# Research on the Reliability of Bridge Structure Construction Process System Based on Copula Theory
## 3 Engineering Applications
### Basic Information of the Algorithm  
Reference [34] analyzed the reliability of the long-span continuous rigid frame bridge system based on the engineering background of the Labajin Super Large Bridge in the Yingjing section of Yahu Expressway in China. The main span of the bridge is a separated variable cross-section continuous rigid frame, and the span combination of the whole bridge is (105 + 2 \(\times\) 200 + 105) m. The main span is 610 m long, the main beam of the superstructure adopts a prestressed single-box single chamber section, and the bridge is a fully prestressed concrete structure. In order to reduce the amount of calculation, reference [34] only took the single span bridge of the main span for calculation. The steps of analyzing the reliability of bridge system in reference [34] can be roughly divided into the following four steps. 1. Use MIDAS/Civil software to establish the finite element analysis model of the whole bridge. 2. Use the modified \(\beta\)-bound method to search for the main failure modes of the bridge structure. 3. Use the improved quadratic series response surface method to solve the failure function in different failure modes. 4. Use the wide boundary method, narrow boundary method, and PENT method to calculate the reliability indexes of the continuous rigid bridge system in the construction stage respectively. The basic information related to the calculations in this paper in reference [34] is shown below. In reference [34], the entire bridge construction process is divided into 28 construction stages. Each construction stage includes four processes: moving hanging baskets, tying reinforcement, placing concrete, and tensioning prestressing. The maximum cantilever construction stage of the middle pier \"T\" structure was selected as an example for the system reliability analysis. For the mechanical characteristics of this continuous rigid bridge, three failure modes were selected: axial compression stability failure and tensile and compressive stress failure during the construction and operation periods. The failure of any section of the main girder or the instability of any pier will lead to the failure of the bridge system, so each failure mode forms a tandem system. The FEA results were used to find the location where the stress in the main girder section is relatively large or small and the tensile and compressive stress failure of the main girder at this location and the stability failure of pier 10 were selected as the main failure modes as shown in the Table 2 below. The functions of each failure mode in the construction phase were obtained by fitting the modified quadratic series response surface method as follows. \[g_{1}(X)=430.966K_{R1}-15.389K_{R1}K_{G}-101.534K_{G} \tag{14}\]  
\[g_{2}(X) =1.96K_{R3}-(0.74-0.28K_{G}+0.29K_{E}+0.078K_{\zeta}-0.12K_{Q1} \tag{15}\] \[-0.13K_{E}^{2}+0.2K_{\xi}^{2}+0.01K_{Q1}^{2})\] (16) \[g_{3}(X) =26.5K_{R2}-(-10.66+19.52K_{G}+4.11K_{E}+5.44K_{\zeta}+1.11K_{Q1}\] \[-2.49K_{\zeta}^{2}-2.22K_{E}^{2}-2.22K_{\zeta}^{2}+0.27K_{Q1}^{2})\] (17) \[g_{4}(X) =26.5K_{R2}-(-0.41+12.67K_{G}-4.78K_{E}-2.78K_{\zeta}-0.19K_{Q1}\] (18) \[+0.99K_{\zeta}^{2}+2.22K_{E}^{2}-2.22K_{\xi}^{2}+0.27K_{Q1}^{2}) \tag{19}\]  
\[g_{5}(X)=26.5K_{R2}-(1.15+8.72K_{G}-0.67K_{E}+0.67K_{\xi}+0.46K_{Q1}+2.49K_{G}^{ 2}) \tag{20}\]  
where \(K_{R1}\) is the uncertainty factor for stability resistance calculations; \(K_{R2}\) is the axial compressive strength uncertainty factor for C60 concrete for the main beam; \(K_{R3}\) is the axial tensile strength uncertainty factor of C60 concrete for the main beam; \(K_{G}\) is the weight uncertainty of the member, \(K_{G}=\frac{G}{GK}\) (\(G\) is the actual member weight, \(G_{K}\) is the standard weight of the member specified in the current code); \(K_{E}\) is the equivalent modulus of elasticity uncertainty factor; \(K_{\xi}\) is the stress pipe friction uncertainty factor; \(K_{Q1}\) is the construction period expressed as the construction load uncertainty factor.