Analysis of a beam cross-section under coupled actions including transversal shear

An efficient, finite elements (FE) based method of cross-sectional analysis of a longitudinally homogeneous beam, with an arbitrary cross section and material model assumed, submitted to tension/compression, bending, torsion, and shear is developed. A new formulation of this classical problem is done as a special case of the homogenization theory. It takes into account 3D strain and stress state, fulfilling equilibrium, compatibility, and constitutive equations at any point, stress-free conditions on lateral surfaces, but preserves control by cross-sectional stress resultant forces and corresponding generalized strains. A source of problems caused by the presence of a shear force is explained and a way of overcoming it is shown. In the paper both variational formulation of the problem and its 2D FE solution are given. Benchmark cases, including shear of: a square, a composite L-section, an orthotropic square, and an elasto-plastic analysis of shearing of a square section, show correctness of the method. Finally, two practical cases of application of the method to an analysis of reinforced concrete beam sections are shown.