Axis symmetric
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This presentation discusses the shape function of axisymmetric elements used in finite element analysis. Axisymmetric elements are 2D elements that can model objects that are symmetric about an axis, like pressure vessels. They have advantages over full 3D modeling like smaller model size, faster computation, and easier post-processing. The presentation covers how axisymmetric elements are defined based on the radial, circumferential and longitudinal directions. It also derives the element stiffness matrix which relates displacements to forces and describes the shape functions which define element deformation based on node positions. Examples of applying axisymmetric elements to problems like pressure vessels and engine parts are provided.
Axis symmetric
- 1. Presentation on Shape Function of Axisymmetric Element PSG COLLEGE OF TECHNOLOGY COIMBATORE-641005 Presented by, GOWSICK C S (16MI34) KARTHIKEYAN K (16MI06) 1st year ME-CIM Department Of Mechanical Engineering PSG College of Technology
- 2. Introduction • Axisymmetric element is an two-dimensional element with 3 nodes and 6 DOF. • When element is symmetry with respect to geometry and loading exists about an axis of the body Application: • Soil masses subjected thick-walled pressure vessels.
- 3. Introduction Advantages – Smaller models (3D to 2D) – Faster execution – Easier post processing (FEA software) To model This ?
- 4. How to model ?
- 5. How to model ?
- 6. Axisymmetric Element • In Triangular tori,each element is symmetric with respect to geometry and loading about z axis. z axis is called the axis of symmetry or the axis of revolution. • Nodal points are I,j,m. • r, Φ, and z indicate the radial, circumferential, and longitudinal direction.
- 7. Examples • Domed pressure vessel • Engine valve stem
- 8. Derivation of the Stiffness Matrix N,M-Mid side nodes
- 9. z axial stress , Φ hoops stress r radial stress
- 10. Derivation of the Stiffness Matrix • The normal strain in the radial direction is then given by • The tangential strain is then given by • The longitudinal normal strain given by • Shear strain in the r-z plane given by
- 11. Properties • Isotropic E≡G≡K≡v (uniform) in x,y,z E.g All metals except mercury • Orthotropic- E≡G≡K≡v varies orthogonal wrt x,y E.g composite fibre, plywood • Anisotropic- E≡G≡K≡v varies non uniformly in x,y,z E.g Rocks
- 12. • Isotropic stress/strain relationship • Step 1-Select Element Type o The element has three nodes with two degrees of freedom per node(that is, ui, wi at node i )
- 13. • Step 2 Select Displacement Functions o The element displacement functions are taken to be
- 14. • The nodal displacements are
- 15. • The general displacement function is then expressed in matrix • Substituting the coordinates of the nodal points
- 16. • Performing the inversion operations
- 17. Shape Function • Interpolation function w.r.t fixed nodes • Input – nodal position • Output - deformation
- 18. • Shape functions • General displacement function
- 19. • Step 3 Define the Strain/Displacement and Stress/Strain Relationships
- 20. Strain Stress Step 4 Derive the Element Stiffness Matrix and Equations
- 21. • Centroid point of element • Surface Forces • Body force
- 23. Problem
- 24. Global martrix
- 34. PROBLEM 2
- 35. PROBLEM 2
- 36. PROBLEM 2
- 37. PROBLEM 2
- 38. Reference • Daryl L. Logan, "A first course in finite element method” •“Introduction to finite element in engineering” by D.Belegundu
- 39. Thank you