Bending of thin-walled beams of open section with influence of shear, part I: Theory

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is based on the Vlasov's general beam theory as well as on the Timoshenko's beam bending theory. The theory is valid for general thin-walled open beam cross-sections. Part I is devoted to the theoretical developments and part II discusses analytical and obtained numerical results. The theory is based on a kinematics assuming that the cross-section maintains its shape and including three independent warping parameters due to shear. Poison's effect is ignored, as well as warping constrains due to shear (as it known, those effect have small and for engineering praxis neglected influence on the stresses and displacements). Closed-form analytical results are obtained for three-dimensional expressions of the normal and shear stresses. Under general transverse loads, reduced to the cross-section principal pole, the beam will be subjected to bending with influence of shear and in addition to torsion due to shear with respect to the cross-section principal pole and to tension/compression due to shear, in the case of non-symmetrical cross-sections. The beam will be subjected to bending with influence of shear (i) in the plane of symmetry under the loads in that plane and in addition to tension/compression due to shear, (ii) in the plane through the principal pole orthogonal to the plane of symmetry under the loads in that plane and in addition to torsion with respect to the principal pole, in the case of the mono-symmetrical cross-sections. The beam will be subjected to bending with influence of shear in the principal planes, in the case of the bi-symmetrical cross-sections. The principal cross-section axes as well as the principal pole are defined by the classical Vlasov's theory of thin-walled beams of open section. The analytical and numerical analyses presented in part II include comparisons with the classical beam theory, Euler-Bernoulli' theory (EBBT) as well as comparisons with the finite element method (FEM).