## 5 Numerical Modelling of Light Embankment Piling
### The Finite Element Model
#### 5.1.1 The Piles  
In FE modelling, piles are typically modelled as mesh element clusters with the actual geometry and properties of the real pile, referred to in this paper as \"volume piles\" (Figure 7d). Pile-soil interaction (toe resistance and shaft friction) is manually applied with an interface, permitting soil arching to be modelled locally along the pile. The interface simulates the thin zone of intensely shearing soil at the contact between the pile outer surface and the soil, as the pile is expected to settle relative to the soil during loading of the pile group. The use of volume piles is limited by the size of a pile in relation to the overall model size. The common practice in FE modelling is to use larger and fewer mesh elements when possible in order to keep the computation time shorter, since every mesh element requires computation. Volume piles require a small ratio between the mesh element size and the pile diameter to maintain sufficient simulation accuracy. The mesh element size is dependent on the model size. In the work presented in this paper, the pile diameter was too small in relation to the model size to generate a time-efficient mesh for volume piles. Instead, each pile was  
Figure 6: Cross-section of the semi-floating pile group model and element mesh. Soil layer thickness for the floating pile group model is in parentheses. Figure 7: Illustration of ways of modelling piles: (**a**) embedded beam with pile head in level with the transition between embankment and subsoil; (**b**) embedded beam with an extension, length \(L_{ext}\), into the embankment with shaft resistance \(\tau_{ext}\); (**c**) embedded beam with a plate fixed to the node at the pile head; and (**d**) volume pile. \(L_{pile}\) is the pile length. modelled as an embedded beam (EB) (Figure 7a-c). An EB consists of a line element (beam) with a stiffness equivalent to, in this case, a timber pile. The pile-soil interface is modelled with a node-to-node interaction in the form of two linear elastic-perfectly plastic springs for each pair of nodes; one elastic-perfectly plastic spring models the gap displacement and the other models slip displacement. In addition, a region of the soil surrounding the EB is given elastic properties, to give an overall behaviour similar to that of a volume pile. This elastic region is divided into two parts: a cylindrical region around the pile shaft and a hemispherical region encasing the pile toe. Both geometrical parts have a diameter equal to that of the modelled pile. In contrast to volume piles, EBs go through the generated mesh, allowing a continuous mesh. Thus, EBs can be used for cases where the pile diameter is small in relation to the model size, saving computation time. The mesh size controls the resolution of springs in the pile-soil interface, not the line element itself. More information about EBs can be found in the study by [PERSON] and [PERSON] [44]. Modelling a GRPSE and piled embankments in general requires realistic modelling of the load transferred directly onto the piles through the arches (\(Q_{head}\)). With no pile caps prescribed for lightly piled embankments [16], \(Q_{head}\) is carried directly by the pile head. As an EB has no true volume, and thus no pile head surface, the axial load is subsequently zero at the pile head (\(Q_{head}=0\)) when modelling in Figure 7a. In addition, any expected moment at the pile head is zero. Transferring a load onto the top node of an embedded beam (in this case the pile head) is done either by applying a point load on the top node or by attaching the top node to a structure, e.g., a plate element. A third option is to extend the pile into the embankment (Figure 7b) according to  
\[Q_{head}=\tau_{ext}\cdot\theta_{ext}\cdot L_{ext} \tag{1}\]  
where \(\tau_{ext}\) is the shaft resistance of the extension, \(\theta_{ext}\) is the circumference of the extension, and \(L_{ext}\) is the extension length.