## 2 Materials and Methods
### Finite Element Model  
Conducting a laboratory test in geotechnical foundation engineering is both a difficult and time-consuming procedure along with quite expensive. Nevertheless, the results acquired from such tests are the closest to those procured from field tests. As a result, establishing a numerical model that is less costly and not time-consuming, as well as being in good consistency with the test results, can provide a significant benefit to geotechnical designers. Therefore, a series of 2D and 3D numerical analyses were performed to study the load-displacement behavior of helical piles and to demonstrate their failure mechanisms. For numerical analyses, PLAXIS 2D (Version 2023.2) [37] and PLAXIS 3D (Version 2023.2) [38] software, based on the finite element method frequently used by researchers [39; 40], were chosen due to some of their advantages. One of the advantages of the software is that it includes the ability to create different complicated material models to reflect soil behaviors; plus, it covers some specific structural elements for foundations, piles, anchors, geogrids, retaining walls, etc. [41]. Two-dimensional and three-dimensional finite element models established for the problem are presented in Figure 4, respectively. In the 2D model, it was enough to define the length (x = 750 mm; half of the length was considered since the axisymmetric model was used) and the depth (y = 1000 mm), whereas in the 3D model, the length (x = 1500 mm), width (y = 1200 mm), and depth (z = 1000 mm) values must be entered into the model. Within the scope of the study, the properties of the investigated loose sandy soil were \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Effective grain size, D\({}_{10}\) (mm) & 0.26 \\ D\({}_{30}\) (mm) & 0.44 \\ D\({}_{60}\) (mm) & 0.78 \\ Coefficient of uniformity, C\({}_{\rm u}\) & 2.97 \\ Coefficient of curvature, C\({}_{\rm c}\) & 0.96 \\ Soil class (USCS) & SP \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the sieve analysis. Figure 3: Grain size distribution curve of the sand sample. introduced to the software, and the soil was assigned to the defined area for 2D modelling and assigned to the defined volume for 3D modelling. Since it is essential to use an appropriate soil model for the prediction of soil behaviors, the numerical analyses were accomplished using the Hardening Soil model which can represent the behavior of different soils, both soft soil and stiff soils [42]. The model was established in the frame of the theory of elasticity, and the material behavior is non-linear before failure, but after failure, it is considered based on Mohr-Coulomb strength parameters [43,44]. Following the definition of soil properties, the properties of the pile element were introduced to the software, and the relevant structural element was assigned to the relevant region. The helical pile element was modelled in both 2D and 3D using the plate element. While in 2D modelling the axisymmetric approach was considered to create the finite element model, in 3D modelling the finite element model was established identically to the physical model test setup. In addition, in 3D modelling, SOLIDWORKS 2016 [45] software was used to create the helix structure of the pile element identically, and then, the pile element was transferred to the PLAXIS 3D software, where the analyses were performed. The pile shaft with an outer diameter (d) of 22 mm, a thickness of 2.5 mm, and a length (L) of 600 mm was created for the straight pile as it was compatible with the test. Helix plates were modelled as well as the shaft for different diameters of helical piles such as D\({}_{2}\) = 60 mm, 80 mm, 100 mm, and 120 mm. The distance between the bottom and top point of the plate (pitch) is related to the diameter of the relevant helix, and they were determined for the lower and upper plates as 0.3D\({}_{1}\) and 0.3D\({}_{2}\), respectively.