Rigid elements often cause a stress concentration. That is why they should not be used in a region where accurate stress is needed. You can use deformable if it gives you a better result. 

I expect the analytical equation does not account for large deflection.

It looks like there is a face on the solid cylinder where no force is applied. This makes the resultant force move from the center of the solid length. Did you account for that in the analytical result?

What is the analytical equation?  

Where are you looking to evaluate the stress?

Is it at the plane where the solid meets the beam? It is recommended to transition from solid to beam at a different location from where you want to evaluate stress.

Is the beam cross-section the same as the solid?

Why are you switching from beam to solid elements?

What is the result if you just use beam elements?