1. MANE 4240 & CIVL 4240 Introduction to Finite Elements
Prof. Suvranu De
FEM Discretization of 2D Elasticity

2. Reading assignment:
Lecture notes
Summary:
FEM Formulation of 2D elasticity (plane stress/strain)
Displacement approximation
Strain and stress approximation
Derivation of element stiffness matrix and nodal load vector
Assembling the global stiffness matrix
Application of boundary conditions
Physical interpretation of the stiffness matrix

3. Recap: 2D Elasticity x y Su ST
Volume (V) u v px py
Xa dV
Xb dV
Volume element dV
Su: Portion of the boundary on which displacements are prescribed (zero or nonzero)
ST: Portion of the boundary on which tractions are prescribed (zero or nonzero)
Examples: concept of displacement field

4. Case 2: Pure shear Case 1: Stretch
Example y 2 2 1 2 3 4 x
For the square block shown above, determine u and v for the following displacements
Case 2: Pure shear y
Case 1: Stretch y
1/2 2 4 2 1 x

5. Solution Case 1: Stretch Case 2: Pure shear
Check that the new coordinates (in the deformed configuration)
Case 2: Pure shear
Check that the new coordinates (in the deformed configuration)

6. Recap: 2D Elasticity
For plane stress
(3 nonzero stress components)
For plane strain
(3 nonzero strain components)

7. Strong formulation
Equilibrium equations
Boundary conditions
1. Displacement boundary conditions: Displacements are specified on portion Su of the boundary
2. Traction (force) boundary conditions: Tractions are specified on portion ST of the boundary
Now, how do I express this mathematically?
But in finite element analysis we DO NOT work with the strong formulation (why?), instead we use an equivalent Principle of Minimum Potential Energy

8. Principle of Minimum Potential Energy (2D)
Definition: For a linear elastic body subjected to body forces X=[Xa,Xb]T and surface tractions TS=[px,py]T, causing displacements u=[u,v]T and strains e and stresses s, the potential energy P is defined as the strain energy minus the potential energy of the loads (X and TS)
P=U-W

9. x y Su ST
Volume (V) u v px py
Xa dV
Xb dV
Volume element dV

10. Strain energy of the elastic body
Using the stress-strain law
In 2D plane stress/plane strain

11. Principle of minimum potential energy: Among all admissible displacement fields the one that satisfies the equilibrium equations also render the potential energy P a minimum.
“admissible displacement field”:
1. first derivative of the displacement components exist
2. satisfies the boundary conditions on Su

12. Finite element formulation for 2D:
Step 1: Divide the body into finite elements connected to each other through special points (“nodes”) py y x v u 1 2 3 4 u1 u2 u3 u4 v4 v3 v2 v1 3 px 4 2 v
Element ‘e’ 1 u ST y x Su x

13. Total potential energy
Potential energy of element ‘e’:
This term may or may not be present depending on whether the element is actually on ST
Total potential energy = sum of potential energies of the elements

14. Displacement at any point x=(x,y) (x3,y3) 3 u3 v4 (x4,y4) v2
Step 2: Describe the behavior of each element (i.e., derive the stiffness matrix of each element and the nodal load vector).
Inside the element ‘e’ v3
Displacement at any point x=(x,y)
(x3,y3) 3 u3 v4
(x4,y4) v2
Nodal displacement vector 4 u4 v u2 v1 2
where
u1=u(x1,y1)
v1=v(x1,y1)
etc u
(x2,y2) y x u1 1
(x1,y1)

15. Recall
If we knew u then we could compute the strains and stresses within the element. But I DO NOT KNOW u!!
Hence we need to approximate u first (using shape functions) and then obtain the approximations for e and s (recall the case of a 1D bar)
This is accomplished in the following 3 Tasks in the next slide

16. TASK 1: APPROXIMATE THE DISPLACEMENTS WITHIN EACH ELEMENT
TASK 2: APPROXIMATE THE STRAIN and STRESS WITHIN EACH ELEMENT
TASK 3: DERIVE THE STIFFNESS MATRIX OF EACH ELEMENT USING THE PRINCIPLE OF MIN. POT ENERGY
We’ll see these for a generic element in 2D today and then derive expressions for specific finite elements in the next few classes
Displacement approximation in terms of shape functions
Strain approximation
Stress approximation

17. TASK 1: APPROXIMATE THE DISPLACEMENTS WITHIN EACH ELEMENT
Displacement approximation in terms of shape functions u v3 y x v 1 2 3 4 u1 u2 u3 u4 v4 v2 v1
Displacement approximation within element ‘e’

