## 1 Introduction  
The initial approach considered in solving an engineering problem can investigate whether any analytical solution is available in the literature for the current problem. In complicated problems where analytical solutions are not sufficient, there is a tendency to employ numerical solutions such as finite element, discrete element, finite difference, or boundary element methods by considering acceptable margins of error [1]. However, in cases where both solutions are inadequate due to the complexity of the problem, performing a laboratory test naturally becomes a necessity. In addition, another reason to conduct a laboratory test is to achieve the validation of numerical solutions. Nevertheless, carrying out a large-scale physical test requires much time, effort, energy, space, and cost compared to the foregoing solutions [2]. Therefore, to keep those parameters at a reasonable level, small-scale laboratory tests are generally performed by reducing the dimensions of the large-scale problem at certain ratios. On the other hand, it is essential to provide geometric, kinematic, and dynamic similarities between small-scale (model) and large-scale (prototype) tests in order for the model to accurately reflect the prototype behaviour [3]. These three types of similarity--geometric, kinematic, and dynamic--represent cases where the dimensions, motions, and forces in all directions corresponding to each other between model and prototype are similar, respectively (Figure 1). For complete similarity between the model and the prototype, all of those similarity cases should be satisfied by the model [4]. In order to provide complete similarity between the model and the prototype, [PERSON]'s pi (\(\pi\)) theorem is a method frequently employed by engineers. In the theory propounded by [PERSON] [5], it is stated that the number of dimensionless pi (\(\pi\)) terms (\(i=n-k\)) is equal to the difference between the number of variables (\(n\)) included in the problem and the maximum number of primary dimensions (\(k\)) belonging to those variables. This theorem offers a useful method to define one variable in terms of several other variables [6]. The first task is to determine the dependent and independent variables involved in the problem [7]. Following that, primary dimensions belonging to those variables should be constituted. According to the SI unit system, the primary dimensions that are frequently encountered in problems specific to geotechnical engineering are mass (\(M\)), length (\(L\)), and time (\(T\)). Consider as an example the mathematical expression that gives the vertical normal stress at a certain depth in a certain volume of soil where the unit weight of soil (\(\gamma=\rho\beta\)) is related to the stress component (\(\sigma\)) via depth (\(z\)). The variables in this example and the primary dimensions of those variables are \(V=\{\sigma,\rho,\beta,z\}=\{\text{normal stress, density of soil, gravity, depth}\}\) and \(D=\{M,L,T\}=\{\text{mass, length, time}\}\), respectively (i.e., \(n=4\) and \(k=3\)). The primary dimensions of the variables can be indicated as follows:  
\[\{\sigma\}=[\text{kg}^{1}\text{m}^{-1}\text{s}^{-2}]=[\text{M}^{1}\text{L}^{- 1}\text{T}^{-2}] \tag{1}\]  
\[\{\rho\}=[\text{kg}^{1}\text{m}^{-3}]=[\text{M}^{1}\text{L}^{-3}] \tag{2}\]  
\[\{g\}=[\text{m}^{1}\text{s}^{-2}]=[\text{L}^{1}\text{T}^{-2}] \tag{3}\]  
\[\{z\}=[\text{m}^{1}]=[\text{L}^{1}] \tag{4}\]  
After performing a series of systematic mathematical operations according to [PERSON]'s pi (\(\pi\)) theorem, the number of pi (\(\pi\)) terms (\(i=n-k=4-3=1\)) is determined and the dimensionless pi (\(\pi\)) term can be obtained as \(\pi_{1}=\sigma/\rho gz\), or \(\pi_{1}=\sigma/\gamma z\). Since the number of variables is limited in this example, the dimensionless pi (\(\pi\)) term is effortlessly calculated.