Reliability-based vector nonstationary random critical earthquake excitations for parametrically excited systems

The study considers the earthquake response of stack-like structures subjected to simultaneous action of random horizontal and vertical earthquake acceleration components. The governing equation of motion in this case is approximated by a set of coupled randomly time varying ordinary differential equations. The components of earthquake accelerations are modelled as nonstationary Gaussian random processes that are obtained by multiplying deterministic modulating functions with partially specified stationary random processes. Specifically, it is assumed that the matrix of power spectral density (psd) functions of the stationary components is not known, while, the variance, average rate of zero crossings, entropy rate and frequency range of interest are taken to be known. The unknown input psd matrix is determined such that the reliability index associated with a specified structure performance function is minimized. The solution procedure employed, combines the theory of Hasofer–Lind reliability indices, response surface modelling and constrained nonlinear optimization tools. The critical input psd matrix so obtained leads to the definition of excitation models that produce the least favorable response, which, at the same time, possess a few of the well known properties of earthquake loads. A numerical example that illustrates the concepts developed with reference to a chimney structure is provided.