# Numerical Simulation of Piles in a Liquefied Slope Using a Modified Soil-Pile Interface Model
## 2 Numerical Simulation
### Numerical Modeling and Input Parameters  
A numerical simulation of the model structure and ground under seismic loading was performed using Fast Lagrangian Analysis of Continua in 2 Dimensions (FLAC2D). This software has the advantage of dealing with the analysis of dynamic and large deformation problems because it uses explicit, finite difference, and Lagrangian analysis methods to solve the differential equations of a continuum. Figure 3 shows the geometry of the numerical model and the mesh on the centrifuge model. The soil container was modeled as an elastic material with high stiffness, and an interface using beam elements was inserted between the container and soil to simulate the separation due to liquefaction. The beam interface had different interaction springs at two sides, with the spring between the beam and the container set as an elastic spring with a sufficient modulus to prevent the soil from penetrating the container. On the other hand, the spring between the beam and the soil was given a friction angle to allow for separation between the soil and the container [37]. The input properties of the interface beam and spring between the container and soil are summarized in Tables 3 and 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Diameter or Width (m)** & **Length (m)** & **Thickness (m)** & **Flexural Stiffness (kN\(\cdot\)m\({}^{2}\))** & **Weight (kN)** \\ \hline Deck & 7.3 & 7.3 & 1.0 & – & 1490 \\ Pile & 0.92 & 20.0 & 0.05 & 7.66 \(\times\) 10\({}^{5}\) & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the model deck and piles (at the prototype scale). Figure 2: Input base motions: (**a**) ramped motion with a 0.2 g maximum and (**b**) a constant 0.1 g amplitude. The boundary conditions of the model consisted of the ground surface, the horizontal boundary of the container, and the bottom boundary of the container. Normal stress was applied to the ground surface to simulate the hydrostatic pressure caused by the reservoir. In order to prevent the reflection of seismic waves, the horizontal and bottom boundaries of the container were set as a free field boundary and a quiet boundary, respectively. To prevent mesh distortion near the ground surface, which had a weak strength due to the low confining stress, 1 kPa of cohesion was applied. The initial pore pressure level and the saturation status of the ground surface were fixed so that the dissipation of the excess pore pressure was not disturbed. Additionally, Rayleigh damping was set on the foundation and soil to control high-frequency noise. The PM4 Sand model was adopted as the liquefiable soil model. This model is a plasticity soil model that determines soil behavior based on the stress ratio and is developed specifically for simulating the liquefaction behavior of sand soil in dynamic analysis conditions [38]. The primary input parameters of the PM4 Sand model are the relative density, shear modulus coefficient, and contraction rate parameters. The relative density was directly applied with the value obtained from experiments (26%). The shear modulus coefficient was determined based on Equations (1)-(4) from [PERSON] et al. [39] for silica sand. The secondary input parameters of permeability, maximum and minimum void ratio, and critical friction \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **Beam–Container** & **Beam–Soil** \\ \hline Normal spring coefficient (N/m/m) & \(1.0\times 10^{9}\) & \(1.0\times 10^{9}\) \\ Shear spring coefficient (N/m/m) & \(1.0\times 10^{9}\) & \(1.0\times 10^{9}\) \\ Friction angle of normal and shear spring (deg) & - & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input properties of the interface spring between the container and soil. Figure 3: The 2D numerical model for the simulation of the centrifuge test of [PERSON] et al. (2023). angle were determined based on the laboratory tests of [PERSON] et al. (2018) for silica sand.