## 4 Parametric Study
### Effect of Relative Density of Liquefiable Soil  
Figure 13 shows the peak envelopes of the bending moment and the pile lateral displacement along the depth both for D\({}_{\text{r}}\) = 30% and D\({}_{\text{r}}\) = 50% when input acceleration of 0.13 g is applied. As the relative density was increased from 30% to 50%, the peak value of the bending moment and the pile lateral displacement was decreased by approximately 45% and 40%, respectively. It would be due to the increases of liquefiable soil stiffness and relatively smaller values of kinematic force induced by soil movement. Usually, liquefaction hardly occurs in deeper depth because of the high confining pressure. The fully liquefied depth was approximately 3 m at D\({}_{\text{r}}\) = 50% and approximately 5 m at D\({}_{\text{r}}\) = 30%. Therefore, as the liquefiable layer became denser, the liquefied depth became shallower and the bending moment and the pile lateral displacement significantly decreased. These results show that soil compaction would be very effective for stabilization of a pile in liquefiable soil. On the other hand, this kind of behavior is wholly different from the dry sand case. Figure 14 shows  
Figure 12: Dynamic pile response envelopes according to the thickness of liquefiable layer: (**a**) Peak bending moment along the depth, (**b**) Maximum lateral displacement along the depth.  
peak envelopes of the bending moment and pile lateral displacement for different relative densities in dry sand ([PERSON] and [PERSON], 2019 [23]). Strict quantitative comparison between two analysis cases would be improper because analysis condition of dry sand case was slightly different from liquefiable case, but qualitative comparison would be worthwhile to investigate critical distinction in trend of dynamic pile behavior according to the soil condition. It can be seen that dynamic pile responses show almost identical values for different relative densities in dry sand. It is significantly different pattern with liquefiable condition. In Figure 15, this phenomenon can be clearly identified by comparing peak values of dynamic bending moment obtained in different soil conditions. In the figure, black line graph with grey dots shows percentage of gap between peak value of dynamic bending moment for D\({}_{\rm r}\) = 30% case and D\({}_{\rm r}\) = 50% case for different soil conditions. The gap in liquefied soil condition shows much larger value than the gap in dry soil condition. It can be demonstrated that kinematic force is much more dominant in liquefiable sand than in dry sand from the results. Moreover, peak value of dynamic bending moment in dry soil condition for D\({}_{\rm r}\) = 50% was larger than that for D\({}_{\rm r}\) = 30%, which is different pattern with liquefied soil condition. Ground displacement tends to be out of phase with the inertial force, restraining the pile stress from increasing in dry soil condition, so it can be identified that D\({}_{\rm r}\) = 50% case exerted larger reaction forces on the pile foundation than D\({}_{\rm r}\) = 30% case in the figure. Based on a series of analyses, important dynamic characteristics as kinematic and inertial interaction for different soil condition were clarified by full-scale numerical simulation.  
Figure 13: Dynamic pile response envelopes according to the relative density in liquefiable condition: (**a**) Peak bending moment along the depth, (**b**) Maximum lateral displacement along the depth.