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Theory of Elasticity: Plane Strain Problems and Boundary Conditions

Explore fundamental principles of elasticity, solving differential equations for plane strain and stress, reducing governing equations from 8 to 3, and understanding compatibility and strain-displacement relations. Study isotropic stress-strain relations and Saint-Venant’s Principle.

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Theory of Elasticity: Plane Strain Problems and Boundary Conditions

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  1. MER200: Theory of Elasticity Lecture 6 TWO DIMENSIONAL PROBLEMS Boundary Conditions Plane Strain Problems MER200: Theory of Elasticity

  2. Fundamental Priniples • Prescribed System of Forces • 3D Elasticity Problem • 6 components of Stress • 6 components of Strain • 3 components of Displacement • Elasticity Equations • 3 equations of Equilibrium • 6 Strain Displacement relations • (6 compatibility conditions) • 6 Stress-Strain relations MER200: Theory of Elasticity

  3. Equilibrium MER200: Theory of Elasticity

  4. Strain-Displacement MER200: Theory of Elasticity

  5. Compatibility MER200: Theory of Elasticity

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

  7. Approach to Solution • Solving Differential Equations • Plane Strain • Plane Stress MER200: Theory of Elasticity

  8. Boundary Conditions • Equilibrium must hold • ABC coincident with body surface • Stress resultants on this surface are T MER200: Theory of Elasticity

  9. Saint-Venant’s Principle MER200: Theory of Elasticity

  10. Plane Strain Problems MER200: Theory of Elasticity

  11. Plane Strain Condition • Prismatic Member • Held between FIXED, smooth, rigid planes • External forces function of x and y only • All cross sections experience identical deformations • Including ends • Frictionless nature of end constraint • Permits x-y deformations • Precludes z displacements, w=0 MER200: Theory of Elasticity

  12. Equations of Elasticity forPlane Strain MER200: Theory of Elasticity

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

  14. Constitutive Equations forPlane Strain MER200: Theory of Elasticity

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

  16. A bar of constant mass Density hangs under its own weight and is supported by a uniform stress as shown. Assume that all stress except the z normal stress vanish. MER200: Theory of Elasticity

  17. Problem questions • Based on the above assumptions, reduce the above 15 equations to seven equations in terms of normal z stress, strain and displacements. • Integrate the equilibrium equations to show what the resultant normal load is. MER200: Theory of Elasticity

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