# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 5 Stability Criteria of the MRP During Construction
### Displacement Criteria
#### 5.2.2 Vertical Displacement Criterion  
The distribution of each vertical displacement data point is depicted in Figure 10. Figure 10a shows that the vertical displacement data points in the upper area of the MRP are above the ramp and the vault data points of the main line tunnel, and the distribution is relatively stable. In Figure 10b, for example, in case 1, the curve change process is stable without any mutation points, indicating that the vertical displacement of the MRP can be uniformly analyzed within the full clear distance of the tunnel. Figure 8: Comparison of actual horizontal displacement and predicted value of \(X1\). The inset shows a Q-Q plot of residuals, where the dots represent sample quantiles and the solid line represents theoretical quantiles under a normal distribution. Figure 9: Numerical distribution of the actual horizontal displacement at the center of the MRP. The vertical displacement of the MRP through the vault settlement of two tunnels was predicted and analyzed via the multiple linear regression method with the same horizontal displacement. The dependent variable was the vertical displacement (\(Y1\)) at monitoring point 2, whereas the independent variables were the crown settlement of the ramp (\(Y2\)) and the crown settlement of the main tunnel (\(Y3\)). The relationship among these variables is shown in Figure 11a. The \(Y1\) data closely matched the regression plane, indicating a good fit for the multiple regression model. The residual analysis revealed that the mean of the residuals was close to zero, indicating an unbiased model. Overall, the multiple linear regression model provided a reasonable fit for the data, and the regression plane effectively illustrated the relationships between \(Y2\), \(Y3\), and \(Y1\). Figure 11: Multiple regression model of the vertical displacement: (**a**) Regression plane; (**b**) Histogram of residuals. Figure 10: Distribution of vertical displacement data points: (**a**) Distribution of vertical displacement data points; (**b**) Trend of vertical displacement with width-span ratio under case 1. \(Y1\) is the vertical displacement of point 2. \(Y2\) is the crown settlement of the ramp tunnel. \(Y3\) is the crown settlement of the main tunnel. The predicted vertical displacement was compared with the monitored vertical displacement, as shown in Figure 12. The predicted values fluctuated on both sides of the real values, showing a certain linear change. To further validate the accuracy of the model fitting, Figure 12 presents the Q-Q plot of the actual vertical displacement versus the predicted values. From the Q-Q plot, it can be seen that the residual points of the actual vertical displacement generally follow a 45-degree diagonal line, indicating that the residuals approximately follow a normal distribution. This suggests that the residuals of the regression model do not significantly deviate from normality, and the model's fit is reasonable. The distributions of the real vertical displacement in the middle rock under various working conditions are shown in Figure 13. The vertical displacement was evenly distributed in the range of \(-7.0\) mm\(\sim\)4.5 mm, with \(-4.0\) mm\(-2.5\) mm accounting for 92.39% of the total displacement. The maximum vertical displacement was \(-7.0\) mm, which served as the control limit. Therefore, the criterion for vertical displacement was obtained via the following formula:  
\[Y1=1.216Y2+0.183Y3+6.069\leq-7.0, \tag{2}\]  
Figure 12: Comparison of actual vertical displacement and predicted value of the vertical displacement of point 2. The inset Q-Q plot shows the residual quantiles (dots) versus the theoretical quantiles under a normal distribution (line), indicating the degree of normality in residual distribution. Figure 13: Numerical distribution of true vertical displacement inside the upper area of the MRP.