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.

中文翻译：

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相关结构可靠性分析 Rackwitz-Fiessler 方法的改进

Rackwitz-Fiessler (RF) 方法被公认为是解决不相关非正态可靠性问题的一种有效方法，它通过将原始非正态变量转换为基于等效正态条件的等效正态变量。然而，当涉及到相关可靠性问题时，这种传统的RF方法往往会被抛弃，因为等效Normal条件的逐点实现特性使得RF方法难以清晰地描述变换变量的相关性。为此，从等概率变换和copula理论的角度对传统的RF方法进行了一些改进。首先，RF方法从原始空间到标准Normal空间的正向变换过程从几何学的角度解释为等概率变换。这一观点使我们可以合理地描述变换变量的随机依赖关系，与纳塔夫变换（NATAF）中的随机依赖关系相同。因此，提出了一种相应的增强射频（EnRF）方法来处理皮尔逊线性相关描述的相关可靠性问题。此外，我们根据copula不变定理和严格递增的等概率变换揭示了RF方法的隐式高斯copula假设。同时，基于仅 copula 的秩相关，例如 Spearman 和 Kendall 相关，引入了两种改进的 RF (IRF) 方法来克服 EnRF 中 Pearson 相关性的潜在缺陷。后来，以NATAF为参考，在Hasofer-Lind可靠性算法中也讨论了上述三种RF方法的计算成本和效率。最后，展示了四个说明性结构可靠性示例，以验证新提出的射频方法的可用性和优势。