## 3 Case Study
### Compared with the FEM
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \(T_{i}\) **[N]** & \(A_{i}\) **[\(m^{2}\)]** & \(F_{i}\) **[N]** & \(L_{i}^{{}^{\prime}}\) **[m]** & \(x_{i}\) **[mm]** & \(X_{i}\) **[mm]** \\ \hline Basic parameter & & \multicolumn{4}{c}{\(E_{n}=2.05\times 10^{11}\) Pa, \(A_{n}=0.0042\) m\({}^{2}\)} & \\ Initial state & 0 & \(1.01\times 10^{6}\) & 27.126 & 27.090 & 0 & 0.76 \\
1 st tension & \(2.03\times 10^{5}\) & \(9.55\times 10^{5}\) & 27.118 & 27.090 & 2.13 & 2.90 \\
1 st unload & \(2.49\times 10^{5}\) & \(8.11\times 10^{5}\) & 27.118 & 27.097 & \(-1.45\) & 1.45 \\
2 nd tension & \(4.06\times 10^{5}\) & \(7.65\times 10^{5}\) & 27.111 & 27.097 & 1.65 & 3.10 \\
2 nd unload & \(4.56\times 10^{5}\) & \(6.09\times 10^{5}\) & 27.111 & 27.104 & \(-1.59\) & 1.52 \\
3 rd tension & \(6.09\times 10^{5}\) & \(5.64\times 10^{5}\) & 27.105 & 27.104 & 1.60 & 3.12 \\
3 rd unload & \(6.59\times 10^{5}\) & \(4.06\times 10^{5}\) & 27.105 & 27.111 & \(-1.60\) & 1.52 \\
4 th tension & \(8.11\times 10^{5}\) & \(3.61\times 10^{5}\) & 27.098 & 27.111 & 1.60 & 3.12 \\
4 th unload & \(8.62\times 10^{5}\) & \(2.03\times 10^{5}\) & 27.098 & 27.118 & \(-1.60\) & 1.52 \\
5 th tension & \(1.01\times 10^{6}\) & \(1.58\times 10^{5}\) & 27.092 & 27.118 & 1.60 & 3.12 \\
5 th unload & \(1.07\times 10^{6}\) & 0 & 27.092 & 27.126 & \(-1.60\) & 1.52 \\ \hline \hline \end{tabular} Note: \(F_{i}\): the internal force of the new hanger; \(T_{i}\): the internal force of the pocket hanging hanger; \(L_{i}\): the unstressed length of the hanger; \(L_{i}^{\prime}\): the unstressed length of the pocket hanging hanger; \(x_{i}\): the displacement in the current case; \(X_{i}\): the accumulative displacement.  
\end{table} 
Table 2: Calculation results of the removal process in different cases during the new hanger installation process. Figure 11: Bridge deck displacement test results at the lower end of the new hanger in different cases. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Type** & **Applicable Parts** & **Modulus of Elasticity [kN/m\({}^{2}\)]** & **Bulk Density [kN/m\({}^{3}\)]** \\ \hline
16 Mn & Arch rib & 2.10 \(\times\) 10\({}^{8}\) & 76.98 \\ OVMLZM7-55 III & Old hangers & 2.05 \(\times\) 10\({}^{8}\) & 78.5 \\ Finished deformed bar & Temporary hangers & 2.06 \(\times\) 10\({}^{8}\) & 100.7 \\ OVMLZM7-55 IV & New hangers & 2.05 \(\times\) 10\({}^{8}\) & 78.5 \\ C50 & Deck & 3.45 \(\times\) 10\({}^{7}\) & 26 \\ Q345 & Main girders and crossbeams & 2.06 \(\times\) 10\({}^{8}\) & 100.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Materials of the model. In the hanger replacement process of half-through arches with a suspended deck by cable hangers, there are two important steps: old hanger cutting and new hanger installation. The finite element simulation is as follows:  
1. In the FEM, the cutting of the old hanger actually involves the simulation of the geometric nonlinearity. It is noted that the mentioned nonlinearity refers to the geometric nonlinearity caused by the cross-sectional area change of the hanger and the non-stress length change of the hanger in the hanger demolition and tension process. It is usually time-consuming to simulate the nonlinearity by the finite element method, and there is no good solution for the simulation of the non-stress length change of the hanger in the finite element method. However, the proposed method in this paper can not only accurately calculate the results, but is also more efficient. 2. In order to simplify, several repeated elements are usually established at the same position of the old hanger. These elements have the same parameters except the cross-sectional area. The cutting process of the old hanger is simulated by activating the hanger corresponding to the area of the construction stage in different construction stages and passivating the hanger of the previous construction stage. It is also important to note that the element needs to be activated along the initial tangential displacement of the member. 3.