## 4 Parametric Study
### Effect of Pile Length on Dynamic Behaviour of Pile  
Another repetitive analyses were carried out for 9 model systems with different pile lengths of 2 m (1.1 T), 4 m (2.2 T), 5 m (2.7 T), 8 m (4.4T ), 9 m (4.9 T), 10 m (5.5 T), 12 m (6.6 T), 18 m (9.8 T), and 23 m (12.6 T) to examine the effect of pile length on the dynamic behavior of the pile. Each pile length was marked as the characteristic length of pile (T) which was defined by the analytical approach performed by [PERSON] and [PERSON] (1960) [35] as Equation (7).  
\[\mathrm{T}\ =\ \sqrt[5]{\frac{EI}{n_{h}}} \tag{7}\]  
where \(n_{h}\) is the soil reaction constant and EI is flexural rigidity of pile.  
Figure 17 depicts the comparison of the maximum lateral pile displacement profiles. When the pile length exceeds 5 T, the maximum lateral displacement profile shows long-pile behavior. In contrast, when the pile length is shortened to below 3 T, the maximum lateral displacement profile shows short-pile behavior which has no negative bending moment. When the pile length is between 3 T and 5 T, the maximum lateral displacement profile shows intermediate behavior. Figure 18 shows the relationship between pile length and rotational depth, where each parameter is normalized by T. In this figure, rotational depth means the distance between the ground surface and the rotation axis of the pile. It is identified that each parameter shows a linear relation with a high correlation coefficient. This means that the pile rotates at a depth of about 70% of the pile length when the pile behaves as a short pile.  
Figure 16: Maximum lateral pile displacement envelopes obtained from three different relative densities. (**a**) Input acceleration: 0.13 g; (**b**) Input acceleration: 0.25 g.  
Figure 19 depicts a comparison of the peak bending-moment profile. The trend in the bending moment was very similar to that of the lateral pile displacement. When the pile length exceeds 5 T, the bending-moment profile shows long-pile behavior. When the pile length is less than 5 T, the bending moment profile of a short pile was appeared. Figure 20 shows the occurrence depth of the peak bending moment according to the pile length. It is identified that the occurrence depth of the peak bending moment increases as the pile length increases when the pile length is shorter than 5 T. This kind of behavior appears when the pile length is no longer than 5 T, and the occurrence depth of the peak bending moment is about 30% of the pile length. However, if the pile length becomes longer than 5 T, the occurrence depth of the peak bending moment no longer increases, but rather is converged. Based on this phenomenon, it can be suggested that the peak bending moment occurs at 30% of the pile length when pile length is no longer than 5 T and at about 30% of 5 T (1.6 T) when the pile length is longer than 5 T. Through the obtained dynamic responses of the pile for different pile lengths, it is demonstrated that dynamic pile responses are significantly different for pile length. It strongly affected not only in response trend according to the depth but also in rotational depth and occurrence depth of the peak bending moment. And it is identified that proposed numerical model can be used to predict the effect of the pile length on the total number of particles.  
Figure 17: Maximum lateral pile displacements for various pile lengths (input acceleration: 0.13 g).  
Figure 18: Relationship between pile length and rotational depth.  
capture complex dynamic characteristics of soil-pile-structure interactive behavior, also can provide some important point of view on practical seismic design of pile foundation.