Introduction to plate and shell finite elements
Objective: To explain concepts of
plate theory and their application to developing finite elements
Plate and shell finite elements are the most widely-used,
as they can model many vehicles and machines: for instance, the hull of
a ship or the body of a car. They are combined with beam elements in many
A plate is a 2D analogy to a beam, that
can bend-out-of-plane, as well as deforming in-plane. Plate theory concentrates
on this out-of-plane behaviour, as the in-plane deformation is that of
a plane stress situation, discussed earlier. In a doubly-curved plate,
in-plane deformation cannot occur without causing out-of-plane deformation,
and vice versa. The term “shell” is commonly used in situations where this
coupling is important.
Why use special plate elements?
A plate can always be modelled with solid elements. However,
to represent bending accurately, the mesh needs to be several elements
deep through the thickness of the plate. As this thickness is small compared
to other dimensions, unless a very large number of elements are used, the
elements have high ratios of their maximum to minimum side lengths. This
causes stiffness coefficients associated with the elements to vary greatly
in magnitude. A plate element avoids having the use several elements through
the thickness. It does so by placing the nodes half way through the thickness,
and implying the displacement variation due to bending, by using rotation
degrees of freedom.
Meaning of the rotation Degrees of Freedom
A nodal rotation about an axis in the plane of a plate is
strictly that of a line through the node, normal to the plate surface.
This rotation of the normal can represent
shear (shear in a plane normal to the surface) the bending being the dominant
Quantities used to represent the bending stress variation in a plate
The Bending Stress Variation
To find a stiffness matrix for a plate element, we need an
appropriate [D] matrix, relating quantities resembling stress and strain.
We could write the stress-strain relations for each layer of thickness
dz and integrate through the thickness. This is in fact necessary with
fibre-composite plates. However, it is usually easier to work with effective
quantities, which represent the linear variation of stress or strain through
the thickness. In the case of stress, bending moments/length measure the
severity of the stress variation. These are output by STRAND7, and other
packages, when you ask for a listing of “Stresses”.
Predicting Surface Stresses From Bending Moments/Length
Consider bending stress, zero on the mid-surface of a plate.
Unit length of plate of thickness t, resembling a rectangular beam of depth
t and unit width. So
Hence given a bending moment per length we can
find bending stress. This is superimposed on any average (or membrane)
stress due to in-plane loading i.e.
A key difference between beams and plates is that plates
can twist, causing shear stress in the plane of the plate, that varies
through its thickness. (Like torsion of a thin-walled beam.)
A linear variation of shear stress through the thickness
of a plate, from txy
to -txy is caused by
twisting moments per length, which act on a piece of plate dx by dy in
size, as shown below. Note there are complementary twisting moments, just
as there are complementary shear stresses on an infinitesimal element of
material. i.e all 4 moments exist together for equilibrium.
Arrows are axes of moments and indicate the sense
dw/dx and dw/dy
Kirchoff Plate Theory
In a thin plate, bending deformation dominates the out-of-plane
behaviour. Transverse shear strain can be neglected (as it is in calculating
deflections of beams). The normal to the surface remains normal, and hence
its rotation equals that of the surface, approximated by its tangent, the
slope of the surface.
Consider a plate in the xy plane with transverse z displacement
w. The rotations are:
Mindlin Plate Theory
Transverse shear strain really varies quadratically through
the thickness of a plate, being zero on both surfaces and a maximum at
mid-thickness (as in a beam of rectangular X-section). If this variation
is ignored, and the shear strain treated as constant, then the change in
the right angle between the surface and a line initially normal to the
Most plate finite elements assume Kirchoff plate
theory, and neglect transverse shear strain. These become inaccurate with
very thick plates, for which the transverse shear strain is significant.
Elements assuming Mindlin plate theory that assume transverse shear does
not vary through the thickness can cope with thicker plates.
Note that in both theories, the normal to the
surface remains a straight line, implying a linear variation through the
thickness of strain components in planes parallel to that of the plate
surface, like the components in planes parallel to that of the plate surface,
like the distribution assumed in beam bending theory. If this is not so
(eg at a joint between plates where there is a stress concentration) then
the results from an analysis using plate elements to be treated with caution.
Quantities Used To Represent The Bending Strain Variation In A Plate
Curvature as a measure of severity of Bending Strain.
The more a plate bends, the worse the bending strain at either
surface, and the higher the curvature. Hence curvature can be used as an
effective “strain”, representing the severity of the bending strain variation
through the thickness of a plate.
Consider Kirchoff plate theory.
For bending about y, the rate of change of slope is curvature
The slope of a plate not only changes in the direction of
the slope, but also in the direction along the plate, normal to the direction
of slope. The result of twist, i.e.,
Twist occurs due to the twisting moments/length introduced
in the previous section, and is associated with shear strain gxy
varying linearly through the thickness.
The in-plane behaviour of a plate is that of a plane stress
situation. The resultant average stresses are called “membrane” stesses
(e.g those in a balloon). Bending stresses and strains are superimposed
on these, as are twisiting moments and shear strain due to twist. These
are associated with out-of-plane deformation. We can write relations between
moments/length and curvature or twist, analogous to the equation for a
beam. M becomes a vector ie.
M = vector of moments/length at a position x,y
in a plate.
Write M = [D] eb
is a vector of effective strains (curvature and twist terms). i.e.
The matrix [D] is then
is Poisson’s ratio, and E is Young’s modulus.
t in this equation is the “bending thickness”
of STRAND7. The factor Et3/12 can be adjusted in NASTRAN.