Report - PEER - University of California, Berkeley

can be used as a random variable of normal distribution. In Fig. 3 (b), the vertical axisis logarithm of the maximum displacement, and thus the linearity of a set of sampledata shows that the probability distribution of logarithm of maximum displacement isnormal. This verifies that the choice of probability distribution (log-normaldistribution) in the context of the derivation presented hereafter is appropriate.Probability1(a)Non-cumulativeCumulativeLN (max. displ.)6543(b)T=0.3 sec. T=0.6 sec. T=0.9 sec.0-2 -1 0 1 2Standard normal variate, s2-2 -1 0 1 2Standard normal variate, sFigure 3. Probability distribution and probability paper; (a) standard normalprobability distribution, (b) log-normal probability paper (S.R.=0.2).3.3 Fast Demand EstimationThe response database is a collection of pre-analyzed responses for a wide range ofstructures in the format of statistical correlation parameters. Therefore, the responseestimation of a structure can be instantly carried out because it entails only readingthe corresponding value from the latter database according to the given responseparameters such as stiffness and strength of the structure. Additionally the latterdatabase enables the analyst to construct vulnerability functions by dealing with onlytwo statistical parameters (mean and standard deviation) instead of massive data froma group of dynamic response history analyses. This feature renders the construction ofvulnerability functions much easier than the conventional methods. Concurrently, thefidelity of information is maintained because the cumulative normal or log-normaldistribution that is used to represent the probability distribution of maximum responsedepends on only the mean and standard deviation of the used random variables.3.3.1 Response ParametersIn order to obtain mean (µ) and standard deviation (σ) of maximum responses usingthe database, the analyst needs to determine the structural idealization type andresponse parameters. In this paper, a bi-linear structural model was utilized, as shownin Fig. 4. The period and the non-dimensional strength ratio (S.R.) are utilized as themost convenient parameters. Strength ratio (S.R.) is defined as the ratio of lateralyield strength (P) to the total weight (W) of the structure. This is a convenientparameter that is used extensively in seismic design practice. It is noteworthy that theapproach presented in this paper is independent of the assumption of bilinear190