## 1 Introduction
As a result, the modulus of subgrade reaction has a constant value for the infinitely rigid foundation, while it varies over the contact area between the infinitely flexible foundation and the soil. The settlement profiles of infinitely rigid and flexible foundations with a uniformly distributed load on an elastic half-space are demonstrated in Figure 2.  
However, no system is infinitely flexible or rigid due to the interaction developed between the soil and the foundation. In the literature, various equations can be found to estimate the modulus of subgrade reaction [2,13, 14, 15, 16]. However, these equations do not consider the effect of soil stratification, the irregularities in the foundation shape, or the irregularities in the load distribution on the modulus of subgrade reaction, so they are insufficient to obtain satisfactory results for many cases in construction practice. In recent studies, [PERSON] et al. [17] introduced a new analytical method reflecting the flexibility of mat foundation and the soil-spring coupling effect. [PERSON] et al. [18] showed, with the help of numerical analyses, that the distribution of the subgrade reaction modulus is non-uniform, so its value at the corners and edges of the mat foundation is higher than that at the center. [PERSON] and [PERSON] [19] proposed an equation describing the spatial distribution of spring stiffness based on the results of three-dimensional finite element analyses of slabs resting on elastic soil. [PERSON] et al. [20] derived an equation to estimate  
Figure 1: Contact pressure for infinitely rigid foundations (**a**) resulting from a centrally acting axial load on an elastic half-space, (**b**) resulting from a centrally acting axial load on an elastoplastic half-space, and (**c**) resulting from an eccentrically acting axial load on an elastoplastic half-space. Figure 2: Settlement profiles of infinitely rigid and infinitely flexible foundations on an elastic half-space. the scale effect on loading plates with diameters of less than 762 mm. [PERSON] and [PERSON] [21] performed plate load tests on layered soils with and without geogrid reinforcement. The results of these tests indicated that the interface effect reduces the subgrade reaction modulus, which appears when the spacing between the plates is less than 1.33 times the plate width. [PERSON] et al. [22] stated, with the help of 3D-numerical analyses, that the interface effect can be ignored when the spacing between the foundations becomes larger than three times the foundation width. [PERSON] et al. [23] studied the distribution of the subgrade reaction modulus using an elastic-perfectly plastic constitutive model. Their work indicated that the subgrade reaction moduli developing under elastic loads decrease from the corner to the center of foundation, while they increase from the corner to the center under the ultimate load. [PERSON] et al. [24] developed a novel prediction model to estimate the subgrade reaction modulus in soft soils improved with floating deep cement mixing columns. [PERSON] et al. [25] indicated with the help of numerical analyses and the Mohr-Coulomb soil model that the modulus of subgrade reaction is strongly affected by the relative rigidity of the foundation depending on the soil stiffness and foundation dimensions. Based on the results of plate load tests and FE analyses, [PERSON] et al. [26] stated that the subgrade reaction modulus for foundations in gravelly soils can be estimated using [PERSON]'s extrapolation method as in clayey soils. In the present work, an approximate procedure is studied based on the relative stiffnesses of soil-foundation systems suggested by DIN--Technical Report 130. To estimate the behavior of soil-foundation systems (rigid or flexible), the limit values of relative stiffness are first determined for raft foundations on elastic soils with the stiffness moduli obtained from one-dimensional consolidation tests by using the finite element analyses.