# Stability Analysis of the Longitudinal Slope Linear Shield Tunnel Excavation Face
## 3 Results and Discussion
### Results and Discussion of Ultimate Support Ratio in Numerical Simulation Results  
The changes observed in the development process of the transparent soil tests indicate that the magnitude of the support force is a critical factor in triggering the transition from local to global instability. To facilitate the description of the relationship between support pressure and excavation face stability, the support ratio is defined as \(\lambda=\text{Pu}/\text{Pu}_{0}\)[24]. In the numerical simulation, a uniformly distributed support force is applied. The support ratio is gradually reduced in a stepwise manner during each calculation cycle. After each reduction, the corresponding limit support ratio of the excavation face is determined based on the control method of the numerical simulation by detecting and recording the results. Figure 8 illustrates the relationship between the horizontal displacement (\(y\)) at the center of the excavation face and the support ratio (\(\lambda\)) for different burial depth ratios. Figure 7: Comparison of surface displacements. As shown in Figure 9, when the support ratio \(\lambda\) ranges from 1 to 0.6, the influence of slope on horizontal displacement is relatively weak, and the center displacement for different slopes tends to converge. However, as the support ratio further decreases, the effect of slope on horizontal displacement becomes more significant, specifically manifesting as an increase in the horizontal displacement at the center of the excavation face with larger slope values. Furthermore, when \(C/D=3\), an increase in the depth-to-diameter ratio leads to a greater magnitude of stress release, resulting in a more pronounced increase in the horizontal displacement (\(y\)) at the center of the excavation face during soil destabilization. This occurs because the reduction in the support ratio directly weakens the ability of the support structure to restrain the surrounding rock, thereby expanding the range of stress redistribution in the surrounding rock and exacerbating the deformation of the excavation face. Figure 8: Relationship between horizontal displacement y of the center of the excavation face and support ratio \(\lambda\). (a) \(C/D=1\); (b) \(C/D=2\); (c) \(C/D=3\). Figure 9 illustrates the relationship between the vertical displacement (\(z\)) and the support ratio (\(\lambda\)) at the excavation face under different burial depth ratios. The study reveals that under varying burial depths, as the support ratio decreases, the slope has a significant impact on the vertical displacement at the center of the excavation face. Specifically, a larger slope results in greater vertical displacement in the soil at the excavation center, a pattern consistent with the variation of horizontal displacement at the same location. Under an upward (elevated) slope condition (\(i>0\)), the soil at the excavation face is more prone to increased vertical displacement. This phenomenon can be attributed to the direct stress exerted by the overlying soil on the excavation face, which facilitates the downward sliding of the soil in front of the excavation. Figure 10 illustrates the relationship between the ultimate support ratio (\(\lambda_{\rm lim}\)) and the slope (\(i\)) for different burial depth ratios. For varying burial depth ratios (\(C/D\)), a clear positive correlation is observed between \(\lambda_{\rm lim}\) and \(i\). As the longitudinal slope increases, the ratio rises significantly, while the rate of increase remains approximately constant. This indicates that an increase in slope exacerbates the sliding tendency of the soil in front of the excavation, thereby reducing the stability of the surrounding soil. Consequently, greater support strength is required to counteract this sliding tendency. Furthermore, as the burial  
Figure 9: Relationship between vertical displacement \(z\) of the center of the excavation face and support ratio \(\lambda\). (**a**) \(C/D=1\); (**b**) \(C/D=2\); (**c**) \(C/D=3\).