## Design of Curved Beams

## Design of Curved Beams

(OP)

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.

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.

## RE: Design of Curved Beams

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/

## RE: Design of Curved Beams

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."

## RE: Design of Curved Beams

## RE: Design of Curved Beams

rb157 is right. it is not linear.

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## RE: Design of Curved Beams

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.

## RE: Design of Curved Beams

Yes, or an upside down arch.

Let's say it is a half-circle.

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

## RE: Design of Curved Beams

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.

## RE: Design of Curved Beams

BA

## RE: Design of Curved Beams

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.

## RE: Design of Curved Beams

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

## RE: Design of Curved Beams

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/

## RE: Design of Curved Beams

## RE: Design of Curved Beams

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.

## RE: Design of Curved Beams

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## RE: Design of Curved Beams

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.

## RE: Design of Curved Beams

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?

## RE: Design of Curved Beams

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/

## RE: Design of Curved Beams

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.

## RE: Design of Curved Beams

## RE: Design of Curved Beams

Quote:

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

Yes, or an upside down arch.

Quote:

A sense of proportion here, please.

How bent is the beam?

Let's say it is a half-circle.

## RE: Design of Curved Beams

The angle through which it is curved is not important. The important criterion is the depth of the section over the radius of curvature. I'm not aware of any firm rules, but if the ratio is less than about 10 the non-linear strain across the section will start to become significant. Greater than that it is OK to use standard section design methods. Either way you obviously need to take account of the shape of the beam in analysing the actions at any section, but modelling the beam as a series of short straight segments is normally a perfectly OK way to do that.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Design of Curved Beams

The ratio in question being Radius/thickness of course!

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Design of Curved Beams

## RE: Design of Curved Beams

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Design of Curved Beams

It would seem that DougJ and I are in agreement on proportions, radius/thick. and analysis modeling methods. I also agree with him that the angle of curvature is not important, but at the change from curved to straight there will be some transition length btwn. the two. I don't recall a magic number for rad./thick, but I would guess that >10 is a real safe break point. Run a few FEA models with rad./thick of 10 and on down, and compare them to our regular straight beam analysis results for bending.

## RE: Design of Curved Beams

BA

## RE: Design of Curved Beams

The wind code requires a point load at midspan for wind load on a cylindrical surface. So there will definitely be some bending moments.

The frame analysis is still just going to give me a bending moment and axial force for design. I'd still want to know how to account for the curvature's effect on the stress distribution. Maybe in this case, I could do something really conservative like take what the straight beam requirement would be and then double it, but that isn't very rational. I'd like to know how to do this somewhat rationally if I'm later faced with a situation where I am going to be pushing a curved beam section to it's limit.

## RE: Design of Curved Beams

That's correct.

## RE: Design of Curved Beams

Now we are talking about a semi-circular wall supported only at the two ends. This is not the same problem at all. There will be significant torsional stresses due to gravity load.

BA

## RE: Design of Curved Beams

I'll say it once again. With this thickness to radius ratio the effect of the curvature on the strain distribution is totally neglible. Bending moments, axial load and shear forces from the frame analysis are all you need.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Design of Curved Beams

@IDS, he wants to know for himself, that's how apprentice engineers become journeymen engineers and journeymen engineers become masters of the profession. If you take my meaning.

Michael.

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

## RE: Design of Curved Beams

I don't know about IDS, but I do not "take your meaning". I believe that this is one of the most confusing threads I have ever read. Why doesn't somebody say something clearly and sensibly so that we can get it all back on track?

BA

## RE: Design of Curved Beams

draw a free body of your wall. the airload goes from being reacted as shear in the beam/wall at the mid-span to being reacted axially at the supporting columns. it's easy to build the internal forces in the beam/wall, yes? (i assume so, since you're mainly interested in the effects of the curvature on the local stresses in the beam/wall).

follow th elinks provided and you'll see the d/R is small and the non-linear effects are negligible.

as others have noted, remember to include the effect of weight.

## RE: Design of Curved Beams

Get on the stick, draw a sketch and clarify your problem, and answer some of our questions. You have a bunch of people here trying to be helpful, but you're doing a damn poor job of explaining what you are really dealing with. After your OP and much back and forth, both DougJ and I are telling you that it would appear that what you are dealing with, would not, should not, be called, a curved beam, and would not be analyzed as a curved beam, it's a circular arch. You can keep asking the same question and our answer won't change, thus DougJ's last post, and I still agree with him, with a radius/thick. of 10'/.5' = 20 you do not have a curved beam problem. I suggested that you copy and study and understand the washington.edu link, and if you wish apply that to your problem, but you will see that it really doesn't move the N.A. an amount worth considering. Divide that darn thing into 15 - 2' beam elements and have at it. It appears to be an arch, so study arch design, however loaded. Don't make us guess, but my guess is that its an arch laying on its side, 20' dia., 6" thick wall standing how high, and loaded by horiz. loads (the wind), thus the curved in the plane of loading, and I said constrained on its lower edge (meaning, on a footing) or is it also cantilevered out into thin air off the columns?

## RE: Design of Curved Beams

BA

## RE: Design of Curved Beams

Rather than take the time required to write a full project description, he tried to isolate that one effect with an unrelated hypothetical structure.

It didn't work because the rest of us were immediately detoured because we, being experienced practical engineers, were uncomfortable with seeing only one facet of an obviously multifaceted problem.

While the nonlinear effect is in the elastic range, he sees that the neutral axis probably moves back to the centroidal axis for the full hinge, but he is wondering what happens when the concrete starts and progresses to fail. I suspect that the same thing happens.

