## 4 Engineering Application and Analysis
### Optimization in Engineering Design  
Optimal tunnel design is crucial for ensuring construction safety and enhancing economic efficiency. An in-depth analysis utilizing the H-B strength criterion for rock mass, theoretical calculations for tunnel lining, and the synergistic interaction between the lining and surrounding rock indicate that the support effectiveness can be optimized by adjusting the lining thickness and timing the support installation appropriately. Using the fault section of the Wufengshan Tunnel as a case study, an objective function based on stress safety, constructed via the least squares method, systematically evaluates various  
Figure 16: Deformation fitting curve of the surrounding rock wall with parameter \(n\) constrained. Figure 17: Changes of surrounding rock stress and plastic zone thickness when parameter c is fixed. combinations of lining thickness and reserved deformation. This approach facilitates the determination of the most opportune moment for support and the optimal lining thickness. During the optimization process, a mechanical model was initially developed to characterize the mechanical behavior of both the lining structure and the surrounding rock. Utilizing an objective function minimization approach, a comprehensive analysis was conducted on 60 combinations of lining inner-to-outer radius ratios ranging from 1.055 to 1.13 (corresponding to 6 different lining thicknesses) and reserved deformations from 0 to 26.9 mm (corresponding to 10 reserved deformations). These calculations were performed using an iterative algorithm with a precision of 0.001. The parameter values within the objective function are detailed in Table 5.  
Tunnel lining and surrounding rock strength-efficiency optimization function:  
\[C(t,\Delta u)=(\frac{2t^{2}p_{\mathrm{i}}}{t^{2}-1}-1.2f_{\mathrm{c}})^{2}+ \left[p_{\mathrm{i}}+\sigma_{\mathrm{ci}}(\frac{m_{\mathrm{b}}p_{\mathrm{i}}}{ \sigma_{\mathrm{ci}}}+s)^{a}-\frac{(1+\sin\varphi)p_{\mathrm{i}}+2c\cos\varphi} {1-\sin\varphi}\right]^{2}, \tag{39}\]  
where \(p_{\mathrm{i}}=\left[-\frac{k}{2\sqrt{m}}+\frac{1}{2}\sqrt{\frac{k^{2}}{m}-4k( \Delta u+n)}\right]^{2}\). The following constraints were applied:  
\[\left\{ \begin{array}{l}\sigma_{\mathrm{max}}\leq\sigma_{\mathrm{cmax}}\\ \sigma_{\mathrm{\theta w}}\leq\sigma_{\mathrm{wmax}}\end{array} \right.. \tag{40}\]  
By inputting the data into the objective function and its constraint conditions, the calculated results under various scenarios are presented in Table 6. Analysis reveals the following trends: with a fixed reserved deformation of 9 mm, the objective function value exhibits a trend of decreasing initially and then increasing as the inner-to-outer radius ratio of the lining increases. Similarly, at an inner-to-outer radius ratio of 1.07, the objective function value decreases to a minimum and then rises with increasing reserved deformation. This suggests that an optimal combination of lining thickness and reserved deformation exists that minimizes the objective function within a certain range of these parameters. Consequently, at a reserved deformation of 9 mm and an inner-to-outer radius ratio of 1.07, the objective function achieves its minimum value of 0.086, with the lining's outer radius being 7.18 m and the lining thickness being 47 cm. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(t\) & \(\Delta u/\)mm & 0 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 26.9 \\ \hline
1.055 & 601.2 & 277.5 & 84.3 & 4.7 & 18.4 & 101.0 & 221.9 & 341.7 & 405.8 & 329.5 \\
1.07 & 419.5 & 172.5 & 38.4 & 0.1 & 37.9 & 128.7 & 244.5 & 351.4 & 405.9 & 347.2 \\
1.085 & 301.1 & 108.6 & 15.3 & 4.5 & 57.1 & 151.2 & 261.4 & 358.2 & 405.9 & 357.6 \\
1.1 & 220.4 & 68.2 & 4.4 & 12.7 & 74.7 & 169.7 & 274.5 & 363.2 & 405.9 & 364.5 \\
1.115 & 163.7 & 42.2 & 0.5 & 22.5 & 90.5 & 185.1 & 285.0 & 367.0 & 406.0 & 369.4 \\
1.13 & 122.8 & 25.5 & 0.6 & 32.7 & 104.7 & 198.1 & 293.5 & 370.1 & 406.0 & 373.0 \\ \hline \hline \end{tabular}
\end{table}