The probability density evolution method for dynamic response analysis of non-linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non-linear stochastic structures is proposed. In the method, the dynamic response of non-linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one-dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark-Beta time-integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single-degree-of-freedom non-linear conservative system and dynamic responses of an 8-storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non-linear stochastic structures are usually irregular and far from the well-known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency