Push over

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.-JapanCooperative 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 theNational 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 theconceptual 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 exactlydetermined by pushover analysis of the structure subjected to lateral forces distributed over the height of the building according to s* = mφn , where m is the n mass matrix and φ n 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 MPAprocedure. 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, p eff,n ( t ) = −s nu g ( t ) , in the modal expansion of the effective earthquake && forces, p eff,n ( t ) = −m ι u g ( t ) . The base shear-roof displacement (Vbn − um ) curve isdeveloped

from a pushover analysis for force distribution s* . This pushover curve is idealized as bilinear n 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 determinedby combining the rno ( n = 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 mostplastic 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 enoughfor 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...