Probabilistic approach should be incorporated into engineering designs due to the presence of uncertainties. Most of the techniques in probabilistic designs involving uncertainty and reliability analyses are originally developed under the assumption that all random variables are normally distributed. When dealing with a problem that involves non-normal random variables, normal transformation are usually implemented so that the techniques developed for normal random variables could be applied. _{0 1 2} ² ₃ ³ _{0 1 2 3}...[ Read more ]

Probabilistic approach should be incorporated into engineering designs due to the presence of uncertainties. Most of the techniques in probabilistic designs involving uncertainty and reliability analyses are originally developed under the assumption that all random variables are normally distributed. When dealing with a problem that involves non-normal random variables, normal transformation are usually implemented so that the techniques developed for normal random variables could be applied.

In this thesis, the third-order polynomial normal transformation (TPNT) technique is studied. The TPNT technique relates an arbitrary random variable X and the standard normal random variable Z in a third-order polynomial form as X=a₀+a₁Z+a₂Z² +a₃Z³ . The polynomial coefficients, a₀ ,a₁ ,a₂ ,a,₃ could be determined from four different parameter estimation methods, namely, product-moments, L-moments, Least-square, and Fisher-Cornish methods. The TPNT technique is less restrictive in the sense that it merely requires the information of the first four statistical moments (product moments or L-moments) or quantiles of a random variable. Therefore, the TPNT technique is applicable even if only samples of the random variables are known, for which the Rosenblatt transformation cannot be applied because it requires the full distribution information.

The monotonically increasing relation between X and Z is required to guarantee the non-decreasing property of the cumulative distribution hnction (CDF) of X. However, this relation could be violated using the TPNT because of the intrinsic feature of third-order polynomial equation. A detailed investigation on the TPNT regarding the monotonicity range by the four parameter estimation methods and their accuracies in describing the statistical information of the non-normal random variables are made. Composite methods are proposed to compromise between the monotonicity range and accuracy. Furthermore, the ability of the TPNT to establish the joint probability density function (PDF) of correlated random variables from their marginal distributions is examined. The results are found to be competitive as compared with the existing models while the computation complexity is reduced significantly.

The applications of the TPNT in various engineering problems, such as uncertainty analysis, reliability analysis, and hydrologic frequency analysis, are proposed and demonstrated in this thesis. Test cases are extracted from literatures to examine the performance of the TPNT in these applications. All the results indicate promising features of the TPNT technique for its computation simplicity as well as comparable accuracy if not better.