Rectangular Plate, large deflections 1

[jrmack]

I am looking into stresses and deflections of relatively thin (span/thickness approx 300) rectangular plates (clamped against rotation and somewhat rigid in the plane of the plate along the perimeter of the plate). I read that when the deflection is in the order of the plate thickness (or above), that some finite element assumptions no longer apply. I want to consider support of uniform load carried by not only plate bending, but also membrane tension. The membrane tension I would be like to consider is the same as what supports a trampoline canvas (and it is not plate bending). Is there an algoithm that will handle this, or a short routine, possibly mathcad? I do have SAP 2000 and know that with non linear large deflection elements that this can be solved, but would rather input my variables (two widths, plate thickness and properties, and uniform pressure).

 

[amaratre]

Hi,
You are right when you say that the normal assumptions wont apply. Can you try modeling it by using a general Finite Element Package? Just a thought.
amar

 

[joekm]

This does sound like a job for a non-linear approach. However, the problem sounds pretty standard. If I understand correctly, it is a membrane clamped at all four ends under a uniform load. I wouldn't be surprised if the answer to that one exist in the literature.

If there is a local university library, I would check there. The specific form governing plates is the bi-harmonic equation. From what I understand, specific solutions to that equation have been the subject of doctoral dissertations. I think there would be a pretty good chance that your solution is out there.

--
Joseph K. Mooney
Director, Airframe Structures - FAA DER
Delta Engineering Corporation

 

[rlnorton]

Timoshenko's "Theory of Plates and Shells" has a chapter on large deflections of plates, with a fair amount of coverage on rectangular plates.

 

[pja]

A plate which is clamped on all 4 sides will exhibit a good deal of nonlinear behavior at deflections which on the other the order of a thickness. Along with the Timoshenko text take a look at a book by Chia which has some approximate close form solutions which you may be able to use.

 

[prex]

In the site below, under Plates -> Membranes -> Rectangular, there is a calc sheet for uniformly stretched and transversely loaded membranes (be warned that it won't work when the aspect ratio is too far from one).
Sorry, a sheet for large deflections of bent rectangular plates is not yet available there.

prex

http://www.xcalcs.com

Online tools for structural design

 

[Tankman650]

Timoshenko has the theory and some good examples.

Roarks Formulas for Stress and strain has a table for most edge conditions that can be programmed in to a spread sheet.

I have the 5th edition and the table is on page 408. The only problem with the table is that it only gives the max. membrane and bending stresses and its location which may not be at the edge but most likely is at the center of the span.

 

[ShaggyPE]

Check out this site:

http://www2.nttc.edu/leads/

Contained in this site are NASA tech briefs.
Look up GSC-13561 and GSC-13367

These are documents with derived formulas for Deflection and Stress in Preloaded Rectangular (and Square) Membrane.

I just finished a problem where I used this information to determine the necessary preload on a thin stretched mylar film. The equations are based on the strain energy theory which is the same as the Timoshenko formulas. To simplify the derivation, Timoshenko formulas assume a Poisson ration of .3. The NASA formulas make no material assumptions.

 

[jrmack]

Tankman650: I am looking at Roark p 408 5th edition. There are a number of edge conditions in the table. What does 'Held, Not Fixed' mean?

Shaggy18VW: I tried to get the two tech briefs you mention but cannot find them on the NASA website. Could you please give me a little more direction?

 

[JStephen]

"Held" means the plate edge can't move in or out in the plane of the plate- held so that it develops membrane stresses. P. 405, paragraph 10.11.

"Fixed" means the edge can't move or rotate in any direction. In this case "not fixed" means the edge can rotate.

"Held, not fixed" also has to mean that the plate is supported vertically at the edges.

 

[jrmack]

My case is simply supported and sliding around the perimeter. The horizontal component required for membrane would come (in my case) from the in plane shear of the plate.

Held not fixed gives the largest deflection out of the table on ROARK 5 p 408. Held and riveted is next. Held and fixed is substantially the least deflection.

If Held, not fixed is rotation but no translation (vertically or horizonatlly). What is held and riveted?

 

[DRDearth]

The problem geometry you describe isn't relatively thin it's THIN. I would guess bending will become membrane tension under small loading. You would need a comprehensive non-linear with large deflection capability to model your problem. But, there are cookbook solutions to simple geometries one can find in the engineering literature.

David R. Dearth, P.E.
Applied Analysis & Technology

 

[prex]

jrmack, it is unclear to me what 'held and riveted' means. It is certainly a condition intermediate between the hinged ('not fixed') and clamped ('fixed') cases, but how far it is unspecified. To clear up the question one should consult the NACA references cited by Roark (that BTW are freely available online at

http://naca.larc.nasa.gov

).
Unfortunately your condition is not provided by Roark (it would be called 'not held, not fixed').
Check from time to time the site below: large deflections of rectangular plates are due to come soon.

prex

http://www.xcalcs.com

Online tools for structural design

 

[prex]

jrmack, I think I understand now what's 'held and riveted': 'held' means, as was already clear, that the edges are not free to move in the plane of the plate and in a direction normal to themselves, 'riveted' that the edges are not free to move in the plane of the plate and along themselves.
However this doesn't change much in the conclusions.

prex

http://www.xcalcs.com

Online tools for structural design

 

[ShaggyPE]

jrmack,
In regards to the Nasa Tech briefs, go to the site listed in the above message. You will need to become a member (free, just fill out the form). Once in, you can download technical articles that appeared in print as NASA Tech Briefs. The above mentioned numbers can be input into the "case number" field. There is a selection as to which format you would like. The one you'll want is: "View electronically a NASA Technical Support Package (TSP) with more detailed information related to this technology if one is available."