# Research on the Reliability of Bridge Structure Construction Process System Based on Copula Theory
## 3 Engineering Applications
### Calculation of System Reliability Indicators  
#### 3.2.1 Establishment of Copula Function  
Analyzing the basic case of the calculation in Section 3.1, it can be seen that the failure of any one mode in the construction process will lead to the failure of the whole structural system. So, the failure modes are in series with each other. The joint failure probability of the bridge structure system construction process failure occurring is shown in the following equation. For simplifying the calculation, this paper only considered the correlation between two failure modes.  
\[\begin{array}{ll}P_{f}&=P\{Z_{1}(X)\leq 0\cup Z_{2}(X)\leq 0\cup\cdots\cup Z_{n }(X)\leq 0\}\\ &=\sum\limits_{i=1}^{n}P\{Z_{i}(X)\leq 0\}-\sum\limits_{1\leq i<j\leq n}^{n}P\{Z_ {i}(X)\leq 0,Z_{j}(X)\leq 0\}\\ &+\sum\limits_{1\leq i<j<k\leq n}^{n}P\{Z_{i}(X)\leq 0,Z_{j}(X)\leq 0,Z_{k}(X) \leq 0\}\\ &+\cdots+(-1)^{n-1}P\{Z_{1}(X)\leq 0,Z_{2}(X)\leq 0,\cdots,Z_{n}(X)\leq 0\}\\ &\approx\sum\limits_{i=1}^{n}P\{Z_{i}(X)\leq 0\}-\sum\limits_{1\leq i<j \leq n}^{n}P\{Z_{i}(X)\leq 0,Z_{j}(X)\leq 0\}\end{array} \tag{21}\]  
Bringing in the arithmetic example in Section 3.1, Equation (21) becomes  
\[\begin{array}{ll}P_{f}&=P\{g_{1}(X)\leq 0\cup g_{2}(X)\leq 0\cup g_{3}(X) \leq 0\cup g_{4}(X)\leq 0\cup\\ &g_{5}(X)\leq 0\cup g_{6}(X)\leq 0\cup g_{7}(X)\leq 0\}\end{array} \tag{22}\]  
The correlation between failure modes was established using the Copula function, and according to Equation (2), it can be seen that the bridge system failure probability is a 7-dimensional Copula function. According to Equation (3), it can be derived that the probability of failure mode occurrence of the binary combined structural system is shown in the following equation.  
\[P(g_{1}(X)\leq 0,g_{2}(X)\leq 0)=C\Big{(}P_{f_{1}},P_{f_{2}};\,\theta_{12} \Big{)} \tag{23}\]  
Treating each failure mode as a probability event. According to the addition formula of multiple probability events in probability theory, the corresponding calculation formula of the failure probability of the bridge system is obtained, in which the joint failure probability of failure criterion is calculated according to Copula theory. Then Equation (22) can be converted into the following equation.  
\[\begin{array}{ll}P_{f}&=\{g_{1}(X)\leq 0\cup g_{2}(X)\leq 0\cup g_{3}(X) \leq 0\cup g_{4}(X)\leq 0\cup\\ &g_{5}(X)\leq 0\cup g_{6}(X)\leq 0\cup g_{7}(X)\leq 0\\ &=P\{g_{1}(X)\leq 0\}+P\{g_{2}(X)\leq 0\}+P\{g_{3}(X)\leq 0\}\\ &+P\{g_{4}(X)\leq 0\}+P\{g_{5}(X)\leq 0\}+P\{g_{6}(X)\leq 0\}\\ &+P\{g_{7}(X)\leq 0\}-C\Big{(}P_{f_{1}},P_{f_{2}}\Big{)}-C\Big{(}P_{f_{1}},P_{f_{3} }\Big{)}-C\Big{(}P_{f_{1}},P_{f_{4}}\Big{)}\\ &-C\Big{(}P_{f_{1}},P_{f_{5}}\Big{)}-C\Big{(}P_{f_{1}},P_{f_{6}}\Big{)}-C\Big{(} P_{f_{1}},P_{f_{7}}\Big{)}-C\Big{(}P_{f_{2}},P_{f_{3}}\Big{)}\\ &-C\Big{(}P_{f_{2}},P_{f_{4}}\Big{)}-C\Big{(}P_{f_{2}},P_{f_{5}}\Big{)}-C\Big{(} P_{f_{2}},P_{f_{6}}\Big{)}-C\Big{(}P_{f_{2}},P_{f_{7}}\Big{)}\\ &-C\Big{(}P_{f_{3}},P_{f_{4}}\Big{)}-C\Big{(}P_{f_{3}},P_{f_{5}}\Big{)}-C\Big{(} P_{f_{3}},P_{f_{6}}\Big{)}-C\Big{(}P_{f_{3}},P_{f_{7}}\Big{)}\\ &-C\Big{(}P_{f_{4}},P_{f_{5}}\Big{)}-C\Big{(}P_{f_{4}},P_{f_{6}}\Big{)}-C\Big{(} P_{f_{4}},P_{f_{7}}\Big{)}-C\Big{(}P_{f_{5}},P_{f_{6}}\Big{)}\\ &-C\Big{(}P_{f_{5}},P_{f_{7}}\Big{)}-C\Big{(}P_{f_{6}},P_{f_{7}}\Big{)}\end{array} \tag{24}\]