Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 8:
FEM FOR PLATES &
SHELLS
1
CONTENTS

INTRODUCTION
 PLATE ELEMENTS
– Shape functions
– Element matrices

SHELL ELEMENTS
– Elements in local coordinate system
– Elements in global coordinate system
– Remarks
Finite Element Method by G. R. Liu and S. S. Quek
2
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.
 Computationally tedious due to more DOFs.
Finite Element Method by G. R. Liu and S. S. Quek
3
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.
 Much software like ABAQUS does not offer plate
elements, only the general shell element.
Finite Element Method by G. R. Liu and S. S. Quek
4
PLATE ELEMENTS

z, w
Consider a plate structure:
y
fz
Middle plane
h
x
Middle
plane
(Mindlin plate theory)
Finite Element Method by G. R. Liu and S. S. Quek
5
PLATE ELEMENTS

Mindlin plate theory:
u ( x, y, z )  z y ( x, y)
Middle
plane
v( x, y, z )   z x ( x, y )
In-plane strain:
ε   zχ
where
 

  x
L 0

 

  x


  y x


χ  Lθ  
 x y

 x   y 
y
 x


0 

 
 y 
 


 y 
Finite Element Method by G. R. Liu and S. S. Quek
(Curvature)
6
PLATE ELEMENTS
w 

 xz    y  x 
Off-plane shear strain: γ     


w

 yz    x 


y 
Potential (strain) energy:
h
h
1
1
T
U e    ε σdAdz    τ T γdAdz
2 Ae 0
2 Ae 0
In-plane stress &
strain
Off-plane shear
stress & strain
 xz 
G 0 
τ    
γ  c s γ

 0 G
 yz 
Finite Element Method by G. R. Liu and S. S. Quek
7
PLATE ELEMENTS
Substituting
ε   zχ
 xz 
G 0 
, τ       0 G  γ  c s γ


 yz 
1 h3 T
1
Ue  
χ cχdA   hγ T c s γdA
2 Ae 12
2 Ae
Kinetic energy:
Substituting
1
Te    (u 2  v 2  w2 )dV
2 Ve
u ( x, y, z )  z y ( x, y)
v( x, y, z )   z x ( x, y )
1
h3 2 h3 2
1
2
Te    (hw   x   y )dA   (dT I d)dA
2 Ae
12
12
2 Ae
Finite Element Method by G. R. Liu and S. S. Quek
8
PLATE ELEMENTS
1
h3 2 h3 2
1
2
Te    (hw   x   y )dA   (dT I d)dA
2 Ae
12
12
2 Ae
where
w
 
d   x  ,
 
 y

h

I 0


0

0
 h3
12
0
Finite Element Method by G. R. Liu and S. S. Quek

0 

0 


 h3 
12 
9
Shape functions

Note that rotation is independent of deflection w
4
4
4
i 1
i 1
i 1
w   N i wi ,  x   N i x i ,  y   N i y i
where
N i  14 (1   i )(1  i )
(Same as rectangular
2D solid)
Finite Element Method by G. R. Liu and S. S. Quek
10
Shape functions
h
4 (1, +1)
(u4, v4, w4,
x4,y4,z4)
2b
2a
1 (1, 1)
(u1, v1, w1,
x1,y1,z1)
w
 
 x   Nd e
 
 y
z, w

3 (1, +1)
(u3, v3, w3,
x3,y3,z3)
where

2 (1, 1)
(u2, v2, w2,
x2,y2,z2)
 N1
N   0
 0
0
N1
0
Node 1
0
0
N1
N2
0
0
0
N2
0
Node 2
0
0
N2
N3
0
0
 w1 
 
 x1 
 y1 
 
 w2 
 
 x2 
 y 2 
de   
 w3 
 x 3 
 
 y 3 
w 
 4
 x 4 
 
 y 4 e
0
N3
0
0
0
N3
Node 3
Finite Element Method by G. R. Liu and S. S. Quek


 displacement at node 1




 displacement at node 2




 displacement at node 3




 displacement at node 4


N4
0
0
0
N4
0
Node 4
0
0 
N 4 
11
Element matrices
h
w
1
T
 
T

(
d
I d)dA
Substitute  x   d  Nd e into e

A
2 e
 
 y

where
1 T 
Te  d e m ed e
2
me  
Ae
T
N I NdA
(Can be evaluated
analytically but in practice,
use Gauss integration)
Recall that:

