Stiffness matrix of simple shell element (Simple reasoning?)

[BennyS]

Hi together,
I struggle a lot with the task to set-up a stiffness matrix for a simple shell element.
In Nastran, I have a 5x5mm flat shell with 1mm thickness (isotropic), represented by a CQUAD4 element.

Can somebody explain an easy way to setup the corresponding stiffness matrix? I already read that its a 24x24 matrix, 12x12 from bending and 8x8 from membrane stiffness, the rotational degrees of freedom are usually 0 but are given a small value to make the matrices invertible. Unfortunately, Ive not found an easy way to create these submatrices, thus the element matrix.

Any advice would be really appreciated

Cheers
Benny

 

[ShellsPlatesMeshes]

Hi Benny,

Derivation of stiffness matrices for shell elements is a pretty complex problem; its solution is usually a trade secret of software companies developing FEA codes. I would suggest you to have a look at the following webpage that provides reference to a book devoted to that subject (the book is supplied with a CD-ROM containing the source codes):

http://members.ozemail.com.au/~comecau/quad_shell.htm

Hopefully you will find that information helpful.

 

[BennyS]

Thank you for your response
I guessed that,
but is it really so hard to set-up the stiffness matrix of a simple 5mm*5mm*1mm shell?

 

[ShellsPlatesMeshes]

Yes, it is hard. The mathematical derivations are not simple, and the implementation requires numerical integration, etc.

 

[klaus.]

To set up a stiffness matrix does not matter on the size ..
Maybe we do not understand what you really want to do !?!?!
If you are using an FE Program ( Like Nastran) why do you care about how to set up the stiffness Matrix ??





best regards
Klaus

 

[BennyS]

Thanks for your responses.
I try to set-up the matrices to check a problem using a rather "analytical" or scientific approach.
I would require something that connects the young's modulus to the stiffness terms in the matrix. I want to show that varying the young's modulus by 1% is influencing the stiffness by 1%.
Is it possible to show that somehow?

Regards
Benny

 

[IDS]

The stiffness matrix for any type of element could be written:

E[k1, k2, ...]

Where factors k1 etc are independent of E, so the stiffness is proportional to E.



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BennyS]

Hi IDS,
thanks for your response.
for ANY type of element?
I found some literature which says that the material properties are included as a "Material property Matrix". (e.g. shell elements) Where also the Poisson's ratio etc are included. For this reason I was not competely sure, if the proportionality is the case for each application.
So, is the young's modulus direct proportional to each DOF of each element?

 

[klaus.]

DOF ( degree of freedom) has absolutely nothing to do with young's modul
I think you are mixing things up without basic knowledge of FE method ...




best regards
Klaus

 

[BennyS]

But the DOF of a Grid Point do have an entry in the stiffness matrix, right? So, rotational DOF1 of grid point 1 is connected to rotational DOF1 of grid point 2 (beam example).So according to my understanding, it is possible that the factors of translational stiffness and rotational stiffness are different.

Thank you all for the effort.

 

[gravityandinertia]

The degrees of freedom have their own vector that represents displacements. In structural FEA, you are essentially solving a system of spring equations, KD=F. Or stiffness*displacement = applied force. The displacement vector, D, is your Degrees-of-Freedom. F is from your applied forces, and K, the stiffness matrix comes from your geometry, element formulations, and material properties. While K isn't directly tied to the DOFs, what links them is K changes based on the mesh size,which changes the size of the D vector and the corresponding number of DOFs.

 

[BennyS]

Thank you gravityandinertia. I already know something about equations of motions, and I was aware about the relation between stiffness matrix and DOF vector.
My problem was that I had in mind, that the kind of element AND the kind of DOF(translational or rotational) play a role for the value of k11..k12..k22..etc.
My aim is still to gain some experience concerning how the stiffness matrix is set-up. To set up a global system stiffness matrix if only beam elements are used is quite easy to understand. The problem is that I wanted to doan analytical comparison or at least provide the theory behind it( concerning the relation between Young's modulus and the stiffness factors of a simple shell element (so K11..K12.. etc))
In my "understanding", if a simple beam element (from grid A to B) is put into focus, the rotational degree of freedom of B has another stiffness than the y-translational DOF of B for example. In the beam theory, the Young's modulus is included directly in the equations for bending etc. (E * I / l^3) etc.
For the shell theory, I just found some matrices that are looking like the equation including the young's modulus :

http://homepages.rpi.edu/~des/4NodeQuad.pdf

(page 6, upper right matrix) The element stiffness matrix is therefore created using this "material matrix". But I would like to have some further information and understanding.

For this reason, I would have liked to have a showcase for the role of the young's modulus in the CQUAD4 of Nastran (for example).

Thank you so far!

 

[gravityandinertia]

I'm confused. You keep mentioning analytical, but really what you seem to be describing is the symbolic representation of the numerical solution. The analytical solution won't deal with DOFs at all. It will be a formula that relates load, material properties (E and poisson ratio), and geometry (thickness, length, etc).

 

[rb1957]

read up of FEA theory ... many texts are available. I think you can get somewhere by considering a simpler membrane first, and extend to plate.

This has been done many times, isn't new, well written up.

As noted above, most (all?) elements in the stiffness matrix are proportional to E.

another day in paradise, or is paradise one day closer ?

 

[WARose]

So, is the young's modulus direct proportional to [the stiffness of] each DOF of each element?

For a quad (having degrees of freedom that include in-plane & out of plane [i.e. flexural rigidity]) the answer to that is: yes. The stiffness is directly proportional to E. And that includes shear stiffness since G is a function of E & Poisson’s ratio.

The only thing I can't remember is the (in-plane) rotational (i.e. "drilling") shape functions.

 

[rb1957]

but aren't drilling freedoms typically suppressed ("autoK6spc") ?

as being not a "real" freedom ?
or because most element coding neglect it ?
(because it is essentially irrelevant to most solutions ?)

another day in paradise, or is paradise one day closer ?

 

[WARose]

[blue](rb1957)[/blue]

but aren't drilling freedoms typically suppressed ("autoK6spc") ?

as being not a "real" freedom ?
or because most element coding neglect it ?
(because it is essentially irrelevant to most solutions ?)

I think a lot of codes did/do. However, I've worked mainly with STAAD over the last 15 years and it has drilling degrees of freedom [dof] for both 4 node plate elements and 8 node solid/"brick" elements.

In the case of the solid elements, STAAD didn't use to have them. (In fact, IIRC, STAAD didn't have any rotational degrees of freedom at the nodes.) This use to make their solids prohibitively stiff (especially for dynamic analysis). Since they have introduced these dof, these elements have become much more flexible and acting closer to reality. (And ergo made their use feasible.)

 

[rb1957]

interesting, i'd've thought that membranes (without bending stiffness) would've been less stiff than plates (with bending stiffness)

another day in paradise, or is paradise one day closer ?

 

[WARose]

[blue](rb1957)[/blue]

interesting, i'd've thought that membranes (without bending stiffness) would've been less stiff than plates (with bending stiffness)

Since the modifications I spoke of, the 8-node solid element [in STTAD] now is less stiff than the flexural stiffness of plates. (At least the last time I checked.)

It use to be, the solid elements in STAAD were ridiculously stiff.

 

[rb1957]

"the solid elements in STAAD were ridiculously stiff" ... like TET4s ?

another day in paradise, or is paradise one day closer ?