar
Es
5 modulus of elasticity of reinforcement
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The strain-producing yield in the reinforcement is
ey
5 fy /Es
5 60/29,000
5 0.00207
The maximum force is then produced in the reinforcement and is given by
Fmax
5 60As kips
In accordance with ACI Section 22.2.2.4.1, the depth of the equivalent rectangular concrete stress
block is
where:
a
5 cb1
b1
5 compression zone factor
5 0.85 as defined in ACI Section 22.2.2.4.3
then:
a
5 24 3 0.85
5 20.4 in
The strain distribution across the section and the forces developed are shown in Figure 4-23.
Figure 4-23 Strain distribution in the section
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Seismic Design of Concrete Structures
All bars from 1 through 8 are in tension and are stressed to the yield stress. Bar 9 has a tensile force of
T9
5 3.625 3 3 3 0.62
5 7 kips
The sum of the tensile forces in the reinforcement is
ST
5 60(12 3 0.31 1 8 3 0.79) 1 7
5 609 kips
The compressive forces in the reinforcement are
C10
5 3.625 3 11.5 3 2.37 5 99 kips
C11
5 3.625 3 16.5 3 2.2 5 132 kips
C12
5 60 3 2.37
5 142 kips
The sum of the compressive forces in the reinforcement is
SC
5 373 kips
The force in the concrete stress block is given by ACI Section 22.2.2.4.1 as
Cc
5 0.85fc9(ah 2 A9)
s
5 0.85 3 4(20.4 3 12 1 15 3 3 2 2 3 2.37 2 2.2)
5 962 kips
The nominal axial load capacity at this strain condition is
Pn
5 Cc 1 SC 2 ST
5 962 1 373 2 609
5 726 kips
The design axial load capacity at this strain condition is
fPn
5 0.65 3 726
5 472 kips
. Pu . . . satisfactory
The nominal moment capacity for this neutral axis depth is obtained by summing moments about the
middepth of the section and is given by
Mo
5 (72 2 20.4/2)Cc 1 69.5T1 1 64.5T2 1 59.5T3 1 45T4 1 27T5 1 9T62 9T7
2 27T8 2 45T9 1 59.5C10 1 64.5C11 1 69.5C12
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5 61.8 3 962 1 69.5 3 142 1 64.5 3 132 1 59.5 3 142 1 45 3 37.2
1 27 3 37.2 1 9 3 37.2 2 9 3 37.2 2 27 3 37.2 2 45 3 7 1 59.5
3 99 1 64.5 3 132 1 69.5 3 142
5 111,916 kip-in
5 9326 kip-ft
The strain in the extreme tension steel is
es
5 0.000125e
5 0.000125 3 117.5
5 0.015
. 0.005
Hence, from ACI Figure R21.2.2b, the section is tension controlled and
f
5 0.9
The design moment capacity is given by ACI Section 21.2.2 as
fMo
5 0.9 3 9326
5 8394 kip-ft
 Mu . . . satisfactory
The wall is adequate with the assumed depth to the neutral axis of c 5 24 inches.
Boundary zone requirements
The theoretical elastic displacement, caused by the code-prescribed design level forces, is given in
Figure 4-22 as
dxe
5 2.2 in
Applying the deflection amplification factor defined in ASCE 7 Table 12.2-1 as Cd 5 5, the total
inelastic design displacement is derived from ASCE 7 Section 12.8.6 as
du
5 Cd dxe /Ie
5 5 3 2.2/1.0
5 11 in
du /hw 5 11/576
5 0.019
. 0.005 . . . satisfies ACI Section 18.10.6.2
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The parameter for determining if special boundary elements are necessary is
lw /600(1.5du /hw) 5 144/(600 3 1.5 3 0.019)
5 8.4 in
,c
Hence, special boundary elements are necessary, extending horizontally a distance from the extreme
compression fiber given by ACI Section 18.10.6.4(a) as the larger of
lz
5 c 2 0.1lw
5 24 2 0.1 3 144
5 9.6 in
or
lz
5 c/2
5 12 in . . . governs
The length provided is
lz
5 15 in . . . satisfactory
The special boundary elements must extend vertically a distance above the base given by ACI Section
18.10.6.2(b) as the larger of
hz
5 Mu /4Vu
5 8400/(4 3 250)
5 8.4 ft
or
hz
5 lw
5 12 ft . . . governs
The length provided is
hz
5 12 ft . . . satisfactory
Boundary zone confinement reinforcement
The spacing of the confinement reinforcement is limited by ACI Section 18.7.5.3 to the minimum
value given by
s
5 hmin /4
5 15/4
5 3.75 in
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or
s
433
5 6db
5 6 3 1.0
5 6 in
or
so
5 4 1 (14 2 hx)/3 . . . ACI Equation (18.7.5.3)
where:
hx
5 15 2 2 3 1.5 2 2 3 0.5 . . . from Figure 4-22
5 11 in
then:
so
5 4 1 (14 2 11)/3
5 5 in
Using #4 hoop reinforcement bars at a spacing of 4 inches on center, and providing 11⁄2-inch clear
cover to the bars, gives a core dimension, measured out-to-out of the hoop reinforcement, of
bc
5 15 2 3 . . . from Figure 4-22
5 12 in
The area of confinement reinforcement required is given by ACI Table 18.10.6.4(f), which is
Ash
5 0.09sbc fc9/fyt
5 0.09 3 4 3 12 3 4/60
5 0.29 in2
A #4 hoop provides an area of confinement reinforcing of
Ash
5 0.40 in2
. 0.29 . . . satisfactory
4.3 Slender wall design
The alternative design method of slender concrete walls11 is an empirical design technique detailed in
ACI Section 11.8. In the design of a slender wall, consideration of P-delta effects is required.
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4.3.1 General requirements
In accordance with ACI Section 11.8.1.1, the following limitations apply:
•
the cross-section must be constant over the height of the wall
•
the wall must be tension controlled for out-of-plane moment effects
•
the design moment strength is governed by
where:
fMn
≥ Mcr
Mcr
5 cracking moment defined in ACI Section 24.2.3.5
5 fr Ig /yt
fr
5 modulus of rupture defined in ACI Equation (19.2.3.1)
5 7.5l( fc9)0.5
l
5 modification factor for lightweight concrete from ACI Table
19.2.4.2
yt
5 distance from centroidal axis of gross section to the extreme tension
fiber
5 h/2 . . . for centrally placed reinforcement
Ig
5 moment of inertia of gross concrete section neglecting reinforcement
5 lw h3/12
•
h
5 overall thickness of wall
lw
5 horizontal length of wall
the factored axial force at the midheight of the section is governed by
Pu
•
≤ 0.06 fc9Ag
the maximum permissible deflection at midheight due to service loads, including secondorder effects, is
Ds
≤ lc /150 . . . lc is the vertical distance between supports
In accordance with ACI Section 11.8.2.1, the wall is analyzed as a simply supported, axially loaded
member subject to an out-of-plane uniformly distributed lateral load, with maximum moments and
deflections occurring at midheight.
An effective area of longitudinal tension reinforcement is used in the design calculations and this is
defined in ACI Section R11.8.3.1 as
Ase,w
5 As 1 Pu h/2fy d
5 (Pu 1 As fy)/fy . . . for one central layer of reinforcement
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Hence, the nominal moment strength is given by
Mn
5 Ase,w fy(d 2 a/2)
The depth of the equivalent rectangular concrete stress block is defined in ACI Section 22.2.2.4.1 as
a
5 Ase,w fy /0.85fc9lw
The depth to the neutral axis is
where:
c
5 a/b1
b1
5 compression zone factor as defined in ACI Section 22.2.2.4.3
4.3.2 Required strength
The factored applied moment, Mu, at the location of maximum moment in the wall must include the
effects of the factored axial loads and eccentricities, the factored lateral load, and the P-delta effects.
As shown in Figure 4-24, the ultimate moment is given by ACI Equation (11.8.3.1a) as
Mu
5 Mua 1 Pu Du
5 Mua /[1 2 5Pulc2/(0.75)(48EcIcr)] . . . from ACI Equation (11.8.3.1d)
where:
Pu
5 factored applied axial load at the location of the maximum moment
5 Pur 1 Puw
Pur
5 factored applied axial load at top of wall
Puw
5 factored weight of wall above the location of the maximum moment
(for a wall hinged top and bottom with a uniformly distributed load, the
maximum moment occurs at midheight)
Mua
5 maximum moment due to factored lateral and eccentric vertical loads
5 wu lc2/8 1 Pure/2
wu
5 factored lateral load
e
5 eccentricity of applied axial load at top of wall
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h
Figure 4-24 Loads on slender wall
and
Du
5 wall displacement at location of maximum moment
5 5Mu lc2/(0.75)48Ec Icr . . . from ACI Equation (11.8.3.1b)
where:
lc
5 vertical distance between supports
Icr
5 moment of inertia of cracked section transformed to concrete
5 nAse,w(d 2 c)2 1 lw c3/3 . . . from ACI Equation (11.8.3.1c)
n
5 modular ratio of elasticity
5 Es /Ec
≥6
Es
5 modulus of elasticity of reinforcing steel
5 29,000 ksi
Ec
5 modulus of elasticity of concrete
5 57,000( fc9)0.5 . . . from ACI Section 19.2.2.1
The factored moment Mu may be determined either by iterative calculation using ACI Equation
(11.8.3.1a) or by direct calculation using ACI Equation (11.8.3.1d).
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4.3.3 Service load deflections
For seismic loads, the appropriate load combination is given by ACI Section R11.8.4 as
u
5 D 1 0.5L 1 0.7E
The maximum permissible deflection at midheight, Ds , due to service vertical and lateral loads is given
by ACI Section 11.8.1.1(e) as
Ds
5 lc /150
For Ma . 2Mcr /3, the midheight service deflection is given by ACI Table 11.8.4.1(b) as
Ds
5 2Dcr /3 1 (Ma 2 2Mcr /3)(Dn 2 2Dcr /3)/(Mn 2 2Mcr /3)
For Ma ≤ 2Mcr /3,
where:
Ds
5 MaDcr /Mcr . . . ACI Table 11.8.4.1(a)
Dcr
5 5Mcr lc2/48Ec Ig . . . ACI Equation (11.8.4.3a)
Dn
5 5Mn lc2/48Ec Icr . . . ACI Equation (11.8.4.3b)
Ma
5 maximum unfactored moment due to service loads, including P-delta
effects, obtained by iteration of deflections
5 Msa 1 Ps Ds
Msa
5 maximum unfactored applied moment due to service loads, not including
P-delta effects
Ps
5 unfactored axial load
Example 4-5
The slender wall of a tilt-up concrete structure is shown in Figure 4-25. The structure has a redundancy
factor of r 5 1.0 and the 5-percent damped, design spectral response acceleration for a period of 0.2
second is SDS 5 0.826g. The service level gravity loads are indicated in the figure and act at an eccentricity of 7 inches with respect to the center of the proposed wall section. Reinforcement consists of
Grade 60 bars and the normalweight concrete cylinder strength is 4000 psi. The structure is assigned
to seismic design category D, and the roof diaphragm may be considered flexible. Determine if the
reinforcement details are satisfactory.
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Roof
Roof
h
24.78
Figure 4-25 Details for Example 4-5
Solution
The applied loads are determined first.
