The flexural vibrations of a ring in its plane are studied by taking shear effect into account, and it is proposed to compute the deviation from the classical frequency formula, which is only valid for very small ratios of the thickness to the diameter. In a second chapter the possibility of extension is included, and in this procedure the frequencies of extensional vibrations are obtained as a by-product. It is seen that the shear effect plays a predominant role in the deviation from the classical formula, and is more important than curvature effects and the rotational inertia. Extensibility influence is found to be very small. An interesting comparison can be made with the similar influence of secondary factors on the flexural vibration's frequencies of a straight beam.
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F. Buckens (1950) studied this question.