## 3 System Reliability Evaluation Method Considering Member Failure Credibility  
In the failure criterion of the existing system reliability assessment, it is considered that the contribution of each slab failure to the failure of the bridge structure system is equal. In fact, the remaining slab will bear the external load when a certain slab fails; however, the load effect of the remaining slab under the external load is also different when the failed slab is located at different spatial positions of the bridge. For example, the external load borne by the adjacent #2 and #3 slabs after the failure of the side slab is larger than that borne by the #2 and #3 slabs after the failure of the middle slab. Our research group concluded that different spatial positions of slabs in bridges would lead to different contributions of member failure to structural system failure of simply supported slab bridges through theoretical analysis and engineering practice. The engineers unanimously confirmed this inference after the investigation of experienced design engineers. Therefore, it can be considered that the contribution of different slab failures to the structural system failure of simply supported slab bridges is different; however, it is difficult to quantitatively express such differences by mechanical theory. This paper proposes to use reliability to express such differences. For hinged slab bridges, the transverse load distribution coefficient can reflect the different spatial positions of slabs and indirectly indicate the contribution of member failure to system failure. Therefore, the transverse distribution coefficient is adopted in this paper to construct the failure credibility index of hinge joints without damage, and its expression is as follows:  
\[Bel_{i}=\frac{m_{i}^{0}}{\max(M)} \tag{14}\]  
where \(Bel_{i}\) is the failure credibility of the \(i\)-th member, \(m_{i}^{0}\) represents the transverse distribution coefficient of the \(i\)-th member without damage, and \(M=[m_{1}^{0},m_{2}^{0},\cdots,m_{n}^{0}]\) represents the vector composed of the transverse distribution coefficient of each main beam without damage. The joint failure probability of the simultaneous failure of multiple members according to the Copula theory [48] is as follows:  
\[P_{f}=P[Z_{1}(X)\leq 0,Z_{2}(X)\leq 0,\cdots,Z_{n}(X)\leq 0]=C\Big{(}P_{fZ_{1 }},P_{fZ_{2}},\cdots,P_{fZ_{n}}\Big{)} \tag{15}\]where \(Z_{i}\) denotes the limit state equation of \(i\)-th members, \(P_{f_{Z_{i}}}\) represents the failure probability of the \(i\)-th member, and \(C(\cdot)\) expresses the copula function. On this basis, member failure credibility is considered in this paper, and the joint failure probability of n members established by the Copula function is expanded in Equation (16). \[ \begin{array}{ll}P_{f,Bel}&=P\{(Z_{1}(X)\leq 0,Bel_{1}),(Z_{2}(X)\leq 0,Bel_{2}), \cdots,(Z_{n}(X)\leq 0,Bel_{n})\}\\ &=C\Big{(}P_{f_{Z_{1}},Bel_{1}},P_{f_{Z_{2}},Bel_{2}},\cdots,P_{f_{Z_{n}},Bel_ {n}}\Big{)},\end{array}  \tag{16}\]  
where \(P_{f,Bel}\) represents the joint failure probability of n members considering failure credibility, and \(P_{f_{Z_{i}},Bel_{i}}\) represents the failure probability of the \(i\)-th member considering the failure credibility. As member failure probability and member failure credibility are independent variables, \(P_{f_{Z_{i}},Beli}=P(Z_{i}(X)\leq 0)\times Bel_{i}\). There are three kinds of hinged slab structure system failure modes: Failure criterion I considers that any one member failure causes the whole system failure; failure criterion II reckons that if two arbitrary adjacent members fail, the system fails; failure criterion III holds that if any three adjacent member fails, the whole system will fail. The authors of this paper proposed a structural system reliability assessment method considering three kinds of system failure criteria based on Copula function calculation [48].