## 5 A Case Study
### Analysis of the Internal Forces and External Deformations  
Figure 12 shows the contour plots of the analytical results for the radial internal forces, circumferential internal forces, and radial deformation of the Ring 3 segment during Stage I. As shown in the figure, the internal forces and deformations exhibited a clear symmetrical pattern, which is consistent with engineering practice. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ring** & **Depth/(m)** & \(p_{1}\)**/kPa** & \(p_{2}\)**Proposed Method/kPa** & \(p_{2}\)**Rankine/kPa** & \(p_{2}\)**Monitored/(kPa)** \\ \hline
22 & 6 & 0.00 & \(-\)38.65 & \(-\)33.16 & - \\
18 & 14 & 0.00 & \(-\)129.92 & \(-\)108.76 & - \\
14 & 22 & 0.00 & \(-\)179.85 & \(-\)184.36 & - \\
10 & 30 & 0.00 & \(-\)217.28 & \(-\)259.96 & - \\
6 & 38 & 0.00 & \(-\)237.32 & \(-\)335.56 & \(-\)190.33 \\
3 & 44 & 0.00 & \(-\)210.36 & \(-\)392.26 & \(-\)175.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Internal and external pressures on the segment at Stage III. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ring** & **Depth/(m)** & \(p_{1}\)**/kPa** & \(p_{2}\)**/kPa** & \(p_{2}\)**Monitored/(kPa)** \\ \hline
22 & 6 & \(-\)72.00 & \(-\)66.00 & - \\
18 & 14 & \(-\)168.00 & \(-\)154.00 & - \\
14 & 22 & \(-\)264.00 & \(-\)242.00 & - \\
10 & 30 & \(-\  
Based on the assumptions of the plane section, linear elasticity, and uncracked concrete, the axial force in the steel bars and the circumferential stress at the section center, as obtained from monitoring, can be calculated using Equation (41):  
\[\sigma^{\prime}_{\mid\varphi}=\frac{N_{s1}+N_{s2}}{2E_{s}}E_{c} \tag{41}\]  
where \(\sigma^{\prime}_{\varphi}\) is the circumferential internal stress; \(N_{s1}\) and \(N_{s2}\) are the monitored axial forces in the steel bars on the outer and inner sides of the segment, respectively; \(E_{s}\) is the elastic modulus of the steel bars; and \(E_{c}\) is the elastic modulus of the reinforced concrete. Figure 13 presents a comparison between the theoretical calculations and the monitored values of the annular internal forces for Rings 3 and 6. As shown in the figure, the relative error between the results obtained from earth pressure theory and the measured values was within 10%, with a correlation coefficient of 0.98, indicating a strong correlation between the analytical and monitored values. The error falls within a reasonable range, verifying the reliability of the proposed analysis. Figure 14 shows the analysis results of internal forces. As observed from Figures 13 and 14, the internal forces were positively correlated with the burial depth. When the internal pressure exceeded the external pressure, the maximum radial stress occurred at the inner arc surface, while the maximum radial deformation and circumferential stress occurred at the outer arc surface. Conversely, when the external pressure was greater than the internal pressure, the pattern was reversed. A large pressure difference between the inside and outside can result in excessive forces on the segment, while excessive internal pressure may cause the segment to undergo tension. In practical engineering, it is important to avoid both large pressure differences and excessive internal pressure. Figure 14: Analysis results of internal force and deformation. (**a**) Radial internal forces; (**b**) circumferential internal forces; (**c**) radial deformation. Figure 13: Comparison of analytical and monitored internal forces for Rings 3 and 6. Figure 15 presents the internal forces and deformation along the depth direction, where the solid line corresponds to the outcomes derived from the calculation method proposed herein, while the dotted line represents the results based on the Rankine earth pressure theory. During the first stage, the internal forces and deformations of the segment increased approximately linearly with burial depth, consistent with the linear growth of soil and fluid pressures acting on the segment at this stage.