Report - PEER - University of California, Berkeley

A group of response time histories obtained from dynamic response historyanalyses with a series of ground motions are summarized in two statisticalparameters; mean and standard deviation of maximum responses. Then, the latterprocess is repeated for a range of earthquake intensities and structural responseparameters to construct the response matrix. The dimension of the response matrixcan be reduced by representing the mean and standard deviation as functions ofearthquake intensities. After this step, the response matrix contains constants ofregression functions that represent mean (µ) and standard deviation (σ) of maximumdisplacement demand. Finally, the response database is constructed by summing theresponse matrices for various earthquake scenarios and structural idealization types.3.2 Probability DistributionThe effect of variability in member capacity on the global response is very smallcompared to that of variability in the ground motion. Therefore, in this paper,earthquake ground motion is considered as the only random variable. As anillustrative example in this paper, a set of artificial ground motions is used. The latterground motions are synthesized to simulate an earthquake event for lowland soilprofile in Memphis, TN, USA and entitled 'Scenario #3' among three scenariosgenerated as a part of Mid-America Earthquake (MAE) Center research project HD-1(Hazard Definition). Scenario #3 consists of ten records simulating an earthquakeevent of magnitude (M w ) 5.5 and a focal depth of 20 km with 84 percentile level (onestandard deviation above the mean value) from the prediction model. Details arediscussed in (Romero et al. 2001).A vulnerability curve is a cumulative conditional probability of structuralresponse exceeding a prescribed limit states for a range of earthquake intensities. Inthis paper, maximum displacement is utilized to represent response of structures andit is assumed to be a log-normally distributed random variable. This means that thelogarithm of the maximum displacement has a normal probability distribution, asshown in Fig. 3 (a). In order to examine the validity of the assumed probabilitydistribution, log-normal probability paper is constructed as shown in Fig. 3 (b). Thethree sets of sample data plotted on the latter probability paper are obtained fromdynamic response history analyses of three different structures that have the samestrength ratio (0.2) but different periods (0.3, 0.6 and 0.9 sec.). Ten records of theearthquake scenario for lowland profile in Memphis with PGA of 0.2g were utilizedfor the analyses. The plotting position of a sample data is determined by calculatingits cumulative probability then its inverse, standard normal variate. The cumulativestandard normal probability of the mth value among the N data (x 1 , x 2 , . . . , x N ,arranged in increasing order) of the logarithm of maximum displacement isdetermined by m/(N+1) and its basis is discussed in (Gumbel 1954).Since the horizontal axis of the probability paper is the standard normal variate, s,which is the inverse of the standard normal cumulative probability, a linearrelationship between the vertical and horizontal axes guarantees that the vertical axis189