D-A-CH TAGUNG 2011 - SGEB

Some applications require assessing the probability that a specific event will occur during acertain condition at least once or that several such events will occur simultaneously. Forexample, there may be an interest in knowing whether the vulnerable elements of a lifelinenetwork (e.g., bridges) are likely to be simultaneously damaged during an earthquake or at leastwhether one element will be damaged (e.g., [14]). To analyse such cases, we considered fivebridges located within the area (Figure 4b). The corresponding characteristics of the bridgeswere taken from Liao and Loh [50]. The damage for bridges was estimated using HAZUSrecommendations (see also [51, 52]) as follows:RCR RCR P [ DS | IM ]Tall DS iii, (12)where RCR T is the total repair cost ratio, RCR i is the repair cost ratio or damage factor (i.e.,the fraction of the replacement cost for the i th damage mode) and P [ DSi | IM ] is theprobability of being in damage state DS i for a given ground motion level IM. Each damage stateexpresses a range of repair cost ratios, and the so-called best mean repair cost ratio or centraldamage factor is used in the calculations. Stergiou and Kiremidjian [52] provided amethodology for modelling the uncertainty in repair cost ratios; however, in this work, we useonly the best mean values.4 RESULTS AND DISCUSSIONWe considered the following parameters of loss distribution: the mean value of lossLOSSMEANNi1LOSSiN , where LOSS i is the total loss value for the i simulation, and N is thetotal number of simulations; the standard deviation of the loss distribution LD; the coefficientof variation CV LOSSLD MEAN; the median value for which the cumulative probabilityfunction equals 0.5; and the particular values of loss with a certain probability of not beingexceeded, e.g., 90% (LP 90 ) or 99% (LP 99 ).Let us first analyse the parameters of loss distribution for two extreme cases: (1) where allthe variability is between earthquakes (σ Τ = 0, ρη = 1.0); and (2) where all the variability iswithin earthquakes (σ η = 0, ρη = 0.0). Figure 5 shows the PDF and CDF plots for the cases, andTable 3 summarises the characteristics of the distribution.ParameterAll of the variability isbetween earthquakesAll of the variability iswithin earthquakesMean value,Mln$Median value,Mln$StandarddeviationCoefficientof variation146 / 395 290 / 550 145 / 390 285 / 55074 / 295 73 / 295 145 / 390 284 / 550200 / 330 535 / 690 13 / 22 35 / 451.36 / 0.84 1.84 / 1.24 0.093 / 0.056 0.124 / 0.082Table 3. Characteristics of total loss distribution for the extreme cases (variants DLC1 / DLC2).16612