## 3 Results
### Stress Concentrations in Partial-Chamfer Hubs (0 \(<\beta<1\))  
A partial-chamfer hub can be defined by three parameters: the chamfer height \(l_{\rm r}\), the chamfer angle, \(\alpha\), and the hub thickness, \(t\). In this study, the influence of each one of them on the stress concentrations at the shaft-hub interface was analyzed. The variations of the SCFs \(K_{\rm r}\), \(K_{\rm t}\), and \(K_{\rm vM}\) with the dimensionless chamfer height \(\beta\) are represented in Figure 12 for diverse values of the chamfer angle with a fixed value of the hub thickness, i.e., \(\gamma=0.5\). Notice that the conventional hub was created either for a null chamfer angle, \(\alpha=0^{\circ}\) (for any value of \(\beta\)) or for a null chamfer height, \(\beta=0\) (for any value of \(\alpha\)). Taking this into account, the variation of the SCF \(K_{\rm r}\), \(K_{\rm t}\), and \(K_{\rm vM}\) with \(\beta\) clearly does not exist, it shown in the horizontal lines for the \(\alpha=0^{\circ}\) case in Figure 12a,c.e. In addition, the values of \(K_{\rm r}\), \(K_{\rm t}\), and \(K_{\rm vM}\) for \(\beta=0\) (null chamfer height, i.e., conventional hub) were the same for all the curves plotted in Figure 12a,c,e, i.e., for any value of the considered chamfer angle. Thus, according to the results, a nonlinear variation of \(K_{\rm r}\) with \(\beta\) was observed for different values of the hub chamfer angle. Thus, a high decrease of radial SCF was obtained for low values of the chamfer height, reaching a uniform horizontal distribution for intermediate and high values of \(\beta\). Therefore, the minimum reduction of the SCF was reached for full-chamfer hubs (\(\beta=1\)), but a similar reduction could be also obtained by using intermediate and high chamfer heights. For the sake of clarity, the transition chamfer height (\(\beta^{\star}\)) is defined as the value of \(\beta\) where the uniform distribution of SCF is reached. According to the results shown in Figure 12a, transition chamfer height was dependent on the chamfer angle. Thus, the value of \(\beta^{\star}\) progressively decreased as the chamfer angle increased. In this way, for high chamfer angles, the uniform distribution was reached for a chamfer height as low as 0.25, whereas for intermediate chamfer angles (\(10^{\circ}<\alpha<15^{\circ}\)) was \(\beta=0.5\). Figure 12b shows a 3D plot where the variations of radial SCF, \(K_{\rm r}\) (\(z\)-axis) with chamfer height, \(\beta\) (\(x\)-axis), and chamfer angle, \(\alpha\) (\(y\)-axis) can be observed. Thus, the curves shown in Figure 12a are in fact sections of the surface shown in the 3D plot for different planes corresponding to a given chamfer angle. Figure 12: Stress concentration factors as a function of chamfer height \(\beta\) for diverse values of chamfer angle \(\alpha\) in partial–chamfer hubs with a hub thickness of \(\gamma\) = 0.5. (**a**) radial stress, (**b**) 3D view, (**c**) tangential stress, (**d**) 3D view, (**e**) von Mises stress, (**f**) 3D view. The line in the plane \(\beta=0\) and the line in the plane \(\alpha=0^{\circ}\) represent the conventional hub; hence, no variation appeared with the other variables. The curve shown in the plane \(\beta=1\) (full-chamfer hub) is the two-linear variation previously outlined in Figure 10a with a sudden decrease of radial SCF as \(\alpha\) increased up to the transition chamfer angle and later followed by a soft decrease. From Figure 12b, the design zone is clearly visualized as the uniform plane zone within the range of the parameters \(\alpha>15^{\circ}\) and \(\beta>0.4\) where the values of \(K_{\rm r}\) were close to 1. Therefore, the stress concentration was reduced, and the design equations are valid for estimating the real stress state. Regarding the tangential SCF, as was discussed above, for \(\beta=0\) (conventional hub) the tangential SCF was the same for all the values of the chamfer angle, \(\alpha\).