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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

4

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