Large deformations of thin curved plane beam of constant initial curvature

A theory for the large deformations of a thin curved plane beam of constant initial curvature is presented, based on the hypothesis that only the longitudinal component of the strain tensor exists in the beam. In this case, five of the six compatibility equations are identically satisfied, while the remaining one requires the axial strain to vary parabolically over the cross-section of the curved beam. The non-linear strain-displacement relations are then solved for two in-plane displacement components in terms of two arbitrary functions of the longitudinal co-ordinate, angle ϑ. It is shown that, as a consequence of our hypothesis, the displacements vary linearly over the cross-sections of the beam. The obtained functional form of the displacement components leads to equilibrium equations of a beam on a deformed configuration expressed in terms of deformation functions. For the special case of a linearly elastic material and circular cantilever beam subjected to conservative and non-conservative point loads these equations have been numerically integrated. A number of numerical examples, including the bending of a C-shaped spring, are presented.