## 2 Stress Analysis Method of the Arch Ring Section
### Analysis of the Initial Stress State of the Arch Ring Section Considering Prestressing Effect
section after prestressing tensioning and the stress formula of the top and bottom flanges of the arch ring section after buckle cable tensioning under the coupling effect of cable force and prestressing are derived. The detailed derivation and analysis process is as follows. **Condition 1:** Prestressing tensioning condition  
As shown in Figure 2, any two arch ring sections are selected for structural analysis, and the stress distribution at the top and bottom flanges of the section after prestressing tensioning is analyzed for sections \(i\) and \(i+1\) in the figure. According to the calculation and analysis model of the arch ring section, the stresses at the top and bottom flanges of \(i+1\) section can be calculated as follows. Axial force:  
\[N_{i+1}=\sigma_{pe_{i+1}}\cdot A_{p_{i+1}}+\sigma^{\prime}_{pe_{i+1}}\cdot A^ {\prime}_{p_{i+1}}+G_{i+1}\cdot\cos\beta_{i+1} \tag{5}\]  
Bending moment:  
\[M_{i+1}=\sigma_{pe_{i+1}}\cdot A_{p_{i+1}}\cdot y_{pn_{i+1}}-\sigma^{\prime}_{pe _{i+1}}\cdot A^{\prime}_{p_{i+1}}\cdot y_{pn_{i+1}}-G_{i+1}\cdot\sin\beta_{i+1 }\cdot\frac{l_{i+1}}{2}-M_{\text{s}_{i+1}} \tag{6}\]  
Eccentric distance:  
\[e_{n_{i+1}}=\frac{M_{i+1}}{N_{i+1}} \tag{7}\]  
The top flanges:  
\[\sigma_{e_{i+1}}=\frac{N_{i+1}}{A_{n_{i+1}}}+\frac{N_{i+1}\cdot e_{n_{i+1}}}{ I_{n_{i+1}}}\cdot y_{n_{i+1}} \tag{8}\]  
The bottom flanges:  
\[N_{p}=\sigma_{pe}\cdot A_{p}+\sigma_{pe}^{\prime}\cdot A_{p}^{\prime} \tag{1}\]0  
The stresses at the top and bottom flanges of the section \(i\) can be calculated as follows. Axial force:  
\[N_{p}=\sigma_{pe}\cdot A_{p}+\sigma_{pe}^{\prime}\cdot A_{p}^{\prime} \tag{1}\]1  
Bending moment:  
\[ \begin{array}{c}M_{i}=\sigma_{pe_{i}}\cdot A_{p_{i}}\cdot y_{pn_{i}}-\sigma^{ \prime}_{pe_{i}}\cdot A^{\prime}_{p_{i}}\cdot y^{\prime}_{pn_{i}}-G_{i}\cdot \sin\beta_{i}\cdot\frac{l_{i}}{2}+T_{i}\cdot\sin\alpha_{i}\cdot l_{i}+T_{i} \cdot\cos\alpha_{i}\cdot\frac{l_{i}}{2}-G_{i+1}\cdot\sin\beta_{i+1}\cdot\left( \frac{l_{i+1}}{2}\right.\\ \left.+l_{i}\right)-M_{\text{s}_{i}}\\ \end{array}  \tag{11}\]  
Eccentric distance:  
\[e_{n_{i}}=\frac{M_{i}}{N_{i}} \tag{12}\]  
Figure 2: Structural calculation model of the arch ring prestressing tensioning working condition. The top flanges:  
\[\sigma_{c_{i}}=\frac{N_{i}}{A_{n_{i}}}+\frac{N_{i}\cdot e_{n_{i}}}{I_{n_{i}}} \cdot y_{n_{i}} \tag{13}\]  
The bottom flanges:  
\[\sigma_{t_{i}}=\frac{N_{i}}{A_{n_{i}}}-\frac{N_{i}\cdot e_{n_{i}}}{I_{n_{i}}} \cdot y_{n_{i}} \tag{14}\]  
**Condition 2:** Buckle cable tensioning condition  
As shown in Figure 3, any two arch ring sections are selected for structural analysis, and the stress distribution at the top and bottom flanges of the arch ring sections under the action of buckle and stress coupling after buckle tensioning is analyzed with the \(i\) and \(i+1\) sections in the figure. The analysis of the calculation model of the arch ring section in Figure 3 is described. The stresses at the top and bottom flanges of section \(i\)+1 can be calculated as follows. Axial force:  
\[N_{i+1}=\sigma_{pe_{i+1}}\cdot A_{p_{i+1}}+\sigma^{\prime}_{pe_{i+1}}\cdot A ^{\prime}_{p_{i+1}}+T_{i+1}\cdot\text{cos}\alpha_{i+1}+G_{i+1}\cdot\text{cos} \beta_{i+1} \tag{15}\]  
Bending moment:  
\[M_{i+1}=\sigma_{pe_{i+1}}\cdot A_{p_{i+1}}\cdot y_{pn_{i+1}}-\sigma^{\prime}_{ pe_{i+1}}\cdot A^{\prime}_{p_{i+1}}\cdot y^{\prime}_{pn_{i+1}}+T_{i+1}\cdot \text{sin}\alpha_{i+1}\cdot I_{i+1}+T_{i+1}\cdot\text{cos}\alpha_{i+1}\cdot \tfrac{h}{2}-G_{i+1} \tag{16}\]  
\[\text{-sin}\beta_{i+1}\cdot\tfrac{I_{i+1}}{2}-M_{s_{i}}\]  
\[\text{Eccentric distance:}\]  
\[e_{n_{i+1}}=\frac{M_{i+1}}{N_{i+1}} \tag{17}\]  
The top flanges:  
\[\sigma_{c_{i+1}}=\frac{N_{i+1}}{A_{n_{i+1}}}+\frac{N_{i+1}\cdot e_{n_{i+1}}}{ I_{n_{i+1}}}\cdot y_{n_{i+1}} \tag{18}\]  
The bottom flanges:  
Figure 2:0  
The stresses at the top and bottom flanges of section \(i\) can be calculated as follows. Axial force:  
Figure 2:1  
Figure 2:2 Simplified mathematical model of main arch ring buckle tensioning condition.