# Three-Dimensional Theoretical Model for Soil Arching with Inclined Slip Surfaces
## 3 Derivation of 3D Theoretical Model for Soil Arching with Inclined Slip Surfaces
\right\}\] \[=\bigg{(}BL+\frac{2(B+L)(H-z)}{\tan\alpha}+\frac{4(H-z)^{2}}{ \tan^{2}\alpha}\bigg{)}\frac{\rm{d}\sigma_{\rm v}}{\rm{d}z}\]  
The boundary condition is the surface load \(q\), so the boundary condition equation is expressed as follows:  
\[z=0,\sigma_{\rm v}=q \tag{14}\]  
According to the boundary condition Equation (14), the only solution of ordinary differential Equation (13) can be determined; that is, to solve the initial value problem of ordinary differential equations. However, the ordinary differential Equation (13) is so complex that it is difficult to give the explicit expression of the solution. Therefore, the fourth-order Runge-Kutta method is used to calculate the vertical stress of soil when the depth is \(z\) by matlab programming. The ordinary differential Equation (13) can be written as:  
\[\frac{\rm{d}\sigma_{\rm v}}{\rm{d}z}=f(z_{\rm n},\sigma_{\rm vn}) \tag{15}\]  
and the calculation formula of the fourth-order Runge-Kutta method is expressed as follows:  
\[\left\{ \begin{array}{l}P_{1}=f(z_{\rm n},\sigma_{\rm vn})\\ P_{2}=f\Big{(}z_{\rm n}+\frac{i}{2},\sigma_{\rm vn}+\frac{i}{2}P_{1}\Big{)}\\ P_{3}=f\Big{(}z_{\rm n}+\frac{i}{2},\sigma_{\rm vn}+\frac{i}{2}P_{2}\Big{)}\\ P_{4}=f(z_{\rm n}+i,\sigma_{\rm vn}+iP_{3})\\ \sigma_{\rm vn+1}=\sigma_{\rm vn}+\frac{i}{6}(P_{1}+2P_{2}+2P_{3}+P_{4})\end{array} \right. \tag{16}\]  
where \(i\) is the length of integral step and \(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\) are integral parameters. An example is used to verify whether the 3D theoretical model for loose earth pressure with inclined slip surfaces is consistent with two-dimensional loose-earth pressure solution mentioned above. Multiply both sides of Equation (13) by \(\tan\alpha/L\), Equation (13) can be written as:  
\[ \begin{array}{l}(\gamma B-2c)\tan\alpha+2\gamma(H-z)+2\sigma_{\rm v}-2 \sigma_{\rm v}\big{(}\cos^{2}\!\alpha+K_{\rm l}\!\sin^{2}\!\alpha\big{)}(\tan \varphi\tan\alpha+1)+\\ 2\frac{2}{L}[(H-z)\big{(}B+2H-2z)]-2\frac{c}{L}[B\tan\alpha+4(H-z)]\\ \left.\begin{array}{l}2\big{(}\cos^{2}\!\alpha+K_{\rm v}\!\sin^{2}\!\alpha \big{)}\Big{[}B+\frac{2(H-z)}{\tan\alpha}\\ -\frac{\sigma_{\rm v}}{L}\Big{\}}+\frac{4\big{(}\cos^{2}\!\alpha+K_{\rm v}\! \sin^{2}\!\alpha\big{)}(H-z)}{\tan\alpha}-2B-\frac{8(H-z)}{\tan\alpha}\\ +2\big{(}\cos^{2}\!\alpha+K_{\rm v}\!\sin^{2}\!\alpha\big{)}\big{[}B\tan\alpha+ 2(H-z)\big{]}\tan\varphi\\ +4\big{(}\cos^{2}\!\alpha+K_{\rm l}\!\sin^{2}\!\alpha\big{)}(H-z)\tan\varphi \end{array}\right\}\\ =\frac{\rm{d}\sigma_{\rm v}}{\rm{d}z}\big{[}B\tan\alpha+2(H-z)\big{]}+\frac{ \rm{d}\sigma_{\rm v}}{\rm{d}z}\frac{2}{L}\bigg{[}B(H-z)+\frac{2(H-z)^{2}}{ \tan\alpha}\bigg{]}\end{array}  \tag{17}\]  
It can be found that Equation (17) has the same form as the 2D ordinary differential Equation (5) when \(L\) tends to infinity. A specific case is also used for validation. In this case, the following parameters \(H=15\) m, \(B=3\) m, \(\varphi=35^{\circ}\), \(c=0\) kPa, \(q=0\) kPa and \(\gamma=18\) kN/m\({}^{3}\) are used for calculation. Inthe 3D example, the value of \(L\) is 1000 m, which is approximately considered to degenerate into a 2D case. The calculation results are shown in Figure 5.  
It can be obtained from the calculation results that the curve of the earth pressure ratio versus depth under different angles of slip surface calculated by a degenerated 3D solution is consistent with the results of the 2D solution; thus, the correctness of the 3D solution can be verified.