## 2 Materials and Methods  
### Governing Equations for Segmented Liniings  
Monolithic reinforced concrete linings are prone to cracking under high internal pressure, which may compromise their load-bearing capacity. By introducing preset seams, the radial displacement of the lining can be released, allowing the internal gas pressure to be effectively transferred to the surrounding rock, thereby fully utilizing the rock mass's bearing capacity. A typical segmented lining structure with preset seams is shown in Figure 1. This structure consists of four main components: a segmented reinforced concrete lining, preset seams, retractable joints, a sliding layer, and sealing elements at the seams. Sealing materials are applied to the inner wall of the working area, and sealing strips are installed at the seams to prevent high-pressure gas leakage. Steel bedding is locally embedded between the flexible sealing materials and the lining to prevent the sealing materials from being pushed into the seams under pressure. The sliding layer ensures that preset seams open under internal pressure and helps redistribute circumferential stresses in the lining. The working mechanism of the segmented lining is as follows:  
(1) Before gas injection, the preset seams remain closed, as shown in Figure 1a. (2) As the working is pressurized, internal pressure increases, causing slippage between the primary and secondary linings. This leads to a gradual opening of the preset seams, allowing for the radial displacement of the lining to be released and redistributing circumferential stress. The internal pressure is thereby effectively transferred to the surrounding rock. Without the sliding layer, significant shear stress would develop between the primary and secondary linings to counterbalance the internal pressure, preventing seam opening and inhibiting radial displacement release. (3) When the internal pressure reaches its maximum, the opening displacement of the preset seams reaches its peak value, \(D_{\text{max}}\), and the radial displacement of the lining is fully released, as shown in Figure 1b. In establishing the governing equations, several simplifications were made for computational efficiency and model tractability:  
Figure 1: Structural components and working mechanism of the segmented lining. (1) The surrounding rock was modeled as an elastoplastic material and assumed to be of uniform quality and isotropic. Spatial variability and mechanical anisotropy caused by bedding planes, joints, or faults were not considered. (2) The retractable joints between segmented linings were represented by linearly elastic elements. This simplification does not account for potential nonlinear deformation, interface slip, or damage-induced degradation that may occur under service conditions. Shear deformation occurs between the initial support and the lining before any sliding takes place at their interface. This mechanical behavior is captured in the model presented in Figure 2. The maximum shear deformation, represented by \(\delta_{\rm m}\), is influenced by the friction coefficient and the shear stiffness of the interface, \(K_{\rm n}\). The model highlights how the interaction between the two structural components contributes to the overall system response. To analyze the stress state of the lining, a small segment is isolated, as shown in Figure 2b. The analysis is carried out in both the radial and circumferential directions. A polar coordinate system is introduced to facilitate the formulation of the corresponding equilibrium equations:  
\[\frac{\partial\sigma_{r}}{\partial r}+\frac{1}{r}\frac{\partial\tau_{\rm \mu\theta}}{\partial\theta}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \tag{1}\]  
\[\frac{1}{r}\frac{\partial\sigma_{\theta}}{\partial\theta}+\frac{\partial\tau _{\rm\mu\theta}}{\partial r}+\frac{2\tau_{\rm\mu\theta}}{r}=0 \tag{2}\]  
The radial (\(\sigma_{r}\)), circumferential (\(\sigma_{\theta}\)), and shear (\(\tau_{\rm\mu\theta}\)) stresses characterize the internal force distribution in the lining.