An efficient estimation of probability of first-passage in a linear system

Reliability analysis of a structure under random vibratory loads involves estimation of the probability of the response exceeding a limit. The classical, brute force approach to such analysis is the Monte Carlo method. However, due to its slow convergence rate, it is often impractical for large-scale engineering structures. In many engineering applications, such as offshore platforms under wave loads, the excitation is represented by Power Spectral Density (PSD) functions. Random time histories of the excitation are generated using a linear combination of sinusoids that are consistent with the PSD of input load. This paper proposes a method that reduces the computational cost of MCs of a linear system with a separable performance function; that is, a function that can be decomposed into parts and calculated independently. The method generates sinusoidal functions of the excitation, finds the system response to each sinusoid, and stores the responses in a database. Then it samples with replacement the sinusoids of the response from the database, finds the system response to the superposition of these sinusoids and checks for failure. This procedure yields a very large number of values of the failure indicator function even from a database with a modest number of sinusoids because it uses sampling with replacement. The efficiency of the proposed approach is demonstrated by estimating the probability of the first excursion in a ten-bar truss model. In this example, the method predicts the probability of failure using less than 0.2 % of the calculated values of the failure indicator function than the standard MCs.