# Three-Dimensional Theoretical Model for Soil Arching with Inclined Slip Surfaces
## 1 Introduction  
Load transformation from the yielding part of soil to an adjacent part is known as the soil-arching effect, which plays an important role in the design of various geotechnical infrastructures. [PERSON]'s trapdoor test was an important milestone in the development of theories on soil arching. [PERSON] then proposed two-dimensional (2D) arching theory through trapdoor tests using horizontal thin-layer differential element analysis method based on the assumption of vertical slip surfaces [1]. After that, many researchers have performed trapdoor test to study the soil-arching effect [2; 3; 4; 5; 6; 7; 8; 9]. [PERSON]'s solution is widely used by practical engineering in filed applications because of its simplicity. However, the assumption that the slip surfaces are vertical may not be suitable for all kinds of soil and engineering conditions. Inclined slip surfaces on a trapdoor have been observed in many experimental studies. [PERSON] [4] and [PERSON] [10] observed non-vertical slip surfaces in their model tests. [PERSON] et al. [7] carried out a series of trapdoor tests. Different slip surfaces were observed in tests including the final slip surface inclined to the outside of trapdoor. [PERSON] et al. [11] conducted circular retracting Trapdoor model testes and the results showed that slip surfaces tend to have a trapezoid shape in shallower cover soils. [PERSON] and [PERSON] [12] carried out a trapdoor modeltest and corresponding numerical simulation. It was found that the final slip surfaces were slightly outward inclined in the tests and simulation results. [PERSON] et al. [13] found three kinds of soil-arching evolution patterns in Trapdoor tests and described the inclination of the slip surfaces. Many scholars have carried out relevant research by using the assumption of inclined slip surfaces. [PERSON] and [PERSON] [14] presented arching theory assuming basin-shaped slip surfaces close to the actual slip surfaces. British Standards [15] also adopted this kind of slip surface model. [PERSON] et al. [16] proposed a method for calculating the loosening earth pressure based on the ellipsoid theory of particle flows using an ellipse slip face. [PERSON] et al. [17] proposed analytical solutions for shallow cohesive soils overlying trench voids under various slip surfaces in the shape of vertical, trapezoid, spiral or arching. Most studies on the soil-arching effect are based on 2D conditions, but in practical engineering the soil-arching effect happens under three-dimensional (3D) conditions. For example, during the process of tunnel excavation, the stress-transfer phenomenon caused by soil-arching happens not only in the transverse section, but also in the longitudinal section. Many experimental and theoretical studies have been performed under 3D conditions. [PERSON] and [PERSON] [18] established a 3D analytical model of a circle-shaped arch for a plane-strain geometry. [PERSON] et al. [19] performed three-dimensional trapdoor experiments to study the mechanical behavior in tunnel excavations. [PERSON] et al. [20] formulated a three-dimensional analytical solution to evaluate stresses in backfilled vertical narrow openings using vertical slip surfaces. [PERSON] [21] deduced the solution of loose earth pressure in the case of 3D inclined slip surfaces, but there were some omissions in the derivation. [PERSON] and [PERSON] [22] presented experimental and numerical studies of the trapdoor problem in 3D with a rectangular trapdoor. [PERSON] and [PERSON] [23] studied the 3D effect of soil arching in piled embankments by discrete element simulation. However, there are still few theoretical studies on the 3D soil-arching effect considering the inclined slip surface. Therefore, a 3D theoretical model was proposed in this study considering inclined slip surfaces. Horizontal thin-layer differential element method was used to obtain the loosening soil pressure. The 3D solution is verified with the traditional analytical solutions in the case of 2D simplification.