Dr. Chris Wilson Mechanical Engineering Tennessee Tech, Cookeville, TN 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 April 3, 2001

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

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

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

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

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

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

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

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

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

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

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

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

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

Plane Stress Model April 3, 2001

Plane Strain Model April 3, 2001

More Plane Strain Model April 3, 2001

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

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

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

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

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

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

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