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This paper introduces an efficient methodology for evaluating the first four central moments of performance functions in structural reliability assessment. The procedure of the proposed method is divided into three major steps. First, a performance function is approximated as the sum of univariate and bivariate functions by the bivariate dimension-reduction method. Next, a new criterion based on importance measure analysis is proposed for dividing the bivariate functions into significant bivariate functions and non-significant bivariate functions, and then the new approximation of the performance function is obtained. Finally, formulas are derived for evaluating the first four central moments of the proposed performance function, in which the first four raw moments of the univariate and bivariate functions can be achieved by a point estimating method in a standard normal space. The effectiveness and efficiency of the proposed method are demonstrated through three numerical examples.

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. [ABSTRACT FROM AUTHOR]

Estimation of statistical moments of performance functions is one of the main topics in structural reliability analysis by moment methods. It is possible to apply the well-developed multiple-dimensional cubature formulas to assess such statistical moments. In this paper, a new criterion is proposed to fulfill the aim. Several efficient cubature formulas are firstly revisited. Then, a new criterion is proposed to select "the best" cubature formula that gives the most accurate statistical moments of performance functions. This criterion is established based on the marginal moments of input random variables. The cubature formula that provides the smallest difference between the estimated values and the exact values of marginal moments is determined as the adopted one for statistical moments of performance functions. Several numerical examples are presented to illustrate the effectiveness of the proposed criterion, in which Monte Carlo simulations are employed for comparison. [ABSTRACT FROM AUTHOR]

Information on the distribution of the basic random variable is essential for the accurate analysis of structural reliability. The usual method for determining the distributions is to fit a candidate distribution to the histogram of available statistical data of the variable and perform approximate goodness-of-fit tests. Generally, such candidate distribution would have parameters that may be evaluated from the statistical moments of the statistical data. In the present paper, a cubic normal distribution, whose parameters are determined using the first four moments of available sample data, is investigated. A parameter table based on the first four moments, which simplifies parameter estimation, is given. The simplicity, generality, flexibility and advantages of this distribution in statistical data analysis and its significance in structural reliability evaluation are discussed. Numerical examples are presented to demonstrate these advantages. [ABSTRACT FROM AUTHOR]