## 2 Experimental Test Description
### \(D\) Numerical Modeling  
In previous researches, a representative elementary volume model, called a quarter of a pile model, was used for investigating the behavior of the piled embankments under cyclic loading since it allowed reducing the calculation time as well as obtaining reliable results [1; 6; 12; 25; 26; 27; 28; 29; 30]. With respect to the laboratory tests, three geometrical configurations were numerically studied including a mattress of 100 mm over a reinforced soil with/without a slab, and a mattress of 50 mm over a reinforced soil with a rigid slab. The rigid inclusion-reinforced soil is placed on a non-deformable steel plate, the boundary at bottom \(z=-0.4\) m is therefore restrained in the \(z\)-direction. Due to the symmetry, the vertical boundaries are fixed in their perpendicular directions (Figure 4). The numerical simulations are performed using the finite element method (FEM) with ABAQUS V6.14 [31]. The discretized elements size is selected to be lower than one-tenth of the wavelength \(\lambda\) to minimize the distortion effect of the wave [14]. The quarter pile, soils, and slab were represented by 5742 volumetric 3C8D elements (8-node linear brick) in 3D mesh, as shown in Figure 5. Concerning the constitutive models, the pile and the slab are represented by the linear elastic model. The compressible soil is modeled by the MCC model. The LTP is characterized by the HYP model. Due to the non-integration of HYP model inside ABAQUS software, the UMAT subroutine of the HYP model developed by [PERSON] et al. (2008) was integrated [17]. The moisture of the compressible soil was kept at 10%, and the loading rate was slowly applied (1 kPa/min). The excess pore water was neglected. The numerical calculations were conducted in drained conditions. \begin{table}
\begin{tabular}{c c c c c} \hline
**Testing Conditions** & **Load Transfer Platform thickness** & **Testing Type** & **Testing Name** & **Number of Load Cycles** \\ \hline \multirow{3}{*}{With a slab} & \multirow{3}{*}{100 mm} & Monotonic & M\_D100 & – \\ \cline{3-5}  & & Cyclic 1 & C1\_D100 & 50 \\ \cline{3-5}  & & Cyclic 2 & C2\_D100 & 50 \\ \hline \multirow{3}{*}{Without a slab} & \multirow{3}{*}{100 mm} & Monotonic & M\_R100 & 50 \\ \cline{3-5}  & & Cyclic 1 & C1\_R100 & 50 \\ \cline{3-5}  & & Cyclic 2 & C2\_R100 & – \\ \hline With a slab & 50 mm & Cyclic 1 & C1\_D50 & 50 \\ \hline \end{tabular}
\end{table} 
Table 2: Testing cases used in the tests of Houda. The LTP is made of granular soil, so the HYP model is suggested to use due to its cyclic modeling abilities. The HYP model requires thirteen parameters, as presented in Table 3. The eight parameters from 1 to 8 are representative of the nonlinear elasto-plastic response. They are determined based on a fitting with experimental triaxial test curves (Figure 6). The five remaining parameters from 9 to 13 are representative of the _intergranular_ strain concept which enable modelling the soil cyclic behavior. A set of parameters proposed by [PERSON] and [PERSON] [16] was successfully applied to study the cyclic response of sands [18; 19; 20; 21; 22]. Due to the insufficient data to calibrate the parameters of the intergranular strain state, the set of parameters given in [16] was used in the numerical analyses. The initial void ratio of the granular layer is set equal to 0.7. All the HYP model parameters for the LTP are listed in Table 3. In addition, the MCC model is chosen for the compressible soil. The fitting between experimental and numerical results of the \(e\)-log(\(p^{\gamma}\)) curve of the oedometer test allows determining the model's parameters (Figure 7). The parameters of the MCC model are illustrated in Table 4. The vertical pile's behavior is assumed to be linearly elastic with the Young's modulus of \(E\) = 72.5 GPa, the Poisson's ratio of \(\gamma\) = 0.20 and the unit weight of \(\gamma\) = 27 kN/m\({}^{3}\). Figure 4: Representative elementary volume model used for 3D numerical analyses. Figure 5: 3D mesh of the quarter of a pile model.