Structural Reliability Analysis Based On Copula Function

Uncertainties of loads, material properties and structure sizes widely exist in engineering structures, reliability method is an effective tool to process uncertainties of these problems. The current reliability methods typically assume that input random variables are independent and can be converted to a standard normal space for the solution. However, under most circumstances, there are correlations among random variables, and these correlations can have considerable effect on the reliability analysis result. Currently, the primary reliability methods to address correlations include the Nataf and Rosenblatt transformations. However, Nataf transformation only considers linear correlations among the variables. Therefore, this transformation can only accurately measure correlations among the variables under some specific sample distribution conditions. Many sample distribution types or joint distribution functions do not follow a Gaussian distribution, and Nataf transformation could lead to considerable errors. Rosenblatt transformation is based on the accurate joint probability distribution function, whereas in actual applications, multidimensional joint probability distribution functions are typically unknown. Thus Rosenblatt transformation has considerable limitations relative to actual application. Therefore, it is very important for the reliability analysis and design of the complex structure to develop a new method to overcome these shortcomings of the above methodsThis article conducts a series of studies on a new mathematical tool in the structural reliability analysis field, i.e., the Copula function, and the main work is as follows:(1) A Copula function based evidence theory model for correlation analysis and corresponding structural reliability method is proposed to deal with reliability design problems with dependent evidence variables. The Copula function is used to describe the correlation between evidence variables, the weight of evidence variable is calculated from the samples to identify the best Copula function. The joint basic probability assignment (BPA) is gained by making the difference between the marginal BPAs using Copula function. Then the reliability interval is gained by calculating the cumulative BPA of the focal elements in the reliable domain.(2) A Vine Copula based structural reliability analysis method is proposed, which is an effective approach for performing a reliability analysis on complex multi-dimensional correlation problems. A joint probability distribution function among multi-dimensional random variables is established using a Vine Copula function, based on which a reliability analysis model is constructed. Two solution algorithms are proposed to solve this reliability analysis model:one is based on Monte Carlo simulation (VC-MCS) and another one is based on the first order reliability method (VC-FORM). Although VC-MCS has low efficiency, it can provide a generalized solution approach and provide an important reference solution for the development of other high-efficiency algorithms. VC-FORM has high efficiency, and can be used to solve real-world engineering problems.(3) The Vine Copula function is introduced into the field of structural system reliability analysis and corresponding structural system reliability analysis method based on Vine Copula function is established. The Vine Copula function is used to describe the dependence among different performance functions, and the system failure probability is calculated using marginal failure probability and Monte Carlo integration. This method provides a general and flexible way of calculating the system failure probability in isolation from the marginal failure probabilities and can capture different dependence among the performance functions.