MANE 4240 & CIVL 4240Introduction to Finite Elements

Four-noded rectangular element

Prof. Suvranu De

Reading assignment:

Logan 10.2 + 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

Finite element formulation for 2D:

Step 1: Divide the body into finite elements connected to each other through special points (“nodes”)

x

y

Su

STu

v

x

px

py

Element ‘e’

3

21

4

y

xvu

1

2

3

4

u1

u2

u3

u4

v4

v3

v2

v1

4

4

3

3

2

2

1

1

v

u

v

u

v

u

v

u

d

Displacement approximation in terms of shape functions

Strain approximation in terms of strain-displacement matrix

Stress approximation

Summary: For each element

Element stiffness matrix

Element nodal load vector

dNu

dBD

dBε

eV

k dVBDBT

S

eT

b

e

f

S ST

f

V

T dSTdVXf NN

Constant Strain Triangle (CST) : Simplest 2D finite element

• 3 nodes per element• 2 dofs per node (each node can move in x- and y- directions)• Hence 6 dofs per element

x

y

u3

v3

v1

u1

u2

v2

2

3

1

(x,y)

vu

(x1,y1)

(x2,y2)

(x3,y3)

332211

332211

vy)(x,N vy)(x,N vy)(x,Ny)(x,v

u y)(x,Nu y)(x,Nu y)(x,Ny)(x,u

Formula for the shape functions are

A

ycxbaN

A

ycxbaN

A

ycxbaN

2

2

2

3333

2222

1111

12321312213

31213231132

23132123321

33

22

11

x1

x1

x1

det2

1

xxcyybyxyxa

xxcyybyxyxa

xxcyybyxyxa

y

y

y

triangleofareaA

where

x

y

u3

v3

v1

u1

u2

v2

2

3

1

(x,y)

vu

(x1,y1)

(x2,y2)

(x3,y3)

Approximation of the strains

xy

u

v

u v

x

y

x

Bdy

y x

332211

321

321

332211

321

321

000

000

2

1

y)(x,Ny)(x,N y)(x,N y)(x,Ny)(x,Ny)(x,N

y)(x,N0

y)(x,N0

y)(x,N0

0y)(x,N

0 y)(x,N

0y)(x,N

bcbcbc

ccc

bbb

A

xyxyxy

yyy

xxx

B

3

3

2

2

1

1

321

321

v

u

v

u

v

u

N0N0N0

0N0N0N

y)(x,v

y)(x,uu

dNu

Approximation of displacements

Element stiffness matrix

eV

k dVBDBT

AtkeV

BDBdVBDB TT t=thickness of the elementA=surface area of the element

Since B is constant

t

A

Element nodal load vector

S

eT

b

e

f

S ST

f

V

T dSTdVXf NN

Class exercise

eTS

ST

SdSTf N

For the CST shown below, compute the vector of nodal loads due to surface traction

x

y

fS3x

fS3y

fS2y fS2x

2 3

1

el S

along

T

SdSTtf

31 32N

py=-1(0,0) (1,0)

Class exercise

el Salong

T

SdSTtf

31 32N

x

y

fS3x

fS3y

fS2y fS2x

2 3

1

py=-1

1

0ST

The only nonzero nodal loads are

1

0 3222 x yalongS dxpNtfy

1

0 3233 x yalongS dxpNtfy

x

xxy

xyy

xy

y

y

yxyy

A

xyyyxyx

A

xba

A

ycxbaN

yalong

1

)(

)1(

0x1

0x1

x1

det

)1(

x1

x1

x1

det

222

231

1

3

2

11

1

33

22

11

11

13311322

0

222322

(can you derive this simpler?)

2

)1)(1(1

0

1

0 3222

t

dxxt

dxpNtf

x

x yalongS y

Now compute

1

0 3233 x yalongS dxpNtfy

4-noded rectangular element with edges parallel to the coordinate axes:

x

y(x,y)

vu

(x1,y1)(x2,y2)

(x3,y3)

1 2

34(x4,y4)

2b

2a

4

1iy)u(x,y)(x,u

iiN

4

1iy)v(x,y)(x,v

iiN

• 4 nodes per element• 2 dofs per node (each node can move in x- and y- directions)•8 dofs per element

Generation of N1:

x

y

1 2

34

2b

2a

N1

l1(y)

l1(x)

1

1

21

21 )(

xx

xxxl

At node 1

has the property

0)(

1)(

21

11

xl

xl

Similarly

41

41 )(

yy

yyyl

has the property

0)(

1)(

41

11

yl

yl

Hence choose the shape function at node 1 as

4241

4

21

2111 4

1)()( yyxx

abyy

yy

xx

xxylxlN

134

243

312

421

4

14

14

14

1

yyxxab

N

yyxxab

N

yyxxab

N

yyxxab

N

Using similar arguments, choose

Properties of the shape functions:

1. The shape functions N1, N2 , N3 and N4 are bilinear functions of x and y

nodesotherat

inodeatyx

0

''1),(Ni

3. Completeness

yy

xx

i

i

4

1ii

4

1ii

4

1ii

N

N

1N

2. Kronecker delta property

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.

