The fragility of a structural component is defined as the conditional probability of the component attaining or exceeding a prescribed limit state for a given set of boundary variables. For most structural components probabilistic models of limit states that properly account for the underlying uncertainties are not available. In this work, such models are developed by use of idealized mechanistic models and available laboratory test data. The Bayesian model assessment approach is used for this purpose. For a given model form this process essentially involves estimation of a set of model parameters and the statistics of a model error term. Once the models are developed, the fragility is computed by use of computational reliability methods. The work described in this article was carried out at the University of California, Berkeley, by PEER faculty participants Professors Armen Der Kiureghian and Khalid Mosalam, and doctoral candidate Paolo Gardoni.
Probabilistic Models
In order to develop predictive capacity models the generic model form
(1)

is used, where is the quantity to be predicted expressed through a variancestabilizing transformation as described below; is a candidate deterministic model (ideally derived from first principles) expressed in terms of a set of measurable variables x; is a correction term for the bias in as a function of the measurable variables and a set of unknown parameters ; and is a random variable with zero mean and unknown variance representing the random error in the model. Selection of a suitable transformation supports the assumption of a normal distribution for with its variance being independent of , and a parsimonious parameterization of . In the present study, the linear form
(2)

is used, where is a set of preselected "explanatory" functions.
The set of unknown parameters of the model is estimated by use of the Bayesian updating rule (Box and Tiao 1992). This involves the formulation of a likelihood function based on the available observations and the selection of a prior distribution. Maximum likelihood estimation and an importance sampling method are used to compute the posterior statistics of the parameters.
Experimental Data and Parameter Estimation
A large body of column test data is available at the University of Washington  PEER website http://maximus.ce.washington.edu/~peera1/. The database contains the results of cyclic lateral load tests of reinforced concrete columns with circular and rectangular cross sections under constant axial load. Included in the database are 109 circular columns, 43 of which failed in shear and 66 that failed either in flexure or in a combined flexureshear mode. This data and noninformative priors were used to estimate the shear and drift capacity models for circular bridge columns. With respect to each capacity the data can be classified into "failure" and "no failure" observations. It is important to note that both types of observations provide information for estimating the model parameters.
Probabilistic Models for Deformation and Shear Capacity
The procedure outlined above has been used to develop deformation and shear capacity models for reinforced concrete circular columns. For the deformation capacity is selected, where denotes the capacity with respect to drift ratio. For the shear capacity is selected, where V is the shear capacity, is the gross crosssectional area, and is the direct tensile strength of concrete. For , stateoftheart deterministic models used in the current practice are used. In this short article details of model formulation are not presented. The interested reader is referred to Gardoni et al. 2000. Figure 1 shows a comparison between the measured and predicted values of the drift capacities for the test columns based on the deterministic and probabilistic models. The failure data are shown as solid dots and the censored data are shown as open triangles. Similarly, Figure 2 shows a comparison between the measured and the predicted values of the normalized shear capacity, .
For a perfect model the failure data should line up along the 1:1 dashed line and the censored data should lie above it. The deterministic models on the left are strongly biased on the conservative side, since most of the failure as well as the censored data are below the 1:1 line. The probabilistic models on the right clearly correct this bias. The dotted lines in these figures delimit the region within one standard deviation of the model error. While the conservatism inherent in the mechanical model might be appropriate for a traditional design approach, for a performancebased design methodology unbiased estimates of the capacity are essential. The Bayesian estimates are unbiased and properly account for the underlying uncertainties.
Fragility Assessment
The capacity models can now be used to assess the fragility function for any circular column with specified geometry and material properties. As an example, Figures 3a and 3b show the estimated fragilities for a representative bridge column in California constructed after 1982 and before 1994. These estimates account for variability in the material properties and the axial load, as well as for the random error inherent in the model. The predictive curves shown indicate the mean fragility, whereas the dashed curves approximately indicate a 70% confidence interval in the fragility estimate due to the uncertainty in the estimated parameters. Figure 4 shows a contour plot of the bivariate fragility function with respect to deformation and shear demand variables.
Current research is aimed at extending this methodology to reinforced concrete bridge systems in order to assess their seismic fragility.
P. Gardoni, A. Der Kiureghian, and K. M. Mosalam
Department of Civil and Environmental Engineering
University of California, Berkeley
References
Gardoni, P., A. Der Kiureghian, K. M. Mosalam. 2000.
Fragility assessment of R/C structural components. In preparation. Box, G. E.
P., and G. C. Tiao. 1992. Bayesian inference in statistical Analysis. Reading,
Mass.: AddisonWesley.