## 4 Calculation Example
### Analysis Model  
The tie-down tension analysis model is shown in Figure 34, including structural columns, compression ring, cable net, CR brackets, tie-downs, and other components. Figure 33: Traction cable forces are almost the same before and after adjusting. Figure 32: Internal forces of axis 1 and axis 15 CR external supports before and after adjusting. A total of 58 tie-downs connecting cable clamps and their corresponding anchorage are installed, among which the axis 35 and axis 51 tie-downs have a herringbone shape considering the actual construction site requirements. The link 10 element (tension-only element) is adopted for the new tie-downs in the model, and its material characteristics are shown in Table 14.  
Results of Synchronous Proportional Tension  
Synchronous equal proportion tension is adopted in this section, which means that the tension rings are pulled down to the design elevation at the same time according to the same proportion from the configuration after traction lifting. According to the analysis results in Section 4.2.1, the maximum distance of the tension ring elevation relative to the design elevation is about 2 m after the cable net is lifted. The integral tension of the tie-downs is divided into five steps, and each step reduces the tension ring by about 0.4 m. The corresponding setting of tie-down tension calculation conditions is shown in Table 15. \"Tie-down tension 20%\" means that the reduction in tie-downs is 20% of the distance between the tension ring elevation and the relative design elevation, and the tie-downs pulled down by 100% means that the tension ring is pulled to the design elevation. During the tensioning of tie-downs, the vertical elevation of the tension ring gradually decreases until the pull-down is completed. The tension ring elevation of the pull-down \begin{table}
\begin{tabular}{c c} \hline \hline
**Calculation Condition** & **State** \\ \hline C1 & Traction lifting completed \\ C2 & Tie-down tension 20\% \\ C3 & Tie-down tension 40\% \\ C4 & Tie-down tension 60\% \\ C5 & Tie-down tension 80\% \\ C6 & Tie-down tension 100\% (pull down tension ring to design elevation, i.e., pull-down is completed) \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Calculation conditions during synchronous proportional tension tension. Figure 34: Tie-down tension analysis model. (a) Whole model; (b) Tie-downs. completion state is between \(-11\) mm and 38.8 mm, basically reaching the designed equilibrium state elevation, as shown in Figure 35.  
The radial deformation Ux of 56 CR nodes under each tension calculation condition through local cylindrical coordinate system was drawn as a deformation curve, which is shown in Figure 36. From C1 to C3, the CR radial deformation at the minor axis expands outward to 144 mm. The CR radial deformation at 45\({}^{\circ}\) is relatively small and gradually decreases, and at the long axis it remains small and stable. From C4 to C6, CR radial deformation gradually decreases and finally reaches about 0, which means that the plane configuration of the CR reaches the designed one when tie-down tension is completed. Figure 37 shows that the maximum equivalent stress of the CR during various tension construction steps gradually increases and finally reaches the maximum value of 144.7 MPa. Figure 36: Radial deformation Ux of CR under various calculation conditions in local coordinate system. Figure 35: Vertical deformation Uz (m) of C6. Nephograms of tie-down forces under each calculation condition are shown in Figure 38, which indicate that the tie-down force at the short axis is largest, followed by the long axis, and the cable force at the quarter is relatively the smallest. When the pull-down is completed, the cable force of the tie-downs reaches a maximum value of 547.6 kN. Figure 37: Maximum equivalent stress curve of CR. Figure 38: Tie-down force nephograms (N): (a) C2, (b) C3 (c) C4, (d) C5, (e) C6.