# Research on the Reliability of Bridge Structure Construction Process System Based on Copula Theory
## 3 Engineering Applications
### Calculation of System Reliability Indicators
#### 3.2.6 Improvement of Computational Theory  
In order to better guide the actual bridge construction project, this paper makes the following improvements to the traditional Copula theory.  
According to the calculation process in Section 3.2.5 of this paper, when establishing the linkage between failure modes, the correlation between two of the individual failure modes is established many times. So, when applying Equation (25), the same indicator is subtracted from other indicators several times when calculating. For example, in \(C\left(P_{f_{3}},P_{f_{4}}\right)\), \(C\left(P_{f_{3}},P_{f_{5}}\right)\), \(C\left(P_{f_{3}},P_{f_{6}}\right)\), and \(C\left(P_{f_{3}},P_{f_{7}}\right)\), the linkage between failure mode 3 and other failure modes is established several times, and in \(C\left(P_{f_{3}},P_{f_{6}}\right)\), \(C\left(P_{f_{4}},P_{f_{6}}\right)\), \(C\left(P_{f_{5}},P_{f_{6}}\right)\), and \(C\left(P_{f_{6}},P_{f_{7}}\right)\), the linkage between failure mode 6 and other failure modes is established several times. Although the links established are between two different failure modes, if we directly use Equation (25), the probability of partial failure of the same failure mode will be subtracted several times, which will lead to a low failure probability, large reliability index, and safe calculation results. Therefore, this paper proposes the following improvement: when subtracting the Copula joint distribution of the correlation between failure modes, the inverse of the number of occurrences of such failure modes should be multiplied as a coefficient before the Copula function. According to Equations (21) and (23), the improved equation is shown in the following equation.  
\[\begin{array}{ll}P_{f}&=P\{Z_{1}(X)\leq 0\cup Z_{2}(X)\leq 0\cup\cdots \cup Z_{n}(X)\leq 0\}\\ &=\sum\limits_{i=1}^{n}P\{Z_{i}(X)\leq 0\}-\sum\limits_{1\leq i<j\leq n}^{n}P \{Z_{i}(X)\leq 0,Z_{j}(X)\leq 0\}\\ &+\sum\limits_{1\leq i<j<k\leq n}^{n}P\{Z_{i}(X)\leq 0,Z_{j}(X)\leq 0,Z_{k}(X) \leq 0\}\\ &+\cdots+(-1)^{n-1}P\{Z_{1}(X)\leq 0,Z_{2}(X)\leq 0,\cdots,Z_{n}(X)\leq 0\} \\ &\approx\sum\limits_{i=1}^{n}P\{Z_{i}(X)\leq 0\}-\sum\limits_{1\leq i<j \leq n}^{n}P\{Z_{i}(X)\leq 0,Z_{j}(X)\leq 0\}\\ &=\sum\limits_{i=1}^{n}P_{f_{Z_{i}}}-\frac{1}{\max-i_{\min}1}\sum\limits_{1 \leq i<j<n}^{n}C\left(P_{f_{Z_{i}}},P_{f_{Z_{j}}}\right)\end{array} \tag{26}\]  
We recalculated the reliability index of the bridge system in the construction process in the example of this paper according to Equation (26). Equation (25) becomes the following equation.  
\[\begin{array}{ll}P_{f}&=P\{g_{1}(X)\leq 0,g_{2}(X)\leq 0,g_{3}(X)\leq 0,g_{4}(X) \leq 0,\\ &g_{5}(X)\leq 0,g_{6}(X)\leq 0,g_{7}(X)\leq 0\}\\ &=P\{g_{1}(X)\leq 0\}+P\{g_{2}(X)\leq 0\}+P\{g_{3}(X)\leq 0\}\\ &+P\{g_{4}(X)\leq 0\}+P\{g_{5}(X)\leq 0\}+P\{g_{6}(X)\leq 0\}\\ &+P\{g_{7}(X)\leq 0\}-\frac{1}{4}[C\big{(}P_{f_{3}},P_{f_{4}}\big{)}-C\big{(}P_{ f_{3}},P_{f_{5}}\big{)}-C\big{(}P_{f_{3}},P_{f_{6}}\big{)}\\ &-C\Big{(}P_{f_{3}},P_{f_{7}}\big{)}-C\big{(}P_{f_{4}},P_{f_{6}}\big{)}-C\big{(} P_{f_{4}},P_{f_{7}}\big{)}\\ &-C\Big{(}P_{f_{5}},P_{f_{6}}\big{)}-C\big{(}P_{f_{5}},P_{f_{7}}\big{)}-C\big{(} P_{f_{6}},P_{f_{7}}\big{)}]\end{array} \tag{27}\]  
Using Equation (27), it was calculated that the failure probability of the system during the construction of the bridge structure was \(4.9895\times 10^{-6}\). With the help of MATLAB software, the system reliability index aw 4.4176. Compared with the calculation results in Section 3.2.5 of this paper, it was found that the improved Copula theory for calculating the reliability of bridge structure system overcomes the problem that the calculation results of the traditional copula theory are partial to safety. The calculation results fell within the boundary range (4.376, 4.52) calculated by the wide boundary method.