Time-dependent reliability analysis with joint upcrossing rates

In time-dependent reliability analysis, an upcrossing is defined as the event when a limit-state function reaches its failure region from its safe region. Upcrossings are commonly assumed to be independent. The assumption may not be valid for some applications and may result in large errors. In this work, we develop a more accurate method that relaxes the assumption by using joint upcrossing rates. The method extends the existing joint upcrossing rate method to general limit-state functions with both random variables and stochastic processes. The First Order Reliability Method (FORM) is employed to derive the single upcrossing rate and joint upcrossing rate. With both rates, the probability density of the first time to failure can be solved numerically. Then the probability density leads to an easy evaluation of the time-dependent probability of failure. The proposed method is applied to the reliability analysis of a beam and a mechanism, and the results demonstrate a significant improvement in accuracy.