Report - PEER - University of California, Berkeley

Probability of Non-Exceedance10.80.60.40.200 5000 10000 15000 20000 25000 30000Connection Repair Cost - $YieldingBucklingFracturingFigure 8. Example loss function for moment resisting steel frames.Loss functions for life losses can be developed by evaluating historical data onthe number of serious and fatal injuries in buildings of different constructioncharacteristics that experienced different levels of damage. Comparisons of this typetypically show that life losses are negligible unless partial or total collapse occurs. Byconvolving the likelihood that persons are in a portion of a building that is subject tocollapse at the time an earthquake occurs, with the statistical rate that collapse hasresulted in various levels of injury in the past, it is possible to develop loss functionsthat relate the probability of serious injury and life loss to the collapse damage state.Once the hazard function, structural and nonstructural response functions, anddamage and loss functions for a building have been formed it is possible to completethe performance assessment process by estimating the risk of the various losses interms meaningful to the different decision makers who must select the desiredbuilding performance that will serve as the basis for .design. The simplest form ofloss prediction consists of determining the expected value of a loss (deaths, dollars, ordown time) given that the structure experiences a specific intensity of ground motionthat may for example, represent an event with a 10% probability of exceedance in 50years. The process starts by determining the conditional probability that structuralresponse, in our previous example interstory drift (δ i ), will be to a given level, if thestructure experiences this level of ground motion intensity. This is determined byevaluating the response function for the structure (e.g., Figure 4) at the given intensitylevel. The next step is to determine the total probability that the structure will bedamaged to each of the possible damage states (DS j ). This is performed byintegrating the conditional probability of experiencing each of the damage statesP(DS j |δ i ) as a function of interstory drift, δ i , e.g., from Figures 5, 6 and 7, with theprobability of experiencing different levels of δ I obtained from the response function.Finally, the expected loss is computed by summing the probable value of the loss(PV(Loss⏐DS j ), e.g., from loss functions such as Figure 8, given the occurrence of adamage state times the total probability of experiencing each of the damage statesP(DS) summed over all possible damage states, or in equation form:99