## 3 Results
### Stress Concentrations in Full-Chamfer Hubs (\(\beta=1\))
For higher chamfer angle values, \(\alpha>15^{\circ}\), the radial stress at the hub edge was lower than the values of such a stress at the inner points of the hub (where the uniform distribution appeared, \(\zeta=0\)) and, consequently, the maximum value of the radial stress distribution \(\sigma_{\rm r}\) is similar to the one given by theoretical equations for these cases. As the chamfer angle increased, a lower radial stress value appeared at the hub's center plane. This can be attributed to a stress redistribution caused by the reduction at the hub surface. So, for high chamfer angles, a loss of the contact pressure was produced because of, on the one hand, the reduction of radial stress at the hub edge surroundings (the  
Figure 7: Hub axial distribution of radial stress at the shaft–hub interface. most significant one) and, on the other hand, the soft reduction of radial stress observed at the hub center (\(\zeta=0\)). However, the axial distribution of tangential stress for a conventional hub (\(\alpha=0^{\circ}\)), \(\sigma_{\rm t}\) (blank circles in Figure 8), progressively increased from the hub's central plane (\(\zeta=0\)) and reached a maximum value nearby the hub edge (\(\zeta=1\)). Later, it decreased until reaching the minimum value at the hub edge. As in the case of the radial stress, the tangential stress at the hub center (\(\zeta=0\)) is similar to the theoretical value given by Equation (3). The effect of the chamfer angle on the tangential stress was the opposite of the effect observed in the radial stress. Thus, the values of the tangential stress at the hub center softly increased as the chamfer angle increased. In addition, the values of the tangential stress at the hub edge (\(\zeta=1\)) progressively increased with the chamfer angle from a minimum value obtained for the conventional hub case, \(\sigma_{\rm t}=149.6\) MPa (without chamfer, \(\alpha=0^{\circ}\)), up to the highest value, \(\sigma_{\rm t,max}=206.6\) MPa, obtained when the chamfer angle was the highest one considered (\(\alpha=30^{\circ}\)). It should be highlighted that for low chamfer angles (\(\alpha<15^{\circ}\)) the maximum stress appeared nearby the hub edge, whereas for high chamfer angles (\(\alpha>15^{\circ}\)) the maximum tangential stress appeared at the hub edge. With regard to the distribution of the von Mises stress for a conventional hub (blank circles in Figure 9), the stress softly increased in the zone out of the stress concentration (\(0<\zeta<0.8\)) and close to the hub edge, the von Mises stress heavily increased, reaching a maximum value at the hub edge. As in the previous distributions, the effect of the chamfer vanished as the distance from the hub edge increased, reaching similar values to the ones predicted by theory for all  
Figure 8: Hub axial distribution of the tangential stress at the shaft–hub interface. Figure 9: Hub axial distribution of von Mises stress at the shaft–hub interface. chamfer angles considered, \(\sigma_{\rm vM}=215\) MPa (Equation (4)). According to the results, as the chamfer angle \(\alpha\) increased, a notable reduction on the stress concentration at the extremity of the hub was achieved (from a peak of 281.7 MPa corresponding to the conventional hub case up to 227.1 MPa for a chamfered hub with the highest chamfer angle \(\alpha=30^{\circ}\)). As a consequence, the von Mises stress was more uniformly distributed throughout the hub-shaft interface. All the effects previously discussed can be better analyzed in terms of the variation of the stress concentration factors (SCFs, \(K_{\rm j}\)) with the chamfer angle \(\alpha\) for diverse chamfer heights \(\gamma\) (Figure 10).