## 5 Numerical Modelling of Light Embankment Piling
### The Finite Element Model
#### 5.1.1 The Piles
Each variable of Equation (1) is set to result in a pre-known value of \(Q_{head}\), giving the correct load onto the pile heads in the FE model. This also allows moment forces in level with the pile head. However, neither the case shown in Figure 7a nor that in Figure 7b generates arching. The lack of arching in the case of Figure 7a was observed by [PERSON] et al. [45] in the FE simulations presented in the same road embankment case as simulated in this paper. For the simulations presented in this paper, the pile head was added as a rigid plate (Figure 7c) to generate arching. The rigid plate is defined as a completely rigid surface element that can move but cannot be deformed. Displacement and rotation of the rigid plate is based on the stiffness of the media connected to it (in this case the EB in the soft subsoil) and its moment of inertia, respectively. The connection between the top and the rigid plate is fully rigid. Thus, by modelling the embedded beams as in Figure 7c, the load is transferred through the pile head onto the pile in a similar fashion as for a volume pile, while also allowing arches to form. The rigid plate was modelled as an octagon to keep the contour as simple as possible for the meshing algorithm. The long diagonal of the octagonal rigid plate was equal to the pile diameter. An interface was added to the top of the rigid plate, with its thickness set equal to the height of an equilateral triangle of stationary soil on top of the pile head, with the apex angle equal to the friction angle of the embankment soil material [46]. To test the method of modelling each pile as an EB with a rigid plate, comparisons of the axial pile load as well as \(Q_{head}\) and the effective major principal stress (\(\sigma_{1}^{\prime}\)) vectors were made for the methods of modelling in Figure 7. A separate simplified model of a single pile in a square pile group with \(s=1.4\) m was simulated. The embankment height (\(H\)) was set to \(2.0\) m, the average height of the embankment in Figure 1, with a unit weight of \(20\)\(\mathrm{kN}/\mathrm{m}^{3}\). For the comparison, the maximum shaft and toe resistance of each EB was set equal to the shear strength of the modelled adjacent soil to equal the maximum shaft and toe resistances of the volume pile. The extension of the embedded beam with extension (Figure 7c) had a length (\(L_{ext}\)) of \(0.4\) m and the shaft resistance (\(\tau_{ext}\)) was set according to Equation (1) to give a load at the pile head (\(Q_{head}\)) equal to the simulated \(Q_{head}\) of the volume pile. The embedded beam with extension (Figure 7b) and plate (Figure 7c) resulted in similar axial pile load over depth. Both techniques gave lower normal force at the toe compared to the volume pile (Figure 7d), while the total axial pile load was approximately equal. Figure 8 shows the direction of the \(\sigma_{1}^{\prime}\)-vectors and the value of \(Q_{head}\) for the four modelling techniques. The simulated \(Q_{head}\) was close to zero for the modelled EB (Figure 8a). For an EB with extension \(Q_{head}\) = 62.2 kN (Figure 8b) and an EB with rigid octagonal plate resulted in \(Q_{head}\) = 61.1 kN (Figure 8c), 2% lower than for the volume pile for which \(Q_{head}\) = 62.3 kN (Figure 8d). The arch formation for an EB came closest to that of a volume pile when adding a rigid plate, as shown by the direction of the \(\sigma_{1}^{\prime}\)-vectors in Figure 8. Based on the total axial load, \(Q_{head}\) and arch formation, an EB with a rigid plate as pile head can be seen as a valid approximation of a volume pile.