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Linear Elasticity – Plane Problems
MCEN 5023/ASEN 5012
Chapter 7
Fall, 2006
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Plane ProblemsPlane Stress and Plane Strain:
Plane stress: All the stress components associate with 3-direction are zero.
0231333 === σσσ 011 ≠σ 022 ≠σ 012 ≠σ
Plane strain: All the strain components associate with 3-direction are zero.
0231333 === eee 011 ≠e 022 ≠e 012 ≠e
03 =F
03 =u
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Plane ProblemsPlane Stress and Plane Strain:
In a plane stress problem, which components of strain are non-zero?
In a plane strain problem, which components of stress are non-zero?
Chapter 5 of textbook
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x1
x2
x3
X3 direction dimension is infinitesimal
Plane Stress
Plane ProblemsPlane Stress and Plane Strain:
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Plane ProblemsPlane Stress and Plane Strain:
x1
x2
x3
The two ends (X3 surfaces) of the bar are fixed
Plane Strain
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Plane ProblemsPlane Stress and Plane Strain:
Dam Tunnel
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Plane ProblemsPlane Stress and Plane Strain:
Crack
In general, if the problem has one dimension is much larger (or smaller) than the other two directions, one should consider plane strain (stress).
Roller
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Plane ProblemsUnder what conditions a problem can be approximated as a plane problem?
Geometry: the cross section of the body is independent of X3 coordinate
Plane Stress
Plane Strain
If two ends are fixed?
Yes
If X3 dimension is very thin?No
Yes
If the acting zoon of load is small as compared with the characteristic length in X1-X2 plane?
No
Yes
No
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Plane ProblemsPlane Stress and Plane Strain:
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Plane ProblemsPlane Stress and Plane Strain:
Plane Stress
2*
1 ν−=
EEv
vv−
=1
*
Plane Strain
( )2121v
EE++
=′ν
ννν+
=′1
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Plane ProblemsPlane Problems:
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Plane Problems
Solutions to Plane Problems
1. Displacement Methods
2. Stress Methods
3. Stress Function (Airy Stress Function)
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Plane Problems
Airy Stress Function
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Plane Problems
Airy Stress Function
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Plane Problems
Airy Stress Function: Short Beam
y
P
x
a
2b ( )( )ybyxaA 23 3−−=φ
P is total shear force on the left side
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Plane Problems
Airy Stress Function: Short Beam
y
P
x
a
2b
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Plane Problems
Airy Stress Function: Short Beam
( )yxaA −= 611σ
( )2212 33 byA −=σ
34hbPA −=
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Shear Stress
-1.4
-0.9
-0.4
0.1
0.6
1.1
0 0.5 1 1.5 2 2.5 3 3.5
Shear Stress
y
Plane Problems
Airy Stress Function: Short Beam
• Parabolic distribution
• Zero at top/bottom surface
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Plane Problems
Polar Coordinates
θ
r
x
y reθe
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Plane Problems
Airy Stress Function: Axisymmetric Problem
( )rφφ =
21
Plane Problems
Pressure Vessel:
ip
op
Inner Diameter: 2aOuter Diameter: 2b
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Plane Problems
Pressure Vessel:
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Pressure Vessel
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1 1.2 1.4 1.6 1.8 2
Distance r
Stre
ss θθσ
maxτ
rrσ−
0,1 == oi pp
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Plane Problems
Application of Pressure Vessel:
Infinite Plate with a small central hold subjecting to equibiaxial tension
q
q
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Plane Problems
Airy Stress Function: Non-axisymmetric Problem
q
q
Stress concentration:
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Plane Problems
Airy Stress Function: Non-axisymmetric Problem
q
q
Stress concentration:
θσ r rrσ
θ
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Plane Problems
Airy Stress Function: Non-axisymmetric Problem
Stress concentration:
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Plane Problems
Airy Stress Function: Non-axisymmetric Problem
Stress concentration:
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Plane Problems
Airy Stress Function: Non-axisymmetric Problem
0,1 21 == pp 1,1 21 == pp 1,1 21 == pp