## 3 Analysis of Active Earth Pressure in Circular Shafts
### Numerical Iterative Solution for Active Earth Pressure  
This study incorporates interlayer shear stress into the cohesive soil mechanics model. Due to the presence of \(\mathrm{d}\sigma_{0}\), it is not possible to obtain an explicit solution for the active earth pressure through integration; therefore, a numerical solution must be employed to derive the active earth pressure. This numerical solution divides the soil into horizontal sections based on the step length \(\Delta z\), as shown in Figure 6. By analyzing the \(i\)-th layer of the horizontal differential soil element, the vertical and radial force equilibrium equations are derived as follows (in Figure 6, the shear stress is considered only in the vertical force analysis). \[\overline{\sigma}_{\rm{v}(i-1)}\cdot S_{ABCD}-\overline{\sigma}_{\rm{wi}} \cdot S_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}-\tau_{\rm{wi}}\cdot S_{ ADD^{\prime}A^{\prime}}-(\sigma_{ri}\cos\beta+\tau_{fi}\sin\beta)\cdot S_{ BCC^{\prime}B^{\prime}}+\Delta G_{i}=0 \tag{21}\]  
\[\overline{\sigma}_{\rm{ahw}i}\cdot S_{ADD^{\prime}A^{\prime}}-\overline{\tau} _{(i-1)}\cdot S_{ABCD}+\overline{\tau}_{i}\cdot S_{A^{\prime}B^{\prime}C^{ \prime}D^{\prime}}-(\sigma_{ri}\sin\beta-\tau_{fi}\cos\beta)\cdot S_{ BCC^{\prime}B^{\prime}}+2\overline{\sigma}_{\rm{a}\theta i}\sin\bigg{(} \frac{\Delta\theta}{2}\bigg{)}\cdot S_{ABB^{\prime}A^{\prime}}=0 \tag{22}\]where the action areas \(S_{ABCD}\), \(S_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\), \(S_{ADD^{\prime}A^{\prime}}\), \(S_{BCC^{\prime}B^{\prime}}\), and \(S_{ABB^{\prime}A^{\prime}}\) of each force can be obtained using Equation (23); \(\Delta G_{i}\) is the self-weight of the horizontal differential soil element of the \(i\)-th layer, determined by Equation (24). Given the small thickness of the horizontal differential soil element, it is assumed that within the thickness \(\Delta z\), the horizontal stress at the wall-soil contact surface and the circumferential stress of the soil element are approximately linearly distributed, which can be calculated using Equation (25). The remaining parameters can be determined using Equation (26), based on the stress relationships derived from the active limit equilibrium condition and the spatial arching effect. \[\left\{ \begin{array}{l}S_{ABCD}=\left[\frac{H-(i-1)\Delta z}{\tan\beta} \right]\left[r_{0}+\frac{H-(i-1)\Delta z}{\tan\beta}\right]\Delta\theta\\ S_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}=\left(\frac{H-\Delta z}{\tan \beta}\right)\left(r_{0}+\frac{H-2\Delta z}{2\tan\beta}\right)\Delta\theta\\ S_{ADD^{\prime}A^{\prime}}=r_{0}\Delta\theta\Delta z\\ S_{BCC^{\prime}B^{\prime}}=\left[r_{0}+\frac{H-(i-1)\Delta z}{\tan\beta} \right]\frac{\Delta\theta\Delta z}{\sin\beta}\\ S_{ABB^{\prime}A^{\prime}}=S_{CDD^{\prime}C^{\prime}}=\left[\frac{2H-(2i-1) \Delta z}{2\tan\beta}\right]\Delta z\\ \end{array} \right. \tag{23}\]  
\[\Delta G_{i}=\gamma\left(\frac{S_{ABCD}+S_{A^{\prime}B^{\prime}C^{\prime}D^{ \prime}}}{2}\right)\Delta z \tag{24}\]  
\[\left\{ \begin{array}{l}\overline{\sigma}_{\text{ahw}i}=\frac{\left[k_{\text{ show}(i-1)}\overline{\sigma}_{v(i-1)}+K_{\text{shw}}\overline{\sigma}_{vi} \right]}{2}\\ \overline{\sigma}_{abi}=\frac{\left[k_{\phi(i-1)}\overline{\sigma}_{v(i-1)}+ K_{\theta}\overline{\sigma}_{vi}\right]}{2}\\ \end{array} \right. \tag{25}\]  
\[\left\{ \begin{array}{l}\overline{\tau}_{i}=K_{\text{rf}}\overline{\sigma} _{vi}\\ \tau_{\text{wi}}=\overline{\sigma}_{\text{ahw}i}\tan\delta+c_{\text{w}}\\ \tau_{\text{ff}}=\sigma_{ri}\tan\varphi+c\end{array} \right. \tag{26}\]  
By combining Equations (21)-(26), the iterative formula for a can be established:  
\[\overline{\sigma}_{vi}=\frac{\alpha_{\sigma(i-1)}\overline{\sigma}_{v(i-1)}- \alpha_{c}c-\alpha_{c_{w}}c_{w}+\Delta G_{i}}{\alpha_{\sigma_{i}}} \tag{27}\]  
where  
\[\alpha_{\sigma(i-1)}=\left[1+KK_{\tau(i-1)}\right]S_{ABCD}-\tfrac{1}{2}K_{ \text{ahw}(i-1)}(K+\tan\delta)S_{ADD^{\prime}A^{\prime}}-\tfrac{1}{2}KK_{\theta (i-1)}\Delta\theta S_{ABB^{\prime}A^{\prime}}\]  
Figure 6: Horizontal differential soil element and force analysis.