## 2 Materials and Methods
### Properties of Foundations
The ratio of \(\mathrm{q_{load}}/\mathrm{q_{contact}}\) is lower than 1.1 for the square-shaped foundations a1 to a3 and b1 to b3 (Figure 10) and lower than 1.15 for the rectangular-shaped foundation d1 with a single uniform load (Figure 11a) when the value of K is lower than 0.001. For foundations d2 and d3 in Figure 11a, which had varying cross-sections, the ratios of \(\mathrm{q_{load}}/\mathrm{q_{contact}}\) are larger than 1.15 for K-values of between 0.0002 and 0.001. The relatively small contact areas of these foundations led to the relatively small settlements for the same uniform load. For the rectangular-shaped foundations c2 and c3 with two uniformly distributed loads, the ratios of \(\mathrm{q_{load}}/\mathrm{q_{contact}}\) are not provided. As demonstrated in Figure 5b,d, a relatively high difference between \(\mathrm{q_{load}}\) and \(\mathrm{q_{contact}}\) arose in these foundations even for K = 0.0001. This resulted from the interaction between the foundation loads, so the deviations became higher with increasing difference between the foundation loads. Figure 10: Ratio of foundation load to contact pressure depending on the relative stiffnesses of the soil–foundation systems: (**a**) foundations a1 to a3; (**b**) foundations b1 to b3. Figure 9: Ratio of max. settlement to min. settlement for foundations d1 to d3 depending on the relative stiffnesses of the soil–foundation systems: (**a**) foundation thickness = 1 m; (**b**) foundation thickness = 0.25 m.  
Finally, it can be stated that the value of K \(<\) 0.001 provided in Ref. [9] is satisfactory only for foundations with regular shapes and a single uniform load. In other cases, a value of K smaller than 0.001 is required to assume the systems as flexible. Comparison of Subgrade Reaction Moduli Obtained from the Finite Element Analyses and Analytical Method Considering the Relative Stiffness of Soil-Foundation Systems  
In the present section, the subgrade reaction moduli obtained from the FE analyses are compared to those from the analytical method by using four case studies, in which the effects of irregularities in foundation shape and in structural loads on the behavior of the soil-foundation systems are examined. Case studies 1 and 2 treat flexible soil-foundation systems, while rigid systems are studied in case studies 3 and 4 according to Refs. [8,9]. The analytical analyses were performed with the help of the software GGU-Settle Version 7.03 [29]. A uniformly distributed building load was first calculated using the ratio of building weight to foundation area. This load was then considered the contact pressure to determine the vertical stress distribution in soil based on the theory of Boussinesq [28]. Assuming that the soil-foundation system behaves infinitely rigidly or flexibly according to Refs. [8,9], the settlements were subsequently calculated based on the theory of elasticity [30]. Here, the thickness of the compressible soil layer was chosen considering the depth of settlement effect according to the German geotechnical standard. Finally, the distribution of the subgrade reaction modulus was determined by using the ratio of the contact pressure to the settlement.