## 3 Modified \"Wedge-Prism\" Theoretical Calculation Model  
The existing research shows that the classical \"wedge-prism\" limit equilibrium model is usually conservative in calculating the limit supporting force of shield excavation face, and the main reason is that the instability mode of excavation face is different from the actual situation [14; 25].  
The reasonable instability mode is the premise of using limit equilibrium method to calculate the ultimate supporting force accurately. In the classical \"wedge-prism\" limit equilibrium model, the unstable \"prism\" has been extended to the ground surface in the limit state, and the height \(H\) of the prism is approximately equal to the distance \(C\) from the top of the tunnel to the ground surface, which is not affected by the buried depth ratio \(C\)/\(D\). [PERSON] [26] found that the tunnel buried depth ratio has a significant influence on the instability mode (mainly reflected in the height of prism) through centrifuge model test on the instability of shield excavation face in sandy soil layer. As shown in Figure 4, when the tunnel depth is relatively small (such as \(C\)/\(D\) = 0.5), the instability zone has expanded to the surface in the limit state, and the height \(H\) of the \"prism\" is equal to the distance \(C\) from the top of the tunnel to the surface. When the depth of the tunnel is relatively large (such as \(C\)/\(D\) = 1 and 2), the instability zone does not extend to the surface (still inside the soil) in the limit state, and the height of the \"prism\" is less than the distance from the top of the tunnel to the surface. It can be seen that when the classic \"wedge-prism\" limit equilibrium model is used to calculate the ultimate supporting force of the shield excavation surface, the value of the \"prism\" height is only applicable to relatively shallow buried tunnels (such as \(C\)/\(D\) = 0.5). When calculating relatively deep buried tunnels (such as \(C\)/\(D\) = 1 and 2), the value of its \"prism\" height is too large. Therefore, in order to calculate the limit supporting force of shield excavation face more accurately, the influence of tunnel buried depth ratio on the height \(H\) of unstable prism should be considered in the modified \"wedge-prism\" theoretical model.  
Figure 3: Stability analysis model of shield excavation face—principal stress direction. (**a**) Side-looking; (**b**) Front view.  
Figure 5 is the revised \"wedge-prism\" theoretical calculation model proposed in this paper considering the influence of burial depth ratio on the instability mode of excavation face. Compared with the classical \"wedge-prism\" limit equilibrium model ([PERSON] 1996), the improvement of the model is that the value of the height of the unstable \"prism\" is related to the buried depth ratio \(C/D\) of the tunnel, rather than the unstable \"prisms\" all extend to the surface in the limit state.  
In order to clarify the value of \"prism\" height in the modified \"wedge-prism\" model, the author statistically gives the height of unstable \"prism\" under different tunnel buried depth ratios in previous  
Figure 4: Instability mode of excavation face in limit state.  
Figure 5: Modified “wedge-prism” theoretical calculation model.  
studies [26; 27; 28; 29; 30; 31; 32] (Table 1). As shown in Table 1, when the tunnel burial depth ratio is relatively small (such as C/D = 0.5), the height of the \"prism\" is equal to the buried depth of tunnel vault. When the tunnel buried depth ratio is relatively large (such as C/D = 1 and 2), the height of \"prism\" is smaller than the buried depth of the tunnel vault, and the height width ratio of prism is H/L = 1.5-2.2, and the average value of H/L is about 1.9. It can be seen that in the modified \"wedge-prism\" model, when the buried depth ratio is relatively small, \(H=C\) is taken; when the buried depth ratio is relatively large, \(H=1.9\) L is approximately taken. In summary, the height of the \"prism\" in the \"wedge-prism\" model is modified to \(H=\min\{\)C, 1.9 L\(\}\).