## 1 Introduction
### Self-Balanced Upper and Lower Piles' Calculation Process
#### 2.2.1 Internal Force Calculation of Self-Balanced Upper Pile  
According to the finite difference relationship between the force and deformation of each unit:  
\[T_{i}=\frac{\left(E_{p}A_{p}\right)_{i}}{\Delta L}(w_{i}-w_{i-1}) \tag{10}\]  
\[T_{i+1}=\frac{\left(E_{p}A_{p}\right)_{i+1}}{\Delta L}(w_{i+1}-w_{i}) \tag{11}\]  
where \(E_{p}\) and \(A_{p}\) stand for the pile unit's elastic modulus and cross-sectional area, respectively. By substituting Equations (6), (8), (10) and (11) into Equation (9), the control equation of the node \(i\) can be described as Equation (12):  
\[\frac{\left(E_{p}A_{p}\right)_{i}}{\Delta L}w_{i-1}-\left[\frac{\left(E_{p}A_{ p}\right)_{i}+\left(E_{p}A_{p}\right)_{i+1}}{\Delta L}+K_{z,i}\right]w_{i}+ \frac{\left(E_{p}A_{p}\right)_{i+1}}{\Delta L}w_{i+1}+K_{z,i}\frac{\sum\limits \limits_{j=0}^{n}l_{y}K_{z,i}w_{j}}{E_{p}l_{y}L_{y}L_{y}L_{y}\Delta L/D+\sum \limits_{j=0}^{n}l_{y}K_{z,i}}=-G_{i}-R_{i} \tag{12}\]  
where  
\[h_{i}=\frac{\left(E_{p}A_{p}\right)_{i}}{\Delta L} \tag{13}\]  
Figure 3: Pile–soil interaction and micro-unit force model. (**a**) Pile–soil interaction model. (**b**) Differential element force analysis.  
\[g_{i}=-\left[\frac{\left(E_{p}A_{p}\right)_{i}+\left(E_{p}A_{p}\right)_{i+1}}{ \Delta L}+K_{z,i}\right] \tag{14}\]  
\[k_{i}=\frac{\left(E_{p}A_{p}\right)_{i+1}}{\Delta L} \tag{15}\]  
\[n_{i,j}=K_{z,i}\frac{\sum\limits_{j=0}^{n}I_{bj}K_{z,j}}{E_{p}U_{p}\Delta L/D+ \sum\limits_{j=0}^{n}I_{bj}K_{z,j}}w_{j} \tag{16}\]  
\[m_{i}=-G_{i}-R_{i} \tag{17}\]  
As demonstrated in Equation (18):  
\[h_{i-1}w_{i-1}+g_{i}w_{i}+k_{i+1}w_{i+1}+\sum\limits_{j=0}^{n}\left(n_{i,j}w_{ j}\right)=m_{i} \tag{18}\]  
Assuming that the load box is installed at \(i=k\) (\(k\leq n\)), when \(k=n\), the load box is installed at the bottom of the pile, and for the upper pile boundary condition, there is \(R_{1}=0\) at the top of the upper pile, as demonstrated in Equation (19):  
\[\frac{\left(E_{p}A_{p}\right)_{1}}{\Delta L}(-w_{0}+w_{1})=0 \tag{19}\]  
At the top load box \(i=k\) of the upper pile, \(R_{k}=-F\), as demonstrated in Equation (20):  
\[\frac{\left(E_{p}A_{p}\right)_{k}}{\Delta L}(-w_{k-1}+w_{k})=-F_{n} \tag{20}\]  
where \(F_{n}\) is the load size value of the load box.  
The governing equation of the upper pile is written in matrix form as demonstrated in Equation (21):  
\[[K_{z}+K_{c}]\{w\}=\{m_{z}\}\quad(0\leq i\leq k) \tag{21}\]  
where \([K_{z}]\) is the upper pile body's stiffness matrix, \([K_{c}]\) is the additional stiffness matrix for coupling effects, and \(\{w\}\) is the axial displacement column vector of the lower pile units' denote, which can be represented as Equation (22):  
\[\{w\}=\left\{w_{0}\quad w_{1}\quad\cdots\quad w_{k-1}\quad w_{k}\ \right\}^{T}, \tag{22}\]  
where \(\{m_{z}\}\) is the external load column vector of the upper pile, which can be expressed as Equation (23):  
\[\{m_{z}\}=\{G_{i}\}+\{Q_{i}\}=\left\{\gamma U\Delta L_{1}\quad\gamma U\Delta L _{2}\quad\cdots\quad\gamma U\Delta L_{k-1}\quad-F_{n}+\gamma U\Delta L_{k}\ \ \right\}^{T} \tag{23}\]  
Therefore, the upper pile governing equation can be expressed as Equation (24):  
\[\left\{\begin{bmatrix}g_{0}&k_{0}\\ h_{1}&g_{1}&k_{1}\\ &\cdots&\cdots\end{bmatrix}\right. \tag{24}\]  
\[\left\{\begin{bmatrix}g_{0}&k_{0}\\ h_{1}&g_{1}&k_{1}\\ &\cdots&\cdots\end{bmatrix}\right. \tag{25}\]  
\[\left.\begin{bmatrix}h_{i}&g_{i}&k_{i}\\ &&\cdots&\cdots\end{bmatrix}\right. \tag{26}\]  
\[\left.\begin{bmatrix}h_{k-1}&g_{k-1}&k_{k-1}\\ &&h_{k}&g_{k}&k_{k}\end{bmatrix}+\left[\begin{bmatrix}&h_{i,j}\\ &\cdots\end{bmatrix}\right]\right\}\left\{\begin{bmatrix}w_{0}\\ w_{1}\\ \vdots\\ w_{i}\\ \vdots\\ w_{k-1}\\ w_{k}\end{bmatrix}=\left\{\begin{bmatrix}m_{0}\\ &m_{1}\\ &\vdots\\ &m_{i}\\ &\vdots\\ &\vdots\\ m_{k-1}\\ &m_{k}\end{bmatrix}\right\} \tag{27}\]