## 4 Research on the Calculation Method of Internal Force of Double-Row Skidding Pile
### Calculation of Internal Forces in Piles Below Shear Zero  
The internal forces and displacements of the pile body below the shear zero point (i.e., BC section) are calculated by the finite difference method, and Figure 9 shows the force diagram of the pile body below the shear zero point. According to the elastic foundation beam theory, the finite difference method is used to calculate the internal forces and displacements below the shear zero point of the double-row pile sliding pile (BC section), and the deflection differential governing equation [28] can be expressed as follows:  
\[EI\frac{\mathrm{d}^{4}y}{dz^{4}}+B_{0}K(z)y=q(z) \tag{9}\]  
where \(EI\)--structural stiffness of the double-row anti-slip pile;  
\(B_{o}\)--calculated width of the pile/m;  
\(K(z)\)--foundation coefficient;  
\(Y\)--horizontal displacement of pile body/m;  
\(Z\)--distance from pile top/m;  
\(q(z)\)--landslide thrust/(kN/m);  
The pile deflection and differential points are shown in Figure 10. According to the finite difference theory, the back-row pile is discretized into n segments on average, and the length of each segment is h. The top of the back-row pile to the bottom of the pile is numbered as \(-2\), \(-1\), \(0\ldots n\), \(n+1\), \(n+2\), and there are a total of \(n+5\) nodes, among which \(-2\), \(-1\), \(n+1\), \(n+2\), are imaginary nodes; the node at the top of the pile is number 0 and the bottom of the pile is n; the node at the sliding surface is number l [29]. The front row of the piles and the back row of the piles take the same equal segmentation; then, the difference equation at any node below the sliding surface of the pile is as follows:  
\[\left\{ \begin{array}{c}y_{i-2}-4y_{i-1}+(6+\frac{B_{0}K(z)\mathrm{d}^{4}}{EI} )y_{i}-4y_{i+1}+y_{i+2}=\frac{q(z)\mathrm{d}^{4}}{EI}\\ (0\leq i\leq l)\\ y_{i-2}-4y_{i-1}+(6+\frac{B_{0}K(z)\mathrm{d}^{4}}{EI})y_{i}-4y_{i+1}+y_{i+2}=0 \\ (l<i\leq n+2)\end{array} \right. \tag{10}\]  
Figure 9: Shear force diagram below zero. According to the structural assumption that the B end is the hinged end, the bending moment and shear force at the top B of the BC section pile in the calculation model is the bending moment \(M_{\text{B}}\) and shear force \(Q_{\text{B}}\) at the bottom B of the AB section pile; then, the boundary condition at the top of the BC section of the slip-resistant pile in the finite difference method is as follows:  
\[\left\{ \begin{array}{l}y_{-1}-2y_{0}+y_{1}=\frac{M_{\text{B}}h^{2}}{EI}\\ y_{-2}-2y_{-1}+2y_{1}-y_{2}=0\end{array} \right. \tag{11}\]  
The bottom of the pile at C is considered as a free end; then, the bending moment and shear force at the bottom of the pile are zero and the boundary condition at the bottom of the pile is as follows:  
\[\left\{ \begin{array}{l}y_{-1}-2y_{0}+y_{1}=\frac{M_{\text{B}}h^{2}}{EI}\\ y_{-2}-2y_{-1}+2y_{1}-y_{2}=0\end{array} \right. \tag{12}\]  
\[[\mathbf{A_{i}}][\mathbf{Y_{i}}]\mathbf{=}[\mathbf{Q_{i}}] \tag{13}\]  
where \([\mathbf{A_{i}}]\) is the coefficient matrix; \([\mathbf{Y_{i}}]\) is the pile displacement matrix; and \([\mathbf{Q_{i}}]\) is the load matrix; where:  
\[[\mathbf{A_{i}}]= \begin{bmatrix}0&1&-2&1&0&\cdots&\cdots&\cdots&0&0&0&0\\ 1&-2&0&2&1&\cdots&\cdots&\cdots&0&0&0&0&0\\ 1&-4&a_{0}&-4&1&\cdots&\cdots&\cdots&0&0&0&0&0\\ 0&1&-4&a_{1}&-4&1&\cdots&\cdots&0&0&0&0&0\\ &&&&\cdots&\cdots&\cdots&&&&\\ &&&&\cdots&\cdots&\cdots&&&&\\ 0&0&0&0&0&\cdots&\cdots&1&-4&a_{n-1}&-4&1&0\\ 0&0&0&0&0&\cdots&\cdots&\cdots&1&-4&a_{n}&-4&1\\ 0&0&0&0&0&\cdots&\cdots&\cdots&&1&-2&1&0\\ 0&0&0&0&0&\cdots&\cdots&\cdots&-1&2&0&-2&1\end{bmatrix}  \tag{14}\]  
\[[\mathbf{Y_{i}}]= \begin{bmatrix}y_{-1}&y_{-2}&y_{0}&\cdots&y_{l}&\cdots&y_{n- 1}&y_{n}&y_{n+1}&y_{n+2}\end{bmatrix} ^{\text{T}} \tag{15}\]  
\[[\mathbf{Q_{i}}]=\frac{H^{4}}{EI}\Big{[}\frac{M_{\text{B}}}{h^{2}}-\frac{2Q_{ \text{B}}}{h}&q_{0}&\cdots&q_{l}&\cdots&0&0&0&0\end{bmatrix}^{\text{T}} \tag{16}\]  
Figure 10: Difference point diagram of pile shear below zero.