Efficient exceedance probability computation for randomly uncertain nonlinear structures with periodic loading

A method is developed to compute low-level response amplitude exceedance probabilities associated with uncertain nonlinear structures with random parameters and deterministic periodic forcing. Emphasis is focused on accurate and efficient computation in the tails of the exceedance probability distribution function associated with the largest possible response of one displacement variable for unspecified forcing frequency and normally distributed parameters. This gives a measure of system reliability when a large amplitude response exceedance of a specified threshold is designated as the mode of failure. The method exploits the First-Order Reliability Method (FORM) in which the failure surface is constructed via the Harmonic Balance Method (HBM). This combined approach is tested on a Duffing oscillator with harmonic forcing and up to three uncertain parameters, for which the frequency of multiple-solution-maximum-amplitude is found directly, and the probability computed via the Hasofer-Lind reliability index. The accuracy of the proposed HBM-FORM, in the tails of the amplitude exceedance probability, is shown for the Duffing example to be acceptably accurate, whereas the efficiency is shown to be around 1000 times faster than Direct Integration and around 200 times faster than Monte Carlo simulation.