Conditional probability Markov chain simulation based reliability analysis method for nonnormal variables

Based on fast Markov chain simulation for generating the samples distributed in failure region and saddlepoint approximation (SA) technique, an efficient reliability analysis method is presented to evaluate the small failure probability of non-linear limit state function (LSF) with non-normal variables. In the presented method, the failure probability of the non-linear LSF is transformed into a product of the failure probability of the introduced linear LSF and a feature ratio factor. The introduced linear LSF which approximately has the same maximum likelihood points as the non-linear LSF is constructed and its failure probability can be calculated by SA technique. The feature ratio factor, which can be evaluated on the basis of multiplicative rule of probability, exhibits the relation between the failure probability of the non-linear LSF and that of the linear LSF, and it can be fast computed by utilizing the Markov chain algorithm to directly simulate the samples distributed in the failure regions of the non-linear LSF and those of the linear LSF. Moreover, the expectation and variance of the failure probability estimate are derived. The results of several examples demonstrate that the presented method has wide applicability, can be easily implemented, and possesses high precision and high efficiency.