Generalized Warping Torsion Formulation

A general formulation for torsional-flexural analysis of beams with arbitrary cross section is presented in a general coordinate system. The theory maintains Vlasov's approach in terms of generalized strains and stresses and yields the same system of differential equations. The common hypothesis of transversely rigid cross section, which overestimates the effective flexural and torsional section stiffness, is replaced by the assumption that stresses in the plane of the cross section are small. The resulting theory reduces to the exact solution of Timoshenko when warping effects are neglected. Shear stresses due to shear forces, warping torsion, and Saint-Venant torsion are determined as the gradient components of a unique potential function. These equations are solved with the finite element method, which also provides the flexural and torsional section stiffness and the shear center. Numerical examples are presented and results are compared with full three-dimensional finite element analyses. The formulation is simple and, in spite of the limitations of the simplifying hypotheses, sufficiently accurate for many engineering applications, bypassing costly three-dimensional finite element analyses.