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1 Linear Elasticity – Plane Problems MCEN 5023/ASEN 5012 Chapter 7 Fall, 2006
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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 3-direction are zero.

33 = 13 = 23 = 0F3 = 0

11 0 22 0 12 0

Plane strain: All the strain components associate with 3-direction are zero.

e33 = e13 = e23 = 0

e11 0

e22 0

e12 0

u3 = 0

2

Chapter 5 of textbook

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?

3

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

5

Plane ProblemsPlane Stress and Plane Strain:

Dam

Tunnel

6

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 X1-X2 plane?

No

If two ends are fixed?

Yes Yes Plane Strain

8

Plane ProblemsPlane Stress and Plane Strain:

9

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 +

10

Plane ProblemsPlane Problems:

11

Plane ProblemsSolutions to Plane Problems 1. Displacement Methods

2. Stress Methods

3. Stress Function (Airy Stress Function)

12

Plane ProblemsAiry Stress Function

13

Plane ProblemsAiry Stress Function

14

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

15

Plane ProblemsAiry Stress Function: Short Beam y

P2b

x

a

16

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

18

Plane ProblemsPolar Coordinates

y

e

er

r

x

19

Plane ProblemsAiry Stress Function: Axisymmetric Problem

= (r )

20

Plane ProblemsPressure Vessel: Inner Diameter: 2a Outer Diameter: 2b

pipo

21

Plane ProblemsPressure Vessel:

22

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

23

Plane ProblemsApplication of Pressure Vessel: Infinite Plate with a small central hold subjecting to equibiaxial tension

q

q

24

Plane ProblemsAiry Stress Function: Non-axisymmetric Problem Stress concentration:

q

q25

Plane ProblemsAiry Stress Function: Non-axisymmetric Problem Stress concentration:

r

rrq

q

26

Plane ProblemsAiry Stress Function: Non-axisymmetric Problem Stress concentration:

27

Plane ProblemsAiry Stress Function: Non-axisymmetric Problem Stress concentration:

28

Plane ProblemsAiry Stress Function: Non-axisymmetric Problem

p1 = 1,

p2 = 0

p1 = 1,

p2 = 1

p1 = 1,

p2 = 1

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