Factored loads
Where the effects of dead load and seismic load are additive, the applicable loading case is given by
ASCE 7 Section 2.3.6, which is
U
5 1.2D 1 1.0E 1 1.0L 1 0.2S
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
Considering a 1-foot width of panel, the factored axial load from the roof, not including roof live load,
is given by
Pur
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826)220 1 1.0 3 0 1 0 1 0
5 300 lb
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The factored axial load from the weight of the wall above the midheight section is
Puw
5 (1.2 1 0.2SDS)D 1 rQE 1 1.0L 1 0.2S
5 (1.2 1 0.2 3 0.826)(150 3 9 3 6/12) 1 1.0 3 0 1 0 1 0
5 922 lb
The total factored axial load at the midheight section is
Pu
5 Pur 1 Puw
5 1222 lb
The factored seismic lateral force on the wall is given by ASCE 7 Section 12.11.1 as
wu
5 0.40ISDS ww
5 0.40 3 1.0 3 0.826(150 3 6/12)
5 24.78 psf
The corresponding factored bending moment at the midheight section is
Muw
5 wu lc2/8
5 24.78 3 172/8
5 895 lb-ft
The eccentricity of the roof load about the wall centerline is
e
5 7 in
The factored bending moment at the midheight section caused by the eccentricity is
Mue
5 Pur e/2
5 300 3 7/24
5 88 lb-ft
The total factored wall moment at the midheight section due to seismic lateral force and the eccentric
roof load is
Mua
5 Muw 1 Mue
5 895 1 88
5 983 lb-ft
5 11.80 kip-in
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Applied factored moment including P-delta effects
The modulus of elasticity of the concrete is given by ACI Section 19.2.2.1 as
Ec
5 57,000( fc9)0.5
5 57,000(4000)0.5/1000
5 3605 ksi
The modular ratio is
n
5 Es /Ec
5 29,000/3605
5 8.0
. 6 . . . satisfies ACI Section 11.8.3.1(a)
For #4 vertical bars at 9-inch spacing, the reinforcement area per foot width is
As
5 0.27 in2
The effective reinforcement area for one central layer of reinforcement is given by
Ase,w
5 (Pu 1 As fy)/fy
5 (1.222 1 0.27 3 60)/60
5 0.29 in2
The depth of the equivalent rectangular stress block, as shown in Figure 4-26, is given by ACI Section
22.2.2.4.1 as
a
5 Ase,w fy /0.85fc9lw
5 0.29 3 60/(0.85 3 4 3 12) . . . for a 1-foot width
5 0.43 in
For a concrete strength of 4000 pounds per square inch, the factor b1 is given by ACI Section 22.2.2.4.3
as
b1
5 0.85
The depth to the neutral axis is given by ACI Section 22.2.2.4.1 as
c
5 a/b1
5 0.43/0.85
5 0.51 in
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Figure 4-26 Strain conditions
From ACI Equation (11.8.3.1c), the moment of inertia of the cracked section is
Icr
5 nAse,w(d 2 c)2 1 lw c3/3
5 8.0 3 0.29(6/2 2 0.51)2 1 12 3 0.513/3
5 14.91 in4
The magnification factor to account for P-delta effects is given by ACI Equation (11.8.3.1d) as
B1
5 1 2 5Pu lc2/(0.75)48Ec Icr
5 1 2 5 3 1.222 3 (17 3 12)2/(0.75 3 48 3 3605 3 14.91)
5 1 2 0.13
5 0.87
The factored applied moment including P-delta effects is given by
Mu
5 Mua 1 Pu Du
5 Mua /B1
5 11.80/0.87
5 13.56 kip-in
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Design strength of section
The nominal moment strength is given by11
Mn
5 Ase,w fy(d 2 a/2)
5 0.29 3 60(3 2 0.43/2)
5 48.46 kip-in
As shown in Figure 4-26, the strain in the tension reinforcement is given by
es
5 ec(3 2 c)/c
5 0.003(3 2 0.51)/0.51
5 0.015
. 0.005
Hence, from ACI Section R21.2.2, the section is tension controlled and ACI Section 11.8.1.1(b) is
satisfied.
In accordance with ACI Section 21.2.1, the strength reduction factor is given by
f
5 0.9
Hence, the design moment strength is given by
fMn
5 0.9 3 48.46
5 43.61 kip-in
. Mu . . . satisfactory
Axial stress
The total factored axial load at midheight of the wall is
Pu
5 1222 lb
The factored axial load stress at midheight of the wall is
0.06fc9Ag 5 0.06 3 4000 3 12 3 6
5 17,280 lb
. Pu
The required limitation of ACI Section 11.8.1.1(d) is satisfied.
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Cracking moment
The moment of inertia of the concrete section about its centroidal axis, neglecting reinforcement, is
Ig
5 bt3/12
5 12 3 63/12
5 216 in4
The modulus of rupture is given by ACI Equation (19.2.3.1) as
fr
5 7.5l( fc9)0.5
5 7.5 3 1.0(4000)0.5
5 474 lb/in2
The distance from the centroidal axis of the gross section to the extreme fiber in tension is obtained
from Figure 4-26 as
yt
5d
5 3 in
The cracking moment is given by ACI Equation (24.2.3.5b) as
Mcr
5 fr Ig /yt
5 474 3 216/3000
5 34.13 kip-in
, fMn
The required limitation of ACI Section 11.8.1(c) is satisfied.
Service level deflection
Where the effects of dead load and seismic load are additive, the applicable loading case for service
load design is given by ACI Section R11.8.4 as
U
5 D 1 0.5L 1 0.7E
5 (1 1 0.2 3 0.7SDS)D 1 1.0L 1 0.7rQE
Considering a 1-foot width of panel, the service level axial load from the roof, not including roof live
load, is given by
Pr
5 (1 1 0.14SDS)D 1 0
5 (1 1 0.14 3 0.826)220 1 0
5 245 lb
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The service level axial load from the weight of the wall above the midheight section is
Pw
5 (1 1 0.14SDS)D
5 (1 1 0.14 3 0.826)(150 3 9 3 6/12)
5 753 lb
The total service level axial load at the midheight section is
Ps
5 Pr 1 Pw
5 998 lb
The service level seismic lateral force on the wall is
w
5 0.7wu
5 0.7 3 24.78
5 17.35 psf
The corresponding factored bending moment at the midheight section is
Mw
5wlc2/8
5 17.35 3 172/8
5 627 lb-ft
The eccentricity of the roof load about the wall centerline is
e
5 7 in
The service level bending moment at the midheight section caused by the eccentricity is
Me
5 Pr e/2
5 245 3 7/24
5 71 lb-ft
The total service level wall moment at the midheight section due to seismic lateral force and the eccentric roof load is
Msa
5 Mw 1 Me
5 627 1 71
5 698 lb-ft
5 8.38 kip-in
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The maximum allowable deflection under service loads is given by ACI Section 14.8.4 as
Ds
5 lc /150
5 17 3 12/150
5 1.36 in
The corresponding moment at this deflection due to service loads including P-delta effects is
Ma
5 Msa 1 Ps Ds
5 8.38 1 0.998 3 1.36
5 9.74 kip-in
, 2Mcr /3 . . . ACI Table 11.8.4.1(a) applies
The midheight deflection, corresponding to the moment, Mcr , is given by ACI Equation (11.8.4.3a) as
Dcr
5 5Mcr lc2/48Ec Ig
5 5 3 34.13(17 3 12)2/(48 3 3605 3 216)
5 0.19 in
From ACI Table 11.8.4.1(a), the deflection for a P-delta moment of Ps Ds is
D
5 Ma Dcr /Mcr
5 9.74 3 0.19/34.13
5 0.054 in
, Ds
Hence, the midheight deflection corresponding to the actual service level moment is less than the maximum allowed and the section is satisfactory.
4.4 Anchorage in concrete
The development of an anchor rod in a concrete wall or footing is determined by the methods given
in ACI Chapter 17, as amended by IBC Section 1905.1.8. Embedment failure modes13 in the concrete
element that must be considered include concrete breakout, pullout, side-face blowout, shear breakout,
concrete pryout, and splitting. These are illustrated in Figure 4-27. Failure modes in the steel anchor
are tensile failure and shear failure.
Cast-in anchors and post-installed anchors have different design requirements. This section covers the
requirements of cast-in headed studs, headed bolts, and hooked bolts.
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In designing anchors for structures assigned to seismic design categories C through F for seismic
loads, the desired failure mode is yielding of the ductile steel element of the anchor. This is achieved
by the provisions of ACI Section 17.2.3 as modified by IBC Section 1905.1.8. The modified procedure
is as follows:
•
In accordance with ACI Section 17.2.3.4.1, where the tensile component of the strength-level
earthquake force is equal to or less than 20 percent of the total factored tensile force, the anchor
design strength is taken as equal to the design strength of an anchor for nonseismic loads.
•
In accordance with ACI Section 17.2.3.4.2, where the tensile component of the strength-level
earthquake force exceeds 20 percent of the total factored tensile force, the anchor is designed
in accordance with ACI Section 17.2.3.4.3 and the design strength is determined from ACI
Section 17.2.3.4.4.
•
Where anchors are designed for the wall anchorage force given by ASCE 7 Section 12.11.2.1,
which is Fp 5 0.4SDS ka IeWp , IBC Section 1905.1.8 adds the following exception to ACI Section 17.2.3.4.2:
Anchors designed to resist wall out-of-plane forces with design strengths equal to or greater
than the force determined in accordance with ASCE 7 Equation 12.11-1 or 12.14-10 shall
be deemed to satisfy Section 17.2.3.4.3(d).
•
ACI Section 17.2.3.4.3(d), as modified by IBC Section 1905.1.8, is:
The anchor or group of anchors shall be designed for the maximum tension obtained from
design load combinations that include E, with E increased by W0. The anchor design tensile
strength shall be calculated from 17.2.3.4.4.
•
Then, in accordance with ACI Section 17.2.3.4.4, the design strength for concrete breakout,
pullout, and side-face blowout is reduced by 25 percent.
•
Where anchor reinforcement is provided in accordance with ACI Section 17.4.2.9, the design
strength of the anchor reinforcement is used instead of the concrete breakout strength.
An exemption to ACI Section 17.2.3.5.3 is provided by IBC Section 1905.1.8 for the concrete breakout strength in shear parallel to an edge of anchor bolts attaching wood sill plates of light-frame wood
structures to foundations or foundation stem walls, provided all of the following are satisfied:
1. The allowable in-plane shear strength of the anchor is determined in accordance with ANSI/
AWC NDS14 Table 12E for lateral design values parallel to grain.
2. The maximum anchor nominal diameter is 5⁄8 inch.
3. Anchor bolts are embedded into concrete a minimum of 7 inches.
4. Anchor bolts are located a minimum of 13⁄4 inches from the edge of the concrete parallel to the
length of the wood sill plate.
5. Anchor bolts are located a minimum of 15 anchor diameters from the edge of the concrete perpendicular to the length of the wood sill plate.
6. The sill plate is of 2- or 3-inch nominal thickness.
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The concrete breakout strength in shear need not be computed for this condition.
An exemption to ACI Section 17.2.3.5.3 is provided by IBC 1905.1.8 for the concrete breakout strength
in shear parallel to an edge of anchor bolts attaching cold-formed steel track of light-frame construction to foundations or foundation stem walls, provided all of the following are satisfied:
1. The maximum anchor nominal diameter is 5⁄8 inch.
2. Anchor bolts are embedded into concrete a minimum of 7 inches.
3. Anchor bolts are located a minimum of 13⁄4 inches from the edge of the concrete parallel to the
length of the track.
4. Anchor bolts are located a minimum of 15 anchor diameters from the edge of the concrete perpendicular to the length of the track.
5. The track is 33 to 68 mil designation thickness.
The concrete breakout strength in shear need not be computed for this condition.
The allowable in-plane shear strength of exempt anchors, parallel to the edge of concrete, shall be
permitted to be determined in accordance with AISI S100 Section E3.3.1.
In light-frame construction of bearing or nonbearing walls, an exemption to ACI Section 17.2.3.5.3(a)
through (c) is provided by IBC Section 1905.1.8 for the shear strength of concrete anchor bolts, of
1-inch diameter or less, attaching sill plate or steel track to foundations, when the design strength of
the anchors is determined in accordance with ACI Section 17.5.2.1(c). This permits the adoption of
a nominal shear strength for the bolts, parallel to an edge, of twice the value given by ACI Equations
(17.5.2.1a) and (17.5.2.1b).
