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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,

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

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