## 1 Introduction
The recognition after the main failure mode of the structure system needs to consider the correlation of each failure mode, effectively integrating the failure mode to calculate the failure probability of the structural system reliability. After identifying the main failure modes of the structural system, it is necessary to consider the correlation of each failure mode to effectively integrate the failure probability of each failure mode to calculate the reliability of the structural system. Therefore, how to find the main failure modes and how to calculate the failure probability are two key problems in reliability analysis. In terms of structural failure mode search, the most commonly used methods are the optimality criterion method, \(\beta\)-Unzipping method, and branched-and-bound method [30]. [PERSON] et al. proposed an optimality criterion method based on the load accumulation to search for major failure modes [31]. [PERSON] et al. established a failure tree by the \(\beta\)-Unzipping method to identify the main failure modes [32]. [PERSON] et al. proposed an improved critical intensity branched-and-bound method considering correlation between failure paths to identify major failure modes, which improved the search efficiency of traditional methods [33]. After the main failure modes are identified, the failure modes are usually combined by series and parallel systems to obtain the failure probability of the bridge structure by accurate calculation. In terms of calculating the failure probability of the system, the interval estimation method (mainly including the narrow bounds method and wide bounds method) and probabilistic network technique method are often used to calculate the failure probability of the system [34]. [PERSON] proposed the wide bounds method to calculate the failure probability of series and parallel systems, ignoring the correlation between failure modes, which could only roughly estimate the failure probability of structural systems [35]. [PERSON] et al. proposed the narrow bounds method considering the correlation between any two failure modes, to determine the upper and lower bounds of the system failure probability [36]. [PERSON] et al. arranged and grouped failure modes according to correlation based on the probabilistic network technique method, and selected m-group-significant failure modes to replace all the major failure modes to calculate the joint failure probability [37]. Although these methods have some applicability to some extent, there are some defects in the excessive failure modes or the strong correlation between them. As Copula functions can not only capture the nonlinear correlation between failure modes, but also construct multiple joint distribution functions, which can easily calculate the joint failure probability of multiple members, they are introduced in the reliability assessment of structural systems. [PERSON] et al. established a theoretical model based on the Copula function and applied it in the reliability analysis of failure-related structural systems [38]. [PERSON] et al. applied the Copula function to the system reliability assessment of simply supported beam bridges, and established the system reliability assessment method of series, parallel, and mixed models [39], which may be convenient and efficient for the structural system assessment of hinged slab bridges with multiple slabs. In engineering practice, some parameters and detection information of bridge structures are difficult to be expressed by random variables, but these parameters and detection information have obvious uncertainty, so scholars introduce fuzzy theory into the risk assessment and reliability assessment of bridge structures. [PERSON] et al. proposed a bridge risk assessment framework under multiple disasters based on fuzzy theory [40]. [PERSON] et al. used fuzzy theory to evaluate the state of reinforced concrete bridges in a corrosive environment [41]. [PERSON] et al.