What is the definition of plates and shells?

[DTS419]

I'm looking for a simple definition or explanation of plates and shells, and how they (plates in particular) differ from slabs. Specifically with application to thin tank walls (concrete), it seems as though the analysis of plates and shells is replacing the traditional slab assumption.
I have on hand Timoshenko's text "Theory of Plates and Shells." However, the problem with this book is that it assumes the reader already has a basic understanding of the definition of plates and shells.
What I need is a simple explanation of the intuitive difference between the behavior of beams and slabs, and plates and shells. Thanks in advance for your help!

 

[Stringmaker]

Plate elements vary from shell elements in that they offer five DOF's at each node rather than the six DOF which shells do. I believe that plate elements eliminate the in-plane rotational DOF. This is great for decreased computational time. However, I have seen cases in modal analyses where plate elements missed modes which shell elements picked up with both meshes being the same size. Missed modes are a huge deal in any sort of dynamics! Aside from that I think plate elements should be fine to use in most static applications where most loads are pressure or other loads normal to the surface and in-plane rotatons are typically negligible.

It'd be wise to do some investigation of your own seeing what/if the two elements yield different results for your applications and how those compare to correlated data.

Good luck!
-Brian

 

[DTS419]

Thanks. Perhaps I should start this way: What is a plate?

 

[UcfSE]

A shell element has all the DOF's that a plate element has. In addition a shell element can take membrane forces. A plate element cannot.

See also thread727-126331. Try searching the site for other related threads.

 

[DTS419]

Ok, what are "membrane" forces?

 

[rb1957]

a thin shell deflects under out-of-plane loads ('cause it isn't stiff enough in bending, unlike a thick plate). this causes the thin shell to stretch in-plane (a bit like a balloon). the resulting stress field is termed "membrane".

basically thick plates react out-of-plane loads by a bending stress, and thin plates by deflecting out-of-plane (and generating membrane stresses/loads/strains). Since we're in the FE forum, thick plates can be analyzed under small displacement assumptions, but thin plates can't (since the displacements are large) and need large displacement models (typically non-linear).

 

[DTS419]

did you learn this stuff at grad school?

 

[rb1957]

actually in my day (long ago) it was undergrad material

 

[JStephen]

The problem is that these terms are not always used consistently.

I think Timoshenko uses plate for a flat surface, and shell for a curved surface, and doesn't use the term "slab". The plate or shell can be thick or thin, can have membrane forces or not, etc. This does not relate to FEM terminology, number of nodes, etc. A lot of the distinction you're looking for is in FEM terminology, not in plate/shell design.

 

[DTS419]

Yeah, that's pretty much my problem. I am only used to designing structural elements via conventional mechanics methods. This matrix-differential equation-computer madness is new to me.

 

[pja]

Kirchhoff plates(thin) can be derived from 3d elasticity using the following kinematic assumptions:
u=u_hat+z * phi
v=v_hat+z * psi
w=w_hat+z * xi

One then assumes the following conditions on the strain:
shear strain tau_xz=0
shear streain tau_yz=0
normal strain eps_zz=0

where the first 2 assumptions are equivalent to saying that fibers which were orignally perpendicular to the mid-plane of the plate stay perpendicular after deformation. The last assumption signifies invariance of the distance of any point from the middle surface.

Using these 3 assumptions one can write the phi,psi and xi in terms of the derivatives of the mid-plane displacements u_hat,v_hat and w_hat.

For Mindlin theory(moderately thick plates) tau_xz and tau_yz are no longer assumed to be zero and therefore phi and psi are independent.

Higher order shear theories use higher order terms( in z) in the power series expansion of w,v and u in z(first equations).

Shells can be derived in a similar manner since shells are plates whose middle surface in the undeformed state is not a plane but a curved surface. Curvilinear coordinates are usually used to derive the shell equations.