Chap_07 Plane Stress and Strain

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MCEN 5023/ASEN 5012 Chapter 7
Linear Elasticity Plane Problems
Fall, 2006
1
Plane ProblemsPlane Stress and Plane Strain: Plane stress: All the stress components associate with 3direction are zero.
33 = 13 = 23 = 0F3 = 0
11 0 22 0 12 0
Plane strain: All the strain components associate with 3direction are zero.
e33 = e13 = e23 = 0
e11 0
e22 0
e12 0
u3 = 0
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Chapter 5 of textbook
Plane ProblemsPlane Stress and Plane Strain: In a plane stress problem, which components of strain are nonzero?
In a plane strain problem, which components of stress are nonzero?
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Plane ProblemsPlane Stress and Plane Strain:
x2
X3 direction dimension is infinitesimal
x1 x3
Plane Stress4
Plane ProblemsPlane Stress and Plane Strain: The two ends (X3 surfaces) of the bar are fixed
x2
x1 x3
Plane Strain
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Plane ProblemsPlane Stress and Plane Strain:
Dam
Tunnel
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Plane ProblemsPlane Stress and Plane Strain:
Roller
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).7
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 Yes NoIf X3 dimension is very thin?
NoIf the acting zoon of load is small as compared with the characteristic length in X1X2 plane?
No
If two ends are fixed?
Yes Yes Plane Strain
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Plane ProblemsPlane Stress and Plane Strain:
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Plane ProblemsPlane Stress and Plane Strain:
E E = 1 2*
v* =
v 1 vPlane Strain
Plane Stress
E = E
1 + 2 = (1 + v )2 1 +
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Plane ProblemsPlane Problems:
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Plane ProblemsSolutions to Plane Problems 1. Displacement Methods
2. Stress Methods
3. Stress Function (Airy Stress Function)
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Plane ProblemsAiry Stress Function
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Plane ProblemsAiry Stress Function
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Plane ProblemsAiry Stress Function: Short Beam y
P2b
P is total shear force on the left side
x
= A(a x )( y 3 3b 2 y )a
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Plane ProblemsAiry Stress Function: Short Beam y
P2b
x
a
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Plane ProblemsAiry Stress Function: Short Beam
11 = 6 A(a x ) y
12 = A(3 y 2 3b 2 )P A= 4hb 317
Plane ProblemsAiry Stress Function: Short BeamShear Stress
1.1
0.6
Parabolic distribution0.1
y
0 0.4
0.5
1
1.5
2
2.5
3
3.5
Zero at top/bottom surface
0.9
1.4
Shear Stress
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Plane ProblemsPolar Coordinates
y
e
er
r
x
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Plane ProblemsAiry Stress Function: Axisymmetric Problem
= (r )
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Plane ProblemsPressure Vessel: Inner Diameter: 2a Outer Diameter: 2b
pipo
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Plane ProblemsPressure Vessel:
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Pressure Vessel2 1.8 1.6 1.4 Stress 1.2 1 0.8 0.6 0.4 0.2 0 1 1.2 1.4 1.6 1.8 2 Distance r
pi = 1, max rr
po = 0
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Plane ProblemsApplication of Pressure Vessel: Infinite Plate with a small central hold subjecting to equibiaxial tension
q
q
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Plane ProblemsAiry Stress Function: Nonaxisymmetric Problem Stress concentration:
q
q25
Plane ProblemsAiry Stress Function: Nonaxisymmetric Problem Stress concentration:
r
rrq
q
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Plane ProblemsAiry Stress Function: Nonaxisymmetric Problem Stress concentration:
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Plane ProblemsAiry Stress Function: Nonaxisymmetric Problem Stress concentration:
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Plane ProblemsAiry Stress Function: Nonaxisymmetric Problem
p1 = 1,
p2 = 0
p1 = 1,
p2 = 1
p1 = 1,
p2 = 1
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