The non-linear theory of the pure bending of prismatic elastic solids

The problem of the bending of a prismatic elastic solid by finite torques under large deformation conditions is considered. Using the semi-inverse method, the initial three-dimensional boundary-value problem of the non-linear theory of elasticity is reduced to a two-dimensional non-linear boundary-value problem for a region in the form of the cross-section of the beam. Two formulations of the problem are given in the cross-section: in terms of the displacements and of the stresses. Stress functions are introduced and a variational formulation of the two-dimensional problem is obtained, based on the supplementary energy principle. An approximate solution of the problem of the strong bending of a beam of rectangular cross-section is found for a semi-linear material and for a Bartenev-Khazanovich material using the Ritz method. (C) 2000 Elsevier Science Ltd. All rights reserved.