Report - PEER - University of California, Berkeley

(noting that the probability that the binary decision variable, DV, is strictly greaterthan zero (given the IM level) is simply the probability that the capacity (as measuredin IM terms) is less than the IM demand, i.e., the cumulative distribution function, F C ,evaluated at the given IM demand level). Here the dispersion measure, β RC , is that ofthe randomness or aleatory or record-to-record variability in the capacity. The plot ofthe F C of a IM-based global instability capacity is shown in Fig. 1 (Krawinkler(2004)). It was developed from incremental dynamic analyses pushed to the “flatline” (Vamvatsikos, (2002)).P(Collapse)10.90.80.70.60.50.40.30.20.10Collapse Capacity of Van Nuys Structure, T=1.5 sec.Median Collase Capacity = 0.45 gDispersion (σ Ln(x) ) = 0.37Individual Data PointsFitted LognormalDistribution50/50 10/50 2/500 0.2 0.4 0.6 0.8 1Sa(T 1 )/gFigure 3. Cumulative distribution function, F, for global instability capacity inIM (S a ) terms. (Krawinkler 2004).When supplemented with two unit median (lognormal) random variables withdispersions β UC and β UH , to reflect epistemic uncertainty in respectively the mediancapacity and the hazard curve frequencies, the limit state frequency above becomes an(epistemically uncertain) lognormal random variable, whose mean estimate is(Cornell (2002), Jalayer (2003)):1 2 2 1 2 2λLimit State= FC( sa) | dλS ( sa) || = λ ( Cˆ)exp(k βRC)exp( k β )(5)∫ a2 UC2Sain which FCis the mean estimate of the CDF of the capacity and the mean estimateof the hazard curve is ˆ 1 2λS( s)= λS( s)exp(βUH( s)). Ĉˆ is the median estimate of theaa2median capacity. The dispersion (standard deviation of the natural log) of the limit2 2 2state frequency is β λ = βUH+ k β . Note that the mean estimate can beLimit StateUCdetermined directly from a mean estimate of the hazard curve and that confidencebands on the limit state frequency can be obtained from these two β parameters and astandard Gaussian table. Finally note that the epistemic uncertainty depends on boththat in the hazard (demand) and that in the capacity, the latter modified by the50