On the First Excursion Probability with Random Threshold

Abstract

The problem of reliability of structures under dynamic random loads has been a subject of considerable research in the last 20 years. Most of the research has been devoted to two general approaches: cumulative damage (fatigue failure) and first passage failures of structures. The latter approach deals usually with one dimensional problems and considers stationary and nonstationary processes with wide and narrowband spectra (e.g.[1–8]). Some generalizations for multidimensional processes are also available in the literature (e.g.[9–11]). For engineering purposes the most interesting generalization seems a multidimensional first passage problem with a random safe domain. Solutions of such the problem would lead to reliability estimations of structures with random load capacity under stochastic dynamic excitations (analogous to routine first order second moment methods known in static analysis). Unfortunately even for simple one dimensional problems the first passage probability can only be approximated (usually numerically). The cost of multidimensional approximations increases rapidly. Thus as a first approximation one can consider a first passage problem with random threshold. This problem has also been previously analyzed by Lange and Friedrich [12]. In what follows classic Rice formula [13] derived e.g. in [14] will be rederived to include the effect of randomness of level which is crossed. Then some approximations for stationary as well as nonstationary processes will be formulated.