Joint First-Passage Probability and Reliability of Systems under Stochastic Excitation

The first-passage probability, describing the probability that a scalar process exceeds a prescribed threshold during an interval of time, is of great engineering interest. This probability is essential for estimating the reliability of a structural component whose response is a stochastic process. When considering the reliability of an engineering system composed of several interdependent components, the probability that two or more response processes exceed their respective safe thresholds during the operation time of the system is an equally essential quantity. This paper proposes simple and accurate formulas for approximating this joint first-passage probability of a vector process. The $n\text{th}$ order joint first-passage probability is obtained from a recursive formula involving lower order joint first-passage probabilities and the out-crossing probability of the vector process over a safe domain. Interdependence between the crossings is approximately accounted for by considering the clumping of these events. The accuracy of the proposed formulas is examined by comparing analytical estimates with those obtained from Monte Carlo simulations for stationary Gaussian processes. As an example application, the reliability of a system of interconnected equipment items subjected to a stochastic earthquake excitation is estimated by linear programming bounds employing marginal and joint component fragilities obtained by the proposed formulas.