An analytic solution for bending of thin-walled laminated composite beams of symmetrical open sections with influence of shear

Abstract

An approximate analytic solution for bending of thin-walled laminated composite beams of symmetrical open sections with influence of shear is presented. Symmetrically laminated beams that possess membrane orthotropy are considered. The classical Euler–Bernoulli’s and Timoshenko’s bending beam theories are augmented by terms which take into account the shear strain in beam mid-surface as well as the warping of the cross section due to shear. Consequently, the beam is subjected to bending with the influence of shear caused by transverse forces acting in the plane of symmetry and in addition to tension/compression due to shear in the case of cross sections with one axis of symmetry. The beam is subjected only to bending with the influence of shear in the case of transverse forces acting in the planes of symmetry. The expressions for the normal stresses and displacements are presented in the closed analytic form. The factor that depends on the fiber orientation is introduced in order to analyze the material influence on shear. Simply supported and clamped beams subjected to distributed transverse load are investigated. The solutions obtained for open sections are also valid for mono-symmetrical closed sections, taking into account the constraint that loads act in the beam longitudinal plane of symmetry/planes parallel to the longitudinal plane of symmetry. Illustrative examples are provided, and the analytic results show a very good agreement with the results obtained by the finite element analysis utilizing three-dimensional shell elements.