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MECH300H Introduction to Finite Element Methods
Lecture 9
Finite Element Analysis of 2-D Problems – Axi-symmetric Problems
Axi-symmetric ProblemsDefinition:
A problem in which geometry, loadings, boundaryconditions and materials are symmetric about one axis.
Examples:
Axi-symmetric Analysis
Cylindrical coordinates:
zzryrx ;sin ;cos
zr , ,
• quantities depend on r and z only• 3-D problem 2-D problem
Axi-symmetric Analysis
Axi-symmetric Analysis – Single-Variable Problem
0),(),(),(1
002211
zrfuaz
zrua
zr
zrura
rr
Weak form:
dswq
rdrdzzrwfwuaz
ua
z
w
r
ua
r
w
e
e
n
),(0 002211
where zrn nz
zruan
r
zruaq
),(),(2211
Finite Element Model – Single-Variable Problem
j jj
u u
Ritz method:
ei
ei
ej
n
j
eij QfuK
1
where ( , ) ( , )j jr z x y
iw
Weak form
11 22 00
e
j je i iij i jK a a a rdrdz
r r z z
where
e
ei if frdrdz
e
ei i nQ q ds
Single-Variable Problem – Heat Transfer
Heat Transfer:
Weak form
1 ( , ) ( , )( , ) 0
T r z T r zrk k f r z
r r r z z
0 ( , )
e
e
n
w T w Tk k wf r z rdrdz
r r z z
wq ds
( , ) ( , )n r z
T r z T r zq k n k n
r z
where
3-Node Axi-symmetric Element
1 1 2 2 3 3( , )T r z T T T
1
2
3
2 3 3 2
1 2 3
3 2
1
2 e
r z r zr z
z zA
r r
3 1 1 3
2 3 1
1 3
1
2 e
r z r zr z
z zA
r r
1 2 2 1
3 1 2
2 1
1
2 e
r z r zr z
z zA
r r
4-Node Axi-symmetric Element
1 1 2 2 3 3 4 4( , )T r z T T T T
1 2
34
a
b
r
z1 2
3 4
1 1 1
1
a b a b
a b a b
Single-Variable Problem – Example
Step 1: Discretization
Step 2: Element equation
e
j je i iijK rdrdz
r r z z
e
ei if frdrdz
e
ei i nQ q ds