Figure 4-27 Concrete embedment failure modes
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4.4.1 Design requirements for tensile loading
Concrete breakout strength
For a single anchor remote from the edges of the concrete element, as shown in Figure 4-28, the failure surface in the concrete is assumed to be a pyramid with its apex at the centerline of the rod at the
bearing contact surface of the head, with the failure surface radiating outward to the surface at a slope
of 1 to 1.5. The projected area of this failure surface on the concrete outer surface is given by ACI
Equation (17.4.2.1c) as
where:
ANco
5 9h2ef
hef
5 effective anchor embedment depth
5 depth from the concrete outer surface to the bearing contact surface of the
head
Figure 4-28 Concrete failure surface for a single anchor
Failure occurs, and concrete breakout results, when the tensile stress on the failure surface exceeds the
tensile strength of the concrete. The nominal concrete breakout strength for a single cast-in anchor in
tension is given by ACI Equation (17.4.2.1a) as
where:
Ncb
5 ANcyed,Nyc,Nycp,N Nb/ANco
ANc
5 projected area of the failure surface for a single anchor as limited by
adjacent free edges
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yed,N
449
5 modification factor for edge effects
5 1.0 for a minimum edge distance of 1.5hef from ACI Section 17.4.2.5
5 0.7 1 0.3ca,min /1.5hef for a minimum edge distance , 1.5hef
yc,N
5 modification factor for cracked concrete from ACI Section 17.4.2.6
5 1.0 for concrete that is cracked at service load levels
5 1.25 for concrete that is uncracked at service load levels
ycp,N
5 modification factor for post-installed anchors from ACI Section 17.4.2.7
5 1.0 for cast-in anchor
Nb
5 basic concrete breakout strength in tension of a single cast-in anchor in
cracked concrete as defined in ACI Section 17.4.2.2
5 16la( fc9)0.5(hef )5/3 for 11 in , hef , 25 in . . . from ACI Equation (17.4.2.2b)
5 24la( fc9)0.5(hef )1.5 for all other values of hef . . . from ACI Equation (17.4.2.2a)
la
5 modification factor for lightweight concrete
5 1.0 for normalweight concrete with cast-in anchor
Where anchor rods are spaced closer than three times their embedment depth, the failure surfaces of
adjacent anchors intersect. The failure surface for such an anchor group is determined by projecting
the failure surface outward from a line through the anchor heads, as shown in Figure 4-29. The projected area of this failure surface on the concrete outer surface, when remote from edges, is given by
ANc
5 3hef (b 1 3hef ) . . . for a single row of anchors
5 (a 1 3hef )(b 1 3hef ) . . . for multiple rows of anchors
≤ nANco
where:
n
5 number of anchors in the group
b
5 distance between outside anchors in the group
a
5 distance between outside anchors in the group perpendicular to b
The nominal concrete breakout strength for a cast-in anchor group in tension is given by ACI Equation
(17.4.2.1b) as
where:
Ncbg
5 ANcyec,Nyed,Nyc,Nycp,N Nb /ANco
ANc
5 projected area of the failure surface for the anchor group as limited by
adjacent free edges
yec,N
5 modification factor for eccentrically loaded anchor groups
5 1.0 for concentrically loaded groups from ACI Section 17.4.2.4
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1.5
1
b+
b
1.5
a
a+
1
b+
b
Figure 4-29 Concrete failure surface for an anchor group
The strength reduction factor for an anchor or anchor group governed by concrete breakout, side-face
blowout, pullout, or pryout strength is given by ACI Section 17.3.3 as
f
5 0.75 . . . where supplementary reinforcement is provided to tie the concrete
failure prism into the structural member
5 0.70 . . . where supplementary reinforcement is not provided
Steel strength of anchor
The design strength of a ductile anchor rod in tension is given by ACI Section 17.3.3 and ACI Equation
(17.4.1.2) as
where:
fNsa
5 0.75Ase,N futa
Ase,N
5 effective cross-sectional area of anchor rod in tension
futa
5 specified tensile strength of the anchor steel
≤ 1.9fya
≤ 125,000 psi
fya
5 specified yield strength of anchor steel
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Ductile bolts include ASTM A307 grade A with a minimum specified tensile strength of 60 kips per
square inch.
The design strength of a brittle steel anchor rod in tension is given by ACI Section 17.3.3 and ACI
Equation (17.4.1.2) as
fNsa
5 0.65Ase,N futa
Pullout strength of anchor in tension
The nominal concrete pullout strength for a single anchor in tension is given by ACI Equation (17.4.3.1)
as
where:
Npn
5 yc,P Np
Np
5 8Abrg fc9 . . . headed bolt or stud
5 0.9eh da fc9 . . . hooked bolt, where 3da , eh , 4.5da
Abrg
5 net bearing area of bolt or stud head
fc9
5 concrete compressive strength
eh
5 distance from outer tip of hooked bolt to inner surface of the shaft
da
5 outside diameter of bolt
yc,P
5 modification factor for cracked concrete from ACI Section 17.4.3.6
5 1.0 for concrete that is cracked at service load levels
5 1.4 for concrete that is uncracked at service load levels
Side-face blowout strength of anchor in tension
Side-face blowout is caused by spalling of the concrete surface adjacent to the head of an anchor that is
close to the face of the concrete. The nominal concrete blowout strength for a single anchor in tension,
with hef . 2.5ca1, is given by ACI Equation (17.4.4.1) as
Nsb
5 160ca1(Abrg fc9)0.5 l a . . . for ca2 ≥ 3ca1
5 (1/4 1 ca2 /4ca1)160ca1(Abrg fc9)0.5 l a . . . for 1.0 ≤ ca2 /ca1 ≤ 3
where:
ca1
5 minimum distance from center of anchor shaft to edge of concrete
ca2
5 distance from center of anchor shaft to edge of concrete perpendicular to ca1
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4.4.2 Design requirements for shear loading
Concrete breakout strength
For a single anchor remote from edges perpendicular to the shear force, as shown in Figure 4-30, the
failure surface in the concrete is assumed to be a half pyramid with a side length of 3c1 and a depth
of c1. The projected area of this failure surface on the concrete outer surface is given by ACI Section
R17.5.2.1 as
where:
AVco
5 4.5c2a1
ca1
5 distance from center of anchor rod to edge of concrete in the direction of
the shear force
ca1
ca1
1.5 ca1
V
1.5 ca1
1.5ca1
1.5
1.0
V
1.5 ca1
3c
a1
ca1
Figure 4-30 Concrete failure surface for a single anchor
The nominal concrete breakout strength for a single cast-in anchor in shear is given by ACI Equation
(17.5.2.1a) as
where:
Vcb
5 AVcyed,V yc,V yh,V Vb /AVco
AVc
5 projected area of the failure surface for a single anchor as limited by corner
influences and member thickness
yed,V
5 modification factor for edge effects
5 1.0 for ca2 ≥ 1.5ca1 from ACI Section 17.5.2.6 where ca2 5 distance from
center of anchor rod to edge of concrete normal to direction of shear force
5 0.7 1 0.3ca2/1.5ca1 . . . for ca2 , 1.5ca1
yc,V
5 modification factor for cracked concrete from ACI Section 17.5.2.7
5 1.0 for concrete that is cracked with no supplementary reinforcement
5 1.4 for concrete that is cracked with a #4 bar or greater between the anchor
and the edge and with the reinforcement enclosed in stirrups
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yh,V
453
5 modification factor for anchors located in a member with ha , 1.5ca1
5 (1.5ca1 /ha)0.5 . . . from ACI Equation (17.5.2.8)
≥ 1.0
ha
5 thickness of member
Vb
5 basic concrete breakout strength in shear of a single anchor in cracked
concrete as defined in ACI Section 17.5.2.2
5 7(le /da)0.2(da)0.5( fc9)0.5 la(ca1)1.5 . . . from ACI Equation (17.5.2.2a)
≤ 9la( fc9)0.5(ca1)1.5
where:
da
5 diameter of anchor
le
5 load-bearing length of anchor for shear
5 hef for anchors of constant thickness
≤ 8da
The nominal concrete breakout strength for an anchor group in shear, as shown in Figure 4-31, is given
by ACI Equation (17.5.2.1b) as
where:
Vcbg
5 AVcyec,V yed,V yc,V yh,V Vb/AVco
AVc
5 projected area of the failure surface for the anchor group as limited by
corner influences and member thickness
yec,V
5 modification factor for eccentrically loaded anchor groups
5 1.0 for concentrically loaded groups from ACI Section 17.5.2.5
ca1
V
ha
s
1.5
1.0
3c
a1 + s
Figure 4-31 Concrete failure surface for an anchor group in shear
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The strength reduction factor for an anchor or anchor group is given by ACI Section 17.3.3 as
f
5 0.75 . . . where supplementary reinforcement is provided to tie the concrete
failure prism into the structural member
5 0.70 . . . where supplementary reinforcement is not provided
Steel strength of anchor
The design strength of an anchor rod in shear is given by ACI Section 17.5.1.2 as
fVsa
5 0.65Ase,V futa . . . headed stud
5 0.65 3 0.6Ase,V futa . . . headed bolt and hooked bolt
where:
Ase,V
5 effective cross-sectional area of anchor rod in shear
Ductile bolts include ASTM A307 grade A with a minimum specified tensile strength of 60 kips per
square inch.
where:
futa
5 specified tensile strength of anchor rod
≤ 125 ksi
≤ 1.9fy
fya
5 specified yield strength of anchor
Concrete pryout strength of anchor in shear
The nominal concrete pryout strength for a single anchor in shear is given by ACI Equation (17.5.3.1a)
as
where:
Vcp
5 kcp Ncb
kcp
5 1.0 . . . for hef , 2.5 inches
5 2.0 . . . for hef ≥ 2.5 inches
Ncb
5 nominal concrete breakout strength for a single anchor in tension as given
by ACI Equation (17.4.2.1a)
5 ANcyed,N yc,N ycp,N Nb /ANco
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4.4.3 Interaction of tensile and shear forces
Where Vua . 0.2fVn and Nua . 0.2fNn, the interaction expression of ACI Equation (17.6.3) applies,
and
Nua /fNn 1 Vua /fVn ≤ 1.2
Where Vua , 0.2fVn , shear effects are neglected, the full design strength in tension is permitted, and
fNn
≥ Nua
Where Nua , 0.2fNn , tension effects are neglected, the full design strength in shear is permitted, and
fVn
≥ Vua
Example 4-6
Check the design of the wall anchorage of the tilt-up concrete structure shown in Figure 4-32. The
structure has an importance factor of Ie 5 1.0; a redundancy factor of r 5 1.0; and the 5-percent
damped, design spectral response acceleration for a period of 0.2 second is SDS 5 0.826g. The building
is assigned to seismic design category D. The normalweight concrete cylinder strength is 4000 psi. The
roof diaphragm may be considered flexible and spans Lf 5 50 feet. Anchor bolts are 1⁄2-inch-diameter
hex head ASTM A307 grade C with a minimum specified tensile strength of 60 ksi. Anchorages are
located at sa 5 8-foot centers and anchor bolts are not torqued. Supplementary reinforcement is not
provided and the concrete may be considered cracked. The weight of the wall is w 5 75 lb/ft2. Shear
on the attachment is negligible and the full design strength in tension is permitted.
ca1
Figure 4-32 Details for Example 4-6
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Solution
Anchor bolts are located 6 inches from the top of the wall, and edge distance and spacing exceed the
minimum values specified in ACI Section 17.7. Hence, side-face blowout and splitting need not be
considered.
The anchor bolts are ductile, and from ACI Section 17.3.3, the strength reduction factors are
f
5 0.75 . . . for tension on a ductile anchor bolt
f
5 0.70 . . . for concrete breakout or pullout without supplemental
reinforcement
The wall anchor force is determined from ASCE 7 Section 12.11.2.1. Hence, in accordance with
IBC-modified ACI Sections 17.2.3.4.2 and 17.2.3.4.3, the strength of the anchorage for concrete
breakout and pullout is reduced by 25 percent.
