# Research on the Reliability of Bridge Structure Construction Process System Based on Copula Theory
## 1 Introduction
The approximate numerical calculation method is used to convert the complex high-dimensional integration into a simple one-dimensional integration problem so as to solve the approximate solution with certain accuracy. The PENT method (Probabilistic Network Evaluation Technique) proposed by [PERSON] et al. is a typical approximate numerical computation method [11]. The basic principle of this method is that the failure modes are divided into several groups according to the closeness of their correlation and the failure mode with the highest failure probability in each group is selected as the representative failure mode, assuming that the representative failure modes are independent of each other. The failure probability of the system is estimated by using the following equation:  
\[P_{f}=1-\prod_{i=1}^{m}\Bigl{(}1-P_{fi}\Bigr{)} \tag{1}\]  
where \(P_{f}\) is the failure probability of the structural system; \(m\) is the total number of representative failure modes; \(i\) is the serial number of the representative failure modes, the value range of \(i\) is \([1,m]\); and \(P_{fi}\) is the failure probability of each representative failure mode. In addition to the above three \"point estimation methods\", many scholars have formulated a series of extensions and improvements on the basis of these methods to make them more relevant to practical engineering applications. [PERSON] [12] proposed the reliability analysis method of series system by Taylor expansion of the cumulative distribution function of multidimensional normal distribution. However, for the reliability calculation of complex practical structural systems, the calculation amount is relatively large. [PERSON] et al. proposed a first-order reliability calculation scheme by converting the non-normal correlation uncertainty vector into an independent standard normal vector to simplify the reliability calculation of tandem systems. [PERSON] [13] proposed a conditional probability method based on the PENT method, which avoids the drawback of the PENT method. The interval estimation method includes the wide and narrow bounds methods. [PERSON] [14] was the first to propose a wide-bound formula for the failure probability of a tandem system. The basic idea is that the failure probability of a structural system lies between the failure probability solved when the failure modes are completely correlated and the failure mode solved when the failure modes are completely independent. Since the wide bounds formula does not consider the correlation between the failure modes, it is generally used to roughly estimate the reliability of the structural system. To address the problem that the wide bounds method is too broad in scope, [PERSON] [15] proposed a narrow-bounded range formula for the failure probability of a structural system. The narrow-bound method considers the probability of failure of two failure modes at the same time, which results in a narrower range than the wide-bound method and is more efficient in calculation. However, it only considers the failure of two failure modes together, and the estimation results are relatively rough. By analyzing domestic and foreign methods for calculating system reliability, it can be seen that the proposed system reliability calculation methods have two major defects. Firstly, the calculation is too cumbersome to be applied to the actual complex engineering structures. Secondly, the correlation between failure modes is not considered comprehensively. Therefore, a method that is easy to calculate and accurately considers the correlation between failure modes is needed to guide the engineering practice. In recent years, Copula theory has developed rapidly in the field of mathematics, providing a new way to establish the joint distribution function of related variables, which was first used in the field of finance. Copula theory was first proposed by [PERSON] [16] in the 1950s and [PERSON]'s theorem laid the foundation for the application and development of Copula theory.