# What exactly is the thin wall assumption in regards to beam cross sections?

What does it mean with respect to calculating stresses on a beam undergoing loading and when should it be applied?

• Which reference are you working from? – Solar Mike Jul 26 '18 at 17:56
• Not really sure what you mean, sorry – skyarex Jul 26 '18 at 18:09

The thin-wall assumption is relevant when calculating stresses due to shear and torsion.

This is because shear and torsion generate shear flow through the cross-section, as seen below: Source

This shear flow is a representation of how the reaction to the applied shear force (or shear force binary pair is equivalent to the applied torsion) "flows" through the section.

For any given segment of the section (for example, each of the sides of the box section above), it's obvious that the total shear stress "going in" is the same as the stress "going out". However, how much of that stress is found near the outer face, near the inner face and around the middle?

Honestly, that gets kind of messy to calculate, so instead the "thin-walled assumption" is made instead. We just assume that the thickness of each of the walls is so small that whatever variance there is through the wall can be ignored. Put another way, it assumes that the shear flow is uniform throughout each of the walls.

As far as I know, there is no formal definition of when a wall can be considered thick or thin, but a common one is $d/t>20$ (seen here on Wikipedia), where $d$ is the section's width or diameter, and $t$ is the wall's thickness. That being said, I've seen the thin-wall assumption used in sections with $d/t=10$ as well.

That Wikipedia article actually has a good description of how much easier thin-walled sections are to calculate than thick-walled ones. Thin-walled sections have closed-form solutions, while thick-walled ones require integration to solve. Fun times!

The example I gave above is of a closed thin-walled section, but the assumption can also be made with open sections (I, L and T sections, for example). Obviously the solution for the shear flow is dependent on the cross-section, but closed sections perform drastically better under torsion than open ones.

I would guess (please provide a reference for your question) that it means you can assume all the nice simplifications which allow the problem to be solved by hand.

The assumtions/simplifications are:

1. The structure must be thin (width: length ≈ 1:100)
2. Bending deflection doesn't exceed the structure's thickness.
3. Perpendicular lines in the structure before bending will stay perpendicular on the neutral axis after bending

You can also look for Bernoulli/Euler. A more advanced beam theory is provided by Timoshenko.

Over at rearchgate a similar question was asked.

• Actually, the only real assumption is #3. The first two are just (very pessimistic, using the numerical criteria given in the answer) rules of thumb to ensure #3 is satisfied. – alephzero Jul 26 '18 at 19:43
• Actually I think you've confused thin-walled beams with slender beams. The comparison between cross-section size and beam length is for slender beams, while the thin-walled assumption is based entirely on cross-section dimensions. – Wasabi Jul 27 '18 at 2:43