## 4 Limiting Support Force of the Tunnel Face
### Verification of Support Force of Shield Face  
Based on the correlation between the horizontal displacement of soil and the uniform stress applied, the ultimate load of the tunnel can be ascertained. As the uniform horizontal thrust exerted on the tunnel face incrementally rises, the soil on the tunnel face undergoes compression and deformation. When the thrust is minimal, the soil on the face actively deforms inward towards the tunnel; conversely, when the thrust is substantial, the soil on the face is passively squeezed and deforms into the surrounding soil. Upon reaching a certain critical thrust, the soil deformation on the tunnel face undergoes a sudden shift from positive (negative) to negative (positive), or there is a sudden increase in the soil deformation on the tunnel face. This critical point is identified as the ultimate support force on the tunnel face. Consequently, during the numerical simulation process, the horizontal displacement of the center point of the tunnel face along the tunnel axis was concurrently monitored under varying horizontal thrust conditions. The monitoring outcomes are illustrated in Figure 29.  
Figure 29: Horizontal displacement of tunnel face center point under different shield thrust: (a) 0–418 kPa; (b) 200–412 kPa. In accordance with the displacement change patterns at the center point of the tunnel face under different horizontal thrusts, as obtained through the FLAC\({}^{\text{3D}}\) numerical simulation method, it can be deduced that when the horizontal thrust applied to the tunnel face reaches 350 kPa, the trend of horizontal displacement at the center point of the tunnel face inverts. Specifically, when the horizontal thrust is below 350 kPa, the horizontal displacement at the center point is positive, indicating that the soil mass on the tunnel face predominantly moves inward towards the tunnel. However, when the horizontal thrust exceeds 350 kPa, the horizontal displacement at the center point becomes negative. At this juncture, the soil mass on the tunnel face is passively compressed and deformed into the earth ahead of the tunnel face under the influence of the horizontal thrust. This observation preliminarily suggests that the limiting support force of the tunnel face exceeds 350 kPa. As shown in Figure 30, as the uniform thrust applied to the tunnel face gradually increases from 0, the soil mass on the tunnel face first moves to the inside of the tunnel, and the displacement of the tunnel face is negative, which is because the stress applied to the face is not enough to support the self-weight of the soil mass on the face, and the face actively departs. When the uniform thrust exerted on the face of the tunnel further increases, the soil mass on the face of the tunnel compresses and degenerates inside the soil mass. At this time, the displacement of the face of the tunnel is positive because the stress exerted on the face can support not only the dead weight of the soil mass on the face but also the dead weight of the soil mass on the face. In addition, it breaks through the cohesive force of the soil, the sliding resistance generated at the middle section of the deformation process, and the passive deformation of the soil's squeezing face. That is, when the stress exerted on the face increases to about 412 kPa, the horizontal displacement value of the tunnel face shows an obvious turning point, as shown in Figure 30 as the intersection points of the red inclined dashed line and the red near-horizontal dashed line. After that, when the uniform thrust applied on the tunnel face increases slightly, the horizontal displacement of the tunnel face increases substantially, as shown by the red dashed horizontal line in Figure 30. Therefore, the stress value that causes significant changes in the deformation state of the tunnel face can be defined as the ultimate support force of the tunnel face, and the ultimate support force of the tunnel face determined by the numerical model in this study is 412 kPa.