1 Finite Element Method FEM FOR PLATES & SHELLS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 8:

1

FFinite Element Methodinite Element Method

FEM FOR PLATES & SHELLS

for readers of all backgroundsfor readers of all backgrounds

G. R. Liu and S. S. Quek

CHAPTER 8:

2Finite Element Method by G. R. Liu and S. S. Quek

CONTENTSCONTENTS INTRODUCTION PLATE ELEMENTS

– Shape functions– Element matrices

SHELL ELEMENTS– Elements in local coordinate system– Elements in global coordinate system– Remarks


INTRODUCTIONINTRODUCTION

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.


PLATE ELEMENTSPLATE 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.



Consider a plate structure:

x

y z, w

h

fz Middle plane

Middle plane

(Mindlin plate theory)



Mindlin plate theory:

( , , ) ( , )

( , , ) ( , )

y

x

u x y z z x y

v x y z z x y

χε zIn-plane strain:

Middle plane

where

yx

y

x

yx

x

y

Lθχ (Curvature)

yx

y

x

0

0

L



Off-plane shear strain:

y

wx

w

x

y

yz

xz

γ

Potential (strain) energy:

zAzAUee A

Th

A

Th

e dd2

1dd

2

100 γτσε

In-plane stress & strain

Off-plane shear stress & strain

γcγτ syz

xz

G

G

0

0



Substituting χε z ,

AhAh

Uee A s

T

A

Te d

2

1d

122

1 3

γcγcχχ

γcγτ syz

xz

G

G

0

0

Kinetic energy: 2 2 21( )d

2 ee V

T u v w V

3 32 221 1

( )d ( )d2 12 12 2e e

Te x yA A

h hT hw A A d I d

( , , ) ( , )

( , , ) ( , )

y

x

u x y z z x y

v x y z z x y

Substituting


PLATE ELEMENTSPLATE ELEMENTS3 3

2 221 1( )d ( )d

2 12 12 2e e

Te x yA A

h hT hw A A d I d

x

y

w

d3

3

0 0

0 012

0 012

h

h

h

Iwhere ,


Shape functionsShape functions

Note that rotation is independent of deflection w

, ,4

1

4

1

4

1iyi

iyixi

ixii

i

NNwNw

)1)(1(41 iiiN where (Same as rectangular

2D solid)


Shape functionsShape functionsh

x e

y

w

Nd

1

1

1

2

2

2

3

3

3

4

4

4

displacement at node 1




x

y

x

y

e

x

y

x

y e

w

w

w

w

dwhere

1 2 3 4

1 2 3 4

1 2 3 4

Node 1 Node 2 Node 3 Node 4

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

N N N N

N N N N

N N N N

N

1 ( 1, 1) (u1, v1, w1,

x1,

y1,

z1)

2 (1, 1) (u2, v2, w2,

x2,

y2,

z2)

3 (1, +1) (u3, v3, w3, x3,

y3,

z3)

2a

4 ( 1, +1) (u4, v4, w4,

x4,

y4,

z4)

2b

z, w


Element matricesElement matrices

Substitute

h

x e

y

w

d Nd into

eeT

eeT dmd 2

1

1( )d

2 e

Te A

T A d I d

where T de

e AAm N I N

Recall that:

3

3

0 0

0 012

0 012

h

h

h

I(Can be evaluated analytically but in practice, use Gauss integration)



AhAh

sAAeee

d][d][12

OTOITI3

BcBcBBk

Substitute

h

x e

y

w

d Nd into potential energy function

from which we obtain

I4

I3

I2

II1 BBBBB

yNxN

yN

xN

jj

j

j

0

00

00IjB

iijj

iijj

by

N

y

N

ax

N

x

N

)1(4

1

)1(4

1

, byax Note:



O4

O3

O2

OO1 BBBBB

0

0Oj

jj

jj

NyN

NxNB

(me can be solved analytically but practically solved using Gauss integration)

A

f

eA

z

e d

0

0T

Nf

For uniformly distributed load,

001001001001zTe abff


SHELL ELEMENTSSHELL 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.


