## 1 Introduction
In this paper, the plastic state that could appear in the soil in the case of very large loads is omitted--the soil, like any other material, then begins to \"flow\". It is assumed that when calculating most structures and foundations on elastic soil, relatively small loads occur on the soil. Therefore, in simpler analyses, the linear-elastic behavior of the soil is acceptable [12]. [PERSON], [PERSON], and [PERSON] [10]; [PERSON] and [PERSON] [13]; [PERSON] and [PERSON] [11]; and [PERSON] [14], based on assumptions for the elastic half-space, gave simplified relations for the spring constants of soil. These are, respectively:  
* The soil spring constant \(K_{x}\), which is related to the horizontal movement (see Figure 1a) \[K_{x}=2G(1+\
u)\sqrt{A}\cdot\beta_{x},\] (1)
* The soil spring constant \(K_{z}\), which is related to the vertical movement (see Figure 1b) \[K_{z}=\frac{G}{1-\
u}\cdot\beta_{z}\sqrt{A},\] (2)
* The soil spring constant \(K_{\phi}\), which is related to the rotational movement (see Figure 1c) \[K_{\phi}=\frac{G}{1-\
u}\cdot\beta_{\phi}\cdot 8 ab^{2},\] (3)where  
\(G\) [Pa]--the shear modulus of soil; \(\
u\) [-]--the Poisson's ratio of soil; \(A\) [m\({}^{2}\)]--the foundation area (\(A=2a\cdot 2b\)); \(2a\) [m]--the length of the foundation (along the axis of rotation in the case of rocking); \(2b\) [m]--the width of the foundation (in the plane of rotation in the case of rocking); and \(\beta_{x},\ \beta_{z},\ \beta_{\phi}\) [-]--the spring coefficients. It should be mentioned that when describing the values characterizing the discussed models (\(K_{x}\), \(K_{z}\), and \(K_{\phi}\)), the authors cited above call them \"constants\", which should be considered imprecise in view of the current knowledge on the nonlinear behavior of the soil medium. However, in order to maintain consistency with the cited literature, this nomenclature was also adopted in this work. The slab geometry designations are shown in Figure 2. The spring coefficients \(\beta_{x}\), \(\beta_{z}\), and \(\beta_{\phi}\) for a rectangular foundation can be determined from the nomogram in Figure 3.  
[PERSON], [PERSON], and [PERSON] [10]; [PERSON] and [PERSON] [13]; [PERSON] and [PERSON] [11]; and [PERSON] [14] reported that the spring constant \(K_{x}\), related to horizontal movement, and the spring constant \(K_{z}\), related to vertical movement, were derived based on the assumptions given in [PERSON]'s paper [9]. In turn, the spring constant \(K_{\phi}\), which corresponds to rotational movement, was derived based on the assumptions of [PERSON]'s papers [12; 15]. It should be emphasized that a spring constant represents a linear relationship between the applied load and the foundation displacement, which in turn implies a linear stress-strain relationship of the soil. The formulas for the spring constants were obtained for rigid footings [10]. These formulas are still used to model the contact between the structure and the subsoil and can be found in the source literature [16; 17; 18; 19]. Figure 1: Spring constants associated with movement: (**a**) horizontal, (**b**) vertical, and (**c**) rotational. Figure 3: Spring coefficients \(\beta_{z},\ \beta_{x}\), and \(\beta_{\phi}\) [-] for rectangular footings with respect to \(\alpha=\frac{a}{b}\) [-]. Figure 2: Designations of rectangular footing geometry. In this paper, based on the assumptions given by [PERSON] [9], the formula for the spring constant \(K_{x}\) and the spring coefficient \(\beta_{x}\) were derived. It was shown that [PERSON]'s ratio \(\
u\) not only occurs in the term of the equation for the spring constant \(K_{x}\) before spring coefficient \(\beta_{x}\), but also in the spring coefficient. It was also indicated that the graph showing the dependence between the spring coefficient \(\beta_{x}\) and \(\alpha=\frac{\alpha}{\beta}\), i.e., the ratio of the length and width of a rectangular footing, which is given, among others, in papers [10; 11; 13; 14; 16] and was calculated for a constant value of [PERSON]'s ratio \(\
u=0.3\).