## 2 Background of Scale Model Similitude Theories
### The Concept of Application of Scale Model Similitude to Soil Mechanics
[PERSON] (1991) [18], also employed \"dimensional analysis\" to develop the functional relation given in Equation (15) for scale model similarity, which is used in shaking table tests to evaluate the effects of soil-pile interaction for several intensities of shaking:  
\[\frac{y}{b}=K\bigg{(}\frac{l}{u_{o}},\frac{\rho_{p}}{\rho_{s}},\frac{EI}{G_{s} u_{o}^{4}},\frac{\omega^{2}u_{o}}{g},\frac{m_{o}}{\rho_{s}u_{o}^{3}}\bigg{)} \tag{15}\]  
In this expression, \(y\) is the lateral displacement of the pile, \(b\) is the pile diameter, \(l\) is the pile length, \(u_{o}\) is the input motion amplitude (applied at the base of the model), \(\rho_{p}\) is the pile density, \(\rho_{s}\) is the soil density, \(EI\) is the flexural rigidity of the pile, \(G_{s}\) is the shear stiffness of the soil (which depends on depth and strain), \(\omega\) is the natural frequency of input motion, \(g\) is the acceleration of gravity, and \(m_{o}\) is the superstructure mass. [PERSON] (1991) found that fulfilling the second and third scaling laws simultaneously is challenging. The former entails identical model and prototype material densities, while the latter yields a prototype-to-model pile flexural rigidity ratio of \(\lambda^{5}\). In order to tolerate imperfect model similarity, tests must be viewed as prototype cases against which analytical simulations can be validated. In addition, it was indicated that test results can be depicted in terms of dimensionless variables in order to facilitate comparisons with data from large-scale tests. [PERSON] (1989) [17], developed a scale model for a shaking table test designed to simulate the constitutive behaviour of saturated soils by considering a tangent modulus method based on the results published by [PERSON] (see Figure 4). A set of scaling relationships for the soil-structure-fluid interaction system subjected to dynamic loading was derived whereby the scaled dynamic problem was defined in terms of geometric, density, and strain scaling factors. The methodology prescribes the geometric (\(\lambda\)) and density (\(\lambda_{\rho}\)) scaling factors; the strain scaling factor (\(\lambda_{e}\)) is then derived using shear wave velocity tests for the prototype and model of the soil, as given in Equation (16):  
\[\lambda_{e}=\frac{\lambda}{\bigg{(}\frac{(V_{S})_{p}}{(V_{S})_{m}}\bigg{)}^{2}} \tag{16}\]  
A counterintuitive consequence in this approach is that quantities for a particular model with the same dimensions may have distinct scaling factors such as length and deflection. Nevertheless, this technique has been validated through comparison with laboratory tests. According to [PERSON] (1989) [17], this approach is only applicable for low-strain circumstances, and the soil particles remain in contact but continue to contribute to the liquefaction issue involving medium to dense sand soil deposits. In another approach, [PERSON] (1989) [19] employed the governing equations for dynamic equilibrium in order to develop scaling factor relationships for the soil model in centrifuge testing. This derivation was further refined by [PERSON] (1997) [20], such that it can be employed for a granular saturated soil subjected to a centrifuge modelling programme or a 1g environment. The expression for the developed relationships is given as:  
\[\bigg{(}1-\frac{x^{*}}{\sigma^{*}},\frac{x^{*}}{t^{*2}}\rho^{*}\bigg{)}\ \rho_{m}\bigg{[}\frac{\partial^{2}u_{lim}}{\partial t_{m}^{2}}+\frac{\partial u _{jm}}{\partial t_{m}}.\frac{\partial}{\partial x_{jm}}\bigg{(}\frac{ \partial u_{lim}}{\partial t_{m}}\bigg{)}\bigg{]}=\bigg{(}1-\frac{x^{*}}{ \sigma^{*}}\cdot X^{*}\bigg{)}\ X_{im} \tag{17}\]  
where \(x\), \(\sigma\), \(t\), \(r\), \(X\), and \({}^{*}\) denote the stress, time, mass density, body force, and prototype-to-model ratio, respectively. In 1997, [PERSON] identified the dynamic behaviour of the scaling soil constitutive problem for 1g environment testing. Additionally, [PERSON] proposed modifications to the material properties of the model.