## 3 Results
### Stress Concentrations in Full-Chamfer Hubs (\(\beta=1\))  
Numerical simulations reveal the stress fields at the hub and shaft in colored maps. From these plots, it is easy to visualize the stress distributions at the hub-shaft interface and the effect of a chamfer hub on the stress field from a qualitative point of view. Figure 6a,b show the von Mises stress fields of a conventional hub and a full-chamfer hub with a chamfer angle of \(\alpha=15^{\circ}\). In these plots, a huge stress concentration was observed at the hub edge of the conventional hubs (right side of the hub, yellow zone, point A in Figure 6a). In addition, a  
Figure 5: Mesh used in FEM simulations: general view showing the applied boundary conditions and the detailed view at the hub edge where stress concentration appears at the contact of the hub–shaft interface. Figure 6: Hub von Mises stress fields obtained by FEM simulations for (**a**) a conventional hub; (**b**) a full–chamfer hub with a chamfer angle of \(\alpha=15^{\circ}\). highly localized effect of stress concentration was also revealed since the zone where the stress concentration appeared at the shaft-hub interface was only noticeable nearby the hub edge (point A), whereas the stress field was uniformly distributed far from the hub edge. However, for full-chamfer hubs, the high-stress zone (yellow zone) disappeared and, consequently, the stress concentration of the conventional hub significantly decreased. As result a more uniform distribution of stress in the axial direction at the interface was obtained. To go deeper in the analysis, the axial distributions of the radial stress (\(\sigma_{\rm r}\)), tangential stress (\(\sigma_{\rm t}\)), and von Mises stress (\(\sigma_{\rm vM}\)) at the shaft-hub interface are plotted in Figures 7-9, respectively, for full-chamfer hubs (\(\beta=1\)), considering different values of the chamfer angle \(\alpha\) within the range from \(\alpha=0^{\circ}\) (conventional hub) to \(\alpha=30^{\circ}\). In these plots the distributions are represented in terms of the dimensionless distance \(\zeta\), defined as \(\zeta=2\,z/l\), where \(\zeta=0\) represents the half cross-section of the hub (\(z=0\), point B in Figure 3) and \(\zeta=1\) represents the hub edge (\(z=1/2\), point A in Figure 3). For the conventional hub (\(\alpha=0^{\circ}\), blank circles in Figure 7), the radial stress, \(\sigma_{\rm r}\), distributions exhibited two clearly defined zones. In the first one (out of the stress concentration), within the range \(0<\zeta<0.8\), the stress was uniformly distributed, reaching a value similar to the one given by theory (Equation (2)). The second zone, placed nearby the hub edge (\(\zeta>0.8\)), showed a high increment of stress, reaching a maximum value at the hub edge (\(\zeta=1\)). Thus, on the one hand, the maximum value at this zone revealed the intensity of the stress concentration and, on the other hand, the local effect of the stress concentration was also estimated as just 20% of the half hub contact length \(l\). For the first zone (\(0<\zeta<0.8\)), the radial stress distributions were quite similar to the theoretically predicted ones for all the considered chamfer angles. However, at the vicinity of the hub edge (\(\zeta>0.8\)) where a high stress concentration appeared, the effect of the chamfer geometry on the stress distribution was highly significant. According to the results, the stress concentration progressively decreased as the hub chamfer angle increased, even for low chamfer angles. Thus, the stress distribution became progressively more uniform, becoming almost plain (i.e., without stress concentrations) for medium of chamfer angle values, \(\alpha=15^{\circ}\).