Stiffness matrix for a 4 node plate in bending (12x12)

[Stazz]

I have looked long and hard for this stiffness matrix and I know it's out there. I know there is a general matrix (just like the ones they give for beams) where the values are functions of L, E, and I and A etc. I can't find anything on the internet that gives these values, I only find mind bogeling ways to derive this matrix which is over my head.

This is the closest I could find in these posts...

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


Ahhhh, I don't want to buy this book, can someone please point me to a page that has this 12 x 12 matrix or please put these values in this post. I know how to derive this using a program by doing the unit displacement stuff but I need a general one that considers all E's I's and L's.

Thanks you!

 

[LeHC]

I put together a formula in the attached picture. I am not sure if it's right. Some one who knows about this, please confirm. The formula if for C0 mindlin plate.

Note that the formula is good for rectangular elements only. For arbitrary quadrilaterals, the kinematic matrix B need to be modified (not included, don't want to discourage you with complicated stuff.)

Hope this helps!

 

[Stazz]

This is exactly what I needed. Thanks! I'm just still trying to figure out what the Nn,x and Nn,y terms are equal to in the Bm matrix. I do have a book that attempts to explain this but I'm getting lost in their "jacobian" matrix stuff and there's a lot of conversions to get these terms. I understand that the Nn terms are just the local cordinates of these node, right? It's just the other terms that I can't understand.

 

[LeHC]

Here is the formula of B for arbitrary quadrilateral (see attached picture). It follows what called iso-parametric formulation. Simply replace the B matrix in the formula I previously posted with this new one. Also, please check it before use (I don't ussually get things right at the first shot.)
The formula is in local plane. You only need to transform the resulted stiffness matrix (not the formula) if you want to plug in 3D structures.
Enjoy the FEA!

LHC

http://cgFEA.com