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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements

1999 Yijun Liu, University of Cincinnati 138

Chapter 6. Solid Elements for 3-D Problems

I. 3-D Elasticity Theory

Stress State:

y

F

x

z

y

yx

yz

zy

zx

z

xz

x

xy

y , v

, u

z, w

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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements

1999 Yijun Liu, University of Cincinnati 139

{ } [ ] )1(, ij

zx

yz

xy

z

y

x

or

==

Strains:

{ } [ ] )2(, ij

zx

yz

xy

z

y

x

or

==

Stress-strain relation:

+=

zx

yz

xy

z

y

x

zx

yz

xy

z

y

x

v

v

vvvv

vvv

vvv

vv

E

2

2100000

02

21

0000

002

21000

0001

0001

0001

)21)(1(

or )3(E =

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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements

1999 Yijun Liu, University of Cincinnati 140

Displacement:

)4(

),,(

),,(

),,(

3

2

1

=

= uu

u

zyxw

zyxv

zyxu

u

Strain-Displacement Relation:

)5(,,

,,,

x

w

z

u

z

v

y

w

y

u

x

v

z

w

y

v

x

u

xzyzxy

zyx

+

=

+

=

+

=

=

=

=

or

( )

( ) notation)tensor(2

1

simply,or

3,2,1,,2

1

,, ijjiij

i

j

j

i

ij

uu

jix

u

x

u

+=

=

+

=

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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements

1999 Yijun Liu, University of Cincinnati 141

Equilibrium Equations:

0

or

,0

)6(,0

,0

, =+

=+

+

+

=+

+

+

=+

+

+

ijij

z

zzyzx

y

yzyyx

x

xzxyx

f

fzyx

fzyx

fzyx

Boundary Conditions (BCs):

)traction(

)7()(,

)(,

jiji

ii

uii

nt

tractionspecifiedontt

ntdisplacemespecifiedonuu

=

==

Stress Analysis:

Solving equations in (6) under the BCs in (7).

p

n

u

)(

+=u

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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements

1999 Yijun Liu, University of Cincinnati 142

II. Finite Element Formulation

Displacement Field:

i

N

ii

i

N

ii

N

iii

wNw

vNv

uNu

=

=

=

=

=

=

1

1

1

)8(

Nodal values

In matrix form:

)9(

)13(

)33()13(

2

2

2

1

1

1

21

21

21

0000

00000000

=

N

N

w

v

uw

v

u

NN

NNNN

w

vu

M

L

L

L

or dNu=

Using relations (5) and (8), we can derive the strain vector

=B d(61) (63N)(3N1)

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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements

1999 Yijun Liu, University of Cincinnati 143

Stiffness Matrix:

)10(= vT

dvBEBk

(3N) (3N6)(66)(63N)

Numerical quadratures are often needed to evaluate the

above integration.

Rigid-body motions for 3-D bodies (6 components):

3 translations, 3 rotations.

These rigid-body motions (singularity of the system of

equations) must be removed from the FEA model to ensure the

quality of the analysis.