Closed-Form Design Quantiles Under Skewness and Kurtosis: A Hermite Approach to Structural Reliability

Abstract

A Hermite-based framework for reliability assessment within the limit state method is developed in this paper. Closed-form design quantiles under a four-moment Hermite density are derived by inserting the Gaussian design quantile into a calibrated cubic translation. Admissibility and implementation criteria are established, including a monotonicity bound, a positivity condition for the platykurtic branch, and a balanced Jacobian condition for the leptokurtic branch. Material data for the yield strength and ductility of structural steel are fitted using moment-matched Hermite models and validated through goodness-of-fit tests. A truss structure is subsequently analysed to quantify how non-Gaussian input geometry influences structural resistance and its associated design value. Variance-based Sobol sensitivity analysis shows that departures of the radius distribution toward negative skewness and higher kurtosis increase the first-order contribution of geometric variables and thicken the lower tail of the resistance distribution. The closed-form Hermite design resistances agree closely with numerical integration results and reveal systematic deviations from FORM estimates, which depend solely on the mean and standard deviation. Monte Carlo simulations confirm these trends and highlight the slow convergence of tail quantiles and higher-order moments. The proposed approach remains fully compatible in the Gaussian limit and offers a practical complement to EN 1990 verification procedures when skewness and kurtosis have a significant influence on design quantiles.