I found two websites with thick cylinder stress calculators on them.
I suppressed the Thermal Load and changed all three materials to middle.
Both calculators showed that the radial, hoop and axial stress for a single material cylinder are correctly computed by ANSYS (no surprise).
Your mesh is coarse, so your results have a small error compared with the exact solution.
If you reduce the radial thickness of your elements, you will reduce the size of the error.

I don't have mathematica so I can't look at the equations for the 3-layer cylinder.
It is possible to write the equations for a 3-layer thick cylinder. I would use MathCAD. 
If you don't have that you could use Excel, but that is harder to find mistakes in coding.