Complete monotonic expression of the fourth-moment normal transformation for structural reliability

Probability distributions of basic random variables are essential for the accurate evaluation of structural reliability. In engineering practice, the probability distributions of some random variables are often unknown and the only available information about these may be their statistical moments. To conduct structural reliability analysis without the exclusion of random variables with unknown probability distributions, the fourth-moment normal transformation (FMNT) has been proposed. However, the applicability of expression of the FMNT has not been sufficiently investigated. Furthermore, the monotonic regions of the FMNT are not defined without which the application of the transformation is inconvenient, or even unreliable in reliability analysis. In the present paper, a complete expression of the FMNT including six cases with different combinations of skewness and kurtosis is derived, and the monotonicity of each case of the FMNT expression is confirmed. Literature suggests that the complete monotonic expression of the fourth-moment normal transformation is the first time to be successfully accomplished up to date. Through the numerical examples, the FMNT is found to be quite efficient for normal transformation and to be sufficiently accurate to include random variables with unknown probability distributions in structural reliability analysis.