The properties of the 1⁄2-inch-diameter hex bolts are13
Ase
5 effective area
5 0.142 in2
Abrg
5 bearing area of head
5 0.291 in2
futa
5 minimum specified tensile strength
5 60 ksi
Applied loads on the anchor
The weight of wall tributary to an anchor is obtained from Figure 4-32 as
Wp
5 sawH 2/2lc
5 8 3 75 3 24.52/[1000(2 3 24)]
5 7.5 kips
The span in feet of the flexible diaphragm is
Lf
5 50 ft
The amplification factor for diaphragm flexibility is
ka
5 1.0 1 Lf /100
5 1.0 1 50/100
5 1.5
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The seismic force on an anchor is given by ASCE 7 Equation (12.11-1) as
Fp
5 0.4SDS ka IeWp
5 0.4 3 0.826 3 1.5 3 1.0 3 7.5
5 3.72 kips . . . governs
The minimum permissible force on one anchor is
Fmin
5 0.2ka IeWp
5 0.2 3 1.5 3 1.0 3 7.5
5 2.25 kips . . . does not govern
, Fp
In accordance with ASCE Section 12.11.2.2.2, steel elements of the anchor system other than anchor
bolts must be designed for the strength design force given by
Tu
5 1.4Fp
5 1.4 3 3.72
5 5.21 kips
Concrete breakout strength in tension
As shown in Figure 4-32, the projection of the failure surface for the anchor group on the concrete
outer surface has an area of
ANc
5 (1.5hef 1 ca1)(s 1 3hef)
5 (1.5 3 5.5 1 6)(6.5 1 3 3 5.5)
5 328 in2
, 2ANco . . . satisfies ACI Section 17.4.2.1
The projection of the failure surface for a single anchor on the concrete outer surface has an area of
ANco
5 9h2ef
5 9 3 5.52
5 272 in2
The basic concrete breakout strength in tension of a single anchor in cracked concrete, as defined in
ACI Section 17.4.2.2, is
Nb
5 24( fc9)0.5(hef)1.5
5 24(4000)0.5(5.5)1.5/1000
5 19.58 kips
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yed,N
5 0.7 1 0.3ca1,min /1.5hef
5 0.7 1 0.3 3 6/8.25
5 0.92
yc,N
5 yec,N 5 ycp,N 5 1.0
The design concrete breakout strength for the anchor group for seismic loading is given by ACI Section 17.2.3.4.4 and ACI Equation (17.4.2.1b) as
0.75fNcbg 5 0.75fANcyec,Nyed,Nyc,Nycp,NNb/ANco
5 0.75 3 0.70 3 328 3 1.0 3 0.92 3 1.0 3 1.0 3 19.58/272
5 11.40 kips
. Fp . . . satisfactory
Pullout strength of anchor in tension
The design concrete pullout strength for a single anchor in tension is given by ACI Section 17.2.3.4.4
and ACI Equation (17.4.3.1) as
0.75fNpn 5 0.75fyc,PNp
where:
Np
5 8Abrg fc9 . . . for a headed bolt
5 8 3 0.291 3 4
5 9.31 kips
yc,P
5 modification factor for pullout strength of anchors for cracked concrete
from ACI Section 17.4.3.6
5 1.0 . . . for concrete that is cracked at service load levels
hence:
0.75fNpn 5 0.75 3 0.7 3 1.0 3 9.31
5 4.89 kips
The concrete pullout strength of the two bolts is
2Npn
5 2 3 4.89
5 9.78 kips
. Fp . . . satisfactory
Strength of anchor rods in tension
The steel strength is based on the effective area of the threaded rod. For a 1⁄2-inch-diameter threaded
rod, the effective area is13
Ase,N
5 0.142 in2
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459
The minimum specified tensile strength of the two ASTM A307 grade A anchor rods is 60 kips per
square inch. Hence, the design strength of the two 1⁄2-inch-diameter ductile anchor rods is given by ACI
Equation (17.4.1.2) as
2fNsa 5 2fAse,N futa
5 2 3 0.75 3 0.142 3 60
5 12.78 kips
. Fp . . . satisfactory
References
1. American Society of Civil Engineers. Minimum Design Loads and Associated Criteria for Buildings and Other Structures: ASCE 7-16. Reston, VA, 2016.
2. International Code Council. 2018 International Building Code. Washington, DC, 2018.
3. American Concrete Institute. Building Code Requirements for Structural Concrete and Commentary: (318-14). Farmington Hills, MI, 2014.
4. American Concrete Institute. Reinforced Concrete Design Handbook SP-17(14). Farmington
Hills, MI, 2014.
5. Structure Point LLC. Concrete Design Software: spColumn. Skokie, IL, 2014.
6. Williams, A. Design of Reinforced Concrete Structures. Dearborn Press. Chicago, IL, 2012.
7. Structural Engineering Association of California. “Reinforced Concrete Structures.” SEAOC Blue
Book: Seismic Design Recommendations. SEAOC. Sacramento, CA, 2009.
8. Fanella, D. A. Design of Low-Rise Concrete Buildings for Earthquake Forces. ICC. Washington,
DC, 2009.
9. Fanella, D. A. “Special Moment Frames.” Structural Engineer, 3, No. 8, (28–33). September
2002.
10. Fanella, D. A. “Structural Walls.” Structural Engineer, 3, No. 10, (32–35). November 2002.
11. American Concrete Institute and Structural Engineers Association of Southern California. Report
of the Task Committee on Slender Walls. Los Angeles, CA, 1982.
12. Simpson Strong-Tie Company Inc. Wood Construction Connectors. Catalogue C-C-2017. Pleasanton, CA, 2017.
13. Cook, R. A. Strength Design of Anchorage to Concrete. Portland Cement Association. Skokie, IL,
1999.
14. American Wood Council. National Design Specification for Wood Construction. ANSI/AWC
NDS-2018. Leesburg, VA, 2018.
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5
Seismic Design of Wood Structures
Nomenclature
A
area
in2
Ao
total area of openings in a perforated shear wall
ft2
b
length of a shear wall or shear wall segment
ft
bs
length of a shear wall or shear wall segment for determining aspect ratio
ft
C
compression chord force
lb
Co
shear capacity adjustment factor
–
E
modulus of elasticity
psi
G
specific gravity
–
Ga
apparent shear stiffness from nail slip and panel shear deformation
kips/in
h
height of a shear wall or shear wall segment
ft
Ke
effective length factor
–
L
dimension of a diaphragm perpendicular to the application of force
ft
SLi
sum of perforated shear wall segment lengths
ft
R
response modification coefficient
–
t
uniform uplift force
lb/ft
T
tension chord force
lb
v
induced unit shear
lb/ft
V
seismic base shear
lb
V
shear force
lb
W
dimension of a diaphragm parallel to the application of force
ft
x
distance from chord splice to nearest support
ft
Seismic and Wind Forces: Structural Design Examples
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Seismic Design of Wood Structures
Symbols
Da
total vertical elongation of wall anchorage system
in
Dc
diaphragm chord splice slip in diaphragm
in
ddia
maximum diaphragm deflection determined by elastic analysis
in
W0
system overstrength factor
–
5.1 General provisions
5.1.1 Building classification
In accordance with IBC1 Section 602, only buildings of construction classification III, IV, or V may be
constructed of wood. For Type III construction, fire-retardant-treated wood framing complying with
IBC Section 2303.2 is required for exterior walls. For Type IV construction, heavy timber construction
is required. Type V construction may utilize any materials permitted by the code. Determination of the
applicable construction classification depends on building height and area limitations, as specified in
IBC Section 503, and on the intended use of the building, as specified in IBC Section 302.
5.1.2 Design methodology
Wood structures may be designed using the allowable stress design method, the load and resistance
factor design method, or the conventional light-frame construction provisions. The load combinations
applicable to the allowable stress design method are specified in ASCE 72 Section 2.4. The load combinations applicable to the load and resistance factor design (LRFD) method are specified in ASCE 7
Section 2.3. The conventional light-frame construction provisions of IBC Section 2308 are prescriptive requirements based on generally accepted practice and are restricted to light wood-frame building
construction with a maximum height of three stories. The method is not permitted, in accordance with
IBC Section 2308.2, for buildings in areas where the basic wind speed exceeds 130 miles per hour, or
for buildings with live load exceeding 40 pounds per square foot. The method is limited by IBC Table
2308.2.1 to buildings with a maximum of two stories in seismic design category C. The method is
limited by IBC Table 2308.2.1 to buildings with a maximum of one story in seismic design categories
D and E.
For the allowable stress design (ASD) method, IBC Section 2306.1 specifies the adoption of the
National Design Specification for Wood Construction3, 4 and Special Design Provisions for Wind and
Seismic.5 In the ASD method, the calculated stress in an element, due to the service level loads, must
not exceed the prescribed allowable stress.
For the LRFD method, IBC Section 2307.1 specifies the adoption of the National Design Specification
for Wood Construction and Special Design Provisions for Wind and Seismic. The LRFD method is
based on limit state principles to determine the maximum load-carrying capacity of a structure. A uniform level of reliability is achieved for all structures by ensuring that the nominal resistance capacity
Seismic and Wind Forces: Structural Design Examples
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of an element multiplied by the appropriate reduction factor is not less than the demand produced by
the factored loads.
5.2 Lateral-force-resisting system
5.2.1 Lateral load path
Figure 5-1 shows the lateral load path in a one-story structure and indicates the individual components
and fastening details required. A continuous load path is necessary to transfer the lateral seismic and
wind forces from the upper portion of the structure to the foundations. Vertical and horizontal structural assemblies are used to provide a lateral-force-resisting system and the assemblies are secured and
interconnected by fasteners.
6
6
6
Figure 5-1 Lateral load path
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Seismic Design of Wood Structures
For conventional light-frame construction, IBC Table 2304.10.1 provides prescribed fastening details
and these are the minimum required for all lightweight wood-frame construction. As required by IBC
Section 2304.10.2, sheathing fasteners shall be driven flush with the surface of the sheathing. Protruding or overdriven nails do not provide the intended shear capacity. As specified by IBC Section
2304.10.6, fabricated fasteners shall be formed from galvanized steel or other approved corrosionresistant material with a minimum thickness of 0.0329 inch.
The horizontal, or nearly horizontal, structural assemblies consist of the roof and floor diaphragms that
transmit lateral forces to the shear walls. Diaphragms are composed of wood structural panels fixed
to wood framing members. Wood structural panels are defined in IBC Section 202 as consisting of
plywood, oriented strand board, and composite panels.
The vertical structural assemblies consist of shear walls and are also composed of wood structural
panels fixed to wood framing members. The imposed lateral forces produce overturning forces and
racking of the shear walls.
5.2.2 Connection details
The connection details required in a two-story, wood-frame structure to ensure the transfer of lateral
forces from the roof and second floor to the foundation are shown in Figure 5-2. The roof diaphragm
shear is transferred to end blocking between framing joists and the boundary nail spacing is obtained
from SDPWS5 Tables 4.2A–4.2D. The shear is transferred from the end blocking to the top plate of
the second-story shear wall, either by proprietary framing anchors or by means of horizontal wood
blocking, as shown. The alternative method shown, using toenails, is not recommended because of
the possibility of splitting caused by close nail spacing or shrinking of the end blocking. In seismic
design categories D, E, and F, SDPWS Section 4.1.7 prohibits the use of toenails where the lateral
force exceeds 150 pounds per linear foot for ASD and 205 pounds per linear foot for LRFD. The nail
spacing required at the shear wall edge is obtained from SDPWS Tables 4.3A–4.3D. The total shear
force at the bottom of the second-story shear wall is due to the self-weight of the shear wall plus the
roof diaphragm shear. This is transferred through the bottom plate to the second-floor end blocking
by nailing. The shear from the second-floor diaphragm is also transferred to the end blocking by the
diaphragm boundary nailing. Similarly, the accumulated forces are transferred to the sill plate of the
first-story shear wall. Finally, anchor bolts transfer the force in the sill plate to the concrete foundation.
