## 6 Solution of the Large-Diameter Pile  
Substituting Equation (3) into Equation (4) and applying a Laplace transform produces  
\[[(V^{\mathcal{P}})^{2}+(v^{\mathcal{P}}rs)^{2}]\frac{2^{\frac{\mu}{D_{1}^{\rm P}} (z,s)}}{\frac{\mu^{2}}{\alpha^{2}}-s^{2}U^{\rm P}(z,s)-\frac{2\pi\
u_{1}}{\rho A ^{\rm P}}(G_{1}^{\rm s}+c_{1}^{\rm s}s)}\sum\limits_{n=1}^{\infty}q_{1n}^{\rm s }\left[\left[-B_{1n}^{\rm s}I_{1}(\delta_{1n}^{\rm s}r_{1})+C_{1n}^{\rm s}K_{1}( \delta_{1n}^{\rm s}r_{1})\right]\cos(h_{1n}^{\rm s}z^{\rm s})\right]=0 \tag{20}\]  
where \(V\mathcal{P}=\sqrt{\mathbb{E}^{\rm P}/\rho^{\rm P}}\), and \(U\mathcal{P}(z,s)\) is the Laplace transform of \(u^{\rm P}(z,t)\).  
Setting \(s=i\omega(i=\sqrt{-1})\), the general solution for Equation (20) is achieved as  
\[U^{\rm P^{\prime}}=D_{1}^{\rm P}\cos(\frac{\omega}{\eta}z)+D_{2}^{\rm P}\sin( \frac{\omega}{\eta}z). \tag{21}\]  
Furthermore, the particular solution for Equation (20) is given by  
\[U\mathcal{P}^{\rm s}=\sum\limits_{n=1}^{\infty}M_{n}^{\rm s}\cos(h_{1n}^{\rm s }z) \tag{22}\]  
where \(D_{1}^{\rm P}\), \(D_{2}^{\rm P}\) and \(M_{n}^{\rm s}\) are undetermined coefficients; \(\eta=\sqrt{\left(V\mathcal{P}\right)^{2}+\left(v^{\rm P}rs\right)^{2}}\).  
Substituting Equation (22) into Equation (20) with rearrangement yields  
\[M_{n}^{\rm s}=\frac{2\pi\
u_{1}q_{1n}^{\rm s}}{\rho^{\rm P}A^{\rm P}}\frac{(G_ {1}^{\rm s}+c_{1}^{\rm s}s)[B_{1n}^{\rm s}I_{1}(q_{1n}^{\rm s}r_{1})-C_{1n}^{ \rm s}K_{1}(q_{1n}^{\rm s}r_{1})]}{\left[\left(V\mathcal{P}\right)^{2}+\left(v ^{\rm P}r_{1}s\right)^{2}\right](h_{1n}^{\rm s})^{2}-\omega^{2}}. \tag{23}\]  
Therefore, the solution for Equation (21) is obtained as  
\[U\mathcal{P}=D_{1}^{\rm P}\cos(\frac{\omega}{\eta}z)+D_{2}^{\rm P}\sin(\frac{ \omega}{\eta}z)+\sum\limits_{n=1}^{\infty}M_{n}^{\rm s}\cos(h_{1n}^{\rm s}z)\,. \tag{24}\]  
Combining Equations (9), (19a), (19b) adn (24) gives  
\[U\mathcal{P}=D_{1}^{\rm P}[\cos(\frac{\omega}{\eta}z)+\sum\limits_{n=1}^{ \infty}\gamma_{n}^{\prime}\cos(h_{1n}^{\rm s}z)\,]+D_{2}^{\rm P}[\sin(\frac{ \omega}{\eta}z)-\sum\limits_{n=1}^{\infty}\gamma_{n}^{\prime\prime}\cos(h_{1n }^{\rm s}z)\,] \tag{25}\]  
where the coefficients of \(\gamma_{n}^{\prime}\) and \(\gamma_{n}^{\prime\prime}\) are provided in Appendix A.  
In terms of Equations (8a) and (8b), the longitudinal impedance at the head of the pile shaft can be expressed by  
\[Z(\theta)=\frac{P(s)}{U\mathcal{P}(z,s)}=\frac{\mathbb{E}^{\rm P}A^{\rm P}}{H} \frac{[1-\left(v^{\rm P}\overline{r_{1}}\theta\eta\right)^{2}/\left(V\mathcal{ P}\right)^{2}]\theta}{\frac{D_{1}^{\rm P}}{D_{2}^{\rm P}}(1+\sum\limits_{n=1}^{ \infty}\gamma_{n}^{\prime})-\sum\limits_{n=1}^{\infty}\gamma_{n}^{\prime \prime}}=\frac{\mathbb{E}^{\rm P}A^{\rm P}}{H}K_{d}^{\prime} \tag{26}\]  
where \(K_{d}^{\prime}=\theta[1-\left(v^{\rm P}\overline{r_{1}}\theta\eta\right)^{2}/ \left(V\mathcal{P}\right)^{2}]/[\frac{D_{1}^{\rm P}}{D_{2}^{\rm P}}(1+\sum \limits_{n=1}^{\infty}\gamma_{n}^{\prime})-\sum\limits_{n=1}^{\infty}\gamma_{n} ^{\prime\prime}]\) is the dimensionless complex stiffness and \(P(s)\) is the Laplace transform of \(p(t)\).  
\(K^{\prime}{}_{d}\) can be rewritten as  
\[K^{\prime}{}_{d}=K_{\rm r}+{\rm i}K_{i} \tag{27}\]  
where \(K_{\rm r}\) and \(K_{i}\) are the true stiffness and equivalent damping; \(R=\frac{k\mathbb{P}}{E\mathcal{P}A^{\rm P}}H\), \(A_{b}=\frac{k\mathbb{P}}{E\mathcal{P}A^{\rm P}}H\), \(\frac{D_{1}^{\rm P}}{D_{2}^{\rm P}}=\frac{\sum\limits_{n=1}^{\infty}\gamma_{n}^ {\prime\prime}\sin(\overline{h}_{1n}^{\rm s})+(R+sA_{b})[\sin(\theta)-\sum\limits_ {