## 1 Introduction  
This paper extends the thin-walled beam theory outlined and derived previously using the geometry of three extruded aluminium alloy channels to US specification [1]. Therein, the point was emphasised that each thin-walled beam structure requires individual consideration depending upon the cross-section and the manner of its support under an imposed loading. What might appear as a simple structure is deceiving from the complex analyses required of its stress state and consequently the likely mode of its failure. It is a testimony to classical elasticity theory that solutions are forthcoming from its linear constitutive relation, coupled with a description of force and torque equilibrium, with strain and displacement compatibility matched to the boundary conditions that prevail. This theory has long been applied by the Engineering Science Data Unit (ESDU) in the UK to develop a library of case studies of lightweight alloy beam structures used in applications where the weight to strength ratio is an issue. Their data sheets have been produced on a variety of related topics including plate buckling within thin I, Z and U sections [2]. The present work has been undertaken to provide further useful design data in this arena. In particular, given the prevalence of the FEM today for stress analysis, these design data can be configured for the validation of an FE stress analysis. A theoretical approach can serve to validate finite element predictions and also provide the benchmark for comparisons with experimental data found from scaled model testing. The present approach adopts the classical theory where it is able provide closed solutions for a specific structure. Fortunately, here, there are many practical applications of thin structures to which the theory can be applied, especially those where minimum weight restrictions are imposed. More recent developments with thin structures include thin-film technology [3] and flexible devices [4]. It is expected that a thin-walled stress analysis would have relevance here in the presence of mechanical loading involving twist, stretch and bend in static and dynamic applications. Previously, in Ref. [1], the flexural shear, bending and torsion of a thin-walled cantilever beam of an open cross-section were derived and considered separately. For an open channel section, with \(x\)-axis symmetry, the axes of loading were referred to shear centre E, centroid G and another arbitrary position, O, at the centre of the web. In the case of an axial torsion, it was shown how both the free-end and constrained-end warping displacements depended upon the axis of torsion coincident with the stated positions. So too does the axial stress depend upon the torsion axis as it is induced by the fixed-end constraint. It was seen that beam flexure involving bending and shear act together only when transverse point forces act upon the shear centre. In this paper, bending moments accompanying transverse forces may be displaced from \(x\) and \(y\) axes at G to parallel axes at E after applying bending theory. It is shown here that any the shift between said axes also involves a _trans-moment_, but this moment does not arise when E and G lie upon a common \(x\)-axis for an axially symmetric channel section. Stress analysis considers the torsion, shear and bending that arise when transverse forces applied at G are displaced to shear centre E. The common centre enables a superposition between similar stress from different sources. A net shear flow is found as a sum of the flows from shear and torsion arising from the applied transverse forces. Accompanying this, a net axial stress also arises as a sum of the axial stress from bending and from the beam's end constraint. The latter induces axial stress by preventing the axial warping displacement that would otherwise occur under torsion when the ends are free.