# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 4 Discussion
Figure 18a illustrates the partial uncentering (differential effect) for the three scenarios and shows how saturation (\(y_{lim}\)) is reached at a lower radius threshold for reduced-diameter wheels, while Figure 18b shows the total uncentering (adding bogie rotation) in the worst case (leading wheelset, outer wheel), but even in this case, flange-rail contact is less aggressive owing to dynamics:
* The results for the three scenarios were obtained for a 1235-MPa normal pressure at the tread contact area with the rail when the wheels run on straight tracks, attaining such a value by adjusting the axle load for each scenario. Pressures existing at the flanges were not equated due to the variability of the force exerted by the rail on the flange on the curve radius, which would make it very difficult to obtain unique axle load values. * It is necessary to limit axle load on reduced-diameter wheels, as their contact area with the rail is reduced as well, and the normal pressure is proportional to the load-area ratio. This reduction in the contact area responds to the decrease in the longitudinal radius \(R_{y2}\), which is proportional to the wheel diameter. With a lower \(R_{y2}\) value, a greater longitudinal relative curvature (\(A\)) is obtained, which diminishes the intersection between the theoretical paraboloids and, as a result, the contact patch size (as \(A\) diminishes, the longitudinal semi-axis (\(a\)) does as well). * Flange wear is between 10 and 1000 times more intense than tread wear, so the former was taken for elaborating the curves. This is due to the fact that lateral radii are very reduced for flange-rail contact (\(R_{x_{1}}=13-36\) mm, \(R_{x_{2}}=36\) mm), opposing tread-rail contact radii (\(R_{x_{1}}=80-300\) mm, \(R_{x_{2}}\rightarrow\infty\)), increasing in turn the relative lateral curvature (\(B\)), which diminishes the intersection between the theoretical paraboloids and, as a result, the contact patch size (as \(B\) diminishes, the longitudinal semi-axis (\(b\)) does as well). This size is smaller than that of the tread contact patch. * Wheel diameter is more influential on tread wear than on flange wear. This owes to the fact that the radius \(R_{y_{2}}\) (proportional to wheel diameter) is dominating, along with the radius \(R_{x_{1}}\) (which is in the same order of magnitude), at the tread (\(R_{y_{2}}\sim R_{x,1}\) and \(R_{y_{1}},R_{x_{2}}\rightarrow\infty\)). In contrast, at the flange, \(R_{y_{2}}\) is not the dominating radius, being dominated by \(R_{x_{1}}\) and \(R_{x_{2}}\) values, which are in a lower order of magnitude (\(R_{x_{1}}\sim R_{x,2}<R_{y_{2}}\) and \(R_{y_{1}}\rightarrow\infty\)), and \(R_{x_{1}}\) and \(R_{x_{2}}\) hold constant independently of the wheel diameter. * RCF is predicted for every flange-rail contact (except for isolated cases where the 355-mm wheel is negotiating a curve with a radius closely below 1850 m, this value being the threshold radius in this case) as a consequence of the high normal pressure (5-7 GPa) at the flange contact area with the rail. Although the contact patch size is smaller than that of the tread contact patch, such a high pressure is withstandable by the material since indentation is elevated (0.2-0.3 mm) and pressure can stack in many layers (isobaric surfaces), as in hydrostatics. * RCF effects can be mitigated by setting a reduced wear depth limit and in the current work, it was due to the hypothesis established. In the real operation, it is the economical factor the one prioritized, which forces to find the trade-off between the crack growth and wear depth limit. * As a consequence of RCF and the fatigue induced during reprofiling (which leaves residual stresses) and also for operational safety reasons, operators' internal regulations forbid eliminating more than 80 mm in diameter for a 920-mm wheel, more than 60 for a 680-mm one, and more than 20 for a 355-mm one. * Last, Table A4 (Appendix C) gathers the RCF and wear results for the three different wheels when negotiating the tightest curve, the one with the 265-m radius.