Report - PEER - University of California, Berkeley

construction materials. Similarly, the limit state period (T Lsi ) of each building classcan be defined as a function of the height (or number of storeys), the geometricalproperties of the building, and the mechanical properties of the construction materials.The uncertainty in ∆ Lsi and in T Lsi is accounted for by constructing a vector ofparameters that collects their mean values and standard deviations. By assigningprobability distributions to each parameter, FORM can be used to find both thecumulative distribution function (CDF) of the limit state displacement capacity,conditioned to a period, and the CDF of the limit state period, which are thencombined to create the joint probability density function of capacity.3.2.1 Probabilistic Modelling of Geometrical PropertiesA given building class within a selected urban area may comprise a large number ofstructures that present the same number of storeys and failure mode, but that featurevarying geometrical properties (e.g., beam height, beam length, column depth,column/storey height), due to the diverse architectural and loading constraints thatdrove their original design and construction. Since such variability does affect in asignificant manner the results of loss assessment studies (see Glaister and Pinho,2003), it is duly accounted for in the current method by means of the probabilisticmodelling described below.Clearly, one could argue that by carrying out a detailed inspection of the buildingstock, such variability could be significantly reduced (in the limit, if all buildingswere to be examined, it could be wholly eliminated), however at a prohibitive cost interms of necessary field surveys and modelling requirements (vulnerability wouldthen be effectively assessed on a case-by-case basis). This epistemic component ofthe geometrical variability of reinforced concrete members has been modelled in thepresent work by means of normal or log-normal probability distribution functions,derived from European building stock data, as described in Crowley et al. (2004).3.2.2 Probabilistic Modelling of Reinforcing Bar Yield StrainMirza and MacGregor (1979) have suggested that once a probabilistic distribution foryield strength has been found, it can be divided by a deterministic value of themodulus of elasticity, which features a very low coefficient of variation, to producethe distribution of the yield strains. These two researchers have also concluded,through a series of experimental parametric studies, that a normal distribution wouldaccurately represent the variability of reinforcement bars’ yield strength, in thevicinity of the mean, whilst a beta distribution correlated well over the whole range ofdata. The coefficient of variation in the yield strength was found to be between 8% -12% when data were taken from different bar sizes from many sources. Morerecently, the Probabilistic Model Code (JCSS, 2001) has also suggested that a normaldistribution can be adopted to model the yield strength of steel. Therefore, a normaldistribution for the steel yield strength (and subsequently yield strain) has beenadopted in the current work.181