## 2 Materials and Methods
### Properties of Foundations
Accordingly, the ratios of the maximal settlements to minimal settlements \(\mathrm{S_{max}}/\mathrm{S_{min}}\) were relatively low for the rigid systems examined in this section. For the foundation a1 with a relative stiffness of K = 1.3909, which is classified as rigid according to Ref. [9], the ratio of \(\mathrm{S_{max}}/\mathrm{S_{min}}\) is equal to 1.01 (Figure 4a). For the other rigid systems with K = 0.1739 (Figures 4c, 5a,c and 6a,c), this ratio was higher than 1.01 due to the relatively low K-value, so it varied between 1.03 and 1.10. The varying ratios of \(\mathrm{S_{max}}/\mathrm{S_{min}}\) for the same value of K can be explained by the effect of foundation geometry and load distribution on the soil settlement. Under a uniformly distributed load of 100 \(\mathrm{kN}/\mathrm{m}^{2}\), two stress zones occurred in the flexible cases with K = 0.0004 and 0.0001. The average values of the contact pressures (q\({}_{1}\) and q\({}_{2}\)), which had a maximum deviation of \(\pm\)12.5% from the foundation load, are illustrated in Figures 4b,d and 6b,d. The relatively high derivations that appeared in the foundations shown in Figure 5b,d are explained by the effect of load distribution on the contact pressure, because these foundations had two uniformly distributed loads in contrast to the other foundations (Figure 3c). It should be noted that a pressure concentration developed on the edges of all soil-foundation systems examined. The pressure concentrations in the rigid systems were higher than those in the flexible systems, in accordance with Equation (3). The width of the zones  
Figure 6: The values of contact pressure q and settlement S for foundations with cross-sections with varying widths and a single uniformly distributed load: (**a**) foundation d2, with K = 0.1739; (**b**) foundation d2, with K = 0.0001; (**c**) foundation d3, with K = 0.1739; and (**d**) foundation d3, with and K = 0.0001.  
with relatively high pressures was equal to 0.5 m, which corresponded to the side length of the triangle elements in the finite element mesh generated in the numerical models. In Figures 7-9, the ratios of the maximal settlements to minimal settlements \(\mathrm{S_{max}/S_{min}}\) are provided depending on the relative stiffnesses of soil-foundation systems K. It can be seen from Figures 7, 8a, and 9a that the ratio of \(\mathrm{S_{max}/S_{min}}\) is lower than 1.1 when the value of K is higher than 0.174. That means that a value of K higher than 0.1, as suggested by [9], is not satisfactory for assuming the soil-foundation system as rigid. Based on the relatively low values of K in Figures 8b and 9b, the ratios of \(\mathrm{S_{max}/S_{min}}\) are relatively high (>1.5). Here, it should be noted that the relatively low value of \(\mathrm{S_{max}/S_{min}}\) = 1.2 for the foundation c3 with K = 0.0027 in Figure 8b results from the fact that this foundation is loaded with a relatively high uniform load of 260 kN/m\({}^{2}\) on its edges and with a lower uniform load of 10 kN/m\({}^{2}\) in its central area, which led to settlements at a similar magnitude. Figure 8: Ratio of max. settlement to min. settlement for foundations c1 to c3 depending on the relative stiffnesses of the soil–foundation systems: (**a**) foundation thickness = 1 m; (**b**) foundation thickness = 0.25 m.  
Figure 7: Ratio of max. settlement to min. settlement for foundations a1 to a3 and b1 to b3 depending on the relative stiffnesses of the soil–foundation systems. The ratios of the foundation load to the contact pressure \(\mathrm{q_{load}}/\mathrm{q_{contact}}\) are provided depending on the K-values in Figures 10 and 11. Here, \(\mathrm{q_{contact}}\) corresponded to the average value of contact pressures in the central area of the foundation, so the relatively high pressures developing on the edge zones with a width of 0.5 m were ignored.