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.

 

[a2mfk]

Josh- its an arch as I understand, not curved in the horizontal plane...

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.

 

[IDS]

Quote: A sense of proportion here, please. How bent is the beam? Let's say it is a half-circle.

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/

 

[IDS]

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 ratio in question being Radius/thickness of course!

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[abusementpark]

OK, it is essentially a curved, half-circle, 6" thick concrete wall that is subject to external wind load. There are vertically spanning columns on each end that act like supports which the wall spans horizontally between. The span/diameter of the wall is 20', which makes the radius 10'.

 

[IDS]

I'd suggest using a frame analysis, on a unit width basis. A uniform wind load, with radial pressure loading, will just put it into compression, but if you apply the load to only part of the structure you will get some bending moments plus axial load. The bending moments should be quite small so you can afford to be quite conservative.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dhengr]

We’ve come full circle, we are essentially back to an arch or a catenary given the proportions and loadings you’re talking about. And, this is what BA and Paddington were talking about, although part of the time they were talking about a specific shape and loading, a uniformly loaded parabola. But, if I understand you, the columns are 20' apart and are vertical, the wall stands vertically, 6" thick, but is in a half circular shape in plan and spans btwn. the columns, and is loaded laterally by wind loads? Is it then constrained on its lower edge? And the loading may not be symmetrical.

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.

 

[BAretired]

Provide a sketch of the structure. Your question cannot be asnwered definitively without some idea as to the geometry.

BA

 

[abusementpark]

I'd suggest using a frame analysis, on a unit width basis. A uniform wind load, with radial pressure loading, will just put it into compression, but if you apply the load to only part of the structure you will get some bending moments plus axial load. The bending moments should be quite small so you can afford to be quite conservative.

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.

 

[abusementpark]

We've come full circle, we are essentially back to an arch or a catenary given the proportions and loadings you're talking about. And, this is what BA and Paddington were talking about, although part of the time they were talking about a specific shape and loading, a uniformly loaded parabola. But, if I understand you, the columns are 20' apart and are vertical, the wall stands vertically, 6" thick, but is in a half circular shape in plan and spans btwn. the columns, and is loaded laterally by wind loads? Is it then constrained on its lower edge? And the loading may not be symmetrical.

That's correct.

 

[BAretired]

I don't get it. I thought we were talking about beams curved in the plane of loading.

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

 

[IDS]

I'd still want to know how to account for the curvature's effect on the stress distribution.

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/

 

[paddingtongreen]

@abusementpark, that Washington link, posted by frv gives the formulas solving for rectangular sections on pages 7&8.

@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.

 

[BAretired]

paddingtongreen,

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

 

[rb1957]

agree with BA ... the OP should have started with "curved walls" and i think we'd've understood much better.

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.

 

[dhengr]

Bemusement:
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?

 

[BAretired]

The problem needs to be clarified if you want any sensible answers. Otherwise, we don't understand the issue.

BA

 

[paddingtongreen]

As I see this, abusementpark has a project that is, or includes, a semicircular wall, loaded by the wind. He is aware that the stress distribution in curved members, loaded in the plane of curvature, is not linear. He wants advice on on how to apply this in practice to his wall.

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.

 

[abusementpark]

OK, I'll post a sketch tomorrow when I gain access to scanner. I thought the descriptions would suffice and didn't intend to ignite a fire here.

I don't get it. I thought we were talking about beams curved in the plane of loading.

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.

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

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.

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?

@abusementpark, that Washington link, posted by frv gives the formulas solving for rectangular sections on pages 7&8.

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.

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.

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?

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)

Correct.

agree with BA ... the OP should have started with "curved walls" and i think we'd've understood much better.

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

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.

I'm glad someone understands.

 

[dhengr]

Abusementpark:
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.

 

[rb1957]

it seems you're willing to accept that the effect is small for your case, and are asking instead, "what if" ? i don't mind someone trying to understand the limits of an analysis; but i think you'll learn more by doing, not just asking.

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.