# Study on the Time Domain Semi Analytical Method for Horizontal Vibration of Pile in Saturated Clay
## 4 Example Analysis
### Pile Foundation Vibration Analysis  
Based on Table 1, add viscosity coefficient \(\eta=1\times 10^{8}\)\(Pa\cdot s\), and use load function Equation (6) to calculate the horizontal vibration response of pile foundation in saturated clay foundation. Figure 6 shows the time-domain response of pile top displacement under different shear modulus \(G_{p}\) values. It can be seen from the figure that as the shear modulus of the pile foundation increases, the amplitude of pile top displacement decreases. When \(G_{p}\) increases to \(6E_{p}\), the displacement amplitude no longer changes, indicating that the Timoshenko beam has converged to the Euler beam. Figure 7 shows the vibration response of pile top displacement under different load frequencies. The figure shows that the larger the load frequency, the smaller the amplitude of pile top displacement. This result is consistent with basic vibration theory [31; 32]. Figure 5: Comparison of pile top displacement with FEM. Figure 6: Influence of shear modulus of pile on the vibration displacement of pile top (\(\omega=2\pi\), \(a=0\)). Figure 8 reflects the influence of soil viscosity order on the response of pile top displacement. The figure shows that the larger the viscosity order, the smaller the amplitude of pile top displacement, and the peak point of displacement moves backward on the time axis, reflecting the delayed effect of soil creep. According to the definition of the Kelvin rheological model, the higher the viscosity order, the stronger the rheological properties. Therefore, the results in Figure 6 indicate that the stronger the rheological properties of the soil, the smaller the displacement response amplitude of the pile foundation vibration and the more obvious the delay effect. Figure 9 further indicates that as the order of soil viscosity increases, the amplitude of pile top displacement decreases almost linearly, and the higher the load frequency, the greater the decreasing rate of displacement amplitude. To further explore the parameter correlation of the displacement delay effect, this article defines delay time \(t_{a}=\bar{t}_{1}-T/4\), where \(\bar{t}_{1}\) is the time corresponding to the first displacement peak and \(T=2\pi/\omega\) is the load period. Figure 10 shows the variation of delay time ta with the viscosity order. It can be seen from the figure that as the viscosity order increases, the delay time correspondingly increases. It is worth noting that the lower the load frequency, the more significant the change in delay time with the viscosity order. Figure 8: Influence of viscosity order on pile top vibration displacement (\(G_{p}=E_{p}\), \(\omega=2\pi\)). Figure 7: Response of pile top displacement under different frequencies (\(G_{p}=E_{p}\), \(\omega=2\pi\)). Figures 11 and 12 show the distribution of pile bending moment \(M_{p}=\partial^{2}u_{p}/\partial z^{2}\) and shear force \(Q_{p}=\partial^{3}u_{p}/\partial z^{3}\) with depth at \(t=T/4\), respectively. Figure 11 shows that as the viscosity order increases, the positive bending moment of the pile decreases while the negative bending moment increases. Figure 12 shows that as the viscosity order increases, both the positive and negative shear forces will increase. However, it should be considered that due to the delay effect, the bending moment and shear force values at \(t=T/4\) cannot represent the peak values at each viscosity order. Therefore, Figures 13-15 show the time-domain response of bending moment and shear force at the characteristic depth. Figure 13 shows that there is also a delay effect in the bending moment of the pile body, and as the viscosity order increases, the amplitude of the bending moment decreases slightly. Figure 14 shows that the shear force at the top of the pile is almost unaffected by the viscosity order, and there is no significant delay effect or amplitude change phenomenon. Figure 15 shows that at larger depths, there is a significant delay in shear force, and the larger the viscosity order, the greater the shear amplitude.