chpt07-fem for plates & shellsnew

8/22/2019 Chpt07-FEM for Plates & Shellsnew

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The Finite Element Method

A Practical Course

FEM FOR PLATES & SHELLS

CHAPTER 7:


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CONTENTS

INTRODUCTION

PLATE ELEMENTS

Shape functions

Element matrices

SHELL ELEMENTS

Elements in local coordinate system

Elements in global coordinate system

Remarks

CASE STUDY


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INTRODUCTION

FE equations based on Reissner-Mindlin plate

theory will be developed.

FE equations of shells will be formulated bysuperimposing matrices of plates and that of 2D

solids.

Computationally tedious due to more DOFs.


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PLATE ELEMENTS

Geometrically similar to 2D plane stress solidsexcept that it carries only transverse loads. Leadsto bending.

2D equivalent of the beam element.

Rectangular plate elements based on Reissner-Mindlin plate theory will be developed

conforming element. Many software like ABAQUS do not offer plate

elements, only the general shell element.


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PLATE ELEMENTS

Consider a plate structure:

z, w

h

z Middle lane

Middle

plane

(Reissner-Mindlin plate

theory)


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PLATE ELEMENTS

Reissner-Mindlin plate theory:

( , , ) ( , )

( , , ) ( , )

y

x

u x y z z x y

v x y z z x y

zIn-plane strain:

Middle

plane

where

yx

y

x

yx

x

y

L (Curvature)

0

0

x

y

y x

Lin which


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PLATE ELEMENTS

Off-plane shear strain:

y

wx

w

x

y

yz

xz

Potential (strain) energy:/ 2 / 2

/ 2 / 2

1 1d d d d

2 2e e

h hT T

eA h A h

U A z A z

In-plane stress &

strain

Off-plane shear

stress & strain

c syz

xz

G

G

0

0

2 /12 or 5/6


8/32

PLATE ELEMENTS

Substituting z ,

AhAh

Uee A

sTA

Te d2

1d

122

1 3

cc

c syz

xz

G

G

0

0

Kinetic energy: 2 2 21

( )d2 e

eV

T u v w V

3 32 221 1( )d ( )d

2 12 12 2e eT

e 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


9/32

PLATE ELEMENTS

3 32 2 21 1( )d ( )d

2 12 12 2e eT

e x yA A

h hT hw A A d I d

x

y

w

d

3

3

0 0

0 012

0 012

h

h

h

Iwhere ,


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Shape functions

Note that rotation is independent of deflection w

,,4

1

4

1

4

1iyi

i

yixi

i

xii

i

NNwNw

)1)(1(41 iiiN where

(Same as rectangular

2D solid)


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Shape 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)(w1,x1,y1)

2 (1, 1)(w2,x2,y2)

3 (1, +1)

(w3,x3,y3)

2

4 (1, +1)(w4,x4,y4)

2

z, w


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Element matrices

Substitute

h

x e

y

w

d Nd into

eeT

eeT dmd 21

1( )d

2 eT

eA

T A d I d

where T de

eA

A m 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)


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Element matrices

AhAh

sAA

eee

d][d][12

OTOITI3

BcBcBBk

Substitute

h

x e

y

w

d Nd into potential energy function

from which we obtain

I

4

I

3

I

2

II1

BBBBB

yNxN

yN

xN

jj

j

j

0

00

00I

jBii

jj

ii

jj

by

N

y

N

ax

N

x

N

)1(4

1

)1(4

1

, byax Note:


14/32

Element matrices

O

4

O

3

O

2

OO1 BBBBB

0

0O

j 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,

001001001001zT

e abff


15/32

SHELL ELEMENTS

Loads in all directions

Bending, twisting and in-plane deformation

Combination of 2D solid elements (membraneeffects) and plate elements (bending effect).

Common to use shell elements to model plate

structures in commercial software packages.


16/32


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)

2

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

x4,y4,z4)

2

, wConsider a flat shell element

4node

3node

2node

1node

4

3

2

1

e

e

e

e

e

d

d

d

d

d

displacement in direction



rotation about -axis



i

i

i

ei

xi

yi

zi

u x

v y

w z

x

y

z

d


17/32


Membrane stiffness (2D solid element):

4node3node

2node

1nodenode4node3node2node1

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

m

e

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

Bending stiffness (plate element):

4node

3node

2node

1nodenode4node3node2node1

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

b

e

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

(2x2)

(3x3)


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4node

3node

2node

1node

000

0

0

000

0

0

000

00

000

0

0

4node

000

0

0

000

0

0

000

00

000

0

0

3node

000

0

0

000

0

0

000

00

000

0

0

2node

000

0

0

000

0

0

000

00

000

0

0

1node

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, arezeros in local

coordinate

system.


19/32


Membrane mass matrix (2D solid element):

1311 12 14

2321 22 24

3331 32 34

4341 42 44

node3node1 node2 node4

node 1

node 2node 3

node 4

mm m m

m mm m m

emm 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

node3node1 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


20/32


4node

3node

2node

1node

000

0

0

000

0

0

000

00

000

0

0

4node

000

0

0

000

0

0

000

00

000

0

0

3node

000

0

0

000

0

0

000

00

000

0

0

2node

000

0

0

000

0

0

000

00

000

0

0

1node

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, arezeros in local

coordinate

system.

(24x24)


21/32

Elements in global coordinate system

TkTK eT

e

TmTM eT

e

e

T

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


22/32

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.


23/32

CASE STUDY

Natural frequencies of micro-motor

l i ( )


24/32

CASE

STUDYMode

Natural Frequencies (MHz)

768 triangular

elements with480 nodes

384 quadrilateral


1280

quadrilateral


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


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CASE STUDY

Mode 1:

Mode 2:


26/32

CASE STUDY

Mode 3:

Mode 4:


27/32

CASE STUDY

Mode 5:

Mode 6:


28/32

CASE STUDY

Mode 7:

Mode 8:


29/32

CASE STUDY

Transient analysis of micro-motor

F

F

F

x

x

Node 210

Node 300


30/32

CASE STUDY


31/32

CASE STUDY


32/32

CASE STUDY

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