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Chapter 9 – Axisymmetric Elements

Chapter 9 – Axisymmetric Elements. Examples of axisymmetric problems: Pressure vessels Cylindrical shaft with notch or filleted step Stresses near a spherical void Hertzian contact between spheres

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Chapter 9 – Axisymmetric Elements

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  1. Chapter 9 – Axisymmetric Elements Examples of axisymmetric problems: Pressure vessels Cylindrical shaft with notch or filleted step Stresses near a spherical void Hertzian contact between spheres Use of axisymmetric elements provides computational efficiency as compared to full 3-D analysis

  2. Axisymmetric Example – Pressure Vessel

  3. Axisymmetric Example – Soil Foundation

  4. Axisymmetric Example – Valve Stem

  5. Axisymmetric Volume Element

  6. Stresses – Axisymmetric Problems Non-zero stresses - r, ,z, rz r = z = 0 ( due to axisymmetry )

  7. Strains – Axisymmetric Problems Non-zero strains - r, ,z, rz r = z = 0 ( due to axisymmetry ) Stress – Strain Relations: 4x4

  8. Formulation of 3 Node Triangular Axisymmetric Element

  9. Axisymmetric Elements – Displacement Fields Displacement fields:

  10. Axisymmetric strain components

  11. Axisymmetric Stress-Strain Relations

  12. Three node triangle(Note: not constant strain) Assumed interpolation: Nodal displacements:

  13. Interpolation functionsNote: same as CST where matrix form:

  14. 4 x 1 4 x 6 6 x 1 Strain-Displacement Relations

  15. Stresses 4 x 1 4 x 4 4 x 6 6 x 1

  16. Element Stiffness Matrix

  17. Methods for Evaluation [k] • Numerical integration • Analytic derivation • Approximate [B] at centroid  []

  18. Distributed Body Forces Can be used to handle stresses induced by centripetal acceleration in rotating bodies

  19. Distributed Body Forces (cont.) where  - angular velocity  - mass density

  20. Distributed Body Forces (cont.) Approximate radial body force using centroid radius Work equivalent concentrated forces:

  21. Distributed Surface Forces At node j:

  22. Distributed Surface Forces (cont.)

  23. Axisymmetric Example Thick Walled Pressure Vessel

  24. Finite element model(coarse mesh – 4 elements) “Section 9.2 solution”

  25. Finite element model(refined meshes)

  26. Results

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