## 3 Calculation Method of the Collaborative Deformation of Rock and Lining Based on the H-B Criterion
### Optimization Methods for Lining Structure and Support Timing  
Tunnel lining projects have a strict control on the reserved deformation and lining thickness, as these parameters significantly influence the synergistic deformation between the surrounding rock and the lining. Under different working conditions, different combinations of reserved deformation and lining thickness result in varying maximum compressive stresses within the lining, as illustrated in Figure 8. For a specified combination of the lining's inner-to-outer radius ratio \(t\) and reserved deformation \(\Delta u\), intersections of planes perpendicular to the \(x\)-axis and \(y\)-axis with the stress surface coincide at a single point, referred to as the control point in Figure 8. Considering the safety of engineering, the maximum compressive stress \(\sigma_{\rm max}\) in the lining must be less than the project control stress \(\sigma_{\rm cmax}\), hence, the control point should lie below the control surface \(z=\sigma_{\rm cmax}\).  
In the concrete lining, at the inner wall where \(r=R_{\rm a}\), the maximum design stress \(\sigma_{\rm cmax}=1.2f_{\rm c}\) is determined according to the Guo-Wang criterion [30; 31], with \(f_{\rm c}\) representing the concrete's uniaxial compressive strength.  
Given a certain ground stress \(p_{0}\), the surrounding rock may experience harmful deformations nearing or reaching a predefined limit if the lining is thin, its elastic modulus is low, or the rock strength is weak. This predefined state is assessed using the rock Mohr-Coulomb damage criterion in this study. Using conversion equation [32], rock mechanical parameters from the H-B strength criterion are translated into cohesion \(c\) and the angle of internal friction \(\varphi\) of the rock. Consequently, the maximum tangential stress [30] that can be sustained at the rock cavity wall are:  
\[\sigma_{\rm wmax}=(1+\sin\varphi)\sigma_{\rm rw}/(1-\sin\varphi)+2c\cos\varphi /(1-\sin\varphi), \tag{37}\]  
where \(\sigma_{\rm rw}\) is the value of radial stress at the wall of the perimeter rock.  
To fully harness the support strength of both concrete lining and surrounding rock, it is most rational in design for the lining and rock to approach their preset failure states simultaneously under limit conditions, thereby minimizing the \"total waste\" of support strength. Consequently, an optimization function for the strength-efficiency of tunnel lining and surrounding rock is constructed based on the least squares method:  
\[C(t,\Delta u)=(\sigma_{\text{max}}-\sigma_{\text{cmax}})^{2}+(\sigma_{\text{w} }-\sigma_{\text{wmax}})^{2}. \tag{38}\]  
Clearly, the optimal design is achieved when the objective function is minimized, indicating that the concrete lining and surrounding rock are simultaneously approaching or reaching the preset failure state. This ideal condition, where the objective function equals zero, represents the theoretical best design; however, it is challenging to attain in practice. The following section presents a detailed discussion with a specific numerical example.