## 4 Influence of Lining Thinning on Rock Pressure in Four-Track HSR Tunnels  
### Numerical Model Development  
A three-dimensional numerical model of an HSR tunnel with lining thinning and thinning defects was established using Abaqus (Version 2022). The model dimensions were 92.7 m in length, 50 m in width, and 100 m in height, as shown in Figure 12a. The surrounding rock mass was assumed to be homogeneous and isotropic, which is a common simplification in tunnel simulations to facilitate computational feasibility and ensure modeling clarity. The rock behavior was simulated using the Mohr-Coulomb elastoplastic constitutive model, while the primary support and secondary lining were modeled as linear elastic materials. Surface-to-surface contact interactions were defined between the primary support, the secondary lining, and the rock. Boundary conditions included normal constraints on all side boundaries and fixed constraints at the model base.  
To facilitate the analysis of different scenarios, thinning s at various locations were assigned reference numbers, as shown in Figure 12b. The region directly influenced by the thinning was defined as the thinning-affected zone, while the surrounding adjacent region was referred to as the active thinning zone, as illustrated in Figure 12c. The loading and simulation conditions used in the numerical study are listed in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Condition** & **Description** & **Control Parameters** & **Rock Grade** \\ \hline
1 & No defect & - & III, IV, V \\ \hline
2 & Lining thinning & \(P_{\text{j}}\), \(L^{\text{arc}}_{\text{j}}\), \(L_{\text{j}}\), \(R_{\text{j}}\) & III, IV, V \\ \hline
3 & Lining degradation + thinning & \(P_{\text{j}}\), \(L^{\text{arc}}_{\text{j}}\), \(L_{\text{j}}\), \(R_{\text{j}}\), t & III, IV, V \\ \hline
4 & Rock degradation + thinning & \(P_{\text{j}}\), \(L^{\text{arc}}_{\text{j}}\), \(L_{\text{j}}\), \(R_{\text{j}}\), t & III, IV, V \\ \hline
5 &
\begin{tabular}{c} Rock degradation + lining \\ degradation + thinning \\ \end{tabular} & \(P_{\text{j}}\), \(L^{\text{arc}}_{\text{j}}\), \(L_{\text{j}}\), \(R_{\text{j}}\), t & III, IV, V \\ \hline \hline \end{tabular}
\end{table}
Table 3: Numerical simulation conditions.  
Figure 12: Numerical model of an HSR tunnel with thinning defects: (**a**) Overall model geometry. (**b**) Thinning position numbering. (**c**) Classification of thinning-affected and active thinning zones.  
While the assumptions of homogeneity and isotropy may not fully capture the inherent heterogeneity and anisotropic behavior of natural rock masses, they remain the most widely adopted simplifications in current engineering practice--particularly when site-specific anisotropic parameters are unavailable. Within the context of this study, which focuses on comparative analysis and parametric evaluation at the engineering scale, the use of these assumptions is considered reasonable and appropriate. In future work, the influence of spatial variability and heterogeneity in rock mass properties will be further considered to improve the accuracy and realism of the simulation results.  
The thinning defect of the tunnel lining was modeled as a sector-shaped body characterized by three key parameters: the thinning length, the thinning-thickness ratio, and the thinning arc length. The mechanical parameters of the tuff were obtained through field sampling and laboratory testing conducted at the XBS Tunnel site. The method used to determine the rock mass parameters is illustrated in Figure 13. The parameters adopted for the numerical simulation are summarized in Table 4.