Estimate of small first passage probabilities of nonlinear random vibration systems by using tail approximation of extreme distributions

• A new method is developed for estimating the small probabilities of first-passage failure of nonlinear structures.
• An extrapolation method is proposed to efficiently estimate the parameters of the SGLDs fitting the nonlinear random responses.
• The tails of the joint distributions of extreme value responses are determined based on the multivariate normal Copula.

Efficient estimate of the small first passage probabilities of nonlinear structures under random excitations is of great importance in structural reliability analysis. Some methods, such as the subset simulation method, tail equivalent linearization method, asymptotic sampling method, Monte Carlo simulation method based on the extreme value theory, etc., have been developed for estimating the probabilities, however, the efficiency of the estimate is still a challenging task. In the present study, a new method is developed to overcome the challenge. The method approximates the tails of the univariate extreme distributions of the responses by using the shifted generalized lognormal distributions, in which the model parameters are estimated by an efficient method called the extrapolation method. Based on the approximate tail distributions and covariances of the extreme responses, the tails of the multivariate extreme distributions of the nonlinear response are determined by using the Nataf model. Finally, from the relationship between the first passage probabilities and the extreme value distributions, the small first passage probabilities of interest can be estimated. The efficiency of the developed method is illustrated by the small first passage probability analyses of a single-degree-of-freedom system subjected to a stationary Gaussian process and two connected electric substation equipment items subjected to earthquake base motion.