1. MANE 4240 & CIVL 4240 Introduction to Finite Elements
Prof. Suvranu De
Four-noded rectangular element

2. Reading assignment:
Logan Lecture notes
Summary:
Computation of shape functions for 4-noded quad
Special case: rectangular element
Properties of shape functions
Computation of strain-displacement matrix
Example problem
Hint at how to generate shape functions of higher order (Lagrange) elements

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

4. 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

5. Constant Strain Triangle (CST) : Simplest 2D finite elementx y u3 v3 v1 u1 u2 v2 2 3 1
(x,y) v u
(x1,y1)
(x2,y2)
(x3,y3)
3 nodes per element
2 dofs per node (each node can move in x- and y- directions)
Hence 6 dofs per element

6. Formula for the shape functions arex y u3 v3 v1 u1 u2 v2 2 3 1
(x,y) v u
(x1,y1)
(x2,y2)
(x3,y3)
where

7. Approximation of displacements
Approximation of the strains

8. Element stiffness matrix
Since B is constant A
t=thickness of the element
A=surface area of the element
Element nodal load vector

9. Class exercise fS3y fS2y fS2x fS3x
For the CST shown below, compute the vector of nodal loads due to surface traction 1 y
fS3y
fS2y
fS2x
fS3x 2 3 x
(0,0)
py=-1
(1,0)

10. Class exercise fS3x fS3y fS2y fS2x x y 2 3 1 py=-1
The only nonzero nodal loads are
(can you derive this simpler?)

11. Now compute

12. 2 dofs per node (each node can move in x- and y- directions)
4-noded rectangular element with edges parallel to the coordinate axes: x y
(x,y) v u
(x1,y1)
(x2,y2)
(x3,y3) 1 2 3 4
(x4,y4) 2b 2a
4 nodes per element
2 dofs per node (each node can move in x- and y- directions)
8 dofs per element

13. Hence choose the shape function at node 1 as
Generation of N1:
At node 1 x y 1 2 3 4 2b 2a N1
l1(y)
l1(x)
has the property
Similarly
has the property
Hence choose the shape function at node 1 as

14. Using similar arguments, choose

15. Properties of the shape functions:
1. The shape functions N1, N2 , N3 and N4 are bilinear functions of x and y
2. Kronecker delta property
3. Completeness

16. 3. Along lines parallel to the x- or y-axes, the shape functions are linear. But along any other line they are nonlinear.
4. An element shape function related to a specific nodal point is zero along element boundaries not containing the nodal point.
5. The displacement field is continuous across elements
6. The strains and stresses are not constant within an element nor are they continuous across element boundaries.

17. The strain-displacement relationship
Notice that the strains (and hence the stresses) are NOT constant within an element

18. Computation of the terms in the stiffness matrix of 2D elements (recap)
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

19. 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

20.  Notice that these formulae are quite general (apply to all kinds of finite elements, CST, quadrilateral, etc) since we have not used any specific shape functions for their derivation.

21. Compute the unknown nodal displacements.
Example
1000 lb
300 psi y 3 4
Thickness (t) = 0.5 in
E= 30×106 psi
n=0.25
2 in 1 2 x
3 in
Compute the unknown nodal displacements.
Compute the stresses in the two elements.
This is exactly the same problem that we solved in last class, except now we have to use a single 4-noded element

22. Write down the shape functions
Realize that this is a plane stress problem and therefore we need to use
Write down the shape functions x y 3 2

23. The nonzero displacements are
We have 4 nodes with 2 dofs per node=8dofs. However, 5 of these are fixed.
The nonzero displacements are u2 u3 v3
Hence we need to solve u2 u3 v3
Need to compute only the relevant terms in the stiffness matrix

24. Compute only the relevant columns of the B matrix

25. Similarly compute the other terms

26. How do we compute f3y 4 3

27. How about a 9-noded rectangle?
Corner nodes y 5 2 1 a a b 9 8 6 x
Midside nodes b 7 3 4
Center node
Question: Can you generate the shape functions of a 16-noded rectangle?
Note: These elements, whose shape functions are generated by multiplying the shape functions of 1D elements, are said to belong to the “Lagrange” family