Nonlinear Response of Structures Subjected to Stochastic Excitations via Probability Density Evolution Method

Stochastic response analysis plays an increasingly important role in assessing the performance and reliability of engineering structures subjected to disastrous dynamic loads such as earthquakes and strong winds. In the past decade, a family of probability density evolution method (PDEM) has been developed where a completely decoupled generalized density evolution equation (GDEE) is proposed. The dimension of GDEE could be arbitrary and in most cases a reduced one-dimensional equation is adequate. This provides an efficient solution for stochastic response of complex engineering structures. In the present paper, PDEM is incorporated with the efficient representation of stochastic processes to implement stochastic dynamic response analysis of multi-degree-of-freedom (MDOF) systems subjected to stochastic excitations. Stochastic dynamic responses of linear and nonlinear base-excited systems are investigated by PDEM, the pseudo-excitation method and the Monte Carlo simulations. The Sobol' sequence is employed to generate representative white noise process, and a high-performance stochastic harmonic function representation with fewer random variables is employed to generate representative time histories of a stochastic process with specified power spectral density model. The involved seismic excitations are modeled by banded white noise, Kanai-Tajimi filtered stationary process, Clough-Penzien filtered stationary process and Clough-Penzien filtered process with non-stationary modulation, respectively. Numerical results reveal that PDEM is feasible for stochastic response analysis of MDOF nonlinear systems subjected to stochastic excitations with fair accuracy and efficiency.

[1]  Yu-Kweng Michael Lin.  Probabilistic Theory of Structural Dynamics , 1976 .

[2]  Ding Wang,et al.  Physical random function model of ground motions for engineering purposes , 2011 .

[3]  Jie Li,et al.  A random physical model of seismic ground motion field on local engineering site , 2012 .

[4]  L. Landau.  Quantum Mechanics-Nonrelativistic Theory , 1958 .

[5]  E. Vanmarcke,et al.  Random Fields: Analysis and Synthesis (Revised and Expanded New Edition) , 2010 .

[6]  M. Moshinsky,et al.  The harmonic oscillator in modern physics , 1996 .

[7]  G. Schuëller.  A state-of-the-art report on computational stochastic mechanics , 1997 .

[8]  Jianbing Chen,et al.  Stochastic modeling of engineering dynamic excitations for stochastic dynamics of structures , 2012 .

[9]  Jintai Chung,et al.  A new family of explicit time integration methods for linear and non‐linear structural dynamics , 1994 .

[10]  Jianbing Chen,et al.  Optimal determination of frequencies in the spectral representation of stochastic processes , 2013 .

[11]  Erik H. Vanmarcke.  Quantum origins of cosmic structure , 1997 .

[12]  Jianbing Chen,et al.  Stochastic Dynamics of Structures , 2009 .

[13]  Michel Loève,et al.  Probability Theory I , 1977 .

[14]  Kathrin Abendroth,et al.  Nonlinear Finite Elements For Continua And Structures , 2016 .

[15]  Erik H. Vanmarcke,et al.  Random Fields: Analysis and Synthesis. , 1985 .

[16]  M. Shinozuka,et al.  Simulation of Stochastic Processes by Spectral Representation , 1991 .

[17]  Moshe Gitterman,et al.  The noisy oscillator : the first hundred years, from Einstein until now , 2005 .

[18]  Weiling Sun,et al.  Stochastic Harmonic Function Representation of Stochastic Processes , 2013 .

[19]  Jianbing Chen,et al.  The principle of preservation of probability and the generalized density evolution equation , 2008 .

[20]  Weiqiu Zhu,et al.  Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation , 2006 .

[21]  Jie Li,et al.  Homogenization-based multi-scale damage theory , 2010 .

[22]  N. Bershad,et al.  Random differential equations in science and engineering , 1975, Proceedings of the IEEE.

[23]  Abraham Pais,et al.  'Subtle Is the Lord...' The Science and the Life of Albert Einstein , 1984 .

[24]  Roger Ghanem,et al.  Partition of the probability-assigned space in probability density evolution analysis of nonlinear stochastic structures , 2009 .

[25]  F. Pillichshammer,et al.  Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration , 2010 .

[26]  Shahram Sarkani,et al.  Random Vibrations: Analysis of Structural and Mechanical Systems , 2003 .

[27]  T. Caughey,et al.  The exact steady-state solution of a class of non-linear stochastic systems , 1982 .

[28]  Jianbing Chen,et al.  Advances of Probability Density Evolution Method for Nonlinear Stochastic Systems , 2011 .

[29]  Farzad Naeim.  Dynamics of Structures, Theory and Applications in Earthquake Engineering, 2nd Edition , 2001 .

[30]  Jianbing Chen,et al.  Dynamic response and reliability analysis of structures with uncertain parameters , 2005 .

[31]  Zhenzhou Lu,et al.  Parametric sensitivity analysis of the importance measure , 2012 .

[32]  Yongbo Peng,et al.  Scheme and application of phase delay spectrum towards spatial stochastic wind fields , 2013 .

[33]  Jianbing Chen,et al.  Improving Point Selection in Cubature by a New Discrepancy , 2013, SIAM J. Sci. Comput..

[34]  Pol D. Spanos,et al.  Karhunen-Loéve Expansion of Stochastic Processes with a Modified Exponential Covariance Kernel , 2007 .

[35]  H. Zhang,et al.  Parameter Analysis of the Differential Model of Hysteresis , 2004 .

[36]  Helmut J. Pradlwarter,et al.  Reliability assessment in structural dynamics , 2013 .