## 5 Conclusions  
In tunnel lining design, it is essential to holistically consider the stress and deformation characteristics of both the tunnel lining and the surrounding rock to accommodate varying rock conditions. Utilizing the H-B strength criterion and elastic-plastic cylinder, we quantitatively describe the stress and deformation within the surrounding rock and lining. By integrating the theory of coordinated deformation between the rock and lining, we establish a theoretical method for calculating the reserved deformation of the tunnel lining. To optimize the supportive strength of both the surrounding rock and the concrete lining, we introduce an objective function based on stress safety, employing the least squares method to determine the optimal combination of lining thickness and support installation time. This approach not only enhances the synergistic support effect between the rock and the lining but also offers a scientific foundation for optimal tunnel engineering design. The key findings are summarized as follows:  
1. A reduction in the support pressure of the surrounding rock and an increase in the excavation radius result in more extensive plastic deformation. On average, the plastic zone thickness increases by 1.95 mm for each 1 cm increment in the excavation radius. The lining's equivalent resistance coefficient \(K\) is a critical parameter influencing the stress distribution within the surrounding rock and the development of the plastic zone. The impact of lining thickness on \(K\) is more pronounced than that of the concrete's elastic modulus, which is a primary factor affecting the lining's equivalent resistance coefficient. 2. The reserved deformation and lining thickness significantly affect the stress in the surrounding rock and the lining. When the reserved deformation increases by 1% of the maximum displacement of the tunnel wall, the maximum lining stress decreases by an average of 351.17 kPa, and the plastic zone thickness increases by an average of 39.52 mm. Under a given reserved deformation, a 1 cm increase in lining thickness \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Concrete Lining Strength Grade** & **C20** & **C25** & **C30** & **C35** & **C40** \\ \hline Optimal combination & (56.2, 12) & (76.9, 9) & (59.2, 9) & (47.0, 9) & (17.5, 12) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimal lining construction combinations for various concrete strength grades. leads to a reduction of 83.2 kPa in the maximum lining stress, an increase of 6.87 kPa in the equivalent stress around the tunnel wall, and a decrease of 4.27 mm in the surrounding rock's plastic zone thickness. 3. For the fault section of the Wufengshan Tunnel, the optimized design parameters have been determined. The recommended timing for support is when the tunnel wall's surrounding rock deforms by 9 mm, equivalent to 33.5% of the maximum wall displacement. The optimized lining thickness is 47 cm, representing a reduction of approximately 36.5% compared to the initial design of 74 cm. This refined design not only minimizes construction costs but also enhances the overall safety of the tunnel lining. The present model does not explicitly incorporate the effects of groundwater conditions on rock mass behavior, which is acknowledged as a limitation. Typically, the influence of groundwater on rock mass behavior is addressed by the modified \(BQ\) value ([\(BQ\)]). Future research should aim to refine the model by more explicitly integrating groundwater conditions, thereby enhancing its predictive accuracy and applicability in cases where groundwater significantly affects rock mass stability and deformation. This improvement will provide a more comprehensive and scientifically robust solution for the optimized design of tunnel engineering projects. C.S. : Investigation, Methodology, Formal analysis, Conceptualization, Funding acquisition, Project administration, Supervision, Validation, Writing--review and editing. Z.Z.