Adaptation of fast Fourier transformations to estimate structural failure probability

Probabilistic analysis with multiple non-normal random variables requires multi-fold integration, for which the closed-form solutions do not exist. Moreover, it is almost impossible to estimate failure probability accurately without the use of numerical integration. A central problem in probabilistic analysis is the computation of the cumulative distribution function of the limit-state. Since the limit-state surface is not available in a closed-form, the convolution theorem is usually thought to be not applicable to the problem. In this paper, a methodology based on function approximations and the convolution theorem is presented to estimate the structural failure probability. The convolution integral is solved efficiently using the fast Fourier transform technique, and the limit-state is approximated using a two-point adaptive nonlinear approximation at the most probable failure point. The proposed technique estimates the failure probability accurately with significantly less computational effort compared to the Monte Carlo simulation. The methodology developed is applicable to structural reliability problems with any number of random variables and any kind of random variable distribution, including normal, log-normal, extreme value, Weibull, etc. The accuracy and robustness of the proposed algorithm is demonstrated by several examples having highly complex, explicit/implicit performance functions. (C) 2002 Elsevier Science B.V. All rights reserved.