# Study of Load Calculation Models for Anti-Sliding Short Piles Using Finite Difference Method
## 2 Structural Calculation Model for Anti-Sliding Short Piles
### Finite Difference Solution for the Anti-Sliding Short Pile Model  
As illustrated in Figure 3, in the application of the finite difference method, it becomes essential to discretize the entire anti-sliding pile into n segments, where each segment possesses a length of \(h\). Nodes are sequentially numbered from the pile head to the pile end as 0, 2, \(\ldots\), \(n\). The node corresponding to the pile body at the sliding surface is designated as \(m\). Furthermore, two virtual nodes are introduced above the pile head and beneath the pile base, marked as \(-2\), \(-1\), \(n+1\), and \(n+2\), resulting in a total of \(n+5\) nodes. Figure 2: Force diagram for anti-sliding short piles. The governing difference equation at any arbitrary node can be formulated as follows:  
\[\frac{EI}{h^{4}}(x_{i-2}-4x_{i-1}+6x_{i}-4x_{i+1}+x_{i+2})+B_{0}K_{j}ihx_{i}=q(z) \tag{5}\]  
For the anti-sliding short pile, the boundary conditions are determined by the bending moment \(M_{0}\) and shear force \(Q_{0}\) applied at the pile head due to the overlying soil pressure. These conditions are specified as follows:  
\[\left\{ \begin{array}{l}x_{-1}-2x_{0}+x_{1}=\frac{M_{0}h^{2}}{EI}\\ x_{-2}-2x_{-1}+2x_{1}-x_{2}=\frac{-2Q_{0}h^{3}}{EI}\end{array} \right. \tag{6}\]  
In the case of an elastic foundation, it is assumed that the pile base of the embedded section of the anti-sliding short pile acts as a free end, with both the bending moment and shear force being zero. Consequently, the boundary conditions at the pile base can be defined as follows:  
\[\left\{ \begin{array}{l}x_{n-1}-2x_{n}+x_{n+1}=0\\ x_{n-2}-2x_{n-1}+2x_{n+1}-x_{n+2}=0\end{array} \right. \tag{7}\]  
When expressing the pile body nodes using a central difference format, Equation (4) can be restated as follows:  
\[\left\{ \begin{array}{l}x_{i-2}-4x_{i-1}+(6+\frac{B_{0}K_{i}h^{4}}{EI})x_{i}- 4x_{i+1}+x_{i+2}=\frac{q(z)h^{4}}{EI}\\ (0\leq i\leq m)\\ x_{i-2}-4x_{i-1}+(6+\frac{B_{0}K_{i}h^{4}}{EI})x_{i}-4x_{i+1}+x_{i+2}=0\\ (m<i\leq n+2)\end{array} \right. \tag{8}\]  
Figure 3: Schematic diagram of finite difference locations for the anti-sliding short pile. By substituting the \(n+1\) nodes along the pile body into Equation (4) and incorporating the four boundary conditions at the pile head and pile base, it becomes feasible to determine the horizontal displacements (\(x_{i}\)) at each node along the pile body. Let:  
\[a_{i}=6+\frac{B_{0}K_{i}\hbar^{A}}{EI} \tag{9}\]  
As a result, the linear equation system for the entire pile matrix of the rear pile can be represented as follows:  
\[[k_{i}][X_{i}]=[Q_{i}] \tag{10}\]  
In the equation, \([k_{i}]\) signifies the coefficient matrix, \([x_{i}]\) denotes the pile body displacement matrix, and \([Q_{i}]\) represents the load matrix. More specifically:  
\[[k_{i}]= \begin{bmatrix}0&1&-2&1&0&\cdots&\cdots&\cdots&0&0&0&0&0\\ 1&-2&0&2&1&\cdots&\cdots&\cdots&0&0&0&0&0\\ 1&-4&a_{0}&-4&1&\cdots&\cdots&\cdots&0&0&0&0&0\\ 0&1&-4&a_{1}&-4&1&\cdots&\cdots&0&0&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&&&&\vdots&\vdots&\vdots&\vdots& \vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&&&&\ddots&\vdots&\vdots&\vdots& \vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&&&&\ddots&\vdots&\vdots&\vdots& \vdots\\ 0&0&0&0&0&\cdots&\cdots&1&-4&a_{n-1}&-4&1&0\\ 0&0&0&0&0&\cdots&\cdots&\cdots&0&1&-4&a_{n}&-4&1\\ 0&0&0&0&0&\cdots&\cdots&\cdots&0&1&-2&1&0\\ 0&0&0&0&0&\cdots&\cdots&\cdots&-1&2&0&-2&1\end{bmatrix}  \tag{11}\]  
\[[X_{i}]= \begin{bmatrix}x_{-2},x_{-1},\cdots x_{n}\vdots x_{m+1},\cdots x_{n+1},x_{n+2}\end{bmatrix} ^{T} \tag{12}\]  
\[[Q_{i}]=\frac{\hbar^{A}}{EI} \begin{bmatrix}\frac{M_{0}}{\hbar^{2}},&-\frac{2Q _{0}}{\hbar},&q_{0},&\cdots,&q_{m},&\cdots&0,&0,&0,&0\end{bmatrix} ^{T} \tag{13}\]  
By incorporating Equations (11)-(13) into Equation (10), a consolidated system of differential equations is established for both the loaded section and the embedded section of the anti-sliding short pile.