Report - PEER - University of California, Berkeley

4.2.2 Modeling the Data Using Standard Probability DistributionsThe data in Figure 1, combined into method of repair groups using combinationMethod Two as discussed in Section 4.2.1, were used to generate fragility functionsdefining the probability that a joint would require at least a particular method of repairgiven a specific value of an EDP. Standard cumulative probability distributionfunctions (CDFs) may be used to define fragility functions. In the current study, threestandard CDFs were calibrated to fit the data using the Method of MaximumLikelihood. The standard distributions considered included• Lognormal distribution: Commonly employed distribution. Requirespositively valued data.• Weibull distribution: Less commonly used distribution. The distributionallows for a stronger influence of extreme-valued data. This is desirable forthe current application where small-demand values are important.• Beta distribution: Less commonly used distribution. Allows for an upper andlower bound to be defined for the distribution, which may be desirable forthe current application.The three CDFs were tested using three standard goodness-of-fit tests to identifya preferred distribution for use in modeling the data:• The Chi-Square test: For accurate results, this test requires that the totalnumber of data points exceed 50. This was not the case here, so the accuracyof these results is questionable.• The Kolmogorov-Smirnov (K-S) test: This test is exact for all sample sizes,but requires that the distribution parameters not be estimated from the data.This was not the case here, so the accuracy of this test also is questionable.• The Lillefors test: This test is exact for all sample sizes and is designed forsituations in which distribution parameters are estimated from the data set.This test is appropriate only for the normal distribution; however, byconsidering the log of the data, this test can be used to evaluate thelognormal distribution. This was done for the current study.Application of these tests indicated that the Beta distribution was not appropriatefor use in modeling the data and that the Weibull and lognormal distributions wereequally good. The lognormal distribution was chosen as the preferred distribution forthis study because of its widespread use in comparison to the Weibull distribution.4.2.3 Proposed Fragility FunctionsFigure 2 shows the proposed fragility functions linking method of repair with EDP forthree of the five EDPs. Given a specific EDP value and a specific method of repair,these models define the probability that joint damage will be such that, at a minimum,the specific method of repair will be required to restore the joint to its originalcondition. Fragility functions are shown only for three of the five proposed EDPsbecause only for these EDPs do the fragility functions have well spaced means andlow coefficients of variation. These factors result in the progression from a relatively218