In The Name O f ALLAH

1. In The Name Of ALLAH

2. Introduction to Finite Elements in Engineering Chapter 6 Axisymmetric Solids Subjected to Axisymmetric Loading 1/20

3. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading INTRODUCTION Three dimensional Problems Two dimensional Problems Axisymmetric Problems Axisymmetric Bodies subjected to Axisymmetric Boundary Conditions. All loadings, supports, geometry, deformations and stresses are independent of the rotational angle θ. 2/20

4. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading q2n-1=u = Displacement in r- direction q2n=w = Displacement in z- direction 3/20

5. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading 6-3 FINITE ELEMENT MODELING: TRIANGULAR ELEMENT Using the three functions N1 , N2, N3 : 4/20

6. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading 5/20

7. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading 6/20

8. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading 7/20

9. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Element Stiffness Matrix Note:The fourth row in B has term of “ Ni /r ” and B is dependent to “ r ” and we will encounter problem to calculate the element stiffness matrix. As a simple approximation, B and r can be evaluated at the centroid of the triangle as follow: 8/21

10. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Body Force Term 9/21

11. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Surface Traction 10/21

12. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Surface Traction 10/21

13. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Example 6.1:Consider a long cylinder of inside diameter 80 mm and outside diameter 120 mm snugly fits in a hole over its full length. The cylinder is subjected to an internal pressure of 2 MPa. Using two elements on the 10 mm length shown, find the displacements at the inner radius. Plane strain 11/21

14. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading For both elements, det J=200 mm2 , Ae =100 mm2 12/21

15. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading By using Elimination approach on assembling the matrices yields: 14/21

16. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Stress calculation 15/21

17. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Temperature Effects 16/21

18. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Cylinder Subjected to Internal Pressure In this problem we need to model only the rectangular region of the length L bound between ri , ro Nodes fixed end are constrained in the z and r directions. Stiffness and force modifications will be made for these nodes. 17/21

19. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Press Fit on a Rigid Shaft Boundary Condition is: 1 2 18/21

20. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Press Fit on a Elastic Shaft 1 2 3 4 Boundary Condition is: 19/21

21. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Bellevile Spring • A rectangular element considered. • An axisymmetric Load P is placed at the top corner. • The bottom supporting corner is constrained in the z direction. • The load-deflection curve is nonlinear. • The stiffness depends on the Geometry. 20/21

22. CHAPTER 6 - Axisymmetric Solid Subject to Axisymmetric Loading Example 6.3:A steel disk flywheel rotates at 3000rpm. The outer Diameter is 24in. And the hole diameter is 6 in. Find the value of thr maximum tangential stress under the following conditions: thickness = 1 in,E=30×106 psi, Poisson’s ratio= 0.3,wt. density=0.283 lb/in3. Considering four-element finite element model. With neglecting gravity load, then we have: 21/21