On bending and twisting of beams, part I: Theory

Abstract

A combined bending and twisting theory for a straight beam under external loading is developed in the article. Typical assumptions of beam theory are made for the stresses. For the displacements an assumption is made, in which the conventional rigid body movement of the cross-section is augmented by an axial displacement made up of four warping functions. Simple boundary value problems, which can be solved using the finite element method, are derived for determining the warping functions. The equations of the combined bending and twisting of the beam are derived using the principle of virtual work. The developed theory can be regarded as a generalization of the bending and twisting theory of open thin-walled beams. It, however, applies to all cross-section types even consisting of different materials and takes into account the effect of shear deformation to the deflections of the beam.