Buckling of circular arches and rings: a general question. 7

[WARose]

Recently I had a couple of projects that involved lifting rings [that were part of equipment] and a couple of arches that were loaded with both in-plane and out-of-plane forces. (By in-plane, I mean forces that are applied radially to an arch or something like a vertical load that can cause snap through buckling. By out-of-plane forces I mean forces coming in from a perpendicular direction to that…..something like wind load.)

I pulled all available resources off my shelf……but was still left with a very important question: almost all the formulas that dealt with arch/ring buckling typically gave buckling values that we caused by in-plane forces. I really couldn’t find anything for out-of-plane loads. Now, one reference I have [i.e. ‘Guide to Stability Design Criteria for Metal Structures’ by: B.G. Johnson, 3rd Edition, p.476-477] had this to say on the subject: “Out-of-plane buckling of rings, arches, and other curved planar members is in every way analogous to lateral-torsional buckling [LTB] of beams and beam-columns.” They proceed to give a number of buckling formulas, but I’m not sure they are applicable to the out-of-plane forces I discuss here.

So I guess what my question is: where can you find allowable for such a situation? Is it just a matter of calculating LTB as per an applicable code? If so, what would you call your unbraced length? For example, in that ring lift scenario I mentioned earlier, we just basically lifted a ring with attached lugs (8 of them around the circumference; it had a 30’ radius). The self-weight of the ring is causing the out-of-plane moment. Obviously the lifting point isn’t restrained torsionally…….but laterally, I would think the arch [think of the ring in pieces] would be stiff enough to prevent lateral translation (but obviously that would require calcs). [What I did there for unbraced length was subdivide the circumference of the ring into 4 pieces; because I reasoned that it would buckle into at least 4 parts.] Same deal with an circular arch (receiving out-of -plane load): at the least, I would think the highest unbraced length would be ¼ of the circumference (if it was a complete circle). Only thing about those approaches is: you kind of counting on the structure to brace itself…..and I always thought that was big no-no.

Anyway, any insight is appreciated…..and be sure to have a good 4th.

 

[JoshPlumSE]

I don't think you will find a one size fits all formula for an allowable load.

There have been some similar discussions on this list related to buckling loads for tapered or stepped columns and such. Where you use a method of successive approximations to give the buckling load. But, I believe that method is better suited for nominally straight members.

My recommendations would be to use the AISC's Direct Analysis method. You can apply some lateral notional loads to create some initial displacements so that the P-Delta analysis will capture the buckling of the arch. This actually works really well provided that your notional loads create an initial displacement that is reasonable close to the correct buckled shape of the strucure. In some programs, you could run an Eigen buckling analysis of the structure.

 

[IDS]

You might find my blog article here useful (or at least interesting)

http://newtonexcelbach.wordpress.com/2011/01/01/buckling-of-rings-and-arches/

It includes a link to a spreadsheet that will do the buckling analysis for an arch or ring. I'd suggest doing an incremental non-linear analysis (including non-linear geometry and material effects) in a 3D finite element or frame analysis program, and use the spreadsheet as a check. Either way make sure to verify your results with at least two independent methods. Also note the comment at the end of the blog article about the effect of asymmetric buckling; it can greatly reduce the buckling load.



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[dhengr]

WARose:
That is a nasty problem and Josh probably has the right idea and the right software to start honing in on the problem. I don’t know of a direct ‘old school’ approach to the solution. As you suggested we have pretty good classical literature on arches loaded in their plane and I’ve done a number of these over the years with varying rise to span ratios. For your lateral loading condition, I would liken the arch to a curved beam, loaded perpendicular to the plane in which the curved beam exists. That might be like a semi-circular balcony fascia beam resisting the gravity loads and fixed against bending and torsion at its supports. And, there is a fair amount of literature on that problem too, for bridges and buildings and the like. Although, I think I always had trouble finding good buckling criteria for that problem too. And, the two should be able to be superimposed to start to get to Josh’s computer approach. I have the same Ed. of Bruce Johnson’s book, and at a quick glance I probably have a few of the ref’s. he cites. I’ll try to see if I can find anything.

 

[WARose]

I appreciate all those that have replied thus far. As several have said: this may lend itself to a computer solution but I’ve always been a bit wary of that as I’ve never used a program to predict such a buckling load. [I remember I tried it on STAAD one time and didn’t even understand the output.] From what I understand, such a solution can be extremely unconservative if certain things aren’t taken into account (initial imperfections, localized buckling, proper safety factor, etc.)……so that’s why I lean towards a “book” solution that can be combined with other results manually.

 

[BAretired]

The problem is too difficult for the average engineer to analyze. For out of plane forces, it is best to provide lateral bracing.

BA

 

[rb1957]

i'd look into FEA to analyze the ring. maybe if the frame is a constant section, uniform load, cantilvered base then you could reasonably easily find the static reactions. i'd wonder if there was something self-cancelling happening with the reactions, so the FEA suggestion.

for an allowable, i'd calc something for crippling, or if stable then fcy

 

[msquared48]

I believe what you are referring to ids called "snap-through" buckling, and can happen with an arch, or similar structure, that is eccentrically loaded. Studied it a little, very little really, in graduate school many years ago.

Mike McCann
MMC Engineering

http://mmcengineering.tripod.com

 

[charliealphabravo]

Sounds like a problem for ANSYS and the wind tunnel procedure or published wind tunnel studies. Depending on size and slenderness the structure may be sensitive to gust effects and resonance at natural frequencies which won't be predicted by any load pattern made up of the static code wind loads. You might be able to get around this with dampeners.

