## 2 Analysis and Derivation of the Inclination Angle of the Sliding Surface Behind the Shaft Wall
### Analytical Model and Basic Assumptions
\[F_{\delta}=\frac{\sin(\beta-\varphi)}{2\pi r_{0}\cos(\beta-\varphi-\delta)}\big{\{} G-[2\pi\cot(\beta-\varphi)]F_{\theta}-[\sin\beta+\cos\beta\cot(\beta-\varphi)]F_{c}-F_{cw}+F_{q} \big{\}} \tag{3}\]  
Since the [PERSON] assumption tends to overestimate the influence of circumferential stress, [PERSON] [17] introduced the circumferential stress coefficient \(\lambda\) (i.e., \(\sigma_{\theta}=\lambda\sigma_{1}\)), suggesting that its value should range between 1 and \(K_{0}\). From the perspective of engineering safety, [PERSON] recommended adopting \(K_{0}\) (where \(K_{0}\) denotes the coefficient of static earth pressure). However, existing studies have demonstrated that as the radial distance from the shaft wall increases, the influence of the circumferential arching effect progressively diminishes, leading to a corresponding decrease in circumferential stress. The magnitude of the circumferential stress coefficient \(\lambda\) is strongly dependent on the radial distance: near the shaft wall, the coefficient approaches 1, whereas near the sliding surface, it tends toward \(K_{0}\)[28; 30]. For computational convenience, it is assumed that the circumferential stress decreases linearly along the radial direction, meaning that the circumferential stress coefficient \(\lambda\) varies linearly with the radial coordinate \(r\):  
\[\lambda=1-(1-K_{0})\Bigg{(}\frac{r-r_{0}}{r_{\beta}-r_{0}}\Bigg{)} \tag{4}\]  
where \(K_{0}\) is the static earth pressure coefficient, and \(K_{0}=1-\sin\varphi\); \(r_{\beta}\) represents the radial coordinates at the sliding surface. The average circumferential stress coefficient \(\overline{\lambda}\) of the horizontal differential element at any height under the active limit state can be determined using Equation (4):  
\[\overline{\lambda}=\frac{1+K_{0}}{2} \tag{5}\]  
Further, the average circumferential stress coefficient \(\sigma_{\text{a}\theta}\) of the horizontal differential unit can be expressed as:  
\[\sigma_{\text{a}\theta}=\overline{\lambda}\sigma_{1}=\bigg{(}\frac{1+K_{0}}{ 2}\bigg{)}(\gamma z+q) \tag{6}\]  
The circumferential force \(F_{\theta}\) can ultimately be obtained from Equation (6):  
\[F_{\theta}=\int_{0}^{H}\sigma_{\text{a}\theta}\bigg{(}\frac{H-z}{\tan\beta} \bigg{)}\;\text{d}z=\frac{(1+K_{0})(\gamma H+3q)H^{2}}{12\tan\beta} \tag{7}\]  
The self-weight \(G\) of the sliding soil mass behind the shaft wall is given by:  
Figure 2.0  
In addition, the values of the other pertinent parameters are summarized as follows:  
\[\left\{ \begin{array}{l}F_{cw}=c_{w}S_{c_{w}}=2\pi c_{w}r_{0}H\\ F_{c}=cS_{c}=\frac{\pi cH(2r_{0}+H\cot\beta)}{\sin\beta}\\ F_{q}=qS_{q}=\pi qH\cot\beta(2r_{0}+H\cot\beta)\end{array} \right. \tag{9}\]  
where \(c_{w}\) and \(c\) represent the cohesion at the wall-soil interface and the cohesion of the soil, respectively \(c_{w}=c(\tan\delta/\tan\varphi)\); \(q\) represent the uniformly distributed load acting on the fill surface; \(S_{c_{w}}\), \(S_{c}\), and \(S_{q}\) denote the action areas corresponding to the forces mentioned above. Based on [PERSON]'s earth pressure theory, Equations (7)-(9) are substituted into Equation (3), and the extremum (i.e., parameter \(\mathrm{d}F_{\mathrm{s}}/\mathrm{d}\beta=0\)) is determined, which can be readily solved using MATLAB (R2023b, The MathWorks, Inc., Natick, MA, USA). Details are omitted here for brevity. Accordingly, the inclination angle of the soil sliding surface under the active limit state is obtained. This angle is influenced by multiple factors, including the internal friction angle of the soil, the wall-soil friction angle, the soil cohesion, the uniformly distributed load on the fill surface, and the shaft radius-to-height ratio. \[\beta=f\Big{(}\varphi,\ \delta,\ c,\ q,\ \frac{r_{0}}{H}\Big{)} \tag{10}\]  
Unlike the traditional planar earth pressure theory, Equation (10) more precisely captures the effect of the radius-to-height ratio on the inclination angle of the sliding surface.