## 2 Background of Scale Model Similitude Theories
As strain is a dimensionless parameter, the scaling relation between the model and prototype strains is 1:1. The densities \(\rho\) of the materials in the model \(\rho_{model}\) and the prototype \(\rho_{prototype}\) are ordinarily applied as a basis for defining the relationship between scale factors \(\mu\) and \(\lambda\). \(\mu\), \(\lambda\), and \(\tau\) can be expressed in accordance with Equations (1)-(3), respectively:  
\[\frac{L_{m}}{L_{p}}=\lambda \tag{1}\]  
\[\frac{M_{m}}{M_{p}}=\mu \tag{2}\]  
\[\frac{T_{m}}{T_{p}}=\tau \tag{3}\]  
In these expressions, the subscripts for \(M\), \(L\), and \(T\) correspond to the model (\(m\)) and prototype (\(p\)), respectively. Following on from this, the ratio of the model density (\(\rho_{model}\)) to the prototype density (\(\rho_{prototype}\)) equals unity when the following criteria is satisfied:  
\[\frac{\rho_{model}}{\rho_{prototype}}=1=\frac{M_{m}}{M_{p}}\cdot\frac{V_{p}}{V_ {m}}\]  
Once \(\frac{M_{m}}{M_{p}}\) is \(\mu\) and \(\frac{V_{p}}{V_{m}}=\frac{1}{\lambda^{3}}\), \(\mu\) can be computed as \(\lambda^{3}\). The time scale factor \(\tau\) can be computed using the equation of the inertial force ratio, in accordance with Equations (4)-(7), where \(A_{m}\) and \(A_{p}\) represent the areas of the model and prototype, respectively, and \(\gamma_{m}\) and \(\gamma_{p}\) are the corresponding model and prototype unit weights:  
\[\left(\frac{M_{m}A_{m}}{M_{p}A_{p}}\right)=\left(\frac{\gamma_{m}}{\gamma_{p} }\right)\cdot\lambda^{3}\cdot\left(\frac{A_{m}}{A_{p}}\right) \tag{4}\]with the weight ratio given as:  
\[\left(\frac{\gamma_{m}}{\gamma_{p}}\right)\cdot\lambda^{3} \tag{5}\]  
The model accelerations must be equal to the prototype accelerations, and thus:  
\[\left(\frac{A_{m}}{A_{p}}\right)=1=\frac{\left(\frac{L_{m}}{T_{D}^{2}}\right)}{ \left(\frac{L_{p}}{T_{p}^{2}}\right)}=\frac{L_{m}}{L_{p}}\frac{T_{p}^{2}}{T_{m} ^{2}}=\lambda\cdot\left(\frac{T_{p}}{T_{m}}\right)^{2} \tag{6}\]  
in which  
\[1=\lambda\cdot\left(\frac{T_{p}}{T_{m}}\right)^{2},\;\lambda=\left(\frac{T_{m }}{T_{p}}\right)^{2} \tag{7}\]  
where \(\tau=\frac{T_{m}}{T_{p}}\). \(\tau\) can then be computed as equal to \(\lambda^{\frac{1}{2}}\). Through the determination of scaling factors for mass (\(\mu\)), length (\(\lambda\)), and time (\(\tau\)) in relation to \(\lambda\), it is possible to establish a comprehensive set of scaling relationships that are dimensionally accurate for all variables under investigation. The earthquake resistance of rock-filled dams and sloping core dams was examined by [PERSON] and [PERSON] (1956) [11], and [PERSON] (1963) [12] respectively, through the utilisation of scale models in their methodology. One limitation of this methodology is that it treats each variable in isolation, disregarding its system function. A methodology of dimensional analysis involves the Buckingham \(\pi\) theorem approach of scale model. According to this theorem, \"_any dimensionally homogeneous equation involving certain physical quantities can be reduced to an equivalent equation involving a complete set of dimensionless products_\" [13]. Consequently, the solution for a studied physical quantity of interest \((X_{1},X_{2},\;\ldots,\;X_{n})\) can be expressed as given in Equation (8):  
\[F(X_{1},X_{2},\;\ldots,\;X_{n})=0 \tag{8}\]  
and stated in the form of \(\pi\), as given in Equation (9):  
\[\frac{L_{m}}{L_{p}}=\lambda \tag{1}\]0  
In this expression, \(\pi\)(s) denotes the autonomous dimensionless products of the physical quantities \(X_{1},X_{2},\;\ldots,\;X_{n}\); the symbol \(\eta\) represents the count of dimensionless products; while \(n\) signifies the quantity of physical variables. The relation between these two terms is expressed in Equation (10):  
\[\frac{L_{m}}{L_{p}}=\lambda \tag{1}\]1  
The process of establishing of individual \(\pi\) terms involves the classification of physical variables into dimensionless quantities. It is imperative to incorporate all of the variables while ensuring that the term \(m\) remains independent.