Report - PEER - University of California, Berkeley

preferably linearized in a piecewise fashion, i.e., they are represented by finitenumber of lines or planes in two- and three-dimensional models, respectively.Once the incremental scale factor is determined, the new value of the cumulativescale factor is calculated from Eq.8 and any response quantity of interest developed atthe end of the (i)’th pushover step is obtained from the generic expression of Eq.14. Ifrequired, increments of modal displacements and modal pseudo-accelerations can becalculated from Eq.6 and Eq.4, respectively. Adding to those calculated at the end ofthe previous step, the new coordinates of all modal capacity diagrams may beobtained simultaneously from Eqs.5.2.4 Estimating Seismic Demand: Peak Response QuantitiesThe above-described pushover-history procedure is repeated until cumulativespectrum scale factor defined by Eq.8 exceeds unity at the end of a given pushoverstep. When such a step is detected, which is indicated by superscript (p), incrementalscale factor corresponding to this final pushover step is re-calculated from Eq.8 as∆F%(p) (p 1)= 1 − F% −(15)Peak value of the generic response quantity is again obtained from Eq.14 for i = p.2.5 Summary of Practical Implementation of IRSAA detailed derivation of IRSA is presented above for the sake of completeness. Notethat the actual practical implementation of the procedure based on lumped plasticitymodel combined with smoothed response spectrum and equal displacement rule isvery simple and transparent. The analysis stages to be applied at each pushover stepof IRSA are summarized in the following:(1) Run a linear response spectrum analysis (RSA) with a sufficient number ofmodes by considering the instantaneous second-order stiffness matrix correspondingto the current plastic hinge configuration. Preferably use a matrix transformationmethod (e.g., Jacobi method) in free-vibration analysis to accommodate singular or(1)negative-definite matrices. Use the same spectral displacements, Sden, at all pushoversteps as seismic input, which are defined only once at the first pushover step as elasticspectral displacements. Alternatively, compatible spectral pseudo-accelerationsdefined at each step by Eq.9 may be used. Obtain all response quantities of interest,(i)r% , by applying an appropriate modal combination rule (e.g., CQC rule — Eq.13).(2) Specialize the generic expression of Eq.14 for the response quantities thatdefine the coordinates of the yield surfaces of all potential plastic hinges, i.e., biaxialbending moments and axial forces in a general, three-dimensional response of a(0)framed structure. Response quantities due to gravity loading are considered as r in(i)the first pushover step. Calculate the incremental scale factor, ∆F % , according to theyield conditions of all potential plastic hinges and identify the new yielded hinge.352