Multivariate log-concave probability density class for structural reliability applications

Abstract

Structural reliability analysis is conventionally based on a description of uncertainty via a joint probability density function (JPDF). This paper builds on an alternative concept of working with a probability distribution class, which is the set of all distributions that satisfy several prior pieces of information. A multivariate probability class is introduced given the first- and second-moment information and the condition on log-concavity of the JPDF, which is versatile enough to cover the majority of multivariate probabilistic models that are typically used in reliability applications. Owing to the strong mathematical properties of this class, it is shown that a reliability analysis in the multidimensional space of uncertainty is reduced to a univariate problem, given the linearity of the failure surface with respect to uncertain parameters. Therefore, a generalization of the Chebyshev inequality for the univariate class of distributions with a log-concave PDF is applied to calculate the upper bound of the probability of failure. The benefit of this method is that fitting a JPDF, particularly with limited amounts of data, is facilitated, yet the method provides a tight but not overly pessimistic estimate of the probability of failure. A bivariate numerical example is provided for demonstration.