## 2 Finite Element Modelling  
### Model Description  
A 5.9 m high embankment and a 0.6 m high pavement were located on a 10 m high subsoil layer, as shown in Figure 1a. The concrete piles with square caps, which were 1 m in width and 0.5 m in thickness, were placed in a square layout through the subsoil. The diameter and spacing of the concrete piles were 0.3 m and 2.5 m, respectively. The geogrid was placed 0.1 m above the base of the embankment. The three-dimensional (3D) unit cell models of gravel and solidified soil embankments were simulated using the finite element package \"Abaqus\" (version 6.16). The 3D unit cell models (see Figure 1b) include one-quarter of the pile and its tributary area [27]. The bottom boundaries of the 3D unit cell models were completely fixed, as shown in Figure 1c. The vertical side boundaries were horizontally fixed due to the symmetry of the models. In the literature, the 3D unit cell model was widely used for the stress analysis of the pile-supported embankment, considering the structural symmetry and computational efficiency [28; 29; 30; 31]. According to [PERSON] and [PERSON] [27], the results of the 3D unit cell model of the pile-supported embankment showed good agreement with those from the full model for pile-supported embankment in settlements, vertical stress and lateral deflection. [PERSON] et al. [30] pointed out that the unit cell model has the ability to capture the behavior of a basic volume cell in a larger 3D model and reduce the complexity; meanwhile, it could provide quality results and computational performance. The water table was at the surface of the soft subsoil, and the initial pore pressures before the embankment construction were assumed to be hydrostatic. After the embankment load was applied, the bottom and top boundaries of the subsoil were set to zero pore pressure, indicating that they could drain freely. The vertical boundaries were treated as impermeable surfaces. The subsoil was discretized with 8-node stress-pore pressure-coupled brick elements (C3D8P), while the piles, embankment, and pavement were discretized with 8-node brick elements (C3D8). The geogrid was simulated using 3-dimensional membrane elements with four nodes and full integration (M3D4), with nonzero stiffness only in the X-Y plane. The mesh of 3D unit cell models is shown in Figure 1d. To validate the model settings and parameters, Model A and Model B of pile-supported gravel embankments reported by [PERSON] and [PERSON] [29] were remodeled and referred to as \"Case A\" and \"Case B\" in this paper, under static and dynamic loads, respectively. The models of solidified soil embankment under static and dynamic loads were performed and referred to as \"Case C\" and \"Case D\", respectively. The summary of the load types and embankment materials is shown in Table 1.  
The interaction between the pile and the subsoil was surface-to-surface contact. The interaction between the geogrid and the embankment was tie interaction. The shear resistance angle (\(\varphi_{i}\)) between the pile and the soft soil in the horizontal plane was calculated, and the coefficient of friction (\(\mu_{i}\)) was calculated according to [PERSON] and [PERSON] [30].  
\[\varphi_{i}=\tan^{-1}\left(\frac{\sin\varphi^{\prime}\times\cos\varphi^{\prime} }{1+\sin^{2}\varphi^{\prime}}\right) \tag{1}\]  
\[\mu_{i}=\tan\varphi_{i} \tag{2}\]  
where \(\varphi^{\prime}\) represents the effective friction angle of the soft soil. In this work, the effective friction angle of the soft soil was 26\({}^{\circ}\).  
\begin{table}
\begin{tabular}{c c c} \hline  & **Load Types** & **Embankment Materials** \\ \hline Case A & Static loads & Gravel \\ Case B & Dynamic traffic loads & Gravel \\ Case C & Static loads & Solidified soil \\ Case D & Dynamic traffic loads & Solidified soil \\ \hline \end{tabular}
\end{table}
Table 1: Load types and embankment materials in 3D unit cell models.