To resist the uplift of the second-story shear wall, the end posts of the second story and the first-story
shear walls are tied together with steel straps. Alternatively, the end posts of the shear walls may be
tied together by a steel rod connected to hold-down supports either bolted or nailed to the end posts.
Seismic and Wind Forces: Structural Design Examples
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6
Figure 5-2 Lateral and vertical force transfer
Seismic and Wind Forces: Structural Design Examples
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5.3 Diaphragms
5.3.1 General requirements
5.3.1.1 Diaphragm function
A plywood diaphragm acts as a horizontal deep beam to collect and transfer lateral forces to the shear
walls. Structural wood panels form the beam web to resist shear force, purlins act as web stiffeners,
and the boundary members normal to the load form the flanges to resist flexural effects.7, 8 Shear
stresses are assumed uniformly distributed across the depth of the diaphragm. The boundary members,
acting as the flange or chord of the diaphragm, may consist of the double top plate of a wood-frame
shear wall, a steel or wood ledger on the inside face of a concrete wall, or steel reinforcement in a
masonry or concrete wall. The contribution of the plywood sheathing to the flexural capacity of the
deep beam is neglected and the chords are assumed to resist the total applied moment by developing
axial forces that provide a couple equal and opposite to the moment. As shown in Figure 5-3, the axial
force in a chord is given by
Fc
5 Ft
5 MD /BD
where:
MD
5 bending moment in the diaphragm
5 wL2/8
BD
5 distance between chord centers
 depth of diaphragm
w
Figure 5-3 Diaphragm action
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5.3.1.2 Blocked diaphragms
The terms blocked diaphragm and unblocked diaphragm are defined in SDPWS Section 2.2 and
shown in Figure 5-4. When all edges of the structural wood panels are supported by and are nailed to
framing members, the diaphragm is termed blocked. This increases the strength of the diaphragm and
may be achieved by spacing purlins and sub-purlins at suitable centers so as to support the edges of a
4 3 8-foot panel. When this framing arrangement is not possible, 23 flat-wise blocking pieces may
be nailed or clipped to the framing members, as shown in Figure 5-4. Connecting the structural wood
panels to framing members around the entire perimeter of the panel prevents buckling of the panel
and provides higher allowable design loads than unblocked diaphragms. SDPWS Table 4.2.4 specifies
the maximum aspect ratio for unblocked wood structural panel diaphragms as 3:1 and for blocked
diaphragms, 4:1.
Figure 5-4 Diaphragm construction
5.3.2 Diaphragm strength
The strength of wood structural panels is controlled by the shear strength of the panel, by nail heads
pulling through the panel face, by nails splitting panel edges, and by buckling of the panel. The nominal unit shear capacity of a plywood diaphragm depends on the sheathing thickness, grade, and orientation; the width of the framing members; the support of the panel edges; and the nail spacing, type,
and penetration. The nominal unit shear capacities for blocked wood structural panel diaphragms are
given in SDPWS Table 4.2A. Separate values are provided for wind or seismic loads, and values for
wind loads are 40-percent higher than for seismic loads. For cases not covered by this table, additional
nominal unit shear capacities may be obtained from SDPWS Table 4.2B for high load diaphragms for
wind or seismic loading.
For the ASD method, allowable unit shear capacity is determined by dividing the tabulated nominal
unit shear capacity by a reduction factor of 2. For the LRFD method, the design unit shear capacity is
determined by multiplying the tabulated nominal unit shear capacity by a resistance factor, fD, of 0.8.
Seismic and Wind Forces: Structural Design Examples
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Seismic Design of Wood Structures
In accordance with SDPWS Section 4.2.7.1.1, the following construction requirements are necessary
for wood structural panel diaphragms:
•
minimum size of panel is 4 3 8 feet, except at boundaries and changes in framing where a
minimum panel dimension of 24 inches is required unless all edges of the undersized panels
are supported by and fastened to framing members or blocking
•
maximum spacing of nails at panel edges is 6 inches
•
where the spacing of supporting framing is less than 48 inches, the maximum nail spacing
along intermediate framing members is 12 inches; otherwise, the maximum spacing is 6 inches
•
nails along intermediate framing members and blocking must be the same size as specified for
panel edge nailing
•
nails are located at least 3⁄8 inch from panel edges
•
a 3-inch nominal or greater framing member is required at abutting panel edges where nails are
spaced at 21⁄2 inches or less or where 10d nails with a penetration exceeding 11⁄2 inches and a
spacing of 3 inches or less are used; otherwise, 2-inch nominal framing members may be used
•
nails at panel edges are staggered where nails are spaced at 21⁄2 inches or less or where 10d nails
with a penetration exceeding 11⁄2 inches and a spacing of 3 inches or less are used
High load diaphragms develop their additional strength by the use of multiple rows of fasteners at
adjoining panel edges and boundaries. Nominal 3- or 4-inch-wide framing members, as specified in
SDPWS Table 4.2B, are required at these locations in order to prevent splitting of the framing members. Boundary and panel edge nailing details are shown in Figure 5-5.
Panel joint
S = tabulated nail spacing
Top plate
5 or 7 equal
spaces
21/2 – 31/2
Framing
1
/2
/8 min
3
/8 min
1
/2
3
21/2
S
Boundary nailing
S
Adjoining panels
Figure 5-5 High load diaphragm nailing details
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The more stringent requirements for high load blocked wood structural panel diaphragms are given in
SDPWS Section 4.2.7.1.2 and these are:
•
minimum size of panel is 4 3 8 feet, except at boundaries and changes in framing where a
minimum panel dimension of 24 inches is required unless all edges of the undersized panels
are supported by and fastened to framing members or blocking
•
maximum spacing of nails at panel edges is 6 inches
•
where the spacing of supporting framing is 32 inches or less, the maximum nail spacing along
intermediate framing members is 12 inches; otherwise, the maximum spacing is 6 inches
•
nails along intermediate framing members and blocking must be the same size as specified for
panel edge nailing
•
nails are located at least 3⁄8 inch from panel edges but not less than the distances shown in Figure 5-5
•
a 3-inch nominal or greater framing member is required at diaphragm boundaries and abutting
panel edges; otherwise, 2-inch nominal framing members may be used
•
nails at diaphragm boundary edges are equally spaced and staggered where nails are spaced at
3 inches or less
Nominal unit shear capacity values are provided in SDPWS Table 4.2C for unblocked wood structural
panels. SDPWS Table 4.2D provides nominal unit shear capacity values for diagonal, double diagonal,
and horizontal lumber sheathing.
The shear stress is assumed uniform over the depth of the diaphragm and the unit shear stress in a
diaphragm is given by
where:
q
5 Q/BD
Q
5 shear force at the section considered
BD
5 depth of diaphragm
For a given plywood panel grade, thickness, orientation, and edge support, the required nail spacing
may be obtained from SDPWS Table 4.2A. Since the shear decreases in a uniformly loaded diaphragm
from the end supports to midspan, the nail spacing may be progressively increased. The strength of a
diaphragm may be increased by increasing the grade and thickness of the plywood, reducing nail spacing, increasing the width of framing members, blocking all panel edges, and staggering panel edges in
the direction of the applied force.
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Seismic Design of Wood Structures
The nominal unit shear capacities given in SDPWS Table 4.2A are based on the use of common nails.
The tabulated shear capacities are also based on framing of Douglas Fir-Larch or Southern Pine. For
framing of other species, these values are multiplied by the adjustment factor
AF
5 1 2 (0.5 2 G)
≤ 1.0
where:
G
5 specific gravity of the framing lumber
Example 5-1
The tilt-up concrete industrial building shown in Figure 5-6 is located on a site with a site classification
D. The seismic importance factor is Ie 5 1.0 and the seismic design category is D. The design spectral
response accelerations are SDS 5 0.826g and SD1 5 0.469g. The weight of the roof is 19 pounds per
square foot and the weight of the concrete bearing walls is 75 pounds per square foot with a compressive strength of 4 kips per square inch. The roof sheathing is 3⁄8-inch-nominal Structural I grade plywood and the roof framing is of Douglas Fir-Larch. Assume the roof diaphragm is flexible and neglect
the effect of wall openings. Draw the required nailing diagram and determine the chord reinforcement
required for north-south seismic loads.
Solution
The aspect ratio of the roof diaphragm is
a
5 256/120
5 2.1
, 4.0 . . . complies with SDPWS Table 4.2.4 for a blocked diaphragm
North-south tributary dead load
The relevant dead load tributary to the roof diaphragm in the north-south direction is due to the north
and south walls and the roof dead load, and is given by
Roof
5 19 3 120
5 2280 lb/ft
North wall 5 75 3 24.52/(2 3 24)
5 938 lb/ft
South wall 5 938 lb/ft
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L
N
W
t=
Figure 5-6 Details for Example 5-1
The total dead load tributary to the roof diaphragm in the north-south direction is
wpx
5 (2280 1 2 3 938)256/1000
5 1064 kips
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Seismic Design of Wood Structures
Redundancy factor
The structure is regular in plan with shear walls on all four sides. In addition, the length of the east and
west shear walls is W 5 120 feet, and the height is h 5 24 feet. Hence, in accordance with ASCE 7
Section 12.3.4.2b, the number of equivalent bays is
n
5 W/h
5 120/24
55
.2
Hence, the building complies with ASCE 7 Section 12.3.4.2b, and the redundancy factor is
r
5 1.0
Seismic parameters
The design spectral response accelerations are given as
SDS
5 0.826g
SD1
5 0.469g
The seismic importance factor is
Ie
5 1.0
The seismic design category is D and specially detailed reinforced concrete shear walls are required
with a response modification factor from Table 1-16 of R 5 5.0.
Fundamental period
The approximate fundamental period is given by ASCE 7 Equation (12.8-7) as
where:
Ta
5 Ct(hn)3/4
Ct
5 0.02 for a tilt-up concrete building
hn
5 roof height
5 24.5 ft
Then, the fundamental period is
Ta
5 0.02(24.5)3/4
5 0.22 sec
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The response spectrum parameter is
TS
5 SD1 /SDS
5 0.469/0.826
5 0.57
. Ta
Seismic design coefficient
Hence, ASCE 7 Equation (12.8-2) governs and the seismic design coefficient is
Cs
5 SDS Ie /R
5 0.826 3 1.0/5.0
5 0.165
In seismic design category D, the strength level design seismic load acting on a diaphragm is given by
ASCE 7 Equation (12.10-1) as
Fpx
5 wpx SFi /Swi
5 Cswpx . . . for a single-story structure
5 0.165 3 1064
5 176 kips
The minimum diaphragm force is specified by ASCE 7 Equation (12.10-2)
Fpx
5 0.2SDS Iewpx
5 0.2 3 0.826 3 1.0 3 1064
5 0.165 3 1065
5 176 kips
From ASCE 7 Equation (12.8-11), the diaphragm force for a single-story building is
Fx
5 Vwx hxk/Swi hik . . . for Ta , 0.5 sec, k 5 1.0, and V is the seismic base shear
5 Cswpx
5 0.165 3 1064
5 176 kips . . . governs
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North-south diaphragm shear
Applying the tributary area method, the strength level shear force along the diaphragm boundaries at
grid lines 1 and 9 is
QE
5 Fpx /2
5 88 kips
The strength level shear force on the diaphragm is given by ASCE 7 Equation (12.4-1) as
E
5 rQE 1 0.2SDS D
5 1.0 3 88
5 88 kips
The required nominal shear capacity is given by SDPWS Section 4.2.3 as
Vn
5 E/0.8
5 110 kips
The required nominal unit shear capacity along the diaphragm boundaries is
vn1
5 Vn /W
5 110 3 1000/120
5 917 lb/ft
The nail spacing may be changed at the beam locations, shown in Figure 5-7, and the unit shear a distance 48 feet from the boundary is given by
vn2
5 vn1 3 80/128
5 573 lb/ft
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917
573
573
917
Figure 5-7 Nailing diagram
Nail spacing
The required nail spacing is obtained from SDPWS Table 4.2A with a case 4 plywood layout applicable and all edges blocked. Framing, at continuous panel edges parallel to the load in the north-south
direction, consists of 31⁄8 3 12-inch glued-laminated beams, and in the east-west direction, consists of
33 purlins. Using 3⁄8-inch Structural I grade plywood and 8d nails with 13⁄8-inch penetration, the nail
spacing required in the two diaphragm zones is given in Table 5-1.
