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MER200: Theory of Elasticity Lecture 8

MER200: Theory of Elasticity Lecture 8. TWO DIMENSIONAL PROBLEMS Stress Functions Examples. Polynomial Solution. Linear combinations of POLYNOMIALS In X and Y Undetermined coefficients of the stress function φ Must satisfy biharmonic equation Must be 2 nd degree or higher

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MER200: Theory of Elasticity Lecture 8

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  1. MER200: Theory of Elasticity Lecture 8 TWO DIMENSIONAL PROBLEMS Stress Functions Examples MER200: Theory of Elasticity

  2. Polynomial Solution • Linear combinations of POLYNOMIALS • In X and Y • Undetermined coefficients of the stress function φ • Must satisfy biharmonic equation • Must be 2nd degree or higher • In order to yield non-zero stress function • Finding polynomials may be laborious MER200: Theory of Elasticity

  3. 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

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

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

  6. Polynomial of the 2nd Degree • Form • Associated stresses • For a rectangular plate • Simple tension: c2≠0 • Double Tension: c2≠0, a2≠0 • Pure Shear: b2≠0 MER200: Theory of Elasticity

  7. Polynomial of the 3nd Degree • Form • Associated stresses • For a rectangular plate • Pure Bending • a3=b3=c3≠0 MER200: Theory of Elasticity

  8. Polynomial of the 4nd Degree • Form • Associated stresses • The Biharmonic Equation is satisfied if MER200: Theory of Elasticity

  9. Polynomial of the 5nd Degree • Form • Associated stresses • The Biharmonic Equation is satisfied if MER200: Theory of Elasticity

  10. Example 1 • Consider the beam shown. Using polynomials, determine expressions for the stresses in the beam MER200: Theory of Elasticity

  11. Polynomial of Order 5 MER200: Theory of Elasticity

  12. Polynomial of Degree 2 MER200: Theory of Elasticity

  13. Polynomial of Order 3 MER200: Theory of Elasticity

  14. Pure Bending MER200: Theory of Elasticity

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