Curved Beam

[wvtech76]

I am designing some curved beams that support floor framing on one side and roof framing on the other side, so they are curved in the plan view and loaded in the plane of the web just like a regular beam. The radius is 130' and the beams vary in length from 18' to 24', so the amount of curvature (measured from the chord) varies with the maximum being around 8.5". Uniform loading of 3.2 klf.

My question: Is there an amount of curvature that can be tolerated and still use conventional "straight beam" formulas? If the curvature needs to be considered to determine the moment and torsion, where can I find formulas that are easy to use and make sense? Everything I have found on line seems to be based upon research projects that are not really applicable and any formula derivation seems to want to be for concentrated loading. Also, most research seems to be on sharply curved members and those loaded opposite to what I have, such as hooks and arches.

Any help would be appreciated.

 

[KootK]

1) I don't know of an in print recommendation but, at 8.5" in 18', I'd say that you're in straight beam territory. Especially with infill framing beams on both sides to soak up any torsion as roll beams.

2) Your main issue is torsion. See the AISC design guide on the subject.

3) The bridge world is a good place to look for info on girders curved in plan.

4) Any chance your detailing might support just using a straight beam? It sounds as though there might be potential.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[wvtech76]

I would go with a straight beam in a heartbeat, but architects are involved and the beam also supports a curved wall. Since I had to space my columns unequally (hence they varying beam lengths) to hit walls and partitions below, using unequal straight length segments would goof up the visual effect, so it needs to stay curved.

I checked the AASHTO Curved Steel Highway specs and it just says that if any girder or beam is curved, the curved specs apply. No minimum curvature was given that would still allow straight beam theory to be used.

 

[Ussuri]

This might also be of some use. Example 6 covers beams curved in plan.

http://www.steelconstruction.info/index.php?title=Special:ImagePage&t=SCI+P281.pdf

 

[wvtech76]

Yes. I found that publication, SCI-P281. Example 6 is very close to my condition, but the example does not make any sense to me. On Sheet 3 of the example, it states to divide the moment by the assumed distance between the flanges to get the force in the flanges. The example (which is in metric) has 500mm as the distance. The moment they give is 253 kNm. 253/0.5=506 They have F=25.6. From there on, they use that value, but where did it come from?

 

[KootK]

It's called the bi-moment method and is described in the AISC design guide that I mentioned. If you want a quantitative basis for when curvature can be ignored, I'd use the bi-moment method to estimate torsional flange stresses. If they're less than 5-10% of the total, forget about it. You're in the enviable position of not having to worry much about lateral torsional buckling which is helpful.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[JStephen]

Roark's Formulas For Stress and Strain has equations for curved beams in it, they're just pretty much a mess.
A second issue is that allowable stresses are not well covered under the regular steel codes.

 

[Lomarandil]

130' radius doesn't seem that loose (in my world).

AASHTO 4.6.1.2b has a criteria for neglecting curvature for major axis effects in I-girder bridges. For a single girder/beam, it boils down to arc span/radius < 0.06 rad. End spans of continuous members get modified by 0.9, interior spans by 0.8. You're a good bit over that limit.

Per AASHTO, you would still have to account for minor axis and torsional effects regardless. The "bi-moment" method (just decoupling the torsion into lateral forces in each flange) is the simplest manner.

AASHTO 6.10.1.6 could be another reference point for the magnitude of your lateral stresses (it's the threshold at which you start magnifying moments for second order effects). Although I think KootK's threshold of 5-10% of total flange stress is about right.

 

[Lomarandil]

Also, NCHRP 725 has some information about the relative accuracy of different analysis methods for curved girders -- I don't remember a direct transition point for ignoring curvature in the analysis, but it may inform your opinion.

 

[wvtech76]

Thanks for all of your responses. I will check the AISC Design Guide, it think it is No. 9 for torsion, and the bi-moment method and go from there.

 

[Ussuri]

I believe the axial force comes from the secondary analysis from the radial forces in the plane of the curvature.

The calculation starts off with the curved beam subject to a vertical UDL. He has analysed it using a frame package to come up with the moment diagram and the values at each of his node points.

From there at each node point he computes an equivalent flange tension/compression (the M/0.5m) which gives the axial load (F) in the flange. He then uses this to determine the radial component (F/R), in the same way you would calculated the pressure on something if wrapped a rope around it and pulled, w = T/R.

Doing this calc at each node location will give him a radial load which he feeds back into the analysis model. This load is now in the plane of the curvature. Re-running the analysis for this load case will give you an additional axial component. I guess if you run the analysis this will be the 25.6kN.