Consistent Treatment of Extensional Deformations for the Bending of Arches, Curved Beams and Rings

Abstract

A consistent treatment of the extensional deformations of thin circular rings subjected to general distributed and concentrated loadings is presented. Coupled Euler equations and consistent boundary condition combinations in terms of the radial and tangential midsurface displacements are obtained for the dynamical problem using Hamilton’s principle. Discontinuity conditions corresponding to discrete application of generalized forces are also provided. For static analysis, the governing equations are transformed into two uncoupled sixth-order differential equations in the radial and tangential displacements, respectively, and the complete solutions are obtained for general loading. Closed-form solutions for displacements and stress resultants are developed for illustrative and comparative purposes relative to specific full ring, curved beam, and arch examples. The results confirm that extensional deformations can play a significant role in the bending of thin curved beams, and arches and that such structures are more readily and consistently analyzed by the present treatise than by Winkler curved-beam theory with its potential for inconsistent approximations toward thinness and resulting possible violation of static equilibrium by the stress resultants calculated from the stress-displacement state.