Chao-Huang Cai

Evaluating the statistical moments of performance functions from the perspective of balancing accuracy and efficiency remains a challenge. This paper proposes an efficient algorithm for evaluating the statistical moments of performance functions. The main procedure of the proposed method consists of three steps. First, based on the bivariate dimension-reduction method, the performance function is approximated by a summation of one-dimensional and two-dimensional functions. Next, according to the criterion of delineating the existence of cross terms, the two-dimensional functions are decomposed as functions including and excluding cross terms. Third, the one-dimensional point estimate method is used to evaluate the statistical moments of the one-dimensional functions and the two-dimensional functions without cross terms, whereas the two-dimensional sparse grid stochastic collocation method is applied to estimate the statistical moments of the two-dimensional functions with cross terms. Several numerical examples are presented to illustrate the efficiency, accuracy, and applicability of the proposed method. The results demonstrate that the proposed method achieves a good balance between accuracy and efficiency and provides a useful tool for evaluating the statistical moments of performance functions.

In structural reliability analysis, the input variables are often nonnormal and correlated. A procedure for efficient normal transformation, i.e., transforming dependent nonnormal random variables into independent standard normal ones, is often required. In general, Rosenblatt transformation is available to realize the normal transformation when the joint probability density function (PDF) of basic random variables is available and Nataf transformation can be used when the marginal PDFs and correlation coefficients are known. However, the joint PDF and marginal PDFs of some random variables are often unknown in practice, and the probabilistic characteristics of these variables are easier to be expressed using the statistical moments and correlation matrix. It is in this regard that the objective of the present paper is to develop a methodology for normal transformation including correlated random variables with unknown joint PDF and marginal PDFs. Based on the third-moment transformation technique for transforming independent nonnormal random variables into independent standard normal ones, the third-moment transformation is further developed for transforming the correlated variables including unknown joint PDF and marginal PDFs into independent standard normal variables. A first-order reliability method for structural reliability analysis including correlated random variables with unknown joint PDF and marginal PDFs is developed based on the proposed transformation. Using the proposed method, the normal transformation and reliability analysis can also be achieved for correlated nonnormal random variables with knowledge of only the statistical moments and correlation matrix. The simplicity and efficiency of the proposed method is further demonstrated through several numerical examples.