Stiffness matrix for Rect Plate & nodes on top

[Dinosaur]

Hello guys. I check this forum regularly and believe there are many knowledgeable fellows here. I am trying to develop a model requiring rectangular plates connected to other elements. Since the other elements are actually attached to the edge of the plates, I think I could avoid some problems if I could find a rectangular plate bending element (20DOF) with the nodes on the top face instead of at mid-plane as you normally find in the literature. Is anyone aware of where I might find such a stiffness matrix? I'm going to do a search on the web, but I doubt it is going to appear in all the noise out there. Thanks in advance for any help. - Ed

 

[ekline]

In FEMAP (for MSC/Nastran solver) I often meshed a surface instead of a midplane. And then in the element definition, I would offset the nodes and stress recovery points by t/2. This had the advantage of making connectivity easier, and avoided the generation of messy midplanes. I'm not sure how that looks in the stiffness matrix, though.

 

[SWComposites]

Offsets from the midplane of a shell element should be used with EXTREME CAUTION, as the results can often be incorrect or not what is intended. For instance axial loads applied to the nodes which are offset from the shell midplane will also induce a bending moment in the element (maybe this is ok and maybe not, depends on what you are trying to model). Offsets can also give pure rubbish for results , as demonstrated in a recent paper:
D. Laird, F. Montoya and D. Malcolm, "Finite Element Modeling of Wind Turbine Blades", AIAA Paper 2005-195.

 

[Dinosaur]

Well, like I said, you fellows are pretty sharp. I am interested in obtaining the stiffness matrix for the reasons identified above; it gets rid of the nasty problem of transforming the section using dummy elements. Equally, I recognize that this is not a trivial problem, and like most FE problems, requires some careful attention.

Unfortunately, I don't have any high powered FE software, so I need the matrix so I can make up the FE model in a Mathcad -or- Matlab type solution.

Fortunately, my loads are normal to the surface of the element, so the axial/moment coupling in the equations is not a problem on the input end of the problem but rather on the post-processing results interpretation part.

Thanks again for any help you can provide. So far it doesn't sound like the information already exists in the litrature, so maybe I'll have to derive the solution myself. Oh joy, oh rapture, unconfined! - Ed

 

[GBor]

Dinosaur,

You may want to look at various laminated plate theories. I would have to think about it for a bit, but you may be able to divide your cross-section into layers as appropriate and generate the stiffness matrix for which you are looking. In other words, if you have a 0.25" (8mm) plate, you may be able to divide it into slices of 0.05" (1.6mm) and calculate the stiffness matrix as if one of the outer slices was the bending plane. Seems like this should fail somewhat since it wouldn't be a valid assumption...it would be like offsetting the nodes, but your stiffness matrix would be built based on the offset.

Hmmmmm...something to think about, at least.

Garland

 

[ThomasH]

In Robert Cook's book "Conceps and applications of Finite Element Analysis" you can find several examples of stiffness matrices. I took a quick look but couldn't find exactly what you are looking for assuming I understand your problem correct.

However, an idea, would it help to look at a brick (solid) element? Then you have 8 nodes but will probably be able to derive matrices for translation and rotation from the 8 nodes to your 4-noded plate. But like I said, it's av very quick idea on a possible procedure to create the matrices if you can't find the in the literature. The idea could prove useless.

Good Luck

Thomas

 

[gwolf]

Dinosaur, I think that you are trying to re-invent the wheel here. If you are an engineer trying to solve a problem then please just get hold of an FE program and do it - it's simple. If you are an interested researcher then good luck, it's a trickey problem and beyond me.

 

[Dinosaur]

I'd love to be able to get a high end FE program and just get the company pay for it, but that isn't the case. I am working on this on the side so I can compare results against another method. Yea, I can construct a model using solids, or plates and rigid dummy elements to link the plates to the other elements, but that requires twice as many nodes and risks creating a singular matrix that cannot be inverted.

I will just have to bite the bullet and buy a huge stack of paper and a few packs of 0.5 mm lead and then take five days vacation and ... maybe I'll get it done in a few weeks.

Thanks for the help. Maybe I'll get lucky. - Ed

 

[ThomasH]

Dinosaur

I think you missunderstood me a little. What I meant is that you use a solid element formulation to create an ekvivalent plate formulation for a plate where the connections are offset. "Twice as many nodes", true but the DOF's should be the same of you do it correctly. Note that a typical plate has 5 or 6 DOF's per node depending on the formulation while a solid element only has 3 DOF's per node.

Another approach would be to get an evaluation of the "high end" software if this is a "once off" situation. But if you don't have experience with this kind of software I'm sure this is a good example to start with.

Good Luck

Thomas

 

[GBor]

Dinosaur,

Not sure what you are actually trying to connect to with your plate elements, but one technique is to extend the row of plate elements onto, say, your bricks, but give them virtually zero stiffness (modulus of 1). This resolves the DOF issues adding one additional line of node and minimal stiffness to your transition.

 

[Dinosaur]

ThomasH,

Given your suggestion, to use bricks instead of plates, are there any compatability issues if a beam element runs along one edge? I know there could be an incompatability between the brick and the beam, I just don't know if it is likely.

If the beam part rotates at each end but experiences almost no displacement, would the brick nodes displace into a shape reasonably consistent with this rotation (e.g. with the beam along the bottom edge, would the top nodes of the brick displace relative to the bottom nodes an amount consistent with the angle of rotation?)

If I were using a plate with a rotational degree of freedom in the proper orientation, then there is no question.