Report - PEER - University of California, Berkeley

practice the localized gravity load collapse mechanisms are often handled throughcomponent fragility functions P(DM|EDP), such as the functions for slab-columnconnection punching failure mentioned previously (Aslani and Miranda, 2003).Where collapse is simulated directly, such as indicated by the end points of theIDA curves for the reinforced-concrete building in Figure 2, the MAF of collapse canbe calculated by integrating the first two terms of (1) as follows:( ) = ∫ P Collapse IM =λ Collapse im i| dλ(IM)IMThe graphical interpretation of this is shown schematically in Figure 5, where the IMhazard and median IDA curves are plotted together with a common IM (= Sa T1 )vertical axis. The median EDP-IDA curve and associated probability distributions arestatistical representations of the IDA data (e.g., Figure 2). The calculation ofλ(Collapse) by (2), is simply the integration of the vertically plotted distribution ofP[Collapse|Sa], shown in Figure 5, with the λ(IM) hazard curve.When localized collapse (or a global collapse triggered by a local failure) isdetected indirectly through a damage function, the integration takes the form:λ Collapse = max P LC EDP = edp P EDP > edp IM im dλ(IM)(3)( ) [ ]∫∫ jj=IM,EDPThe first integral can be visualized by the integration of the λ(Sa) hazard curve withthe horizontal distribution of P[IDR| Sa], shown in Figure 5. The result of thisintegration is a mean annual frequency exceedence curve for EDP, λ(EDP>Y). Thiscurve is plotted on the left side of Figure 6, alongside a set of component damageprobability curves, P[DM = dm i | IDR = idr i ], with the two graphs associated by theircommon vertical EDP axis. The component damage curves of Figure 6 are similar tothe ones shown previously in Figure 4b, only in this case they pertain to structuralcomponents where the final damage state, e.g., DM 3 , corresponds to a local collapseFigure 5. Integration of IM hazard withEDP response.i(2)(LC) condition. The secondintegration in (3) is performed overthe full range of EDP for everystructural component that has aDM associated with collapse.Assuming that the failure of anyone such component is severeenough to be deemed “collapse,”the resulting MAF of collapse isdetermined by the maximumlikelihood of collapse in any oneelement — hence, the “max”notation in (3).23