Elastic buckling of plates - plasticity correction factor?

[LtBuzzkill]

Hello.

I am doing buckling analysis of a flat plate under compressive load. The plate is rather thick, so the critical buckling allowable I get from the Euler-derived formulas in my text books gives me an allowable above the proportional limit of the material. No problem, I can just go to my textbook solution of plotting a correction curve based on the Ramberg-Osgood relationship between elastic stress and the tangent-modulus of the material. Enter the correction curve with my Euler stress, bounce it off the curve, and read my corrected value to the left. I'll call this the "textbook method." Easy-peasy, right?

However, my office has a precedent of dividing the Euler stress by Young's modulus to calculate the slope of a line drawn from the origin to the compressive-modulus curve for the material from MMPDS. Intersect the curve, then read off the stress level to the left and there's your corrected allowable. I'll call this the "slope-intercept method." I have no idea of the basis of this method or its validity. Can't find a reference for it anywhere, including several structures manuals that I have access to. But I am forced to use it. Does anyone else use this method, or can anyone help me understand why it is acceptable?

I know there are various theories of buckling, and various ways of calculating the plasticity correction factor depending on boundary conditions. The older methods in NACA TN 3781 and Bruhn seem particularly tedious to me. The method I described above comes from Michael Niu's book, where he states it can be used for any edge fixity conditions. Jean-Claude Flabel also makes reference to a plotted curve used to find the corrected stress based on a correction factor. You'll find example curves that look like the one I plotted in each of those books. So I am convinced this is a common, textbook solution, and it is fairly quick and easy to do.

The slope-intercept method gives me values close to what I get from the textbook method, but I prefer using methods for which I can produce a reference, and that I understand.

LBK

 

[rb1957]

if your in-house method works (compares reasonably with other analytical approaches) then you're good to go.

Reference it as "in-house" method as used in report ABC, or someway so that your people can follow where it came from and so "oh yes, our method of plasticity, sure".

It does sound like a neat method ... you have a slope lower than E so you intersect the material stress/strain curve somewhere beyond yield ... cool !

another day in paradise, or is paradise one day closer ?

 

[dik]

When I make 'design note' sheets using excel or SMath, I usually include alternative measures so that I have a reasonable output using the various measures, and, as new ones come along, I append these to my original design sheets.

Dik

 

[LiftDivergence]

One method I particularly like for plastic buckling of a flat plate is presented in Peery Sections 14.12 and 14.13. He references Cozzone and Melcon which is also mentioned in Bruhn. It requires computation of an indexing parameter which relies on the Secant Yield stress for a slope of 0.70E. Sounds like the second method you describe above is working with a secant but it may be missing something. The other methods I am familiar with including the one in Peery I mentioned all rely on non-dimensional curves of some kind. I guess I'd have to see an example.

A paper that might be of interest to you:

Cozzone, F.P., and M.A. Melcon, "Non-Dimensional Buckling Curves - Their Development and Application"

The Handbook of Structural Stability, Part 1 which you mentioned references this as well as Ramberg & Osgood, which is NACA TN 902 which you should be able to find. How does your method differ from the classic three parameter method of Ramberg & Osgood?



Keep em' Flying
//Fight Corrosion!

 

[Sparweb]

Our friends at Boeing also have something to say on the matter: BDM 6520.
"Inelastic reduction factor" is the term they use.
The factors are dependent on the edge fixity.

STF

 

[LtBuzzkill]

It's been a while since I made the original post, so I thought I owed everyone who responded an update.

When I wrote the original post my nose was out of joint because I was excited about having found a powerful, referenceable method for correcting buckling allowables for plasticity, only to have it pedantically rejected. But having compared the results from both methods (I did combine them in one Excel sheet so they can be compared in the future, thanks Dik) I've found them to be no more than 1 or 2 ksi away from each other for the stress levels I'm looking at. Besides, the slope-intercept method does give a handy way to deal with clad sheet when the buckling strength is low (the cladding adds to the strength at low stress levels, and using the compressive tangent-modulus curve is the only way I have found to account for that). So I decided that this is not the hill I want to die on.

