Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 323

§ 5. Beams and Plates: The Kirchhoff Plate

A plate is a thin continuum subject to applied forces which - in contrast to the
case a membrane - are orthogonal to the middle surface. We distinguish between
two cases. The Kirchhoff plate leads to a fourth order elliptic problem. Usually it
is solved using nonconforming or mixed methods.
The Mindlin-Reissner plate (that is also called Reissner-Mindlin plate or
Mindlin plate) involves somewhat weaker hypotheses. It is described by a differen-
tial equation of second order, and so at first glance its numerical treatment appears
to be simpler. However, it turns out that the calculations for the Mindlin-Reissner
plate are actually more difficult, and that the problems plaguing the Kirchhoff plate
are still present, although concealed. In particular, the Mindlin plate tends to shear
locking, and using standard elements leads to poor numerical results.
The analogous reduction of thin membranes, i.e., of membranes with one
very small dimension, leads to beams.
After introducing both plate models, we turn our attention first to a discussion
of the Kirchhoff plate, and in particular to the clamped plate .

The Hypotheses

We consider a thin plate of constant thickness t whose middle surface coincides
with the (x, y) -plane. Thus, = ω × ( −

2 . We suppose that
the plate is subject to external forces which are orthogonal to the middle surface.

t t

2 , + 2 ) with ω ⊂ R

5.1 Hypotheses of Mindlin and Reissner.
H1. Linearity hypothesis. Segments lying on normals to the middle surface are
linearly deformed and their images are segments on straight lines again.
H2. The displacement in the z -direction does not depend on the z -coordinate.
H3. The points on the middle surface are deformed only in the z -direction.
H4. The normal stress σ 33 vanishes.
Under hypotheses H1-H3 the displacements have the form

u i (x,y,z) =− zθ i (x, y), for i = 1 , 2 ,

( 5 . 1 )

u 3 (x,y,z) = w(x,y).

We call w the transverse displacement or (normal) deflection , and θ = (θ 1 ,θ 2 ) the

rotation .