## 3 Results
### Stress Concentrations in Full-Chamfer Hubs (\(\beta=1\))
In addition, the curves included in the sections of the surface for \(\alpha=0^{\circ}\) reveal the influence of the hub thickness \(\gamma\) in conventional hubs (curve A) and, in a similar way, the section for the highest chamfer angle considered (\(\alpha=30^{\circ}\)) reveals the variation of the previously discussed loss of \(K_{\rm r}\) obtained for high chamfer angles (curve B). Figure 10: Stress concentration factors as a function of the chamfer angle, \(\alpha\), for different values of the hub edge \(\gamma\). (**a**) radial stress, (**b**) 3D view, (**c**) tangential stress, (**d**) 3D view, (**e**) von Mises stress, (**f**) 3D view. Furthermore, the non-linear variation of the transition chamfer angle, \(\alpha^{\star}\), with the chamfer height \(\gamma\) was also observed (curve C). Regarding curve A, as the hub thickness \(\gamma\) increased, the radial SCF increased too, reaching a maximum value for \(\gamma\) = 0.5, and later \(K_{\rm r}\) softly decreased up to \(\gamma\) = 1. On the other hand, curve B showed a progressive decrease of the radial SCF as \(\gamma\) increased from \(K_{\rm r}\) =1 for the low chamfer thickness (\(\gamma\) < 0.375) up to \(K_{\rm r}\) = 0.82 obtained for the high hub thickness (\(\gamma\) = 1). Finally, the transition chamfer angle (curve C) had a non-linear variation with \(\gamma\). To go deeper in the analysis, taking into account the relevance of the variations of the transition chamfer angle for the design of chamfer hubs, the variation of such a parameter with the hub thickness is plotted in Figure 11. Thus, for the low hub thickness (\(\gamma\) < 0.5), the transition chamfer angle increased from \(13^{\circ}\) to \(15^{\circ}\). For the medium hub thickness (0.5 < \(\gamma\) < 0.75), \(\alpha^{\star}\) does not depend on \(\gamma\) and, finally, \(\alpha^{\star}\) decreased for the high hub thickness (\(\gamma\) = 1). This result is quite interesting since, by using the curve shown in Figure 11, the optimal chamfer angle can be selected for a given hub thickness. In this way, the optimal chamfer angle is the same, \(\alpha\) = \(15^{\circ}\), for the most common values of the hub thickness, 0.5 < \(\gamma\) < 0.75, making the design of chamfer hubs much easier. The value of tangential SCF for conventional hubs (\(\alpha\) = 0\({}^{\circ}\)) was almost the same for all hub thicknesses considered and it was slightly higher than 1, (\(K_{\rm t}\) = 1.07). The variation of the tangential SCF, \(K_{\rm t}\), linearly increased with the hub chamfer angle as shown in Figure 10c for all the hub thicknesses analyzed. However, the stress concentration level was lower than the one observed in the radial stress. In addition, the slopes of the linear variation of tangential SCF were dependent on hub thickness. Therefore, as the hub thickness increased, the slope of the linear variation also increased. Consequently, the highest values (\(K_{\rm t}\) = 1.46) were reached for high hub thicknesses and high chamfer angles. Figure 10d shows the 3D plot of variation of \(K_{\rm t}\) (\(z\)-axis) with the chamfer angle, \(\alpha\) (\(x\)-axis) and hub thickness, \(\gamma\) (\(y\)-axis). In this plot, the lines shown in Figure 10c are the cuts of the surface for a given hub thickness. The 3D plot (Figure 10d) reveals, on the one hand, the negligible variation of \(K_{\rm t}\) with \(\gamma\) for conventional hubs (plane \(\alpha\) = 0\({}^{\circ}\)) with an almost straight horizontal line (curve A) and, on the other hand, the non-linearly increasing trend of tangential SCF with the hub thickness for the highest chamfer angle considered (curve B). Therefore, according to these results, the use of high hub thicknesses and high chamfer angles is not recommended since the SCF increased up to 1.46 or, in other words, it increased by 36%. Finally, regarding the von Mises SCF, a progressive reduction of such variable was observed as the chamfer angle increased up to a value close to 1 (\(K_{\rm vM}\) = 1.05) for high chamfer angles (\(\alpha\) > 20\({}^{\circ}\)).