## 4 Discussion  
The values of the spring coefficient \(\beta_{x}\), which were calculated from the relationship derived in the paper (Formula (9)) for \(\
u=0.3\), are analogous to the values given in the literature, among others in papers [10,11,13,14,16], as can be seen in Figure 4. This proves that the relationships for this spring coefficient were derived correctly. Figure 5 and Tables 1 and 2 present the influence of different values of Poisson's ratio \(\
u\) on the \(\beta_{x}\) coefficient. When comparing the obtained values of the spring coefficient \(\beta_{x}\) with the values of the spring coefficient \(\beta_{x}\) calculated at a constant value of \(\
u=0.3\) (see Tables 1 and 2), it can be seen that the differences in the obtained results are greater for a Poisson's ratio \(\
u<0.3\). Moreover, it can be seen that this difference increases for an increasing value of the ratio of the sides of the rectangular footing \(\alpha=\frac{\delta}{b}\). It is the largest (of around 11.5%) for \(\alpha=10\). For a Poisson's ratio \(\
u>0.3\), the differences in the obtained results are smaller; however, for the ratio of the sides of the footing \(\alpha=\frac{\delta}{b}=10\), the difference is still large, i.e., about 7.8%. It can also be noticed that for the Poisson's ratio \(\
u<0.3\), the values of the spring coefficient \(\beta_{x}\) obtained from the relationship derived in this paper (Formula (9)) are higher than the values of the spring coefficient \(\beta_{x}\) given in the literature (in, among others, papers [10,11,13,14,16]), i.e., calculated at a constant value of \(\
u=0.3\) (compare Figure 5 and Table 1). In turn, for the Poisson's ratio \(\
u>0.3\), the values of the spring coefficient \(\beta_{x}\) were lower than the values of the spring coefficient \(\beta_{x}\) given in the above-mentioned literature (compare Figure 5 and Table 1). It was only for a Poisson's ratio close to 0.5 and the ratio of the footing approaching \(\alpha=\frac{\delta}{b}=0.5\) that the value of \(\beta_{x}\), obtained from the relationship derived in the paper (Formula (9)), was slightly (0.9%) higher than the one obtained from the calculations for a constant value of \(\
u=0.3\) (compare Figure 5 and Table 1). The tendency for the initial decrease and then increase in the dimensionless spring coefficients with the change in the \(\alpha\) coefficient, i.e., the ratio of the sides of a rectangular foundation, results from the complex interaction of several factors related to the distribution of displacements and stresses in the soil and the geometry of the foundation. Foundations closer to a square shape show larger displacements because the soil resistance is then the least effective. This occurs when the soil has difficulty generating lateral resistance across the entire width. Therefore, the value of the spring coefficient decreases. With the increase in the \(\alpha\) coefficient, more soil begins to counteract the movement (the soil offers strong resistance) and the displacement decreases, and thus the value of the spring coefficient increases. Based on the values of the elasticity coefficient \(\beta_{x}\) obtained from numerical calculations (see Tables 3 and 4), it can be observed that for small ratios \(\alpha=\frac{\delta}{b}\) and Poisson's ratio values in the range \(\
u=0.1\)-0.3, the agreement was obtained at the level of 3-10% with respect to the \(\beta_{x}\) values obtained from analytical formulas. However, for larger values of Poisson's ratio \(\
u=0.4\)-0.5, the agreement is at the level of 10-20%. With increasing ratio \(\frac{\delta}{b}\) and Poisson's ratio \(\
u=0.5\), the difference between the results obtained from numerical and analytical calculations increases to about 28%. Figures 8-10 show the three-dimensional dependence of the spring constant \(K_{x}\) for different values of 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 \(\alpha=\frac{\delta}{b}\)).