As to my note to IDS, when I was young, I didn't accept assurances such as IDS's that all would be well, I wanted to find out for myself so I would have the assurance for the rest of my career, in addition to the experience of the research itself.

Having said all this, I will probably prove to be quite wrong.

Michael.

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

## RE: Design of Curved Beams

The wall is supported at its base, that's why I didn't mention anything about gravity loads.

OK I understand that, but what if the thickness to radius ratio was such that the effect of the curvature on the strain distribution wasn't negligible. How would account for in an ultimate strength design?

It gives formulas for the strain distribution, which is non-linear. However, that doesn't quite tell the story for ultimate strength design. For example, concrete ultimate strength is based on the assumption of linear strain.

I'm sorry but what is the difference? Unless I am missing something here, no matter what my geometry and supports conditions are, there won't be the case of pure compression because the loading is not uniform... right?

Correct.

Well, for design, I was thinking you would analyze a 1' horizontal strip like a beam.

I'm glad someone understands.

## RE: Design of Curved Beams

One of the problems I have with the newer editions of the codes and all this LRFD, and ultimate strength design, and the like, is the difficulty I have in getting from my Strength of Materials or Theory of Elasticity understanding of the problem to the final design formulas and equations in the code for that problem. What with all the intervening voodoo; load factors, resistance factors, factors more than unity and some less than unity, some in the numerator, and some in the denominator, some if the sun is shining, etc. etc. etc...... which lie btwn. to two realms.

For your problem, a 10' radius circular arch, 6" thick, I would find the moments, shears and axial loadings by whatever method you like, for an arch and as a function of the various load conditions. Then I would treat those just as you would for a simple straight beam, in the material you are using and too that mat'ls. code. The tightly curved beam theory as outlined in the Washington.edu link does not apply to your case, it does not cause a significant enough N.A. shift to make a difference, and you should be able to see this by doing the r = h/ln(ro/ri) math. You will not have only compression in your arch, because it is not parabolic (which is a different geometry consideration than the tightly curved beam geometry), and also the loading is not uniform over its span length. The unit width strip seems like a reasonable member to look at. But, just to throw a new monkey wrench into the gears, this is a two way slab. Arched in one direction, but with one arch stiffness at the free top edge of the wall and quite a different arch stiffness where the footing acts as a deep beam arch edge stiffener; and then in the vert. direction it is free at the top, and cantilevered off the footing at the bottom of the wall, and you still won't tell us how high the wall is. You guys use FEA and computer programs to analyze and design everything these days and it seems FEA would be quite appropriate here to account for the arch and cantilever, two directional, action.

If, at some point in time, you are dealing with a beam with a tight enough curve to bring the stress change development, N.A. shift, as shown in the washington.edu link into play, might you not do the following: simple straight beam bending moment implies f = Mc/I; curved beam theory implies fi = Mci/Aeri & fo = Mco/Aero, their equations (7); now you've determined your load factored moment by regular methods, and I would just adust that moment by the ratio {(c/I)/(ci/Aeri)} for the inner fiber or by the ratio {(c/I)/(co/Aero)} for the other fiber. These adjusted moments would be plugged into the std. code equations once to look at inner fiber conditions and a second time to look at the outer fiber conditions. These are fairly quick, shootin from the hip thoughts, and I'll need to ruminate on them for a while, or listen for other ideas. If you actually run into the problem, lets broach the subject again, and I'll dig out my files, and see how I might adjust what I did then to try to conform to today's code. Again, my work was in steel not concrete, so a steel structure would be my easiest starting point.

## RE: Design of Curved Beams

Assuming you're talking about a masonry/concrete wall (without rebar) i'm guessing you're more concerned with tension stresses, yes? then read the posted links and see what would it take to get tension stresses in excess of the typical bending distribution. what some of geometry would give you excessive compression stresses ?

But then your OP talks about steel, and plastic stresses. so i'd start with the links and load my problem untill the results exceed the yield stress and then wonder how the material would react.

## RE: Design of Curved Beams

- Get the strain distribution for the particular radius/thickness ratio, from the link posted earlier.

- Apply the ultimate strain specified in the applicable code (usually 0.003 or 0.0035) at the compressive face.

- Assume a neutral axis position and calculate strains and stresses across the section using published stress/strain curves for steel and concrete (linear/ perfectly plastic for the steel, and parabolic/ plastic with no tensile strength for the concrete makes it easy and is perfectly adequate for design purposes, but you can use something more precise if you want to).

- Integrate the stresses to find the nett force.

- Adjust theneutral axis position and repeat until the nett force is equal to the applied axial force at the section.

- Take moments about the section centroid. That's the section ultimate moment capacity.

- Apply the capacity reduction factor in accordance with the code, and compare with the applied (factored up) design bending moment.

- Compare with a conventional analysis and let us know if the difference is more than a few percent.

Because the steel is likely to be well past yield, and most of the concrete in compression will also be past yield, the effect of the non-linear strain is likely to be very small, but for a sufficiently thick section (or small radius) it may become significant.

Doug Jenkins

Interactive Design Services

http://newtonexcelbach.wordpress.com/

## RE: Design of Curved Beams

## RE: Design of Curved Beams

There is one fundamental issue that has been missed due to the fact that things went off in the wrong direction ( a better description at the beggining would have given much more appropriate comments from the start).

If this is precast cladding then then needs to be provision for thermal movent and if so then arching action will be out of the question. You will then have to treat it as a beam.

Wind suction plus eccentric dead load will give your maximum support reaction at the top.

Even if it is fully restrained at the ends then wind suction will induce tension in the member.