I did some reading up on how to correct for plasticity using other structures manuals, as well as the three-parameter method in NACA TN 902 (thanks, LD), so I do feel better informed about how the aerospace world outside our office does this. While I still think our method is too simple to be true after reading up on other methods (including those that consider edge fixity), and it still bugs me that I can't find a published reference for it, I figure it's good enough for now.

Thanks to all for your input. Keep it coming, if you have more to add.

LBK

 

[sdm919]

Hi LBK,

Regarding the method you mentioned in your first post: If I understand your method, I used it in school years ago, although haven't done it since. However, it is given as Figure 14-20 page 423 in the book "Theory and Analysis of Flight Structures" by Robert M. Rivello. (By the way, besides Bruhn, this is one of my favorite Aircraft Structures books).

Stress divided by modulus is strain, and the line you draw from the origin represents that strain. Where this line intersects the compressive modulus curve gives the stress (vertical axis) and modulus (horizontal axis) for that strain. So if the compressive modulus curves you are using are Tangent Modulus curves (they probably are), this amounts to your using the Tangent modulus in your buckling analysis, and your plasticity correction coefficient would correspond to eta = Et/E. I think the curves in MIL-HDBK-5 are tangent modulus curves, but you should find out what the ones you are using really are.

In general, using Et is considered the right way to go for columns. For plates, there are several plasticity correction factors that have been derived depending on loading and boundary conditions. See NACA-TN-3781, part I of Handbook of Structural Stability, or again Rivello's book. For columns, the correction factor is Et/E, for shear/torsion dominated it is Es/E, where Es=secant modulus. Plates under compression are a combination of Et and Es. A general factor for plates was derived by Stowell and is given as eqn. 15-43 in Rivello's book. However, we know that the tangent modulus is the most conservative, and the secant modulus is the least conservative and all the other methods, being some sort of weighted averages of these 2 are somewhere in between. So you are safe using tangent modulus in all cases if you don't want to go to the extra effort.

Does this make sense?

 

[sdm919]

Hi again,

One more point I want to highlight: the Ramberg-Osgood formulation is a way to represent the stress strain curve analytically, but does not directly give you the plasticity correction factor. It does give you the tangent modulus (Et) and if you want the secant modulus (Es) at a specific point, and these parameters can be used to calculate the specific plasticity correction factor you want.

So, if you are using Ramberg-Osgood only, without any specific equations for the plasticity correction factor, you are probably replacing E with Et in your buckling formula, which is the correct plasticity correction factor for columns, but is conservative for plates. Plates have a 2D effect, even if loaded uniaxially, which makes them retain their stiffness better than a 1D column when they go plastic.

Fig. C5.7 in Bruhn has the plasticity factors for flanges, where torsion dominates, and it is governed by Es/E. Fig. C5.8 in Bruhn is the Stowell factor, which is a combination of Et and Es. He gives the equations for the plasticity correction factors for these cases in the caption to the figure, even if he does not give it in the text itself.

Sorry for the double post.

 

[LtBuzzkill]

Thank you sdm, for your input.

Thanks for confirming there is some validity to finding the intersection of the strain line and the tangent-modulus curve. I had not found that method anywhere. Now I'm a bit more comfortable using it. I haven't followed through and compared the result with using a correction factor from a combination of Et and Es from a structures manual yet, so I don't know yet how conservative it is. I may have to find out for myself at some point.

As for my Ramberg-Osgood method, I am using an equation for the plasticity correction factor. So it is not a straight-up substitution of Et for E in the buckling formula. I'd like to know how Niu came up with that equation since he doesn't seem to give a source for it. But for the time being I have a method that works.

Thanks to all for the responses.

LBK

 

[sdm919]

Hi LBK,

There is a NACA report showing a comparison of the different (7 altogether) plasticity correction factors on one graph.

A Unified Theory of Plastic Buckling of Columns and Plates
by Elbridge Z. Stowell
NACA-TN-1556
or
NACA-TR-898
(2 slightly different versions of same report)

Just for my curiosity, which Niu equation are you referring to?

Thanks,
sdm919

 

[LtBuzzkill]

Hi, sdm.

Sorry for the late reply. The plastic reduction for compression is eta = (Et/E)^0.5. For shear stress it would be eta = (Gt/G)^0.5. Et and Gt are defined by the Ramberg-Osgood relation.

Thanks,
LBK