Theory of Flexural Shear, Bending and Torsion for a Cantilever Channel

Abstract

Thin-walled open-section beams, such as channel beams, exhibit a highly coupled mechanical behaviour when subjected to transverse loading, bending, and torsion. This coupling is mainly associated with non-uniform shear flow, the offset between the centroid and the shear centre, and the development of restrained warping in cantilever configurations. Accurate prediction of the resulting stresses and deformations therefore requires a unified analytical treatment that goes beyond classical uncoupled beam theories. In this study, a comprehensive theoretical formulation is developed for the analysis of flexural shear, bending, Saint-Venant torsion, and warping torsion in thin-walled cantilever channel beams. Closed-form expressions are derived for the internal force resultants, shear flow distribution, torsional rotation, and warping-induced stresses using classical thin-walled beam theory and elasticity principles. The influence of load eccentricity with respect to the shear centre and the effect of warping restraint at the fixed end are explicitly incorporated. The applicability of the proposed formulation is demonstrated through two independent numerical examples involving different loading conditions, sectional geometries, and eccentricities. The numerical results show that bending stresses generally dominate the structural response, while web shear stresses remain moderate. However, the study also confirms that even small load eccentricities can generate significant torsional effects, and that restrained warping near the fixed support may substantially contribute to the maximum longitudinal stresses. The presented unified analytical framework provides a reliable and physically transparent tool for the assessment and design of thin-walled cantilever channel beams subjected to combined flexural, shear, and torsional loading. The results are of direct relevance to mechanical, civil, and aerospace engineering applications where open-section beams are widely used.