# Study on the Time Domain Semi Analytical Method for Horizontal Vibration of Pile in Saturated Clay
## 3 Equation Solving
### Solution of Pile Foundation Vibration Equation  
Performing Laplace transformation on Equations (13) and (14), we obtain:  
\[\begin{array}{l}G_{p}S\bigg{(}\frac{\partial\dot{u}_{p}}{\partial z}-\frac{ \partial^{2}\dot{u}_{p}}{\partial z^{2}}\bigg{)}+\rho_{p}Ss^{2}\dot{u}_{p}+ \dot{q}=0\\ G_{p}S\bigg{(}\dot{\phi}_{p}-\frac{\partial\dot{u}_{p}}{\partial z}\bigg{)}+ \rho_{p}I_{p}s^{2}\dot{\phi}_{p}-E_{p}I_{p}\frac{\partial^{2}\dot{\phi}_{p}}{ \partial z^{2}}=0\end{array} \tag{34}\]  
By using Equation (34), it can be obtained that:  
\[\begin{array}{l}\frac{\partial\dot{u}_{p}}{\partial z}=\frac{\partial^{2} \dot{u}_{p}}{\partial z^{2}}-\frac{\rho_{p}s^{2}}{C_{p}}\dot{u}_{p}-\frac{ \dot{d}}{C_{p}S}\\ \frac{\partial\dot{u}_{p}}{\partial z}=\dot{\phi}_{p}+\frac{\rho_{p}I_{p}s^{2}} {C_{p}S}\dot{\phi}_{p}-\frac{E_{p}I_{p}}{C_{p}S}\frac{\partial^{2}\dot{\phi}_{ p}}{\partial z^{2}}\end{array} \tag{35}\]  
Taking the partial derivative of \(z\) on both sides of Equation (34) and then substituting Equation (35) to compile the decoupled pile foundation motion equation:  
\[\begin{array}{l}\frac{\partial^{4}\dot{u}_{p}}{\partial z^{4}}+C_{1}\frac{ \partial^{2}\dot{u}_{p}}{\partial z^{2}}+C_{2}\hat{u}_{p}-\frac{1}{C_{p}S}\frac{ \partial^{2}\dot{q}}{\partial z^{2}}+\bigg{(}\frac{\rho_{p}s^{2}}{C_{p}S_{p}}+ \frac{1}{E_{p}I_{p}}\bigg{)}\hat{q}=0\\ \frac{\partial^{4}\dot{u}_{p}}{\partial z^{4}}+C_{1}\frac{\partial^{2}\dot{\phi}_ {p}}{\partial z^{2}}+C_{2}\hat{\phi}_{p}+\frac{1}{E_{p}I_{p}}\frac{\partial \dot{q}}{\partial z}=0\end{array} \tag{36}\]The general solution of the fourth-order ordinary differential Equation (36) is [25]:  
\[\left.\begin{aligned} \hat{u}_{p}&=D_{1}\cos(\chi_{1}z)+D_{2} \sin(\chi_{1}z)+D_{3}\text{ch}(\chi_{2}z)+D_{4}\text{sh}(\chi_{2}z)+Q_{1}\\ \hat{\varphi}_{p}&=D_{5}\cos(\chi_{1}z)+D_{6}\sin( \chi_{1}z)+D_{7}\text{ch}(\chi_{2}z)+D_{8}\text{sh}(\chi_{2}z)+Q_{2}\end{aligned}\right\} \tag{37}\]  
where sh(0) represents a hyperbolic sine function, and \(D_{1}\)-\(D_{8}\) are undetermined coefficients; \(Q_{1}\) and \(Q_{2}\) are a set of special solutions to Equation (34). In Equations (36) and (37), the expressions for \(\chi_{1}\), \(\chi_{2}\), \(C_{1}\), \(C_{2}\), \(Q_{1}\), and \(Q_{2}\) can be found in Appendix A.  
From the relationship neutralized by Equation (37), it can be obtained that:  
\[\left.\begin{aligned} D_{5}&=\eta_{1}D_{1}, \quad D_{6}=\eta_{2}D_{2}\\ D_{7}&=\eta_{3}D_{3},\quad D_{8}=\eta_{4}D_{4}\end{aligned}\right\} \tag{38}\]  
where \(\eta_{1}\)-\(\eta_{4}\) are known coefficients, and the specific form is shown in Appendix A.  
Using the pile-soil contact coupling condition Equation (19) and the pile end boundary condition Equations (21) and (22), five boundary equations were constructed to determine the undetermined coefficients \(B_{1}\) and \(D_{1}\)-\(D_{4}\). At this point, the displacement field, stress field, and pore pressure of the pile-soil system have been determined. Combining the initial condition Equation (15) for Laplace inversion, a time-domain numerical solution is obtained. To ensure the accuracy of the long-term inversion, the Crump method [26; 27] is used for Laplace inversion in this paper.