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MER200: Theory of Elasticity Lecture 13

Explore 2D elasticity problems in polar coordinates, equilibrium equations, strain-displacement relations, Hooke's Law, compatibility equations, axisymmetric solutions, stress analysis, and case studies. Learn key concepts and solve engineering problems efficiently.

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MER200: Theory of Elasticity Lecture 13

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  1. MER200: Theory of Elasticity Lecture 13 2D Problems in Polar Coordinates MER200: Theory of Elasticity

  2. 2D Problems in Polar Coordinates • Polar coordinates or cylindrical coordinates are special cases of curvilinear orthogonal coordingates MER200: Theory of Elasticity

  3. Equations of Equilibrium • Radial • Circumferential MER200: Theory of Elasticity

  4. Strain-Displacement Relations • Radial • Circumferential • Shear Strain MER200: Theory of Elasticity

  5. Hooke’s Law - Plane Stress MER200: Theory of Elasticity

  6. Hooke’s Law – Plane Strain MER200: Theory of Elasticity

  7. Rotation about the z-axis MER200: Theory of Elasticity

  8. Compatibility Equations in terms of Stress Function • BiHarmonic Equation • Stress Components MER200: Theory of Elasticity

  9. Axisymmetric Problems • Body is symmetrical about z-axis • Applied forces and/or displacements are symmetrical about z-axis • Stress and displacement independent of θ • Derivatives with respect to θ vanish • Stress functions independent of θ MER200: Theory of Elasticity

  10. Axisymmetric Plane Problems • Bi-Harmonic Equation • Substitution reduces equation to a DE with Constant Coefficients MER200: Theory of Elasticity

  11. Solution and Stresses • General Solution • Resulting Stresses MER200: Theory of Elasticity

  12. CASE 1: Simply Connected Region • In order that the stresses remain FINITE at the origin • The stresses reduce to MER200: Theory of Elasticity

  13. CASE 2: Multiply Connected • A circular cylinder with a concentric circular hole • Compatibility Equations are not sufficient to guarantee single valued displacements MER200: Theory of Elasticity

  14. Example 1 • Consider a hollow cylinder which is subjected to the prescribed radial displacements • ur(a)=u0 • ur(b)=0 • Determine an expression for the displacements and stresses MER200: Theory of Elasticity

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