## 3 Analytical and Numerical Analysis of the Influence of Poisson's Ratio on the Obtained Results  
Based on Equation (9), it can be seen that the dimensionless spring coefficient \(\beta_{x}\) not only depends on \(\alpha=\frac{\pi}{b}\), but also on [PERSON]'s ratio \(v\). Additionally, it should be noted that in Formula (8), [PERSON]'s ratio \(v\) not only appears in the \(\beta_{x}\) coefficient, but also in the term of the equation before this coefficient. In turn, in the literature (among others, in papers [10; 11; 13; 14; 16]) the same form of the equation for spring constant \(K_{x}\) as in Equation (8) is given, but the nomogram for reading the \(\beta_{x}\) value is only given for one constant value, i.e., \(v=0.3\) (see Figures 3 and 4). Therefore, a reader wishing to calculate the \(K_{x}\) values based on the formula and nomogram presented in the above-mentioned literature may unknowingly obtain a certain error. This chapter presents the results of numerical analyses that take into account different values of [PERSON]'s ratio \(v\), and its influence on both the dimensionless spring coefficient \(\beta_{x}\) and the spring constant \(K_{x}\). Figure 4: Spring coefficient \(\beta_{x}\) [-] for [PERSON]’s ratio \(v=0.3\) [-] for rectangular footings with respect to \(\alpha=\frac{\pi}{b}\) [-]; — Formula (9), - - after [PERSON] and [PERSON], 1967 [13]. In Figure 5 is a nomogram for the dimensionless spring coefficient \(\beta_{x}\), plotted for different values of [PERSON]'s ratio \(\
u\), which can be used to more accurately determine the value of \(K_{x}\). Table 1 provides the \(\beta_{x}\) values for several exemplary values of the \(\alpha\) and \(\
u\) coefficients. Table 2 shows the percentage change in the spring coefficient \(\beta_{x}\) for different values of Poisson's ratio \(\
u\) in relation to the spring coefficient value \(\beta_{x}\) calculated at a constant value of \(\
u=0.3\). This change was calculated from the following relationship:  
\[\Delta\beta_{x}=\frac{\beta_{x}(\
u)-\beta_{x}(\
u=0.3)}{\beta_{x}(\
u=0.3)}[ \%]. \tag{10}\]  
For the purpose of numerically determining the spring coefficient \(\beta_{x}\), Finite Element Method (FEM) three-dimensional (3D) models of the concrete foundation footing and the soil (Figure 6) were built using the Abaqus FEA software v. 2017 [20]. The foundation footing and the soil layer beneath it are represented by general-purpose eight-node linear hexahedral elements of type C3D8. The interlayer interaction between the top surface of \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\
u\)} & \multicolumn{11}{c}{\(\alpha\)} \\ \cline{2-13}  & **0.5** & **0.8** & **1** & **1.5** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline
0.1 & 1.050 & 1.017 & 1.011 & 1.016 & 1.032 & 1.071 & 1.112 & 1.151 & 1.188 & 1.223 & 1.255 & 1.286 & 1.316 \\
0.2 & 1.026 & 0.987 & 0.978 & 0.978 & 0.989 & 1.022 & 1.057 & 1.092 & 1.125 & 1.156 & 1.186 & 1.214 & 1.241 \\
0.3 & 1.014 & 0.968 & 0.956 & 0.949 & 0.956 & 0.983 & 1.014 & 1.045 & 1.074 & 1.103 & 1.130 & 1.155 & 1.180 \\
0.4 & 1.013 & 0.959 & 0.943 & 0.930 & 0.933 & 0.953 & 0.979 & 1.006 & 1.033 & 1.059 & 1.083 & 1.107 & 1.129 \\
0.5 & 1.023 & 0.959 & 0.939 & 0.919 & 0.917 & 0.931 & 0.952 & 0.976 & 1.000 & 1.023 & 1.045 & 1.067 & 1.088 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spring coefficient \(\beta_{x}\) for rectangular footings with respect to \(\alpha=\frac{\alpha}{\beta}\) and Poisson’s ratio \(\
u\). Figure 5: Spring coefficient \(\beta_{x}\) [-] for rectangular footings with respect to \(\alpha=\frac{\alpha}{\beta}\) [-] for different Poisson ratios \(\
u\) [-]. the soil layer and the bottom surface of the foundation footing was modeled as a standard contact surface-to-surface with friction between the contact surfaces. Furthermore, normal behavior with \"hard\" contact pressure overclosure and tangential behavior with so-called \"rough\" friction were used.