18. We’ll derive specific expressions of the shape functions for different finite elements later

19. TASK 2: APPROXIMATE THE STRAIN and STRESS WITHIN EACH ELEMENT
Approximation of the strain in element ‘e’

21. Compact approach to derive the B matrix:

22. Stress approximation within the element ‘e’

23. TASK 3: DERIVE THE STIFFNESS MATRIX OF EACH ELEMENT USING THE PRINCIPLE OF MININUM POTENTIAL ENERGY
Potential energy of element ‘e’:
Lets plug in the approximations

24. Rearranging
From the Principle of Minimum Potential Energy
Discrete equilibrium equation for element ‘e’

25. Element stiffness matrix for element ‘e’
For a 2D element, the size of the k matrix is 2 x number of nodes of the element
Question: If there are ‘n’ nodes per element, then what is the size of the stiffness matrix of that element?
Element nodal load vector
STe e
Due to body force
Due to surface traction

26. If the element is of thickness ‘t’
For a 2D element, the size of the k matrix is 2 x number of nodes of the element dA
dV=tdA
Element nodal load vector t
Due to body force
Due to surface traction

27. The properties of the element stiffness matrix
1. The element stiffness matrix is singular and is therefore non-invertible
2. The stiffness matrix is symmetric
3. Sum of any row (or column) of the stiffness matrix is zero! (why?)

28. Computation of the terms in the stiffness matrix of 2D elements
The B-matrix (strain-displacement) corresponding to this element is
We will denote the columns of the B-matrix as x y
(x,y) v u 1 2 3 4 v4 v3 v2 v1 u1 u2 u3 u4 u1 v1 u2 v2 u3 u4 v3 v4

29. The stiffness matrix corresponding to this element is
which has the following form u1 v1 u2 v2 u3 u4 v3 v4
The individual entries of the stiffness matrix may be computed as follows

30. For this create a node-element connectivity chart exactly as in 1D v3
Step 3: Assemble the element stiffness matrices into the global stiffness matrix of the entire structure
For this create a node-element connectivity chart exactly as in 1D v3
Element #1 3 u3
ELEMENT
Node 1
Node 2
Node 3 1 2 3 4 v1 v4 1 u1 u4 4 v2 y x
Element #2 u2 2 v u

31. Stiffness matrix of element 1u2 v2 u3 v3 u1 u2 v1 v2 u3 v3 u4 v4 u2 v2 u3 v3 u4 v4
There are 6 degrees of freedom (dof) per element (2 per node)

32. é ù ê ú ê ú ê ú K = ê ú ê ú ê ú ê ú ê ú ë û Global stiffness matrix u1v1 u2 v2 u3 v3 v4 u4 é ù u1 ê ú v1 ê ú u2 ê ú v2 K = ê ú u3 ê ú ê ú v3 ê ú u4 ê ú ë û v4 8 8
How do you incorporate boundary conditions?
Exactly as in 1D

33. Finally, solve the system equations taking care of the displacement boundary conditions.

34. Physical interpretation of the stiffness matrix
Consider a single triangular element. The six corresponding equilibrium equations ( 2 equilibrium equations in the x- and y-directions at each node times the number of nodes) can be written symbolically as x y u3 v3 v1 u1 u2 v2 2 3 1

35. Choose u1 = 1 and rest of the nodal displacements = 03 y 2 x
Hence, the first column of the stiffness matrix represents the nodal loads when u1=1 and all other dofs are fixed. This is the physical interpretation of the first column of the stiffness matrix. Similar interpretations exist for the other columns

36. = “Force” at d.o.f ‘i’ due to unit displacement at d.o.f ‘j’
keeping all the other d.o.fs fixed
Now consider the ith row of the matrix equation
This is the equation of equilibrium at the ith dof

37. Consistent and Lumped nodal loads
Recall that the nodal loads due to body forces and surface tractions
These are known as “consistent nodal loads”
1. They are derived in a consistent manner using the Principle of Minimum Potential Energy
2. The same shape functions used in the computation of the stiffness matrix are employed to compute these vectors

38. Traction distribution on the 1-2-3 edge px= p py= 0 b
p per unit area
Example y
Traction distribution on the edge
px= p
py= 0 1 b 2 x b 3
We’ll see later that N1 N2 N3

39. The consistent nodal loads arey 1
pb/3 b 2
4pb/3 x b 3
pb/3

40. The lumped nodal loads arey 1
pb/2 b 2 pb x b 3
pb/2
Lumping produces poor results and will not be pursued further

41. Summary: For each element
Displacement approximation in terms of shape functions
Strain approximation in terms of strain-displacement matrix
Stress approximation
Element stiffness matrix
Element nodal load vector