Bars under Torsional Loading: A Generalized Beam Theory Approach

In this paper both the static and dynamic analyses of the geometrically linear or nonlinear, elastic or elastoplastic nonuniform torsion problems of bars of constant or variable arbitrary cross section are presented together with the corresponding research efforts and the conclusions drawn from examined cases with great practical interest. In the presented analyses, the bar is subjected to arbitrarily distributed or concentrated twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. For the dynamic problems, a distributed mass model system is employed taking into account the warping inertia. The analysis of the aforementioned problems is complete by presenting the evaluation of the torsion and warping constants of the bar, its displacement field, its stress resultants together with the torsional shear stresses and the warping normal and shear stresses at any internal point of the bar. Moreover, the construction of the stiffness matrix and the corresponding nodal load vector of a bar of arbitrary cross section taking into account warping effects are presented for the development of a beam element for static and dynamic analyses. Having in mind the disadvantages of the 3D FEM solutions, the importance of the presented beamlike analyses becomes more evident. 1. Introduction In engineering practice, we often come across the analysis of members of structures subjected to twisting moments. Curved bridges, ribbed plates subjected to eccentric loading, or columns laid out irregularly in the interior of a plate due to functional requirements are the most common examples. When the warping of the cross section of a member is not restrained, the applied twisting moment is undertaken from the Saint-Venant [1] shear stresses. In this case the angle of twist per unit length remains constant along the bar. However, in most cases arbitrary torsional boundary conditions are applied either at the edges or at any other interior point of the bar due to construction requirements. This bar under the action of general twisting loading is leaded to nonuniform torsion, while the angle of twist per unit length is no longer constant along it. The consequences of restrained warping were first presented by Marguerre [2]. Although there is extended literature on the Saint-Venant uniform torsion problem for homogeneous isotropic cylindrical bars with simply or multiply connected cross sections [3–8], the extensive use of structural elements subjected to torsional loading necessitates a reliable, accurate, and