I've resolved problems with shaft failures from both bending and torsional stresses using either or both approaches. In the OP"s case there is a tremendous bending moment imposed on shaft by the roll movement and one hell of a torsional load if a bearing seized. The formal analysis has always been a point of contention among engineers involved as it is never as straight forward as it seems. I was involved with this type failure long before the advent of computers and FEA so my approach could be called Edisonion though in reality was culmination of a lot of rudimentary analysis, understanding of the failure modes, application of some principles of mechanics and materials.
Sans any formal analysis for bending failures I tend to go with the elliptical radius for the reduction of stress concentration at any change in section. I like the change in diameter if as in the above case where there is potential for higher torsional loads on the hub flange. The change in diameter (step) can also be used to get weld area out of the highly highly stressed area. I believe that this approach has worked so well is because the original designers were never very far off. Some that worked I would really like to go back and do a formal analysis.
Anecdotal:
The Hub From Hell:
In the manufacture of a nonwoven fabric there are two sets of what are called Crown Rolls, a roll that is forced to operate with an induced bow. These rolls are approximately 14" in diameter and 12' long and are mounted 3 ft off the floor on stands. All these rolls have bolted on hub flanges, 2 piece welded 4140, and you can imagine there unknown forces from unknown directions acting on the hub and shaft during every revolution.
Under our operating conditions the original hub flanges sheared the mounting fasteners, hex head grade 5, We reworked the HF's to take SHCS, We went with our plant standard , H-11 at 220,000 ultimate TS. The manufacture of this products requires the use of HCL and very shortly we started have failures of the H-11, You would hear a bang and the bolt heads would come off like bullets and make a trip around the area. We rebolted with standard SHCS at around 160,000 ultimate. It took around 4 weeks and bang when enough went the roll went bouncing across the floor. We again reworked the flanges to used more socket heads capscrews where we drew the standard back to 120,000 UTS.
This worked then we started having preferential corrosion to the weld metal, so we clad the highly stressed area with Hastelloy C. imperfect but better. Then we started having fatigue failures on the shaft All the king's calculations were saying this shaft was more than adequate as the stresses were low. We upped the diameter as much as possible as we didn't want to raise the strength level due to H2 attack from government. This was causing so much concern that a select committee from corporate engineering and two consultants were brought to our site. The conclusion after a week that the existing hub was more than satisfactory. Before they left I presented them with a newly broken hub shaft. Broken shafts and bearing failures, another story, were about to put us out of the nonwoven business. I convinced management to go with a test of a 3 step Hastelloy C shaft welded to a many bolt 4140 flange. This worked extremely well and despite the astronomical cost they went across the house.
When everything was under o semblance of control a young engineering, who I will put in the class of one of the sharpest I've worked with started work with our group. During a period while we were a little slack I gave him the hub shaft problem. His first iteration was that everything looked fine and there should be no problem. He didn't present his work at the time and said he wanted to work some more on it. All of a sudden he was ready to leave the company, another long story, and on his last day he showed me a stack of computation papers with all type of force diagrams and stated he knew why the shafts were mechanically failing, discounting the H2 exposure. I never got to see the answer.
