## 3 Calculation of Stress and Displacement in the Elastic-Plastic Zone
### Calculation of Yield Deformation of the Rock Mass in the Plastic Zone  
The strength parameters of the surrounding rock vary with the softening coefficient in the plastic region. For this reason, previous studies have provided widely divergent solutions for the softening behavior of rock mass. [PERSON] [5] used the finite difference method to obtain the yield deformation of the soft rock mass surrounding a tunnel. [PERSON] and [PERSON] [24] subsequently improved [PERSON]'s calculation method and proposed a numerical method with the increment of radial stress in the strain-softening zone represented as a series of concentric rings. As shown in Figure 3, the present study adopted the method by [PERSON] and [PERSON] to uniformly divide the radial stress in the plastic zone into \(n\) concentric rings. The normalized radius of the inner and outer boundaries of the \(i\)th annulus are \(\rho_{(i-1)}=\frac{r_{(i-1)}}{R_{p}}\) and \(\rho_{(i)}=\frac{r_{(i)}}{R_{p}}\). The increment in radial stress of each ring can be obtained as:  
\[\Delta\sigma_{r}=\frac{pi-\sigma_{R}}{n} \tag{25}\]  
Therefore, the radial stress of the \(i\)th radius can be expressed as:  
\[\sigma_{r(i)}=\sigma_{r(i-1)}+\Delta\sigma_{r} \tag{26}\]  
Substituting Equation (26) into Equation (11), the following is obtained:  
\[\sigma_{\theta(i)}=\sigma_{r(i)}+H\Big{(}\sigma_{r(i)},\eta_{(i-1)}^{p}\Big{)} \tag{27}\]  
Figure 3: Spatial distribution of the normalized radius division of the plastic zone. therefore:  
\[\Delta\sigma_{\theta(i)}=\sigma_{\theta(i)}-\sigma_{\theta(i-1)}\]  
The elastic strain of the rock mass in the plastic zone in terms of stress with consideration of the initial hydrostatic stress \(\sigma_{0}\) can be expressed as:  
\[\left( \begin{array}{c}\varepsilon_{r(i)}^{\epsilon}\\ \varepsilon_{\theta(i)}^{\epsilon}\end{array} \right)=\frac{1+v}{E}\bigg{[} \begin{array}{cc}1-v&-v\\ -v&1-v\end{array} \bigg{]}\left( \begin{array}{c}\sigma_{r(i)}-\sigma_{0}\\ \sigma_{\theta(i)}-\sigma_{0}\end{array} \right) \tag{28}\]  
According to Hooke's Law, the increase in elastic strain can be calculated as:  
\[\left( \begin{array}{c}\Delta\varepsilon_{r(i)}^{\epsilon}\\ \Delta\varepsilon_{\theta(i)}^{\epsilon}\end{array} \right)=\frac{1+v}{E} \bigg{[} \begin{array}{cc}1-v&-v\\ -v&1-v\end{array} \bigg{]}\left( \begin{array}{c}\Delta\sigma_{r(i)}\\ \Delta\sigma_{\theta(i)}\end{array} \right) \tag{29}\]  
By substituting Equation (29) into the equilibrium equation [Equation (18)], the following is obtained:  
\[\frac{\sigma_{r(i)}-\sigma_{r(i-1)}}{\rho_{(i)}-\rho_{(i-1)}}+\frac{\sigma_{r (i)}-\sigma_{\theta(i)}+\sigma_{r(i-1)}-\sigma_{\theta(i-1)}}{\rho_{(i)}+ \rho_{(i-1)}}=0 \tag{30}\]  
The \(i\)th normalized radius is obtained as:  
\[\rho_{(i)}=\frac{\sigma_{\theta(i)}+\sigma_{\theta(i-1)}-2\sigma_{r(i-1)}}{ \sigma_{\theta(i)}+\sigma_{\theta(i-1)}-2\sigma_{r(i)}}\rho_{(i-1)} \tag{31}\]  
During the yield stage of the surrounding rock, total strain of each concentric ring can be decomposed into elastic strain and plastic strain components as:  
\[\left( \begin{array}{c}\varepsilon_{r}\\ \varepsilon_{\theta}\end{array} \right)=\left( \begin{array}{c}\varepsilon_{ r}^{\epsilon}\\ \varepsilon_{\theta}^{\epsilon}\end{array} \right)+\left( \begin{array}{c} \varepsilon_{r}^{p}\\ \varepsilon_{\theta}^{p}\end{array}