The increased load is dues to stress concentrations, which are the result of the way your part is constrained. Whenever there is an absolute "fixed" constraint, such things will arise. Hand calcs don't account for stress concentrations.
Stress should go up as element size decreases. If elements are too large, stress errs to the low side. Reduce element size in regions with high stress gradients until stress stops increasing drastically.
thanks guys. So you're saying I should constrain it differently, adjust the mesh or just accept that I'll be getting errors around the restrained face? Also, that still doesn't answer #3.... why is the length of the rod influence the stress? Should be the same calculations with an increase in equations, right? or am I way off?
How do your hand calcs match up in the middle of the rod? I expect exact.
Why is the fixed base showing high stresses?
Because it is fully fixed. Poisson's effect causes a 3D state of stress to be created as the tensile elongation along the long axis causes a contraction along perpendicular axes. Because it is fully fixed on the end, i.e. there is no compliance in a fixed restraint, as you decrease mesh size the stresses at the constraints will continue to rise because you have an impossible situation.
In other words you have welded the end of the rod to an infinitely stiff base.
Correct constraints on the base would be sliding on the face, two corner nodes in one direction in the end face plane and then one corner node perpendicular to that.
And just to add to Pauls answer, I attach a picture so you can see the result with his suggested constraints. (I understood that you have the Xpress version that only supports the "fixed geometry" constraint). Result is 2000 psi regardless of lengths as expected.
Restraints are tricky and I believe that Solidworks should have more focus on this matter in the simulation course. My only rule of thumb is to try to avoid the fixed geometry as long as possible.
And just for completeness, setting the stress plot to 0-2000psi would show a solid color. The blotchiness is due to rounding at the nodes from the exact answer at the Gauss points.
The alternative for Xpress is to learn how to apply St. Vennant's Principle.