Axisymmetric

Axisymmetric finite element analysis

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

2. LEARNING OBJECTIVES Basic concepts To derive the Axisymmetric element stiffness matrix [K] Strain-Displacement matrix [B]and Stress strain matrix [D] T emperature Effects G alerkin 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 F ollowing 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 side AB 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]= S tress 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]

27. 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/