Dynamic reliability assessment of nonlinear structures using extreme value distribution based on L-moments

Abstract In this paper, an efficient method using the linear moments (L-moments) is proposed for dynamic reliability assessment of nonlinear structures subjected to non-stationary ground motions. The proposed method comprises three main steps. First, the extrema of structural responses are obtained from the random function-spectral representation model (RFSRM) and nonlinear time history analysis, in which the principal points are first combined with the RFSRM and a sampling strategy is suggested to determine the final samples of extrema by deleting close values. Second, after the samples of extrema are obtained, the first four L-moments can be estimated from their definitions. Finally, using the first four L-moments of extrema of structural responses, the probability distribution of extrema is approximated by a cubic normal distribution based on the L-moments and the corresponding failure probability of a nonlinear structure is calculated. Three numerical examples are presented to demonstrate the efficiency, accuracy, and usefulness of the proposed method for the dynamic reliability assessment of nonlinear structures, and the conventional spectrum representation method and subset simulation method are used for comparison.

[1]  Yan-Gang Zhao,et al.  Monotonic Expression of Polynomial Normal Transformation Based on the First Four L-Moments , 2020 .

[2]  P. Spanos,et al.  Response and First-Passage Statistics of Nonlinear Oscillators via a Numerical Path Integral Approach , 2013 .

[3]  Allen I. Fleishman.  A method for simulating non-normal distributions , 1978 .

[4]  Armen Der Kiureghian,et al.  TAIL EQUIVALENT LINEARIZATION METHOD FOR NONLINEAR RANDOM VIBRATION , 2007 .

[5]  Jianbing Chen,et al.  The Number Theoretical Method in Response Analysis of Nonlinear Stochastic Structures , 2007 .

[6]  Zhao-Hui Lu,et al.  Unified Hermite Polynomial Model and Its Application in Estimating Non-Gaussian Processes , 2019, Journal of Engineering Mechanics.

[7]  F. Kong,et al.  Wavelet-Galerkin approach for power spectrum determination of nonlinear oscillators , 2014 .

[8]  Jinghai Gong,et al.  Estimate of small first passage probabilities of nonlinear random vibration systems by using tail approximation of extreme distributions , 2016 .

[9]  M. Shinozuka,et al.  Digital simulation of random processes and its applications , 1972 .

[10]  S. Rice.  Mathematical analysis of random noise , 1944 .

[11]  Jun Xu,et al.  Efficient reliability assessment of structural dynamic systems with unequal weighted quasi-Monte Carlo Simulation , 2016 .

[12]  Pol D. Spanos,et al.  Galerkin scheme based determination of first-passage probability of nonlinear system response , 2014 .

[13]  Jianbing Chen,et al.  A GF-discrepancy for point selection in stochastic seismic response analysis of structures with uncertain parameters , 2016 .

[14]  M. Stein.  Large sample properties of simulations using latin hypercube sampling , 1987 .

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

[16]  Yan-Gang Zhao,et al.  First passage probability assessment of stationary non-Gaussian process using the third-order polynomial transformation , 2018, Advances in Structural Engineering.

[17]  Jianbing Chen,et al.  The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters , 2007 .

[18]  E. Vanmarcke.  On the Distribution of the First-Passage Time for Normal Stationary Random Processes , 1975 .

[19]  Yan-Gang Zhao,et al.  Fourth-Moment Standardization for Structural Reliability Assessment , 2007 .

[20]  Kaichao Zhang,et al.  Reliability analysis for structures with multiple temporal and spatial parameters based on the effective first-crossing point , 2017 .

[21]  J. J. Coleman.  Reliability of Aircraft Structures in Resisting Chance Failure , 1959 .

[22]  J. R. Wallis,et al.  Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form , 1979 .

[23]  Masanobu Shinozuka,et al.  Simulation of Multivariate and Multidimensional Random Processes , 1971 .

[24]  M. Grigoriu.  Crossings of non-gaussian translation processes , 1984 .

[25]  J. Beck,et al.  Estimation of Small Failure Probabilities in High Dimensions by Subset Simulation , 2001 .

[26]  Y. Peng,et al.  Random function based spectral representation of stationary and non-stationary stochastic processes , 2016 .

[27]  Wei Yang,et al.  Closed-Form Solution to First Passage Probability for Nonstationary Lognormal Processes , 2016 .

[28]  Ying Min Low,et al.  A new distribution for fitting four moments and its applications to reliability analysis , 2013 .

[29]  Arvid Naess,et al.  Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method , 2008 .

[30]  Alice Zoppé,et al.  Principal points of univariate continuous distributions , 1995 .

[31]  Zhao-Hui Lu,et al.  A flexible distribution and its application in reliability engineering , 2018, Reliab. Eng. Syst. Saf..

[32]  Jianbing Chen,et al.  The equivalent extreme-value event and evaluation of the structural system reliability , 2007 .

[33]  M. Shinozuka,et al.  Simulation of Seismic Ground Motion Using Stochastic Waves , 1989 .

[34]  H. Cramér.  On the intersections between the trajectories of a normal stationary stochastic process and a high level , 1966 .

[35]  Zhangjun Liu,et al.  Probability Density Evolution of a Nonlinear Concrete Gravity Dam Subjected to Nonstationary Seismic Ground Motion , 2018 .

[36]  Phaedon-Stelios Koutsourelakis.  A note on the first-passage problem and VanMarcke's approximation - short communication , 2007 .

[37]  George Deodatis,et al.  Non-stationary stochastic vector processes: seismic ground motion applications , 1996 .

[38]  S. Winterstein.  Nonlinear Vibration Models for Extremes and Fatigue , 1988 .

[39]  P. Spanos,et al.  Hilbert Transform–Based Stochastic Averaging Technique for Determining the Survival Probability of Nonlinear Oscillators , 2019, Journal of Engineering Mechanics.

[40]  Yeou-Koung Tung,et al.  Investigation of polynomial normal transform , 2003 .

[41]  S. Winterstein,et al.  Comparing L-Moments and Conventional Moments to Model Current Speeds in the North Sea , 2011 .

[42]  Jun He.  Approximate Method for Estimating Extreme Value Responses of Nonlinear Stochastic Dynamic Systems , 2015 .

[43]  P. Spanos,et al.  Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements , 2016 .

[44]  E. Cornish,et al.  The Percentile Points of Distributions Having Known Cumulants , 1960 .

[45]  Y. Wen.  Method for Random Vibration of Hysteretic Systems , 1976 .

[46]  Zhangjun Liu,et al.  Probability density evolution analysis of stochastic nonlinear structure under non-stationary ground motions , 2019, Structure and Infrastructure Engineering.