Abstract A finite element solution is developed for a circular ring or curved beam. A polynomial displacement function is used. The solution gives high accuracy for an element which subtends an angle as great as 90°. Both inextensional and extensional deformations are considered. The loads may be either concentrated or distributed. It is shown that failures which have occurred in earlier attempts to use polynomial displacement functions for curved elements are due not to the neglect of rigid body motions as frequently stated in the literature, but to the neglect of coupling between normal and tangential displacements.