Deflection and Moments for Plate (Two Simply Supported Edges and Two Free Edges)

[CWEngineer]

I am trying to look for a fourier series formula that calculates the deflection and moments for plate (simply supported on two opposite edges & free on the two other edges). I looked at Thoery of Plates and Shells by Timoshenko and Szilard and was not able to find it there. Does anyone know of a reference that might have this information?

Thanks

 

[msquared48]

Roark and Young "Formulas for Stess and Strain".

Mike McCann
MMC Engineering

http://mmcengineering.tripod.com

 

[rb1957]

Ynot FEA ?

 

[structSU10]

What makes this different than a beam? What is your loading that makes it differ?

I found a PDF of a book 'Formulas of stress, strain, and structural matrices' (pilkey, walter) that has a chapter on plates, found it via google i think.

 

[CWEngineer]

This is a thin plate. I looked at Roark and Young "Formulas for Stress Strain", and was not able to find it there.

 

[aggman]

To me for the question of thin verses thick to even be considered we have to establish if the ends are restrained against lateral movement. If the ends are free to translate then once the plate fails in bending then it will collapse since axial tension can not be held. If the ends are restrained against translation then it falls somewhere between a pure beam and a perfectly flexible cable. This is covered under Table 12, Chapter 7 of Roarks Formula's. (6th edition). Since you are supported on two opposite edges and no adjacent edges the plate can only function in one way bending.

 

[JStephen]

If the loading is uniform, it should just be a wide beam. There is an adjustment factor required in deflection, which I think is mentioned in Roark's book. (The normal beam theory assumes that the compression and tension flanges can freely expand/contract in the lateral direction, and this is no longer true with a plate or wide flat bar.)

 

[CWEngineer]

I am looking into a rectangular plate, that is uniformly loaded, simply supported at two oppositeve edges and free at the to other edgers. Yes, right now I am looking into pure bending. Table 12 in Chapter 7 is for Beams, would this be applicable to a thin plate? Also, looking for moment equations. Appreciate your help.

 

[BAretired]

For a uniform load, beam equations apply except that EI should be replaced with D where D = Eh³/12(1-[ν]²). In the above, h is thickness and [ν] (Greek letter nu) is Poisson's Ratio.



BA

 

[aggman]

A plate is nothing more than a beam when spanning in only one direction. Any differences in actual results based on the "plate" verses "beam" are negligible for practical results.

 

[JoshPlumSE]

The reference I used in college was AC Ugural's "Stresses in Plates and Shells". I remember a thorough treatment of Navier's solution (which uses an infinite fourier series) for simply supported rectangular plates. Could that be the type of solution you are looking for?

Though, like others have pointed out, for your situation I'm not sure that the solution differs at all from a beam solution.

 

[CWEngineer]

Yes that is the type of solution I was looking for. I also have looked at AC Ugural's book, but the plate with my boundary conditions (two edges simply supported and the other two free) is also not there.

 

[slickdeals]

This is a one way plate like other contributors have mentioned. Treat it as unit width member to get forces, moments.

One other reference for plate moments is published by Moody, do a google search for a free copy.

 

[BenAustralia]

Excuse the potentially ignorant question, but at what point do the equations used for beams not become applicable? If the plate deflects enough so that its predominantly in tension, can beam formulas be used?

 

[BAretired]

When a plate has large deflections such that it starts acting as a shell, beam formulas will not do. See Chapter 13, Theory of Plates and Shells.

BA

 

[prex]

Here, from the first site below, you'll find a solution to this problem, obtained with a polynomial approximation through energy minimization. Though it is not a direct solution of the differential equation, it can be used to check the very tiny difference between beam and plate behavior in this particular boundary condition.

prex
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[IDS]

When a plate has large deflections such that it starts acting as a shell, beam formulas will not do. See Chapter 13, Theory of Plates and Shells.

But the standard solutions for flat plates with other support configurations also assume small deflections. If these solutions are regarded as acceptable, then the standard beam solution is acceptable for the case of a slab simply supported at opposite ends, with a load over the full width. In fact I presume that is why books such as Roark's do not give a separate solution for that case.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

It depends on what you mean by "simply supported". If you mean hinge/roller, I would agree that the beam formula works regardless of deflection. But BenAustralia was asking about the situation where the plate is predominantly in tension. This can only happen if the supports are hinge/hinge in which case, beam theory does not apply.

BA

 

[JoshPlumSE]

I think the procedures described in AC Ugural's book would still work though. The formulas should just become a lot simpler. Our professor had us do something similar... Use Navier's solution for slightly different boundary condition (or for an SS plate on an elastic foundation).

 

[IDS]

Just to clarify, BA and I were addressing different questions. The OP appears to be asking about simply supported conditions with either a uniform load or full width edge load, in which case there is no reason (that I can see) not to use the standard beam equations, rather than worrying about a Fourier series.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/