Random Vibration and Dynamic Reliability Analyses for Nonlinear MDOF Systems under Additive Excitations via DPIM

In this paper, a novel direct probability integral method (DPIM) is proposed to address stochastic response and dynamic reliability analyses of nonlinear multidegree-of-freedom (MDOF) systems subjected to additive random excitations. First, the probability density integral equation (PDIE) with a Dirac delta function is established to characterize the randomness propagation of the MDOF system based on the principle of probability conservation. DPIM decouples the PDIE and governing differential equation of the system and efficiently solves PDIE by using the partition of probability space and smoothing of the Dirac function to achieve the probability density function of stochastic response. Moreover, the equivalent relation among the PDIE, Fokker-Planck-Kolmogorov (FPK), and Chapman-Kolmogorov-Smoluwski (CKS) equations in the Markov system is also derived, demonstrating the applicability of DPIM for the Markov system. Then, the first-passage dynamic reliability of the MDOF system under additional excitations is assessed by introducing equivalent extreme value mapping of the stochastic process. Finally, two examples of nonlinear MDOF systems subjected to filtered Gaussian white noise and nonstationary seismic excitation are investigated, respectively. Comparing the calculated results for nonlinear MDOF systems with those using the path integral method and Monte Carlo simulation indicates the high accuracy and efficiency of DPIM. Specifically, the numerical example of a 9-story frame building with a nonlinear hysteretic model and soil-structure interaction (SSI) indicates that as the stiffness of soil decreases, the dynamic reliability of frame building is gradually reduced.