Closed form solutions for shear deformable thin-walled beams including global and through-thickness warping effects

Abstract The present study develops a theory for the static analysis of thin-walled members with general asymmetric cross-sections. The theory captures global and through-thickness warping, in addition to shear deformation effects. Starting with the principle of stationary potential energy, the governing equilibrium equations are developed along with the possible boundary conditions. The theory leads to seven field equations in the axial displacement, lateral and transverse displacements, bending rotation angles, angle of twist, and warping deformation. Unlike conventional non-shear deformable thin-walled beams which predict a torsional response that is uncoupled from the flexural responses, the differential equations obtained herein are fully coupled when applied to beams with asymmetric cross-sections. Closed-form solutions are formulated for beams with asymmetric, monosymmetric, point symmetric, and doubly symmetric cross-sections. Comparisons with the predictions of previous thin-walled beam theories/finite elements and shell finite element solutions verify the validity of the present theory and quantify the effects of shear deformation, through-thickness warping, and the coupling arising from to shear deformation. Numerical examples illustrate the partial coupling effects arising in the special cases of monosymmetric, point symmetric, and doubly symmetric sections.