Report - PEER - University of California, Berkeley

PEER has put forward PBSA methodologies which can be represented by its“framing equation”,∫∫∫λ( DV)= G ( DV | DM)| dG(DM | EDP)| | dG(EDP | IM)| | dλ(IM)|(1)which is described in some detail 4 and applied in this workshop in Deierlein (2004),Krawinkler (2004), and Miranda (2004). Suffice it to say here that the integral isdesigned to isolate a pair-wise sequence of four (generally vector-valued) randomvariables representing ground motion intensity (IM), structural responses such asMIDR and peak floor accelerations (EDP), damage states (DM) and finally “decisionvariables” (DV) such as lives and dollars. The pair-wise sequences presume that thevariables are only simply coupled, or that each variable is, to use the word fromabove, “sufficient” with respect to those before it in terms of its prediction of thoseafter it. For example, it assumes that P[DM =x|EDP = y and IM = z] = g(y) and notof 5 y and z. In the context of PBSA it permits the specialist in each subject (e.g., costestimation) to deal only with prediction of costs from given damage states withoutworrying about what ground motion or structural deformations caused the damage.Binary limit state analysis (such as assessment of collapse frequency) can be thoughtof as a special case when the DV is scalar and binary, DV = 1 being the collapseevent. This formulation contains as special cases most of the common limit state andloss estimation schemes. One such is that using “fragility curves” which typicallyrepresent the probability of some binary limit state (collapse, severe economic loss,etc.) as a function of ground parameter (IM) such as PGA. Such a result is obtained ifthe second two integrals are collapsed leavingλ ( DV = 1) = ∫ GDV| IM(0 | IM ) | dλ(IM ) | in which G DV|IM (0|IM) is the fragility curveresulting from using one or methods (e.g., Monte Carlo simulation) to find theprobability of the limit state (collapse, for example) as a function of, say, PGA or S a .We shall use this in an example below. (Note that the probability that the binary limitstate variable is 1 is the probability that it is strictly greater than 0.)Even with the simplifications inherent in the PEER framing equation thespecification of the necessary probability distributions and the “propagation” of thoseuncertainties (i.e., the numerical computation of the integral) can be a daunting task.The specification of the aleatory and epistemic uncertainties requires inputting jointdistributions such as the G EDP|IM and G DM|EDP when for, say, detailed PBSAeconomic loss estimation, the number of relevant EDP’s may be a vector of at leasttwo to four per floor (e.g., peak IDRs and floor accelerations) and each potential lossproducingelement in the PBSA model of the building (structural members, partitions,expensive laboratory equipment, etc.) may in principal deserve a random variable4 The G functions are complementary cumulative distribution functions, i.e., the probability that a randomvariable strictly exceeds the argument. The absolute values of their derivatives are probability densityfunctions in the continuous case.5 To the probabilistic this is a kind of Markovian dependence.47