## 3 Mechanical Model of the Existing Pile under Adjacent Pile Driving
### Response of Existing Pile Based on Winkler Model  
Based on the Winkler foundation model, The concept model for the response of the existing pile caused by adjacent pile driving is shown in Figure 4. The penetrated depth and diameter of the new pile are respectively \(L\) and \(d_{0}\); the depth and diameter of the existing pile are \(H\) and \(D_{0}\), respectively. From Equation (5), the lateral displacement can be obtained. Under the existing pile, the soil displacement would be limited. In this section, the soil is assumed as homogeneous and continuous. The adjacent pile and soil have deformation compatibility. The final lateral displacement is equal to the squeezing displacement, which can be expressed as \(\omega(z)\). Based on the Winkler foundation model, the lateral earth pressure can be expressed as:  
\[p(z)=k_{\mathrm{s}}u_{\mathrm{r}} \tag{6}\]  
where \(p\) is the lateral pressure in the unit's existing pile, and \(k_{\mathrm{s}}\) is the lateral modulus in the existing pile, which can be expressed as [41]:  
\[k_{\mathrm{s}}=0.65\bigg{(}\frac{E_{\mathrm{s}}D_{0}^{4}}{E_{\mathrm{p}}I_{ \mathrm{p}}}\bigg{)}^{\frac{1}{12}}\cdot\frac{E_{\mathrm{s}}}{\left(1-\
u \right)^{2}} \tag{7}\]  
where \(E_{\mathrm{s}}\) is deformation modulus of pile-side soil; \(E_{\mathrm{p}}I_{\mathrm{p}}\) is the flexural rigidity of section in existing pile; and \(\
u\) is the Poisson's ratio. Figure 4: Equivalent simulation of static-pressure pile penetration. Figure 3: Analysis workflow of stress and deformation of used piles. According to the deflection curve equation, the unit length of the existing pile can be seen as an elastic foundation beam. The governing equation of displacement including the existing pile and adjacent pile can be expressed as:  
\[E_{\text{P}}I_{\text{P}}\frac{d^{4}\omega(z)}{dz^{4}}+k_{\text{s}}D_{0}\omega(z)= D_{0}p(z) \tag{8}\]  
For the existing pile: if the pile is a friction pile, the pile tip can be seen as a free end; if the pile is end-bearing, the pile tip can be considered as a fixed end. Because the superstructure has been removed, the top of the existing pile can be seen as a free end, and the bending moment and shear stress are assumed as zero. The direct solving of Equation (8) only applies to homogeneous soils. For layered soils, the difference method should be used. The existing piles are divided into \(n\) equal segments along the length of the existing piles, and the length of each segment is \(h\). Equation (8) can be given as:  
\[\frac{1}{h^{4}}\omega_{i-2}-\frac{4}{h^{4}}\omega_{i-1}+\bigg{(}\frac{6}{h^{4} }+\frac{E_{\text{s}}D_{0}}{E_{\text{P}}I_{\text{P}}}\bigg{)}\omega_{i}-\frac{4 }{h^{4}}\omega_{i+1}+\frac{1}{h^{4}}\omega_{i+2}=\frac{p_{i}D_{0}}{E_{\text{P}} I_{\text{P}}} \tag{9}\]  
Then, Equation (9) can be written as:  
\[\alpha\omega_{i-2}-\beta\omega_{i-1}+\gamma\omega_{i}-\beta\omega_{i+1}+ \alpha\omega_{i+2}=\frac{p_{i}D_{0}}{E_{\text{P}}I_{\text{P}}} \tag{10}\]  
where:  
\[ \begin{bmatrix}\alpha\\ \beta\\ \gamma\end{bmatrix} = \begin{bmatrix}1&0\\ -4&0\\ 6&1\end{bmatrix} \cdot \begin{bmatrix}\frac{1}{h^{4}}\\ \frac{E_{\text{P}}D_{0}}{E_{\text{P}}I_{\text{P}}}\end{bmatrix}  \tag{11}\]  
On the basis of physical relationships, the turning angle, pile bending moment and shear stress can be written as:  
\[\theta_{i}=\bigg{(}\frac{dw}{dz}\bigg{)}_{i}=\frac{1}{2h}(\omega_{i-1}-\omega_ {i+1}) \tag{12}\]  
\[M_{i}=EI\bigg{(}\frac{d^{2}\omega}{dz^{2}}\bigg{)}_{i}=\frac{EI}{h^{2}}(\omega _{i-1}-2\omega_{i}+\omega_{i+1}) \tag{13}\]  
\[Q_{i}=EI\bigg{(}\frac{d^{3}\omega}{dz^{3}}\bigg{)}_{i}=\frac{EI}{2h^{3}}( \omega_{i-2}-2\omega_{i-1}+2\omega_{i+1}-\omega_{i+2}) \tag{14}\]  
The lateral displacement of the existing pile can be obtained as follows:  
\[\{\omega\}_{(n+1)\times 1}=\{K\}_{(n+1)\times(n+1)}^{-1}\cdot\{P\}_{(n+1)\times 1} \tag{15}\]  
Based on the boundary conditions, the stiffness matrix can be derived. Case 1:  
If the existing pile is a friction pile, the pile top and pile tip are free. The boundary can be expressed as:  
\[