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

© 1999 Yijun Liu, University of Cincinnati

151

Solids of Revolution (Axisymmetric Solids):

Baseball bat shaft

Apply cylindrical coordinates:

( x, y, z)

⇒

(r,

θ

, z)

θ

r, u

z,w

z, w

θ

r, u

θ

σ

z

σ

rz

τ

r

σ

r

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

© 1999 Yijun Liu, University of Cincinnati

152

Displacement field:

(

)

component

ntial

circumfere

No

)

,

(

,

)

,

(

−

=

=

v

z

r

w

w

z

r

u

u

Strains:

)

21

(

)

0

(

,

,

,

,

=

=

∂

∂

+

∂

∂

=

∂

∂

=

=

∂

∂

=

θ

θ

θ

γ

γ

γ

ε

ε

ε

z

r

rz

z

r

z

u

r

w

z

w

r

u

r

u

Stresses:

)

22

(

2

2

1

0

0

0

0

1

0

1

0

1

)

2

1

(

)

1

(



















































−

−

−

−

−

+

=























rz

z

r

rz

z

r

v

v

v

v

v

v

v

v

v

v

v

v

E

γ

ε

ε

ε

τ

σ

σ

σ

θ

θ

d

θ

r

(r+u)d

θ

rd

θ

u

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

© 1999 Yijun Liu, University of Cincinnati

153

Axisymmetric Elements:

)

23

(

∫

=

V

T

dz

d

rdr

θ

B

E

B

k

or

)

24

(

)

(det

2

)

(det

1

1

1

1

2

0

1

1

1

1

η

ξ

π

θ

η

ξ

π

d

d

r

d

d

d

r

T

T

∫∫

∫∫∫

− −

− −

=

=

J

B

E

B

J

B

E

B

k

r, u

3

2

4

1

ξ

η

r, u

2

3

1

1

2

3

3-node element (ring)

4-node element (ring)

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

© 1999 Yijun Liu, University of Cincinnati

154

Applications:

•

Rotating Flywheel:

Body forces:

)

force

nal

gravitatio

(

)

force

inertial

l/

centrifuga

radial

equivalent

(

2

g

f

r

f

z

r

ρ

ω

ρ

−

=

=

z

ω

angular velocity (rad/s)

r

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

© 1999 Yijun Liu, University of Cincinnati

155

•

Cylinder Subject to Internal Pressure:

•

Press Fit:

ring ( Sleeve) shaft

p

0

r

0

2

)

(

r

p

q

π

=

0

r

i

r

δ

+

i

r

“i” “o”

MPC

u

u

i

o

⇒

=

−

δ

:

i

r

r

at

=

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

© 1999 Yijun Liu, University of Cincinnati

156

•

Belleville (Conical) Spring:

This is a geometrically nonlinear (large deformation)

problem and iteration method (incremental approach) needs to
be employed.

p

p

δ

z

δ

r