# Three-Dimensional Theoretical Model for Soil Arching with Inclined Slip Surfaces
## 6 Conclusions  
Based on [PERSON]'s 2D arching theory based on the assumption of a vertical slip surfaces, the derivation of a 2D and 3D theoretical model for soil arching with inclined slip surfaces is presented in this paper. Assuming soil mass as a homogeneous, isotropic and semi-infinite Mohr-Coulomb material, the thin-layer differential element method is adopted to obtain the loosening soil pressure. The 3D solution is verified with the traditional solution in the case of 2D simplification. Several cases have been conducted to investigate the effects of the buried depth ratio of the loose area, the length of loose area, soil parameters, and lateral earth pressure coefficient on the loose-earth pressure. The results of the proposed solution were then compared with test results. Several conclusions can be drawn, as follows:  
1. The 2D solution with inclined slip surfaces can degenerate into the classical [PERSON]'s solution when the angle of inclined slip surfaces is 90 degrees. Additionally, the 3D solution with inclined slip surfaces can degenerate into 2D when the length of the loose zone trends to be infinite, which can verify the correctness of the solution in this paper.  
2. The loose-earth pressure is far less than the self-weight earth pressure due to the existence of the soil-arching effect for both frictional soil and cohesive soil. The earth pressure ratio at each depth increases with the decreases of the angle of inclined slip surfaces. If the angle of the slip surfaces is close to 0, the slip surface is approximately parallel to the bottom, which means no soil-arching effect will happen.  
3. The earth pressure ratio decreases with the increase of the buried depth ratio. For shallow excavation, the change of the angle of the slip surface has little effect on the loose earth pressure. However, for deep excavation, the change of the angle of the slip surface has great effect on the loose-earth pressure. Using traditional [PERSON]'s solution with the assumption of vertical slip surfaces will underestimate the value of the loose-earth pressure for deep excavation condition. It will cause it to be unsafe in design, which needs to be paid attention to in engineering practice.  
4. The soil-arching effect is closely related to soil parameters. The larger the internal friction angle and cohesion of soil, the smaller the value of the loose-earth pressure. The smaller the earth pressure coefficient is, the closer the calculated result is to the self-weight earth pressure. If the earth pressure coefficients in the length and width directions of the excavation loose area are different, the influence of the earth pressure coefficients in the two directions on the results also depends on the aspect ratio of the excavation loose area.  
Figure 16: Comparison of tests results ([PERSON] et al. [19]) and 3D solution under \(H\)/B = 4.  
5. For the 3D soil-arching effect with a small aspect ratio, the 2D solution will underestimate the soil-arching effect and overestimate the loose-earth pressure, resulting in conservative design. The 2D assumption can be approximately used instead of the 3D assumption when the aspect ratio \(L/B>10\).  
6. The loose-earth pressure results by 3D solution in this paper are in good agreement with the experimental results. Compared with the traditional 2D solution, the solution in this paper can analyze and calculate the soil-arching effect more accurately.  
Conceptualization, Q.C.; Project administration, Q.C.; Software, B.Z.; Validation, C.X.; Writing--original draft, L.L.; Writing--review & editing, Q.C. and X.G. All authors have read and agreed to the published version of the manuscript.  
The financial supports from National Natural Science Foundation of China (NSFC Grant No. 52008373) and Natural Science Foundation of Zhejiang Province of China (No. Q22E080445) are greatly acknowledged.  
Not applicable.  
The authors declare no conflict of interest.