# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 5 Stability Criteria of the MRP During Construction
### Plasticity Criteria of the MRP
#### 5.4.2 Plasticity Criteria Analysis of the MRP  
To evaluate the size and influence of the plastic zone across different clear distances and working conditions, the plastic zone index \(C\) was used. This index combined the horizontal length of the plastic zone (\(L\)), the clear distance (\(W\)), and the maximum equivalent plastic strain (PEEQ\({}_{\text{max}}\)) on the section, as shown in Figure 18. The value of index \(C\) was calculated via Formula (4). A larger equivalent plastic strain corresponded to a larger plastic zone and a smaller clear distance, making the MRP less stable. This index measures the size of the plastic zone and its effect. \[C=\text{PEEQ}_{\text{max}}\times\frac{L}{W}, \tag{4}\]  
Figure 18: Schematic diagram of the composition of index \(C\) in the plastic zone of the MRP. By considering the relationship between the displacement index and the plastic zone, the plastic zone index can be calculated via Formula (4). The displacement index was determined on the basis of the vault settlement values (measuring points 5 and 6) and clearance convergence values (measuring points 3, 7, 4, and 8) of the two tunnels. The plastic zone index distributions under various working conditions and different clear distances can be extracted from these data. In Figure 19a, the overall distributions of the plastic zone index, vault settlement values, and horizontal clearance convergence values of the MRP are shown, corresponding to the variation curve of condition 1 with the width-span ratio in Figure 19b. The overall variation in the displacement curve was not significant, but the plastic zone index \(C\) curve experienced a sudden change at a width-span ratio of 0.30, after which it became flat and stable. Therefore, analyzing the data of all clear-distance ranges together was not accurate. To conduct multiple regression analysis, the width-span ratio of 0.11-1.26 was split into two sections: 0.11-0.30 and 0.30-1.26. The output variable was the plastic zone index \(C\), which was calculated via Formula (4). The input variables included the settlement of the ramp vault (\(\gamma 2\)), the settlement of the mainline tunnel vault (\(Y3\)), the convergence of the ramp horizontal clearance (\(X2\)), and the convergence of the mainline tunnel horizontal clearance (\(X3\)). By multiple regression analysis, the fitting parameters in Table 6 were obtained, with an R\({}^{2}\) value of 0.776 in the width-span ratio range of 0.11-0.30, closely matching the measured residual value. In the width-span ratio range of 0.30-1.26, the R\({}^{2}\) value was 0.637, with some deviation from the actual value. Figure 20 shows the comparison between the actual values and the predicted values of the plastic zone index \(C\) calculated via the fitting formula. In Figure 20a, the predicted \begin{table}
\begin{tabular}{c c c} \hline
**Clear Distance** & **1.60 m-4.50 m** & **4.50 m-17.6 m** \\ \hline
**Width–Span Ratio** & **0.11–0.30** & **0.30–1.26** \\ \hline Regression formula & \(10^{3}C=-1.756Y2-4.909Y3\) & \(10^{3}C=-0.531Y2+0.119Y3\) \\  & \(+1.095X2-0.041X3-55.276\) & \(+0.252X2-0.065X3-2.318\) \\ R\({}^{2}\) & 0.776 & 0.637 \\ \hline \end{tabular}
\end{table} 
Table 6: Multiple regression formulas of the plastic zone index \(C\). Figure 19: Distribution of the index of the plastic zone in the MRP: (**a**) the distribution of the index data points of the plastic zone; (**b**) the trend of the index of the plastic zone changing with the width–span ratio (W/B) in case 1.  
values were generally higher than the actual values, indicating a high degree of fit but a conservative estimation. In Figure 20b, the predicted values and the actual values varied among the different data points. The latter segment shows a greater degree of fit, whereas the former segment needed adjustments based on the numerical characteristics. The distribution of the predicted values in Figure 20b reveals that there was a significant numerical fluctuation before the 30 th data point.