Plate Element Stiffness Matrix 2

[capper]

I am hav ing some trouble with a program I am writing and I just want to verify my element matrix with an eternal source so I can strike that off my list as a possible source of error. - does anyone know of a web site that I could get a look at this matrix??

Thanks in advance,
Colin Caprani

 

[Qshake]

Try West Point Academy Department of Civil Engineering. I recall that they had a program that would assemble a stiffness matrix. Of course, this external source has the additional error of understanding the software and using it properly. I think it was called Cadre or something similar.

 

[Qshake]

Whoops! Boy did I read your post incorrectly. Your looking for a plate element stiffness matrix. I thought it was a stiffness matrix for an assemblage of beam elements. Disregard that last post!

 

[austim]

Maybe Qshake is not as wrong as all that.

CadrePro version 3 has 2D plate elements (membrane stresses only), and you should be able to download that in a limited node version. (ref

http://www.cadreanalytic.com).

You may be just too late to download a beta copy of version 4 which has plate bending elements (but the beta version dies a death after some time early this month - it may already be dead).

 

[capper]

Thanks austim but it is the actual matrix that CADRE would have used in their program that I am looking for. I have lots of theory stuff but it is just the result I need - I can't really re-invent the wheel on this one!!

Thanks anyway

Colin Caprani

 

[Qshake]

Be sure that you've exhausted the search with CADRE as the the program there I was referring to was called STACKER and it produced the actual stiffness matrix. Unfortunately it was for the wrong application; not plate elements. Be sure that the subroutine for plates doesn't exist too.

 

[austim]

Hi again capper,

If you cannot get the plate stiffness matrix printed directly from Cadre (I have never actually used Cadre, I just have it in reserve for that special need), then I suggest that you try something along the following lines (going back to very first principles, it never hurts):

Use a series of Cadre analyses with a single plate element as your model.

In turn, restrain all but one of the degrees of freedom, and apply a unit deformation at the remaining DOF. (ie if you have a rectangular plate element, you will need 12 separate anlyses)

You can check your own results by comparing them with the output values of all the restraint forces at the restrained nodes.

 

[capper]

Thanks Qshake, I am trying it now!#

Austim, that is a good idea, but i need a matrix to check results against!! Doing what you said will give the values of the entries in the matrix, but I would need to do a lot of analyses to deterrmine the expressions that lead to that value, and 12x12 = 144 values & expressions!!!

Thanks though,

 

[austim]

Hi, yet again, capper.

Now you have ME baffled!

In your inital post you referred to 'my' element matrix, so I have taken that to mean that you already have your expressions sorted out, and merely want confirmation of them. My procedure would give you some numerical confirmation of your current formulae, and would show up any gross errors.

The task is not quite as onerous as you suggest - yes the matrix is 12*12, but due to its symmetry, there are only 78 different expressions (12 on the diagonal, 66 in each triangle). There, doesn't that make you feel better about it already?

If you really want to compare your formulae with those from another source, then of course you still have to compare 78 formulae. For that exercise I would start with a good text book and chase up any papers on plate stiffness that it refers to.

 

[capper]

austim,
apologies, not the clearest explanation! I just want independant verification. It is the very papers, etc. that you mention that I was hoping someone might know where there are some on the web. And of course you are right about the symmetry! Thanks for the help

 

[emil]

This is easy, I did it before. There is a relatively simple way to extract a stiffness matrix from any commercial finite element software. The procedure is base on the physical meaning of the stiffness matrix. One column of a stiffness matrix represents a set of reactions to impose an unit deflection along the diagonal coordinate and keep all other deflections to zero. The procedure goes like this:
1. Take any FE package and make a model of one rectangular plate element.
2. Constrain all nodes, except the vertical displacement at one node.
3. Apply unit displacement at the free node.
4. Solve the problem.
5. Now, the reactions are the elements in the first column the stiffness matrix.
6. Repeat this 12 times for all degrees of freedom, and you have the complete stiffness matrix.
(I hope this was the question.)

 

[austim]

Hi, yet again, capper.

I make no claim to be a real expert in finite element theory, so the following might well be complete bunkum. Let's see what others say about that?

I have a nasty feeling that you may be trying for an impossible result, since there is not any single unique formulation for the stiffness matrix of ANY shaped finite element. The stiffness matrix depends on assumptions regarding the displacement function used in its development. Hence there have been debates about the relative merits of different formulations used by various authors on the subject.

For each assumed displacement function you will get a different stiffness matrix. All of these may be 'correct', but they will, in general, not give you identical results.

So, even if you use the 'Austim/emil' method of extracting the numeric values from a FE package, it would be little more than good luck if they exactly corresponded with your own values.

As for books/papers -

"Structural Analysis - A.Ghali and A.M. Neville" published by Chapman and Hall, (my copy was reprinted in 1983) has a good chapter on FE development (co-authored by Y.K. Cheung), with stiffness matrices for in-plane triangles and bending rectangles. It also has a number of (hopefully useful) references to other sources.

A web search for O. Zienkiewicz and Y.K. Cheung could well give you some useful references.

Good luck with your endeavours.

 

[austim]

PS,

The in-plane matrix also depends on the type of problem that you wish to analyse.

On the one hand, if you assume a state of Plane Stress, then your matrix will be applicable to steel plates, etc.

If you assume a state of Plane Strain, then you can apply your program to structures where the dimension out of plane is much larger than the dimansions of the elements (eg slices through dams, bearing pedestals, etc).

(That said, I have to admit that I haven't the faintest idea what state of stress/strain is assumed by the software that I have used for the past ten years or so, and have no certain way of finding out).

 

[Dinosaur]

I wish I had read your post a long time ago, but for the curious . . .
The stiffness matrix for a rectangular plate in bending (12DOF) is given on pp.116-120 of "Theory of Matrix Structural Analysis" by J.S. Przemieniecki, published by Dover Publications, Inc., 1968 & 1985.
Good Luck. - Ed

 

[capper]

Well this post is a few years old now!

I know a good bit more about the problem these days and the contributions from the other guys is spot on - usually modern FE programs will numerically calculate the element stiffness matrices on the fly based on theory in Zienkiewicz's book for example. There are many types of element and many possible shape functions for each element. That said, it is very useful to have an explicit matrix for ease of programming, shall we say, less commercially orientated software. So thanks for your reference Dinosaur - very useful.