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# Local buckling of thin-wall tubes in bending2

## Local buckling of thin-wall tubes in bending

(OP)
For thin-walled square and rectangular tubes, I'm trying to find a formula for the loss in bending strength due to local buckling in the compression 'flange' and side walls.

I've heard of heuristics like keeping (width/t) < 50 to avoid dimpling of the walls; I'm looking for something more specific.

For torsion of thin-walled round tubes (steel & aluminum), I found an approximate formula in Roark and Young:
'Loss Factor' = 1600 /(( (OD/t) - 2)^2 + 1600)
(p 318 of 5th ed... formula is from 1935)
but they don't have the equivalent for square & rectangular tubes in bending.  According to that formula, there is a 20% loss in torsion strength when OD/t = 22.  And 50% loss when OD/t = 42.  Quite a lot for some common tubing.

### RE: Local buckling of thin-wall tubes in bending

It is my understanding that all of the standard round and square tubes are sized such that local buckling is not going to be the mode of failure.  I am not certain of that, but I believe it to be so.  Perhaps others would like to comment.

BA

### RE: Local buckling of thin-wall tubes in bending

AISC Steel Construction Manual 13th Edition, Chapter F, Section 7 - This will give the equations you need.

See the front of the book where there give HSS geometric properties.  At the end of that section they tell you how do determine in the shape compact, non-compact, or slender.

### RE: Local buckling of thin-wall tubes in bending

abusementpark,

The question is not whether the shape is compact, non-compact or slender.  The question is about local buckling of thin walled HSS members.  I do not believe local buckling is a problem with commonly available HSS sections, but if anyone has any more info on this, please let us know.

BA

### RE: Local buckling of thin-wall tubes in bending

Perhaps the OP is concerned about the local buckling which occurs when tubes are rolled into tightly curved shapes.  For normal bending members, local buckling is not a problem.

### RE: Local buckling of thin-wall tubes in bending

In Canada, steel sections fall into Class 1, 2, 3 or 4.  Class 1 sections can attain the full plastic moment and subsequent redistribution of the bending moment.  Class 2 sections can attain plastic moment but need not allow for subsequent redistribution.  Class 3 sections permit attainment of the yield moment.  Class 4 sections generally have local buckling as the limit state.

In the particular case of HSS members:

Class 1    b/t <= 420/√Fy
Class 2    b/t <= 525/√Fy
Class 3    b/t <= 670/√Fy

In Canada, we use the SI system, so Fy is expressed in MPa.  For HSS, Fy would normally be 350 MPa, so a Class 3 HSS requires a b/t ratio less than or equal to 35.8.

BA

### RE: Local buckling of thin-wall tubes in bending

#### Quote:

abusementpark,

The question is not whether the shape is compact, non-compact or slender.  The question is about local buckling of thin walled HSS members.  I do not believe local buckling is a problem with commonly available HSS sections, but if anyone has any more info on this, please let us know.

AISC defines flange local buckling and web local buckling as applicable limit states for hollow structural sections (HSS) in bending.  However, you need to the classification of the shape (compact, non-compact, or slender) to determine in the limit states are applicable.

I forget how it is precisely defined, but I know that if it is a compact section, then local buckling isn't an issue.

### RE: Local buckling of thin-wall tubes in bending

abusementpark,

I apologize...you are correct about compact and non-compact sections.  If by slender you mean slenderness of the wall then that would be correct too.  I took it to mean slenderness of the column.  You can give me a big fat raspberry.

BA

### RE: Local buckling of thin-wall tubes in bending

BAretired,

Haha. No problem.

Cheers.

### RE: Local buckling of thin-wall tubes in bending

Hollow sections are generally self stiffening as regards flat plate surfaces buckling, either because of their curvature (round shapes) or because there is a nearby corner stiffener.  I'm of the same mind as BA, and believe we are generally pretty safe against local buckling when we use compact shapes, in our terminology, and many commonly available HSS fall into this group.

If we could weed through all the machinations the code writers have gone through, what with all the factoring up and down for a bunch of probabilistic considerations, etc. (I hesitate to use the term 'bunch of Junk,' although that's about what I really think, because some of those people are my friends, when I'm not out buying new editions of their latest work) and  including accounting for the differences btwn. SI and US units; I suspect that the Canadian classes 1, 2 & 3 and their b/t's, and the AISC's compact, non-compact, and slender; would lead to BA's explanation and terminology 'fully plastic moment and redistribution, attain plastic moment, and attain yielding moment.'  I further suspect that they would both lead back to the same testing and theoretical work if any of us could follow the math any more.  Our (AISC's) compact, non-compact and slender are all mixed together in beam/column stability and buckling theory, lateral torsional buckling theory, unsupported lengths, and if you finally account for and protect against the former possibilities, you are working at a stress level were localized plate buckling can still be an issue.  But, this has now become a cookbook recipe exercise over last fifteen years or so, rather than an engineering understanding of the whole problem and concept.  And, I'm certainly not suggesting that I can really understand it all any longer.

