## 4 Findings and Discussion
### Determination of Dimensionless Pi (\(\pi\)) Terms  
This section focuses on obtaining dimensionless pi (\(\pi\)) terms belonging to the problem investigated in this study by considering the physical variables of a problem and the primary dimensions of those variables. For that purpose, the first task is to determine the physical variables of the relevant problem and the primary dimensions of the variables (Figure 5). The mechanical behaviour of soil depends highly on stress levels and should be provided between the model and prototype [19]. Thus, the plan is to ensure this criterion between model and prototype while selecting the variables. The selected variables and the primary dimensions of those variables are V = {\(q\), \(s\), \(t\), \(A\), \(\sigma\), \(D\), \(g\)} and D = {\(M\), \(L\), \(T\)}, respectively (i.e., \(n\) = 7 and \(k\) = 3). The variables of \(q\), \(s\), \(t\), \(A\), \(\sigma\), \(D\), and \(g\) refer to compression load, settlement, thickness of the foundation, area of the foundation, vertical stress under the foundation, diameter of foundation, and gravity, respectively. However, [PERSON]'s ratios of foundation and clay soil are not included in the dimensional analysis, since they are already dimensionless variables. In addition to this, some intrinsic characteristics of clay soil and rigid foundation such as [PERSON]'s modulus and unit weights are not included in the dimensional analysis, since the same values have been considered in small-scale and large-scale investigations. Therefore, the primary dimensions of the physical variables of the problem under consideration can be indicated as given below. \[V=\{q,s,t,A,\sigma,D,g\};\,n=7 \tag{17}\]  
\[D=[\text{M}^{1}\text{L}^{1}\text{T}^{-2},\text{L}^{1},\text{L}^{1},\text{L}^{ 2},\text{M}^{1}\text{L}^{-1}\text{T}^{-2},\text{L}^{1},\text{L}^{1}\text{T}^{ -2}];\,k=3 \tag{18}\]  
\[M=[1\;0\;0\;0\;1\;0\;0] \tag{19}\]  
\[L=[1\;1\;1\;2-1\;1\;1] \tag{20}\]  
Figure 5: Physical variables selected for clay soil under the rigid foundation. \[T=[-2\ 0\ 0\ 0\ -2\ 0\ -2] \tag{21}\]  
After determining the variables and their primary dimensions as given above, the dimensionless pi (\(\pi\)) terms belonging to the problem can be calculated as given in Table 6 by employing the MATLAB program, and the dimensionless pi (\(\pi\)) terms can be obtained as \(\pi_{1}=q/(\sigma D^{2})\), \(\pi_{2}=s/D\), \(\pi_{3}=t/D\), and \(\pi_{4}=A/D^{2}\). When several arrangements are carried out, the following relationships can be established between small-scale (model) and large-scale (prototype). \[\pi_{1}=q/(\sigma D^{2})\to q_{m}/(\sigma_{m}{D_{m}}^{2})=q_{p}/(\sigma_{p}{D _{p}}^{2})\to q_{p}=q_{m}(\sigma_{p}{D_{p}}^{2}/\sigma_{m}{D_{m}}^{2}) \tag{22}\]  
\[\pi_{2}=s/D\to s_{m}/D_{m}=s_{p}/D_{p}\to s_{p}=s_{m}(D_{p}/D_{m}) \tag{23}\]  
\[\pi_{3}=t/D\to t_{m}/D_{m}=t_{p}/D_{p}\to t_{p}=t_{m}(D_{p}/D_{m}) \tag{24}\]  
\[\pi_{4}=A/D^{2}\to A_{m}/{D_{m}}^{2}=A_{p}/{D_{p}}^{2}\to A_{p}=A_{m}({D_{p}} ^{2}/{D_{m}}^{2}) \tag{25}\]  
Using these dimensionless pi (\(\pi\)) terms and numerical data of the small-scale physical test, the settlement values corresponding to the compression loads can be predicted for different large-scale cases, as seen in Table 7, based on the physical test reported by [PERSON] et al. (2012). For instance, consider Case 1: all that needs to be done is to decide on the diameter of foundation (D = 2 m). Following that, other variables (compression load, settlement, thickness of foundation, and area of foundation) can be determined employing the foregoing equations. Also in this study, the variables of vertical stresses under the foundation and gravity are the same for the model and prototype behaviours. Table 7 presents only the ultimate compression load-settlement values for various foundation diameters.