Report - PEER - University of California, Berkeley

The main difficulty in assigning a probability distribution to the yield strength ofthe steel used in a group of buildings, however, is the possibility that different gradeshave been used, which would lead to a distribution with multiple peaks and troughs(see Crowley et al., 2004). One approach to solve this problem could be to calculatethe probability of failure for the building class given each possible steel grade, usingthe normal distribution to model the dispersion for each grade, and then to compute aweighted average of failure, knowing or judging the use of each steel grade within thebuilding class. The validity of such an approach would become questionable,however, if different steel grades were often used within individual buildings.3.2.3 Probabilistic Modelling of Limit States Threshold ParametersDymiotis et al. (1999) have studied the seismic reliability of RC frames usinginterstorey drift to define the serviceability and ultimate structural limit states. Theyhave found that a lognormal distribution may be used to describe the variability ininterstorey drift for both limit states. Therefore, the variability in non-structural limitstates, defined in this work as a function of interstorey drift limiting values, will berepresented by means of lognormal distributions, using the mean drift ratios that havebeen suggested by Crowley et al. (2004).Kappos et al. (1999), on the other hand, report the ultimate concrete strainreached in 48 tests of very well-confined RC members. A simple statistical analysis ofthis data shows that it would appear that in the case of limit state sectional strains alognormal distribution is also able to describe the variability of these parameters.Hence, and since for the structural limit states it is the sectional steel and concretestrains that define respective boundaries, it would appear that a lognormal distributionmay also be applied to describe the variability in these limit state parameters. Again,the mean values suggested by Crowley et al. (2004) are employed, in tandem withassumed coefficients of variation.3.2.4 Probabilistic Modelling of Scatter in Empirical RelationshipsA number of empirical relationships have been used to derive the functions ofdisplacement capacity and period that have been presented in Section 2. Theseinclude empirical expressions for the plastic hinge length members and the yieldcurvature of RC members, all of which are discussed in Glaister and Pinho (2003),and an additional empirical parameter employed in the formula derived by Crowleyand Pinho (2004) to relate the height of a building to its yield period. All of theaforementioned relationships rely on empirical coefficients to relate one set ofstructural properties to another, as for example the coefficient of 0.1 in the yieldperiod vs. height equation, T y = 0.1H T . The mean value and standard deviation ofthese coefficients have been taken from the studies carried out to derive thoseformulae, with a normal distribution being used to model the respective dispersion.182