Analytical approximation for stationary reliability of certain and uncertain linear dynamic systems with higher‐dimensional output

An analytical approximation for the calculation of the stationary reliability of linear dynamic systems with higher-dimensional output under Gaussian excitation is presented. For systems with certain parameters theoretical and computational issues are discussed for two topics: (1) the correlation of failure events at different parts of the failure boundary and (2) the approximation of the conditional out-crossing rate across the failure boundary by the unconditional one. The correlation in the first topic is approximated by a multivariate integral, which is evaluated numerically by an efficient algorithm. For the second topic some existing semi-empirical approximations are discussed and a new one is introduced. The extension to systems with uncertain parameters requires the calculation of a multi-dimensional reliability integral over the space of the uncertain parameters. An existing asymptotic approximation is used for this task and an efficient scheme for numerical calculation of the first- and second-order derivatives of the integrand is presented. Stochastic simulation using an importance sampling approach is also considered as an alternative method, especially for cases where the dimension of the uncertain parameters is moderately large. Comparisons between the proposed approximations and Monte Carlo simulation for some examples related to earthquake excitation are made. It is suggested that the proposed analytical approximations are appropriate for problems that require a large number of consistent error estimates of the probability of failure, as occurs in reliability-based design optimization. Numerical problems regarding computational efficiency may arise when the dimension of both the output and the uncertain parameters is large.