## 3 Results and Discussion
### Case-Study: Simulation of the Test of a Farm Tractor
#### 3.7.3 Quasi-Static Test (Considering Also the Tyres Deflection)  
As previously explained, starting from the results of the quasi-static test, illustrated in Section 3.7.2, it is now possible to recalculate the locus of the positions of \(CoG^{\prime}\) and, hence, the graphs of the distribution of the weight on the four wheels of the tractor so as to keep into account also the deflection of the tyre. The main characteristics of the tyres equipping the tractor, reported in the bulletin [49] using a technical information format compliant with ETRTO standard [51], are shown in Table 6.  
To proceed, it is necessary to know the perpendicular and lateral stiffness of the tyres (quantified respectively by the perpendicular and lateral elastic coefficients), e.g., by performing some experiments with proper equipment or finding the values of the elastic coefficients in the literature. It is worth noting that the inflation pressure has a very high influence on the tyre stiffness, and, hence, on the deflection of a tyre [52], so a pressure level has to be stated. In this case, since it would be interesting to have a greater knowledge of the risks that occur when the tractor is used in a working environment and in the worst possible condition, i.e., on a sloping hillside and with the tyres at the minimum possible inflating pressure, it is assumed the execution of a quasi-static test with the tyres inflated at a pressure of 1.0 bar (100 kPa). This value is equal to the lower limit of the pressure range that is normally used in field operations, with the aim of increasing the footprint area and limiting the soil compaction. At such low pressures, the tyre stiffness is very low and the risk of lateral rollover is high, so it is surely a case to investigate. For example, with the indicated pressure of 1.0 bar, a 14.9-30 tyre, i.e., a wheel with a rim diameter that is intermediate between the two diameters of the considered tractor, has a value of 230 kN-m\({}^{-1}\) (i.e., 230 N\(\cdot\)mm\({}^{-1}\)) for the tyre elastic coefficient along the vertical direction [53]. To have two values more suitable for the agricultural tyres referred to in the present case, it is possible to interpolate the experimental load-deflection curves reported in [54] for the 420/85 R24 and 460/85 R34 tyres at the inflating pressures of 1.0 bar. By doing so it is possible to obtain the following coefficients: \(3.87\times 10^{-3}\) mm\(\cdot\)N\({}^{-1}\) (equivalent to an elastic coefficient of 258 N\(\cdot\)mm\({}^{-1}\)) for the 420/85 R24 tyre (very similar to the 14.9-24 tyre), and \(3.58\times 10^{-3}\) mm\(\cdot\)N\({}^{-1}\) (equivalent to an elastic coefficient of 279 N\(\cdot\)mm\({}^{-1}\)) for the 460/85 R34 tyre (very similar to the 18.4-34 tyre). The ratio between the lateral stiffness and the perpendicular stiffness of a tyre has a value highly variable with the characteristics of the tyre itself (specifically, of the carcass) and with the geometry of the tyre section (in particular, the sidewalls height and the rim diameter). Observing the values of that ratio reported \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\begin{tabular}{c} **Tyre** \\ **Denomination** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Rim** \\ **Diameter** \\ **(in.)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Load** \\ **Index (-)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Overall** \\ **Diameter** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Load** \\ **Capacity** \\ **(kgg)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Inlation** \\ **Press (bar)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Speed** \\ **Index (-)** \\ \end{tabular} } \\ \hline
14.9–24 & 24 & 126 & 1245 & 1700 & 1.6 & 600 \\
18.4–34 & 34 & 144 & 1645 & 2800 & 1.6 & 775 \\ \hline \end{tabular}
\end{table} 
Table 6: Characteristics of the tyres equipping the “New Holland TL 90 4 WD” farm tractor.