1. MECH593 Introduction to Finite Element Methods
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: y z
𝑡≪𝐿
𝑡≪𝑊 x

6. Plane Stress and Plane Strain
Plane Strain - Thick Plate:
𝑡≫𝐿
𝑡≫𝑊 z y x
Plane Stress:
Plane Strain:
Replace E by and by

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

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

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

10. Finite Element Formulation for Plane Elasticity
Let
where
and

11. Constant-Strain Triangular (CST) Element
Let

12. Constant-Strain Triangular (CST) Element
- A mesh could be too stiff y P x P
- Mesh locking y II A I x

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

14. 4-Node Rectangular Element for Plane Stress Analysis
Let

15. 4-Node Rectangular Element for Plane Stress Analysis
For Plane Strain Analysis:
and

16. Loading Conditions for Plane Stress Analysis6 5 B 3 4 A 1 2

17. Evaluation of Applied Nodal Forces

18. Evaluation of Applied Nodal ForcesY y 6 5 B x 3 4 A 1 2 X

19. Element Assembly for Plane Elasticity5 6 B 3 4 3 4 A 1 2

20. Element Assembly for Plane Elasticity1 2 3 4 6 5 A B

21. Imposing Boundary Conditions1 2 3 4 6 5 A B

22. Comparison of Applied Nodal Forces

23. 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

24. Plane Stress – Example 2

25. Plane Stress – Example 3

26. Evaluation of Strains

27. Evaluation of Stresses
Plane Stress Analysis
Plane Strain Analysis

28. Isoparametric Elements
Example 1:
Physical domain (physical element)
Reference domain (master element) h h 3 4 4 3 x x y 1 1 2 2 x

29. Isoparametric Elements
Example 2:
Physical domain (physical element)
Reference domain (master element) h h 3 3 1 y 1 x x 2 2 x
Connection with shape functions
expressed in area coordination

30. Isoparametric Elements
Example 2:
Physical domain (physical element)
Reference domain (master element) h h 3 3 6 5 5 6 x 1 y x 4 2 1 4 2 x

31. Isoparametric Elements
An element is an isoparametric element if the same shape functions are employed to approximate geometry as well as the unknown variables.
Stiffness matrix and force vector calculation:

32. Isoparametric Rectangular Elements
where

33. Higher Order 2-D Isoparametric Elements

34. Gaussian Quadrature Formula for Triangles