## 3 Mechanical Model of the Existing Pile under Adjacent Pile Driving
### Response of Existing Pile Based on Pasternak Foundation Model  
The Winkler foundation model fails to account for soil continuity. To this end, based on the Pasternak foundation model, a mathematical model for reflecting the response of the existing pile is proposed in this section. Another mechanical model of the existing pile under new pile penetration based on the Pasternak foundation model is proposed, as shown in Figure 5.  
The existing pile is seen as a rectangular beam with a width of \(D_{0}\) and a stiffness of \(E_{\text{p}}I_{\text{p}}\). If the pile and surrounding soil satisfy deformation-coordinated condition, the differential equation for the equilibrium based on the Pasternak foundation model can be given as:  
\[E_{\text{p}}I_{\text{p}}\frac{d^{4}\omega}{dz^{4}}-GD_{0}\frac{d^{2}\omega}{dz^{ 2}}+k_{\text{s}}D_{0}\omega=p(z)D_{0} \tag{36}\]  
where \(G\) is the shear modulus, which can be expressed as:  
\[G=\frac{E_{\text{s}}t}{6(1+\
u_{\text{s}})} \tag{37}\]  
Considering the shear stress, the lateral earth pressure can be expressed as:  
\[p(z)=k_{\text{s}}u_{r}(z)-Gu^{\prime\prime}r(z) \tag{38}\]  
where \(u_{r}\)\"(z) is the second-order derivative of the squeezing displacement.  
Equation (36) cannot be solved directly for layered soil; it needs to be solved using the difference method. Thus, Equation (36) can be written in difference form:  
\[\begin{array}{c}\frac{1}{h^{4}}w_{i-2}+\left(-\frac{4}{h^{4}}-\frac{GD_{0}}{ E_{\text{p}}I_{\text{p}}I^{2}}\right)w_{i-1}+\left(\frac{6}{h^{4}}+2\frac{GD_{0}}{ E_{\text{p}}I_{\text{p}}I^{2}}+\frac{kD_{0}}{E_{\text{p}}I_{\text{p}}}\right)w_{i}\\ +\left(-\frac{4}{h^{4}}-\frac{GD_{0}}{E_{\text{p}}I_{\text{p}}I^{2}}\right)w_ {i+1}+\frac{1}{h^{4}}w_{i+2}=\frac{D_{0}}{E_{\text{p}}I_{\text{p}}^{2}}p_{i} \end{array} \tag{39}\]  
Similarly, Equation (39) can be rewritten as:  
\[\alpha\omega_{i-2}+\beta\omega_{i-1}+\gamma\omega_{i}+\beta\omega_{i+1}+ \alpha\omega_{i+2}=\frac{p_{i}D_{0}}{E_{\text{p}}I_{\text{p}}} \tag{40}\]  
where:  
\[\begin{bmatrix}\alpha\\ \beta\\ \gamma\end{bmatrix}=\begin{bmatrix}1\\ -4&-1\\ 6&2\end{bmatrix}\left\{\frac{1}{h^{4}}\begin{array}{c}\frac{GD_{0}}{E_{\text {p}}I_{\text{p}}I^{2}}\end{array}\frac{k_{\text{p}}D_{0}}{E_{\text{p}}I_{ \text{p}}I_{\text{p}}}\right\}^{\text{T}} \tag{41}\]  
The lateral displacement of the existing pile can be given as:  
\[\left\{\omega\right\}_{(n+1)\times 1}=\left\{K\right\}_{(n+1)\times(n+1)}^{-1} \cdot\left\{P\right\}_{(n+1)\times 1} \tag{42}\]  
(i). For a friction pile, the stiffness matrix can be written as:  
Figure 5: Pile–soil interaction model for Pasternak foundation.  
\[\{K\}=\begin{bmatrix}4\alpha+2\beta+\gamma&-4\alpha&2\alpha\\ 2\alpha+\beta&\gamma-\alpha&\beta&\alpha\\ \alpha&\beta&\gamma&\beta&\alpha\\ &\ddots&\ddots&\ddots&\ddots&\ddots\\ &&\alpha&\beta&\gamma&\beta&\alpha\\ &&&\alpha&\beta&\gamma-\alpha&2\alpha+\beta\\ &&&2\alpha&-4\alpha&4\alpha+2\beta+\gamma\end{bmatrix} \tag{43}\]  
(ii). For an end-bearing pile, the stiffness matrix can be written as:  
\[\{K\}=\begin{bmatrix}4\alpha+2\beta+\gamma&-4\alpha&2\alpha\\ 2\alpha+\beta&\gamma-\alpha&\beta&\alpha\\ \alpha&\beta&\gamma&\beta&\alpha\\ &\ddots&\ddots&\ddots&\ddots&\ddots\\ &&&\alpha&\beta&\gamma&\beta\\ &&&\alpha&\beta&\gamma&\beta\\ &&&\alpha&\beta&\gamma+\alpha\end{bmatrix} \tag{44}\]