Stochastic Mainshock–Aftershock Simulation and Its Applications in Dynamic Reliability of Structural Systems via DPIM

A novel approach for nonlinear stochastic dynamic analysis is proposed and illustrated with nonlinear building structures subjected to mainshock–aftershock sequences. First, a stochastic seismic sequence model with stochastic parameters was established, and its generation method was derived based on the source–path–site mechanism. Then, the representative point sets of seismic parameters could be chosen based on generalized F-discrepancy, and the correlation between the mainshock and aftershock parameters could be determined by using Copula theory. Finally, the stochastic dynamic response was obtained by solving the probability density integral equation (PDIE). Furthermore, the first-passage dynamic reliability could be obtained by the direct probability integral method (DPIM) combined with the absorbing condition approach. This novel approach was used to obtain stochastic dynamic results for four structures subjected to stochastic seismic sequences, which were compared to those using Monte Carlo simulation (MCS) and probability density evolution method (PDEM) to demonstrate the proposed method's correctness and efficiency. Additionally, the influence of aftershocks on nonlinear structures is explained from the perspective of probability for the first time.