Table 5-1 Nail spacing requirements
Zone
1
2
Diaphragm boundaries
21⁄2 inches
6 inches
Continuous panel edges
21⁄2 inches
6 inches
Other edges
4 inches
6 inches
Intermediate members
12 inches
12 inches
Nominal shear provided, plf
1200
600
Nominal shear required, plf
917
573
The required 21⁄2-inch nail spacing is accommodated at diaphragm boundaries in the 33 ledger and at
continuous panel edges in the 31⁄8-inch glued-laminated beam.
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East-west tributary dead load
The total dead load tributary to the roof diaphragm in the east-west direction is
wpx
5 (19 3 256 1 2 3 938)120/1000
5 809 kips
East-west diaphragm shear
Applying the tributary area method, the strength level shear force along the diaphragm boundaries at
grid lines A and D is
QE
5 Fpx /2
5 0.2 3 0.826 3 1.0 3 809/2
5 67 kips
The required nominal unit shear capacity along the diaphragm boundaries is
vn
5 QE /(L 3 0.8)
5 67 3 1000/(256 3 0.8)
5 327 lb/ft
, vn2
Hence, the nail spacing determined for the north-south seismic direction governs.
Chord reinforcement
The strength level bending moment at the midpoint of the north and south boundaries due to the northsouth seismic force is
MD
5 Fpx L/8
5 176 3 256/8
5 5632 kip-ft
The corresponding strength level chord force is
Ft
5 MD /W
5 5632/120
5 46.9 kips
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477
Using Grade 60 reinforcement at the top of the concrete walls, the area of reinforcement required is
given by ACI9 Section 21.2 as
As
5 Ft /(f 3 fy)
5 46.9/(0.9 3 60)
5 0.87 in2
Providing two #7 bars gives an area of
As
5 1.20 in2
. 0.87 in2 . . . satisfactory
5.3.3 Diaphragm deflection
Diaphragm deflection, in inches, is determined by SDPWS Equation (4.2-1), which is
ddia
5 Dbending deflection 1 Dshear deflection 1 Dchord-splice slip
5 5vL3/8EAW 1 0.25vL/1000Ga 1 S(Dc x)/2W
where:
v
5 maximum unit shear due to strength level design loads in the direction
under consideration, lb/ft
L
5 diaphragm length, ft
E
5 elastic modulus of the chords, psi
A
5 area of chord cross section, in2
W
5 diaphragm width, ft
Ga
5 apparent diaphragm shear stiffness from nail slip and panel shear
deformation, kips/in (from SDPWS Tables 4.2A through 4.2D)
S(Dc x)
5 sum of individual chord-splice slip values on both sides of the diaphragm,
each multiplied by its distance to the nearest support
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The first term in the expression accounts for bending deflection, the second term for shear deflection,
and the third for chord-splice slip. The assumptions used in deriving the expression are:8
•
the diaphragm is simply supported
•
the diaphragm is uniformly nailed
•
the diaphragm is completely blocked
•
the diaphragm depth and width are constant and the diaphragm is without openings
•
the diaphragm is uniformly loaded
Example 5-2
For the tilt-up concrete building shown in Figure 5-6, determine the total inelastic deflection of the
diaphragm.
Solution
In accordance with ASCE 7 Section 12.8.6, deflections are calculated using the strength level code-prescribed design forces.
Hence, the required nominal unit shear values previously determined are multiplied, as shown in Figure 5-8, by the factor
l
5 0.8
Bending deflection
The chord consists of two #7 bars with an area of
A
5 1.20 in2
The elastic modulus of the chords is given by ACI Section 20.2.2.2 as
E
5 29,000,000 psi
The bending deflection is given by
Dbending deflection 5 5vL3/8EAW
5 5 3 734 3 2563/(8 3 29,000,000 3 1.20 3 120)
5 1.84 in
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734
458
458
734
Figure 5-8 Strength level shear values
Shear deflection
For 3⁄8-inch-thick Structural I grade plywood with 8d nails and 33 framing, the diaphragm shear stiffness is obtained from SDPWS Table 4.2A as
Ga
5 9 kips/in
The shear deflection is given by
Dshear deflection
5 0.25vL/1000Ga
5 0.25 3 734 3 256/1000 3 9
5 5.22 in
Chord-splice slip
There is no slip in the reinforced concrete chord.
Diaphragm deflection, in inches, is determined by SDPWS Equation (4.2-1), which is
ddia
5 Dbending deflection 1 Dshear deflection 1 Dchord-splice slip
5 1.84 1 5.22 1 0
5 7.06 in
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5.3.4 Diaphragm flexibility
5.3.4.1 General considerations
The determination of diaphragm flexibility is dependent on the relative deformations of the diaphragms and shear walls in a structure. A building with wood-frame diaphragms may not necessarily
be considered a flexible structure, as this determination depends on the stiffness of the shear walls. A
building with a wood-frame roof and concrete or masonry shear walls will behave as a flexible structure because the walls are highly rigid. A structure with a wood-frame roof and wood shear walls may
not necessarily be considered flexible, as this depends on the relative deformations of the diaphragms
and shear walls.
5.3.4.2 Design methodology
Because the design of a structure is affected by its classification as either a rigid or a flexible structure,
a determination of the classification is necessary at the commencement of the design.
In accordance with ASCE 7 Section 12.3.1.1, diaphragms constructed of wood structural panels or
untopped steel decking are considered flexible, provided either of the following two conditions are
met:
•
the vertical elements of the structure are steel- or composite-braced frames or concrete,
masonry, steel, or composite shear walls
•
the structure is a one- or two-family residential building
In structures of light-frame construction, diaphragms constructed of wood structural panels or untopped
steel decking are also considered flexible, provided all of the following conditions are met:
•
toppings of concrete or similar materials are not placed over wood structural panel diaphragms
except for nonstructural toppings not greater than 11⁄2 inches thick
•
each line of vertical elements of the lateral-force-resisting system complies with the allowable
story drift of ASCE 7 Table 12.12-1
5.3.4.3 Flexible diaphragm
A diaphragm that does not satisfy the conditions listed in Section 5.3.4.2 is considered flexible, in
accordance with ASCE 7 Section 12.3.1.3, when the midpoint displacement of the diaphragm, under
lateral load, exceeds twice the average story drift. This is illustrated in Figure 5-9. The diaphragm may
then be modeled as a simple beam between end supports, and the distribution of loading to the supports is independent of their relative stiffness and is proportional to the tributary areas supported. The
diaphragm has insufficient stiffness to distribute torsional moments.
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Figure 5-9 Diaphragm flexibility
5.3.4.4 Rigid diaphragm
In accordance with ASCE 7 Section 12.3.1.2, diaphragms of concrete slabs or concrete-filled metal
deck with span-to-depth ratios of three or less in structures that have no horizontal irregularities are
considered rigid. The diaphragm is sufficiently stiff to distribute torsional moments and allowance
must then be made for the additional forces created by torsional effects with the diaphragm and supports assumed to undergo rigid body rotation. The distribution of loading to the supports is proportional to their relative stiffness and is independent of the tributary areas supported.
In accordance with IBC Section 1604.4, a diaphragm is rigid for the purpose of distribution of story
shear and torsional moment when the lateral deformation of the diaphragm is less than or equal to two
times the average story drift.
Example 5-3
For the tilt-up concrete building shown in Figure 5-6, determine if the diaphragm may be considered
flexible. The weight of the concrete bearing walls is ww 5 75 pounds per square foot. The length of the
east and west walls is W 5 120 feet.
Solution
To apply the criteria of ASCE 7 Section 12.3.1.3, the deflection of the shear walls on the east and west
ends of the building is required.
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Force acting on the shear wall
Assume that the total 120-foot length of wall on each end acts as a shear wall and that the wall cantilevers from the base. The forces acting on the shear wall consist of the force applied at roof level
from the diaphragm plus the shear force due to the wall self-weight. The force from the diaphragm at
strength level value is
QE
5 88 kips
The wall self-weight is
Ww
5 ww hW
5 0.075 3 24.5 3 120
5 220.5 kips
The force due to the wall self-weight, which acts at the midheight of the wall, is
QW
5 CsWw
5 0.165 3 220.5
5 36 kips
For the purpose of determining the in-plane deflection of the wall, 50 percent of QW may be assumed
as acting at the top of the wall. The equivalent force at the top of the wall is then
QT
5 QE 1 QW /2
5 88 1 36/2
5 106 kips
Shear wall deflection
The rigidity of a cantilever concrete wall is derived as the reciprocal of the deflection of the wall due
to a unit load applied at the top edge. This deflection is given by
where:
d
5 dF 1 dS
dF
5 deflection due to flexure
5 4(H/L)3/Et . . . for a cantilever
dS
5 deflection due to shear
5 3(H/L)/Et
H
5 height of wall
5 24.5 ft
L
5 length of wall
5 120 ft
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E
483
5 elastic modulus of the concrete
5 57,000( fc9)0.5
5 3605 ksi . . . for fc9 5 4000 psi from ACI Section 19.2.2.1
t
5 wall thickness
5 6 in
The rigidity of the shear wall is determined as indicated in Table 5-2.
Table 5-2 Rigidity of shear wall
H
L
4(H/L)3 5 Et dF
3(H/L) 5 Et dS
Et (dF 1 dS) 5 Et d
R/Et
24.5
120
0.034
0.613
0.647
1.546
The actual rigidity of the shear wall is
R
5 1.546Et
5 1.546 3 3605 3 6
5 33,440 kips/in
The deflection of the shear wall is
dxe
5 QT /R
5 106/33,440
5 0.003 in
The total inelastic displacement is given by ASCE 7 Equation (12.8-15) as
dx
5 Cd dxe /Ie . . . where Cd 5 5.0, as given in Table 1-16
5 5.0 3 0.003/1.0
5 0.015 in
, 0.5 3 diaphragm deflection
Hence, the diaphragm is flexible.
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5.3.5 Subdiaphragm requirements
5.3.5.1 Crossties
To distribute the out-of-plane anchorage forces developed by concrete and masonry walls and prevent
the walls separating from the diaphragm, continuous crossties are provided between the diaphragm
chords on opposite walls. In seismic design categories C through F, as stipulated in ASCE 7 Section
12.11.2.2.1, the continuous ties must be additional to the diaphragm sheathing. The diaphragm sheathing is not considered effective for providing the continuity required.
5.3.5.2 Subdiaphragms
To reduce the number of continuous, full-depth ties required, subdiaphragms as defined in ASCE 7
Section 11.2 are used to span between the continuous ties,10, 11 as shown in Figure 5-10. The subdiaphragm must be designed for all criteria prescribed for the main diaphragm, with the anchor ties
running the full depth of the subdiaphragm to provide full transfer of the anchorage force by development into the sheathing. The subdiaphragm must act independently to transfer the wall anchorage
force from the anchorage ties to the continuous, full-depth ties in the main diaphragm. A maximum
aspect ratio of 2.5 is prescribed for the subdiaphragm. Toenails may not be used to provide anchorage
in seismic design categories C through F, nor shall ledgers be used in cross-grain bending or tension.
