Report - PEER - University of California, Berkeley

An efficient demand model relating maximum displacement U max (or drift ratio)and Sa(T 1 ) is used in the last term of the kern in Equation 7. The middle term inEquation 7 is derived using simulation (Mackie 2004b). The first kern term inEquation 7 was computed in Method B. Using a DV limit state of 25% of trafficvolume reduction, the four methods are compared in Figure 6. The values of ζ, thelognormal parameter that describes the dispersion of the model, of the four methodsare 0.96, 0.80, 0.73, and 0.46, respectively. Therefore, while even Method D hasfairly high uncertainty, its prediction of the median value is better than the directapplication of Equation 5.Figure 6. Comparison of Methods A through D for decision fragilities.6. COLLAPSE-RELATED DECISIONThe collapse-prevention limit state used in this paper is a combination of the loss oflateral and vertical load capacity limit states shown in Table 1 (Mackie 2004a). Abridge would be considered closed, i.e., in collapse-prevention limit state, if thelateral load carrying capacity had been reduced by 25% or the vertical load carryingcapacity had been reduced by 50%. Thus, the remaining traffic volume crossing thisbridge is zero. These values are used as an example and will be changed once moredata becomes available. The damage fragility surfaces lateral (longitudinal) andvertical directions are presented in (Mackie 2004b). For the purposes of this example,the direct method (Method A) was used in order to maintain consistency between thelateral and vertical directions. Both of the limit states are plotted in Figure 7 alongwith the probability of closure, defined as the union of the two damage limit states.The probability of the union was approximated as a series system with thecorrelation coefficient computed using the response load carrying data for the lateraland vertical directions. As would be expected, the correlation (ρ = 0.85) betweenvertical and lateral loss of load carrying capacity is high. The probability was62