## 2 Analysis and Derivation of the Inclination Angle of the Sliding Surface Behind the Shaft Wall  
### Analytical Model and Basic Assumptions  
Figure 1 presents the analytical model of soil sliding behind the circular shaft wall, where the shaft has a radius \(r_{0}\) and a depth \(H\). The fundamental assumptions employed in the calculations are as follows:  
(1) Although numerous experimental studies have demonstrated that the sliding surface of soil at the active limit state is curved, existing research indicates that the theoretical earth pressure values calculated using a curved sliding surface and those based on the  
Figure 1: Analytical model of the sliding surface behind the circular shaft wall. Coulomb planar sliding surface differ only slightly (approximately 6%). Moreover, the results obtained with the Coulomb planar sliding surface are generally more conservative [12]. Considering computational complexity and practical engineering applicability, this study assumes that the sliding surface is planar; specifically, the slip line is taken as a straight line passing through the heel of the wall, with an inclination angle of \(\beta\). (2) The shaft wall is assumed to be rough, the backfill behind the shaft wall consists of clay, the backfill surface is horizontal, and the external load is uniformly distributed. Additionally, the influence of interlayer shear stress within the soil is considered. (3) The displacement mode of the shaft wall is assumed to be translational. Establishment of the Equilibrium Equation and Determination of the Sliding Surface Inclination Angle  
The entire sliding soil mass behind the shaft wall is taken as the research object (see Figure 1), and its vertical and horizontal force components are illustrated in Figure 2. In the figure, \(F_{\varphi}\) denotes the force exerted by the soil outside the sliding surface on the sliding soil mass, with an angle \(\varphi\) relative to the normal of the sliding surface (i.e., the internal friction angle of the soil), and the radial coordinate at the point of action is \(r_{1}\). \(F_{\delta}\) represents the force applied by the shaft wall to the sliding soil mass, forming an angle \(\delta\) with the horizontal plane (i.e., the wall-soil friction angle). \(F_{\mathrm{cw}}\) and \(F_{\mathrm{c}}\) are the resultant cohesive forces at the wall-soil interface and along the sliding surface, respectively. \(F_{\mathrm{q}}\) is the resultant force generated by the uniformly distributed load on the backfill surface, \(G\) represents the self-weight of the sliding soil mass, and \(F_{\theta}\) is the resultant force of the circumferential stress acting on the sliding soil mass. Based on the analysis of Figure 2, the static equilibrium equations for the sliding soil mass in the vertical and horizontal directions can be expressed as:  
\[2\pi\big{[}F_{\delta}\sin\delta\cdot r_{0}+F_{\varphi}\cos(\beta-\varphi) \cdot r_{1}\big{]}+F_{c}\sin\beta+F_{\mathrm{cw}}=G+F_{q} \tag{1}\]  
\[F_{\delta}\cos\delta\cdot r_{0}\mathrm{d}\theta+2F_{\theta}\sin\!\left(\frac {\mathrm{d}\theta}{2}\right)+F_{c}\cos\beta\cdot\frac{\mathrm{d}\theta}{2\pi} =F_{\varphi}\sin(\beta-\varphi)\cdot r_{1}\mathrm{d}\theta \tag{2}\]  
Since the value of \(\mathrm{d}\theta\) is relatively small, it can be reasonably approximated as \(\sin\!\left(\frac{\mathrm{d}\theta}{2}\right)\approx\frac{\mathrm{d}\theta}{ 2}\). By further combining Equations (1) and (2), the following expression can be obtained:  
Figure 2: Stress analysis diagram of soil after shaft wall sliding. (**a**) Vertical direction; (**b**) horizontal direction.