## 4 Design of the Soil and Pile Models
### Design of the Pile Model
#### 4.2.1 Pile-Modelling Criteria  
There are a number of pile properties that contribute to the principal pile response modes, such as the slenderness ratio \(L/d\) (where \(L\) and \(d\) are the pile length and diameter, respectively), flexural rigidity \(EI\) (where \(E\) and \(I\) are the pile modulus of elasticity and moment of inertia, respectively), yield behaviour, ductility, moment-curvature relationship, elastic critical buckling load \(P_{cr}\), \(d/t\) ratio (where \(t\) is the pile wall thickness), pile natural frequency, and the relative ground/pile rigidity. According to the scaling methodology described in Section 3, a similarity relationship for the geometry must be applied. The model properties maintain the pile slenderness and relative contact surface area, which accurately replicate the relative spacing and group interaction of the pile group in the scale model. The pile moment-curvature relationship is a principal modelling criterion because it comprises the two significant parameters--flexural rigidity and yield behaviour --that enable the fully nonlinear response of the pile under lateral loading conditions to be described. The commonly adopted approach to achieve this involves designing piles that can respond within the elastic range while utilising the ductile behaviour of the columns in the superstructure, as suggested by [PERSON] et al. in 2012, [PERSON] et al. (2012). The philosophy behind this approach is that it is easier to detect and repair damage to the above-ground parts of structures than it is to address damage to the subsurface parts. By utilising this method, it is possible to model the lateral dynamic response range of the pile. Scaling the pile's flexural rigidity \(EI\) and ensuring that the yield point is equal to or higher than the prototype yield point are the two ways to accomplish this. The model can accurately replicate the soil-pile kinematic and inertial interaction by scaling the resistance properties of the soil appropriately. The soil behaviour captures the nonlinear cyclic response of the pile. The axial response of end-bearing piles is primarily influenced by the soil properties in the bearing layers, which govern the soil-pile interaction. The friction and cohesion between the soil and pile shaft as well as the elastic deformation of the pile are secondary factors that affect the axial response of an end-bearing pile. However, these factors are primarily significant for soft soil deposits. The static axial capacity of the pile is a crucial factor in determining the inertial load that the pile can carry despite the fact that axial loading is dynamic. Pile radiation damping has two behavioural components. The first component is the inherent dynamic characteristics of the pile. This refers to the pile's ability to generate energy that can be radiated into the surrounding soil. The second component is the energy propagation away from the pile, which depends on the relative soil-pile stiffness. The relative soil-pile stiffness parameter can be automatically scaled from the prototype to the model by consistently scaling the soil and pile elastic properties. Achieving the modelling criterion for the inherent pile dynamic characteristics is a complex task. The optimisation of the natural frequency \(\omega\) of an end-bearing pile can be achieved by utilising an expression that describes the natural frequency of a cantilever rod as a function of the rod's mass \(m\). This expression is given in Equation (19) and was developed by [PERSON] and [PERSON] in 1995, [PERSON] and [PERSON] (1995). \[\omega=3.516\sqrt{\frac{EI}{mL^{4}}} \tag{19}\]  
As previously discussed, by scaling the pile geometry and \(EI\), we can determine that the scaling factor for the mass per unit length of the model pile is \(\lambda^{2}\) (refer to Table 2). It is important to examine this scaling criterion to ensure that the application conditions can accommodate conventional materials and other modelling constraints that are necessary for producing an accurate scale model.