Bending Stress of Curved Non-Symmetrical Beams

[nargomech]

Hi,
I am amazed by the lack of formulas to calculate bending stresses of curved beams that are frequently used, this is, when the plane of loading and the plane of curvature is the same, but this is not a plane of symmetry for the section, being structural angles (with plane of bending and loading parallel to a leg) the most common case, and C channels bent the hard way a second common example. Even the classic Roark’s book (Formulas for Stress and Strain, by Young and Budynas, 7th Edition, Chapter 9), only includes symmetrical sections, and the same happens with twelve other books about mechanics of solids and strength of materials I have checked (including two by Timoshenko, but I haven’t found his Strength of Materials), and with Marks' Handbook for MEs. The only non-symmetrical sections I have found, are in a non-common book: Handbook of Formulas for Stress and Strain, by William Griffel, 1966 (Amazon and B&N sell this one for $195, when the new Roark is only $85). There, he says that he solved Timoshenko’s integral for 12 shapes, so I guess that Timoshenko doesn’t have the angle (which is the section I am looking for) either. Well, the four non-symmetrical sections he has are not common at all, they are all based on a 1/4 of circle, either solid or what remains from a square after cutting off that quarter circle (I could use this one to approximate the angle I have).

Would somebody have formulas for curved structural angles ? If you have it, please, don’t refer to another difficult-to-obtain book or journal paper, please scan those pages and upload them in this website (I guess that any publication would be so old, that the copyright is already expired, and people upload whole books all the time anyway). In exchange, I offer to do the same with some pages that somebody maybe interested in, just mention the topic, and I will do my best to find something.

 

[Cockroach]

This is were FEA comes in so handy.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[desertfox]

Hi nargomech

I can't open the files at this link but it might help I don't know:-

http://www.steeltools.org/STEELTOOL...umentKey=f174a7b0-6a9b-4bb7-a024-dd6933645621

 

[prex]

I don't think the lack of symmetry in the section influences the change of stresses due to curvature.
As you probably know, a graphical method exists to treat sections of any shape: this consists in modifying the width b of the section by the ratio R/r (R=radius of curvature at centroid, r=local radius of curvature) and calculating the area of the modified section. So for this method there's no difference between, e.g., a channel and a I section.
I think you can use the formula for a T-beam for your angle.

prex

http://www.xcalcs.com

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[nargomech]

Thanks Coachroach, desertfox, and prex for your feedback. I am sorry for the delay to reply, but I have been swamped the last 2 months. Actually, I followed the last line of prex's advise, based on the fact that the inertia of a T section and the inertia of an angle are the same, so I could use Roark's formulas. The axis I am referring to, is parallel to the horizontal part of the "T", in other words, normal to the axis of symmetry of the T, and the legs of the angle are of course oriented parallel to the T in this comparison. Fortunately for me this axis, where both sections have identical inertia, was the relevant axis in my case.