Lecture Notes: Introduction to Finite Element Method |
Chapter 6. Solid Elements for 3-D Problems |
Chapter 6. Solid Elements for 3-D Problems
I. 3-D Elasticity Theory
Stress State:
y
z
y , v
σy
τ yx
τ yz
τzy
τzx
σz
z, w
F
x
τxy
σ x
τxz |
x, u |
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© 1997-2002 Yijun Liu, University of Cincinnati |
138 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 6. Solid Elements for 3-D Problems |
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σ x |
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σ |
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σ |
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σ={σ }= |
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τ xy |
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τ |
yz |
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τ zx |
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Strains: |
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ε x |
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ε y |
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ε |
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ε= {ε }= |
γ |
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xy |
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γ yz |
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γ zx |
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Stress-strain relation: |
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σ |
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1 − v |
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σ y |
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v |
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σz |
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(1 |
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τxy |
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+ v)(1 − 2v) |
0 |
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τ |
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yz |
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τzx |
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or [σij ]
or [εij ]
v |
v |
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0 |
1 − v |
v |
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0 |
v |
1 − v |
1 |
0 |
0 |
0 |
− 2v |
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2 |
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0
1 − 2v
2
0
(1)
(2)
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ε |
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εx |
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ε |
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γxy |
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γ |
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yz |
1 − 2v |
γzx |
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2 |
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or |
σ = Eε |
(3) |
© 1997-2002 Yijun Liu, University of Cincinnati |
139 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 6. Solid Elements for 3-D Problems |
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Displacement: |
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u(x, |
y, z) |
u |
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1 |
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(4) |
u = v(x, y, z) |
= u2 |
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w(x, y, z) |
u3 |
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Strain-Displacement Relation:
ε |
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= |
∂u , |
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ε |
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∂v |
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, ε |
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= |
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∂w |
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x |
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y |
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∂y |
z |
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∂z |
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∂x |
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= |
∂v |
+ |
∂u |
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γ |
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= |
∂w |
+ |
∂v |
, γ |
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= |
∂u + |
∂w |
(5) |
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xy |
∂x |
∂y |
yz |
∂y |
∂z |
xz |
∂x |
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∂z |
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or |
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1 |
∂ui |
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∂u j |
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(i, j =1,2,3) |
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εij |
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2 |
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+ |
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∂x j |
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∂xi |
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or simply,
εij |
= |
1 |
(ui, j +u j,i ) |
(tensor notation) |
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2 |
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© 1997-2002 Yijun Liu, University of Cincinnati |
140 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 6. Solid Elements for 3-D Problems |
Equilibrium Equations: |
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∂σx |
+ |
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∂τxy |
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+ |
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∂τxz |
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+ f x = 0 , |
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∂x |
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∂y |
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∂z |
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∂τ yx |
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∂σ y |
+ |
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∂τ yz |
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+ f y = 0 , |
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∂x |
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∂y |
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∂z |
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∂τzx |
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∂τzy |
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∂σz |
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+ fz = 0 , |
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∂y |
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∂x |
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∂z |
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or |
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σij, j + fi |
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Boundary Conditions (BC’s): |
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ui = ui , |
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on |
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Γu ( specified displacement ) |
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on |
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Γσ |
( specified traction ) |
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ti = ti , |
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(traction ti =σij |
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n j |
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p
n
Γσ
Γ( =Γu +Γσ )
Γu
Stress Analysis:
(6)
(7)
Solving equations in (3), (5) and (6) under the BC’s in (7) provides the stress, strain and displacement fields (15 equations for 15 unknowns for 3-D problems). Analytical solutions are difficult to find!
© 1997-2002 Yijun Liu, University of Cincinnati |
141 |