Solving the geometric reliability index for a case involving multivariate random variables in the original physical space

Abstract

An effective interpretation of the complex reliability theory will promote its application in engineering practice. Based on an algorithm for solving the geometric reliability index for cases involving two or three random variables in the original physical space, an advanced algorithm for multivariate random variables is further developed by using a hypersphere model and inverse Nataf transformation. The main components of the algorithm include the definition of limit state surface, the construction of a hypersphere model, the determination of probability density iso‐contour points, the identification of limit state of these points, and the definition of an actual reliability index. Several typical practical problems associated with two to five variables are analyzed, and the results are compared with those obtained using the first‐order reliability method (FORM) and the copula‐based sampling method (CBSM), thus validating the accuracy of the proposed algorithm. With the help of the powerful and open‐source R language platform, the visualization of a reliability index in low dimensions and the solvability in high dimensions are implemented, which makes the interpretation of reliability theory more intuitive.