JN Reddy Plane Elasticity : 1

The Finite Element Method

Plane Elasticity

Read: Chapter 11 CONTENTS

• Governing equations • Weak form formulation• Finite element models using

the weak form• Triangular and rectangular

elements• Shear locking • Modeling aspects and discussion

JN Reddy

displacement, stress function, temperature, velocity potentialu u u u- - - -

REVIEW OF PLANE ELASTICITY

Plane Strain Problems

e e e( , , ), ( , , ), 0

0, 0, 0xz yz zz

u u x y t v v x y t w= = = = = =

Plane Stress Problems( , , ), ( , , ),

( , , ),

0, 0, 0

xx xx yy yy

xy xy

xz yz zz

x y t x y t

x y t

s s s s

s s

s s s

= =

=

= = =

2-D Elasticity: 2

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GOVERNING EQUATIONS

Strain-displacement relations

Cauchy’s formula

x xx x xy y

y xy x yy y

t n n

t n n

s s

s s

= +

= +

Stress-strain relations(orthotropic)

2-D Elasticity: 3

Stress equations of motion

2

2

2

2

xyxx xx

xy yy yy

ufx y t

uf

x y t

σσ ρ

σ σρ

∂∂ ∂+ + =

∂ ∂ ∂

∂ ∂ ∂+ + =

∂ ∂ ∂

11 12

12 22

66

00

0 0 2

xx xx

yy yy

xy xy

C CC C

C

=

σ εσ εσ ε

2xx yy xyu v u vx y y x, ,ε ε ε∂ ∂ ∂ ∂= = = +

∂ ∂ ∂ ∂x

z

y

xzσzzσ

yzσ

yyσ

zyσ

xyσxxσ

zxσ

yxσ

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GOVERNING EQUATIONS (continued)

Plane-strain constitutive relations

Plane-stress constitutive relations

2-D Elasticity: 4

1 12 21 2 12 2111 22

12 12 12 21 12 12 12 21

12 212 66 12 12 2 21 1

12 12 21

1 11 1 2 1 1 2

1 2

ν ν ν νν ν ν ν ν ν ν ν

ν ν νν ν ν

( ) ( ),( )( ) ( )( )

, ,( )

E EC C

EC C G E E

− −= =

+ − − + − −

= = =− −

1 2 12 211 22 12

12 21 12 21 12 21

66 12 12 2 21 1

1 1 1ν

ν ν ν ν ν ν

ν ν

= = =− − −

= =

, ,( ) ( ) ( )

,

E E EC C C

C G E E

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DOMAIN DISCRETIZATION (MESH)

2-D Elasticity: 5

x

y

Γ

Γ

Ω e

e

nα

Ω

iuiv

iDisplacement degrees of

freedom at a typical node i

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Weak Forms of the Equations

2-D Elasticity: 6

11 12

12 22

66

xx

yy

xy

u vc cx yu vc cx yu vcy x

s

s

s

¶ ¶= +

¶ ¶¶ ¶

= +¶ ¶æ ö¶ ¶ ÷ç= + ÷ç ÷ç ÷¶ ¶è ø

0 = heΩew1 ρu− ∂σxx

∂x− ∂σxy

∂y− fx dA

= heΩe

ρw1u+∂w1∂x

σxx +∂w1∂y

σxy − w1fx dA

− heSew1tx dS

0 = heΩew2 ρv − ∂σxy

∂x− ∂σyy

∂y− fy dA

= heΩe

ρw2v +∂w2∂x

σxy +∂w2∂y

σyy − w2fy dA

− heSew2ty dS

tx = he (σxxnx + σxyny) , ty = he (σxynx + σyyny)

w1 ∼ δu, w2 ∼ δv

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FINITE ELEMENT MODEL using weak form

2-D Elasticity: 7

Finite element approximation

Finite element model

u ≈n

j=1

uej(t)ψej (x, y), v ≈

n

j=1

vej (t)ψej (x, y)

