Little help to construct the 4 node-quad shell stiffness matrix

[Ronankeating]

hi all,

Actually I'm looking for little help and kinda confirmation, in order to verify that I understood the logic of construing the stiffness matrix. I got the logic of how to construct the stiff matrix for bending and membranes to some level, although FEM books suggests to simply combine both in order to get shell elements, I couldn't figured out how to do it.


I think the most important part is, analytically determine the correlation between the unknowns(depending on what you trying to achieve). And the rest is just to put that knowns into the elasticity theorem, potential, kinetic energy, matrix manipulation etc..

For example, If bending was questioned,I'm supposed to find the w,θ_x,θ_y so I'll start with trying to express the 3D principal deflections(u,v,w ) as function of that unknowns which is been expressed by u(x,y,z) = zθy(x,y), v(x,y,z) =− zθy(x,y).
I presume that those are called the basis functions ??
If it's, what are the basis functions for 6 DOF shell element?


For shell elements confusing part is, I'll have 2D shape function(Nj = 1/4(1 + ξj ξ)(1 + ηj η) ) and with 6DOFs(u,v,w,θx,θy,θz) there will be a derivatives for ∂/∂z order of shape function. Since I don't have corresponding term in shape function for that (z dimension) how am I supposed to get the derivatives of it?


Your comments will be appreciated,

 

[AlephZero]

For a flat element the bending and membrane stiffness are indepedent, just like the bending, axial, and torsional stiffness of a straight beam are indepedent.

So you can formulate each one separately and sum the stiffnesses.

If you want to formulate a warped (non-planar) 4 note shell element, that's a completely different can of worms, and people have been writing papers about how best to do it (with varying amounts of success) for most of the last 50 years!

 

[Ronankeating]

membrane matrix is 8x8
bending matrix is 12x12 so I get 20x20 but
4 node shell is supposed to be 24x24
So what are supposed to be the rest of rows, cols of stiff. matrix, zero ??

Additionally, if the shell is with high-order but remains planar, then the forementioned procedure is valid and only the size of matrix will change?

Thanks in advance,

 

[AlephZero]

membrane matrix is 8x8
bending matrix is 12x12

Those sizes are in the element coordinate system (i.e. for a rectangular element x and y are parallel to the sides and z is normal to the element).

When you transform into the global coordinate system, in general you will get 24x24 matrix with all 6 degrees of freedom at each mesh point.

So what are supposed to be the rest of rows, cols of stiff. matrix, zero ??

For the element formulation you seem to be decribing, the global 24x24 matrix will only have rank 20-6 = 14, because none of the shape functions referred to the "drilling rotations" about the element z axes.

When you create the global stiffness matrix, this rank-deficiency might just "go away" because different elements are at different orientations in space, or you might have to constrain out variables that don't have any stiffness. But that is an issue about how to solve the global equations, not how to formulate the elements,

Additionally, if the shell is with high-order but remains planar, then the forementioned procedure is valid and only the size of matrix will change?

Yes, there may be more nodes and/or more degrees of freedom per node, but the principle is the same - for small displacement linear analysis, anyway

 

[Ronankeating]

ok thanks in advance,

One final question, the FEM books suggest to apply transformation matrix for resulting K_e M_e, F_e matrices. AFAIU this is been done to get the global location (like it's in 3d rod element) and since I'm providing the global coordinates for each individual finite element do I have to apply the transformation matrix?

And what's that mass matrix used for? To include self loading of element and input for time-dependent problems ?


Regards,

 

[AlephZero]

ok thanks in advance,

One final question, the FEM books suggest to apply transformation matrix for resulting K_e M_e, F_e matrices. AFAIU this is been done to get the global location (like it's in 3d rod element) and since I'm providing the global coordinates for each individual finite element do I have to apply the transformation matrix?

Since the element formulation "defines" what its own element coordinate system is, then the element has to do the corresponding transformation to a global coordinates.

Or at the minimum, the element has to define the directions of the element coodinate system at each node of the element, so the transformation can be done somewhere else in the complete FE code.

And what's that mass matrix used for? To include self loading of element and input for time-dependent problems ?

Applying loads caused by weight, the centripetal acceleration of a rotating structure, etc.

Also any type of dynamic analysis - vibration, transient dynamic response, steady state frequency response, etc.