# Study on the Time Domain Semi Analytical Method for Horizontal Vibration of Pile in Saturated Clay
## 4 Example Analysis  
### Algorithm Verification  
To achieve comparative verification, the soil model in this paper was degraded to linear elastic saturated soil, and the pile foundation model was degraded to the Euler beam. According to beam theory, when the shear modulus of Timoshenko beams is sufficiently large, the cross-section does not undergo shear deformation, and Timoshenko beams degenerate into Euler beams [28]. Reference [29] used the Euler beam and linear elastic saturated soil models to provide the displacement of the pile top in saturated soil under a triangular impact load. The soil model is degraded into elastic saturated soil by taking the viscosity order \(a=0\), and the Timoshenko beam is degraded into the Euler beam by taking the pile foundation shear modulus \(G_{p}=10E_{p}\). The other calculation parameters are the same as those in reference [29] (see Table 1). Meanwhile, the load \(P(t)\) is defined as a triangular impact load, and its time-domain and Laplace transform domain expressions are:  
\[\left. \begin{aligned} P(t)&=2P_{0}\frac{t}{t_{0}}H( \frac{t_{0}}{2}-t)+2P_{0}\Big{(}1-\frac{t}{t_{0}}\Big{)}\Big{[}H(t-\frac{t_{0} }{2})-H(t-t_{0})\Big{]}\\ \hat{P}(s)&=\frac{2P_{0}}{t_{0}s^{2}}\Big{(}1-e^{-t_ {0}s/2}\Big{)}^{2}\end{aligned} \right\} \tag{39}\]  
where \(t_{0}\) is the duration of the load action, and \(H(0)\) represents the Heaviside step function. Substitute Equation (39) into the system equation and calculate the time-domain solution of the pile top displacement through MATLAB(2018) programming. Figure 3 shows a comparison of the calculation results of the pile top displacement, and the results show that the degenerate solution in this paper is in good agreement with the literature solution (dimensionless displacement \(u_{p}^{*}=u_{p}/r_{0}\) and dimensionless time \(t^{*}=t/t_{0}\) in the figure). From Figure 3, it can also be seen that when considering the viscous rheological behavior of the soil (\(a=0.5\)), the peak displacement of the pile top significantly decreases, and the deformation recovery rate after loading slows down, which is consistent with the conclusion obtained by [PERSON] [30] through clay consolidation experiments. To further verify the reliability of the proposed method, a comparison was made between the proposed method and the finite element method for the horizontal vibration model of pile foundations in fractional-order saturated clay. Based on COMSOL(6.0)multiphysics field analysis software, a finite element model of soil was constructed under the solid mechanics porous media seepage module (as shown in Figure 4). The stress-strain relationship of the soil skeleton was described using the fractional order Kelvin Voigt constitutive model, and an infinite element domain was used at the radial boundary to make the soil model consistent with the theoretical model in this paper. Under the structural mechanics module, Timoshenko beams are used to simulate pile foundations, and the pile-soil interface is a coupled contact. Figure 5 shows the comparison between the solution proposed in this paper and the finite element solution. The figure shows that the results of the two can be well matched when the load frequency is small, and when the load frequency is high (\(\omega=4\pi\)), the finite element calculation results show instability. Due to the use of the time integration \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{H}\)/\(\mathbf{m}\) & \(\mathbf{r_{0}}\)/\(\mathbf{m}\) & \(\mathbf{E_{p}}\)/GPa & \(\mathbf{\rho_{p}}\)/(\(\text{kg}\cdot\text{m}^{-3}\)) & \(\mathbf{E_{s}}\)/MPa & \(\mathbf{v}\) \\ \hline
20 & 0.5 & 28.5 & 2400 & 5.5 & 0.25 \\ \(n\) & \(k_{f}\)/(m\(\cdot\text{s}^{-1}\)) & \(\rho_{s}\)/(kg\(\cdot\text{m}^{-3}\)) & \(\rho_{f}\)/(kg\(\cdot\text{m}^{-3}\)) & \(\alpha\) & \(M\)/GPa \\
0.4 & \(2.0\times 10^{-7}\) & 2700 & 1000 & 1.0 & 4.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of calculation parameters [29]. Figure 4: Model of finite element analysis.