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Visualizing

Visualizing. Plane Stress and Plane Strain. Dr. Chris Wilson Mechanical Engineering Tennessee Tech, Cookeville, TN. Outline. 3-D Nature of Stress Plane Stress Plane Strain Comparison of Plane Stress and Plane Strain Rules of Thumb Common Questions. Stress in General.

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Visualizing

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  1. 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

  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

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

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

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

  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

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

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

  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

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

  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).

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

  16. Plane Stress Model

  17. Plane Strain Model

  18. More Plane Strain Model

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

  20. Mild Notch: 2-D Models Plane Strain Plane Stress Focus on sxx

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

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

  23. Sharp Notch in Thin Section 3-D Solution

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

  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?

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