## 2 Calculation Methods
### Seismic Fragility Analysis of Bridge Systems Based on the Copula Technique  
The bridge structure is a series system composed of all the structural members of the bridge. Therefore, to evaluate the failure probability of the overall structure of the bridge, it is necessary to consider the seismic fragility evaluation of each of its components. The reliability theory is used to estimate the failure probability of the structure under each damage state:  
\[P_{fs}=P[g_{1}(X)\leq 0,g_{2}(X)\leq 0,\ldots g_{i}(X)\leq 0] \tag{4}\]  
where, \(P_{fs}\) is the failure probability of the bridge structural system, \(g_{1}(X),g_{2}(X),g_{i}(X)\) and \(g_{i}(X)\) are the functions of failure probability for each component of the bridge system. The probability of the simultaneous failure of all components is:  
\[ \begin{array}{l}P[g_{1}(X)\leq 0,g_{2}(X)\leq 0,\ldots g_{i}(X)\leq 0]\\ =P\{F_{1}[g_{1}(X)]\leq F_{1}(0),F_{2}[g_{2}(X)]\leq F_{2}(0),\ldots,F_{i}[g_{ i}(X)]\leq F_{i}(0)\}\\ =C[F_{1}(0),F_{2}(0),\ldots,F_{3}(0)]=C\left(P_{f1},P_{f2},\ldots,P_{fi}\right) \end{array}  \tag{5}\]Then, the failure probability of the bridge system considering the correlation of components is obtained:  
\[ \begin{array}{ll}P_{fs}&=P[g_{1}(X)\leq 0\cup g_{2}(X)\leq 0\ldots\cup g_{i}(X) \leq 0]\\ &=P_{f1}+P_{f2}+\ldots+P_{fi}-C\Big{(}P_{f1},P_{f2},\ldots P_{fi}\Big{)}\end{array}  \tag{6}\]  
where, \(P_{f1},\ P_{f2},\ P_{fi}\) are the failure probabilities of each component of the bridge system; \(C\) is the copula function. The seismic fragility of the bridge system can be obtained by substituting the seismic fragility of the bridge components into Equation (3). According to Equation (6), the copula function is the key to solving the fragility of the bridge system. The above method can easily determine the seismic fragility of each component using IDA calculation results. However, it is difficult to calculate the seismic fragility of two or more components because of the interaction between components. This is because the joint probability distribution of multiple components cannot be directly established by the marginal distribution function of a single component. Therefore, this paper will use the copula technique to analyze the random seismic fragility of the bridge system. The copula technique provides a common method for describing the nonlinear correlation between random variables. The copula function is defined as an n-dimensional probability distribution on \([0,1]^{n}\), which is expressed as [30]:  
\[C(u_{1},u_{2},\ldots u_{n})=P(U_{1}\leq u_{1},U_{2}\leq u_{2},\ldots U_{n}\leq u _{n}) \tag{7}\]  
where, \(u_{1},u_{2},\ldots u_{n}\) is the sample set of uniform random variable \(U\). Given that \(F_{i}(x_{i})\) is an edge probability distribution, its joint probability distribution is represented by \(F(x_{1},x_{2},\ldots x_{n})\). Then, the relationship between \(F_{i}(x_{i})\) and \(F(x_{1},x_{2},\ldots x_{n})\) can be established by the copula function. \[F(x_{1},x_{2},\ldots x_{n})=C(F_{1}(x_{1}),F_{2}(x_{2}),\ldots F_{n}(x_{n}))=C (u_{1},u_{2},\ldots u_{n}) \tag{8}\]  
If \(F_{i}(x_{i})\) (\(i=1,\ldots,n\)) are all continuous functions, then the copula function \(C(u_{1},u_{2},\ldots u_{n})\) is unique. It can be seen that the copula function connects the joint distribution of variables with their edge distribution. As a result of the copula function, the joint distribution function can be expressed explicitly with the edge distribution function. Although it is the connection function of the edge distribution, the form of the copula function is not limited by the edge distribution function. \[P(X_{1},X_{2},\ldots X_{n}|IM)=C(P(x_{1}),P(x_{2}),\ldots P(x_{n})|IM) \tag{9}\]  
where, \(P(X_{i})(i=1,\ldots,n)\) represents the seismic fragility of a single component, which can be obtained from Equation (3), and \(n\) represents the number of components damaged at the same time. By substituting Equation (9) into Equation (6), the seismic fragility of the bridge system can be expressed as:  
\[