A reliability analysis method based on the intersection area division of hypersphere and paraboloid

Abstract

In terms of the approximate failure probability of the limit state function (LSF), First Order Reliability Method (FORM) directly solves the failure probability through the most probable point (MPP), but it can only achieve high accuracy in the LSF with low linearity and non-linearity. Second Order Reliability Method (SORM) performs second-order expansion on the basis of MPP to improve reliability accuracy. However, the approximation may be wrong, and even has a large error in high-dimensional nonlinear problems. In order to get more precise approximation of SORM, this paper proposes an area partition method based on the Intersection Area Division Method (IADM) of Hypersphere and Paraboloid. IADM constructs a hypersphere and an approximate parabolic function of the LSF at MPP. The intersection area of the hypersphere and the approximate paraboloid is defined as the failure region from the geometric of the approximate paraboloid function, and the area of the failure region is calculated by the spherical integral. Finally, the approximation of the failure probability is determined by integrating the ratio of the area to the area of the hypersphere. Several examples proved that IADM is more stable compared with SORM which avoids the generation of wrong solutions and improves the accuracy.