## 4 Design of the Soil and Pile Models
### Model Pile Development
Figure 11 illustrates the moment-curvature relationship for the aluminium model being tested at the prototype scale. It was observed that this relationship surpassed the yield behaviour within the desired range for the target prototype. As mentioned earlier, this outcome was acceptable since the pile was anticipated to exhibit an elastic response. A model pile material that accurately incorporated the scaled moment-curvature relationship was developed to be \"optimal\". This material had a wall thickness of \(5.08\times 10^{3}\) m, an elastic modulus of 20,685 N/mm\({}^{2}\), and a yield stress of 13.8 N/mm\({}^{2}\). To capture the ideal model behaviour, this material must be manufactured due to its unavailability in the market. A four-point loading test was physically examined in the reference case study, and this was replicated numerically in the current study using an Abaqus FEA model and the COM624P program with a 1.8288 m long and \(50.8\times 10^{-3}\) m diameter section of an aluminium tube with a wall thickness of \(711\times 10^{-9}\) mm to corroborate the model pile moment-curvature relationship. In the physical tests, foil strain gauges were positioned on the tube's compression and tension sides and were read at each increment of loading, and the moments and curvatures were determined based on the measured strain data. This aspect was considered when the results were input into the numerical models. The experimental moment-curvature relationship for the scale model pile is presented in Figure 12 along with the numerical data from the FEA model and the theoretical curves, all at the model scale. The agreement within the elastic response range was outstanding, and the test pile failed through a buckling mechanism that closely resembled the one depicted in the FEA and the theoretical yield point. The failure load was 725 N, which imposed a 386 Nm bending moment on the element. These tests showed that the aluminium tube was an appropriate model pile for the testing programme of the scale model. Table 4 lists the input parameters of the prototype pile and soil, whilst Table 5 provides the computed properties and targeted values of the prototype pile and soil. Finally, Table 6 presents the soil and pile models' input parameters, computed properties, and percentage of result deviation from target values. Figure 11: Lower- and upper-bound moment–curvature relationships for the prototype pile. * Aluminum tube dimensions were a diameter of 0.0508 m and a wall thickness of 0.00071 m. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Prototype Input Parameters** & **Symbol** & **Value** & **Units** \\ \hline \hline Pile outer diameter & OD & 0.4064 & m \\ \hline Pile wall thickness & \(t_{wall,p}\) & \(12.7\times 10^{-3}\) & m \\ \hline Pile length & \(L\) & 13.4112 & m \\ \hline Pile density & \(\rho\) & 7700 & kg/m\({}^{2}\) \\ \hline Soil shear strength & \(\mathrm{S_{u}}\) & 25.0 & kN/m\({}^{2}\) \\ \hline Shear wave velocity & \(V_{s}\) & 111.25 & m/s \\ \hline Steel Young’s modulus & \(\mathrm{E_{steel}}\) & 200 & GPa \\ \hline Concrete Young’s modulus & \(\mathrm{E_{concrete}}\) & 28 & GPa \\ \hline Soil Young’s modulus & \(\mathrm{E_{soil}}\) & 28.728 & MPa \\ \hline Soil shear modulus & \(\mathrm{G_{s}}\) & 163.8 & MPa \\ \hline Percentage of concrete \(EI\) contribution & EI & 50 & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input parameters of prototype pile and soil. Figure 12: Theoretical and experimental moment–curvature relationships for dimensions of the aluminium tube model pile [2].