Buckling in Curve Plate

[tjwalker11]

What would be the appropriate model for the buckling of a curved plate? I'm looking to understand buckling in thin films (0.005" thick polyester) when it is in a curved shape, like a half cylinder. This for printing press style handling systems, where the product is wrapped around a rolling cylinder, but in some cases the cylinder would be undercut to prevent contact in the middle, like a sprocketed roller in an old film projector, but on a bigger and wider scale.

 

[JoeTank]

I'm not sure I can visualize this thing. Your post topic title mentions a curved plate, but it sounds like you have a cylinder to deal with. As-noted, I'm not sure I have the right image here yet. Could you give a bit more info?

Steve Braune
Tank Industry Consultants

http://www.tankindustry.com

 

[tjwalker11]

OK. Polyester films are coated for many applications, running in long continuous strips, 0.005" thick, 12" wide, 1000s ft long (give or take an order of magnitude on each). The film is run through a series of "rollers", with the wrap angle on a given roller of 10 to 180 degrees depending on the "web path" moving from roller 1 to 2 to 3 and so on. These rollers support the web in the curved shape.

The moving web is controlled to run at a given speed, but also a machine direction tension, typically 1 lbf per in of width. This tension pulled over the cylindrical roller creates a pressure on the cylinder of Tension/Width/Radius (psi).

In some cases, to it is desirable not to contact the center of the web (it is contact sensitive to scratching or contamination). In these cases, an undercut roller, a cylinder with small diameter in the center, is used, so the curved, tensioned film is only supported at the two ends. The problem is understanding what conditions this is unstable, where the cylinder created by the polyester film will buckle rather that hold its cylindrical shape. Radius, unsupported width, tension, modulus, thickness, supported width, all seem to be factors.

This doesn't fit Euler's buckling criteria, since it is not axially loaded. Also, the slenderness ratio (L/r) places most cases in the "short column" catagory. Lastly, an axial loading model will show larger diameters support more load, where I know from practice, that small diameters are stiffer for this scenario.

Does this give you a better picture?

 

[JoeTank]

I see your question more clearly now. I was thinking the the rotating drum was your problem. I have no experience with the polyester film buckling issue. Hope someone else can help you out on this one.

Steve Braune
Tank Industry Consultants

http://www.tankindustry.com

 

[cab1990]

Hello tj,

Buckling of curved plates has on occasion been termed buckling of the barrel vaulted roof. A google search on this term will produce many hits. However, your thin film has no bending stiffness, and if wrinkling is involved, you will probably need to use finite element analysis to solve the problem.

cab

 

[JoeTank]

This may be a bit of a long shot, but you may wish to check out the NASA publications website. They have an excellent publication on buckling of thin shell space vehicle structures. As a tank and vessel designer I have used it often to address non-Code issues of design. It is loaded with buckling information, including curved panels. You may have to hunt around a bit to find it, but I eventually stumbled into it a couple years ago. Do a Google search, that's how I originally found the NASA archive documents.

Steve Braune
Tank Industry Consultants

http://www.tankindustry.com

 

[insideman]

A wild thought: Would providing a set of air pressure jets to provide the missing support help?

 

[tjwalker11]

insideman, your thought isn't so wild. There are standard designs to float a tensioned web over a positively pressurized perforated cylinder, however, for this application, that is more complicated than we would like to use. I'm seeking the limits of undercut roller use, to know when more complicated designs are or are not needed. tjw

 

[prex]

As I suppose you want a rough insight into the phenomenon, I'll propose to you the following treatment.
A formula in the Roark gives the critical radial pressure over a curved thin panel with straight edges simply supported. The panel should be very long (take l>10r), so that end supporting conditions are non influent.
It is:

Et³(&[ignore]pi[/ignore];²/&[ignore]alpha[/ignore];²-1)
p= -------------
12r³(1-&[ignore]nu[/ignore];²)

where:
E=elastic modulus
t=thickness
&[ignore]alpha[/ignore];=half opening angle (=&[ignore]pi[/ignore];/2 for a half cylinder)
r=radius
&[ignore]nu[/ignore];=Poisson's ratio (=0.5 for elastomers)

The condition of support at straight edges is quite close to yours, as the straight portions of the film coming out of the curved part act somehow as supports (they allow only for sideways displacements).
Note also that, as one would expect, that formula reduces to that of a full long cylinder for &[ignore]alpha[/ignore];=&[ignore]pi[/ignore];/2 (due to symmetry a full cylinder behaves as two facing half cylinders).

Now it is necessary to transform the uniform radial pressure of the formula into the film tension of your setup. This is simply done by calculating the edge reactions of the curved panel in the direction normal to the tangent.
This is simply

T=pr(1-cos&[ignore]alpha[/ignore];)

By combining both equations you should get a quite good estimate of what you are looking for. Of course you will also need to take a margin of safety with respect to the critical value: a factor of 3 is normal in metal construction, but a value of 5 could be more appropriate for you.

prex

http://www.xcalcs.com

Online tools for structural design

 

[arto]

Look at ASME paper #84-APM-4 "Nonlinear Bending and Collapse of Long, Thin, open section Beams and Corrugated Panels" by R.C.Benson. Analysis of "tape measure blade" type geometry, might be applicable here.

