Circular Plate Problem

I've attempted to read up some on a problem I have, but am essentially unable to go forward. Maybe someone here can give me a direction.

I'm attempting to solve a Kirchhoff Plate problem. A circular plate is simply supported near the edge, but also has a single support at the center.

To a pseudo-first-order, the plate is flat on both sides, but in reality is axisymmetric; not necessarily the same on both sides. The thickness is on the order of 1/40 the diameter, so the shear mentioned by Selke is probably not a problem, I think. The solution given by Kirstein and Woolley seems close, even if it doesn't include shear, provided I can take the limit as n->infinity. However, it only has ring of supports.

The material is, I think, quite isotropic and the downward pressure is uniform (air pressure) across the surface.

I could deal with a solution in either one of two forms. Ideally, I would like to express the solution as a function of the height, relative to the edge support, of the center support. However, I could also deal with a solution where the upward pressure of the center support is expressed as a fraction of pi*q*c^2 (where q is the original downward pressure per unit area and c is the plate semi-diameter).

However, I will also need more precise form of confirmation, I think in some sort of FEA tool that I can use to construct a model. Any suggestions in this direction (hopefully close to freeware) would be appreciated.

Reply to
Richard F.L.R.Snashall
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-- OK.

How is it axisymmetric but not the same on both sides?

Are you working this as an axisymmetric plate?

-- That's either the radius or (ro - ri).

Check for a solution in Roark's Formulas for Stress and Strain under circular plates. That should do it or get you very close.

You may be able to find something here.

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Reply to
Jeff Finlayson

Actually Richard, I assume you have consulted in some detail the book Roark's Formulas for Stress and Strain as mentioned also by Jeff. If not, the chapter on circular plates in this book is where you should start off first.

"Jeff F>

Reply to
Lion

looks like a hyperstatic-problem is it just a support in the middle or is it a rigid connection ? if the support has a smal radius, it doesn't mind

so axisemetric but the thickness is a function of r ? the median plane is not flat ? or what do you mean ?

never heard of Kirstein and Woolley

that's good to know, it does simplify a lot

what you have to do is solve 2 problems :

1) a plate with a circular support and the air pressure it will give a solution of how much the plate will give in de center in function of the pressure u(p) 2) a plate with a circular support (in the other way) and a force F in the center it will give a solution of u(F)

then there is the support in the center that will also compress a little under the force F w(F) so w(F) = u(p) - u(F)

you don't need finite elements for that

Reply to
jan hauben

It would be a simple support ball.

The median plane would not quite be flat. However, the deviation is typically small.

NBS (predecessor of NIST). They wrote articles on plate deflection in '67 and '68.

I think I see this now -- it results my artificially forcing the plate to remain on the outer supports. I am assuming, from the other responses, that this Roarke book will have the solutions to these (I cannot find both on the web.); I have it ordered (expensive little thing, isn't it?;-).

Reply to
Richard F.L.R.Snashall

the contact will not remain a point then you can calculate the contactsurface with the Hertzian tension-formula if the support ball is a lot harder then the plate and the displacement remains small the contactpoint will do

since the tension is equivalent with 1/t^2 the error will be small if the deviation is small you can estimate the error

there are no doubt analytical solutions for the two problems i mentioned since the plate is symetrical there is only shear on the intersectionplanes where r is contrant and just like with a beam is its contribution small in conparison with the normal tension

a good book very important i find websites (if you are able to find one) not very relayeble on those 'in dept' problems

Reply to
jan hauben

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Reply to
intechphil

I'm assuming you meant to the specify Plate_Circular_Uniform and Plate_Circular_Central spreadsheets. These are the ones I have been directing my efforts. I'm having a problem with the Plate_Circular_Central method (#16 in the flat plate section of the seventh edition).

The last term, after combining a few things is:

- ( W*a^2 / (16*pi*D ) ) ( (r/a)^2 ln( (r/a)^2 ) )

That part seems to have some unbounded derivatives near the plate center. Is there a way around this?

Reply to
Richard F.L.R.Snashall

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