Structural System Reliability Analysis Based On Multivariate Copula Function

Uncertainties widely exist in external loads,material properties,and structural sizes of engineering structures and systems,affecting the structural stability and probably resulting in structural failures.Therefore,reliability analysis is necessary for quantifying and dealing with the uncertainties.Usually in structural systems,there are multiple failure modes which are mutually dependent,making the structural system reliability analysis quite complicated.Current analysis methods typically assume that the dependence among the failure modes is linear,and then use the linear correlation coefficient to model and quantify the dependence.However,when th e dependence is not linear or the tail dependence exists,the linear model,or the correlation coefficient,may lead to large errors.Therefore,in this thesis,we propose to introduce the Copula function to model the nonlinear dependence.The main contributions are as follows.(1)A new class of two-dimensional Copula functions is constructed.Based on the properties of some Copula functions with invariant dependence structures under univariate truncation,a new two-dimensional conditional Copula function can be constructed by considering the Nested Archimedean Copula function under univariate truncation.(2)A new method to construct multivariate Copula functions is proposed,providing an effective tool for the modeling of multid imensional dependency.Firstly,the joint distribution function of multi-dimensional random variables is decomposed based on the D-Vine model,thus transforming the joint distribution function into multiple products of some two-dimensional Copula functions of the original variables or of their conditional variables.Secondly,the maximum likelihood estimation method and the Akaike information criterion(AIC)are employed to statistically analyze the samples and hence determine the optimal two-dimensional Copula functions between the variables.Finally,the analytical expr ession of the multivariate Copula function is determined.(3)A structural system reliability analysis method based on the multivariate Copula function is proposed to solve system reliability problems involving multiple failure modes with different kinds of dependencies.In the proposed method,the Copula function is employed to model the nonlinear and tail dependence among failure modes,and therefore the proposed method is more accurate than those methods which only capture the linear part of the dependence w ith the linear correlation model.