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.
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