# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 6 Discussion  
Following the aforementioned research, separate evaluations were performed on the stability of the medium rock inclusions on the basis of displacement, stress, and the plastic zone. To assess the state of the MRP inclusions during the construction process, three evaluation criteria must be considered. Consequently, the comprehensive criteria, represented by Formula (6) and derived from the unified Formulas (1)-(5), were developed. When these criteria were used, it was necessary to define the clear-distance width between the middle and intercalated rock, determine the width-span ratio, identify the stress criteria formula and the plastic zone criteria formula corresponding to the width-span ratio range, and finally obtain the values of \(X1\), \(Y1\), \(S1\), and \(C\) on the basis of the deformation data acquired from the vault settlement and horizontal clearance convergence of the two tunnels. Figure 22: Numerical distribution of index \(C\) in the plastic zone of the MRP. \[\left\{ \begin{array}{l}\left\{\begin{array}{l}H=0.296X2-0.230X3-0.190\leq-1.21 \text{ mm }\left(\text{Width-span ratio }0.11\,\sim 1.26\right)\\ V=1.216Y2+0.183Y3+6.069\leq-7.0\text{ mm }\left(\text{Width-span ratio }0.11\,\sim 1.26\right)\\ \end{array}\right\}\\ \left\{\begin{array}{l}S=(-1.002Y2+0.457Y3+0.147X2-0.368X3+0.416)\times 1.184 \text{ (Width-span ratio }0.11\,\sim 0.48)\\ S=(-1.002Y2+0.457Y3+0.147X2-0.368X3+0.416)\times 1.000\text{ (Width-span ratio }0.48\,\sim 0.84)\\ S=(-1.002Y2+0.457Y3+0.147X2-0.368X3+0.416)\times 1.184\text{ (Width-span ratio }0.84\,\sim 1.26)\\ \end{array}\right\}\leq 4.80\text{ MPa },\\ \left\{\begin{array}{l}10^{3}C=(-1.756Y2-4.909Y3+1.095X2-0.041X3-55.276)\times 1.000\text{ (Width-span ratio }0.11\,\sim 0.30)\\ 10^{3}C=(-0.531Y2+0.119Y3+0.252X2-0.065X3-2.318)\times 1.326\text{ (Width-span ratio }0.30\,\sim 0.48)\\ 10^{3}C=(-0.531Y2+0.119Y3+0.252X2-0.065X3-2.318)\times 1.000\text{ (Width-span ratio }0.48\,\sim 1.26)\\ \end{array}\right\}\leq 10^{-2}\end{array} \right.\end{array}\right. \tag{6}\]  
If any of the four values exceeded the corresponding limit value when the rock mass parameters of the section for stability analysis were close to the initial parameters of the surrounding rock, as shown in Table 4, the MRP was deemed to have lost its stability. Furthermore, if the section grade of the MRP was higher than that of the rock mass corresponding to the initial parameters of the surrounding rock, according to Figure 4, the cohesion corresponding to the limit state also increased, resulting in an increase in the stress value at the center of the intermediate rock mass. Therefore, when evaluating the stability of this type of rock mass, the primary factors were the \(X1\), \(Y1\), and \(C\) values obtained from the displacement and plastic zone criteria, whereas the stress criteria served as a supplementary factor. When the stress value exceeded the limit value alone, the instability of the rock mass cannot be determined. On the other hand, if any of the \(X1\), \(Y1\), and \(C\) values exceeded the corresponding limit values, the MRP was considered unstable. Additionally, if the section grade of the MRP was lower than that of the rock mass corresponding to the initial parameters of the surrounding rock, according to Figure 4, the cohesion and elastic moduli corresponding to the limit state also decreased. Consequently, this led to increased vertical displacement in the upper region and horizontal displacement in the middle region of the intermediate rock mass, as well as decreased stress in the center of the MRP. In this scenario, when assessing the stability of this type of MRP, the \(S1\) and \(C\) values obtained from the stress criteria and plastic zone criteria were the primary factors, with the displacement criteria \(X1\) and \(Y1\) serving as supplementary factors. Notably, when only the displacement values \(X1\) and \(Y1\) exceeded the limit value, the instability of the MRP could not be determined.