# Middle Rock Pillar Stability Criteria for a Bifurcated Small Clear-Distance Tunnel
## 5 Stability Criteria of the MRP During Construction
### Stress Criteria of the MRP  
To prevent rock mass fracture caused by excessive stress in the MRP when the displacement does not reach the limit value, it is necessary to establish stress-related criteria and improve the overall criteria system. Considering that the site conditions usually required measurement of vault settlement and clearance convergence, monitoring stress was challenging because of large data fluctuations and monitoring difficulties. Therefore, the area of maximum stress concentration in the MRP, specifically internal measurement point 1 in the central region, was selected as the stress monitoring point. The displacement was measured by the vault settlement value (measuring points 5 and 6) and clearance convergence value (measuring points 3, 7, 4, and 8) of the two tunnels, and relevant data were extracted. Figure 14a shows the overall distribution of the data points of the stress at the center point of the MRP, the vault settlement of the two tunnels, and the convergence value of the horizontal clearance. The stress concentration was distributed mainly between 2 MPa and 4 MPa. Case 1 in Figure 14b demonstrates that as the clear distance increased, the stress value consistently decreased, whereas the displacement change trend was not significant. To quantify and describe the relationship between stress value and displacement monitoring variables, the multiple linear regression method was employed for data analysis. Considering the stress value at monitoring point 1 as the output (S1), the settlement of the ramp vault as \(Y2\), the settlement of the mainline tunnel vault as \(Y3\), the convergence of the ramp horizontal clearance as \(X2\), and the convergence of the mainline tunnel horizontal clearance as inputs, the regression formula was \(S1=-1.002Y2+0.457Y3+0.147X2-0.368X3+0.416\), \(\text{R}^{2}=0.618\). Figure 15 compares the predicted and actual stress values, with the \(x\)-axis representing data points corresponding to different clear-distance ranges. The results indicate that the fitting effect was satisfactory within the range of 100-200 data points, where the corresponding clear-distance range was 6.7 m-11.8 m, and the range of W/B was 0.48-0.84. However, the predicted stress was lower than the monitored value for clear distances between 1.6 m and 6.7 m (data points 1-99), reducing prediction accuracy. Conversely,  
Figure 14: Stress distribution of the central points of the MRP: (**a**) the distribution of the central stress; (**b**) the changing trend of the central stress with the width–span ratio in case 1.  
the predicted stress exceeded the monitored value for clear distances of 11.8 m to 17.6 m (data points 201-322), indicating a conservative bias in the prediction results. Therefore, the auxiliary coefficient should be adjusted for these two ranges. Considering the relatively stable distribution of values within these ranges, a correction coefficient of 1.184 was applied to data points 1-99, whereas a correction coefficient of 0.810 was used for data points 201-322. The distributions of the predicted revised value and Q-Q plots are shown in Figure 15. The predicted values are closely aligned with the monitoring stress. The Q-Q plot provides a visual representation of the normality of the residuals, which is crucial for assessing the validity of the regression model. Compared with the Q-Q plots of the unrevised predicted and actual values, the revised Q-Q plots show a more desirable alignment, and the residual distribution is closer to the normal distribution, indicating that the corrected model exhibits higher accuracy and reliability in fitting the actual data. By recalculating the regression, the relationship between the actual stress and the predicted revised value resulted in an improved R\({}^{2}\) of 0.818, indicating a better alignment with the actual stress value. Figure 16 displays the distribution of actual stress values at the center under various working conditions.