## Appendix B Transfer Matrix Coefficient for Piles in the Passive Segment  
Rewriting Equation (20) into differential equation form reads  
\[\left\{ \begin{array}{l}\frac{dy_{pi}}{dz_{pi}}=\varphi_{pi}\\ \frac{dy_{pi}}{dz_{pi}}=\frac{M_{pi}}{E_{P}}+\frac{b_{1}}{\kappa A_{c}c_{c}} \Big{(}\overline{q}_{pi}-\overline{k}_{pi}y_{pi}\Big{)}\\ \frac{dM_{pi}}{dz_{pi}}=\Big{(}1+\frac{2\overline{R}_{pi}}{\kappa A_{c}c_{c}} \Big{)}Q_{pi}-\Big{(}\overline{N}_{pi}+2\overline{R}_{pi}\Big{)}\varphi_{pi}+ \overline{N}_{pi}\varphi_{IN}\\ \frac{dQ_{pi}}{dz_{pi}}=b_{1}\Big{(}\overline{q}_{pi}-\overline{k}_{pi}y_{pi} \Big{)}\end{array} \right. \tag{17}\]  
where \(y_{pi}\), \(\varphi_{pi}\), \(M_{pi}\), \(Q_{pi}\), \(N_{pi}\), \(z_{pi}\) are the horizontal deformation, axis rotation angle, bending moment, shear force and depth at the bottom of the \(i\)th discrete passive segment for the pile shaft. Equation (17) can be simplified as the following matrix function:  
\[\frac{dS_{pi}}{dz_{pi}}=A_{pi}S_{pi}+f_{pi} \tag{18}\]  
In which \(S_{pi}\), \(A_{pi}\) and \(f_{pi}\) are matrix variables, as follows:  
\[S_{pi}= \begin{bmatrix}y_{pi}&\varphi_{pi}&M_{pi}&Q_{pi}\end{bmatrix} ^{T} \tag{19}\]  
\[A_{pi}= \begin{bmatrix}0&1&0&0\\ -\frac{b_{1}\overline{k}_{pi}}{\kappa A_{c}c_{c}}&0&1/EI_{p}&0\\ 0&-\Big{(}\overline{N}_{pi}+2\overline{R}_{pi}\Big{)}&0&1+\frac{2\overline{R}_ {pi}}{\kappa A_{c}c_{c}}\\ -b_{1}\overline{k}_{pi}&0&0&0\end{bmatrix}  \tag{20}\]  
\[f_{pi}= \begin{bmatrix}0&\frac{b_{1}\overline{q}_{pi}}{\kappa A_{c}c_{c}}& \overline{N}_{pi}\varphi_{IN}&b_{1}\overline{q}_{pi}\end{bmatrix} ^{T} \tag{21}\]  
By assuming \(L\big{[}S_{pi}\big{(}z_{pi}\big{)}\big{]}=F_{pi}(s_{pi})\) and \(L\big{[}f_{pi}\big{(}z_{pi}\big{)}\big{]}=g_{pi}\big{(}s_{pi}\big{)}\) (where the symbol \(L\) represents the Laplace transformation; \(F_{pi}\) and \(g_{pi}\) are the Laplace functions; and \(s_{fi}\) is the corresponding variable of Laplace function), Equation (A23) can be converted to a Laplace function as follows:  
\[L\Bigg{(}\frac{dS_{pi}}{dz_{pi}}\Bigg{)}=L\big{(}A_{pi}S_{pi}+f_{pi}\big{)} \tag{22}\]  
\[F_{pi}\big{(}z_{pi}\big{)}=\big{(}I\times s_{pi}-A_{pi}\big{)}^{-1}S_{p(i-1)}+ \big{(}I\times s_{pi}-A_{pi}\big{)}^{-1}g_{pi}\big{(}s_{pi}\big{)} \tag{23}\]  
In which \(g_{pi}\big{(}s_{pi}\big{)}\) is expressed as  
\[g_{pi}\big{(}s_{pi}\big{)}= \begin{bmatrix}0&\frac{b_{1}\overline{q}_{pi}}{ \kappa A_{c}c_{c}\varphi_{pi}}&\overline{N}_{pi}\varphi_{IN}&\frac{b_{1} \overline{q}_{pi}}{s_{pi}}\end{bmatrix} ^{T} \tag{24}\]  
The Laplace inverse transformation of Equation (A26) is expressed as  
\[S_{pi}\big{(}z_{pi}\big{)}=L^{-1}\Big{[}\big{(}I\times s_{pi}-A_{pi}\big{)}^{ -1}\Big{]}S_{p(i-1)}+L^{-1}\Big{[}\big{(}I\times s_{pi}-A_{pi}\big{)}^{-1}g_{ pi}\big{(}s_{pi}\big{)}\Big{]} \tag{25}\]  
Let \(T_{pi}=\overline{N}_{pi}-2\overline{R}_{pi}\), \(D_{pi}=T_{pi}+\Big{(}b_{1}\overline{k}_{pi}EI_{p}/\kappa A_{c}G_{c}\Big{)}\), \(J_{pi}=1+\Big{(}2\overline{k}_{pi}/\kappa A_{c}G_{c}\Big{)}\), \(\lambda_{pi}=D_{pi}/4E_{I}\), \(\eta_{pi}=b_{1}\overline{k}_{pi}/4 EI_{p}\),  
\[\Big{(}I\times s_{fi}-A_{fi}\Big{)}^{-1}\]  
can be expressed as\[\big{(}I\times s_{pi}-A_{pi}\big{)}^{-1}=\left[