Elements in local coordinate systemElements in local coordinate system

1 ( 1, 1) (u1, v1, w1,

x1,

y1,

z1)

2 (1, 1) (u2, v2, w2,

x2,

y2,

z2)

3 (1, +1) (u3, v3, w3, x3,

y3,

z3)

2a

4 ( 1, +1) (u4, v4, w4,

x4,

y4,

z4)

2b

z, w Consider a flat shell element

4 node

3 node

2 node

1 node

4

3

2

1

e

e

e

e

e

d

d

d

d

d

displacement in direction



rotation about -axis



i

i

iei

xi

yi

zi

u x

v y

w z

x

y

z

d



Membrane stiffness (2D solid element):

4 node

3 node

2 node

1 nodenode4 node3 node2 node1

44

34

24

14

43

33

23

13

42

32

22

12

41

31

21

11

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

me

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

Bending stiffness (plate element):

4 node

3 node

2 node

1 nodenode4 node3 node2 node1

44

34

24

14

43

33

23

13

42

32

22

12

41

31

21

11

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

be

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

(2x2)

(3x3)



4 node

3 node

2 node

1 node

000

0

0

000

0

0

000

0

0

000

0

0

4 node

000

0

0

000

0

0

000

0

0

000

0

0

3 node

000

0

0

000

0

0

000

0

0

000

0

0

2 node

000

0

0

000

0

0

000

0

0

000

0

0

1 node

44

44

34

34

24

24

14

14

43

43

33

33

23

23

13

13

42

42

32

32

22

22

12

12

41

41

31

31

21

21

11

11

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

e

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k0

0k

k

(24x24)

Components related to the DOF z, are

zeros in local coordinate system.



Membrane mass matrix (2D solid element):

1311 12 14

2321 22 24

3331 32 34

4341 42 44

node3 node1 node2 node4

node 1

node 2

node 3

node 4

mm m m

m mm m me

mm m m

mm m m

mm m m

m mm m m

mm m m

mm m m

Bending mass matrix (plate element):

1311 12 14

2321 22 24

3331 32 34

4341 42 44

node3 node1 node2 node4

node 1

node 2

node 3

node 4

bb b b

b bb b be

bb b b

bb b b

mm m m

m mm m m

mm m m

mm m m



4 node

3 node

2 node

1 node

000

0

0

000

0

0

000

0

0

000

0

0

4 node

000

0

0

000

0

0

000

0

0

000

0

0

3 node

000

0

0

000

0

0

000

0

0

000

0

0

2 node

000

0

0

000

0

0

000

0

0

000

0

0

1 node

44

44

34

34

24

24

14

14

43

43

33

33

23

23

13

13

42

42

32

32

22

22

12

12

41

41

31

31

21

21

11

11

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

b

m

e

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m0

0m

m

Components related to the DOF z, are

zeros in local coordinate system.

(24x24)


Elements in global coordinate systemElements in global coordinate system

TkTK eT

e

TmTM eT

e

eT

e fTF

3

3

3

3

3

3

3

3

T0000000

0T000000

00T00000

000T0000

0000T000

00000T00

000000T0

0000000T

T

zzz

yyy

xxx

nml

nml

nml

3T

where


RemarksRemarks

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.


CASE STUDYCASE STUDY

Natural frequencies of micro-motor


CASE CASE STUDYSTUDY

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

8 14.86 12.45 12.17



Mode 1:

Mode 2:



Mode 3:

Mode 4:



Mode 5:

Mode 6:



Mode 7:

Mode 8:



Transient analysis of micro-motor

F

F

F

x

x

Node 210

Node 300







Documents

1 Finite Element Method FEM FOR PLATES & SHELLS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 8: