Warping displacement of Timoshenko beam model

A refined Timoshenko beam model which takes into account warping of the cross sections is presented. The model extends Saint-Venant's theory of uniform torsion to the generic loading of a beam in a coherent manner. The use of kinetic and kinematic assumptions, virtual work expression of full elasticity problem, and the principle of virtual work in derivation brings the presentation to the usual context of engineering models. A new definition of the warping displacement in terms of a variational problem is one of the outcomes. Boundary value problems for the stretch, shear, bending, and torsion loading modes are also given. Analytic and numeric solutions to warping displacements, refined constitutive equations, and correction factors for circular, rectangle, triangle, and open annular cross sections of isotropic material are compared with expressions in literature. As an interesting detail, shear correction factors are found to be purely geometrical quantities, which contradicts many findings in literature.