h

I 0


0

Finite Element Method by G. R. Liu and S. S. Quek
0
 h3
12
0

0 

0 


 h3 
12 
12
Element matrices
h
w
 
Substitute  x   d  Nd e into potential energy function
 
 y
from which we obtain
h3 I T I
ke  
[B ] cB dA   h[B O ]T c s B O dA
Ae 12
Ae

B  B1
I
I
B
I
2
B
I
3
B
I
4

N j
0
0
 N j x 


B Ij  0 N j y
0

0 N j x  N j y 


N j 
1

  i (1   i )
x
 x 4a
N j N j  1


(1   i  ) i
y
 y 4b
Note:   x a ,   y b
Finite Element Method by G. R. Liu and S. S. Quek
13
Element matrices

B O  B O1
B O2
B 3O
B O4

0
N j x
B 
N j y  N j
O
j
Nj
0 
(me can be solved
analytically but practically
solved using Gauss
integration)
 fz 
 
f e   N T  0  dA
Ae
0
 
For uniformly distributed load,
f eT  abf z 1 0 0 1 0 0 1 0 0 1 0 0
Finite Element Method by G. R. Liu and S. S. Quek
14
SHELL ELEMENTS

Loads in all directions
 Bending, twisting and in-plane deformation
 Combination of 2D solid elements (membrane
effects) and plate elements (bending effect).
 Common to use shell elements to model plate
structures in commercial software packages.
Finite Element Method by G. R. Liu and S. S. Quek
15
Elements in local coordinate system
Consider a flat shell element
4 (1, +1)
(u4, v4, w4,
x4,y4,z4)
d e1  node 1
d  node 2
 
d e   e2 
d e3  node 3
d e 4  node 4
2b
 ui  displacement in x direction
 v  displacement in y direction
 i
 w  displacement in z direction
d ei   i 
rotation about x-axis
 xi 
 yi 
rotation about y -axis
 
rotation about z -axis
 zi 
2a
1 (1, 1)
(u1, v1, w1,
x1,y1,z1)
Finite Element Method by G. R. Liu and S. S. Quek
z, w

3 (1, +1)
(u3, v3, w3,
x3,y3,z3)

2 (1, 1)
(u2, v2, w2,
x2,y2,z2)
16
Elements in local coordinate system
Membrane stiffness (2D solid element):
(2x2)


k em  



node1
m
k 11
k m21
m
k 31
k m41
node2
m
k 12
k m22
m
k 32
k m42
node3
m
k 13
k m23
m
k 33
k m43
node4
m
k 14
k m24
m
k 34
k m44
 node 1
 node 2

 node 3

 node 4
node4
b
k 14
k b24
k b34
k b44
 node 1
 node 2

 node 3

 node 4
Bending stiffness (plate element):
(3x3)


k be  



node1
b
k 11
k b21
k b31
k b41
node2
b
k 12
k b22
k b32
k b42
node3
b
k 13
k b23
k b33
k b43
Finite Element Method by G. R. Liu and S. S. Quek
17
Elements in local coordinate system
node
1









ke  











node
2

node
3

node
4

m
m
k 11
0 0 k 12
b
0 k 11
0 0
0 0
0 0
0
b
k 12
0
0
0
0
m
k 13
0
0
0
b
k 13
0
0
0
0
m
k 14
0
0
0
b
k 14
0
0
0
0
k m21 0 0 k m22
0 k b21 0 0
0
0
0 0
0
k b22
0
0 k m23
0 0
0 0
0
k b23
0
0 k m24
0 0
0 0
0
k b24
0
0
0
0
m
m
k 31
0 0 k 32
0 k b31 0 0
0
0
0 0
0
k b32
0
m
0 k 33
0 0
0 0
0
k b33
0
m
0 k 34
0 0
0 0
0
k b34
0
0
0
0
k m41 0 0 k m42
0 k b41 0 0
0
0
0 0
0
k b42
0
0 k m43
0 0
0 0
0
k b43
0
0 k m44
0 0
0 0
0
k b44
0
0
0
0






















 node 1




 node 2




 node 3


Components
related to the
DOF z, are
zeros in local
coordinate
system.


