Report - PEER - University of California, Berkeley

terms of global and local bridge performance. An approach for defining collapse interms of observed damage and decision limit states is presented here. While it wouldbe possible to arbitrarily assign a traffic volume (decision) loss limit state to thecollapse prevention state, it is more intuitive to use a combination of damage limitstates. This combination involves both observable damage to bridge components andloss of overall bridge function.4. COMPONENT-LEVEL DECISION: REPAIR COSTThe probabilistic seismic demand model for this case was formulated previously(Mackie 2003). This PSDM relates Sa(T 1 ), and IM, and drift ratio of the column in thelongitudinal direction, an EDP. Simulation using cloud analysis was performed usingOpenSees (http://opensees.berkeley.edu/) to obtain the PSDM. Assuming a lognormaldistribution, determination of the two unknown parameters in Equation 1 yields a=-4.18 and b=0.885. The CDF curves, obtained by integrating over the full range of IMvalues, describe the demand fragility. Plots of these fragility curves and more detailedrepair cost examples are presented elsewhere (Mackie 2004a).Transition from demand to damage fragility is done using damage data observedin experiments. Given experimental data points in the PEER Structural PerformanceDatabase (Berry 2003), column damage states were regressed versus column designparameters using conventional linear regression. The resulting equation can be used topredict the mean (or median) EDP at which a specified level of damage was observed.CDFs can then be developed using an assumption about the statistical variation of thedata to describe the probability of exceeding a damage state (DM), given a demandlevel (EDP). Alternatively, parameterized non-linear regressions may yield moresuitable equations for describing the mean relationship between demand and damage.Such equations exist for bar buckling and cover spalling in bridge columns (Berry2003).The total probability theorem used to formulate the expression for damagefragility (Equation 3) can be used to convolve the damage model and the demandmodel. The result is a traditional damage fragility curve that shows the probability ofexceeding a damage limit state as a function of ground motion IM. This damagefragility is shown in Figure 1 for the three DM limit states used in the damage model,namely spalling, bar buckling and column failure. Such a set of fragility curves can beimmediately used to assess the change in the probability of exceedance of a limit statewith the change of ground motion intensity. For example, a design scenarioearthquake has an expected intensity of Sa(T 1 ) = 1000 cm/s 2 . The probability ofspalling is 1.0, but the probability of bar buckling is only 0.88. Similarly, theprobability of column failure is slightly less at 0.75.The component-level damage most directly implies repair cost, a direct costeconomic decision variable. Alternatively, repair time could be considered as adecision variable, as it may be a more relevant decision variable for important arteriesin a transportation network than repair cost.57