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Finite Element Method
FEM FOR PLATES &
CONTENTS
INTRODUCTION
PLATE ELEMENTS
– Shape functions
– Element matrices
SHELL ELEMENTS
– Elements in local coordinate system
– Elements in global coordinate system
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INTRODUCTION
FE equations based on Mindlin plate theory will be developed.
FE equations of shells will be formulated by
superimposing matrices of plates and those of 2D solids.
PLATE ELEMENTS
Geometrically similar to 2D plane stress solids except that it carries only transverse loads. Leads to bending.
2D equilvalent of the beam element.
Rectangular plate elements based on Mindlin plate theory will be developed – conforming element.
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PLATE ELEMENTS
Consider a plate structure:
x y
z, w
h
fz Middle plane
Middle plane
PLATE ELEMENTS
Mindlin plate theory:
( , , ) ( , )
In-plane strain:
Middle plane
χ (Curvature)
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PLATE ELEMENTS
Off-plane shear strain:
Potential (strain) energy:
z
In-plane stress & strain
Off-plane shear stress & strain
PLATE ELEMENTS
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PLATE ELEMENTS
3 3
2 2
2
1 1
( )d ( )d
2 e 12 12 2 e
T
e x y
A A
h h
T
hw A
d I d Ax
y w
d
3
3
0 0
0 0
12
0 0
12
h
h
h
I
Shape functions
Note that rotation is independent of deflection w
,
,
4
1 4
1 4
1
i y i i
y i
x i i
x i
i i
N N
w N
w
) 1
)( 1
(
4
1
i i
i
N
where (Same as rectangular
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Shape functions
h
displacement at node 1
displacement at node 2
displacement at node 3
displacement at node 4
Element matrices
Substitute
h
x e
y
w
d Nd into
e e T
e e
T d m d
2 1
1
( )d
2 e
T e
A
T
d I d A
where T d
e
e
A A
m N I N
Recall that:
3
3
0 0
0 0
12
0 0
12
h
h
h
I
(Can be evaluated
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Element matrices
A
Substitute
h
d Nd into potential energy function
from which we obtain
Element matrices
analytically but practically solved using Gauss
integration)
A
For uniformly distributed load,
1 0 0 1 0 0 1 0 0 1 0 0
z T
e abf
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SHELL ELEMENTS
Loads in all directions
Bending, twisting and in-plane deformation
Combination of 2D solid elements (membrane effects) and plate elements (bending effect).
Elements in local coordinate system
Consider a flat shell element
4 node
3 node
2 node
1 node
4
rotation about -axis rotation about -axis rotation about -axis
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Elements in local coordinate system
Membrane stiffness (2D solid element):
4 node
3 node
2 node
1 node node4
node3
node2
node1
Bending stiffness (plate element):
4 node
3 node
2 node
1 node node4
node3
node2
Elements in local coordinate system
4 node
3 node
2 node
1 node
0 node
node
node
node
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Elements in local coordinate system
Membrane mass matrix (2D solid element):
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Bending mass matrix (plate element):
Elements in local coordinate system
4 node
3 node
2 node
1 node
0 node
node
node
node
Components related to the DOF z, are zeros in local coordinate system.
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Elements in global coordinate system
Remarks
The membrane effects are assumed to be
uncoupled with the bending effects in the element level.
This implies that the membrane forces will not
result in any bending deformation, and vice versa.
For shell structure in space, membrane and
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CASE STUDY
CASE
STUDY
Mode
Natural Frequencies (MHz)
768 triangular elements with
480 nodes
384 quadrilateral elements with
480 nodes
1280 quadrilateral elements with
1472 nodes
1 7.67 5.08 4.86
2 7.67 5.08 4.86
3 7.87 7.44 7.41
4 10.58 8.52 8.30
5 10.58 8.52 8.30
6 13.84 11.69 11.44
7 13.84 11.69 11.44
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CASE STUDY
Mode 1:
CASE STUDY
Mode 3:
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CASE STUDY
Mode 5:
CASE STUDY
Mode 7:
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CASE STUDY
Transient analysis of micro-motor
F F F
x x
Node 210
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