## 3 Implied Prototype Scaling Methodology
### Scale-Modelling Factors  
[PERSON] et al. (2019) [21] identified various inter-related interaction modes that characterise the response of a seismic soil-pile-structure interaction system. These modes include the free-field soil site response mode, the soil-pile lateral kinematic interaction mode, the soil-pile axial response mode, and the pile cap radiation damping mode. The variables associated with each interaction mode are presented in Table 1. In this context, the objective of utilising the scale-modelling technique is to achieve what has previously been defined as dynamic similarity, whereby the scale model and prototype are subjected to comparable forces. The program relies on dimensional analysis as the fundamental principle for achieving scale model similarity. The scaling parameters are evaluated based on three primary testing criteria. Initially, the evaluation must be conducted within a 1g environment, wherein comparable model and prototype accelerations are observed. The density of the model soil and prototype soil must be comparable. This phenomenon addresses an additional element of the scaling relationships. Furthermore, it is imperative that the testing medium primarily consists of saturated clay, as its undrained stress-strain behaviour restricts pressure dependence and consequently streamlines the constitutive scaling prerequisites. By defining the scaling conditions for the density and the acceleration, the mass, length, and time scaling factors can all be presented in terms of the geometric scaling factor \(\lambda\). Then, a correct set of dimensionally accurate scaling relationships (i.e., ratio of prototype to model) for the studied variables can be derived. The scaling relationships for all of the potential variables contributing to the primary modes of system response were obtained according to the technique above and are listed in Table 2. A geometric scaling factor \(\lambda\) equal to 8 was selected as a benchmark for the dimensional scaling procedure, in accordance with the recommendations elsewhere [17]. A comprehensive set of scaling relationships for a soil-structure interaction system subjected to dynamic loading condition was then derived. The scaled dynamic problem was entirely defined in terms of geometric scaling factor \(\lambda\)The corresponding scaling relationships for the variables that contributed to the primary system response modes were the density, acceleration, length, force, shear wave velocity, stress stiffness, time, strain, modulus, frequency, and flexural rigidity. Accordingly, when the scaling factor of the shear wave velocity equalled \(\lambda^{\frac{1}{2}}\), the scale model met the Cauchy condition. The calculated strain scaling factor, in accordance with [PERSON] (1989), equalled unity. Consequently, the corresponding set of scaling relationships agreed with the values developed in the current study, as presented in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**No. ** & **Interaction Mode (SSPSI)** & **Variables** \\ \hline I. & Free-field & \(\left[V_{s}(Z),\,\rho(Z),\,\right.\) \\  & & modulus of degradation and damping \((Z)\left]_{soil}\right.\) \\ \hline II. & Soil–pile lateral kinematic interaction & I, \(\left[EI,\,\text{length, diameter, fixity}\right]_{\text{pile}}\), \(\left[\left(\text{stress–strain behaviour}\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  
The implementation of the scaling relationships and the creation of the components of the model soil, pile, and superstructure system are addressed in the subsequent sections of this paper.