## 4 Numerical Results and Discussions
### Discussion  
In this section, the effects of disturbance degree (\(\eta\)), disturbance range (\(\Delta r\) = \(r_{N}-r_{0}\)), and the length of the pile (\(H\)) on the dynamic responses of piles are investigated. Figure 4 shows the influence of the disturbance degree \(\eta\) on the kinematic response and amplification factors of piles. The kinematic response factor was generally smaller than 1. That means the pile response was generally smaller than the soil response. Moreover, it can be seen that the maximum kinematic response factor was several times larger than minimum kinematic response factor. That means the kinematic factor is mainly dependent on incident frequency. From Figure 4a, the kinematic response factor was generally smaller for the strengthened case (\(\eta>1\)) than for the homogeneous one. On the contrary, the kinematic response factor was generally larger for the weakened case (\(\eta<1\)) than for the homogeneous one. Obvious resonance can be seen in Figure 4b. There was a significant amplification effect around the resonance frequencies and the resonance frequencies were almost unchanged for different disturbance degrees. Moreover, the strengthened case led to a larger amplification factor than the homogeneous one at the first resonance frequency. On the contrary, the weakened case led to a smaller amplification factor than the homogeneous one at the first resonance frequency. In the high frequency domain, the amplification factor tended to be stable and the pile embedded in the weakened soil led to a larger amplification factor. This was due to the seismic response, which was amplificated for softer soil. The results indicate the resistant earthquake properties of piles increase with the increase in the stiffness of the adjacent soil. Figure 4: The effects of the disturbance degree (\(\eta\)) on the kinematic response and amplification factors of piles (\(\rho_{p}\) = 2500 kg/m\({}^{3}\), \(E_{p}\) = 36 GPa, \(r_{0}\) = 0.5, \(v_{j}\) = 0.4, \(\beta_{j}\) = 0.2, \(G_{N+1}\) = 36 MPa, \(H\) = 20 m, \(\triangle r\) = 0.2 m). (**a**) Kinematic response factor; (**b**) Amplification factor. Figure 3: Comparisons of the kinematic response factor between the present solution and that of [PERSON] and [PERSON] (2008), [PERSON] et al. (2010) and [PERSON] et al. (2019). Figure 5 shows the influence of the disturbance range \(\triangle r\) on the kinematic response factor of piles. Figure 5a represents the case of \(\eta=0.5\), and Figure 5b represents the case of \(\eta=1.5\). Similarly, the kinematic response factor was generally smaller than 1. It is noted that there was nearly no influence of the disturbance range on the dynamic response of piles. This could be due to only a small region of soil around pile being affected during pile-sinking. Compared with the disturbance range, the disturbance degree plays a dominant role in kinematic response of piles. Comparing Figure 5a with 5b, the kinematic response factor was generally larger for the weakened soil case than in the strengthened case. Figures 6 and 7 show the influence of pile length \(H\) on the kinematic response and amplification factors of piles. Figure 6 represents the case of \(\eta=0.5\), and Figure 7 represents the case of \(\eta=1.5\). As shown, the kinematic factors fluctuated as the frequency increase. It can be concluded from Figure 6a that the kinematic response factor was generally smaller for larger pile length, overall. On the contrary, the amplification factor at the first resonance frequency was generally larger for larger pile length, overall. In the high frequency domain, short piles generally suffer a larger amplification factor. It is obvious that the normalized resonance frequencies were almost unchanged for different pile length. Since the normalized frequency was the function of the pile length, the real resonance frequencies were different for different pile length.