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Appendix B Modal Pushover Analysis Pacific Earthquake Engineering Research Center A Modal Pushover Analysis Procedure to Estimate Seismic Demands for Buildings: Theory and Preliminary Evaluation Anil K. Chopra University of California Berkeley Rakesh K. Goel California Polytechnic State University San Luis Obispo A report on research conducted under grant no. CMS-9812531 from the National Science Foundation: U.S.-Japan Cooperative Research in Urban Earthquake Disaster Mitigation PEER 2001/03 JAN. 2001 A Modal Pushover Analysis Procedure to Estimate Seismic Demands for Buildings: Theory and Preliminary Evaluation Anil K. Chopra University of California Berkeley Rakesh K. Goel California Polytechnic State University San Luis Obispo A report on research conducted under Grant No. CMS-9812531 from the National Science Foundation: U.S.-Japan Cooperative Research in Urban Earthquake Disaster Mitigation PEER Report 2001/03 Pacific Earthquake Engineering Research Center College of Engineering University of California Berkeley January 2001 i ABSTRACT The principal objective of this investigation is to develop a pushover analysis procedure based on structural dynamics theory, which retains the conceptual simplicity and computational attractiveness of current procedures with invariant force distribution, but provides superior accuracy in estimating seismic demands on buildings. The standard response spectrum analysis (RSA) for elastic buildings is reformulated as a Modal Pushover Analysis (MPA). The peak response of the elastic structure due to its nth vibration mode can be exactly determined by pushover analysis of the structure subjected to * lateral forces distributed over the height of the building according to smnn= φ , where m is the mass matrix and φn its nth-mode, and the structure is pushed to the roof displacement determined from the peak deformation Dn of the nth-mode elastic SDF system. Combining these peak modal responses by modal combination rule leads to the MPA procedure. The MPA procedure is extended to estimate the seismic demands for inelastic systems: First, a pushover analysis determines the peak response rno of the inelastic MDF system to individual modal terms, pseff,nn()tt=− u&&g ( ) , in the modal expansion of the effective earthquake forces, pmeff,n ()tt=− ι u&&g (). The base shear-roof displacement (Vubn− m ) curve is developed * from a pushover analysis for force distribution sn . This pushover curve is idealized as bilinear and converted to the force-deformation relation for the nth-“mode” inelastic SDF system. The peak deformation of this SDF system is used to determine the roof displacement, at which the seismic response, rno , is determined by pushover analysis. Second, the total demand, ro , is determined by combining the rnno ()=1, 2,K according to an appropriate modal combination rule. Comparing the peak inelastic response of a 9-story SAC building determined by the approximate MPA procedure with rigorous nonlinear response history analysis (RHA) demonstrates that the approximate procedure provides good estimates of floor displacements and story drifts, and identifies locations of most plastic hinges; plastic hinge rotations are less accurate. The results presented for El Centro ground motion scaled by factors varying from 0.25 to 3.0, show that MPA estimates the response of buildings responding well into the inelastic iii range to a similar degree of accuracy when compared to standard RSA for estimating peak response of elastic systems. Thus the MPA procedure is accurate enough for practical application in building evaluation and design. Comparing the earthquake-induced demands for the selected 9-story building determined by pushover analysis using three force distributions in FEMA-273, MPA, and nonlinear RHA, it is demonstrated that the FEMA force distributions greatly underestimate the story drift demands, and the MPA procedure is more accurate than all the FEMA force distributions methods in estimating seismic demands. However, all pushover analysis procedures considered do not seem to compute to acceptable accuracy local response quantities, such as hinge plastic rotations. Thus the present trend of comparing computed hinge plastic rotations against rotation limits established in FEMA-273 to judge structural performance does not seem prudent. Instead, structural performance evaluation should be based on story drifts known to be closely related to damage and can be estimated to a higher degree of accuracy by pushover analyses. iv ACKNOWLEDGMENT This research investigation is funded by the National Science Foundation under Grant CMS- 9812531, a part of the U.S.-Japan Cooperative Research in Urban Earthquake Disaster Mitigation. This financial support is gratefully acknowledged. v CONTENTS ABSTRACT...................................................................................................................................... iii ACKNOWLEDGMENT .......................................................................................................................v TABLE OF CONTENTS ................................................................................................................... vii 1. INTRODUCTION.........................................................................................................................1 2. ONE-STORY SYSTEMS ..............................................................................................................3 2.1 Equation of Motion........................................................................................................3 2.2 System and Excitation Considered ................................................................................5 2.3 Response History Analysis ............................................................................................6 2.4 Pushover Analysis .........................................................................................................6 3. ELASTIC MULTISTORY BUILDINGS..........................................................................................9 3.1 Modal Response History Analysis.................................................................................9 3.2 Modal Response Spectrum Analysis ...........................................................................12 3.3 Modal Pushover Analysis ............................................................................................12 3.4 Comparative Evaluation of Analysis Procedures ........................................................13 3.4.1 System and Excitation Considered ..................................................................13 3.4.2 Response History Analysis ..............................................................................16 3.4.3 Modal Pushover Analysis ................................................................................22 4. INELASTIC MULTISTORY BUILDINGS ....................................................................................27 4.1 Response History Analysis .........................................................................................27 4.2 Uncoupled Modal Response History Analysis ............................................................28 4.2.1 Underlying Assumptions and Accuracy ..........................................................33 4.2.2 Properties of nth-“mode” Inelastic SDF System .............................................34 4.2.3 Summary..........................................................................................................36 4.3 Modal Pushover Analysis ............................................................................................36 4.3.1 Summary..........................................................................................................37 4.4 Comparative Evaluation of Analysis Procedures ........................................................38 4.4.1 Uncoupled Modal Response History Analysis ................................................38 4.4.2 Modal Pushover Analysis ................................................................................41 4.4.3 Modal Pushover Analysis with Gravity Loads ................................................48 5. COMPARISON OF MODAL AND FEMA PUSHOVER ANALYSES .............................................55 5.1 FEMA-273 Pushover Analysis ....................................................................................55 5.2 Comparative Evaluation...............................................................................................55 6. CONCLUSIONS ........................................................................................................................65 vii REFERENCES..................................................................................................................................69 APPENDICES A: UNCOUPLED MODAL RESPONSE HISTORY ANALYSIS............................................................71 B: MODAL PUSHOVER ANALYSIS.................................................................................................81 C: FEMA FORCE DISTRIBUTION CALCULATION........................................................................85 viii 1 Introduction Estimating seismic demands at low performance levels, such as life safety and collapse prevention, requires explicit consideration of inelastic behavior of the structure. While nonlinear response history analysis (RHA) is the most rigorous procedure to compute seismic demands, current structural engineering practice uses the nonlinear static procedure (NSP) or pushover
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