## 3 Analysis of Active Earth Pressure in Circular Shafts
### Vertical Arching Effect  
Due to the presence of the vertical arching effect (i.e., the soil arch effect in the vertical plane), the direction of the minor principal stress in the soil is no longer horizontal but deflects to a certain extent. The trace of the minor principal stress within a soil element on the same horizontal layer forms a curve, as shown in Figure 4, which can typically be approximated as an arc [4]. According to the limit equilibrium condition, the deflection angles of the major principal stress at the wall-soil contact surface (point A) and at the sliding surface (point B) are given as follows:  
\[\left\{ \begin{array}{l}\theta_{A}=\frac{1}{2}\left[\pi+\delta-\arcsin\left( \frac{\sin\delta}{\sin\varphi}\right)\right]\\ \theta_{B}=\frac{1}{2}\left(\frac{\pi}{2}-\varphi\right)+\beta\end{array} \right. \tag{11}\]  
Figure 4: Analytical model of the vertical arching effect of soil behind the shaft wall. The stress Mohr circle of the soil behind the shaft wall is translated leftward by \(\sigma_{0}\) (i.e., \(c\cot\varphi\)) units, as shown in Figure 5. The stress coordinates of any point in the equivalent coordinate system satisfy the following relationship:  
\[\left\{ \begin{array}{l}\sigma^{\prime}=\sigma+c\cot\varphi\\ \tau^{\prime}=\tau\end{array} \right. \tag{12}\]  
Furthermore, as shown in Figure 4, for the horizontal differential element \(ABB^{\prime}A^{\prime}\) at depth \(z\) under the active limit equilibrium state, assuming that the angle between the major principal stress direction at any point \(C\) and the horizontal plane is \(\theta\), the horizontal stress \(\sigma^{\prime}_{\text{hC}}\), vertical stress \(\sigma^{\prime}_{\text{vC}}\), and interlaminar shear stress \(\tau^{\prime}_{\text{C}}\) at this point are given as follows:  
\[\left\{ \begin{array}{l}\sigma^{\prime}_{\text{hC}}=\sigma^{\prime}_{1} \left[\frac{1+\sin\varphi\cos(2\theta)}{1+\sin\varphi}\right]\\ \sigma^{\prime}_{\text{vC}}=\sigma^{\prime}_{1}\left[\frac{1-\sin\varphi\cos (2\theta)}{1+\sin\varphi}\right]\\ \tau^{\prime}_{\text{C}}=\sigma^{\prime}_{1}\left[\frac{\sin\varphi\sin(2 \theta)}{1+\sin\varphi}\right]\end{array} \right. \tag{13}\]  
By substituting Equation (11) into Equation (13), the horizontal stress (i.e., the horizontal active earth pressure) at point \(A\) on the wall-soil contact surface can be obtained as follows:  
\[\sigma^{\prime}_{\text{ahw}}=\sigma^{\prime}_{1}\bigg{[}\frac{1+\sin\varphi \cos(2\theta_{A})}{1+\sin\varphi}\bigg{]} \tag{14}\]  
Furthermore, the average vertical stress \(\overline{\sigma}^{\prime}_{\text{v}}\) on the horizontal differential element \(ABB^{\prime}A^{\prime}\) can be expressed as follows:  
\[\overline{\sigma}^{\prime}_{\text{v}}=\frac{\int_{\theta_{A}}^{\theta_{B}} \sigma^{\prime}_{\text{vC}}R\sin\theta\mathrm{d}\theta}{L_{z}}=\sigma^{ \prime}_{1}\bigg{[}1-\frac{2\sin\varphi(\cos^{3}\theta_{A}-\cos^{3}\theta_{B} )}{3(1+\sin\varphi)(\cos\theta_{A}-\cos\theta_{B})}\bigg{]} \tag{15}\]  
where \(L_{z}\) is the width of the upper surface of the horizontal differential element \(ABB^{\prime}A^{\prime}\), and \(R=\frac{L_{z}}{\cos\theta_{A}-\cos\theta_{B}}\);  
Similarly, the average shear stress \(\overline{\tau}^{\prime}\) on the horizontal differential element \(ABB^{\prime}A^{\prime}\) can be obtained as follows follows:  
\[\overline{\tau}^{\prime}=\frac{\int_{\theta_{A}}^{\theta_{B}}\tau^{\prime}_{ \text{C}}R\sin\theta\mathrm{d}\theta}{L_{z}}=\sigma^{\prime}_{1}\bigg{[}\frac {2\sin\varphi(\sin^{3}\theta_{B}-\sin^{3}\theta_{A})}{3(1+\sin\varphi)(\cos \theta_{A}-\cos\theta_{B})}\bigg{]} \tag{16}\]  
Figure 5: Schematic diagram of the coordinate translation method.