Keywords: thin-walled members, open sections, Vlasov theory, normal stresses, shear stresses, interaction relations.

The Vlasov thin-walled beam theory [1] leads to four differential equations of equilibrium. These equations relate the cross-section lateral, transverse, and longitudinal displacement fields and the angle of twist of the cross-section. In general, the four equilibrium conditions are coupled. Conventional solution techniques attempt to eliminate the coupling between the differential equations of equilibrium. This is realized by adopting an orthogonal system of coordinates. (i.e., adopting the centroidal principal axes, and adopting the line joining the section shear centre and principal sectorial origin as a fixed radius). While this approach successfully uncouples the equations of equilibrium, it generally suffers from a number of drawbacks including: a) the necessity to perform coordinate transformations, b) the difficulties in handling cross-section discontinuities where members undergo stepwise cross-sectional changes (e.g., members with holes, flange copes, the presence of longitudinal stiffening plates, etc.), c) the boundary conditions may remain or become uncoupled after transformation, and; d) the adoption of an orthogonal system of coordinates may necessitate resolving the internal and external forces along the orthogonal coordinate systems. As a result, the use of the conventional solution technique has been limited to very simple applications. A recent study [2,3] developed a general solution of the Vlasov thin-walled beam theory under non-orthogonal coordinate systems in a finite element context. The new solution scheme offers a number of modelling advantages when compared to conventional solutions based on orthogonal coordinates. The objective of the current paper is to complement the recent study [2,3] by developing general expressions for normal and shearing stresses under a non-orthogonal coordinate systems.

The paper starts by formulating the possible boundary conditions for thin-walled beams under a non-orthogonal coordinate system naturally arising from the principle of virtual work. Subsequently, the paper reviews established solution procedures for displacement field variables under (a) the direct solution of equilibrium conditions, and (b) finite element solutions. Consideration is given to solutions based on orthogonal and non-orthogonal coordinates.

The discussion is extended to include stress computation schemes. Expressions for normal and shearing stresses are developed to relate the normal and shearing stresses to stress-resultants. These are formulated under general non-orthogonal coordinate systems. The well known stress expressions under conventional orthogonal coordinate systems are recovered as a special case from the new expressions.

The new normal and shearing stress expressions are combined using the von-Mises yield criterion to develop a general interaction equation valid for general beams of open cross-section. The interaction equation is applicable in the elastic range of deformation and is quite general. It involves eight stress resultants: biaxial bending, biaxial shear, normal forces, the St. Venant twisting moment, warping twisting moments, and bimoments.

1

V.Z. Vlasov, "Thin Walled Elastic Beams", Second Edition, Israel Program for Scientific Translations, Jerusalem, Israel, 1961.

2

R.E. Erkmen and M. Mohareb, "Non-Orthogonal Solution for Thin-Walled Members - A Finite Element Formulation", Can. J. Civil Eng., Vol. 33(4), 421-433, 2006. doi:10.1139/L05-116

3

R.E. Erkmen and M. Mohareb, "Non-Orthogonal Solution for thin-walled members - Applications and Modelling Considerations", Can. J. Civil Eng., Vol. 33(4), 440-450, 2006. doi:10.1139/L06-027

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