## 3 Formulation and Solution of Governing Equations
### The Solution to the Equations for the Surrounding Soil and Core Soil of the Pile
\[q_{n2}=\int_{0}^{2\pi}\left[\left({\sigma_{r2}}^{\prime}-P_{2}\right)\cos \theta-\tau_{r\theta 2}\sin\theta\right]\big{|}_{r=r_{2}}\cdot r_{2}d\theta \tag{20}\]4  
\[q_{n2}=\int_{0}^{2\pi}\left[\left({\sigma_{r2}}^{\prime}-P_{2}\right)\cos \theta-\tau_{r\theta 2}\sin\theta\right]\big{|}_{r=r_{2}}\cdot r_{2}d\theta \tag{20}\]5  
\[q_{n2}=\int_{0}^{2\pi}\left[\left({\sigma_{r2}}^{\prime}-P_{2}\right)\cos \theta-\tau_{r\theta 2}\sin\theta\right]\big{|}_{r=r_{2}}\cdot r_{2}d\theta \tag{20}\]6  
Equation (25) is a fourth-order ordinary differential equation. By substituting Equations (21) and (22), the general solution is obtained as:  
\[ \begin{array}{l}\overline{u}=K_{1}\cos(\alpha_{1}z)+K_{2}\sin(\alpha_{1}z)+ K_{3}ch(\alpha_{2}z)+K_{4}sh(\alpha_{2}z)\\ +\sum\limits_{n=1}^{\infty}\varepsilon_{1n}A_{1n}ch(g_{n}z)+\sum\limits_{n=1} ^{\infty}\varepsilon_{2n}C_{1n}ch(g_{n}z)\end{array}  \tag{27}\]  
\[ \begin{array}{l}\alpha_{1}=\sqrt{\frac{B+\sqrt{B^{2}-4C}}{2}},\alpha_{2}= \sqrt{\frac{-B+\sqrt{B^{2}-4C}}{2}},\varepsilon_{1n}\frac{\Theta A_{n}}{g_{n} ^{4}+B_{\theta}^{2}+C},\varepsilon_{2n}\frac{\Theta C_{n}}{g_{n}^{4}+B_{ \theta}^{2}+C}\\ \Theta=\frac{g_{n}^{2}}{k^{\prime}G_{p}A}-\frac{1}{E_{p}I_{p}}\left(\frac{ \rho_{p}I_{p}s^{2}}{k^{\prime}G_{p}A}+1\right),\end{array} \]  
where \(K_{1}\), \(K_{2}\), \(K_{3}\), and \(K_{4}\) are undetermined constants. Based on the full contact condition between the pile and the soil (Equations (12) and (13)), the following relationship holds:  
\[ \begin{array}{l}K_{1}\cos(\alpha_{1}z)+K_{2}\sin(\alpha_{1}z)+K_{3}ch(\alpha _{2}z)+K_{4}sh(\alpha_{2}z)\\ +\sum\limits_{n=1}^{\infty}\varepsilon_{1n}A_{1n}ch(g_{n}z)+\sum\limits_{n=1} ^{\infty}\varepsilon_{2n}C_{1n}ch(g_{n}z)=\sum\limits_{n=1}^{\infty}X_{n}A_{1 n}ch(g_{n}z)\end{array}  \tag{28}\]  
\[ \begin{array}{l}K_{1}\cos(\alpha_{1}z)+K_{2}\sin(\alpha_{1}z)+K_{3}ch(\alpha _{2}z)+K_{4}sh(\alpha_{2}z)\\ +\sum\limits_{n=1}^{\infty}\varepsilon_{1n}A_{1n}ch(g_{n}z)+\sum\limits_{n=1} ^{\infty}\varepsilon_{2n}C_{1n}ch(g_{n}z)=\sum\limits_{n=1}^{\infty}Y_{n}C_{1 n}ch(g_{n}z)\end{array}  \tag{29}\]  
\[ \begin{array}{l}X_{n}=-\{\beta_{11}[K_{0}(\beta_{11}r_{1})+K_{2}(\beta_{11}r _{1})]+\beta_{12}B_{1}[K_{0}(\beta_{12}r_{1})+K_{2}(\beta_{12}r_{1})]\}/2+B_{2 }K_{1}(\beta_{13}r_{1})/r_{1}\\ Y_{n}=\{\beta_{21}[I_{0}(\beta_{21}r_{2})+I_{2}(\beta_{21}r_{2})]+\beta_{22}D_{1 }[I_{0}(\beta_{22}r_{2})+I_{2}(\beta_{22}r_{2})]\}/2+D_{2}I_{1}(\beta_{23}r_{2} )/r_{2}\end{array} \]  
By multiplying both sides of Equation (28) and (29) by \(ch(g_{m}z)\) and integrating over the interval [0,L], the following result is obtained:  
\[A_{1n}=\frac{-2M_{n}Y_{n}\big{(}g_{n}^{4}+Bg_{n}^{2}+C\big{)}}{L[\Theta A_{n}Y_ {n}+\Theta C_{n}X_{n}-X_{n}Y_{n}(g_{n}^{4}+Bg_{n}^{2}+C)]} \tag{30}\]  
\[C_{1n}=\frac{-2M_{n}X_{n}\big{(}g_{n}^{4}+Bg_{n}^{2}+C\big{)}}{L[\Theta A_{n}Y_ {n}+\Theta C_{n}X_{n}-X_{n}Y_{n}(g_{n}^{4}+Bg_{n}^{2}+C)]} \tag{31}\]  
\[M_{n}=K_{1}M_{n1}+K_{2}M_{n2}+K_{3}M_{n3}+K_{4}M_{n4}\]  
\[M_{n1}=\frac{g_{n}\cos(\alpha_{1}L)sh(g_{n}L)}{g_{n}^{2}+\alpha_{1}^{2}},\ M_{n2}= \frac{g_{n}\sin(\alpha_{1}L)sh(g_{n}L)+\alpha_{1}}{g_{n}^{2}+\alpha_{1}^{2}}\]  
\[M_{n3}=\frac{g_{n}ch(\alpha_{2}L)sh(g_{n}L)}{g_{n}^{2}-\alpha_{2}^{2}},\ M_{n4}= \frac{g_{n}sh(\alpha_{2}L)sh(g_{n}L)+\alpha_{2}}{g_{n}^{2}-\alpha_{2}^{2}}\]Therefore, the solution to Equation (25) can be expressed as:  
\[