## 3 Calculation Method of the Collaborative Deformation of Rock and Lining Based on the H-B Criterion
### Calculation Method of Lining Displacement Based on Thick-Walled Cylinder Theory  
The lining of a circular tunnel typically exhibits a relatively uniform thickness, which necessitates the application of thick-walled tube theory for analysis. The inner radius of the thick-walled cylinder, corresponding to the inner radius of the tunnel lining, is denoted as \(R_{\text{a}}\), while the outer radius, which is the outer radius of the tunnel lining, is denoted as \(R_{\text{b}}\), as illustrated in Figure 3. An infinitesimal unit \(rd\theta dr\) is taken out at a distance of \(r\) from the center of the cylinder, and \(R_{\text{a}}<r<R_{\text{b}}\). According to the principles of elastoplasticity, the stress and displacement at any point within the lining can be computed under varying internal and external pressure conditions. The radial and hoop stresses at any given point are calculated using Equations (17) and (18):  
\[\sigma_{\text{r}}=\frac{R_{\text{a}}^{2}(R_{\text{b}}^{2}-r^{2})}{r^{2}(R_{ \text{b}}^{2}-R_{\text{a}}^{2})}p_{\text{a}}-\frac{R_{\text{b}}^{2}(R_{\text{ a}}^{2}-r^{2})}{r^{2}(R_{\text{b}}^{2}-R_{\text{a}}^{2})}p_{\text{b}}, \tag{17}\]  
\[\sigma_{\text{0}}=-\frac{R_{\text{a}}^{2}(R_{\text{b}}^{2}+r^{2})}{r^{2}(R_{ \text{b}}^{2}-R_{\text{a}}^{2})}p_{\text{a}}+\frac{R_{\text{b}}^{2}(R_{\text{ a}}^{2}+r^{2})}{r^{2}(R_{\text{b}}^{2}-R_{\text{a}}^{2})}p_{\text{b}}. \tag{18}\]  
The radial displacement \(u\) at any point within the lining is detailed in Equation (19):  
\[u=\frac{1+\mu}{E}\left[\frac{(1-2\mu)r^{2}+R_{\text{b}}^{2}}{r(t^{2}-1)}p_{ \text{a}}-\frac{R_{\text{b}}^{2}+(1-2\mu)t^{2}r^{2}}{r(t^{2}-1)}p_{\text{b}} \right]. \tag{19}\]  
In the formula, \(t=R_{\text{b}}/R_{\text{a}}\); \(E\) and \(\mu\) are the elastic modulus and Poisson's ratio of the thick-walled cylinder, respectively; the meanings of the other symbols remain the same as previously defined. Collaborative Deformation Calculation of the Rock Lining Considering Stress Release Rate of the Rock  
The synergistic deformation of the surrounding rock and lining under various stress release rates following tunnel excavation is depicted in Figure 4. After the excavation of the tunnel, if the surrounding rock is stabilized with a support pressure of \(p_{i}\), the displacement \(u_{\text{w}}\) of the surrounding rock can be determined using Equation (16). Figure 3: Differential element in thick-walled cylinder. Due to the complexity of the expression relating the tunnel wall displacement \(u_{\rm w}\) to the support pressure \(p_{\rm i}\) in Equation (16), direct computation would be extremely challenging and an inefficient use of computational resources. To simplify the reflection of this relationship, we employed a quadratic function fitting method to approximate the theoretical solution of Equation (16), leading to the derivation of the fitting equation: Equation (20). This equation has been effectively utilized for subsequent derivations, analysis, and calculations. The fitting schematic diagram is shown in Figure 5. This approach ensures a high degree of fit, making it applicable to various rock mass types and tunnel environments, effectively capturing the complex relationships and variations in tunnel deformation. \[p_{\rm i}=m\cdot(u_{\rm w}+n)^{2}, \tag{20}\]  
Here, \(m\) represents the deformation characteristics of the surrounding rock, that is, the difficulty of deformation when subjected to support pressure \(p_{\rm i}\); and \(n\) is the maximum deformation of the surrounding rock of the tunnel wall under certain conditions obtained by fitting. For the traffic tunnel lining, the inner wall is usually not subjected to stress, hence \(p_{\rm a}=0\) and \(p_{\rm b}=p_{\rm i}\) are substituted into Equation (19).