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.

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

22. EXAMPLE
Bulb
Drilling platform

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