1. MECH4450 Introduction to Finite Element Methods Chapter 6 Finite Element Analysis of Plane Elasticity

2. Review of Linear Elasticity Linear Elasticity: A theory to predict mechanical response of an elastic body under a general loading condition. Stress: measurement of force intensity with 2-D

3. Review of Linear Elasticity Traction (surface force) : Equilibrium – Newton’s Law

4. Review of Linear Elasticity Strain: measurement of intensity of deformation Generalized Hooke’s Law

5. Plane Stress and Plane Strain Plane Stress - Thin Plate: Conditions: (1) in plane loading (2) thickness is much smaller than lateral dimensions Hooke’s law: Similarly

6. Plane Stress and Plane Strain Plane Strain - Thick Plate: Plane Stress: Plane Strain: Replace E by and by Conditions: (1) in plane loading (2) thickness is much larger than lateral dimensions Hooke’s law:

7. Equations of Plane Elasticity Governing Equations (Static Equilibrium) Constitutive Relation (Linear Elasticity) Strain-Deformation (Small Deformation)

8. Specification of Boundary Conditions EBC: Specify u(x,y) and/or v(x,y) on  NBC: Specify t x and/or t y on  where is the traction on the boundary  at the segment ds.

9. Weak Formulation for Plane Elasticity where are components of traction on the boundary 

10. Finite Element Formulation for Plane Elasticity Let where and

11. Constant-Strain Triangular (CST) Element for Plane Stress Analysis Let

12. Constant-Strain Triangular (CST) Element for Plane Stress Analysis

13. 4-Node Rectangular Element for Plane Stress Analysis Let

14. 4-Node Rectangular Element for Plane Stress Analysis For Plane Strain Analysis: and

15. Loading Conditions for Plane Stress Analysis A B

16. Evaluation of Applied Nodal Forces x’ y’ A

17. Evaluation of Applied Nodal Forces

18. Element Assembly for Plane Elasticity A B 1 2 3 4 3 4 6 5

19. 1 2 3 4 6 5 A B

20. Imposing Boundary Conditions 1 2 3 4 6 5 A B

21. Comparison of Applied Nodal Forces

22. Discussion on Boundary Conditions Must have sufficient EBCs to suppress rigid body translation and rotation For higher order elements, the mid side nodes cannot be skipped while applying EBCs/NBCs

23. Plane Stress – Example 2

24. Plane Stress – Example 3

25. Evaluation of Strains

26. Evaluation of Stresses Plane Stress Analysis Plane Strain Analysis