## 5 Secondary lining Support Characteristic Curve  
### Establishment of Numerical Model of Secondary lining  
#### 5.1.1 Model Size and Convergence Verification  
The schematic of the model size is illustrated in Figure 8a, which is optimized according to the surrounding rock grade with different thicknesses. For Grade III surrounding rock, the thickness values are 0.5, 0.6, and 0.8 m; for Grade IV surrounding rock, the thickness values are 0.8, 1.0, and 1.2 m; and for Grade V surrounding rock, the thickness values are 0.8, 1.0, 1.2, and 1.35 m. The secondary lining is modelled using solid elements, and cable elements are used to simulate the internal and external layers of reinforcement in the secondary lining. The protective layer thickness of the internal and external reinforcement is 5 cm, and the interface thickness of the reinforcement is 0.55 times the diameter of the reinforcement [47]. To avoid solution errors caused by inconsistent model grids, a detailed secondary lining grid model is pre-established within the overall model.  
During the solution process, only the initial stress storage of the original rock is needed, which is then normalized to obtain the stress distribution around the tunnel. Subsequently, the surrounding rock units are deleted, leaving only the secondary lining units, and the internal and external layers of reinforcement are generated. The secondary lining calculation model (Figure 8b) is generated. Since there are six hydraulic tunnels in the tunnel group, considering the symmetry of the model, only the support characteristic curves of Tunnel 1, Tunnel 2, and Tunnel 3 are calculated here. Since the stress distribution of the surrounding rock for Tunnels 1-3 is slightly different, the above method can be used to independently construct the secondary lining models for Tunnels 1-3. The actual loading process is as follows: firstly, the uneven distribution of the original rock stress around the tunnel necessitates the extraction of the stress distribution characteristics of the original rock, and the corresponding surrounding rock resistance is applied based on these characteristics; secondly, the surrounding rock resistance is applied step by step gradually, and the lining deformation corresponding to each step is calculated, assuming a load increment of 10 kPa/step; and finally, the secondary lining support characteristic curve is drawn with the maximum deformation of the secondary lining set to 1.0 m as the solution control condition.  
In addition, the stability of the finite element model to phenomena such as sensitivity to finite element dimensionality, approximation type, and material damage settings, particularly in the cracking zone, was thoroughly assessed. A mesh sensitivity analysis was performed to evaluate how the model's results were affected by mesh refinement. It was found that, upon mesh refinement, no artificial localization of deformations occurred. The convergence of the results was systematically checked by monitoring the displacement and stress values between iterations, with the relative changes kept within a predefined tolerance to confirm that convergence had been achieved. Additionally, when issues of convergence arose with coarser meshes, the model settings were adjusted, and different mesh densities and element types were tested to ensure stable and consistent results. These checks and adjustments confirm that the model's stability and the convergence of the results were adequately addressed during the analysis. Finally, 960 six-node hexahedral elements were used to model the secondary lining concrete, while 384 anchor rod elements were used to simulate the double-layer reinforcement, with 192 anchor rod elements representing the single-layer reinforcement.