# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 5 Stability Criteria of the MRP During Construction  
### Numerical Cases Design  
To assess the stability of intercalated rock based on criteria, it is necessary to determine the lower limit of the stability value for intercalated rock. If it exceeds this value, it can be considered stable, and the parameters of the surrounding rock at each level fall within a certain range. To characterize the variation characteristics of the plastic zone in the MRP, adjustments were made to the parameters of grades III-IV based on the previous section. Using the previous criteria, a cohesion value adjustment unit of 0.01 MPa was selected, and the idea and method of strength reduction for the cohesion value were employed to find the limit state close to the failure of the MRP, thereby establishing the parameters. At this juncture, the cohesion value was determined to be 1.32 MPa, as indicated in Table 4. The initial state values of the surrounding rock in Table 4 represent the parameters of the surrounding rock before excavation after achieving in situ stress balance. During the excavation of the face, it is necessary to reduce the modulus of the excavated rock area to achieve the joint action of support and surrounding rock. Additionally, to account for the influence of factors such as large-scale mechanical work, blasting, and groundwater on the mechanical state of the intercalated rock, the plastic index of the MRP mass gradually decreased after each rock mass was excavated during the construction stage. The reduction  
Figure 4: Relationship between density, elastic modulus, and cohesion. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Elastic Modulus Gradient**} & \multicolumn{3}{c}{**Other Fixed Parameters**} \\ \hline \multirow{2}{*}{**Elastic Modulus/(GPa)**} & \multicolumn{2}{c}{**Cohesion**} & \multirow{2}{*}{**Density/(kg/m\({}^{3}\))**} & \multirow{2}{*}{**Poisson**} & \multicolumn{1}{c}{**Friction**} \\  & \multicolumn{1}{c}{**/(MPa)**} & & & \multicolumn{1}{c}{**/(\({}^{\circ}\))**} \\ \hline
2.4 & 1.2 & & & & \\
3.6 & 1.4 & & & & \\
4.8 & 1.4 & & & & \\
6.0 & 1.4 & & & & \\
7.2 & 1.5 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Elastic modulus and the corresponding convergence value of cohesion. range is displayed in Figure 5. Because cohesion was the main factor, it was reduced by 70%, so the cohesion of the MRP was reduced to 0.93 MPa. To ensure safety control during the construction period of the small clear-distance tunnel, a total of 14 numerical simulation working conditions were designed, as shown in Table 5. These conditions consisted of the basic working conditions and various comparative working conditions. In particular, case 14 focused on the impact of the built-up tunnel crossing above. By analyzing the stress, displacement, and plastic zone under different working conditions, combined with the width-span ratio (W/B) in the rock classification, comprehensive stability criteria for the rock were developed under different criteria and influence factors. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Density/(kg/m\({}^{3}\))**} & \multicolumn{2}{c}{**Elastic**} & \multirow{2}{*}{**Poisson**} & \multirow{2}{*}{**Cohesion**/**(MPa)**} & \multirow{2}{*}{**Friction/(\({}^{\circ}\))**} \\  & & & **Modulus/GPa** & & & \\ \hline Initial state of surrounding rock & 2200 & 3.6 & 0.25 & 1.32 & 20 \\ Strength reduction in the MRP & 2200 & 3.6 & 0.25 & 0.93 & 20 \\ Modulus softening of excavated rock mass & 2200 & 1.2 & 0.3 & 1.32 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of surrounding rock simulation. Figure 5: Monitoring points of the MRP: (**a**) range of the MRP; (**b**) the layout of monitoring points for the MRP. W is the clearance between the two tunnels. B is the width of the tunnel.