Time-Variant Reliability for Systems with Non-monotonic Limit-State Functions

Abstract

Most engineering time-variant reliability problems are the result of component degradation and stochastic loading. The resultant failure modes, and their resultant limit-state functions, produce limit-state surfaces with unpredictable temporal trajectories that may exhibit a combination of increasing and decreasing failure probabilities. In many cases the trajectories are monotonic so that failure increases predictably: in other cases, this is not so. In this paper we present the discrete-time set theory derivation for non-monotonic situations wherein the limit-state surface may recede to provide, what only appears to be, ever decreasing failure probability. The presence of both monotonic and non-monotonic limit-state functions can be easily detected by a parametric polar plot of the most-likely failure points in standard normal space. The polar plot reveals the temporal limit-state surfaces that need to be retained to represent the system limit-state surfaces at any time instant. The minimum set herein is called the extreme limit-state surface. The impact of the work is that the cumulative distribution function (cdf) can be provided with a minimum of failure and safe events. This in turn gives rise to several solution options such as the multi-normal integral method or a special Monte Carlo simulation that obviates the tedious marching-out routine. A series system and a parallel system show the efficacy of the theory.