Finite bending of non-slender beams and the limitations of the Elastica theory

Abstract

The problem of slender solids under finite bending has been addressed recently in Lanzoni and Tarantino (2018). In the present work, such a model is extended to short solids by improving the background formulation. In particular, the model is refined by imposing the vanishing of the axial force over the cross sections. The geometrical neutral loci, corresponding to unstretched and unstressed surfaces, are provided in closed form. Two approximations of the models are obtained linearising both the kinematics and the constitutive law. It is shown that the approximations of the model, corresponding to the Euler Elastica formulation, can lead to significant values of the axial stress resultants despite pure bending conditions. For a generic form of compressible energy function, a nonlinear moment–curvature relation accounting for both material and geometric nonlinearities is provided and then specialised for a Mooney–Rivlin material. The obtained results are compared with simulations of 3D finite element models finding good agreement. The normalisation of the moment–curvature relation provides the dimensionless bending moment as a function of the Eulerian slenderness of the solid. This dimensionless relation is shown to be valid for any aspect ratio of the bent solid and, in turn, it highlights the limitations of the Elastica arising in case of large deformations.