Perturbation solutions are found for buckling of thin elastic circular rings under nonuniform pressures. The solution leads to the same results obtained before by numerical methods. Two cases of pressure distribution of the form p = p0(1 + ε cos 2mθ) and p = p0[1 + ε(cos 2θ + v cos 4θ)] are considered. The analysis yields equations to determine the ring curvature at any angle and pressure.

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