Plates and Shells This course explores the following topics: derivation of elastic and plastic stress-strain relations for plate and shell elements; the bending and buckling of rectangular plates; nonlinear geometric effects; post-buckling and ultimate strength of cold formed sections and typical stiffened panels used in naval architecture; the general theory of elastic shells and axisymmetric shells; buckling, crushing and bending strength of cylindrical shells with application to offshore structures; and the application to crashworthiness of vehicles and explosive and impact loading of structures. The class is taught during the first half of term.
Trang 1MEEM4405 Introduction to Finite Element Analysis
7 Plates and Shells
7.1 Plate Formulation
• Plates may be considered similar to
beams, however:
– Plates can bend in two directions and twist
– Plates must be flat (or else they are shells)
• For thin plate on z = 0 plane, with thickness
t, and neglecting shear strain:
γyz= γzx= 0
y
w
z
v
x
w
z
u
y
x
w
w
∂
∂
−
=
∂
∂
−
=
= ( , )
Trang 2MEEM4405 Introduction to Finite Element Analysis
Thin Plate Formulation
• A differential slice from plate:
Thin Plate Formulation
• For the thin plate, we assume σz= 0 Therefore:
Trang 3MEEM4405 Introduction to Finite Element Analysis
Thin Plate Formulation
• These stresses give rise to moments:
• Maximum stresses are therefore given by:
2
2
2
6
, 6
,
2 since
6
t
M
t
M
t
z t
M
xy
xy
y
y
x x
x
x
=
=
=
τ
σ
σ σ
σ
This is similar to the beam formula, but since the plate is very wide we have a situation similar to plain
strain Flexural rigidity D=EI=Et3 /12
with EI=Et3 /12, but since strain it is very wide (like
Thin Plate Formulation
• This is similar to the beam formula, but
since the plate is very wide we have a
situation similar to plain strain
• For a unit width beam, flexural rigidity
D=EI=Et3/12
• For a unit width plate, flexural rigidity
D=EI/(1-ν2)=Et3/[12(1-ν2)]
• This thin plate theory is also called the
“Kirchhoff” plate theory
Trang 4MEEM4405 Introduction to Finite Element Analysis
Mindlin Plate Theory
• Mindlin Plate Theory assumes that
transverse shear deformation occurs
Mindlin Plate Theory
• The deformations and strains are therefore
given by:
Trang 5MEEM4405 Introduction to Finite Element Analysis
Large Displacements and Membrane
Forces
• A beam with fixed supports will exhibit “string
action” axial forces as shown.
• If we consider both string action and bending
stresses, a beam can carry a distributed load of:
Large Displacements and Membrane
Forces
• A similar situation arises with plates,
however linear plate elements are not set
up to handle “membrane” forces
• If w/t is large (e.g greater than 0.1), a
non-linear analysis must be performed using
elements that handle membrane forces
• In general, however, tensile membrane
forces will have a stiffening effect and
compressive membrane forces will
decrease stiffness
Trang 6MEEM4405 Introduction to Finite Element Analysis
7.2 Plate Finite Elements
• Plate elements must be able to show constant σx,
σy and τxy at each z level to pass a patch test
They must pass the test for constant M x , M y and
M xy.
• Kirchhoff elements can be implemented with 12
dof elements.
• However, they are awkward to use because of
the question of how to handle the twist dof.
Plate Finite Elements
• Mindlin plate elements are more common
• The displacement interpolation is given by:
• N i can be the same shape functions as for
Q4 and Q8 quadrilateral elements
Trang 7MEEM4405 Introduction to Finite Element Analysis
Plate Finite Elements
• “Discrete Kirchhoff Elements” are also
available as triangular elements
Support Conditions
• Support Conditions are similar to those for
beams:
For Mindlin plates, do not restrain θn, to avoid
accuracy problems.
θn , M n– rotation and moment normal to edge
θs , M s– rotation and moment perpendicular to edge
Trang 8MEEM4405 Introduction to Finite Element Analysis
Test Cases
• For plate elements, patch tests and single
element tests include the cases shown
• Many element formulations perform poorly
for these tests
7.4 Shells and Shell Theory
• Shell elements are different from plate
elements in that:
– They can be curved
– They carry membrane and bending forces
• A thin shell structure can carry high loads if
membrane stresses predominate
• However, localized bending stresses will
appear near load concentrations or
geometric discontinuities
Trang 9MEEM4405 Introduction to Finite Element Analysis
Shells and Shell Theory
• Localized bending stresses
Shells and Shell Theory
• For a cylindrical shell of radius R and
thickness t, the localized bending dies out
after a distance λ:
• Membrane stresses do not die out
Trang 10MEEM4405 Introduction to Finite Element Analysis
7.5 Shell Finite Elements
• The most simple shell elements combine a
membrane element and bending element
E.g combining plane stress and plate
elements
• These elements are flat
• When flat elements, it is important that
elements are not all coplanar where they
meet at a node
Shell Finite Elements
• Curved shell elements can be derived from
shell theory
• Isoparametric shell elements can also be
obtained by starting with a solid element
and reducing degrees of freedom
• Thin shell behavior varies widely between
formulations and should be tested before
use
Trang 11MEEM4405 Introduction to Finite Element Analysis
Shells of Revolution
• In axisymmetric problems, shells resemble
beam elements
• Conical elements have problems similar to
flat shell elements