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MER200: Theory of Elasticity Lecture 6

MER200: Theory of Elasticity Lecture 6. TWO DIMENSIONAL PROBLEMS Plane Stress Problems Stress Functions. Equilibrium. Strain-Displacement. Compatibility. Isotropic Stress-Strain Relations. Plane Strain Condition. Thin Plate Load uniformly distributed over thickness

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MER200: Theory of Elasticity Lecture 6

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  1. MER200: Theory of Elasticity Lecture 6 TWO DIMENSIONAL PROBLEMS Plane Stress Problems Stress Functions MER200: Theory of Elasticity

  2. Equilibrium MER200: Theory of Elasticity

  3. Strain-Displacement MER200: Theory of Elasticity

  4. Compatibility MER200: Theory of Elasticity

  5. Isotropic Stress-Strain Relations MER200: Theory of Elasticity

  6. Plane Strain Condition • Thin Plate • Load uniformly distributed over thickness • Load parallel to the plane of the plate • Normal and Shearing stresses on the faces of the plate are zero • σz=τxz= τxz=0 • Stresses with z components through the thickness closely approximated by 0. • Fz=0 • Fx=Fx(x,y), Fy=Fy(x,y) MER200: Theory of Elasticity

  7. Equations of Elasticity forPlane Stress MER200: Theory of Elasticity

  8. Reducing Governing EquationsFrom Eight to Three • Starting with Compatibility MER200: Theory of Elasticity

  9. Constitutive Equations forPlane Strain MER200: Theory of Elasticity

  10. Three Equations • Compatibility in terms of Stress • Equilibrium MER200: Theory of Elasticity

  11. Stress Functions • Plane Stress Compatibility • Plane Strain Compatibility MER200: Theory of Elasticity

  12. Body Force is Conservative • Potential Function V exists • V causes compatibility equations to reduce to one equation with one dependent variable MER200: Theory of Elasticity

  13. Airy’s Stress Function φ • Definitions • Φ implies that equilibrium equations are identically satisfied MER200: Theory of Elasticity

  14. Compatibility equations can now be written • For Plane Stress • Biharmonic operator • Biharmonic equation MER200: Theory of Elasticity

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