buckling analysis for cylindrical ring

Dear Members Hello

I am conducting buckling analysis for cylindrical ring (Short and thick cylinder) with Lateral Pressure.

I reviewed the basic books ("Buckling of Bars, Plates, and Shells" and "Theory of Elastic Stability" – Timoshenko) and all I could find is buckling of rectangular planes in the axial direction or buckling of cylindrical shells which buckles in the radial direction.

Important note: Real testing presented buckling in the axial direction (as plane buckling), load is continuous (the ring is inserted into tube using axial force).

I was hoping to find the solution without FEM programs using acceptable assumptions, approximated solution is also accepted since safety factor of about 2 will be taken.

 thanx for your help