# Study on the Optimization of Pile Length of Micropiles in Soil Landslides
## 2 Methods
### Calculation Method for Static Design of Micropiles
Generally, the anchorage length of micropiles in the sliding body is equal to the thickness of the sliding body at the pile location h\({}_{1}\). When h\({}_{1}\) is less than Lu, the anchorage length in the sliding body is insufficient, and a connecting beam should be installed at the top of the pile for reinforcement. (3) Differential control equation for monopile by elastic coefficient method For soil landslides, the m-method is commonly used to calculate the pile's force using the elastic coefficient method. However, for some dense clay and stone-mixed soils, the k-method is sometimes used. Consequently, this study introduces two elastic pile calculation models, namely the \"m-m method\" and the \"m-k method\", specifically tailored for soil landslides, with the aim of providing a valuable reference for the design and computation of micropiles in all types of soil landslide scenarios. As shown in Figure 3, the total length of the micropile is H, with a loaded section length of h\({}_{1}\) above the sliding surface and an embedded section length of h\({}_{2}\) below the \begin{table}
\begin{tabular}{c c c c} \hline
**Pile Position** & **Rear Pile** & **Middle Row Pile** & **Front Row Piles** \\ \hline Allocation ratio & 0.411 & 0.348 & 0.241 \\ \hline \end{tabular}
\end{table} 
Table 1: Table of landslide thrust distribution coefficients. sliding surface. To ensure the universality of the sliding force distribution form, the sliding force shared on the micropile is assumed to have a trapezoidal distribution, with a load distribution degree of p0 at the pile top and a load distribution degree of pA at the sliding surface. When p0 = pA, it is a rectangular distribution, and when p0 = 0, it is a triangular distribution. The diameter of the micropile is d, and its bending stiffness is EI. Based on the m-method and k-method in the specification, we integrate the procedures of m-method and k-method in the process of application and propose the m-method and m-k-method. 1. The \"m-m\" method  
The \"m-m\" method treats the sliding bed soil and the sliding body soil equally and calculates both the upper and lower sides of the sliding surface using variable foundation coefficients. The foundation coefficient above the sliding surface is assumed to be k1 = m1y (m1 is the proportionality coefficient of the foundation coefficient of the loaded soil layer). The foundation coefficient below the sliding surface is k2 = k + m2y (m2 is the proportionality coefficient of the foundation coefficient of the embedded soil layer, and k is the foundation coefficient at the sliding surface). Then, as shown in Figure 2, under the combined action of the sliding force and soil resistance, the micropile undergoes bending deformation in its loaded section. The bending differential equation for the loaded section is  
\[EI\frac{d^{4}y}{dz^{4}}+b_{0}m_{1}zy=p_{0}+(p_{A}-p_{0})\frac{z}{h_{1}} \tag{3}\]  
The aforementioned equation is a fourth-order variable coefficient homogeneous differential equation, which can be solved through various methods such as the power series method, numerical integration method, and finite difference method. For the purpose of facilitating programming calculation, this study employs the finite difference method for solving the above equation. When employing the finite difference method for solution, initially, the micropile is discretized into smaller segments, wherein the total length H is divided into n segments of length h, and subsequently assigning numerical labels to the nodes within each individual segment. For convenient programming calculation (to avoid negative and zero subscripts), as shown in Figure 3c, for the loaded section of the pile, the top node of the pile is numbered 3, the node of the pile at the sliding surface is numbered X, and two virtual nodes are set at  
Figure 3: Finite difference method calculation diagram. (**a**) Calculation diagram of elastic pile. (**b**) Ground to Sliding Surface Calculations. (**c**) Slip surface to pile base calculation.