 

[tjwalker11]

arto, can you tell me where I can get a copy of Benson's ASME paper...what conference / journal / year was this from?

tjw

 

[Sparweb]

TJ,

A few questions, if you don't mind, because I'd like to picture a few more details.

What percentage of the sheet is supported vs. unsupported middle?
Is there a sharp edge transition to the undercut middle part of the rollers?
Do you control friction with lubricants?
Is the roller perfectly cylindrical?
Is the buckling provoked by misalignment of the rollers?
Is the film at room temperature?
Is the film homogeneous? (ie., is this one sheet of the same material, or laminates of several?

Again, excuse the interrogation, but you have piqued my interest.


"Simplicate, and add more lightness" - Bill Stout
Steven Fahey, CET

 

[tjwalker11]

Steven, Happy to feed your curiosity...

What percentage of the sheet is supported vs. unsupported middle? (This would usually be a low percent, say 0.25-1" on either side of a 8-14" wide web, but I'm wondering what the limits are...one of the reasons to find a model for this.)

Is there a sharp edge transition to the undercut middle part of the rollers? (Likely yes, but the transition edge would be tapered or rounded. I've been thinking to myself how this aspect of the design increases the shear stress at the transition and may greatly change the design limits.)

Do you control friction with lubricants?
(No, this is usually a dry friction, steel-polyester, aluminum-paper, and such. Rollers are usually metal, or rubber covered metal cylinders.)

Is the roller perfectly cylindrical?
(Traditionally, yes. However, some subtle tapers are used in special applicationns, usually less than 1-3 mils per foot taper.)

Is the buckling provoked by misalignment of the rollers?
(This can promote and create shear related buckling and is well model by work from the Okla. St. U Web Handling Research Center. I'd like to ignore that here.)

Is the film at room temperature?
(It can be, for drying solution coatings, but usually room temp.)

Is the film homogeneous?
(Mostly homogenous papers or films, sometime with a thin coating, at least what I'm concerned about.)

Think on.

tjw

 

[Sparweb]

I understand it is the thin coating, then, that you're trying to protect, by supporting only the substrate at the edges.
Have you tried tapered rollers? I'm thinking that a taper that slopes away on each side of the 12" gap would tend to stretch the film across the gap slightly. If the edges contacting the roller aren't critical it may not matter how you blend from the major OD to the minor one. The tapering may make the sheet want to travel side to side, perhaps far enough to slip off one support. You'll need a keeper of some sort. Hmm, as soon as it touches a keeper it might just start wrinkling up again... Vibration (depending on the speed of the system) will get the sheet spanning from roller "A" to roller "B" wobbling about, which complicates your analysis, if you want to take it that far.
A rough attempt at illustration:

keeper taper groove taper keeper
----------- ---------
|___/| |\___|
-------------------

Okay, your shaft is pretty complicated now. This is the typical effect of my ideas. Thankfully, that's all I've come up with.


"Simplicate, and add more lightness" - Bill Stout
Steven Fahey, CET

 

[tjwalker11]

This "bow tie" shaped roller is commonly used as a anti-wrinkle design. However, when I design the geometry for anti-wrinkle, I try not to shift so much stress and strain to the edges that the middle goes to zero.

What I'm interested in now is moving beyond this limitation to intentionally moving the center to zero and totally unsupported. Yes, a challenge it is.

I appreciate your insight. I'm curious how a thin shell engineer understands the "spreading" effect, since it is a novel interaction of a moving web over non-cylindrical rollers. Have you worked in the converting field before, or is this trick well-known enough in design of belts and pulleys? tjw

 

[Sparweb]

Apart from standing in front of a lathe and getting slapped with a strip of emery paper, I can't claim any especially relevant personal experience.

Are you saying you want zero tension across the sheet (along the roller axis)? Uh-oh - you'll be relying on the stiffness of the substrate, for sure.

Is the delicate coating on both sides of the film, or just one? Can you pass the sheet through double-rollers? (Where the idler is a full cylinder)

---->--
O)O The idler is on the outside of the turn.
----<--

As for analysis of thin shells, Prex already got you going on the right track, IMO. Do you have a copy of Roark's Formulas for Stress and Strain? Beware the end-restraint assumptions in any analysis you do, because the edges of your sheet aren't simply-supported by the roller, nor are they fixed, either.



&quot;Simplicate, and add more lightness&quot; - Bill Stout
Steven Fahey, CET

 

[tjwalker11]

SparWeb (Steven),

Thanks for your input. I've ordered a Roark's (didn't have one, but should of).

As to your Qs:
1. Product can be coated on either side, but one-sided is the concern here.
2. Yes, you can do this, but the difficult goal is turning the web contact-sensitive side in, which you may need to do occasionally.

Thanks for the warnings on end constraints. I want to get a model to get started, but expect some experiments will be the &quot;proof in the pudding&quot;.

tjw

 

[arto]

Benson's Article was in Trans. ASME Jnl of Applied Mechanics,from the 1984 PVP ConferenceSan Antonio,june 17-21 1984 Your local Engrg College Library might have it.