Torsion of a thick-walled arbitrary open section bar

[FEAnalyst]

TL;DR Summary

How to calculate maximum shear stress and twist angle of a thick-walled arbitrary open cross-section (e.g. I, T, L, C) bar subjected to torsion ?

Hi,

I analyzed various open cross-section (I, T, L, C) bars using FEA to find maximum shear stress and twist angle. Now I would like to compare these results with approximate hand calculations. Just out of curiosity. However it's not easy to do it analytically. I know about a method of dividing open section into several rectangles and calculating and then summing their torsion constants. Unfortunately the books say that this method applies to thin-walled sections and I would like to solve some thick-walled bars. So is there any other approximate method I could use or maybe the one mentioned above can be used even for thick-walled bars ? Of course I realize that results won't be exact. If there's no approximate method then do you know about any examples for these sections with the use of differential equations ?

Thanks in advance for your help

 

[Chestermiller]

Can you please give an example of the kind of shape you are analyzing?

 

[FEAnalyst]

Sure, here's a picture showing sections I would like to analyze:

They are all typical open profiles often used for bars. Thus I'm a bit surprised that it's so hard to find any examples of torsion calculations for these profiles.

Source of images: https://structx.com/geometric_properties.html

 

[JBA]

If you go to your above reference and click on each of the pictures, it will open a page with the applicable formulas you are seeking for that beam configuration.

 

[FEAnalyst]

Yes, among the properties listed on this website there are some related to torsion - polar moment of inertia and torsional constant. However I don't think that they can be used to calculate max shear stress and twist angle in this kind of section like it's done for circular section: $\tau_{max}=\frac{Tr}{J_{z}}$ and $\theta=\frac{TL}{JG}$ ($T$ - torque, $r$ - distance from central axis, $J_{z}$ - polar moment of inertia, $J$ - torsional constant, $G$ - shear modulus). These formulas apply to circular sections only due to warping of non-circular shaft.

 

[JBA]

If you google the subject you will see that material on this is related to the safe application of such shapes and is therefore simplified by factors and graphs. On the technical side there many references under "torsional stress of structural beams", none of which lends itself to a quick and easy calculation.

 

[FEAnalyst]

I do realize it won't be easy but I would like to try solving it analytically just out of curiosity. If the method for thin sections won't give meaningful results here then I will at least take a look at the approaches involving the use of differential equations. Or maybe hydrodynamic analogy may help. I will be glad if you share some resources that you know about.

I also wonder what is the purpose of these polar moments of inertia and torsional constant listed on the aforementioned website. Maybe they can be used somehow, but not with the regular equations for circular section. However there might be a way to account for warping in calculations utilizing these constants.

 

[JBA]

At this point all I can offer, and you may have already seen, is page 12 of:
http://people.virginia.edu/~ttb/torsion.pdf

 

[FEAnalyst]

Thanks, I used the formulas designed for thin-walled bars to solve the examples that were previously analyzed with FEA and I've found out that there's a very good agreement. Apparently these methods are not limited to thin-walled bars only. Or maybe my examples were close enough to this approximation.