## 2 Theoretical Analysis  
All pile tops are connected by an inclined arc crown beam along the slope surface, which is projected to the horizontal plane but still regarded as the arc beam for the convenience of calculation. The calculation mode can be simplified to a two-hinged arch in light of the research findings and the security of calculation results [28; 29]; only the interactive radial horizontal concentrated force between the pile top and the crown beam was regarded as the redundant force. Additionally, the horizontal thrust at the arch foot and the radial horizontal concentrated force were regarded as the redundant force. However, the bending moment and the torsional moment between the pile top and the crown beam were not considered. In this study, the acting force between the anti-slip pile and the crown beam was taken as the redundant force, and then calculation models of the crown beam and anti-slip pile were established, respectively. At the same time, the internal force of the crown beam and the displacement of the anti-slip pile were analyzed theoretically. According to the displacement coordination condition of the connection between the crown beam and the pile top, the typical equations were presented to solve the redundancy force, and then the bending moment and axial force of the crown beam were calculated. More detailed calculations can be found in a previous study [25], which were analyzed as follows. The arc crown beam was divided into two parts for the calculation model of a two-hinged arch. The basic system with the horizontal thrust (\(\overline{X}_{1}=1\)) at the arch foot was taken as the redundant constraint, as shown in Figure 1. The bending moment (\(\overline{M}_{p1}\)), the shear force (\(\overline{Q}_{p1}\)), and the axial force (\(\overline{N}_{p1}\)) at an arbitrary point p of the crown beam are expressed by Equations (1)-(3):  
\[\overline{M}_{p_{1}}=-y_{p} \tag{1}\]  
\[\overline{Q}_{p1}=-\sin\varphi_{p} \tag{2}\]  
\[\overline{N}_{p1}=\cos\varphi_{p} \tag{3}\]  
where \(y_{p}\) is the abscissa of point \(p\); \(\varphi_{p}\) is the tangent azimuth angle of point \(p\). However, the radial horizontal force (\(\overline{X}_{j_{1}}=1\)) acts alone on the crown beam. The left half span and the right half span need to be calculated separately according to the position of the radial horizontal force. The radial horizontal force is applied on the left half span of the crown beam, as shown in Figure 2. Only the left half span of the crown beam was taken as an instance for analysis during the theoretical calculation process. \(\overline{l}_{\overline{x}_{j-A}}\) is the moment arm of \(\overline{X}_{j_{l}}\) on the left support A, and it can be expressed by Equation (4). The vertical support  
Figure 1: The basic system of two-hinged arch. reaction at position A and B can be calculated, respectively, as shown in Equations (5) and (6). The horizontal support reaction at position A is calculated using Equation (7). \[\overline{I}_{\overline{X}_{ji-A}}=\frac{x_{ji}\cos(\alpha_{A-\overline{X}_{ji}}- \varphi_{ji})}{\cos\alpha_{A-\overline{X}_{ji}}} \tag{4}\]  
\[\overline{Y}_{A}=-\overline{Y}_{B}-\overline{X}_{ji}\cos\varphi_{ji} \tag{5}\]  
\[\overline{Y}_{B}=-\frac{\overline{X}_{ji}\overline{I}_{\overline{X}_{ji-A}}}{l} \tag{6}\]  
\[\overline{X}_{A}=\overline{X}_{ji}\sin\varphi_{ji} \tag{7}\]  
where \(x_{ji}\) is an abscissa of acting point \(p\); \(\alpha_{A-\overline{X}_{ji}}\) is the plane-coordinate azimuth of the left support A from an acting point \(p\); \(\varphi_{ji}\) is the tangent azimuth angle; \(\overline{X}_{ji}\) is the radial horizontal concentrated force; \(l\) is the span. With no consideration of the influence from the horizontal support reaction, Equations (8) and (9) show the bending moment (\(\overline{M}_{p_{ji}}^{0}\)) and the shear force (\(\overline{Q}_{p_{ji}}^{0}\)) at an arbitrary point \(p\) of the crown beam within an interval of \(0\leq x\leq x_{ji}\).