# Analysis on Response of a Single Pile Subjected to Tension Load Considering Excavation Effects
## 4 Difference Numerical Method to Solve the Load Transfer Model  
### Difference Scheme of Load-Transfer Method  
By solving Equation (6), the pile response after excavation can be obtained. However, Equation (6) is a nonlinear second-order differential equation, and it is hard to obtain an analytical solution. Therefore, the load-transfer method combined with numerical difference is adopted in this paper. Firstly, the pile was discretized into \(n\) segments to facilitate the numerical analysis, as shown in Figure 5. The length of each segment is \(h=L/n\). The nodes of each segment are numbered from the top to the base of the pile. Two virtual nodes, numbered -1 and n+1, are added to create boundary conditions. When \(n\) is large enough, linear axial force distribution can be assumed to simplify the analysis. Figure 4: Hyperbolic load-transfer curve for pile shaft during pile loading. In the excavation unloading stage, substituting Equation (1) into Equation (6), the load transfer differential equation of a single pile can be rewritten into:  
\[\frac{d^{2}\omega_{p1,i}}{dz_{i}^{2}}-\lambda_{1,i}(\omega_{p1,i}-\omega_{s0,i}- \omega_{s\intercal 1,i})=0,\ i=0,1,2,\ldots,n \tag{19}\]  
where \(\lambda_{1,i}\) represents the secant stiffness of the pile-soil interface, \(\lambda_{1,i}=\frac{\pi d\tau_{p1,i}}{E_{p}A_{p}\omega_{s0,i1}}\). The second-order central difference formula is adopted to discretize the pile displacement \(\omega_{p1,i}\), which can be expressed as follows:  
\[\frac{d^{2}\omega_{p1,i}}{dz_{i}^{2}}=\frac{1}{h^{2}}\big{(}\omega_{p1,i-1}-2 \omega_{p1,i}+\omega_{p1,i+1}\big{)},\ i=0,1,2,\ldots,n \tag{20}\]  
At the end of the excavation unloading, the pile head is free to move. Below the neutral level, the pile heave is larger than the soil heave. Hence, the pile base is not contacted with the soil, and the pile head is also free to move. The virtual nodes are used to determine the boundary conditions of the pile. The boundary conditions of the uplift piles can be described as:  
\[\omega_{p1,-1}=\omega_{p1,1} \tag{21}\]  
\[\omega_{p1,n-1}=\omega_{p1,n+1} \tag{22}\]  
Equations (21) and (22) represent that the pile head and base are free to move. Substituting Equation (20) into Equation (19), the load transfer differential equation of a single pile in the excavation unloading stage can be written as:  
\[\mathbf{K}_{1}\omega_{p1}=\mathbf{F}_{1} \tag{23}\]  
where \(\omega_{p1}\) is the node displacement vector of the pile in the excavation unloading stage; \(\mathbf{K}_{1}\) and \(\mathbf{F}_{1}\) are the node stiffness matrix and node force vector, respectively. The node stiffness matrix \(\mathbf{K}_{1}\) and node force vector \(\mathbf{F}_{1}\) are given in Appendix A. The pile response in the pile loading stage is also evaluated by the load-transfer method. The discretization pattern shown in Figure 5 is adopted. The bound conditions are that the pile head displacement is known and the pile base is free to move:  
\[\omega_{p2,0}=\omega_{p0} \tag{24}\]  
\[\omega_{p2,n-1}=\omega_{p2,n+1} \tag{25}\]  
Figure 5: Schematic of discretization of a pile for numerical difference analysis. where \(\omega_{p0}\) is the pile head displacement. The load transfer differential equation of a single pile can be expressed as follows:  
Figure 4:0  
Figure 4:1  
The second-order central difference formula is also adopted to discretize the pile displacement \(\omega_{p2}\), which can be expressed as follows:  
Figure 4:2  
Taking the pile displacement \(\omega_{p2,i}\) as the variable, the load transfer differential equation of a single pile can be rewritten as the matrix form:  
Figure 4:3  
where \(\boldsymbol{\omega}_{p2}\) is the node displacement vector of the pile; \(\mathbf{K}_{2}\) and \(\mathbf{F}_{2}\) are the node stiffness matrix and node force vector, respectively. The node stiffness matrix \(\mathbf{K}_{2}\) and node force vector \(\mathbf{F}_{2}\) are also given in Appendix A.