If this is for a stiffener ring for tanks under external pressure, you can reference "Theory and Design of Tubular Stell strucutres" authored by Troitsky and published by JF Lincoln Foundation. If this is what you are looking for I can make an attachment of the relevant section.
Roarks gives the following equation for buckling of a ring under external pressure:
p’ = 3EI/r^3
However, they’re only considering in-plane buckling, so I’m not sure if this helps you. Best of luck.