Report - PEER - University of California, Berkeley

Pinto et al., 2004). The probability of being in a particular damage band may then beobtained from the difference between the bordering limit state exceedanceprobabilities.3.1 Probabilistic Treatment of the DemandThe cumulative distribution function of the displacement demand can be found usingthe median displacement demand values and their associated logarithmic standarddeviation at each period. The cumulative distribution function can be used to obtainthe probability that the displacement demand exceeds a certain value (x), given aresponse period (T Lsi ) for a given magnitude-distance scenario.The displacement demand spectrum that might be used in a loss estimation studycould take the form of a code spectrum or else a uniform hazard spectrum derivedfrom PSHA for one or more annual frequencies of exceedance. Both of these optionshave drawbacks in being obtained from PSHA wherein the contributions from allrelevant sources of seismicity are combined into a single rate of occurrence for eachlevel of a particular ground-motion parameter. The consequence is that if the hazard iscalculated in terms of a range of parameters, such as spectral ordinates at severalperiods, the resulting spectrum will sometimes not be compatible with any physicallyfeasible earthquake scenario. Furthermore, if additional ground-motion parameters,such as duration of shaking, are to be incorporated – as they are in HAZUS (FEMA,1999), in the definition of the inelastic demand spectrum – then it is more rational notto combine all sources of seismicity into a single response spectrum but rather to treatindividual earthquakes separately, notwithstanding the computational penalty that thisentails. Another advantage of using multiple earthquake scenarios as opposed toPSHA is the facility of being able to disaggregate the losses and identify theearthquake events contributing most significantly to the damage.The approach recommended therefore is to use multiple earthquake scenarios,each with an annual frequency of occurrence determined from recurrencerelationships. For each triggered scenario, the resulting spectra are found from aground-motion prediction equation. In this way, the aleatory uncertainty, asrepresented by the standard deviation of the lognormal residuals, can be directlyaccounted for in each spectrum. The cumulative distribution function of thedisplacement demand can then be compared with the joint probability densityfunctions of displacement capacity and period (Section 3.2), and the annualprobability of failure for a class of buildings can be found by integrating the failureprobabilities for all the earthquake scenarios (see Crowley et al., 2004).3.2 Probabilistic Treatment of the CapacityAs has been presented previously, the limit state displacement capacity (∆ Lsi ) of eachbuilding class can be defined as a function of the fundamental period (T Lsi ), thegeometrical properties of the building, and the mechanical properties of the180