Axis symmetric derivation and example problem.

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