## 3 Analytical and Numerical Analysis of the Influence of Poisson's Ratio on the Obtained Results
The adopted contact method permits some relative motion of the contact surfaces after a certain threshold value is exceeded. Due to the nonlinear boundary conditions used in the model, such as contact and friction, the large displacement formulation was used in the static analysis. The self-weight of the soil layer was omitted in the analysis. The following material and geometric parameters were assumed for the calculations:  
* Soil: \(E_{\text{s}}\) = 100 MPa (Young's modulus), \(v\) = 0.1, 0.2, 0.3, 0.4, 0.5 [-] (Poisson's ratio),
* Concrete: \(h\) = 0.5 m, \(2a\times 2b\) = 1 \(\times\) 1, \(2\times\) 1, \(4\times\) 1, \(8\times\) 1 [m], \(E_{\text{cm}}\) = 31 GPa, \(v\) = 0.2 [-], \(\rho\) = 2400 kg/m\({}^{3}\). In order to numerically determine the spring coefficient \(\beta_{x}\), a horizontal (with respect to the X-axis) unit force was applied to the foundation footing model, which caused displacements (see Figure 7) in the soil and the foundation footing (the base) level. Based on the calculated displacements with respect to the X-axis and the unit force, the spring constant was calculated (see Equation (8)), and then the dimensionless spring coefficient \(\beta_{x}\) value (see Equation (9)) was obtained. Figure 6: Geometry and mesh of finite element model. Figure 7: Displacements in \(X\) direction for one of the analyzed cases. Table 3 shows the spring coefficient \(\beta_{x}\) with respect to \(\alpha=\frac{\theta}{b}\) and Poisson's ratio \(\
u\) for analytical and numerical calculations. Table 4 shows the percentage change in the spring coefficient \(\beta_{x}\). This change was calculated from the following relationship:  
\[\Delta\beta_{x}=\frac{\beta_{x}^{Num.}-\beta_{x}^{Anal.}}{\beta_{x}^{Num.}}[ \%]. \tag{11}\]  
Figure 8 shows the dependence between the spring constant \(K_{x}\) [MN/m] and the shear modulus of soil \(G\) [MPa] and the foundation area \(A\) [m\({}^{2}\)] for several different values of \(\alpha=\frac{\theta}{b}\) [-] as well as for different Poisson ratios \(\
u\) [-]. These solutions were obtained from the analysis of Equation (8). Figure 9 shows the dependence between the spring constant \(K_{x}\) [MN/m] for rectangular footings with respect to the foundation area \(A\) [m\({}^{2}\)] and \(\alpha=\frac{\theta}{b}\) [-] for different values of the shear modulus of soil \(G\) [MPa] and for different Poisson ratios \(\
u\) [-]. Figure 10 shows the dependence between the spring constant \(K_{x}\) [MN/m] for rectangular footings with respect to the shear modulus of soil \(G\) [MPa] and the Poisson ratios \(\
u\) [-] for different values of the foundation area \(A\) [m\({}^{2}\)] and \(\alpha=\frac{\theta}{b}\) [-]. Figure 11 presents contour nomograms showing the values of the spring constant \(K_{x}\) [MN/m] with respect to the shear modulus of soil \(G\) [MPa] and the foundation area \(A\) [m\({}^{2}\)] for several different values of \(\alpha=\frac{\theta}{b}\) [-] and for several different Poisson ratios \(\
u\) [-]. The values of the spring constant \(K_{x}\) [MN/m] can be easily read from these nomograms. The black continuous line shows the values of the spring constant \(K_{x}\) [MN/m], which was determined from Formula (8). The red dashed line shows the values of the spring constant determined for the dimensionless parameter \(\beta_{x}\) at a constant value of Poisson's ratio \(\
u=0.3\), i.e., for the parameter \(\beta_{x}\) given in, among others, papers [10; 11; 13; 14; 16]. Figure 12 presents contour nomograms that show the values of the spring constant \(K_{x}\) [MN/m] for rectangular footings with respect to foundation area \(A\) [m\({}^{2}\)] and value \(\alpha=\frac{\theta}{b}\) [-] for different values of the shear modulus of soil \(G\) [MPa] and Poisson's ratio \(\
u\).