4

4

3

3

2

2

1

1

B

44332211

4321

4321

xy

v

u

v

u

v

u

v

u

y)(x,Ny)(x,Ny)(x,Ny)(x,N y)(x,N y)(x,Ny)(x,Ny)(x,N

y)(x,N0

y)(x,N0

y)(x,N0

y)(x,N0

0y)(x,N

0y)(x,N

0 y)(x,N

0y)(x,N

xyxyxyxy

yyyy

xxxx

y

x

The strain-displacement relationship

yyxxyyxxyyxxyyxx

xxxxxxxx

yyyyyyyy

abB

13243142

3412

1234

0000

0000

4

1

Notice that the strains (and hence the stresses) are NOT constant within an element

  

The B-matrix (strain-displacement) corresponding to this element is          

We will denote the columns of the B-matrix as       

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

x

y

(x,y)

v

u

1 2

34v4

v3

v2v1

u1u2

u3

u4

  1 2 3 4

1 2 3 4

1 1 2 2 3 3 4 4

N (x,y) N (x,y) N (x,y) N (x,y)0 0 0 0

N (x,y) N (x,y) N (x,y) N (x,y)0 0 0 0

N (x,y) N (x,y) N (x,y) N (x,y) N (x,y) N (x,y) N (x,y) N (x,y)

x x x x

y y y y

y x y x y x y x

u1 v1u2 v2

u3 u4v3 v4

1 1

1

1

11

N (x,y)0

N (x,y)0 ; ; and so on...

N (x,y)N (x,y)

u v

xB B

y

yx

 

eV

k dVBDBT

11 12 13 14 15 16 17 18

21 22 23 24 25 26 27 28

31 32 33 34 35 36 37 38

41 42 43 44 45 46 47 48

51 52 53 54 55 56 57 58

61 62 63 64 65 66 67 68

71 72 73 74 75 76 77 78

81 82 83 84 85 86 87 88

k k k k k k k k

k k k k k k k k

k k k k k k k k

k k k k k k k kk

k k k k k k k k

k k k k k k k k

k k k k k k k k

k k k k k k k k

u1

v1

u2

v2

u3

u4

v3

v4

u1 v1u2 v2 u3

u4v3v4

The stiffness matrix corresponding to this element is

which has the following form

1 1 1 1 1 2

1 1 1 1

T T T11 12 13

T T21 21

B D B dV; B D B dV; B D B dV,...

B D B dV; B D B dV;.....

e e e

e e

u u u v u uV V V

v u v vV V

k k k

k k

The individual entries of the stiffness matrix may be computed as follows

 

 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. 

 

Example

x

y

2

34

1

300 psi1000 lb

3 in

2 inThickness (t) = 0.5 inE= 30×106 psi=0.25

(a) Compute the unknown nodal displacements.(b) 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

Realize that this is a plane stress problem and therefore we need to use

psiE

D 72

10

2.100

02.38.0

08.02.3

2

100

01

01

1

Write down the shape functions

6

)3(

4

164

16

)2(

4

16

)2)(3(

4

1

134

243

312

421

yxyyxx

abN

xyyyxx

abN

yxyyxx

abN

yxyyxx

abN

x y

0 0

3 0

3 2

0 2

We have 4 nodes with 2 dofs per node=8dofs. However, 5 of these are fixed.The nonzero displacements are

u2 u3v3

Hence we need to solve

yfv

u

u

kkk

kkk

kkk

33

3

2

333231

232221

131211

0

0

Need to compute only the relevant terms in the stiffness matrix

2 2 2 3 2 3

3 2 3 3 3 3

3 2 3 3 3 3

T T T11 12 13

T T T21 22 13

T T T31 22 13

B D B dV; B D B dV; B D B dV

B D B dV; B D B dV; B D B dV

B D B dV; B D B dV; B D B dV

e e e

e e e

e e e

u u u u u vV V V

u u u u u vV V V

v u v u v vV V V

k k k

k k k

k k k

u2 u3 v3

6

06

)2(

02

2

2x

y

y

N

x

N

Bu

6

06

03

3

3x

y

y

N

x

N

Bu

6

6

00

3

33

y

x

x

Ny

NBv

Compute only the relevant columns of the B matrix

2 2

T11

3 28 7 5 2

0 0

7

B D B dV

20.5 (0.1067 10 0.533 10 )( ) 3.33 10

6

0.656 10

e u uV

x y

k

yx dxdy

Similarly compute the other terms

How do we compute f3y

ySy ff3

10003

lb

dxx

dxNtf

x

x edgealongS y

2252

3150

3)300)(5.0(

)300(

3

0

3

0 4333

4 3

36 223

343

xxyNN

yy

edge

lbffySy 12251000

33

x

ya a 12

3 4

b

b

Corner nodes

224223

222221

2

)(

2

)(

2

)(

2

)(

2

)(

2

)(

2

)(

2

)(

b

yby

a

xaxN

b

yby

a

xaxN

b

yby

a

xaxN

b

yby

a

xaxN5

6

7

89

Midside nodes

2

22

2822

22

7

2

22

2622

22

5

2

)(

2

)(

2

)(

2

)(

b

yb

a

xaxN

b

yby

a

xaN

b

yb

a

xaxN

b

yby

a

xaN

Center node2 2 2 2

9 2 2

a x b yN

a b

How about a 9-noded rectangle?

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