Nonstationary seismic responses of structure with nonlinear stiffness subject to modulated Kanai–Tajimi excitation

Abstract

A simple structure under earthquake excitation is modeled as a single-degree-of-freedom system with nonlinear stiffness subject to modulated Kanai–Tajimi excitation. The nonstationary responses including the nonstationary probability densities of the system responses and the statistical moments are obtained in semi-analytical form. By applying the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker–Planck–Kolmogorov(FPK) equation governing the nonstationary probability density of the amplitude is derived. Then, the solution of the FPK equation is approximately expressed by a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients are solved from a set of linear first-order differential equations. Thus, the nonstationary probability densities of the amplitude and the state responses as well as the statistic moments of the amplitude are obtained. Finally, two types of the modulating functions, i.e. constant function and exponential function, are considered to give some semi-analytical formulae. The proposed procedures are checked against the Monte Carlo simulation. The effects of the structure natural frequency and the intensity of the excitation as well as the ground stiffness on the system responses are discussed. It should be pointed out that the proposed method is good for broadband excitation and light damping.