# Method to Control the Deformation of Anti-Slide Piles in Zhenzilin Landslide
## 3 Methodology and Validation  
Methods for Anti-Slide Pile Calculations in Two Bedrocks with Different Strengths below the Slip Surface  
The anti-slide pile can be divided into an elastic pile and a rigid pile. When \(\beta\times\text{h}\) is \(<1.0\) it belongs to a rigid pile; otherwise, it belongs to an elastic pile [36]. The h is the embedded section length of the pile. The pile deformation coefficient (\(\beta\)) is calculated using the following formula [34]:  
\[\beta\ =\ \left(\frac{k\cdot B_{\text{p}}}{4 EI}\right)^{\frac{1}{4}} \tag{1}\]  
where \(k\) denotes the modulus of subgrade reaction of the bedrock for the anti-slide pile under the action of lateral forces determined by the compressive strength of the rock [36]; \(E\) denotes Young's modulus of the pile, \(I\) denotes moment of inertia of the pile, and \(B_{p}\) denotes calculation width of the pile (\(B_{p}=\text{b}+1\), in meter, for rectangular section, \(b\) is the width of the pile section). If taking \(k=\) 100,000 kPa/m, \(E=\) 30,000,000 kPa, \(B_{p}=\) 3 m, \(I=\) 4.5 m\({}^{4}\), and \(h=\) 7m; \(\beta=\) 0.154 m\({}^{-1}\) and \(\beta\times\text{h}=\) 1.08. Thus, as regards engineering applications, the behavior of most of the anti-slide piles are elastic in accordance with Equation (1). For bedrocks containing two layers of rocks with different properties (see Figure 5a), the lateral load acting on the pile below the sliding surface can be expressed using two equations as following [36]:  
\[\left\{ \begin{array}{lcl}EI\frac{d^{4}x}{dy}+xk_{1}B_{p}&=&0&(0\leq y\leq y_{1 })\\ EI\frac{d^{4}x}{dy}+xk_{2}B_{p}&=&0&(y_{1}<y\leq h)\end{array} \right. \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Geotechnical Classification** & **Moisture Content** & **Density** & **Cohesion** & **Internal Friction** \\
**Classification** & **\(\omega\) (\%)** & **\(\rho\) (g/cm\({}^{3}\))** & **c (kPa)** & **Angle \(\Phi\) (\({}^{*}\))** \\ \hline Soil & 23.96 & 1.93 & 2.35 & 23.98 \\ Mudstone & 0.13 & 2.5 & 150 & 33 \\ Limestone & 0.1 & 2.7 & 700 & 40 \\ \hline \multirow{2}{*}{**Geotechnical Classification**} & **Young’s Modulus** & \multicolumn{2}{c}{**Natural**} & **Saturated** \\  & **E (\(\times\) 10\({}^{4}\) MPa)** & **Poisson’s Ratio** & **Compressive Strength (MPa)** & **Compressive Strength (MPa)** \\ \hline Soil & - & - & - & - \\ Mudstone & 2 & 0.25 & 12.4 & 8.31 \\ Limestone & 5 & 0.25 & 38.59 & 59.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental parameters of geotechnical materials in the survey area. Figure 4: Core samples of mudstone and limestone in site investigation area (H). where \(x\) is the deflection of the pile, \(y\) is the depth measured from the slip surface, \(k_{1}\) is the modulus of the subgrade reaction of the first layer of rock below the slip surface, and \(y_{1}\) is the thickness of the first rock; \(k_{2}\) is the modulus of subgrade reaction of the second layer of rock below the slip surface; \(h\) is the embedded section length of the anti-slide piles. The other parameters are the same as above. By solving the two equations described in Equation (2), the following expressions can be obtained [31]:  
\[\left\{ \begin{array}{ll}M_{\rm yi}&=\ -4{\rm x}_{i-1}\beta_{i}^{2}Elq \varphi_{3i}-4\theta_{i-1}\beta_{i}Elq\varphi_{4i}+M_{i-1}\varphi_{1i}+\frac{Q _{i-1}}{\beta_{i}}\varphi_{2i}\\ Q_{\rm yi}&=\ -4{\rm x}_{i-1}\beta_{i}^{3}Elq\varphi_{2i}-4\theta_{i-1}\beta_{i}^{2}Elq \varphi_{3i}-4M_{i-1}\beta_{i}\varphi_{4i}+Q_{i-1}\varphi_{1i}\\ \theta_{yi}&=\ \beta_{i}\Big{(}-4{\rm x}_{i-1}\varphi_{4i}+\frac{\theta_{i-1}}{ \beta_{i}}\varphi_{1i}+\frac{M_{i-1}}{\beta_{i}^{2}El}\varphi_{2i}+\frac{Q_{i -1}}{\beta_{i}^{3}El}q\varphi_{3i}\Big{)}\\ x_{yi}&=\ {\rm x}_{i-1}\varphi_{1i}+\frac{\theta_{i-1}}{\beta_{i}}\varphi_{2i}+ \frac{M_{i-1}}{\beta_{i}^{2}El}q\varphi_{3i}+\frac{Q_{i-1}}{\beta_{i}^{3}El}q \varphi_{4i}\end{array}