Report - PEER - University of California, Berkeley

Results indicate that 33% of the respondents would likely close the bridge at leastbriefly (> 1 day) if they observed spalling. This figure increased to 100% for barbuckling. The data yield the following discrete probabilities:P(DV|Spalling = True) = 0.33P(DV|Bar Buckling = True) = 1.00The fragility functions used to develop P(DM|EDP) shown in Figure 5 cannow be combined with the above discrete probabilities to determine the probability ofclosing the bridge given a demand estimate, as follows:∞2P ( DV | EDP) = ∫ P( DV | EDP) dP( EDP) = ∑ P( DV | DM ) P( DM | EDP)(4)−∞i=1The resulting probability distribution is also shown on the right in Figure 5. Onefinal step remains. This involves integrating the seismic hazard curve into Equation(4). But before incorporating the hazard curve into the picture, it is necessary to findthe probability of obtaining a dv given the EDPs resulting from a set of IMs (in a nonannualfrequency format) (to distinguish between IM and dλIM):∞dP( ) ( )( EDP | IM )P DV > dv | IM = ∫ P DV | EDPdEDP(5)0dEDPP(EDP|IM) is evaluated assuming a lognormal distribution:⎛ ⎛ ⎞ ⎞⎜ ⎜edpln ⎟ ⎟⎜b( )⎟( ) ⎜⎝ a imP EDP > edp | IM = im = 1− Φ⎠⎟(6)⎜ σlnedp|im⎟⎜ ⎟⎝ ⎠Using the total probability theorem, the probability of closure given the seismichazard curve implies:∞dv( ) ( )( IM )P DV > dv | λ IM = ∫ P DV > dv | IM dIM(7)0dIMEquation (7) is evaluated numerically to obtain the annual probability ofclosure. The probability of closure in n (n=50) years is given by( 1−P( DV dv)) nP = 1−>(8)4. IMPACT OF MODELING DECISIONSThe probabilistic methodology outlined in the previous section is applied in theevaluation of the simulation model of the I-880 viaduct. EDPs were computed for tenground motions for each hazard level. The EDP vs. IM relationship, as shown inFigure 4, was established for each variation of a model variable. The P(DV|IM)71