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

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 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 then analysed to quantify how non-Gaussian input geometry influences structural resistance and its corresponding design value. Variance-based Sobol sensitivity analysis demonstrates that departures of the radius distribution towards negative skewness and higher kurtosis increase the first-order contribution of geometric variables and thicken the lower tail of the resistance distribution. Closed-form Hermite design resistances are shown to agree with numerical integration results and reveal systematic deviations from FORM estimates, which rely solely on the mean and standard deviation. Monte Carlo simulation studies 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.