Report - PEER - University of California, Berkeley

(i.e., an element in the DM vector) to represent its continuous (e.g., maximum crackwidth in an RC joint, Pagni (2004)) or multi-level discrete (e.g., partition damagestate, Miranda (2004)). In short, these vectors are very large, and even the proposedlimitation to pair-wise joint distributions implies numerous modeling decisions,simplifications, and numerical input parameter estimates. Simplifications used inpractice and research to date include (1) “lumping” many loss elements in a singlerepresentative one (e.g., all partitions on a given floor), which is equivalent toassuming perfect dependence among them; (2) limiting the dependence of each DMelement to a single EDP element (e.g., the partition damage state on floor j dependsonly on the IDR in floor j), and (3) second-moment level modeling (e.g., regressionsof DM on EDP or DV on DM, etc.), which is equivalent to limiting probabilisticdependence specification to simply a correlation coefficient. Further, specification ofepistemic uncertainty implies that similar kinds of specification be provided for, at aminimum, the parameters in the aleatory probabilistic models, e.g., second-momentcharacterization of the (uncertain) mean values of all the EDP’s for a given IM level(e.g., Baker (2003)). This might reflect epistemic uncertainty in the engineeringmodels of nonlinear dynamic behavior adopted in the structural analyses (e.g., Ibarra(2003)). Limited experience and data will insure that there are research opportunitiesin all these directions for years to come.The numerical assessment of the probabilistic PBSA model can be conducted in avariety of technical ways which need not be the primary focus of the modeler/analyst.These include analytical or numerical integration, Monte Carlo (“dumb” and/or“smart”), first-order, second-moment methods, FORM or SORM, etc., plusappropriate hybrids of two or more of these methods (e.g., Baker (2003), Porter(2004)). For example, the nature of the highly nonlinear and detailed dynamicanalyses of MDOF frames suggests that the IM to EDP step will defy formal randomvibration analysis and always require random sampling of accelerograms andnumerical dynamic analysis (e.g., Monte Carlo perhaps coupled with regression orresponse surface analysis), as suggested in the three options outlined above. Theuncertainty in the structural parameters may be included by Monte Carlo within thesemultiple runs, perhaps with special experimental designs, or again by responsesurface methods and/or FOSM methods. Like these uncertainties in structuralparameters and models, limited data may always limit the effective specification ofdamage and economic loss data to little more than second moment specifications andassociated methods.3.2 ExamplesIn this section we present two simple examples of application of the PEER framingequation for the structural limit state case. The first is formal and the second includesnumerical results; both employ analytical integration to “solve” the equation. Whilesuch solutions require the adoption of certain simplifying assumptions they canprovide simplicity and transparency.48