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MER200: Theory of Elasticity Lecture 13

MER200: Theory of Elasticity Lecture 13. 2D Problems in Polar Coordinates. 2D Problems in Polar Coordinates. Polar coordinates or cylindrical coordinates are special cases of curvilinear orthogonal coordingates. Equations of Equilibrium. Radial Circumferential.

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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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