Engineering books and lectures typically make two things clear:
Maximum bending stresses (normal) develop at the beam surfaces, that is, at the greatest distance from the neutral axis.
Shear stresses on the other hand have their maximum at the neutral axis.
Recently we were given a beam with a simple loading condition:
Here you could find the shear stresses at the surface using the flexure formula, and the shear formula for the centroid axis.
This would yield 4.x in as the required diameter with the bending stresses as the limiting factor, which was also the most common answer among students.
However, if you plot the bending stresses at the surface in Mohr circle and rotate the stress element 45 degrees, you're left with a shearing stress at the surface that requires 12.88 in diameter in order to not exceed the requirements.
This seems the most logical to me, and I don't see any reason to differentiate between the shear stresses developed at the centroidal axis and the surface shear due to the rotated stress element.
Am I correct in this assumption and calculation? If / if not, why?