Modeling multivariate distributions using Monte Carlo simulation for structural reliability analysis with complex performance function

The simulation of multivariate distributions has not been investigated extensively. This article aims to propose Monte Carlo simulation (MCS)-based procedures for modeling the joint probability distributions and estimating the probabilities of failure of complex performance functions. Two approximate methods, namely method P and method S, often used to construct multivariate distributions with given marginal distributions and covariance, are introduced. The MCS-based procedures are proposed to simulate the theoretical multivariate distributions or approximate multivariate distributions constructed by methods P and S, which are further used to compute the probabilities of failure of complex performance functions. Four illustrative examples with known theoretical joint probability distributions are investigated to examine the accuracy of the proposed procedures in modeling the multivariate distributions and estimating the probabilities of failure. The results indicate that the bivariate distributions can be effectively simulated by the proposed procedures, which can evaluate the reliability of complex performance functions efficiently. These provide a useful tool for solving the reliability problems with complex performance functions involving correlated random variables under incomplete probability information. The performance of the simulation procedures associated with the two approximate methods highly depends on the level of probability of failure, the form of performance function, and the degree of correlation between variables.