Thin plate buckling - out-of-plane deflection magnitude?

[bugbus]

Hi guys,

I'm dealing with a situation where a thin plate is subjected to an in-plane stress and expected to buckle, which is no concern for the strength of the structure - but I am interested to know how to calculate the theoretical out-of-plane deflection of the plate so I can check if this will cause any unforeseen problems.

The critical stress is calculated from the below equation with kc = 4.0 (both longitudinal edges are pinned):

With t = 6 mm, v = 0.25, E = 200,000 MPa and s = 1930 mm ---> critical stress is 6.8 MPa. This is confirmed by a quick FE model:

The out-of-plane deflection shown above (generated by the linear buckling model) does not have any physical significance, as far as I know - only useful in terms of relative deflection (or 'shape').

Instead, I performed a nonlinear analysis on the plate and monitored the out-of-plane deflection vs applied stress. As you can see below, the plate remains perfectly flat until the buckling load is reached (at about 7 MPa), and then suddenly forms the buckled shape.

Is there any way that the out-of-plane deflection can be calculated for a given stress (greater than the critical buckling stress)? I would like to verify the FE results with something theoretical ideally.

Thanks!

 

[rb1957]

why surprised by the FEM results ? there'd be no out-of-plane deflection until the instability developed. A column (ok, a perfect column) is straight until it buckles, and then it isn't !

if you want anything like "truth", then you'll need some test results. I suspect there is little out there, there may be an equation in Roark. I suspect you'll need to do your own testing. If possible, I'd interest a nearby university/college ... sounds like a good bachelor's thesis ... testing, and modelling.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[ThomasH]

Hi

If you apply a realistic imperfection you should be able to get out-of-plane deformations. Nothing is perfectly flat or perfectly straight.

Thomas

 

[centondollar]

I would not trust that FEA result. If you do a non-linear buckling analysis, there needs to be some initial imperfection (or small lateral load) to cause a gradual increase in deflections that culminates in a sudden increase (buckling) of deflection. With the von Kárman non-linearity, the plate should also shorten axially, and this axial shortening should be coupled with the out-of-plane deflection. Does your model show axial shortening prior to and after buckling?

Is there any way that the out-of-plane deflection can be calculated for a given stress (greater than the critical buckling stress)? I would like to verify the FE results with something theoretical ideally.

For plates, there is no simple hand-calculation approach.
One estimate is given by the secant formula (unit width beam). Another estimate is given by Rayleigh-Ritz for a unit width beam with compressive force and the curvature term expanded up to 2nd order.

 

[ThomasH]

Perhaps I should clarify .

I would start with a linear buckling analysis with the correct loading and boundary condition. Then I would use that buckling shape as base for the imperfections. If you assume a reasonable amplitude for the imperfections and use a non-linear analysis including large deformations and plasticity (if applicable) you should be able to get usable results. The end result depends on how accurate the initial imperfections are.

I would not be surprised if there is a theoretical solution in Timoshenko's "Theory for Plates and Shells" but I don't have it where I am now.

Thomas

 

[rb1957]

Some good comments.

yes, initial imperfections ... but then that asks how large and what shape ?

yes, I won't trust the FEA results.

material shortening ? what about poisson effects ? (stretching on the transverse axis ?)

and what sort of buckling are you looking at ? elastic or permanent ? are you looking to see effects under service (fatigue) loads, or limit load, or ultimate design loads (just before failure) ?

yes, would expect it to be in Tim O'Sheko if anywhere.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[SWComposites]

If my memory is correct, NASA Langley has published various reports/papers over the years with analytical solutions for flat plate post-buckling. There is also a section on plate post-buckling in Timoshenko and Gere, Theory of Elastic Stability.

 

[JStephen]

I don't remember that coming up in Timoshenko's stuff. Most of the theoretical buckling information is oriented towards determining the buckling loads, not post-buckling behavior. In addition to Theory of Plates and Shells, Timoshenko also had "Theory of Elastic Stability" (and note the "elastic" in the title). The first reference was written/updated in 1959 if I remember correctly, and the second in the 1930's.
You might also consider how this gets built. If it's welded, weld distortion may be comparable to the buckling behavior you're trying to evaluate.

If you have a situation where the deflection is unrelated to the internal forces, you might get an estimate of deflection by taking a strip in one direction or the other, assuming length along the strip remains constant, and see what curvature is required to match the end deflections. Example: If you have thin plates attached to large beams where the beams are actually carrying the load rather than the plates.

 

[centondollar]

"Theory of Elastic Stability", 2nd edition, chapter 9.13 details an energy-based method for determining post-buckling displacement of a rectangular plate loaded in uniaxial compression, with lateral (transverse to loading direction) displacement of the plate prevented or free. Not sure if those formulas are of much use in practical design, though, since plate panels are often stiffened, supported in various ways, loaded by shear or irregular load and so on.

 

[rb1957]

NACA/NASA have done a lot of work on shear buckled structures.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[HTURKAK]

Buckling of plates is not end of the game. You should look docs. for POST BUCKLING ..

Two of them ,


-Stability Theory and Its Applications to Structural Mechanics ( by Clive L. Dym )

- Handbook of Thin Plate Buckling and Postbuckling (by Bloom F. )










Tim was so learned that he could name a
horse in nine languages: so ignorant that he bought a cow to ride on.
(BENJAMIN FRANKLIN )

 

[bugbus]

Great suggestions, thanks all. I will check these out and report back with my results