# 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.

Yes, you are right and this is typical behavior of most noncompact cross-sections and also most of the trusses. I have actually seen some twisted steel hollow sections as part of the roof truss of a warehouse after storm damage.

They collapse suddenly with very little warning if stressed beyond service load. this is the trade-off between strength and ductility. They can take the load they have been designed to take just as well as the compact sections with the same second moment of area but if they are exposed to stresses higher than the predesignated they don't transition into a ductile elastoplastic model, they collapse explosively.

if you research aircraft engineering textbooks and research articles you will likely find some answers. Because the airplane frame is mostly thin hollow sections.

Within the realm of linear elasticity theory, it probably wouldn't be too difficult to model buckling + bending analytically. Column bucking is a simple boundary value ODE problem, so you can derive an analytical expression for the beam's deflection curve and use that in your integration for the area moment of inertia.

Modeling nonlinear behavior like yielding would be more tricky, but from a design standpoint, keeping loading within elastic limits should be sufficient to reduce the risk of yielding.

You have correctly recognized it is the buckling phenomenon that causing the member to fail before reaches its tensile capacity, which means, due to slenderness effect, the condition fcr < fc < fy has occurred. In which fc is the compressive stress of the member, fcr is the limiting Euler Buckling Stress, fcr = \PiEI/(KL)^2, KL is the effective (unbraced) length between supports, and K is a factor that depends on the rotational restraint at the ends of an unbraced length (see any design textbook for the approximation of K values). For further study/understanding on this issue/topic, I suggest to review the papers written by Dr. Joseph A. Yura on "Structural Stability".