## 5 Conclusions  
In this paper, the formula for spring constant \(K_{x}\) and spring coefficient \(\beta_{x}\) were derived, thereby demonstrating that the spring coefficient \(\beta_{x}\) depends on Poisson's ratio \(\
u\). It was also shown that the graph presenting the dependence between the spring coefficient \(\beta_{x}\) and the length-width ratio of a rectangular footing, i.e., \(\alpha=\frac{\alpha}{b}\), which was presented, among others, in papers [10; 11; 13; 14; 16], was plotted for a constant value of the Poisson's ratio \(\
u=0.3\) (see Figures 3 and 4). It was shown that taking into account the Poisson's ratio \(\
u\) of a given soil significantly affects the value of the spring coefficient \(\beta_{x}\), and thus affects the value of the spring constant \(K_{x}\). The differences between the \(\beta_{x}\) and \(K_{x}\) values obtained from the calculations based on the formulas derived in this paper and from the calculations based on the formulas and nomogram for the spring coefficient \(\beta_{x}\) given in, among others, papers [10; 11; 13; 14; 16] (i.e., \(\beta_{x}\) calculated for the constant value \(\
u=0.3\)) reached 8-11%. These differences were obtained by assuming the Poisson's ratio \(\
u=0.5\) and \(\
u=0.1\), and the coefficient \(\alpha=10\), for the calculations.  
This paper also provides useful nomograms (see Figures 11-13), which, with known soil parameters (the shear modulus of soil \(G\) and Poisson's ratio of soil \(\
u\)) and foundation geometry (foundation area \(A\) and the ratio of length and width of a rectangular footing \(\alpha=\frac{\alpha}{b}\)) can be used to easily read the value of the spring constant \(K_{x}\). These nomograms can, therefore, be useful in engineering calculations for quick and preliminary estimation values of spring constant \(K_{x}\).  
Based on the results of the numerical Finite Element Method (FEM) calculations (see Tables 3 and 4), it can be assumed that the proposed simplified approach to determining the spring coefficient yields good results (agreement at the level of 3-10% for small ratio \(\alpha=\frac{\alpha}{b}\) and Poisson's ratio values in the range of \(\
u=0.1\)-0.3, and agreement of 10-28% for higher values of the analyzed parameters) in relation to the complex, three-dimensional, numerical FEM model.  
The presented method in the article, the simplified method of modeling the interaction of the structure with the soil, due to its computational efficiency, can be used in preliminary engineering calculations or for the verification of complex numerical models. It should be borne in mind that the analytical methods are desirable for the correct verification of the currently used numerical methods.  
The horizontal stiffness of the subsoil has a crucial impact on structural displacements, especially on flexible soils--greater stiffness limits movement, while lower stiffness increases it, potentially negatively affecting resistance and leading to damage. Incorrect estimation of this stiffness results in a less effective design in terms of safety, usability, and economics.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Acknowledgments:** Calculations were carried out using resources provided by the Wroclaw Centre for Networking and Supercomputing ([[https://www.wcss.pl/](https://www.wcss.pl/)]([https://www.wcss.pl/](https://www.wcss.pl/))).  
**Conflicts of Interest:** The author declares no conflicts of interest.