## 3 Formulation and Solution of Governing Equations
### The Solution to the Equations for the Surrounding Soil and Core Soil of the Pile
\begin{array}{l}\overline{u}=K_{1}\bigg{[}\cos(\alpha_{1}z)+\sum\limits_{n=1}^{ \infty}\gamma_{11}ch(g_{n}z)+\sum\limits_{n=1}^{\infty}\gamma_{21}ch(g_{n}z) \bigg{]}\\ \\ +K_{2}\bigg{[}\sin(\alpha_{1}z)+\sum\limits_{n=1}^{\infty}\gamma_{12}ch(g_{n}z) +\sum\limits_{n=1}^{\infty}\gamma_{22}ch(g_{n}z)\bigg{]}\\ \\ +K_{3}\bigg{[}ch(\alpha_{2}z)+\sum\limits_{n=1}^{\infty}\gamma_{13}ch(g_{n}z) +\sum\limits_{n=1}^{\infty}\gamma_{23}ch(g_{n}z)\bigg{]}\\ \\ +K_{4}\bigg{[}sh(\alpha_{2}z)+\sum\limits_{n=1}^{\infty}\gamma_{14}ch(g_{n}z) +\sum\limits_{n=1}^{\infty}\gamma_{24}ch(g_{n}z)\bigg{]}\\ \\ \gamma_{11}=\theta_{1}M_{n1},\gamma_{12}=\theta_{1}M_{n2},\gamma_{13}=\theta_ {1}M_{n3},\gamma_{14}=\theta_{1}M_{n4},\gamma_{21}=\theta_{2}M_{n1},\gamma_{22 }=\theta_{2}M_{n2},\gamma_{23}=\theta_{2}M_{n3},\\ \\ \gamma_{24}=\theta_{2}M_{n4},\theta_{1}=-2A_{n}Y_{n}\Theta/L\Big{[}\Theta A_{ n}Y_{n}+\Theta C_{n}X_{n}-X_{n}Y_{n}\Big{(}\theta_{n}^{4}+B\theta_{n}^{2}+C \Big{)}\Big{]},\\ \\ \theta_{2}=-2C_{n}X_{n}\Theta/L/\Big{[}\Theta A_{n}Y_{n}+\Theta C_{n}X_{n}-X_{n }Y_{n}\Big{(}\theta_{n}^{4}+B\theta_{n}^{2}+C\Big{)}\Big{]}\\ \\ \end{array} \]  
Similarly, based on Equation (26), the rotation angle of the pile is given by:  
\[ \begin{array}{l}\overline{\theta}=K_{1}\bigg{[}\chi_{1}\sin(\alpha_{1}z)+ \sum\limits_{n=1}^{\infty}\gamma^{\prime}{}_{11}sh(g_{n}z)+\sum\limits_{n=1}^ {\infty}\gamma^{\prime}{}_{21}sh(g_{n}z)\bigg{]}\\ \\ +K_{2}\bigg{[}\chi_{2}\cos(\alpha_{1}z)+\sum\limits_{n=1}^{\infty}\gamma^{ \prime}{}_{12}sh(g_{n}z)+\sum\limits_{n=1}^{\infty}\gamma^{\prime}{}_{22}sh(g _{n}z)\bigg{]}\\ \\ +K_{3}\bigg{[}\chi_{3}sh(\alpha_{2}z)+\sum\limits_{n=1}^{\infty}\gamma^{ \prime}{}_{13}sh(g_{n}z)+\sum\limits_{n=1}^{\infty}\gamma^{\prime}{}_{23}sh(g _{n}z)\bigg{]}\\ \\ +K_{4}\bigg{[}\chi_{4}ch(\alpha_{2}z)+\sum\limits_{n=1}^{\infty}\gamma^{ \prime}{}_{14}sh(g_{n}z)+\sum\limits_{n=1}^{\infty}\gamma^{\prime}{}_{24}sh(g _{n}z)\bigg{]}\\ \\ \gamma^{\prime}{}_{1i}=\Lambda\gamma_{1i},\gamma^{\prime}{}_{2i}=\Lambda \gamma_{2i},\Lambda=-\Big{(}g_{n}^{4}+B\theta_{n}^{2}+C+\rho_{p}As^{2}\Theta \Big{)}/\big{(}k^{\prime}G_{p}Ag_{n}\Theta\big{)}+g_{n},\ i=1,2,3,4.\\ \end{array}  \tag{33}\]  
By substituting Equations (32) and (33) into Equation (19a), the resulting expression is obtained:\(\chi_{1}=-\chi_{2}=-\big{[}\rho_{p}s^{2}/\big{(}k^{\prime}G_{p}\alpha_{1}\big{)}+ \alpha_{1}\big{]}\chi_{3}=\chi_{4}=-\rho_{p}s^{2}/\big{(}k^{\prime}G_{p}\alpha _{2}\big{)}+\alpha_{2}\). Based on Equations \(M(z)=E_{p}I_{p}\partial\theta/\partial z\) and \(Q(z)=k^{\prime}G_{p}A(\theta-\partial u/\partial z)\), the bending moment and shear force of the pile can be expressed as:  
\[ \begin{array}{l}\overline{M}=E_{p}I_{p}K_{1}\bigg{[}\chi_{1}\alpha_{1}\cos( \alpha_{1}z)+\sum\limits_{n=1}^{\infty}\gamma^{\prime}_{11}g_{n}ch(g_{n}z)+ \sum\limits_{n=1}^{\infty}\gamma^{\prime}_{21}g_{n}ch(g_{n}z)\bigg{]}\\ \\ +E_{p}I_{p}K_{2}\bigg{[}-\chi_{2}\alpha_{1}\sin(\alpha_{1}z)+\sum\limits_{n=1}^ {\infty}\gamma^{\prime}_{12}g_{n}ch(g_{n}z)+\sum\limits_{n=1}^{\infty}\gamma^{ \prime}_{22}g_{n}ch(g_{n}z)\bigg{]}\\ \\ +E_{p}I_{p}K_{3}\bigg{[}\chi_{3}\alpha_{2}ch(\alpha_{2}z)+\sum\limits_{n=1}^ {\infty}\gamma^{\prime}_{13}g_{n}ch(g_{n}z)+\sum\limits_{n=1}^{\infty}\gamma^{ \prime}_{23}g_{n}ch(g_{n}z)\bigg{]}\\ \\ +E_{p}I_{p}K_{4}\bigg{[}\chi_{4}\alpha_{2}sh(\alpha_{2}z)+\sum\limits_{n=1}^ {\infty}\gamma^{\prime}_{14}g_{n}ch(g_{n}z)+\sum\limits_{n=1}^{\infty}\gamma^{ \prime}_{24}g_{n}ch(g_{n}z)\bigg{]}\\ \\ \end{array}  \tag{34}\]  
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