## 4 Engineering Application and Analysis
### Analysis of Calculated Results for Rock-Lining Synergy
#### 4.3.2 Effect of Concrete Elastic Modulus on Surrounding Rock of Tunnel  
The concrete elastic modulus, a fundamental mechanical property of the lining, directly influences the restraint exerted on the surrounding rock, subsequently impacting the stress distribution within the rock and the plastic zone development. Figure 14 compares and analyzes the effects of different concrete elastic moduli on the plastic zone thickness and tunnel wall stress, elucidating the specific mechanisms by which elastic modulus variations affect stress relief in the surrounding rock and the evolution of plastic zones.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\Delta u\) = 18 mm & \(\Delta u\) = 22 mm \\ \hline Radial deformation pressure of surrounding rock wall \(p_{\text{1}}\) (kPa) & Increase by 1.88 kPa & Increase by 0.144 kPa \\ Maximum lining hoop compressive stress \(\sigma_{\text{max}}\) (kPa) & Decrease by 83.2 kPa & Decrease by 16.5 kPa \\ Lining outer wall displacement \(u_{\text{b}}\) (mm) & Decrease by 0.019 mm & Decrease by 0.004 mm \\ Lining inner wall displacement \(u_{\text{a}}\) (mm) & Decrease by 0.019 mm & Decrease by 0.004 mm \\ Hoop stress \(\sigma_{\text{0w}}\) (kPa) at the tunnel rock wall & Increase by 12.9 kPa & Increase by 2.04 kPa \\ Equivalent stress \(\sigma_{\text{vw}}\) (kPa) at the tunnel rock wall & Increase by 6.87 kPa & Keep unchanged \\ Thickness of plastic zone of surrounding rock \(H\) (mm) & Decrease by 4.27 mm & Decrease by 0.44 mm \\ \hline \hline \end{tabular}
\end{table}
Table 4: Effect of incremental 1 cm increases in lining thickness on lining and surrounding rock.  
Figure 13: Stress and deformation in lining and surrounding rock for different lining thicknesses.  
Comparing Figures 12 and 14 reveals similar trends in stress and plastic zone thickness variations under different concrete elastic moduli and lining thicknesses:  
1. Figure 12 indicates that a 62.27 cm reduction in concrete lining thickness leads to a 5.97 MPa decrease in hoop stress and a 4.97 MPa reduction in equivalent stress at the tunnel wall, with a corresponding decrease of 3.21 mm in \(\Delta u\) when the plastic zone thickness is zero. These reductions suggest that for every 1% decrease in lining thickness, there is an average decrease of 0.53% in hoop stress and 0.55% in equivalent stress, with \(\Delta u\) decreasing by 0.25% under zero plastic zone thickness conditions.
2. In comparison, Figure 14 shows a more concentrated curve distribution, where a 7 GPa decrease in concrete elastic modulus results in a 0.92 MPa reduction in hoop stress and a 0.79 MPa decrease in equivalent stress at the tunnel wall, with a corresponding \(\Delta u\) reduction of 0.23 mm under zero plastic zone thickness. This translates to a 0.30% decrease in hoop stress and a 0.31% decrease in equivalent stress for every 1% reduction in the concrete's elastic modulus, with \(\Delta u\) decreasing by 0.06% under the same plastic zone conditions.  
The data indicate that lining thickness exerts a more pronounced effect on the stress in the surrounding rock and the thickness of the plastic zone than the concrete's elastic modulus. This suggests that in tunnel lining design, greater emphasis can be placed on optimizing lining thickness over the concrete's elastic properties.