Evaluation of the first-passage probability of non-stationary non-Gaussian structural responses with linear moments and copulas

Abstract

The evaluation of the first-passage probability of non-stationary non-Gaussian structural responses remains a great challenge in the field of random vibrations. In the present paper, a novel method is proposed for evaluating this first-passage probability, whose main contribution is to construct the joint probability density function (PDF) of the structural response and its derivative process under the consideration of their non-Gaussianities and nonlinear correlations. Cubic polynomial models of Gaussian process are developed to characterize the non-Gaussianities of the structural response and its derivative process, whose polynomial coefficients at each instant time are explicitly determined from their corresponding first four linear moments. These linear moments are accurately evaluated using a proposed method combining Sobol sequence with polynomial smoothing. The marginal PDFs and cumulative distribution functions (CDFs) of the structural response and its derivative process are then derived from these polynomial models. Based on Akaike information criterion (AIC) and the marginal CDFs, the optimal copula function at each instant time is selected to capture the linear/nonlinear correlation between the structural response and its derivative process. And thus, the joint PDF is constructed and the first-passage probability is evaluated. The applicability of the proposed method is validated by several numerical examples. It can be concluded that the proposed method provides satisfactory results in evaluating the linear moments, fitting probability distributions, and estimating the first-passage probabilities of non-stationary non-Gaussian structural responses.