Steel shaft broke at a shoulder although the stress was far below yield: is this fatigue?

That is a fatigue failure, and the static safety factor says nothing about it. The belt pull has a fixed direction while the shaft rotates, so each point on the surface goes from tension to compression and back once per revolution. At 1450 rpm that is 87,000 stress reversals per hour, and three months at eight hours a day is around 60 million cycles.

The fracture surface confirms it. The smooth area with curved lines (beach marks) is the crack growing slowly across the section over weeks; the small rough patch is the last piece breaking suddenly when it could no longer carry the load. A small final zone means the nominal stress was low, consistent with your 90 MPa.

Two things the calculation left out explain the numbers. First, the sharp corner is a stress concentration. For a shoulder with a very small fillet radius the factor is around 3, so the local stress at the corner is nearer 270 MPa than 90 MPa.

Second, the limit to compare with is the endurance limit, not yield. For steel, a polished test specimen in rotating bending survives indefinitely at roughly half the tensile strength, 300 MPa here. A real machined shaft of this size does worse: typical correction factors of about 0.8 for surface finish and 0.9 for size bring it to roughly 215 MPa. With 270 MPa at the corner against 215 MPa, a finite life is the expected result.

The most effective fix is geometry. Give the shoulder the largest fillet radius the bearing allows. With a radius of about a tenth of the smaller diameter, 2 mm here, the stress concentration factor in bending falls to roughly 1.6 or 1.7, for a local stress of about 150 MPa, comfortably below the estimated endurance limit.

If the bearing's corner chamfer will not clear a 2 mm radius, use a spacer ring between bearing and shoulder, or an undercut relief groove with a generous radius. Finish the fillet smoothly, since tool marks running around the shaft are small notches themselves, and do not let a keyway run out into the shoulder.

Then reduce the load, which is often higher than assumed. The bending moment at the step is the belt force times the overhang, so moving the pulley as close to the bearing as possible helps in direct proportion: half the overhang, half the stress. Check the belt tension as well. Belts are very commonly over-tightened, and the shaft load from the two belt strands can be several times the force that transmits the torque.

Changing to a stronger steel is less effective than it sounds. The endurance limit rises with tensile strength, but stronger steels are more notch sensitive, so with a sharp corner much of the gain is lost. Surface treatments that leave compressive residual stress, such as shot peening or rolling the fillet, give a real improvement.

The fracture face can tell you more if you look closely before it rusts. Find where the beach marks radiate from: that is the origin, and it will be at the surface in the corner of the step. A single origin suggests modest stress concentration; many small origins around the circumference, separated by little radial steps called ratchet marks, indicate a severe one, which is what a near-sharp corner produces.

Photograph both halves in good side lighting and keep them dry. Comparing the origin position with the keyway, the fillet and any corrosion or fretting marks from the bearing fit often shows that more than one stress raiser was involved, and the replacement design should deal with each of them.