Rackwitz–Fiessler (RF) method is well accepted as an efficient way to solve the uncorrelated non-Normal reliability problems by transforming original non-Normal variables into equivalent Normal variables based on the equivalent Normal conditions. However, this traditional RF method is often abandoned when correlated reliability problems are involved, because the point-by-point implementation property of equivalent Normal conditions makes the RF method hard to clearly describe the correlations of transformed variables. To this end, some improvements on the traditional RF method are presented from the isoprobabilistic transformation and copula theory viewpoints. First of all, the forward transformation process of RF method from the original space to the standard Normal space is interpreted as the isoprobabilistic transformation from the geometric point of view. This viewpoint makes us reasonably describe the stochastic dependence of transformed variables same as that in Nataf transformation (NATAF). Thus, a corresponding enhanced RF (EnRF) method is proposed to deal with the correlated reliability problems described by Pearson linear correlation. Further, we uncover the implicit Gaussian copula hypothesis of RF method according to the invariant theorem of copula and the strictly increasing isoprobabilistic transformation. Meanwhile, based on the copula-only rank correlations such as the Spearman and Kendall correlations, two improved RF (IRF) methods are introduced to overcome the potential pitfalls of Pearson correlation in EnRF. Later, taking NATAF as a reference, the computational cost and efficiency of above three proposed RF methods are also discussed in Hasofer–Lind reliability algorithm. Finally, four illustrative structure reliability examples are demonstrated to validate the availability and advantages of the new proposed RF methods.
Midpoint Derivative-Based Closed Newton-Cotes Quadrature