## 2 Theoretical Analysis
### Governing Differential Equations and Transfer-Matrix Coefficient for Pile in the Passive Segment  
Similarly, for the pile in the passive segment of the pile shaft, an independent local coordinate system on the \(i\)th discrete segment with subscript \(p\) (Figure 5a) is established and, discretizing it into \(n_{p}\) parts with length \(h_{p}=\left(L_{p}/n_{p}\right)\), the average axial force \(\overline{N}_{pi}\) of each discrete segment can be evaluated by  
\[\overline{N}_{pi}=N_{t}+f_{0}L_{0}+f_{1}(2i-1)h_{p}/2 \tag{16}\]Presuming that the average passive load \(\overline{q}_{pi}\) and the average vertical friction stiffness coefficients \(\overline{R}_{pi}\) on the \(i\)th discrete segment are constant and equal to the mean value of those on the corresponding upper and lower cross-section, namely  
\[\overline{q}_{pi}=\big{[}q_{p}\big{(}ih_{p}-h_{p}\big{)}+q_{p}\big{(}ih_{p} \big{)}\big{]}/2 \tag{17}\]  
\[\overline{R}_{vpi}=\big{[}R_{vp}\big{(}ih_{p}-h_{p}\big{)}+R_{vp}\big{(}ih_{p} \big{)}\big{]}/2 \tag{18}\]  
Moreover, by taking into account that the average coefficient of horizontal foundation resistance \(\overline{K}_{pi}\) surrounding the \(i\)th pile segment is constant, the determination of corresponding value is performed on the basis of the mean value theorem of integrals:  
\[\overline{k}_{pi}=m\int_{(i-1)h_{p}}^{ih_{p}}\!\!\big{(}z_{0}+z_{p}\big{)}^{n }dz_{p}/h_{p}=\frac{m\big{[}\big{(}z_{0}+ih_{p}\big{)}^{n+1}-\big{(}z_{0}+(i-1 )h_{p}\big{)}^{n+1}\big{]}}{(n+1)h_{p}} \tag{19}\]  
For the \(i\)th pile segment in the passive segment shown in Figure 5b, the differential equation is expressed as  
\[ \begin{array}{l}\frac{d^{4}y_{pi}}{dz_{pi}^{4}}+\frac{1}{El_{p}}\frac{d}{dz_ {pi}}\big{[}\overline{N}_{pi}\frac{dy_{pi}}{dz_{pi}}\big{]}-\frac{\overline{q }_{IN}}{El_{p}}\frac{d}{dz_{pi}}\big{[}\overline{N}_{pi}\frac{dy_{pi}}{dz_{pi}} \big{]}\\ +2\overline{R}_{vpi}\frac{d^{4}y_{pi}}{dz_{pi}^{2}}-\frac{(\overline{R}_{pi}El _{p}+KA_{c}G_{c})}{KA_{c}G_{c}El_{p}}b_{1}\big{(}\overline{q}_{pi}-\overline{ K}_{pi}y_{pi}\big{)}=0\end{array}  \tag{20}\]  
where the notation of \(p\) indicates the passive segment for the pile shaft. Similarly, for the piles in the passive segment, the transfer-matrix equation for the \(i\)th discrete passive segment of the pile shaft can be formulated as (Appendix B)  
\[S_{pi}=U_{pi}S_{pi(i-1)} \tag{21}\]  
\[\text{where}\,S_{pi}=\big{[}y_{pi}\ \ \ \varphi_{pi}\ \ \ M_{pi}\ \ \ Q_{pi}\ \ \ 1\big{]}^{T}\,\text{and}\,S_{p(i-1)}=\big{[}y_{pi(i-1)}\ \ \ \varphi_{pi(i-1)}\ \ \ M_{p(i-1)}\ \ \ Q_{p(i-1)}\ \ 1\big{]}^{T}\]  
are all 5-by-1 transposed matrices and represent the pile horizontal deformation, axis rota  
Figure 5: Discretization of passive segment and element analysis for a segment. (**a**) Discretization of passive segment. (**b**) Element analysis for a segment. tion angle, bending moment and shear force for the \(i\)th upper and \(i-1\)th lower discrete passive segment for the pile shaft, respectively. In addition, in which \(U_{pi}\) is the transfer matrix coefficient of the \(i\)th pile slice of the pile in the passive segment. By setting \(T_{pi}=\overline{N}_{pi}+2\overline{R}_{pi}\), \(\lambda_{pi}=D_{pi}/4E_{p}\), \(D_{pi}=T_{pi}+\left(b_{1}\overline{k}_{pi}EI_{p}/\kappa A_{c}G_{c}\right)\), \(J_{pi}=1+\left(2\overline{R}_{pi}/\kappa A_{c}G_{c}\right)\), \(\eta_{pi}=b_{1}\overline{k}_{pi}/4 EI_{p}\), \(\beta_{pi}^{2}=\left(b_{1}\overline{k}_{pi}/4 EI_{p}\right)^{1/2}\), \(\kappa_{pi}=\left(\beta_{pi}^{2}-\lambda_{pi}\right)^{1/2}\), \(\xi_{pi}=\left(\beta_{pi}^{2}+\lambda_{pi}\right)^{1/2}\), and \(\omega_{x,pi}=\pm\left(\kappa_{pi}\pm\xi_{pi}\right)\) (where \(x\) equals 1, 2, 3, 4, respectively, and the symbol \"\(i\)\" refers to imaginary, \(\kappa_{pi}\) and \(\xi_{pi}\) are real part and imaginary part), the transfer coefficient matrix for the \(i\)th passive segment \(U_{pi}\) can be generated by incorporating with the Laplace transformation  
\[U_{pi}=