Reliability Analysis Using Parabolic Failure Surface Approximation

Abstract

In the second-order reliability method the failure surface is approximated by a general quadratic surface in the neighborhood of the design point. In this paper this general quadratic surface is further approximated by a parabolic surface. Several methods are proposed to obtain the probability content associated with this parabolic failure surface. It is assumed that the basic random variables are Gaussian. The proposed methods can be broadly grouped into: (1) nonasymptotic approximate methods, (2) exact methods, and (3) asymptotic distribution methods. Most of these methods result in a closed-form expression for the failure probability. For nonasymptotic approximations, a least-square approach and an optimal point expansion method using approximate probability density functions of a quadratic form in Gaussian random variables have been proposed. It is shown that such approximations give accurate results without significant numerical effort. Exact results, however, require greater numerical effort. The new asymptotic result is derived for the case when the number of random variables approaches infinity. Several numerical examples are provided to compare the proposed results with existing equivalent results and Monte Carlo simulations.