# Static and Dynamic Analysis of Non-Pneumatic Tires Based on Experimental and Numerical Methods
## 3 Static Analysis Using FE Method
### Results and Model Validation  
In order to validate the methodology, the results of the FEA were compared with the experimental ones. The comparison was carried out on two different planes: a qualitative comparison that involves the deformation shapes of the experimental and FE model, and a quantitative comparison that includes the following: the Force and Radial Stiffness vs. Displacement diagrams that globally analyze the mechanical behavior of the tire; the DIC results that locally study the deformation of the spokes; the contact patch comparison that focuses on the contact phenomenon of the tread elements in terms of pressure. The qualitative comparison between simulation and experimental results is shown in Figure 9, where the deformed shape of the FE model is overlapped with the real deformed shape, obtained with a 3500 N load with 0\({}^{\circ}\) camber. In order to quantitative validate the model behavior, in Figure 10a, the Force vs. Displacement diagram of the FEA is shown, which is in good agreement with the experimental one. For a wide range of displacements, the reaction force is quite the same for both with a diverging behavior only present for the smallest and largest displacements. At the smallest displacements, the divergences can be explained by the errors that affect the measurement instruments when they acquire the displacement signals at low loads whilst, for the largest displacements, the differences can be explained by the highly nonlinear behavior of the material that constitutes the tire and does not allow to match the exact force trend for the entire range of deformation. For what concerns the Radial Stiffness vs. Displacement diagrams are shown in Figure 10b; further, in this case, the FE model is in good agreement, replicating the softening spring characteristic behavior of a N  
Figure 9: Comparison of the deformed shapes. radial stiffness of the tire decreases with an increasing displacement applied to center of the rim. The spokes deformations were compared with the ones obtained from the application of the DIC method. It was performed on three pairs of spokes but, in this paragraph, we present the results of only one to show the methodology applied. On each spoke, five reference points (I-V) were chosen to investigate the deformation in specific points. In Figure 11a, the deformed shape of the NPT is shown with the red markers representing the position of the points of interests after the deformation, whilst the green before, both on the tire and on a coordinate axis system placed on a plane normal to the optical axis of the camera sensor and with the origin corresponding to the undeformed position of the first marker. The same was done in the FE model, querying the displacement of the nodes that in the model represent the markers, as shown in Figure 11b. In Table 3, we report the values of the displacement of each marker, along the x and y direction of the coordinate axis system previously defined and shown in Figure 11a, obtained experimentally with the DIC tool and numerically with FEA and compared with the calculus of the percentage error, found as follows:  
Figure 10: (**a**) Force vs. Displacement comparison diagram. (**b**) Radial Stiffness vs. Displacement comparison diagram. \[\Delta y_{\textit{s}error}=\frac{\Delta y_{\textit{experimental}}-\Delta y_{ \textit{FEM}}}{\Delta y_{\textit{experimental}}}\times 100,\ \ \ \ \Delta x_{\textit{s}error}=\frac{\Delta x_{\textit{experimental}}-\Delta x_{ \textit{FEM}}}{\Delta x_{\textit{experimental}}}\times 100 \tag{1}\]  
The largest displacement in the X direction was experienced by the III point, since it is placed in the middle of the spokes where the bending is higher. Instead, the largest displacement in the Y direction was found in the first points, being directly under the load application.