Axisymmetric finite element analysis

1. PRESENTED BY
RAJ KUMAR S
P S G COLLEGE OF TECHNOLOGY
FINITE ELEMENT ANALYSIS IN
MECHANICAL DESIGN

2. LEARNING OBJECTIVES
 Basic concepts
 To derive the Axisymmetric element stiffness matrix [K]
 Strain-Displacement matrix [B]and
Stress strain matrix [D]
 Temperature Effects
 Galerkin Approach
 Problem--pessure vessel using the stiffness method.
 Practical Applications of axisymmetric elements.

3. INTRODUCTION
 We consider a special two-dimensional element called the
axisymmetric element with 3 nodes and 6 DOF.
 When element is symmetry with respect to geometry and loading
exists about an axis of the body
 We begin with the development of the stiffness matrix for the
simplest axisymmetric element, the triangular torus, whose vertical
cross section is a plane triangle.
 We then present the longhand solution of a thick-walled pressure
vessel to illustrate the use of the axisymmetric element equations.

4. Axisymmetric Elements
 Problem involving 3-Dimensional axisymmetric solid of revolution subjected to
axisymmetric loading reduce to simple two dimensional problem.
 Total symmetry about the z-axis all deformations and stress are independent of the
rotational angle Φ .
 Two dimensional problem in rz defined on the revolving area.
 z axis is called the axis of symmetry or the axis of revolution
r- radial directions
Z- longitudinal direction
Φ- circumferential direction

5. EXAMPLE
The axisymmetric problem of stresses
acting on the barrel under an internal
pressure loading.
The axisymmetric problem of an engine
valve stem can be solved using the
axisymmetric element.

6. MERITS OF AXISYMMETRIC ELEMENTS
Following practical considerations:
1. Fabrication : axisymmetric bodies are usually easier to manufacture compared to
the bodies with more complex geometries. Eg pipes, piles, axles, wheels, bottles,
cans, cups, nails, etc.
2. Strength : axisymmetric configuration are often more desirable in terms of
strength to weight ratio because of the favorable distribution of the material.
3. Multipurpose : hollow axisymmetric can assume a dual purpose of both structure
as well as shelter, as in a containers, vessels, tanks, rockets, etc.

7. Axisymmetric element with node
1,2and 3
Axisymmetric derivation shape function

8. Body force
Nodal displacement
Displacement vector U

9. Triangular element has two degrees of freedom at each node

10. Co – factors of matric D

11. Area of the triangle can be expressed as r, z co- ordinates
of the nodes

12. where

13. STRESS STRAIN MATRIX [D]
• let u and w denote the displacements in the radial
and longitudinal directions, respectively.
• The sideAB of the element is displaced an
amount u, and side CD is then displaced an amount
in the radial direction.
The normal strain in the radial direction
For axisymmetric deformation behaviour , that the tangential displacement v is equal to zero. Hence, the
tangential strain is due only to the radial displacement.

14. The longitudinal normal strain is given by
shear strain in the r-z plane given by
Summarizing the equations we get,

15. STRESS STRAIN RELATIONSHIP FOR AN ISOTROPIC BODY

16. Shear modulus is given by
The stress is
The stresses can be represented in the matrix form as
STRESS STRAIN MATRIX
(OR)
CONSTITUTIVE MATRIX
FOR GENERAL ISOTROPIC BODY

17. STRESS STRAIN MATRIX
(OR)
CONSTITUTIVE MATRIX
FOR AXSYMMETRIC BODY

18. Strain-Displacement matrix [B]
Displacement function of axisymmetric triangular element is given by
The strain components are
Radial strain,

19. Circumferential strain,
Longitudinal strain,
Shear strain,
Arrange equation in matrix form

20. Partial differential equation
where

21. substitute
The above equation in the form of
[B]= strain-Displacement matrix
[B]=
Coordinate z

22. Assemblage of element stiffness matrix [K]
Stiffness matrix ,[K]
Stiffness matrix ,[K]
Where,
Coordinate , r
A= area of triangular matrix = ½ (b*h)
[B]=Strain-Displacement matrix
[D]=Stress strain matrix

23. Temperature Effects
when the free expansion is prevented in a axisymmetric
element, the change in temperature causes stress in the element
let T be the rise in temperature and the alfa be
the co efficient of thermal expansion. The thermal force vector
due to rise in temperature is given by
For axisymmetric triangular element

24. GALERKIN APPROACH
virtual displacement field is given by
Virtual strain is given by
For axisymmetric problem, Galerkin variation form is given by

25. In the above equation, the first term presenting the internal virtual work
Internal virtual work

26. Internal virtual work ,
The above equation is the form of,
Stiffness matrix, [K]

31. REFERENCE
• Timoshenko S P and Goodier J N “Theory of Elasticity”Tata McGraw Hill Publications
• Logan D L, “A First Course in the Finite Element Method”,Thomson Learning
• Some online resources
https://nptel.ac.in/content/storage2/courses/112104116/ui/Course_home_26.htm
https://nptel.ac.in/courses/105/105/105105041/