# Failure in Hollow, Thin Beam due to Collapse of Cross Section

I'm having trouble figuring out how to predict the failure of a hollow channel due to bending. If I'm correct, for a beam with a static second moment of area, $$I$$, the following equation could be used to find the bending moment needed to cause yielding in a ductile material: $$\sigma _{b}=-\frac{My}{I}$$

However, for a square, hollow extrusion (for instance) with relatively very thin walls, it seems likely that the side walls (perpendicular to the torque vector) would first buckle and start to bow outwards, bringing the top and bottom walls (parallel to the torque vector) closer together. This would "squish" the cross section and significantly reduce its second moment of area as a function of the applied torque.

In other words, if you applied a bending moment to both a hollow beam and a solid beam with equal second moments of area, I hypothesize that the hollow beam would yield under a lesser bending moment due to the deflection in the lateral walls of the cross section.

That being said, I've searched to no avail for any method to model the premature failure of a hollow beam. I would like to be able to relate failure due to yielding to the applied bending moment for an arbitrary hollow, thin cross section, and I would appreciate any help. Since this scenario involves buckling within bending, I anticipate that it would be complex to model analytically. For that reason, I would be content with any sort of approximation, such as introducing a modification to the value of $$I$$ in the above equation.