# Numerical Simulation of Piles in a Liquefied Slope Using a Modified Soil-Pile Interface Model
## 4 Parametric Study
### Effect of the Slope Inclination  
Figure 12 illustrates the depth profile of the residual lateral displacement of soil at three different locations, namely, the (a) slope toe, (b) slope center, and (c) slope crest. The profile at the slope toe exhibited a sharp increase and reached its maximum at the ground surface. On the other hand, the profiles at the slope center and crest showed maximum values at a depth of approximately 5D with a bulging shape, which was attributed to the failure of liquefied slope. D denotes the diameter of the pile. Regardless of the location, an increase in slope inclination led to an increase in lateral displacement of the soil. Moreover, the increase in slope inclination resulted in a greater increase in lateral displacement of soil closer to the ground surface than at deeper depths.  
The normalized depth of the ground without residual displacement did not consistently increase with the slope angle but instead became shallower after the angle of 21.8 degrees. This observation was attributed to the residual lateral displacement of the ground resulting from the liquefaction-induced slope failure. The failure surface of the liquefied slope progressively extended into deeper ground as the slope inclination increased from 12.5 to 21.8 degrees. However, beyond 21.8 degrees, a notable change occurred as the center of rotation of the failure surface started to shift toward the slope crest. This phenomenon can be attributed to the heightened instability of the slope at steeper angles.  
Figure 11: Comparison of the pile-bending moment between the numerical and experimental results (case 3).  
Figure 13 and 14 illustrate the profiles of the residual bending moment of two different groups of piles, namely, group-up (located near the slope crest) and group-down (located near the slope toe), and the residual bending moment at the pile head and toe for various slope inclinations, respectively.  
Figure 12: Profiles of the residual lateral displacement of soil at the slope (**a**) toe, (**b**) center, and (**c**) crest.  
When the slope inclination was less than 20 degrees, the moment at the pile head and toe were nearly symmetric, and there was no significant difference between those of group-up and group-down. However, when the slope inclination exceeded 20 degrees, the residual moment at the pile toe was considerably greater than that at the pile head. [PERSON] et al. [48] reported that in a centrifuge test for 5 \(\times\) 5 group piles in a 27-degree liquefied slope, the maximum bending moment at the pile toe was greater for piles located near the slope toe than for piles located near the slope crest.  
The neutral point, where the bending moment becomes zero, was observed at a distance of 12 m from the pile head, irrespective of the slope inclination and pile location. As the slope inclination increased, both above and below the neutral point, the moment showed a tendency to increase. This phenomenon occurred because the boundary condition at the head of the pile allowed for horizontal movement but restricted rotation. Under this fixed end beam with a sinking support condition, the bending moments at the head and toe were theoretically equal in magnitude and opposite in direction [49]. In addition, [PERSON] and [PERSON] [50] reported that opposite bending moments occurred at the top and bottom boundaries of the liquefied layer in a series of centrifuge tests with a pile in sloping ground.  
Figure 14: Residual bending moment profiles of (**a**) group-down and (**b**) group-up.  
Figure 13: Residual bending moment according to the slope inclination.