An exponential normal transformation using the first three moments with application to structural reliability analysis

In practical engineering, the probability distributions of random variables are often unknown, making normal transformations using Rosenblatt or Nataf transformations unfeasible. To address this issue, the third-moment transformations based only on the first three moments have been proposed for structural reliability analysis. However, the expressions for these third-moment normal transformations involve the square root, denominator term, or logarithmic term, which introduce mathematical constraints and restrict their applicability. To overcome these mathematical constraints, this study proposes an explicit exponential third-moment normal transformation that covers a broad class of random variables. The determination of the proposed normal transformation begins with a predefined exponential form, from which the specific expression is derived based on the Taylor series expansion and the existing normal transformations. Subsequently, the coefficients of this expression are estimated through a trial-and-error process using numerous datasets, which are generated by averaging the results of existing normal transformations across a range of skewness levels. The five numerical examples demonstrate that the proposed exponential normal transformation not only extends the range of applications but also outperforms existing third-moment normal transformations in terms of accuracy.