Dual Power Transformation and Yeo-Johnson Techniques for Static and Dynamic Reliability Assessments

This paper addresses key challenges in the static and dynamic reliability analysis of engineering structures, particularly the difficulty in accurately estimating large reliability indices and small failure probabilities. For static reliability problems, a dual power transformation is employed to transform the performance function into a form approaching a normal distribution. The high-order unscented transformation is then applied to compute the first four moments of the transformed performance function. Subsequently, the fourth-moment method is used to calculate large reliability indices, offering a novel improvement over traditional methods such as FORM and SORM. For dynamic reliability problems, the low-discrepancy sampling technique is integrated to efficiently compute structural responses under random seismic excitation, improving computational efficiency for complex dynamic systems. The Yeo-Johnson transformation is introduced to normalize the extreme values of dynamic responses, and the first four moments of the transformed extreme values are statistically evaluated. Additionally, a third-order polynomial transformation (TPT) is applied to approximate the probability density function, leading to the derivation of the probability of exceedance (POE) curve. The optimal transformation parameters for both the dual power and Yeo-Johnson transformations are determined using the Jarque-Bera (JB) test. Four numerical examples, coupled with Monte Carlo simulation, validate the proposed framework's accuracy and efficiency, providing a robust tool for static and dynamic reliability analysis. This unified approach represents a significant advancement by integrating novel transformations and fourth-moment method, providing a powerful and efficient tool for static and dynamic reliability analysis of engineering structures.