# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 5 Stability Criteria of the MRP During Construction
### Displacement Criteria  
#### 5.2.1 Horizontal Displacement Criterion  
The displacement values of the monitoring points were extracted under various working conditions after tunnel penetration. Monitoring point 1 was an internal horizontal displacement monitoring point located in the central region of the MRP. Monitoring points 3 and 7 were the horizontal displacement monitoring points for the right and left arch waists of the ramp tunnel, whereas monitoring points 4 and 8 were the horizontal displacement monitoring points for the left and right arch waists of the mainline tunnel. The convergence of the horizontal headroom of the two tunnels applied in the field can be determined by comparing the horizontal displacement of the two tunnel arches. The distribution of horizontal displacement data points under different working conditions was obtained statistically, as shown in Figure 6.  
The distribution of horizontal displacement data under various conditions is illustrated in Figure 6. As shown in Figure 6a, convergence data points for the mainline tunnel were predominantly positive, indicating a bulging tendency at both arch waists. This bulging at the left arch waist caused compression in the MRP, influencing its boundary shape. In contrast, the ramp tunnel's horizontal convergence data were mostly negative, suggesting an inward bulging of both arch waists, which increased surrounding rock pressure on the ramp's lining support system. The horizontal displacement at monitoring point 1 was also negative, indicating an overall horizontal shift towards the ramp side in the MRP's  
Figure 6: Distribution of horizontal displacement data points: (**a**) distribution of horizontal displacement data points; (**b**) trend of horizontal displacement with width–span ratio under working condition 1. \(X1\) is the horizontal displacement of point 1. \(X2\) is the horizontal clearance convergence of the ramp tunnel. \(X3\) is the horizontal clearance convergence of the main tunnel. central region. Notably, some anomalous data points of \(-2\) mm and \(+3\) mm were observed under case 8, which indicated an unstable state and failure points in the surrounding rock. These points corresponded to an unstable state and represented failure points under the surrounding rock level. Assessing the stability of the intermediate rock involved selecting deformation values close to the unstable state. As seen in Figure 6b, the displacement trend displayed by the three curves shows a gradual decrease in stability as the width-span ratio increased, with a sharp change observed in the 0.2 to 0.3 range. However, the transition for the intermediate rock was relatively smooth, suggesting that horizontal displacement within the complete clear-distance range can be effectively predicted by the convergence value between the two tunnels. The abovementioned horizontal displacement data points were analyzed via the multiple linear regression method. The horizontal displacement \(X1\) at monitoring point 1 was considered the dependent variable, whereas the horizontal clearance convergence of ramp \(X2\) and the horizontal clearance convergence of mainline tunnel \(X3\) were treated as the independent variables. The multiple regression equation derived from the multiple linear regression equation is shown in Figure 7a. The data of \(X1\) closely align with the regression plane, indicating a good fit of the multiple regression model. The analysis of residuals shows that the mean value of the residuals is close to zero, suggesting an unbiased model, as shown in Figure 7b. Overall, the multiple linear regression model provides a reasonable fit for the data, and the regression plane effectively demonstrates the relationships between \(X2\), \(X3\), and \(X1\). After eliminating the unstable data points in condition 8, a comparison diagram between the actual horizontal displacement and the predicted value of monitoring point 1 is presented in Figure 8. The predicted value closely matches the actual value, with most data points located below the actual point.