1. Dr. Chris Wilson Mechanical Engineering Tennessee Tech, Cookeville, TN
Visualizing
Plane Stress and Plane Strain
Dr. Chris Wilson
Mechanical Engineering
Tennessee Tech, Cookeville, TN

2. Outline 3-D Nature of Stress Plane Stress Plane Strain
Comparison of Plane Stress and Plane Strain
Rules of Thumb
Common Questions
April 3, 2001

3. Stress in General
Stress (and Strain) is a 3-D Quantity Defined at a Point in a Deformed Body
Some Special Cases
Triaxial Stress
2-D or Plane Stress (and Strain)
Pure Shear
Uniaxial Stress
April 3, 2001

4. 3-D Nature of Stress Craig, Mechanics of Materials, Wiley, 1996
April 3, 2001

5. 3-D Stress-Strain Equations
E = Modulus of Elasticity (Young’s Modulus)
n = Poisson’s Ratio
April 3, 2001

6. 3-D Stress-Strain Equations
Multiplied out...
G = Shear Modulus
April 3, 2001

7. What is Plane Stress? Thin plate
Uniform load distributed over thickness
sz, txz, tyz zero on both faces
Thin so sz, txz, tyz are zero throughout
Ugural & Fenster, Advanced Strength and Applied Elasticity,
3rd Ed., Prentice Hall, 1995
April 3, 2001

8. Another Plane Stress Sketch
Bickford, Mechanics of Solids, Irwin, 1993
April 3, 2001

9. Plane Stress Example
Gere, Mechanics of Materials, 5th Ed., Brooks/Cole, 2001
April 3, 2001

10. Plane Stress Equations
Note that the choice zeroing out the z-stresses is arbitrary
(could zero out the x-stresses or the y-stresses).
April 3, 2001

11. What is Plane Strain?
Long prismatic member held between fixed, smooth rigid planes
Lateral loading in x-y plane (e.g., cylinder under pressure)
All cross sections experience identical deformation
ez, gxz, gyz zero throughout
Ugural & Fenster, Advanced Strength and Applied Elasticity,
3rd Ed., Prentice Hall, 1995
April 3, 2001

12. Using Poisson’s Effect to Help
Popov, Mechanics of Materials, 2nd Ed., Prentice-Hall, 1976
April 3, 2001

13. Plane Strain Equations
Note that the choice zeroing out the z-strains is arbitrary
(could zero out the x-strains or the y-strains).
April 3, 2001

14. Plane Stress & Strain Compared
Gere, Mechanics of Materials, 5th Ed., Brooks/Cole, 2001
April 3, 2001

15. Using Finite Element Models
Avoid FEM Jargon
FEM Concerns
Avoid Element Outlines
Avoid Symmetry Models
Avoid Free Meshing
High Gradients at Fixed Ends
Surface Effects in 3-D Models
Mesh Convergence
April 3, 2001

16. Plane Stress Model
April 3, 2001

17. Plane Strain Model
April 3, 2001

18. More Plane Strain Model
April 3, 2001

19. Mild Notch: 3-D Model
Focus
on sxx
April 3, 2001

20. Mild Notch: 2-D Models Focus on sxx Plane Strain Plane Stress
April 3, 2001

21. Mild Notch: 3-D Model
Focus
on szz
April 3, 2001

22. Mild Notch: Plane Strain vs. 3-D
3-D Sliced Twice
Focus on szz
April 3, 2001

23. Sharp Notch in Thin Section
3-D Solution
April 3, 2001

24. Rules of Thumb D W L >> W & D Plane Stress If W  D
Plane Strain If W  5D (2D for Notch)
3-D If D < W < 5D (2D for Notch)
April 3, 2001

25. Some Common Questions Mohr’s Circle or Mohr’s Circles?
Is ezz  0 for Strain Gages?
What about sxx  txz  txz  0?
szz  0  ezz  0?
ezz  0  szz  0?
Stress/Strain at Point or Whole Field?
April 3, 2001