Design of Curved Beams 2

[abusementpark]

For the design of beams that are curved in the plane of the loading, how do you account for the effect the curvature has on the strain distribution?

Both steel and concrete beam design provisions are based on ultimate strength methodologies using a linear strain assumption. Since, curved beams do not have a linear strain relationship, how do you check the beams for flexure? Is there a significant difference?

For steel, I was thinking that since flexural strength is based on a plastic stress distribution the effect of the curvature would only make a difference at lower stress levels, prior to ultimate, but enough redistribution can occur to bring it to a plastic stress distribution.

For reinforced concrete, I am less sure since all of the ductility checks are based linear strain.

 

[IDS]

Since, curved beams do not have a linear strain relationship, how do you check the beams for flexure? Is there a significant difference?

Curved beams do have a linear strain relationship. At least they do under circumstances where a straight would have a linear strain relationship; i.e. away from supports and point loads.

They can be designed for combined flexure and axial load.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

try googling "curved beam stress istribution" ... there's a link to "washington.edu" that looked relevant.

i remember that the stress distribution is very non-linear ... "The distribution of the stress in the case of curved beam is non-linear (Hyper- bolic) because of the neutral axis is initially curved."

 

[a2mfk]

A beam that is curved in the plane of loading- like an arch for an extreme example?

 

[frv]

IDS..

rb157 is right. it is not linear.

http://courses.washington.edu/me354a/Curved Beams.pdf

 

[paddingtongreen]

A sense of proportion here, please.
How bent is the beam?
A parabolic beam has no bending moment (I know it is impractical to load it with a UDL, but theoretically) the load stress is completely axial. A straight beam has a linear bending relationship and no axial. Any change in relationship must be gradual and depend on the rate and type of curvature.



Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[abusementpark]

A beam that is curved in the plane of loading- like an arch for an extreme example?

Yes, or an upside down arch.

A sense of proportion here, please.
How bent is the beam?

Let's say it is a half-circle.

A parabolic beam has no bending moment (I know it is impractical to load it with a UDL, but theoretically) the load stress is completely axial

I'm not sure what you are talking about here. Something similar to a compression ring?

 

[paddingtongreen]

I assumed uniform linear loading along the span.

If you bend the beam to a parabola curve, all of the load in the beam becomes axial instead of flexural.

For a hanging curve, think of a catenary, all axial, no flexural effect. It's not circular, but it is partway there.

The stress distribution, may not be linear but it also may not be large.
I confess, the idea of a beam in a half circle worries me, it isn't a beam any more, it is an arch.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[BAretired]

You can bend the beam in any configuration you like, but if the ends are supported on a hinge and roller, it is a simple beam and subject to the shears and bending moments of same.

BA

 

[abusementpark]

You can bend the beam in any configuration you like, but if the ends are supported on a hinge and roller, it is a simple beam and subject to the shears and bending moments of same.

I agree. However, the stresses that bending moment puts on the beam cross-section are different for curved beams. So I want to know how you are supposed to account for that in design.

 

[BAretired]

abusementpark,

Why are the bending stresses different for curved beams than for straight beams?

If a parabolic arch is uniformly loaded and is hinge supported at each end, the member is in uniform compression. If the load is not uniform or if the shape is not parabolic, there will be some bending as well.

If one of the supports is changed from a hinge to a roller, the arch becomes a beam whose moment is identical to that of a simple beam. The moment at any section is given by the simple beam formula and the stress at any section is given by elementary strength of materials theory. Am I missing something here?

BA

 

[IDS]

IDS.. rb157 is right. it is not

linear.

http://courses.washington.edu/me354a/Curved Beams.pdf

The link gives an analysis for a thick ring, I wouldn't call it a beam. A straight deep beam doesn't have a linear stress distribution either.

For a curved thin member you find the bending moment and axial force at any section from static equilibrium, then design the section in the same way that you would for a thin straight beam.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

another thought, is the beam formed (ie bent) from a straight beam ? are the manufacturing stresses relieved in some way, or are they still in the beam ?

 

[csd72]

If it is on roller supports then the shape of the beam is irrelevant for the maximum bending moment, it is always the same for any simple span under a given load.

Yes the moment stress distribution is different for significantly curved beams but in most structural cases the curvature is minor and can be treated basically as a straight beam.

Deflection will increase due to the increase in beam length though this may/may not be a problem due to the fact that the beam is effectively precambered.

spread of supports may be the major issue for deflection.

There has been some very good articles on this including one in the october 2009 issue of modern steel construction.

 

[slickdeals]

See attached.

http://fsel.engr.utexas.edu/about/events/steer_2010/seminar/ragan.pdf

 

[dhengr]

I first struggled with the curved beam problem back in the early 70's, and we were doing it long hand at that time, and here’s what I found to work and be quite predictable. Over the years these methods were confirmed with FEA, strain gaging, and load testing and deflection measurements, all in surprising good agreement with my hand calcs. As several people have mentioned, you do find the moments and shears and axial loads, on the beam, just as we always have from our Statics and Strength of Materials learnin. Thus, as we always have, you can find the moment and shear at any location along the length of the beam or other member (hook for example). And, in the straight portions of the beam you proceed just as we always have with our std. treatment from Strength of Materials: Ix, Sx, distance from the neutral axis, etc. But the actual stresses are significantly complicated by the curvature, see FRV’s “washington.edu” link and study its development, that’s right on the money. The curvature causes the neutral axis (N.A.) to shift toward the center of curvature, the more so the smaller the radius of curvature, and the N.A. no longer coincides with the centroid for the bending only problem, and the normal stresses are no longer linear as they move away from the N.A., unlike our std. formulation for a straight beam. I’ve always used superposition for axial loads, just algebraically adding P/A, although that seems to get adjusted a bit too, when you check the problem with FEA.

