## 2 Materials and Methods
### Properties of Foundations  
The analyses were performed for 18 foundations concerning the geometry and structure loads. The same values of Young's modulus E\({}_{\text{f}}\) = 31,000 MPa, Poisson's ratio \(\
u_{\text{f}}\) = 0.2, and the unit weight \(\gamma_{\text{f}}\) = 25 kN/m\({}^{3}\) were assigned to the foundation material. The foundations shown in Figure 3 can be classified into four groups:In the first group, the square-shaped foundations with different sizes (5 m \(\times\) 5 m, 10 m \(\times\) 10 m, and 20 m \(\times\) 20 m) were studied (Figure 3a). All three foundations had the same thickness of d = 0.5 m, and they were loaded with a uniformly distributed load of \(p\) = 100 kN/m\({}^{2}\). The relative stiffnesses of these soil-foundation systems were calculated according to Equation (2) and listed in Table 1.  
Figure 3: Foundations examined in the numerical analyses: (**a**) square-shaped foundations with varying side lengths, (**b**) square-shaped foundations with varying thicknesses, (**c**) rectangular-shaped foundations with varying load distributions and thicknesses, and (**d**) foundations with varying shapes and thicknesses. In the second group of foundations, shown in Figure 3b, the distributions of the subgrade reaction moduli were determined for the square-shaped foundations with different thicknesses (d = 0.25, 0.5, and 1 m). All three foundations had the same side length of 10 m and were loaded with a uniformly distributed load of \(p=100\)\(\mathrm{kN/m^{2}}\). The relative stiffnesses of these soil-foundation systems are provided in Table 2.  
The effect of load distributions on the subgrade reaction modulus was examined by using a rectangular foundation (10 m \(\times\) 20 m) with two different thicknesses (d = 0.25 m and 1 m) in the third group (Figure 3c). The uniformly distributed loads (\(p_{1}\) and \(p_{2}\)) on foundations c2 and c3 were chosen such that no eccentricity appeared, so the total load on both foundations had the same value (20,000 kN) as on foundation c1. In the last group (Figure 3d), the differently shaped foundations with two different thicknesses (d = 0.25 m and 1 m) were loaded with a uniformly distributed load of \(p=100\)\(\mathrm{kN/m^{2}}\). The relative stiffnesses of the soil-foundation systems in Figure 3c,d can be taken from Table 3 for d = 1.0 m and 0.25 m. It should be noted that foundations c2 and c3 have varying cross-sections, so the equation suggested by DIN--Technical Report 130 [8] cannot be directly applied to these foundations. Thus, the largest size of this foundation was used as L in Equation (2). \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Stiffness Modulus of Soil**} & \multicolumn{2}{c}{**Relative Stiffness of the Soil–Foundation System K (-)**} \\ \cline{2-4}  & **Square** & **Square** & **Square** \\
**Soil E\({}_{\mathrm{s}}\) (MPa)** & **Foundation** & **Foundation** & **Foundation** \\  & **(5 m \(\times\) 5 m \(\times\) 0.5 m)** & **(10 m \(\times\) 10 m \(\times\) 0.5 m)** & **(20 m \(\times\) 20 m \(\times\) 0.5 m)** \\ \hline
2.5 & 1.3909 1 ** & 0.1739 1  & 0.0217 2  \\
10 & 0.3477 1  & 0.0435 2  & 0.0054 3  \\
40 & 0.0870 2  & 0.0013 3  \\
160 & 0.0217 2  & 0.00027 3  & 0.0004 4  \\ \hline \end{tabular}
\end{table} 
Table 1: Relative stiffnesses of the soil–foundation systems shown in Figure 3a. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Stiffness Modulus of Soil**} & \multicolumn{2}{c}{**Relative Stiffness of the Soil–Foundation System K (-)**} \\ \cline{2-4}  & **Square** & **Square** & **Square** \\
**E\({}_{\mathrm{s}}\) (MPa)** & **Foundation** & **Foundation** & **Foundation** \\
10 & 0.0054 3  & 0.3477 1  \\
40 & 0.0013 2  & 0.0109 3  \\
10 & 0.0004 5  & 0.0027 4  \\ \hline \end{tabular}
\end{table} 
Table 2: Relative stiffnesses of the soil–foundation systems shown in Figure 3b. As shown in Figure 2, the uniformly distributed foundation load corresponds to the contact pressure in infinitely flexible systems, while a uniform settlement develops in infinitely rigid systems.