Where the wall anchor spacing exceeds 4 feet, the wall must be designed to span between the anchors,
in accordance with ASCE 7 Section 12.11.2.
b
N
d
Figure 5-10 Subdiaphragm details
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Example 5-4
For the tilt-up concrete building shown in Figure 5-6, determine a suitable subdiaphragm layout for
north-south seismic forces and calculate the design force in the continuous crossties.
Solution
As shown in Figure 5-10, the 31⁄8 3 12-inch glued-laminated beams at a spacing of 8 feet provide
the wall anchorage locations. The 32 3 20-foot area bounded by the concrete walls and the 51⁄8 3
21-inch glued-laminated beams on grid lines B on the north side and F on the south side and by the
63⁄4 3 24-inch glued-laminated, continuous, full-depth crossties on the east and west sides is selected
as the subdiaphragm. The 51⁄8 3 21-inch glued-laminated beams on grid lines B and F constitute the
subdiaphragm chords.
Aspect ratio
The subdiaphragm aspect ratio is
b/d
5 32/20
5 1.6
, 2.5 . . . satisfactory
Anchorage force
The strength level pull-out force on one anchor was determined in Example 4-6 as
Fp
5 3720 lb
The service level anchorage force at roof diaphragm level is
p
5 0.7Fp /s
5 0.7 3 3720/8
5 326 lb/ft
Subdiaphragm ties
The 31⁄8 3 12-inch glued-laminated subdiaphragm ties transfer the strength level anchorage force of
3720 pounds into the subdiaphragm. The service level stress produced in the tie by the anchor force is
Ft
5 Fc
5 0.7Fp /A . . . where A is the cross-sectional area of the 31⁄8 3 12-inch tie
5 0.7 3 3720/37.5
5 69 psi
This is additional to the bending stress due to dead load, and in accordance with NDS Table 2.3.2, a
load duration factor of 1.6 is applicable for load combinations that include seismic forces.
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Subdiaphragm sheathing stress
The design service level unit shear in the subdiaphragm is
q
5 pb/2d
5 326 3 32/(2 3 20)
5 261 lb/ft
, 600 . . . satisfactory
Hence, the capacity of the nailing in the main diaphragm in zone 2 is adequate.
Crosstie force
The function of the subdiaphragm is to transfer the wall anchorage force into the continuous 63⁄4 3
24-inch glued-laminated crossties. The design service level force in the continuous crossties at a spacing of 32 feet is given by
Pt
5 pb
5 326 3 32
5 10,432 lb
To provide continuity between the north and south walls, hinge connectors are required between grid
lines C and D and between grid lines D and E, as shown in Figure 5-10.
Subdiaphragm chords
The chord force in the subdiaphragm is resisted by the 51⁄8 3 21-inch glued-laminated beams on grid
lines B and F. The design service level force in the 51⁄8 3 21-inch glued-laminated subdiaphragm
chords is given by
Ft
5 pb2/8d
5 326 3 322/(8 3 20)
5 2086 pounds
Because the exterior concrete walls also act as chords, additional reinforcement is required to supplement the reinforcement provided for the main diaphragm chords. Using Grade 60 reinforcement, the
additional area of reinforcement required is given by ACI Section 21.2 as
A9
5 Ft /(f 3 fy)0.7
5 2086/(0.9 3 60,000 3 0.7)
5 0.055 in2
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The total area required, including the main diaphragm chord reinforcement, is
AT
5 A9 1 A
5 0.055 1 0.87
5 0.925 in2
Two #7 bars are provided, giving an area of
As
5 1.20 in2 . . . satisfactory
5.3.6 Design of collectors
5.3.6.1 General requirements
A collector is defined in ASCE 7 Section 11.2 as a diaphragm element, in line with the applied force,
that collects and transfers diaphragm shear forces to the vertical shear walls. As shown in Figure 1-36,
where shear walls are discontinuous or reentrant corner irregularities are present in a building, collector elements or drag struts are required to ensure deformation compatibility and prevent localized
tearing of the diaphragm. The drag strut transfers the shear originating in the unsupported portion of
the diaphragm to the shear wall.
5.3.6.2 Collector design forces
For structures assigned to seismic design category C, D, E, or F, collector elements and their connections are designed in accordance with ASCE 7 Section 12.10.2. The collector design forces are the
maximum of the three following conditions:
•
forces resulting from application at each level of the design lateral force, Fx , calculated from
ASCE 7 Equations (12.8-11) and (12.8-12), which give
Fx 5 Vwx hxk/Swi hik
The value of Fx is determined using load combinations 6 and 7 with overstrength factor W0 of
ASCE 7 Section 2.3.6.
•
forces resulting from the application at each level of the diaphragm design force, Fpx , calculated from ASCE 7 Equation (12.10-1) as
Fpx 5 wpxSFi /Swi
The value of Fpx is determined using load combinations 6 and 7 with overstrength factor W0 of
ASCE 7 Section 2.3.6.
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•
Seismic Design of Wood Structures
forces resulting from the application at each level of the minimum diaphragm design force, Fpx ,
given by ASCE 7 Equation (12.10-2) as
Fpx 5 0.2SDS Iewpx
The value of Fpx is determined using basic load combinations 6 and 7 of ASCE 7 Section 2.3.6.
The following exception is permitted by ASCE 7 Section 12.10.2.1:
•
For structures braced entirely by wood light-frame shear walls, collector elements and their
connections need only be designed for forces resulting from the application at each level of the
diaphragm design force, Fpx , calculated from ASCE 7 Equation (12.10-1) as
Fpx 5 wpx SFi /Swi
The value of Fpx is determined using basic load combinations 6 and 7 of ASCE 7 Section 2.3.6.
The following increase in forces is required by ASCE 7 Section 12.3.3.4:
•
For structures assigned to seismic design category D, E, or F and having a horizontal structural
irregularity of Type 1, 2, 3, or 4 or a vertical structural irregularity of Type 4, the design force
determined from ASCE 7 Section 12.10.1.1 is increased by 25 percent for collectors and their
connections. Where the design force is calculated using the seismic load effects, including the
overstrength factor of ASCE 7 Section 2.3.6, the 25 percent increase is not applied.
5.3.6.3 Load combinations
Using the ASD method, the force is determined using load combinations 8 and 9 with overstrength
factor W0 of ASCE 7 Section 2.4.5 as
where:
F
5 (1.0 1 0.14SDS)D 1 0.7W0QE
F
5 (1.0 1 0.105SDS)D 1 0.525W0QE 1 0.75L 1 0.75S
D
5 dead load
L
5 floor live load
QE
5 strength level effect of horizontal seismic forces
SDS
5 5-percent damped, design spectral response acceleration, for a period of
0.2 second
W0
5 structure overstrength factor given in ASCE 7 Table 12.2-1 and tabulated
for an abbreviated number of structures in Table 1.16
5 amplification factor to account for the overstrength of the structure in the
inelastic range
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Where the effects of gravity and seismic loads counteract, load combination 10 of ASCE 7 Section
2.4.5 is applicable, which is
F
5 (0.6 2 0.14SDS)D 1 0.7rQE
5.3.6.4 Design method
As shown in Figure 5-11, the collector on grid line B between grid lines 2 and 3 transfers the shear in
the flexible diaphragm to the discontinuous shear wall between grid lines 1 and 2. The unit shear in the
diaphragm on each side of the shear wall and collector is given by
qD
5 W/2l
The unit shear in the discontinued shear wall is given by
qW
5 22W/l
–
–
Figure 5-11 Collector details
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The net shear over the length of the collector is
qC
5 2qD
5 2 3 W/2l
5 W/l
The net shear over the length of the shear wall is
qWn
5 2qD 1 qW
5 2 3 W/2l 2 2W/l
5 2W/l
The maximum drag force occurs at the connection of the collector to the shear wall and is given by
F
5 l 3 qC
5 l 3 W/l
5W
Example 5-5
The tilt-up concrete industrial building, shown in Figure 5-12, is located in Orange County, California,
on a site with a site classification D. The maximum considered earthquake response accelerations are
SDS 5 0.826g and SD1 5 0.469g. The weight of the roof is 19 pounds per square foot and the weight
of the concrete walls is 75 pounds per square foot. The roof sheathing is 3⁄8-inch-nominal Structural I
grade plywood and the roof framing is of Douglas Fir-Larch. Assume the roof diaphragm is flexible
and neglect the effect of wall openings. Determine the maximum force that can be delivered to the
collector.
Solution
From Example 5-1, the governing strength level seismic force on the diaphragm is
Fpx
5 176 kips
The service level seismic force on the diaphragm is
W
5 0.7 3 176
5 123 kips
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Figure 5-12 Details for Example 5-5
The net shear over the length of the collector is
qC
5 W/4l
5 123/(4 3 60)
5 0.513 kips/ft
The maximum drag force occurs at the connection of the collector to the shear wall and is given by
F
5 l 3 qC
5 60 3 0.513
5 30.8 kips
The overstrength factor for a bearing wall structure with specially detailed reinforced concrete shear
walls is obtained from Table 1-16 as 2.5. However, from the footnote to ASCE 7 Table 12.1-1, this may
be reduced by 0.5 for a building with a flexible diaphragm. Hence,
W0
5 2.0
The maximum design service level force at the connection of the collector to the shear wall is given by
Fmax
5 F 3 W0
5 30.8 3 2.0
5 61.6 kips
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5.4 Shear walls
5.4.1 General requirements
5.4.1.1 Shear wall function
A shear wall is defined in SDPWS Section 2.2 as a wall designed to resist lateral forces parallel to
the plane of the wall. A shear wall acts as a vertical cantilevered diaphragm in transferring lateral
forces from a horizontal diaphragm to the foundation. The construction details for a typical plywood
sheathed shear wall are shown in Figure 5-13. The plywood sheathing forms the web of the cantilever
to resist shear force, vertical studs act as web stiffeners, and the end studs form the flanges to resist
flexural effects. As in the case of a horizontal diaphragm, the design capacity of a shear wall depends
on the thickness and grade of the plywood sheathing; the width of the framing members; support of
the panel edges; and the spacing, penetration, and type of nail used. This capacity has been determined
experimentally.12
A blocked shear wall is defined in SDPWS Section 2.2 as a shear wall in which all adjacent panel edges
are fastened to either common framing members or common blocking. Nominal unit shear capacity
values are provided in SDPWS Table 4.3A for wood structural panel shear walls with sheathing on one
side only, all panel edges blocked, and minimum 2-inch nominal framing or blocking members. Separate values are provided for wind or seismic loads, and values for wind loads are 40 percent higher than
for seismic loads. The tabulated values are based on the use of common nails or galvanized box nails.
For the ASD method, allowable unit shear capacity is determined by dividing the tabulated nominal
unit shear capacity by a reduction factor of 2. For the LRFD method, the design nominal unit shear
capacity is determined by multiplying the tabulated nominal unit shear capacity by a resistance factor,
fD , of 0.8.
In accordance with SDPWS Section 4.3.7, the following construction requirements are necessary for
wood structural panel shear walls:
•
minimum size of panel is 4 3 8 feet, except at boundaries and changes in framing
•
maximum spacing of studs is 24 inches
•
where stud spacing is less than 24 inches or panel thickness is greater than 7⁄16 inch, the maximum nail spacing along intermediate framing members is 12 inches; otherwise, the maximum
spacing is 6 inches
•
nails along intermediate framing members must be the same size as specified for panel edge
nailing
•
nails are located at least 3⁄8 inch from panel edges
•
maximum nail spacing at panel edges is 6 inches
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In accordance with SDPWS Table 4.3A Note 2, the tabulated nominal unit shear values for 7⁄16-inch
structural panels may be increased to the values given for 15⁄32-inch structural panels with the same
nailing, provided the studs are spaced at a maximum of 16-inch centers or the panels are placed with
the long dimension across the studs.
