1

Chapter 6. Plane Stress / Plane Strain Problems

Element types:

Line elements (spring, truss, beam, frame) – chapters 2-5

2-D solid elements – chapters 6-10

3-D solid elements – chapter 11

Plate / shell elements – chapter 12

2

2-D Elements

Triangular elements – plane stress/plane strain:

CST – “constant strain triangle” – chap. 6

LST – “linear strain triangle” – chap. 8

Axisymmetric elements – chap. 10

Isoparametric elements – chap. 11

4-node quadrilateral element (linear interpolation)

8-node quadrilateral element (quadratic interpolation)

3

Plane stress

0

0

0

0

yzxzz

xy

y

x

4

Plane Strain

0 yzxzz

0

0

0

0

0

yzxz

z

xy

y

x

5

2-D Stress States

Matrix form:

xy

y

x

x

6

Principal Stresses

yxp

yxyx

xy

xy

2tan

2

1

22,

1

22

21

7

Displacements and Strains

Displacement field

Strains

),(

),(

yxv

yxuu

xy

y

x

xyyx x

v

y

u

y

v

x

u

,,

8

Stress-Strain Relations

Recall:

E – Young’s modulusn - Poisson’s Ratio

G – Shear modulus

D

9

Stress-Strain Relations (cont.)

Plane stress

Plane strain

Note, in both cases

xy

y

x

xy

y

xE

2

100

01

01

1 2

xy

y

x

xy

y

xE

2

2100

01

01

211

GE

D

1233

10

Derivation of “Constant Strain Triangle” (CST) Element Equations

Note – x-y are global coordinates (will not need to transform from local to global

m

m

j

j

i

i

v

u

v

u

v

u

d

Step 1 – Select element type

11

Displacement Interpolation

yaxaa

yaxaa

yxv

yxu

654

321

),(

),(

Assume “bi-linear” interpolation – guarantees that edges remain straight => inter-element compatibility

12

Displacement Interpolation (cont.)

As before, rewrite displacement interpolation in terms of nodal displacements (see text for details)

where

mmjjii

mmjjii

vyxNvyxNvyxNyxv

uyxNuyxNuyxNyxu

),(),(),(),(

),(),(),(),(

yxA

N

yxA

N

yxA

N

mmmm

jjjj

iiii

2

12

12

1

13

Displacement Interpolation (cont.)

and

mm

jj

ii

jmi

mji

mjmji

yx

yx

yx

A

xx

yy

yyyx

1

1

1

2

1

14

Displacement Interpolation (cont.)

dN

v

u

v

u

v

u

NNN

NNN

v

u

m

m

j

j

i

i

mji

mji

000

000

15

Displacement Interpolation (cont.)

yxA

N

yxA

N

yxA

N

mmmm

jjjj

iiii

2

12

12

1

Graphically:

16

Step 3 – Strain-Displacement and Stress-Strain Relations

xy

y

x

xyyx x

v

y

u

y

v

x

u

,,

From which it can be shown

dB

v

u

v

u

v

u

A

m

m

j

j

i

i

mmjjii

mji

mji

000

000

2

1

17

Strain-Displacement Relations (cont.)

• Note – the strain within each element is constant (does not vary with x & y)

• Hence, the 3-node triangle is called a “Constant Strain Triangle” (CST) element

18

Stress-Strain Relations

dBD

D

3x1 3x3 3x6 6x1

19

Step 4 – Derive Element Equations

BSpp U

which will be used to derive

BDBtAk T6x6 6x3 3x3 3x6

20

Derive Element Equations (cont.)

Strain energy:

21

Derive Element Equations (cont.)

Potential energy of applied loads:

22

Derive Element Equations (cont.)

Potential energy:

23

Derive Element Equations (cont.)

Substitute

to yield

24

Derive Element Equations (cont.)

Apply principle of minimum potential energy

To obtain

25

Derive Element Equations (cont.)

Element stiffness matrix

26

Steps 5-7

5. Assemble global equations

6. Solve for nodal displacements

7. Compute element stresses (constant within each element)

27

Example – CST element stiffness matrix

28

CST Element Stiffness Matrix

where[B] – depends on nodal coordinates[D] – depends on E,

See text for details

BDBtAk T

29

Body and Surface Forces

Replace distributed body forces and surface tractions with work equivalent concentrated forces.

{ fb } { fs }

30

Work Equivalent Concentrated Forces – Body Forces

For a uniformly distributed body forces Xb and Yb:

3

At

Y

X

Y

X

Y

X

f

f

f

f

f

f

f

b

b

b

b

b

b

my

mx

jy

jx

iy

ix

b

31

Work Equivalent Concentrated Forces – Surface Forces

3

0

2/

0

0

0

2/

At

pLt

pLt

f

f

f

f

f

f

f

smy

smx

sjy

sjx

siy

six

b

For a uniform surface loading, p, acting on a vertical edge of length, L, between nodes 1 and 3:

32

Example 6.2

33

Example 6.2 - Solution

in

d

d

d

d

y

x

y

x

6

4

4

3

3

10

1.104

7.663

2.4

6.609

4.2

301

1005

xy

y

x

4.2

2.1

995

xy

y

x

Element 1 Element 2

34

In-class Abaqus Demonstrations

• Example 6.2• Finite width plate with circular hole

(ref. “Abaqus Plane Stress Tutorial”)

35

Chapter 7 - Practical Considerations in Modeling; Interpreting Results; and

Examples of Plane Stress/Strain Analysis

Discussion of Example 6.2:

36

Example 6.2 - discussion