Report - PEER - University of California, Berkeley

2.3 Pushover-History AnalysisSubstituting Eq.6 into Eq.1 leads to the following expression for the displacementvector increment in the n’th mode at the (i)’th pushover step:∆u = u% ∆ F % ;(i) (i) (i)n nu% = Φ Γ S(10)(i) (i) (i) (1)n n xn denAlthough IRSA is a displacement-controlled procedure, utilizing Eqs.3,9 staticequivalentseismic load vector increment corresponding to the displacement vectorincrement given in Eq.10 may be written for an alternative load-controlled process:∆f = f% ∆ F % ; f%= MΦΓ S(11)(i) (i) (i) (i) (i) (i) (i)Sn Sn Sn n xn ain(i)in which Sainis the compatible inelastic spectral pseudo-acceleration defined by Eq.9.Note that in previous papers on the development of IRSA (Aydinoglu 2003,2004), a different form of scale factor, namely the inter-modal scale factor had beenused in pushover-history analysis. In the present paper, the above-given incrementaland cumulative scale factors are directly used in the subsequent development.Now, the increment of a generic response quantity of interest, such as theincrement of an internal force, a displacement component, a story drift or the plasticrotation of a previously developed plastic hinge etc, may be written as(i) (i) (i)∆ r = r% ∆ F %(12)in which r% (i) is defined through a modal combination rule, such as CompleteQuadratic Combination (CQC) rule asNsNs(i) (i) (i) (i)∑∑ mρmn nm=1 n=1r% = ( r% r% )(13)(i)where N s denotes the total number of considered in the analysis and r%nrefers to theresponse quantity obtained from u% (i)ndefined in Eq.10 or alternatively from f %(i)Sndefined in Eq.11. ρ (i)mnis the cross-correlation coefficient of the CQC rule. Thus,generic response quantity at the end of the (i)’th pushover step can be estimated as(i) (i 1) (i) (i 1) (i) (i)r = r −−+ ∆r = r + r% ∆ F %(14)Note that each pushover step involves the formation of a new hinge, for which anincremental scale factor is calculated. In order to identify the next hinge and toestimate the response quantities at the end of the (i)’th pushover step, the genericexpression given in Eq.14 is specialized for the response quantities that define thecoordinates of the yield surfaces of all potential plastic hinges, i.e., biaxial bendingmoments and axial forces in a general, three-dimensional response of a framedstructure. In the first pushover step, response quantities due to gravity loading areconsidered as r(0) . Considering the yield conditions, the section that yields with the(i)minimum positive incremental scale factor, ∆F % , helps identify the new hinge. Inorder to avoid iterative operations in hinge identification process, yield surfaces are351