 

[WARose]

I believe what you are referring to ids called "snap-through" buckling, and can happen with an arch, or similar structure, that is eccentrically loaded. Studied it a little, very little really, in graduate school many years ago.

I always thought that snap-through buckling happened because of loads applied in the plane of the arch (the typical example being a point load at its peak)……..the type of buckling I refer to here is more analogous to a curved beam [i.e. a bridge girder]. That situation (by everything I’ve read) is covered by the lateral torsional buckling [LTB] equations available in most of the code we use (while also accounting for the torsion such an analysis shows)…….but what prompted my op was: using such equations discounted the stiffness [in-plane] that an arch/ring had, ergo you’d think it would be have a higher LTB value than just considering it a curved beam spanning from x to y.

 

[JoshPlumSE]

WARose -

You bring up a great point.... If you use a computer / FEM solution to model something like this then how do you use hand calcs to verify the accuracy of the analysis tool?

Honestly, this is something that we should be teaching in engineering school now. But, we're not. Graham Powell had a series of articles in Structure magagize (or structural engineer, I always get those two confused) where he talked about this gap in current engineering education.

Roark's formulas for stress and strain has a chapter for elastic stability calculations. I believe there are some arches in there. I'd still say that you want to use the Direct Analysis method. Because that does a better job of accounting for inelastic buckling. But, the formulas in Roark's may give you a starting point for verifying that your software modeling techiques will give you accurate results.

 

[WARose]

You bring up a great point.... If you use a computer / FEM solution to model something like this then how do you use hand calcs to verify the accuracy of the analysis tool?

And that's one of the reasons I've never used a program to predict a buckling mode.

Roark's formulas for stress and strain has a chapter for elastic stability calculations. I believe there are some arches in there.

I know. In fact I used some of them in my analysis.....the only problem is [like I said in my OP]: they are mainly for in-plane/radial loads. While it does give buckling loads for in-plane and out-of-plane considerations: again, its just considering in-plane loads.

 

[JoshPlumSE]

Well, computers can be useful sometimes. ;-)

Use Roark's and other hand calc methods to validate that your program is giving you essentially the same results as you would predict. Then you should be able to have more confidence in the programs ability to work on a theoretically similar problem that you cannnot do by hand.

 

[IDS]

Well, computers can be useful sometimes.

Use Roark's and other hand calc methods to validate that your program is giving you essentially the same results as you would predict. Then you should be able to have more confidence in the programs ability to work on a theoretically similar problem that you cannnot do by hand.

I agree. Many people talk as though a "hand calculation" will always be more reliable than a computer analysis, but the fact is in situations like this hand calculations can give grossly misleading results as well, and a carefully set up computer analysis will pick up failure modes missed in the standard hand calcs, and can also be extended to situations with no standard hand calc solutions.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[ishvaaag]

I did 11 years or so ago a worksheet based in a 1974 text of the ETSAM. Covers a simple case where lateral buckling of arches is contemplated; it is for a Tee section of concrete arch. I attach a printout of the Mathcad 2000 professional worksheet.

I also vaguely think to remember -have not checked yet- the Steel Designer's Handbook had info on lateral buckling of Steel Arches (apart from in plane). I think I also did some worksheets following it but have not met them today; if I find I may post also these.

 

[ishvaaag]

Quite likely the source for the buckling factor in the provided worksheet is german (DIN); or otherwise of the culture of professor Arangoá, that was a good old one, vested in elasticity, mathematics, and matrix analysis (and surely more that I can't know).

 

[ishvaaag]

Maybe the worksheets I will be looking for are from Galambos IV or V (and so I haven't found yet) ... tomorrow will try, time of going to bed. Nite.

 

[WARose]

Thanks for posting ishvaaag. I will dig into it when I get a chance.

One interesting thing that jumps out at me right away is the usage of curved beam formulas for stress calculation. I tried them for some of my problems, but the geometry of some of my situations (and the size of the arches/rings themselves) showed the stress were insignificant compared to the straight bar formula.

 

[ishvaaag]

In the end the parts I ported to worksheets were for in-plane loads, I found the worksheets. However, effectively, I found that the reference I used was Galambos IV

Guide to Stability Design Criteria for Metal Structures
Thodore V. Galambos
Wiley Interscience 1.988

and this book starting p. 593 has sections for out of plane stability of arches, both circular and parabolic, braced arches and requirements for bracing systems and other relevant sections for ultimate strength of arches, vertical and lateral load cases included. Since one of these is for bridges, this points to that the books on superstructure for bridges also should have also related info, maybe the Xhantakos one.

 

[ishvaaag]

Effectively

Theory and Design of Bridges
Petros P. Xanthalos
Wiley Interscience, 1994

has sections 10.8 and 10.19 dealing with buckling, out of plane in similar way to Galambos above. One should then look AASHTO etc just in case.

For ordinary structures amenable to static analysis, the Direct Analysis Method with proper initial imperfections etc should give a reasonable estimate of what to expect.

However, a slender ring dangling in the air on weak inertia axis, and subject to brusque hanger inertial loading movements is certainly a dynamic case and charliealphabravo rightly points to better and higher tools. Even with them one will need to fine tune the loadings to experiment or need be reasonably conservative to cover a somewhat typified case. Other than that we will be at some unspecified level of risk.