Probability Density Evolution Analysis of Seismic Response of Nonlinear Structures With Random Parameters

Abstract

Abstract. The performance and reliability of engineering structures under extreme disastrous dynamic excitations (e.g. earthquakes) is of great concern due to the strong nonlinear mechanical behaviors as well as the uncertainty involved in both structural properties and external excitations. Great efforts have been devoted into this field in the past INTRODUCTIONDue to the randomness of the time and magnitude of earthquakes, it is possible for all the practical engineering structures to enter the phase of exhibiting nonlinear behaviors [1,2]. Therefore, it is of great importance to take into account the coupling of nonlinearity and randomness in assessing the performance and reliability of a practical structure rationally. In the past few decades, though the stochastic dynamics of structures, including the random vibration and the stochastic structural analysis, was extensively investigated, yielding a variety of approaches such as the Monte Carlo simulation (MCS) [3], the random perturbation method [4,5], and the orthogonal polynomials expansion [6,7], ect., great challenges are still encountered in the stochastic dynamical analysis of nonlinear structures of practical engineering interest [8]. Moreover, these methods are mainly aimed at obtaining the first second order statistics of structural responses rather than higher order statistics or probability density functions, which are of crucial importance in the reliability evaluation of nonlinear structures [9,10]. From the perspective of physical stochastic systems, a family of probability density evolution method (PDEM), which is capable of capturing the instantaneous probability density functions of stochastic responses of nonlinear structures, has been developed by Li and Chen in the past decade [11]. Based on the random event description of principle of preservation of probability, a generalized density evolution equation (GDEE), of which the dimension is free from the dimension of the original dynamical system, is derived and solved by incorporating the embedded physical equations [12].Two possible paths, i.e., the ensemble evolution path and the point evolution path, could be employed in the numerical solution of PDEM [12]. In the point evolution based approach, a smart selection of representative points is of paramount importance. Great endeavors have been devoted into the optimal selection of representative point set [13,14]. In the present paper, the generalized F-discrepancy (GF-discrepancy) is employed in the determination of representative points. A numerical example of a nonlinear frame-shear structure is presented. The stochastic seismic responses and the corresponding probabilistic information, including probability density functions, are obtained and analyzed. Then the evaluation of reliabilities under different threshold levels are carried out so that the accuracy and efficiency of PDEM could be verified. Problems to be further studied are discussed.