Calculating axle strength

I need to determine the size and steel grade of axle needed to given a certain load capacity.

- The axle is supported by a pair of self-aligning bearings near the ends, with the wheel beyond that.
- Currently the axle is 25 dia (1") and the centre of the wheel is 50mm beyond the centre of the bearing.

Peak load will be when the laden trolley is placed on the ground/rails and given the usage I don't see fatigue failure as an issue, hence a fair assumption seems to consider only the static case, as a simply supported beam. In my previous company the design would be handed over to the structural team and likely under gone FEA analysis, however I am now in a much smaller outfit and that option is not open to us, so I am looking for some pointers on appropriate hand calculations and or assumptions.

Daniel