## 4 Engineering Application and Analysis
### Analysis of Calculated Results for Rock-Lining Synergy
#### 4.3.1 Effect of Lining Thickness on Tunnel Lining and Surrounding Rock Interaction  
As a critical parameter in tunnel engineering design, lining thickness significantly influences the stability and deformation characteristics of the surrounding rock due to its varying resistance capabilities. Figure 12 presents a detailed description of the interrelation between the lining's stress distribution, the displacement of its inner and outer walls, the stress around the tunnel wall, the plastic zone thickness, and the reserved deformation. The analysis results are summarized below:  
1. Figure 12a illustrates that the two curves are characterized by a nonlinear descending section followed by an ascending section. As depicted in Figure 12b, the displacements of both the inner and outer walls of the lining exhibit a nonlinear decrease, with the inner wall displacement \(u_{\text{a}}\) consistently being slightly greater than the outer wall displacement \(u_{\text{b}}\). Under identical conditions of \(\Delta u\), an increase in the inner-to-outer radius ratio \(t\) results in a nonlinear rise in the surrounding rock pressure on the lining's outer wall and a corresponding decrease in the maximum compressive stress on the lining's inner wall. The displacement curves of the inner and outer walls progressively converge, indicating that thicker linings exhibit enhanced structural stiffness and stability. This effectively restrains the surrounding rock, reduces the pressure on the lining's inner wall, and stabilizes the mechanical response within the lining structure.
2. The initial segments of the two curves in Figure 12c exhibit a similar linear decline, while the curve in Figure 12d presents a distinct parabolic shape. An increased inner-to-outer radius ratio \(t\) results in steeper gradients for the hoop stress \(\sigma_{\text{dw}}\) and the equivalent stress \(\sigma_{\text{vw}}\) around the tunnel wall, with the parabola's left endpoint shifting to the right. This indicates that thicker lining structures more effectively confine stress dissipation in the surrounding rock, thereby reducing the likelihood of the rock entering a plastic state.  
In the Wufengshan Tunnel lining design scheme, represented by the curve for \(t=1.115\) in Figure 12, for every 1% increase in the maximum peripheral rock wall displacement \(\Delta u\), the maximum compressive stress \(\sigma_{\text{max}}\) in the hoop direction of the lining's inner wall decreases by an average of 351.17 kPa, and the plastic zone thickness increases by an average of 39.52 mm. Theoretical calculations indicate that the optimal timing for support initiation is when the peripheral rock wall displacement reaches 8.18 mm, which is approximately 30.4% of the maximum wall displacement. Analysis from Figure 12d shows that the plastic zone begins to form in the surrounding rock when \(\Delta u\) reaches 16.50 mm, about 61.3% of the maximum wall displacement. Therefore, the most effective support installation time is between 30.4% and 61.3% of the maximum peripheral rock wall displacement.  
To thoroughly analyze the concrete effects of lining thickness on the stress and deformation of both the lining structure and the surrounding rock, Figure 13 presents two distinct cases of reserved deformation for analysis.  
Figure 13 illustrates the impact of increased lining thickness on the internal stresses and displacements within the lining, as well as the stress development and plastic zone expansion in the surrounding rock for a given \(\Delta u\). As lining thickness increases, the maximum compressive stress within the lining is substantially reduced, along with the displacements of both the inner and outer walls. Concurrently, the growth of stress at the tunnel wall in the surrounding rock is effectively mitigated, and the thickness of the plastic zone is significantly decreased. Table 4 presents detailed quantitative data on these effects, offering crucial insights for tunnel design optimization.  
Figure 12: Impact of reserved deformation on surrounding rock and lining.