# Bearing Capacity of a Concrete Grouting Pad on the Working Surface of a Highway Tunnel Shaft
## 4 Parameter Sensitivity Analysis
### Effect of Shaft Radius \(r_{2}\) on Bearing Capacity \(q_{\rm u}\)  
Figure 8 illustrates the impact of the effect of shaft radius \(r_{2}\) on \(q_{\rm u}\) under different \(b\) value conditions. The results indicate that \(q_{\rm u}\) decreases as the \(r_{2}\)/\(c\) value increases, while the \(b\) value remains constant. The bearing capacity of the grouting pad increases with an increase in the \(b\) value for a given value of \(r_{2}\)/\(c\). Assuming a constant thickness, the bearing capacity of the grouting pad is inversely proportional to the ratio of the diameter to the area of the bottom loaded area. When the value of \(b\) is set to 1, the grouting pad \(q_{\rm u}\) increases by 15% to 87% compared to when \(b\) is set to 0.5. Similarly, when \(b\) is set to 1, \(q_{\rm u}\) increases by 0.5-1.3 times compared to when \(b\) is set to 0.75. These results indicate a significant increase in the grouting pad \(q_{\rm u}\) when considering the contribution of the middle principal stress. When \(b\) is set to 0.5 and 1, \(q_{\rm u}\) increases by 15-87%. When \(b\) is taken as 0.75 and 1, the \(q_{\rm u}\) increases by 0.5-1.3 times. After considering the contribution of medium principal stress, the \(q_{\rm u}\) of the grouting pad significantly increased. Specifically regarding construction practice, reflection of the working surface high-pressure grouting holes should be placed at the edge of the grouting pad to reflect the working surface. The grouting length should be reasonably controlled to avoid high-pressure grouting in the surrounding holes. It is important to reduce the diffusion area of the grouting pad pulp at the bottom to minimise the possibility of punching shear failure. Therefore, the construction of grouting should follow the principle of \"subdivision, time interval, and jump hole\".  
When \(c\) = 3 and \(b\) = 1, we analysed the effect of \(r_{2}\)/\(r_{1}\) on the bearing capacity \(q_{\rm u}\). Figure 9 shows a non-linear relationship between \(r_{2}\)/\(r_{1}\) and \(q_{\rm u}\). As \(r_{2}\)/\(r_{1}\) increases, \(q_{\rm u}\) first decreases  
Figure 8: Effect of the \(r_{2}\)/\(c\) ratio on \(q_{\rm u}\).  
Figure 7: Parameter \(q_{\rm u}\)–\(b\) relationship curve.  
slowly, then experiences a sudden increase, and finally decreases again. This means that larger \(r_{2}/r_{1}\) ratios do not necessarily result in larger \(q_{\rm u}\) values. Instead, \(q_{\rm u}\) reaches its relative maximum value at a certain ratio of \(r_{2}/r_{1}\). This point aims to guide engineering practice by recommending the use of a uniform yield criterion for designing grouting pads. It is important to check whether the \(r_{2}/r_{1}\) ratio is close to 1.25 to verify the calculation's reasonableness.  
The influence of \(r_{2}/c\) on the value of \(r_{1}\) was calculated using the iterative method, as shown in Figure 10. The value of \(r_{1}\) exhibits a three-stage trend of decreasing, then increasing, and finally decreasing again as \(r_{2}/c\) increases. When \(r_{2}/c\) is less than 1.375, \(r_{1}\) is the largest when \(b\) = 1 and the smallest when \(b\) = 0.5. When \(r_{2}/c\) is between 1.375 and 1.571, the maximum \(r_{1}\) is obtained when \(b\) = 0.75, and the minimum \(r_{1}\) is obtained when \(b\) = 0.5. When \(r_{2}/c\) is greater than 1.833, \(r_{1}\) is the minimum when \(b\) = 1.