## 4 Conclusions  
In the present work, an approximate procedure to estimate the subgrade reaction moduli of soil-raft foundation systems is studied based on the relative stiffnesses of soil-foundation systems suggested by DIN--Technical Report 130. The following results can be drawn from the analyses performed in the present study:  
1. Concerning the estimation of the behavior of soil-foundation systems: - Assuming the soil-foundation systems are rigid, a value of K \(\geq\) 0.174 is sufficient for foundations with symmetrical shapes and symmetrical distributions of uniformly distributed loads. In this case, the ratio of S\({}_{\rm max}\)/S\({}_{\rm min}\) is lower than 1.1. - Assuming the soil-foundation systems are flexible, a value of K \(\leq\) 0.001 is satisfactory only for foundations with regular shapes and a single uniform load. In this case, the ratio of q\({}_{\rm load}\)/q\({}_{\rm contact}\) is lower than 1.1 and 1.15 for the square-shaped and rectangular-shaped foundations, respectively.
2. Concerning the determination of subgrade reaction modulus:  
Figure 21: Results of the FE analysis for case study 4: (**a**) distribution of settlement; (**b**) distribution of subgrade reaction modulus.  
For soil-foundation systems with K \(\geq\) 0.174, the subgrade reaction moduli obtained from the conventional analytical method are about 1.5 to 2.0 times higher than those in the FE analyses considering the soil-foundation interaction;
* For soil-foundation systems with K \(\leq\) 0.0004, both the analytical and the FE methods yield similar values of subgrade reaction moduli. The ratio of k\({}_{\text{analytical}}\) to k\({}_{\text{fem}}\) varies between 1.05 and 1.15.  
The findings provided above enable geotechnical engineers to estimate the subgrade reaction modulus for raft foundations in the pre-design phase without the input of many structural loads, so the ratios of k\({}_{\text{analytical}}\) to k\({}_{\text{fem}}\) in the central areas of the raft foundations of typical residential buildings vary between about 1.0 (flexible) and 2.0 (rigid), depending on the relative stiffnesses of soil-foundation systems calculated according to DIN--Technical Report 130.  
The FE analyses indicated that the subgrade reaction modulus developing in the edge zones with a width of between 0.5 m and 2 m is higher than that in the central area of a foundation. The excessive increase in the subgrade reaction modulus in the edge zones can reach up to about 8 times of that in the central area of foundation.  
Further studies are required for industrial buildings with locally high column loads.  
Conceptualization, S.K. and S.T.; methodology, S.K. and S.T.; software, S.K.; validation, S.K., formal analysis, S.T.; investigation, S.K. and S.T.; resources, S.K.; data curation, S.T.; writing--original draft preparation, S.T.; writing--review and editing, S.K. and S.T.; visualization, S.T.; supervision, S.K.; project administration, S.K. All authors have read and agreed to the published version of the manuscript.  
The article processing charge was funded by Jade University of Applied Sciences in Germany.  
Not applicable.  
The data are contained within the article.  
The authors declare no conflicts of interest.