I had two younger engineers do masters theses on this subject, in effect refining and confirming my long hand approach, although I never got any extra credit for my initial work on the matter. I should have used the subject for my own thesis, but what did I know I was just a working stiff, doing my job. Similar to the radial stresses which can cause problems at the haunch of a lam beam, three hinged church arch, you get some significant radial stresses at the curved beam areas. The stresses are at least biaxial; including what we normally think of as the normal bending stresses, axial stresses and their nonlinear components, our regular shear stresses and these induced radial stresses. For example: the outer fiber stresses or flange stresses in the tension flange, at the larger radius surface have components forcing the flange toward the centroid of the section, while these tension forces or stresses at the smaller radius surface tend to pull the flange away from the centroid of the section or pull the flange away from the webs of the member as it makes the curve. Compressive normal stresses cause the opposite pulling or pushing affects on the larger or smaller radius surfaces. These forces and stresses must be accounted for in flange to web bearing or in the welds which joint the flange and the web. I never looked at the problem in a conc. member, but it would undoubtedly lead to more longit. reinf’g. and most certainly more shear and confinement reinf’g.

None of this has any effect on Paddinton’s or BA’s comments about parabolic arch uniformly loaded having a uniform compression or the catenary cable, and the like, which are kinda different animals or conditions than what the OP asked about, I believe. I think IDS is right about conditions at point loads, supports and transitions, although I just kinda rationalized those away, as we usually do on normal beams. Rb1957 & Csd72 wondered that residual stresses and forming stresses should be kept in mind, although in the normal range structural work they don’t seem to cause much trouble and sometimes actually help localized stress conditions. Certainly, heat straightening and cold cambering don’t usually cause us much heart burn. I more often heat treated after fab. to control movement due to residual stresses during machining, than because of a concern about stresses. Abusementpark wondered about ultimate strength methods and plastic stress distribution and today’s codes in his OP. Irrespective of today’s codes the N.A. shift and the multi axial stress conditions causes yielding to occur much sooner than normal straight beam theory would suggest, particularly in the curved regions. Designs older than mine were showing unacceptable deflections and no one could explain why. It turned out the deflection was caused by the start of a classic hinge rotation in the area of the beam curvature, but nobody thought of that, or really knew how to measure it. The cambers, shapes and deflection in the rest of the structures looked fine.

 

[abusementpark]

dhengr,

Thank you for the post. You seem pretty well-versed in the subject. Do you of any technical references that outline design guidelines for curved steel and/or concrete members?

I did a couple Google search and didn't find anything.

Do you have any personal recommendations on simple ways to account for these effects in design?

 

[IDS]

abusementpark - what sort of radius/depth ratio are you talking about? If it is greater than about 10 the effect of the non-linear stress distribution across the section is negligible, and if you are doing an ultimate limit state design the only effect will be that the ULS moment will be reached at a slighly smaller curvature; the ULS design moment for any specified axial load will be exactly the same.

I have designed literally thousands of concrete arch structures and many of these have been modelled both with plate elements and beam elements, giving very similar results. I have always used the procedures I described above: i.e. find the bending moments and axial loads using standard structural design methods then design the section for combined axial load and flexure assuming a linear strain distribution at any section.

At supports you can use the same methods as for a straight beam; i.e. use a conservative shear design complying with the applicable code, or a strut and tie analysis.

The attached graph shows the difference in stress across an arch section between the "exact" stress and assuming a linear strain distribution. In both cases the material was treated as linear-elastic in compression and tension. The arch was a semi-circle of 5 metres radius and 0.5 m thick, loaded under self weight and hinged at the base. The maximum difference in stress is less than 4%.



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dhengr]

Abusementpark:
As DougJ asked, tell us more about the structure you are considering; length, depth, width, loading, type of material you are considering, etc., so we have some idea a proportions, types of loads and their extent, etc. I never had much luck welding conc. so there’s no sense in my talking about welding if you’re using conc. Other than my one sentence on conc., the last sentence in para. 2 of my above post, it sounds like I should defer to DougJ to give you the lowdown on conc. elements and this curved beam problem. You guys have a big advantage today with the use of FEA for you analysis of the structure, and that is the way I would analyze your structure and find the stresses and forces, using a tighter mesh around the curved areas and beam elements in the straight portions. Other than to getting a good handle on the concept, the Theory of Elasticity slant on the problem, and generally how the curved region of the beam works, I would not use my old methods of doing the analysis any longer. Hide your edress in your answers, someone at something dot something and I’ll get back to you. Make a copy of that “washington.edu” link and study and understand it, that was about the end result of my investigation and it gets you the bending stresses in the curved portion. Then we can talk about how you handle these forces and stresses dependant upon your actual structure.

Once you give us a clue about what your structure looks like and how it’s loaded, I’ll comment more. I’ll really have to dig to find those files, I think I still have them, but they might be 30 - 40 years deep in the files. I was doing those problems for 5 or 6 yrs. by hand before my young assistants came along, and I really haven’t done that problem in many years. You really shouldn’t need help finding the stresses and forces with FEA, you may need some help in interpreting them, and in knowing what to do to account for them in your fabricating, welding, etc. or in your sizing and location of reinf’g. in the case of conc. beams.

 

[JoshPlumSE]

No one has mentioned designing the curved beam for torsion... Are you all just ignoring torsion design, or is it too basic of a concept to bring up? I would think that (at least for steel design) the torsional stresses would be a big deal and would be relatively difficult to calculate for curved beams. At least when it involves sections subject to warping (i.e. wide flange beams or channels).