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