I have a system consisting of a hollow, rectangular cantilever beam (pictured below). I'd like to write the equilibrium stress equations à la the Cauchy Stress notation.

How do I account for the hollow section in the equation? How is it represented in a free-body diagram of the system?

simple schematic of the cross sectional area of the beam.

  • $\begingroup$ A good textbook? $\endgroup$ – Solar Mike Jul 15 '19 at 18:03
  • $\begingroup$ You can find the theory for a solid rectangle in a good textbook (e.g. Timoshenko's advanced books, not his basic "strength of materials" ones). Warning: it's complicated! In principle you can extend this to a hollow rectangle, but there will be singularities in the stress field (i.e. "infinite" stress) at the reentrant corners. This is probably PhD level work, except that it is not particularly interesting as "pure research" and doesn't have much practical use either. $\endgroup$ – alephzero Jul 16 '19 at 11:47
  • $\begingroup$ @alephzero It's funny that you say that this is PhD level work, because that's exactly the context for this question: I'm trying to model this type of system for a paper I'd like to publish. But, the focus of the publication isn't on the math necessarily, so an FEM analysis with Abaqus may suffice. $\endgroup$ – platypusomnibus Jul 16 '19 at 19:14
  • $\begingroup$ If you're going to rely on FEM, why both with the Cauchy equations? FEM is shorthand for "too hard to derive analytically". $\endgroup$ – Wasabi Jul 19 '19 at 2:20
  • $\begingroup$ Love the acronym. I figured that the equations would be a nice compliment to the FEM model. I’ve seen some papers derive analytical expressions that are compared to FEM analyses. But I think I’ll just stick with the computer work now. $\endgroup$ – platypusomnibus Jul 20 '19 at 13:29

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