The generalized first-passage probability considering temporal correlation and its application in dynamic reliability analysis

In the traditional up-crossing rate approaches, the absence of consideration for correlation among crossing events often results in significant inaccuracies, particularly in scenarios involving stochastic processes with high autocorrelation and low thresholds. To fundamentally address these issues and limitations, the probability density function of the first passage time represented by the high-dimensional joint probability density function was investigated, and the equiprobable joint Gaussian (E-PHIn) method is proposed to prevent the redundant counting of the same crossing event. The innovation of the developed method is that it accounts for the correlation among different time instances of the stochastic process and allows for direct integration to derive the first-passage probabilities. When dealing with stochastic processes with unknown marginal distributions, the method of moments was introduced, complementing the E-PHIn method. Meanwhile, corresponding dimensionality reduction strategies are offered to improve computational efficiency. Through theoretical analysis and case studies, the results indicate that the conditional up-crossing rate represents the probability density function of the first-passage time. The E-PHIn method effectively addresses the first-passage problem for stochastic processes with either known or unknown marginal probability density functions. It fills the gap in traditional up-crossing rate approaches within the domain of nonlinear dynamic reliability. For the example structures, the E-PHIn method demonstrates higher accuracy compared to traditional point-based PDEM. Compared to MCS, the E-PHIn method significantly improves analytical efficiency while maintaining high precision for low-probability failure events.