## 2 Theoretical Analysis
### Governing Differential Equations and Transfer-Matrix Coefficient for Pile in the Active Segment
\begin{array}{ccc}\sum\limits_{x=1}^{4}\frac{\omega_{x,a}^{x,a}+4 \lambda_{ai}-\frac{h\Gamma_{ii}}{\omega_{x,a}^{x,a}+2\lambda_{ai}}}{4\omega_{x,a} ^{x,a}+2\lambda_{ai}}&\sum\limits_{x=1}^{4}\frac{\omega_{x,a}\omega_{x,a}^{x,a} \omega_{ai}}{\lambda_{ai}(\omega_{x,a}^{x,a}+2\lambda_{ai})}&\sum\limits_{x=1}^ {4}\frac{\lambda_{ai}\omega_{x,a}\omega_{x,a}^{x,a}\omega_{ai}}{\lambda_{ai}( \omega_{x,a}^{x,a}+2\lambda_{ai})}&\chi_{1,ai}\\ \sum\limits_{x=1}^{4}\frac{-h\Gamma_{ii}\left(\omega_{x,a}^{x,a}+\lambda_{ai} \right)}{4\omega_{x,a}(\omega_{x,a}^{x,a}+2\lambda_{ai})}&\sum\limits_{x=1}^{4 }\frac{\omega_{x,a}^{x,a}\omega_{x,a}\omega_{x,a}^{x,a}\omega_{ai}}{\lambda_{ ai}(\omega_{x,a}^{x,a}+2\lambda_{ai})}&\sum\limits_{x=1}^{4}\frac{\lambda_{ai} \omega_{x,a}\omega_{x,a}^{x,a}\omega_{ai}}{\lambda_{ai}(\omega_{x,a}^{x,a}+2 \lambda_{ai})}&\chi_{2,ai}\\ \sum\limits_{x=1}^{4}\frac{-h\Gamma_{ii}\left(-\frac{\Gamma_{ii}}{\omega_{x,a} ^{x,a}+\lambda_{ai}}\right)}{4\omega_{x,a}(\omega_{x,a}^{x,a}+2\lambda_{ai})}& \sum\limits_{x=1}^{4}\frac{-(\tau_{ai,a}^{x,a}+h\Gamma_{ii})\epsilon^{x,a} \omega_{x,a}\omega_{ai}}{4\omega_{x,a}(\omega_{x,a}^{x,a}+2\lambda_{ai})}&\sum \limits_{x=1}^{4}\frac{\left(\omega_{x,a}^{x,a}+\frac{h\Gamma_{ii}}{\omega_{x,a}^{x,a}}\right)}{4(\omega_{x,a}^{x,a}+2\lambda_{ai})}&\sum\limits_{x=1}^{4} \frac{\left(\omega_{x,a}^{x,a}+\frac{h\Gamma_{ii}}{\omega_{x,a}^{x,a}}\right) \epsilon^{x,a}\omega_{ai}}{4\omega_{x,a}(\omega_{x,a}^{x,a}+2\lambda_{ai})}& \chi_{3,ai}\\ \sum\limits_{x=1}^{4}\frac{-h\Gamma_{ii}\left(\omega_{x,a}^{x,a}+4\lambda_{ai} -\frac{h\Gamma_{ii}}{\omega_{x,a}^{x,a}}\right)}{4\omega_{x,a}(\omega_{x,a}^{x,a}+2\lambda_{ai})}&\sum\limits_{x=1}^{4}\frac{-h\Gamma_{ii}\epsilon^{x,a} \omega_{x,a}\omega_{ai}}{4(\omega_{x,a}^{x,a}+2\lambda_{ai})}&\sum\limits_{x=1 }^{4}\frac{-h\Gamma_{ii}\epsilon^{x,a}\omega_{x,a}\omega_{ai}}{4(\omega_{x,a}^ {x,a}+2\lambda_{ai})}&\chi_{4,ai}\\ 0&0&0&1\end{array} \right] \tag{30}\]  
With  
\[\chi_{1,ai}=\frac{4\lambda_{ai}\overline{N}_{ai}\varrho_{IN}}{ \omega_{x}}\sum\limits_{x=1}^{4}\frac{\omega_{x,a}\omega_{x,a}^{x,a}}{4\omega _{x,a}(\omega_{x,a}^{x,a}+2\lambda_{ai})} \tag{31}\] \[\chi_{2,ai}=\frac{4\lambda_{ai}\overline{N}_{ai}\varrho_{IN}}{ \omega_{x}}\sum\limits_{x=1}^{4}\frac{\omega_{x,a}^{x,a}\omega_{x}}{4(\omega_{ x,a}^{x,a}+2\lambda_{ai})}\] \[\chi_{3,ai}=\overline{N}_{ai}\varrho_{IN}\left(\omega_{x,a}^{2}+ \frac{h\Gamma_{ii}}{\kappa\Lambda\Lambda\epsilon_{i}\epsilon_{i}}\right)\sum \limits_{x=1}^{4}\frac{\omega_{x,a}^{x,a}\omega_{ai}}{4(\omega_{x,a}^{x,a}+2 \lambda_{ai})}\] \[\chi_{4,ai}=\overline{N}_{ai}\varrho_{IN}\left[-1+\left(\omega_{x,a}^{2}+4\lambda_{ai}\right)\sum\limits_{x=1}^{4}\frac{\omega_{x,a}^{x,a} \omega_{ai}}{4(\omega_{x,a}^{x,a}+2\lambda_{ai})}\right]\]  
Then, in view of the response continuity for independent discrete segment, the following relationship holds true:  
\[S_{an_{a}}=U_{an_{a}}U_{a(n_{a}-1)}\cdots U_{a2}U_{a1}S_{a0}=U_{a}S_{a0} \tag{32}\]  
in which the notation \(U_{a}\) is the total transfer coefficient matrix in the active segment; \(S_{a0}=\left[y_{pn_{p}}\quad\varphi_{pn_{p}}\quad M_{pn_{p}}\quad Q_{pn_{p}} \quad 1\right]^{T}\) is the mechanical deformation parameter matrix of the interface between passive and active segments, and \(S_{an_{a}}=\left[y_{an_{a}}\quad\varphi_{an_{a}}\quad M_{an_{a}}\quad Q_{an_{a} }\quad 1\right]^{T}\) is that at the bottom of the pile shaft.