## 4 Design of the Soil and Pile Models
### Model Pile Development  
An iterative process must be employed to determine the flexural rigidity \(EI\) and natural period in the pile design process using an appropriate laboratory test and numerical model for validation. According to the reference case study's pile boundary condition, an equivalent cantilever rod can be used to identify the targeted principal pile parameters. A geometric scaling factor of 8 is used to develop the scaling requirements for the soil and pile models. Depending on the target model \(EI\) and the scaled pile outer diameter, the moment of inertia can be computed according to two conditions; that is, for a solid and a thin-walled tube, respectively. The corresponding lower and upper bound modulus of elasticity can then be determined. The pile density is computed for both of these bounds, which then impart the scaled modes of vibration. The computed densities may range widely between two different values for solid and thin-walled sections, and these values may not be realistic for practical sections. In addition, the material type must be investigated to determine that it is appropriate as a model pile material. In the reference case study, the target \(EI\) equalled 2.3 kNm\({}^{2}\) and the pile outer diameter was fixed at 50.8 mm, therefore the lower and upper bound modulus of elasticity corresponding to solid and thin tubes were 6894.76 and 68947.6 N/mm\({}^{2}\), respectively. The corresponding pile material densities were 295.30 kg/m\({}^{2}\) and 573.4 kg/m\({}^{2}\), respectively. Following a review of suitable materials, an aluminium 6061 T-6 alloy was selected, and the pile was designed as a thin-walled section to satisfy the \(EI\) criterion. Accordingly, the scale model pile was fabricated with a scaled flexural rigidity \(EI\) of 2.420 kNm\({}^{2}\) and an \(L/d\) ratio of 36. The thickness of the pile wall was 0.71 mm, which is a commercially available pipe size. It is important to note that thin-walled tubes may be susceptible to local buckling, but the buckling load \(P_{cr}\) was found to be within the allowable range. The theoretical moment-curvature ratio of the trial pile model must be compared to that of the prototype to ensure that the representation is appropriate. The lower and upper bound prototype scenarios were determined by employing yield stresses in the steel pipe pile of 345 and 483 N/mm\({}^{2}\), respectively, with 0% and 100% concrete \(EI\) contributions that represented fully intact and fully cracked concrete sections, respectively. The lower and upper bound moment-curvature relationships demonstrated at a prototype scale were defined according to the pile analysis code COM624P [31]. The results were calibrated against a four-point loading test finding performed by Caltrans on a 0.61 m concrete-filled steel pipe pile [32]. The moment-curvature relationship of the pile model can be identified using several modern approaches, and SE:MC is one of the computer programs offering this sort of analysis. SE:MC is a powerful tool for structural design and research in which moment-curvature analysis is needed and is based on a strain-compatibility technique. A numerical FEA approach was adopted for the validation process using the Abaqus software to simulate the four-point loading test. The method proposed by [PERSON] (1951), ref. [33] was adopted for the determination of the moment-curvature relationship of the pile. The bending capacity equation for a ductile beam with a circular cross-section is defined as follows:  
\[M=2\;r^{3}\int_{0}^{1}\beta\;\sigma\;\bar{\zeta}\;d\bar{\zeta} \tag{20}\]  
where \(r\) is the radius, \(\beta\) represents the width of the cross section at the ordinate divided by \(r\), \(\bar{\zeta}\) represents the ordinate from the neutral axis divided by \(r\), and \(\sigma\) is the stress at the ordinate of the cross section. An elastic-perfectly plastic stress-strain idealisation was used to model the nominated aluminium material.