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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1998 Yijun Liu, University of Cincinnati 152
Displacement f ield:
( )componentntialcircumfere No),(,),( −== v z r ww z r uu
Strains:
)21()0(,
,,,
==∂
∂
+∂
∂
=
∂
∂==
∂
∂=
θθ
θ
γ γ γ
εεε
z r rz
z r
z
u
r
w
z
w
r
u
r
u
Stresses:
)22(
2
21000
010
010
01
)21()1(
−−
−
−
−+=
rz
z
r
rz
z
r
vvv
vv
vvv
vv
E
γ
ε
ε
ε
τ
σ
σ
σ
θθ
d θr (r+u)d θ
rd θ
u
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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1998 Yijun Liu, University of Cincinnati 153
Axisymmetr ic Elements:
)23(∫ =V
T dz d rdr θBEBk
or
)24()(det2
)(det
1
1
1
1
2
0
1
1
1
1
ηξπ
θηξπ
d d r
d d d r
T
T
∫ ∫
∫ ∫ ∫
− −
− −
=
=
JBEB
JBEBk
r ,u 3 2
41
ξ
η
r,u
2
3
11
2
3
3-node element (ring) 4-node element (ring)
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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1998 Yijun Liu, University of Cincinnati 154
Applications:
• Rotating F lywheel:
Body forces:
)forcenalgravitatio(
)forceinertiall/centrifugaradialequivalent(2
g f
r f
z
r
ρ
ωρ
−=
=
ω angular velocity (rad/s)
r
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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1998 Yijun Liu, University of Cincinnati 155
• Cylinder Subject to I nternal Pressure:
• Press F it:
ring ( Sleeve) shaft
p 0
r
02)( r pq π=
0r
ir
δ+r
“i” “o”
MPC
uu io
⇒
=− δ
:i
r r at =
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Lecture Notes: Introduction to Finite Element Method Chapter 6. Solid Elements
© 1998 Yijun Liu, University of Cincinnati 156
• Bellevil le (Conical) Spring:
This a geometrically nonlinear (large deformation) problem
and iterations need to be employed (Use incremental approach).
p z
r
p
