## FEM derivation for plates on elastic foundation ?

Hi All,

I'm familiar with the theory of how the soil can represented by means of FEM and how soil-structure interaction is being established. Most of books are pointing out at the Winkler -springs implemented mostly for beams and quadrilaterals (4 node) FEM elements. There are obviously other methods e.g. two-parameter modified Vlasov foundation which in my idea is best approximation. If it isn't the best, at least it's better than Winkler.

The approach for deriving the stiffness matrices for Vlasov foundaiton are identical to those of Reissner-Mindlin plates, where transverse shear is being taken into account. In order to, represent the vertical and horizontal spring coeff. of soil, corresponding spring coefficients should be found through strain energy equation.

The displacements are given as :
u(x,y,z), v(x,y,z), w(x,y,z)=w(x,y)*$\phi(z)$

Where $\phi(z)$ is mode shape but not the one obtained from eigen-mode analysis, simply, mode shape that gives the variation of the deflection in the z-direction. For example
$\phi(0)$=1.0, $\phi(H)$=0

Obtainig the spring coefficients is:
for vertical spring coeffcient k=$\int^{H}_{0}E*(∂\phi/∂z)^2*dz$
for horizontal spring coefficient 2t = $\int^{H}_{0}G*(\phi)^2*dz$

Stiffness matrix for vertical direction is formed by:
Minimizing the strain energy by respect to each component of displacement vector.
(Uk)e=1/2$\int[w(x,y)^T*k*w(x,y)*dA]$
and
[Kek]ij=∂(Uk)e2 / ( ∂2wi*∂2wj) similar procedure is valid horizontal stiffness matrix

Having those in hand,
I wish to addopt that for triangular FEM element which has the shape function as : Ni =ai +bi*x+ci*y where ai, bi, and ci are known from nodal coordinates.
My question is, how can that be addopted to the energy equations and how am I supposed to get derivates for dz where I don't have the any representation for Z-direction in my shape function for triangular element?

Your helps will be appreciated!

Regards,
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 Recognitions: Gold Member Science Advisor How about doing it in 2D by borrowing elements from plane stress/strain or carrying out the whole exercise in 3D (going for example for wedge or tetrahedral elements, would probably be almost simplest for bilinear bricks)?
 I haven't done FE in a while and admittedly have never done any work on elastic foundations for FEM, so I will probably not be of too much help here. However, my thought is: do you think you really can do it in 2-D? It is a 3-D problem, with deflection in the vertical direction and a simple bilinear plane stress/plane strain element is really intended to model something that doesn't deflect in the direction perpendicular to the plane (although I may have gone wrong here, correct me if I have - as I said, I haven't done this in a while). Maybe a shell element could do here, or as PerennialII pointed out, doing it all in 3-D.