 node 4


(24x24)
Finite Element Method by G. R. Liu and S. S. Quek
18
Elements in local coordinate system
Membrane mass matrix (2D solid element):
node1 node2 node3 node4

m em  




m
m11
m m21
m
m12
m m22
m
m31
m m41
m
m32
m m42
m
m13
m m23
m
m33
m m43
m
m14
m m24
m
m34
m m44
 node 1
 node 2

 node 3

 node 4
Bending mass matrix (plate element):
node1 node2 node3 node4

mbe  




b
m11
mb21
b
m12
mb22
mb31
mb41
mb32
mb42
b
m13
mb23
mb33
mb43
b
m14
mb24
mb34
mb44
 node 1
 node 2

 node 3

 node 4
Finite Element Method by G. R. Liu and S. S. Quek
19
Elements in local coordinate system
node
1









me  











node
2

node
3

node
4

m
m
m11
0 0 m 12
b
0 m 11
0 0
0
0
0 0
0
b
m 12
0
0
0
0
m
m 13
0
0
0
b
m 13
0
0
0
0
m
m 14
0
0
0
b
m 14
0
0
0
0
m m21
0 0 m m22
0 m b21 0 0
0
0
0 0
0
m b22
0
0 m m23
0 0
0 0
0
m b23
0
0 m m24
0 0
0 0
0
m b24
0
0
0
0
m
m
m 31
0 0 m 32
0 m b31 0 0
0
0
0 0
0
m b32
0
m
0 m 33
0
0
0
0
0
m b33
0
m
0 m 34
0
0
0
0
0
m b34
0
0
0
0
m m41
0 0 m m42
0 m b41 0 0
0
0
0 0
0
m b42
0
0 m m43
0 0
0 0
0
m b43
0
0 m m44
0 0
0 0
0
m b44
0
0
0
0






















 node 1




 node 2




 node 3


Components
related to the
DOF z, are
zeros in local
coordinate
system.


 node 4


(24x24)
Finite Element Method by G. R. Liu and S. S. Quek
20
Elements in global coordinate system
K e  TT k e T
Me  T meT
T
where
Fe  T T f e
l x
T3  l y
l z
T3
0

0

0
T
0

0
0

 0
mx
my
mz
0
0
0
0
0
0
T3
0
0
0
0
0
0
T3
0
0
0
0
0
0
T3
0
0
0
0
0
0
T3
0
0
0
0
0
0
T3
0
0
0
0
0
0
T3
0
0
0
0
0
0
0
0 
0

0
0

0
0

T3 
nx 
n y 
n z 
Finite Element Method by G. R. Liu and S. S. Quek
21
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
bending effects are actually coupled (especially
for large curvature), therefore finer element mesh
may have to be used.
Finite Element Method by G. R. Liu and S. S. Quek
22
CASE STUDY

Natural frequencies of micro-motor
Finite Element Method by G. R. Liu and S. S. Quek
23
Natural Frequencies (MHz)
Mode
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
8
14.86
12.45
12.17
Finite Element Method by G. R. Liu and S. S. Quek
CASE
STUDY
24
CASE STUDY
Mode 1:
Mode 2:
Finite Element Method by G. R. Liu and S. S. Quek
25
CASE STUDY
Mode 3:
Mode 4:
Finite Element Method by G. R. Liu and S. S. Quek
26
CASE STUDY
Mode 5:
Mode 6:
Finite Element Method by G. R. Liu and S. S. Quek
27
CASE STUDY
Mode 7:
Mode 8:
Finite Element Method by G. R. Liu and S. S. Quek
28
CASE STUDY

Transient analysis of micro-motor
F
Node 210
x
x
F
Node 300
F
Finite Element Method by G. R. Liu and S. S. Quek
29
CASE STUDY
Finite Element Method by G. R. Liu and S. S. Quek
30
CASE STUDY
Finite Element Method by G. R. Liu and S. S. Quek
31
CASE STUDY
Finite Element Method by G. R. Liu and S. S. Quek
32