[M11] [0][0] [M22]

uv +

[K11] [K12][K21] [K22]

uv =

F 1F 2

K11ij = he

Ωec11

∂ψi∂x

∂ψj∂x

+ c66∂ψi∂y

∂ψj∂y

dA

K12ij = K

21ji = he

Ωec12

∂ψi∂x

∂ψj∂y

+ c66∂ψi∂y

∂ψj∂x

dA

K22ij = he

Ωec66

∂ψi∂x

∂ψj∂x

+ c22∂ψi∂y

∂ψj∂y

dA

F 1i = heΩe

ψifx dA+ heSe

ψitx dS

F 2i = heΩe

ψifydx dy + heSe

ψity dS

[ ] [ ] M K FΔ Δ+ =

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FINITE ELEMENTS

2-D Elasticity: 8

1

2

3

2u

1u

1v

3v

3u

2v

1

2

3

4

5

6

2

67

82

3

4

1

5

2

3

4

1

1v

1u

4v

4u

3v

3u2v

2u

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K11ij = he

Ωec11

∂ψi∂x

∂ψj∂x

+ c66∂ψi∂y

∂ψj∂y

dA

Numerical Values of Typical Element Matrices

[Ke] =a116a

⎡⎢⎣2b −2b −b b−2b 2b b −b−b b 2b −2bb −b −2b 2b

⎤⎥⎦+ a226b

⎡⎢⎣2a −2a −a a−2a 2a a −a−a a 2a −2aa −a −2a 2a

⎤⎥⎦11[ ]K 66c h11c h

K22ij = he

Ωec66

∂ψi∂x

∂ψj∂x

+ c22∂ψi∂y

∂ψj∂y

dA

[Ke] =a116a

⎡⎢⎣2b −2b −b b−2b 2b b −b−b b 2b −2bb −b −2b 2b

⎤⎥⎦+ a226b

⎡⎢⎣2a −2a −a a−2a 2a a −a−a a 2a −2aa −a −2a 2a

⎤⎥⎦22[ ]K 22c h66c h

2-D Elasticity: 9

JN Reddy Plane Elasticity: 10

Eigenvalue Problems

D DM K F + =

Assume periodic solution

0wD D i te=

( ) 20 0 ,l l wDM K Qe e e e− + = =

( ) 20 0 ,l l wM K U Q− + = =

Assembled equations

JN Reddy Plane Elasticity: 11

TIME APPROXIMATIONS

D DM K Fe e e e e+ =

( ) g

a a

d d

d a a

D D D D

D D D D D D

21 ,12

1 , , 1, (1 )

s s s s

s s s s s s

t t

t

+

+ +

= + += + = - +

Semidiscrete FE model

Newmark scheme (second-order equations)

Fully discretized model

( )D

D D D

K F K K M

F F M

1 1 1 1 1 13

1 1 13 4 5

ˆ ˆ ˆ,

ˆ

s s s s s s

s s s s s s

a

a a a

+ + + + + +

+ + +

= = +

= + + +

JN Reddy 2-D Elasticity: 12

Static Condensation Procedure

( )( )

11 2

12 2

23

11

1

( )( )( )

y x xy x xy x x

= -= += -

1 1 2 2 3( ) ( ) ( ) ( )u x u uy x y x ay x= + +

1 1 2 2

1 2 2 1

2 2

( ) ( )x x x

x x x x

y x y x

x

= +

+ -= + 21

1( )y x 2( )y x

x

3( )y x

1x x= 2x x=

2

11 1 2 2

11 12 13 1 1 1

21 22 23 2 2 2

31 32 33 3

0

0

( ) ( )x

x

d u duEA f u dx u x P u x Pdx dx

K K K u f PK K K u f PK K K f

dd d d

a

æ ö÷ç= - - -÷ç ÷çè ø

é ù ì ü ì ü ì üï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ïï ï ï ï ï ïê ú = +í ý í ý í ýê ú ï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ïï ï ï ï ï ïë û î þ î þ î þ

ò

JN Reddy 2-D Elasticity: 13

Static Condensation Procedure

( ) ( )

11 12 13 1 1 1

21 22 23 2 2 2

31 32 33 3

133 3 31 1 32 2 33 3 31 1 32 2

0

K K K u f PK K K u f PK K K f

K f K u K u K f K u K u

a

a a-

é ù ì ü ì ü ì üï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ïï ï ï ï ï ïê ú = +í ý í ý í ýê ú ï ï ï ï ï ïï ï ï ï ï ïê ú ï ï ï ï ï ïï ï ï ï ï ïë û î þ î þ î þ

= - - = - -

( ) ( )

( ) ( )

( ) ( )

13

3

13

1 13

,

ˆ ˆor

uu u uu

u

uu u u uu u u u

uu u u u u uu u

K fK f

K f

K K f

aaa a

a aa

a a aa a

a aa a a aa

aa

a

-

-

- -

é ù ì üì ü ï ïï ïï ï ï ïê ú = = -í ý í ýê ú ï ï ï ïï ïî þ ï ïë û î þ

+ = + - =

é ù- = - =ê úë û

K K Fu K uK

K u K F K u K K u F

K K K u F K K u F

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BEHAVIOR OF THE LINEAR PLANE ELASTICITY ELEMENT

εxx

2-D Elasticity: 14

Pure bending deformation

Linear elementDeformation(incorrect)

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Elimination of Shear Locking in Linear Elements

Basic idea: Start with the standard expansion in terms ofbilinear shape functions for a typical linear quadrilateralelement, quadratic modes are added as follows:

( ) ( ) ( )4 6

1 5

, , ,x h y x h y x he eh j j i i

j i= =

= + u αD

where are the standard bilinear interpolationfunctions and

( 1,2,3,4)i iy =

( ) ( )2 25 6, 1 , , 1y x h x y x h h= − = −

are the incompatible modes; are generalized displacements (node-less variables).

