# Three-Dimensional Theoretical Model for Soil Arching with Inclined Slip Surfaces
## 2 Derivation of 2D Theoretical Model for Soil Arching with Inclined Slip Surfaces  
[PERSON]'s formulation [1] is based on the equilibrium of a thin horizontal differential element of a soil mass. Several assumptions were made in [PERSON]'s formulation. The soil was assumed as homogeneous, isotropic and semi-infinite Mohr-Coulomb material. The slip surfaces between the loose area and stable area were assumed as vertical. Moreover, the shear strength on slip surfaces were fully mobilized. [PERSON]'s formulation is shown as follows:  
\[\sigma_{\rm v}=\frac{B\gamma-2c}{2K\tan\varphi}\Big{(}1-e^{-2K\tan\varphi(z/B) }\Big{)}+q\cdot e^{-2K\tan\varphi(z/B)} \tag{1}\]  
where \(\sigma_{\rm v}\) is vertical stress, \(z\) is the depth of soil differential element, \(\gamma\) is the unit weight of soil, \(c\) is the cohesive of soil, \(\varphi\) is the internal friction angle of soil, \(B\) is the width of loose zone induced by excavation, \(K\) is the coefficient of earth pressure. [PERSON] and [PERSON] [14] made an attempt to consider slip surfaces closer to the actual slip surfaces in the analysis in 2D condition as shown in Figure 1 The forces acting on the soil differential element in 2D condition can be seen in Figure 2.  
Assuming that the failure of slip surface follows Mohr-Coulomb failure criterion, a vertical equilibrium equation of the soil differential element can be obtained according to the vertical force balance. \[ \begin{array}{l}\gamma\Big{[}B+\frac{2(H-z)}{\tan\alpha}-\frac{\mathrm{d}z}{2 \tan\alpha}\Big{]}\mathrm{d}z=(\sigma_{\mathrm{v}}+\mathrm{d}\sigma_{\mathrm{v }})\left[B+\frac{2(H-z-\mathrm{d}z)}{\tan\alpha}\right]\\ -\sigma_{\mathrm{v}}\Big{[}B+\frac{2(H-z)}{\tan\alpha}\Big{]}+2(c+\sigma_{ \mathrm{n}}\tan\varphi)\mathrm{d}z+2\sigma_{\mathrm{n}}\cos\alpha\frac{ \mathrm{d}z}{\sin\alpha}\end{array}  \tag{2}\]  
where \(\alpha\) is the angle between slip surface and horizontal direction, \(\sigma_{\mathrm{n}}\) is the normal stress perpendicular to slip surface. According to relevant test results [4; 7; 10], \(\alpha\) is usually not very small. Therefore, the second-order infinitesimal in Equation (2) can be ignored. By simplifying and ignoring the second-order infinitesimal, it can be obtained from Equation (2):  
\[\gamma B-2c+\frac{2\gamma(H-z)}{\tan\alpha}+\frac{2\sigma_{\mathrm{v}}}{\tan \alpha}-2\sigma_{\mathrm{n}}\bigg{(}\tan\varphi+\frac{1}{\tan\alpha}\bigg{)}= \frac{\mathrm{d}\sigma_{\mathrm{v}}}{\mathrm{d}z}\bigg{[}B+\frac{2(H-z)}{\tan \alpha}\bigg{]} \tag{3}\]  
According to the equilibrium conditions at both ends of the differential element:  
\[\sigma_{\mathrm{n}}=\sigma_{\mathrm{v}}\Big{(}\cos^{2}\alpha+K\mathrm{sin}^{2 }\alpha\Big{)} \tag{4}\]  
Substitute (4) into (3) will obtain an ordinary differential equation:  
\[ \begin{array}{l}\frac{\mathrm{d}\sigma_{\mathrm{v}}}{\mathrm{d}z}\left[B\tan \alpha+2(H-z)\right]=(\gamma B-2c)\tan\alpha+2\gamma(H-z)+2\sigma_{\mathrm{v}} \\ -2\sigma_{\mathrm{v}}\left(\cos^{2}\alpha+K\mathrm{sin}^{2}\alpha\right)\left( \tan\alpha\tan\varphi+1\right)\end{array}  \tag{5}\]  
Figure 1: Inclined slip surfaces along with soil differential element subjected to stresses in 2D condition. Figure 2: Forces acting on the soil differential element in 2D condition. It can be found that when \(\alpha=90^{\circ}\) in Equation (5), the [PERSON]'s formulation shown in Equation (1) will be derived. Defining \(P\), \(Q\), \(R\) and \(S\) as follows:  
\[P=-2/\tan\alpha \tag{6a}\] \[Q=B+2H/\tan\alpha\] (6b) \[R=\left(\gamma B-2c\right)+2\gamma H/\tan\alpha\] (6c) \[S=\left(\cos^{2}\alpha+K\sin^{2}\alpha\right)\left(P-2\tan\varphi \right)-P \tag{6d}\]  
The general solution of the differential equation can be obtained as follows:  
\[\sigma_{\rm v}=\lambda(Pz+Q)^{S/P}+\frac{PS\gamma z+PQ\gamma-PR+RS}{S(P-S)} \tag{7}\]  
\(\lambda\) is the parameter in the solution process which can be obtained by boundary conditions. The boundary condition is shown as follows:  
\[z=0,\sigma_{\rm v}=q \tag{8}\]  
The parameter \(\lambda\) can be solved according to the boundary condition.