# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 4 Stability Criteria of the MRP Before Construction  
Assessing and evaluating the stress and deformation of rock during the design process was crucial for ensuring the safe control of early small clear-distance tunnels. This evaluation ensured that the deformation and stress of the rock remained within acceptable limits, allowing safe excavation. By adhering to these criteria, the construction process can be dynamically managed to maintain safety. To study the stability changes in small clear-distance tunnels under various parameters, the grade of the surrounding rock was focused on and quantified the mechanical properties of the MRP. The mechanical properties of the surrounding rock were influenced mainly by cohesion [13]. When the stability criteria of the rock surrounding a tunnel were calculated, it was crucial to determine the failure limit state of the MRP first. The cohesion in the failure limit state of the MRP was determined via the SRM. In the SRM, cohesion was obtained by gradually increasing the weakening coefficient of the original rock and the cohesive properties of the soil. This value was then substituted into finite element numerical software for iterative trial calculations until the program stopped solving due to calculation convergence. At this point, the rock mass was considered to be in the ultimate state of instability, and the cohesion obtained was the cohesion in the ultimate state of tunnel stability. For the purposes of simulation, a standard construction condition known as the bench method was selected. The main line tunnel was excavated first at a staggered distance of 15 m, serving as the simulation model. The strength reduction method was used to vary the cohesion from 0.2 MPa to 2.0 MPa to explore the failure process of tunnel stability. Figure 3 illustrates the variation process with different cohesion values. The clear-distance value, \(d\), at the non-convergence position, steady state, and clear distance \(d\) were represented on the right axis of the histogram. The length of the main line tunnel excavation, \(L\), was represented on the main axis of the folded line graph. A non-convergence state was indicated by a numerical value of 1, whereas a convergence state was represented by 2. In the steady state, the cohesion ranged from 1.4 MPa to 2.0 MPa, and the length of the mainline tunnel excavation reached a maximum value of 110 m, indicating the completion of the excavation of the two tunnels. On the other hand, a cohesion range of 0.2 MPa to 1.3 MPa indicated a non-convergence state, with the excavation of the two tunnels not yet \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Density** & **Elastic Modulus** & **Poisson** & **Cohesion** & **Friction** \\  & **/(kg/m\({}^{3}\))** & **/(GPa)** & **Poisson** & **/(MPa)** & **/(\({}^{\circ}\))** \\ \hline Grade III surrounding rock & 2200–2300 & 6–10 & 0.25–0.30 & 2.0 & 39–50 \\ Primary support & 2500 & 30 & 0.20 & / & / \\ Secondary lining support & 2500 & 32.5 & 0.15 & / & / \\ Temporary support & 7900 & 150–200 & 0.30 & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters used in the numerical analysis (Mohr–Coulomb model). completed. Consequently, under specified conditions of density and elastic modulus, a stable critical state existed within the cohesion range of 1.3 MPa to 1.4 MPa. When the cohesion was less than 1.32 MPa, the calculation did not converge, but when the cohesion ranged from 1.32 MPa to 2.0 MPa, the calculation converged. Thus, a cohesion value of 1.32 MPa represented the failure limit of the MRP. Therefore, in the design stage, the stability state of the tunnel can be determined in the early stage by comparing the cohesion value of the rock mass with the change process of the cohesion in Figure 3. However, the density and elastic modulus of the rock are also key factors influencing rock deformation and construction conditions.