## 3 Results and Discussion
### Predictions of the Tyre Footprint Dimensions
#### 3.2.2 Experimental Data Processing  
An ANOVA showed that any variation of Young's modulus (i.e., of the tyre elasticity/stiffness) has no statistically-significant effect on the estimated longitudinal dimension of the tyre footprint (_L axis_), obtained from the FEM simulations. The factors of influence are (Table 5): tyre inflation pressure (_numeric factor_), Poisson's ratio (_numeric factor_), and type of tyre (_categorical factor_). Then, the RSM was applied to the FEM data to develop a numerical model of the longitudinal axis as a function of the tyre type, the inflating pressure (in bar), known from experimentation, and Poisson's ratio, obtained from the simplified FE model at the end of the model-tuning process based on the experimental data. Starting from a quadratic model (in accordance with the analysis \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Sum of Squares** & **df** & **Mean Square** & **F Value** & **_p-_Value Prob \textgreater{} F** \\ \hline Model & 5.253E + 005 & 4 & 1.313E + 005 & 19,853.24 & \textless{}0.0001 \\ _A-Pressure_ & _219.74_ & 1 & _219.74_ & 33.22 & \textless{}0.0001 \\ _B-E_ & _3.12_ & 1 & _3.12_ & _0.47_ & _0.4986_ \\ _C-v_ & _701.25_ & 1 & _701.25_ & _106.00_ & \textless{}0.0001 \\ _D-Type_ & _21,813.54_ & 1 & _218,13.54_ & _3297.38_ & \textless{}0.0001 \\ Residual & 165.39 & 25 & 6.62 & & \\ Cor Total & 5.255E + 005 & 29 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: ANOVA on the values of the longitudinal dimension of the tyre footprint referred to the four possible factors of influence. Figure 10: Longitudinal and transversal dimensions of the FEM footprint axis of the 420/85 R24 tyre with different equivalent Young’s moduli (_E_; column on the left) and Poisson’s ratios (_v_; column on the right): (**a**) \(E\pm 5\%\), inflation pressure of 1.0 bar; (**b**) \(v\pm 5\%\), inflation pressure of 1.0 bar; and of the 460/85 R34 (_c_) \(E\pm 5\%\), inflation pressure of 1.0 bar; (**d**) \(v\pm 5\%\), inflation pressure of 1.0 bar. results), a multilinear model was obtained (through a backward elimination process). The polynomial model is represented graphically in Figure 11 and it is composed of two equations (\(p\): tyre inflation pressure in bar; \(\
u\): equivalent Poisson's ratio) which differ depending on the tyre type, as visible in the following Equation (2) (adjusted R\({}^{2}\) > 0.99):  
\[\left\{ \begin{array}{l}420/85\;\text{R24}:\textit{Axis}=+532.2-22.4\cdot p+354.1\cdot\
u\\ 460/85\;\text{R34}:\textit{Axis}=+796.4-22.4\cdot p+354.1\cdot\
u\end{array} \right. \tag{2}\]  
The effect of an increase in Poisson's ratio toward the value of 0.50 (perfectly incompressible solid) is to extend the longitudinal dimension of the tyre footprint. Vice versa, as the pressure increases, the longitudinal dimension of the footprint of both the tyres shortens, as expected. The type of tyre only has an influence on the intercept value (Figure 11). This is principally due to the fact that the longitudinal dimension of the tyre footprint is determined mainly by the radius index. The same approach used for the longitudinal dimension of the tyre footprint was also followed for the transversal dimension (\(T\)-axis), obtained from the FEM simulations. The ANOVA showed that any variation of the inflation pressure and of Young's modulus (i.e., on the tyre equivalent elasticity/stiffness) has no statistically significant effect on this dimension (Table 6). Both these parameters have the corresponding \(p\)-values above the threshold of 0.05. Therefore, the only factors of influence are Poisson's ratio (_numeric factor_) and type of tyre (_categorical factor_). Then, the RSM was applied to the FEM data to develop a numerical model of the transversal axis as a function of the tyre type and of Poisson's ratio (obtained from the simplified FE model, as a result of the iterative process of tuning). This model is a quadratic model and it is represented graphically