Hokie>>   Look back at BA's excellent post on Newmark's Methods, since your last post there.  We have to improve our applications and interviewing methods, re: old dogs.

### RE: Local buckling of thin-wall tubes in bending

(OP)
Thanks, all, for the very helpful replies!  I checked the formulas against a range of square and rectangular tubes (up to 4" square... I'm an mech engr) and almost none fell into the Slender or Class 4 category.  The only exceptions were a few thin walled 4130 tubes, both because the walls are thin and the yield is high.

### RE: Local buckling of thin-wall tubes in bending

A very valid question on what we have called 'the coke can effect' when the wall thickness is small compared to the diameter then this can really be an issue.

As long as the section meets the wall slenderness limits stated in the code then those formulii can be used but for more slender walls you will need to use more stringent formulii.

Blodgetts would probably be a very good referenece for this, I am sure I saw a formula in there. Unfortunately I do not have access to my notes on this at the moment.

### RE: Local buckling of thin-wall tubes in bending

"Theory of Elastic Stability" by Timoshenko and Gere goes to some length in developing expressions to determine the critical buckling stress of cylindrical shells.  After that, Article 11.4 states:

#### Quote:

It is seen that in all cases failure has occurred at a stress much lower than theory predicts.  In not one case was the ultimate stress more than 60% of the theoretical.
And later:

#### Quote:

On the basis of existing experimental data, an empirical formula for calculating the ultimate strength of cylindrical shells under axial compression was developed.
So it would appear that the buckling of very thin cylindrical shells is not well understood, even by the experts.

BA

### RE: Local buckling of thin-wall tubes in bending

Back up a little bit and educate me then some:

A thin-walled member is loaded in bending, and, if overloaded, will fail somehow.

Isn't the OP asking whether a HSS will fail by local buckling first, rather than some other mode of failure?   If so, how would invoking the steel design codes tell you (me!) how it will fail if overloaded?

Robert

### RE: Local buckling of thin-wall tubes in bending

And, back to the first question, once a thin-walled tube has begun failure by buckling, its shape has been been permanently lost by the kink, and it will keep twisting about that failure point.

unlike stretching, for example, when a member will simply keep stretching under additional load; once something has kinked at a single point, the whole member has zero strength against bending.

The rest of the tube may be still straight and capable of carrying a load, but the structure has failed - will continue to fail - as soon as the first point on the wall of the tube kinks.   It (the structural member) will keep bending at that location.

### RE: Local buckling of thin-wall tubes in bending

#### Quote:

Back up a little bit and educate me then some:

A thin-walled member is loaded in bending, and, if overloaded, will fail somehow.

Isn't the OP asking whether a HSS will fail by local buckling first, rather than some other mode of failure?   If so, how would invoking the steel design codes tell you (me!) how it will fail if overloaded?

The steel design codes give equations for the nominal bending strength (prior to applying a factor of safety) of rectangular HSS based on different limit states (yielding, flange local buckling, web local buckling). Guidance is given based on the geometric properties the HSS in question as to whether or not each limit state is applicable.  So, based on the code criteria, you can determine what the controlling failure mechanism will be.

One caveat being that the equations for local buckling are most likely somewhat conservative and those sections that are barely slender enough to be governed by local buckling per the code may actually reach yield in an experimental investigation.  I can't attest to the accuracy or level of conservatism in the steel code expressions for local buckling in HSS, but they usually err of the side of safety for failure mechanisms that aren't precisely defined by theory, particular for non-ductile failure mechanisms like buckling.  I have seen this in the American steel code for other types of local buckling failures in beams.

### RE: Local buckling of thin-wall tubes in bending

#### Quote:

So it would appear that the buckling of very thin cylindrical shells is not well understood, even by the experts.

Well, it has been well established that the elastic buckling strength for any structural application is a dream when it comes to the real world behavior. The elastic buckling strength is based on the critical assumptions of a pure axial load acting a member with no imperfections.  In many instances, it is difficult to come close to achieving this even under the most idealized lab settings.  Any initial imperfections are immediately exacerbated once loading begins.  I would imagine these effects are particularly significant for a thin cylindrical shells.

Researchers at the time of Timoshenko and Gere's book did not have the tool of finite element analysis to account for these effects.  I'd imagine that by now, someone has investigated these features for thin cylindrical shells through finite elements analysis and been able match their analysis more closely with experimental results.

### RE: Local buckling of thin-wall tubes in bending

Equation (9-7) attached gives an expression for σcr for a simply supported plate of thickness h and width b.  The factor k varies with a/b ratio (a is length of plate) and may be taken as 4.0 when a/b is very large.

The ratio b/h can be calculated from Eq. (9-7) for σcr = Fy (see highlighted text).

BA

### RE: Local buckling of thin-wall tubes in bending

Buckling of (really) thin tubes under compression and bending is treated expensively in the AISI Cold Formed Steel Design Manual.

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