Nominal unit shear capacity values are provided in SDPWS for other sheathing materials besides wood
structural panels. SDPWS Table 4.3B provides nominal unit shear capacity values for wood structural
panels applied over 1⁄2-inch or 5⁄8-inch gypsum wallboard or gypsum sheathing board. SDPWS Table
4.3C provides nominal unit shear capacity values for gypsum and Portland cement plaster shear walls.
Gypsum and Portland cement plaster shear walls are not permitted in seismic design categories E and
F. SDPWS Table 4.3D provides nominal unit shear capacity values for lumber sheathing. Diagonal
lumber sheathing and double diagonal lumber sheathing are not permitted in seismic design categories
E and F. Horizontal and vertical lumber sheathing are not permitted in seismic design categories D, E
and F.
An unblocked shear wall is defined in SDPWS Section 2.2 as a shear wall that has fasteners at boundaries and vertical framing members only. Blocking between vertical framing members at adjacent
panel edges is not provided. Unblocked wood structural panel shear walls are permitted up to a maximum height of 16 feet, with a maximum aspect ratio of 2:1, and a maximum nail spacing of 6 inches
at panel edges. The nominal unit shear capacity, vub , of an unblocked shear wall is given by SDPWS
Equation (4.3-2) as
where:
vub
5 vbCub
Cub
5 unblocked shear wall adjustment factor
vb
5 nominal unit shear capacity of a blocked wood structural panel shear wall
with a stud spacing of 24 inches and nail spacing of 6 inches at panel edges
The unblocked shear wall adjustment factor is given in SDPWS Table 4.3.3.2, which is reproduced in
Table 5-3.
Table 5-3 Unblocked shear wall adjustment factor
Nail spacing, in
Stud spacing, in
Supported edges
Intermediate framing
12
16
20
24
6
6
1.0
0.8
0.6
0.5
6
12
0.8
0.6
0.5
0.4
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b
h
Figure 5-13 Shear wall details7
5.4.1.2 Aspect ratio
To ensure satisfactory deflection of the shear wall, SDPWS Table 4.3.4 imposes a maximum heightwidth ratio on wood structural panel blocked shear walls of 31⁄2:1. For wood structural panels with
aspect ratios (bs /h) greater than 2:1, the nominal shear values in SDPWS Table 4.3A are multiplied by
the aspect ratio factor 1.25 2 0.125h/bs . The shear wall height, h, is defined in SDPWS Section 2.3 as
the clear height from top of foundation to bottom of diaphragm framing above. The shear wall width
is defined in SDPWS Section 2.3 as the horizontal sheathed dimension of wall.
For unblocked wood structural panel shear walls, the maximum aspect ratio permitted is 2:1.
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5.4.1.3 Summing shear capacities
Where sheathing of the same material and of equal shear capacity is applied to both faces of the shear
wall, the nominal shear capacity for the wall may be taken as twice the value permitted for one side, in
accordance with SDPWS Section 4.3.3.3. Where the shear capacities are not equal and the sheathing
materials are dissimilar, the nominal shear capacity is taken as the maximum value given by twice the
permitted capacity for the side with the lower capacity or equal to the permitted capacity for the side
with the higher capacity.
However, for wind design, the combined nominal shear capacity of shear walls sheathed with a combination of wood structural panels, hardboard panel siding, or structural fiberboard on one side and gypsum wallboard on the opposite side may be taken as the sum of the sheathing capacities of each side.
In accordance with SDPWS Section 4.3.3.3.2, summing shear capacities of dissimilar sheathing materials applied to the same face is not permitted.
In accordance with SDPWS Section 4.3.3.4, summing shear capacities of dissimilar sheathing materials applied to the same wall line is not permitted. Where shear walls in a line, with aspect ratios greater
than 2:1, are sheathed with wood structural panels, the nominal shear capacities may be combined,
provided that the shear capacities are multiplied by the factor (2bs /h). Where multiplied by (2bs/h), the
capacities need not be reduced by the aspect ratio factor.
5.4.1.4 Framing members
The width of the nailed face of framing members and blocking is required by SDPWS Section 4.3.7.1
to be 2 inches nominal or greater at adjoining panel edges, except that a 3-inch nominal or greater
width at adjoining panel edges and staggered nailing at all panel edges are required where any of the
following conditions exist:
•
nail spacing is 2 inches or less at adjoining panel edges
•
10d common nails having penetration into framing members and blocking of more than 11⁄2
inches are spaced at 3 inches on center or less at adjoining panel edges
•
required nominal unit shear capacity on either side of the shear wall exceeds 700 lb/ft in seismic design categories D, E, and F
Where the width of the nailed face of framing members is required to be 3 inches nominal, two framing members that each have 2-inch nominal thickness are permitted to be used, provided they are
connected with fasteners designed to transfer the induced shear between members. Where fasteners
connecting the two framing members are spaced less than 4 inches on center, they must be staggered.
Where panels are applied on both faces of a shear wall and nail spacing is less than 6 inches on center
on either side, panel joints are required by SDPWS Table 4.3A Note 6 to be offset to fall on different
framing members. Alternatively, the width of the nailed face of framing members is required to be 3
inches nominal or greater at adjoining panel edges and nails at all panel edges are to be staggered.
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Double end posts are usually provided in order to accommodate the bolts or nails in the hold-down.
Similarly, to provide continuity for the top plate and to provide overlapping at intersections, a double
plate is customarily used, and in accordance with IBC Section 2308.5.3.2, a minimum splice length
of 4 feet must be provided between the two plates with not fewer than eight 16d nails on each side of
the joint.
5.4.1.5 Sill plates
Sill plates are required by IBC Section 2308.5.3.1 to be 2-inch-nominal thickness or larger. IBC Section 2308.5.3.1 requires the sill plate to have a minimum width not less than that of the wall studs
in order to provide a nailing surface for the wall sheathing and to reduce the perpendicular-to-grain
compressive stress in the plate.
IBC Section 2304.12.1.4 requires the sill plate to be of treated wood, or wood naturally resistant to
decay, when located on a concrete foundation in direct contact with earth.
5.4.1.6 Shear walls supporting concrete or masonry walls
Because of excessive deflections, in accordance with SDPWS Section 4.1.5, plywood sheathed shear
walls must not be used to resist lateral forces contributed by concrete or masonry construction in
structures exceeding two stories in height. In addition, for two-story buildings of concrete or masonry
construction, the following limitations are imposed:
•
Shear walls and diaphragms must have all edges blocked and shear walls in the two stories
must align.
•
Story-to-story wall heights must not exceed 12 feet.
•
Story drift must not exceed the limit of ASCE 7 Table 12.12-1.
•
In the lower story, the minimum thickness of plywood permitted is 15⁄32-inch.
•
Diaphragms shall not be designed to transmit lateral forces by torsional force distribution.
•
Diaphragms shall not cantilever past the outermost supporting shear wall.
5.4.1.7 Overturning restraint
Because of the light weight of timber-frame construction, it is usually necessary to provide holddowns at the ends of plywood sheathed shear walls to resist overturning. Overturning restraint is
determined using service level load combinations. An adequate bearing length is required for the bolts
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in the hold-down, and this is achieved with a double end post. Slip between the hold-down and the end
post may cause failure of the nails connecting the sheathing to the sill plate. To reduce slip of the holddown, bolt holes shall be a maximum of 1⁄16-inch oversize and bolts shall be properly tightened. Slip
may be further reduced by using hold-downs with a predeflected seat screwed to the end post. Where a
single hold-down is attached to one side of the end post, an eccentric connection results. This produces
flexural stress in the end post, which must be analyzed at the net section.13
5.4.1.8 Anchor bolts
Transfer of the lateral force from the shear wall to the foundation is achieved by anchor bolts in the sill
plate. IBC Section 2308.3.1 requires these to be not less than 1⁄2-inch diameter, embedded a minimum
of 7 inches into the concrete foundation, and spaced a maximum of 6 feet apart. A bolt shall be located
not more than 12 inches or less than 4 inches from each end of the shear wall and a minimum of two
bolts is required. In seismic design category E, IBC Section 2308.3.1.2 requires anchor bolts to be
not less than 5⁄8-inch diameter. The allowable design single shear value of an anchor bolt connecting a
wood member to concrete is given in NDS Table 12E.
To reduce cross-grain bending in the sill plate, SDPWS Section 4.3.6.4.3 requires a steel plate washer
not less than 0.229 3 3 3 3 inches in size under each nut. As shown in Figure 5-14, the plate washer
must extend to within 1⁄2 inch of the edge of the sill plate on the sheathed side where the nominal unit
shear capacity of the sheathing exceeds 400 lb/ft for wind or seismic loads. To enable this, the hole in
the plate washer is diagonally slotted with a width up to 3⁄16 inch larger than the bolt diameter and a slot
length not exceeding 13⁄4 inches. A standard cut washer is placed between the plate washer and the nut.
Sheathing
½″ max
Standard
cut washer
3″ x 3″ x 0.229″
plate washer with
1¾″ slot
Sill plate
Anchor bolt
Figure 5-14 Sill plate with plate washer14
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Standard cut washers may be used where anchor bolts are designed to resist shear only and all of the
following requirements are met:
•
the shear wall is designed as an individual full-height wall segment with required uplift anchorage at shear wall ends sized to resist overturning, neglecting the dead load stabilizing moment
•
the shear wall aspect ratio, h:b, does not exceed 2:1
•
the nominal unit shear capacity of the shear wall does not exceed 980 lb/ft for seismic, or 1370
lb/ft for wind
It has been determined from a testing program that the yield strength of the wood sill plate governs
over the strength of the concrete in the foundation.15 This is reflected in IBC Section 1905.1.8, which
provides an exemption to ACI Section 17.2.3.5.3 for determining the concrete breakout strength of
anchor bolts in shear. The concrete breakout strength in shear parallel to an edge of anchor bolts
attaching wood sill plates of bearing or nonbearing walls of light-frame wood structures to foundations
or foundation stem walls need not be computed, provided all of the following are satisfied:
1. The allowable in-plane shear strength of the anchor is determined in accordance with NDS
Table 12E for lateral design values parallel to grain.
2. The maximum anchor nominal diameter is 5⁄8 inch.
3. Anchor bolts are embedded into concrete a minimum of 7 inches.
4. Anchor bolts are located a minimum of 13⁄4 inches from the edge of the concrete parallel to the
length of the wood sill plate.
5. Anchor bolts are located a minimum of 15 anchor diameters from the edge of the concrete,
perpendicular to the length of the wood sill plate.
6. The sill plate is of 2-inch or 3-inch nominal thickness.
5.4.1.9 Openings in shear walls
Where openings occur in a shear wall, special design provisions are specified by SDPWS Section
4.3.5. Three different techniques are available for designing shear walls with openings, as shown in
Figure 5-15, and these are:
•
The segmented approach7 considers each full-height segment of the wall as a separate shearresisting element and ignores the stiffening effect of sheathing above and below the openings.
Hold-downs are necessary at the ends of each segment.
•
The perforated shear wall method16 considers the wall capacity as a percentage of the capacity
of a solid wall. Force transfer around the openings is neglected and this provides the lowest
estimate of the wall capacity of the three methods. Hold-downs are required only at the ends
of the overall shear wall.
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•
499
The force transfer method17 considers the force transfer around openings and provides the
highest estimate of the wall capacity of the three methods. Hold-downs are required only at the
ends of the overall shear wall.
Figure 5-15 Shear wall design
5.4.2 Shear wall strength
The nominal shear capacity of a wood structural panel shear wall depends on the thickness and grade
of the wood structural panel; the width of the framing members; support of the panel edges; and the
spacing, penetration, and type of nail used. This capacity has been determined experimentally.12 Nominal unit shear values are provided in SDPWS Table 4.3A for walls with sheathing on one side only, all
panel edges blocked, and a minimum of 2-inch-nominal framing members.
Example 5-6
The wood shear wall shown in Figure 5-16 is located in a buildin