# Study of Load Calculation Models for Anti-Sliding Short Piles Using Finite Difference Method
## 2 Structural Calculation Model for Anti-Sliding Short Piles
### Analysis of the Forces on Anti-Sliding Short Piles  
The primary thrust of the landslide is concentrated in the upper region of the slip surface, causing significant bending deformation in the anti-sliding short piles, particularly near the slip surface. Near the top of the pile, deformation is relatively minimal due to the soil's restraining effect, resulting in nearly parallel movement. The anchoring section below the slip surface is located within the sliding bed and experiences minimal deformation. Considering the bending deformation characteristics of anti-sliding short piles and stress patterns obtained from prior experiments, the distribution of landslide thrust can assume various shapes, such as a rectangle, triangle, or trapezoid. To facilitate calculations, this study simplifies the landslide thrust as a triangular shape. The JANBU method is utilized to calculate the landslide thrust \(q(z)\)[38]. Figure 1 illustrates the forces acting on the blocks. In this figure, \(E_{i}\) and \(E_{i+1}\) denote the inter-block normal forces, \(T_{i}\) and \(T_{i+1}\) represent the inter-block shear forces, \(W_{i}\) is the block's self-weight, and \(Q_{i}\) is the horizontal action force. Additionally, \(N_{i}\) and \(S_{i}\) correspond to the total normal (comprising effective normal force and pore pressure) and shear forces at the base of the block. The remaining symbols are displayed in the diagram. The equilibrium conditions for forces and moments are provided below.  
Transverse,  
\[E_{i+1}=E_{i}+U_{i,i-1}+Q_{i}+N_{i}\sin\alpha_{i}+U_{i}\sin\alpha_{i}-U_{i,i+1}- S_{i}\cos\alpha_{i} \tag{1}\]  
Longitudinal,  
\[T_{i+1}=T_{i}+S_{i}\sin\alpha_{i}+N_{i}\cos\alpha_{i}+U_{i}\cos\alpha_{i}-W_{i} \tag{2}\]  
The final derived formula for calculating the landslide thrust is as follows:  
\[P_{i}=\sqrt{E_{i+1}{}^{2}+T_{i+1}{}^{2}} \tag{3}\]  
In this formula, \(P_{i}\) signifies the thrust of the landslide, while \(E_{i+1}\) and \(T_{i+1}\) represent the resulting forces acting in the transverse and longitudinal directions of the block.  
Figure 1: Force diagram for the blocks.  
In the load-bearing section, the anti-slip short pile experiences bending moments and shear forces at the sliding surface, effectively acting as a fixed support, while the pile head remains unconstrained, serving as a free end. Therefore, the loaded section is treated as a cantilever beam, and calculations for cantilever beams are applied. Below the sliding surface, there is the anchorage section, treated as an elastic foundation beam. Depending on the type of soil strata and following elastic foundation beam theory, various methods such as the'm' method, '\(K\)' method, 'C' method, and others are used for calculations. To derive formulas, this paper utilizes the '\(m\)' method to determine the resistance of the sliding mass and the foundation coefficient, which is calculated as \(K_{\text{j}}(z)=mz\) Here, '\(m\)' represents the calculation coefficient, and '\(z\)' denotes the depth, taking into account soil foundation conditions. Specific stress conditions are illustrated in Figure 2.  
Utilizing the Euler-Bernoulli beam theory, we can express the unified differential governing equation for both the loaded and embedded sections of the anti-sliding short pile as follows:  
\[EI\frac{d^{4}x}{dy^{4}}x+B_{0}K_{\text{j}}x=q(z) \tag{4}\]  
In this equation, \(EI\) represents the stiffness of the anti-sliding short pile structure. \(B_{0}\) denotes the calculated width of the pile. \(K_{j}\) represents the foundation coefficient. \(x\) signifies the horizontal displacement of the pile. \(y\) denotes the distance from the pile head. \(q(z)\) represents the landslide thrust in \(\text{kN}/\text{m}\).