# Study on the Optimization of Pile Length of Micropiles in Soil Landslides
## 2 Methods
### Optimization of Micropiles Length Based on Numerical Simulation  
#### 2.4.1 Mathematical Model for Pile Length Optimization  
Based on the previous discussion on the design calculation method of micropiles, it is evident that precise design calculations for the cross-section of micropiles can be achieved by employing thrust sharing theory and elastic foundation beam theory. However, when determining the length of micropiles, only anchorage requirements have been taken into account. Experimental results indicate that the anchoring characteristics of micropiles gradually manifest as they approach failure. Therefore, the length of micropiles determined solely based on anchorage requirements obviously has great potential for optimization. Thus, it is necessary to conduct optimization design research on the length of micropiles to improve the design theory and method of micropiles. The pile length optimization design proposed in this study is a type of structural optimization design. Generally speaking, structural optimization is the process of solving the variables to be optimized under given design parameters, satisfying the constraints, and obtaining the optimal solution of the objective function. Its mathematical model includes the following aspects:  
1. Design Variables  
In the optimization process, it is often necessary to first specify the design values of some parameters and then adjust the values of other design parameters to achieve the optimization goal. These adjustable design parameters are called design variables. For the micropile length optimization carried out in this study, the design variable is set as the length of the micropiles, denoted as PL. Then we have:  
\[PL=[PL_{1},PL_{2},\cdots PL_{n}] \tag{12}\]  
Where denotes the length of the first row of micropiles along the direction of landslide movement, denotes the length of the second row of micropiles, and so on up to the nth row of micropiles. 2. Objective Function  
The objective function, sometimes called the performance function, is an index used to measure the goodness of the design solution. In this study, the optimization objective \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multicolumn{3}{c}{**Test Results**} & \multicolumn{3}{c}{**Numerical Simulation Results**} \\ \cline{2-7}  & **Measured** & **Multiply the** & **Measured** & **Measured** & **Multiply the** & **Measured** \\  & **Static Load** & **Amplification Factor** & **Dynamic Load** & **Static Load** & **Amplification Factor** & **Dynamic Load** \\ \hline Rear Pile & 55.55 & 124.99 & 109.48 & 54.41 & 122.42 & 115.88 \\ Middle row pile & 47.05 & 105.86 & 100.44 & 53.6 & 120.60 & 87.00 \\ Front row piles & 32.68 & 73.53 & 41.02 & 35.53 & 79.94 & 67.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison table of dynamic load calculation of each row of micropiles (unit: kPa). function is \(PL_{max}-PL_{min}\leq\varepsilon\) to screen out the optimal length of pile length, which represents the cost of construction. Optimization means selecting the best pile length that meets the conditions by algorithm under the premise of satisfying the safety factor so as to reduce the cost of the project. The mathematical relationship between the objective function and the design variables is calculated using a reasonable geotechnical constitutive model and the finite difference method, which will be described in detail later in the study. Constraints  
Constraints are necessary conditions for obtaining a convergent solution in optimization design. The pile length optimization carried out in this study needs to satisfy the following two constraints:  
1. The constitutive equation, boundary conditions, and initial conditions are required for the finite difference solution of the micropile treatment landslide model. 2. The range constraint of the design variable, which is the length of the micropile, needs to be satisfied. The value of the design variable should meet the condition: \(PL_{\text{min}}\leq PL\leq PL_{\text{max}}\).