Closed-Form Solution to First Passage Probability for Nonstationary Lognormal Processes

Abstract

In time-dependent reliability analysis, the calculation of the mean outcrossing or upcrossing rate of a stochastic process from a safe domain or barrier level based on the Rice formula continues to present serious challenge to researchers in the field. Furthermore, the derivation of closed-form analytical solutions to the first passage probability for nonstationary processes has not been very successful except for Gaussian process. The intention of this paper is to drive a closed-form solution for the calculation of mean upcrossing rate of a scalar nonstationary lognormal process from a barrier level. The applicability of this new solution is illustrated in a time-dependent reliability analysis of corrosion-induced concrete cracking. It is found that the results of first passage probability calculated from the derived closed-form solution are in very good agreement with those from Monte Carlo simulation and safety index methods. A merit of the derived solution is that it eliminates unrealistic negative values for inherently positive values of physical properties as a normal distribution would otherwise assume. The paper concludes that the derived closed-form solution for lognormal processes can predict the first passage probability with accuracy. Accurate prediction of first passage probability is of significance in preventing failures of engineering structures during their lifetime.