# Study on the Optimization of Pile Length of Micropiles in Soil Landslides
## 2 Methods
### Calculation Method for Static Design of Micropiles
both ends of the loaded section of the pile, numbered 1, 2 and \(X+1,X+2\), respectively. Then, for any node i in the loaded section of the pile, the control difference equation is:  
\[\frac{EI}{h^{4}}(y_{i-2}-4y_{i-1}+6y_{i}-4y_{i+1}+y_{i+2})+b_{0}m_{1}(i-3)hy_{i}= p_{0}+(p_{A}-p_{0})\frac{(i-3)h}{h_{1}} \tag{4}\]  
The same holds true for the embedded pile segment below the sliding surface, which can be considered a horizontally loaded pile subjected to shear and bending moments at the pile head. The differential equation governing its deflection is:  
\[EI\frac{d^{4}y}{dz^{4}}+b_{0}(k+m_{2}z)y=0 \tag{5}\]  
Using the same discretization approach as the loaded pile segment, as shown in Figure 3, for the embedded pile segment, the node at the pile bottom is numbered 3, the node at the sliding surface is numbered \(X^{\prime}\), and two virtual nodes are placed at both ends of the embedded pile segment, numbered as 1, 2 and \(X^{\prime}+1,X^{\prime}+2\), respectively. The governing difference equation for any node j on the embedded pile segment can then be expressed as:  
\[\frac{EI}{h^{4}}\big{(}y_{j-2}-4y_{j-1}+6y_{j}-4y_{j+1}+y_{j+2}\big{)}+b_{0} \{k+m_{2}[h_{2}-(j-3)h]\}y_{j}=0 \tag{6}\]  
2 The\"m-k\" method  
The \"m-k\" method distinguishes between the sliding soil and the soil of the sliding body, employing the variable foundation coefficient method above the sliding surface and the constant foundation coefficient method below it. Let k\({}_{1}\) = m\({}_{1}\)y be the foundation coefficient for the loaded soil segment above the sliding surface, where m\({}_{1}\) is the proportionality coefficient of the soil foundation coefficient; let k\({}_{2}\) be the foundation coefficient for the anchored soil segment below the sliding surface, which is a constant value. The differential equation governing the loaded segment is identical to that used in the m-m method Equation (3), and the same discretization approach is applied to the pile body, resulting in an equivalent difference equation for any node on the loaded segment as shown in Equation (4) of the m-m method. Below the sliding surface, the embedded pile segment can be considered as a horizontally loaded pile subjected to shear and bending moments at the pile head, and its deflection differential equation is:  
\[EI\frac{d^{4}y}{dz^{4}}+b_{0}k_{2}y=0 \tag{7}\]  
Similarly, by discretizing and numbering the embedded pile segment, the governing difference equation for any node j on the embedded segment can be expressed as:  
\[\frac{EI}{h^{4}}\big{(}y_{j-2}-4y_{j-1}+6y_{j}-4y_{j+1}+y_{j+2}\big{)}+b_{0}k_ {2}y_{j}=0 \tag{8}\]  
By differentiating the aforementioned control points and incorporating corresponding boundary conditions, as well as continuity conditions for internal forces, displacements, and rotations at the sliding surface, a recursive formula can be derived to iteratively solve for node displacement along the pile body. In order to save space in this chapter, the differential method for solving the displacement equation will be detailed in the subsequent algorithm implementation section. (4) Solution of Internal Forces in Micropiles  
Based on the aforementioned difference control equation, the displacement of all nodes along the pile body can be calculated using a computer. The corresponding distributions of pile bending moment, shear force, and soil pressure can be obtained through the utilization of the subsequent recursive formula. Bending moment:\[\text{Bending moment}:\ M_{i}=\frac{EI}{h^{2}}(y_{i+1}+y_{i-1}-2y_{i}) \tag{9}\]  
\[\text{Shear force}:\ Q_{i}=\frac{EI}{2h^{3}}(y_{i+2}-2y_{i+1}+2y_{i-1}-y_{i-2}) \tag{10}\]  
\[EI\frac{d^{4}y}{dz^{4}}+b_{0}m_{1}zy=p_{0}+(p_{A}-p_{0})\frac{z}{h_{1}} \tag{3}\]0  
In the equation, k represents the foundation coefficient. When the \"m method\" is used for calculation, k = mz; when the \"k method\" is used for calculation, k is a constant.