( 5,6)=ei iα

The generalized displacements associated with the incompatiblemodes are unique to each element and must be eliminated via“static condensation”.

2-D Elasticity: 15

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Elimination of Shear Locking in Linear Elements (continued)

=

FK K0K K

α

α αα α

ΔΔ Δ

Δ

Δ0K K F, K Kα α ααα αΔΔ Δ ΔΔ Δ+ = + =or

ed

Ω= TK B CB x

We begin with the standard definition of the stiffness matrix:

where [ ]1 2 3 4 5 6B = B B B B | B B and 5

6

α

α=α

Static condensation: Solving the second equation for , weobtain ( ) 1

K Kαα αα-

-= DD

α

Substituting into the first equation, we arrive at

1ˆ ( )K K K K Kα αα αΔΔ Δ − Δ−=K FD= where2-D Elasticity: 16

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•

•

• • • x

y

θ

312 4

6

1 2 3

7

0p••75

6

8

14

•

0 32

3p hL

23

32L

0 32

6p hL

••

0ps

2332L

•

• •

0 32 0 328 15

0 32 0 328 16

0 32 0 326 11

0 32 0 326 12

cos cos6 6

sin sin6 6

cos cos3 3

sin sin3 3

x

y

x

y

p hL p hLF F

p hL p hLF F

p hL p hLF F

p hL p hLF F

θ θ

θ θ

θ θ

θ θ

=- =-

=- =-

=- =-

=- =-

MODELING ASPECTS:Load Calculation

•

•

• x

y

1

56

1 2 3

4

7 •

••7

56

8

1

0 32

6p hL

0 32

3p hL

0 32 cos3

p hL θ

0 32 cos6

p hL θ

0 32 sin6

p hL θ

2 4

3

11

1

1 1

2-D Elasticity: 17

JN Reddy

1 2 1 211 55 33 12 56 34

1 113 51 15 23 61

1 2 2 5 619 53 35 88 22 66 44

2 5 6 2 5 677 11 55 33 8 2 6 4

,

, 0,

,

,

K K K K K K

K K K K K

K K K K K K K

K K K K F F F F

= + = +

= = =

= + = + +

= + + = + +

MODELING ASPECTS:Assembly of Element coefficients

•

•

• x

y

1

56

1 2 3

4

7 •

••75

6

8

1

0 32

6p hL

0 32

3p hL

0 32 cos3

p hL θ

0 32 cos6

p hL θ

0 32 sin6

p hL θ

2 4

3

11

1

1 1

11

2

3

•

••(5,6) (1,2)

(3,4)

1

2

5•

••(1,2) (3,4)

(9,10)

1

Global Element2-D Elasticity: 18

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MODELING ASPECTS:Bending of a cantilever beam

030msi, 0.25, 10 in., 2 in., 150psiν τ= = = = =E a b

0τ

y

a

8×2 mesh 16×4 mesh

0 0N/m at yt h xτ= − =2-D Elasticity: 19

h

bx

y

a

0τ2N/m x

2-D Meshes

Beam versus elasticity model

JN Reddy Plane Elasticity: 20

DISCUSSION PROBLEMS

1 2 69 GPa, 0.333, 26 GPa, 0.01 mn= = = = =E E G hPlane stress

One quadrant of the domain is used in the finite element analysis (isotropic plate of thickness h)

x

p0 (N/m2)

2b

a

y

• •

1 2 3 4 5 611

1615141312

8 1091 23

•h

t0 = 0.3 N/m2

7 43

002. m

004. m 004. m 004. m

••

•

••

•

• •

•

•

•

• •

• •(1, 2) (5, 6)(13, 14)

(31,32)

(15, 16)

(27, 28)

JN Reddy

SUMMARY

In this lecture we have covered the following topics:

• Review of plane stress and plane strain• Governing equations of plane elasticity• Finite element models using the weak form• Static condensation• Incompatible elements and shear locking • Discussion problems

2-D Elasticity: 21