## 3 Derivation of the Theoretical Formula of the Surrounding Rock Loose Zone
### Theoretical Derivation of the Stress Method
#### 3.1.1 Derivation of Stress Relaxation Range  
In order to analyze the basic theory of the loose zone, the following basic assumptions are made. For a tunnel with a circular cross-section, the surrounding rock is infinite, uniform, continuous, and isotropic. The initial stress field of the surrounding rock (without gravity) is uniform. The tunnel size and surrounding rock stress do not change along the tunnel length direction, and the strain and displacement along the tunnel length direction are zero; that is, the tunnel length is infinite, and the surrounding rock of the tunnel cross-section is in a state of plane stress. The excavation and support of the tunnel are completed instantaneously, and the tunnel support is close to the surrounding rock.  
Based on [PERSON]'s solution and the Mohr-Coulomb yield criterion, the expression for the plastic zone of the surrounding rock can be deduced through a geometric equation, equilibrium differential equation, and continuity condition, which is called the modified Fenner formula or the Castelner formula. The radius of the plastic zone is calculated as follows [36]:  
\[R_{p}=r_{t}\Big{[}\frac{(P_{0}+c\cot\varphi)(1-\sin\varphi)}{P_{1}+c\cot \varphi}\Big{]}^{\frac{1-\sin\varphi}{28\sin\varphi}} \tag{16}\]  
where \(P_{0}\) is the initial in situ stress; \(P_{i}\) is the support reaction; \(c\) and \(\varphi\) are the cohesion and internal friction angle, respectively; \(r_{t}\) is the equivalent circle radius of the chamber section, which can be calculated by \(r_{t}=(H+B)/4\); and \(H\) and \(B\) are the height and width of the chamber section, respectively.  
Two parameters related to the grade of surrounding rock are introduced to improve the formula of the plastic zone radius, which makes the formula more concise and suitable for engineering calculations. The parameters are \(q_{uc}=2c\cdot\cos\varphi/(1-\sin\varphi)\), which is the uniaxial compressive strength of the surrounding rock, and \(K_{\varphi}=(1-\sin\varphi)/2\sin\varphi\), which is the internal friction angle coefficient of the surrounding rock. In addition, \(P_{0}\) is  
Figure 3: Effect of chamber excavation on rock mass: (**a**) Crushing effect of blasting on rock mass of tunnel face, (**b**) crushing effect of blasting on surrounding rock mass.  
the initial in situ stress, \(P_{i}\) is the support reaction, and \(c\) and \(\varphi\) are cohesion and internal friction angle, respectively, of rock and soil mass. The equation for the radius of the plastic zone becomes:  
\[R_{p}=r_{t}\Big{[}\frac{P_{0}+K_{\varphi}q_{\mathrm{ac}}}{P_{i}+K_{\varphi}q_{ \mathrm{ac}}}\Big{(}\frac{2K_{\varphi}}{2K_{\varphi}+1}\Big{)}\Big{]}^{K_{ \varphi}} \tag{17}\]  
Finally, according to previous research [37], the tangential stress at the outer boundary of the surrounding rock stress loose zone surrounded by the plastic zone is considered to be equal to the original rock stress, so the expression of the loose zone can be obtained as follows:  
\[R_{b1}=r_{t}\Big{[}\frac{P_{0}+K_{\varphi}q_{\mathrm{ac}}}{P_{i}+K_{\varphi}q_ {\mathrm{ac}}}\Big{(}\frac{K_{\varphi}}{K_{\varphi}+1}\Big{)}\Big{]}^{K_{ \varphi}} \tag{18}\]  
The thickness of the stress loose zone \(L_{b1}\) is:  
\[L_{b1}=R_{b1}-r_{t}=r_{t}\Big{[}\frac{P_{0}+K_{\varphi}q_{\mathrm{ac}}}{P_{i} +K_{\varphi}q_{\mathrm{ac}}}\Big{(}\frac{K_{\varphi}}{K_{\varphi}+1}\Big{)} \Big{]}^{K_{\varphi}}-r_{t} \tag{19}\]