## 3 Calculation Method
### Establishment of Calculation Method  
#### 3.2.1 Calculation When the Sliding Surface Is Horizontal  
Figure 3 shows the simplified calculation diagram of double rows of buried piles when the sliding surface is horizontal. Figure 3a shows the general analysis model; line OA is the initial horizontal sliding surface, and line segments OB and BD are the potential sliding  
Figure 2: Schematic diagram of landslide reinforcement by double rows of submerged piles. surface over the top. The angle \(\beta\) formed by line segments OA and OB is called the initial rupture angle (\(\beta=45^{\circ}-\phi/2\)). The extended line segment OB intersects at point Q at the front pile, and the angle \(\beta_{1}\) formed by the line BD and BQ is called the secondary fracture angle. H\({}_{1}\) represents the length of the rear pile above the sliding plane, h\({}_{2}\) represents the length of the front pile above the sliding plane, H is the thickness of the sliding body, L is the distance between point O and point A, a is the distance between the front and back piles and b is the width of the pile, \(c\) as the cohesion coefficient and \(\phi\) as the friction angle. Figure 3b shows the loading model of mass POBF, where E\({}_{0}\) is the landslide thrust at line OP, E\({}_{1}\) is the soil resistance generated by mass FBDE (direction parallel to line OB), W\({}_{1}\) is the dead weight of mass POBF and N\({}_{1}\) is the supporting force of mass OAB (direction perpendicular to line OB). Figure 3c shows the loading model of mass FBDE. E\({}_{1}\)' is the landslide thrust transferred by mass POBF to the FB surface (equal in magnitude and opposite in direction to E\({}_{1}\)), W\({}_{2}\) is the dead weight of mass FBDE and N\({}_{2}\) is the supporting force on the BD surface (direction perpendicular to line BD). (1) h\({}_{1}\) calculation when the sliding body does not slide along the passive earth pressure rupture surface OB. Force analysis is carried out on the overtop slider OBFP. The soil body weight of the wedge OBFP is W\({}_{1}\), the landslide thrust on the OP interface is E\({}_{0}\), the resistance generated by the wedge BDEF is E\({}_{1}\) (direction parallel to OB), the supporting force of the soil OAB is N\({}_{1}\) (direction perpendicular to OB fracture plane) and the friction force is N\({}_{1}\)\(\times\) tan\(\phi\). The cohesion is \(\frac{c\hbar_{1}}{\sin\beta}\). Figure 3: Simplified calculation diagram of the row of buried piles (when the sliding surface is horizontal) (**a**). General analysis model; (**b**). loading model of mass POBF; (**c**). loading model of mass FBDE. Given the design safety factor, the landslide thrust E\({}_{0}\) acting on the OP surface can be easily solved using the transfer coefficient method [21], and the relevant solving process is shown in Equations (1) and (2):  
\[E_{i}=W_{i}\text{sin}\,\theta_{i}-W_{i}\text{cos}\,\theta_{i}\frac{\text{tan}\, \varphi_{i}}{F_{s}}-l_{i}\frac{c_{i}}{F_{s}}+\Psi_{i}E_{i-1} \tag{1}\]  
\[\Psi_{i}=\text{cos}(\theta_{i-1}-\theta_{i})-\text{sin}\,(\theta_{i-1}-\theta _{i})\frac{\text{tan}\,\varphi_{i}}{F_{s}} \tag{2}\]  
In Equations (1) and (2), \(E_{i}\) represents the remaining sliding force of bar \(i\). \(E_{i-1}\) represents the remaining sliding force of bar \(i-1\). \(\Psi_{i}\) represents the sliding plane inclination of bar \(i\). \(\theta_{i-1}\) represents the sliding plane inclination of bar \(i-1\). \(W_{1}\) represents the dead weight of bar \(i\). \(l_{i}\) represents the sliding plane length of bar \(i\), and \(c_{i}\) represents the cohesive force of bar \(i\). \(\psi i\) is the residual sliding force transfer coefficient between bars \(i-1\) and \(i\). \(F_{s}\) is the design safety factor. It should be noted that other classical limit equilibrium methods can also be used to solve the sliding force, such as [PERSON] method [25], [PERSON] method [26], [PERSON] method [27], etc. The Morgenstern-Price method assumes that there is a proportional relationship between tangential force and normal force between bars.