Report - PEER - University of California, Berkeley

ln( EDP) a b ln ( IM )= + (1)Probabilistic capacity, or damage, models have also been the subjects of previousresearch (Berry 2003). Experimental observations of damage to structural componentscan be used to generate damage fragility curves conditioned on measures of response(EDPs). These curves are usually specified at discrete damage limit states (DM),therefore making a closed form mathematical model impossible. However, usingreliability techniques for both structural component damage and for bridge-level lossof function, it is possible to describe a damage model in the same lognormal form asthe demand model (Equation 2).ln( DM) c dln( EDP)= + (2)The PEER framework then provides a convenient methodology for generatingboth annual frequencies of exceeding discrete DM limit states and damage fragilitycurves:[ ] [ ]LSP ⎡⎣DM > dm | IM = im⎤⎦ = ∫ P DM | EDP dP EDP | IM dedp (3)Using Equations 1 and 2, the two terms in the kern of Equation 3 are simplylognormal CDFs and PDFs, respectively. In Equation 3, it is assumed that there is nodependence between successive terms in the integral. For example, DMs in the firstterm of the kern are conditioned on EDP values only, without considering the IMs.Finally, to discuss decision limit states, loss models need to be developed thatrelates the damage states (DM) to decision variables (DVs). Once again, thisrelationship can be discrete, such as in current seismic performance criteria, or it canbe continuous. For simplicity, the loss model is also assumed to have lognormal formln( DV) e f ln ( DM)= + (4)The PEER framework then provides a simple extension to produce decisionfragility curves.[ ] [ | ] [ ] [ ]P DV IM = ∫∫ P DV DM dP DM EDP dP EDP IM dedp ⋅ddm(5)3. LIMIT STATESLimit states for highway bridges are formulated at two levels: the component and thesystem. The component level addresses the affect of bridge structural component’sdamage on the post-earthquake response strategy. Specifically, components are55