Report - PEER - University of California, Berkeley

science building. The lateral load-resisting system of this building consists of coupledshear walls in the transverse direction and perforated shear walls in the longitudinaldirection. A numerical model was developed in OpenSees (2003) for this structure,using a representative 2D section of the building along the transverse direction[Figure 4(a), (Lee and Mosalam, 2002)]. The building has a reasonable amount ofnonlinearity contributed through coupling beams connected to elastic, rigid shearwalls. The first and second modal periods of the numerical model were determined as0.28 and 0.64 seconds, respectively. Nonlinear time history analyses were performedusing a modified Newton-Raphson solution strategy.3.3 Probabilistic FormulationThe approach adopted uses the 32 ground motions propagated through the RCbuilding, to generate 224 ground and floor level motions for construction of thefragility curves. The bench-top acceleration is determined using experimental valuesof bench dynamic behavior. Bench-top acceleration time histories are then consideredas input and for the different coefficients of friction of the equipment considered, withtheir uncertainty in mean and standard deviation, the absolute maximum displacementand velocity relative to the bench are determined. Engineering judgment must then beapplied in the selection of limit states for the DMs considered. Upon analyses of theresults, if the limit state is exceeded, then the probability of exceeding that limit stateis unity and if the limit state is not exceeded then the probability is zero. This processis continued for each of the EDP values. To develop the fragility curves, theframework of probability theory is applied, with the underlying assumption that theprobability of exceeding a particular limit state is a lognormal distribution. Shaketable results conducted on these types of equipment, indicate that with increasinginput accelerations, the dispersion in terms of response displacement of the equipmentincreases (Hutchinson and Ray Chaudhuri, 2003; Konstantinidis and Makris 2003).These results, combined with χ 2 goodness-of-fit tests, which indicate lowsignificance levels, substantiate the selection of a lognormal distribution. Theprobability of exceeding a particular limit state is therefore given by:⎛ ln( ai/ m)⎞F(a ) = Φ⎜⎟(6)i⎝ σ ⎠where if peak horizontal floor acceleration (PHFA) is selected as the EDP, F ( a i) =probability of exceeding a particular limit state for a given PHFA a ,im and σ = themedian and log-standard deviation of the lognormal distribution, respectively, andΦ (x) = the value of the standard normal for the variable x. Provided the median andlog-standard deviation of the lognormal distribution are evaluated, for each a ionemay determine the probability that a particular limit state has been exceeded.202