Classical Flexural-Torsional Buckling Beam Model

Although the beam model is the classical model of structural mechanics, applied for centuries, and at least for decades even in buckling problems, its application for thin-walled members can be regarded as only an approximation. Two important approximations are involved: (i) transverse stresses are neglected and (ii) transverse plate flexure is neglected.

To asses the consequence of neglecting the transverse stresses, let us consider the flexural buckling of Example (a). Either the column buckles about its minor or major axis. The majority of the elastic strain energy develops from membrane strains/stresses. If the plates that make up the member are slender enough, the plane stress assumption is certainly more reasonable than the 1D stress assumption, which means that the transverse stresses (Poisson effect) cannot be neglected. Thus, from this aspect the beam model under-estimates the column rigidity. Transverse flexure, though small in extent,
always takes place during flexural buckling. This may easily be demonstrated in a regular FSM or FEM analysis. It is obvious that transverse flexure provides additional flexibility to the column, consequently, from this aspect the beam model over-estimates the column rigidity. Thus, the beam model involves two competing approximations which nearly compensate each other so that the global buckling load provided by the beam model shows good agreement with more sophisticated models (e. g. FSM, see below) for most of practical cases, see Figure 7b.