## Appendix B Transfer Matrix Coefficient for Piles in the Passive Segment
\begin{array}{cc}\frac{\Big{(}s_{pi }^{2}+4\lambda_{pi}\beta_{pi}^{2}+s_{pi}^{4}\Big{)}s_{pi i}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2} \beta_{pi}^{2}+s_{pi}^{4}}&\frac{s_{pi}^{2}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2 }\beta_{pi}^{2}+s_{pi}^{4}}&\frac{s_{pi}/EI_{p}}{4\eta_{pi i}+4\lambda_{pi}s_{pi }^{2}\beta_{pi}^{2}+s_{pi}^{4}}\\ -b_{1}\bar{\mathcal{F}}_{pi}\Big{(}\frac{\beta_{pi}^{2}}{\kappa_{A}\kappa_{C}}+ \frac{\beta_{pi}^{2}}{\kappa_{C}}\Big{)}^{2}}&\frac{s_{pi}^{3}}{4\eta_{pi i}+4 \lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{s_{pi}^{2}/EI_{p}}{4\eta _{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{I_{pi}s_{pi}/EI_ {p}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}\\ \frac{b_{1}\bar{\mathcal{F}}_{pi}\Big{(}\frac{\gamma_{pi}}{\kappa_{A}\kappa_{C} }-I_{pi}\Big{)}s_{pi i}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi }^{4}}&\frac{-\left(b_{1}\bar{\mathcal{F}}_{pi}|_{p}+T_{pi}s_{pi}^{2}\right)}{4 \eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{s_{pi}^{2} /EI_{p}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac {I_{pi}s_{pi}/EI_{p}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi }^{4}}\\ \frac{b_{1}\bar{\mathcal{F}}_{pi}\Big{(}\frac{\gamma_{pi}}{\kappa_{A}\kappa_{C} }-I_{pi}\Big{)}s_{pi i}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi }^{4}}&\frac{-\left(b_{1}\bar{\mathcal{F}}_{pi}|_{p}+T_{pi}s_{pi}^{2}\right)}{4 \eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{s_{pi}^{2} /EI_{p}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac {I_{pi}\Big{(}s_{pi}^{2}+4\lambda_{pi}s_{pi}^{2}+s_{pi}^{4}\Big{)}}{4\eta_{pi i }+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}\\ -b_{1}\bar{\mathcal{F}}_{pi}\Big{(}\frac{\beta_{pi}^{2}+4\lambda_{pi}s_{pi}^{2 }}{\kappa_{A}\kappa_{C}}\Big{)}&-b_{1}\bar{\mathcal{F}}_{pi}|_{p}\bar{\mathcal{ F}}_{pi}|\\ \frac{-b_{1}\bar{\mathcal{F}}_{pi}\Big{(}\frac{\gamma_{pi}}{\kappa_{A}\kappa_{C} }-\frac{\beta_{pi}^{2}}{\kappa_{A}\kappa_{C}}\Big{)}}{4\eta_{pi i}+4\lambda_{pi} s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{-b_{1}\bar{\mathcal{F}}_{pi}|_{p} /EI_{p}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{ \left(s_{pi}^{2}+4\lambda_{pi}\right)s_{pi i}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^ {2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{\left(s_{pi}^{2}+4\lambda_{pi}\right)s_{pi i}}{4 \eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}&\frac{\left(s_{pi }^{2}+4\lambda_{pi}\right)s_{pi i}}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi }^{2}+s_{pi}^{4}}\\ \end{array}  \tag{100}\]  
Simultaneous Equations (101) with (102)  
\[\big{(}I\times s_{pi}-A_{pi}\big{)}^{-1}g_{pi}(s_{pi})=\left[ \begin{array}{ ccc}\frac{1}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}\bigg{[}b_{1} \bar{\mathcal{F}}_{pi}\Big{(}\frac{s_{pi}}{\kappa_{A}\kappa_{C}}+\frac{I_{pi}}{ EI_{p}s_{pi}}\Big{)}+\frac{\bar{\mathcal{F}}_{pi}\eta_{pi}\alpha_{pi}}{EI_{p}}\\ \frac{1}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}\bigg{\{}b_ {1}\bar{\mathcal{F}}_{pi}\Big{[}\frac{\beta_{pi}^{2}}{\kappa_{A}\kappa_{C}}+ \frac{I_{pi}}{EI_{p}}\Big{)}+\frac{\bar{\mathcal{F}}_{pi}\eta_{pi}\alpha_{pi}}{ EI_{p}}\bigg{]}\\ \frac{1}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}\bigg{\{}b_ {1}\bar{\mathcal{F}}_{pi}\Big{[}\Big{(}\frac{s_{pi}^{2}}{\kappa_{A}\kappa_{C}} -\frac{I_{pi}}{EI_{p}}\Big{)}+\frac{\bar{\mathcal{F}}_{pi}\eta_{pi}\alpha_{pi}}{ EI_{p}}\bigg{]}\\ \frac{1}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}\bigg{\{}b_ {1}\bar{\mathcal{F}}_{pi}\Big{[}\Big{(}\frac{s_{pi}^{2}}{\kappa_{A}\kappa_{C}} -\frac{I_{pi}}{EI_{p}}\Big{)}-\frac{\bar{\mathcal{F}}_{pi}\eta_{pi}\alpha_{pi}}{ EI_{p}}\bigg{]}\\ \frac{1}{4\eta_{pi i}+4\lambda_{pi}s_{pi}^{2}\beta_{pi}^{2}+s_{pi}^{4}}\bigg{\{}b_ {1}\bar{\mathcal{F}}_{pi}\Big{[}\Big{(}\beta_{pi}^{2}+4\lambda_{pi}s_{pi}^{2}\beta _{pi}^{2}\Big{)}-\frac{4\eta_{pi}\beta_{pi}}{\kappa_{A}\kappa_{C}}\Big{]}- \frac{4\eta_{pi}\beta_{pi}\alpha_{pi}}{s_{pi}}\bigg{\}}\end{array}