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Comprehensive Study on Elasticity Theory in Two-Dimensional Problems

Explore stress functions, polynomial solutions, Airy’s stress function, and polynomial examples in elasticity theory for biharmonic equations and associated stresses in various degrees. Dive into compatibility equations and examples for practical applications.

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Comprehensive Study on Elasticity Theory in Two-Dimensional Problems

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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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