# Finite Element Method: Plane Stress and Plane Strain

## Full text

(1)

(2)

.

### plane strain

: The strain state when normal strain

### e

z , which is perpendicular to the plane

,

xz

yz

.

(3)

### Stress and strain in 2-D

Stresses in 2-D Principal stress and its direction

x y xy

1

2 2

2

2 2

x y x y

xy

x y x y

xy

max

min

p

xy

x y

### = -

(4)

Displacement and rotation of plane element x- y

x y xy

x y xy

(5)

### { } s = D [ ]{ } e

Stress-strain matrix(or material composed matrix) of isotropic material for plane stress(

z

xz

yz

)

2

### n n

Stress-strain matrix(or material composed matrix) of isotropic material for plane deformation (

z

xz

yz

)

### n

(6)

General steps of formulation process for plane triangular element Step 1: Determination of element type

Step 2: Determination of displacement function

Step 3: Definition of relations deformation rate – strain and stress-strain Step 4: Derivation of element stiffness matrix and equation

Step 5: Introduction a combination of element equation and boundary conditions for obtaining entire equations

Step 6: Calculation of nodal displacement

Step 7: Calculation of force(stress) in an element

(7)

i j m

i i j j m m

(8)

1 2 3

4 5 6

1 2 3

4 5 6

1 2 3 4 5 6

(9)

1

2

3

i i i

j j j

m m m

1 2 3

1 2 3

1 2 3

i j m

i i

j j

m m

1 2 3

-1

i j m

i j m

i j m

-

1

i

j

m

j

m

i

m

i

j

### g g g

i j m j m j i m i m m i j i j

i j m j m i m i j

i m j j i m m j i

### = - = - = -

(10)

After calculation of

### [ ] x

-1 , the equation

-1

### u

can be expressed as an extended matrix form.

i j m

i j m

i j m

i j m 1

2

3

Similarly,

i j m

i j m

i j m

i j m 4

5

6

### u u u

Derivation of displacement function

(

### u

can also be derived similarly)

i j m

i j m

i j m

i j m

1

2

3

(11)

i i i i j j j j

m m m m

i i i i j j j j

m m m m

i i j j m m

i i j j m m

i i i i

j j j j

m m m m

(12)

i i j j m m

i i j j m m

i j m

i j m

i i j j m m

i j m

i j m

i

j

m

(13)

i

j

m

i i

i

(14)

x y xy

### xN uN uN uN uNuNu

x i i j j m m i x i j x j m x m

,

, , ,

i x i i i

i

,

j x

j

m x

m

,

,

i

i j

j m m

(15)

### yxAuuu

i i j j m m

i i i i j j j j m m m m

i j m

i j m

i i j j m m

i i j j m m

i j m

i j m

i

i i

i i

j

j j

j j

m

m

m

m m

(16)

x y xy

x y xy

### [ ] ® { } s = D B d [ ][ ]{ }

(17)

Using minimum potential energy principle.

Total potential energy

p

p

i

i

j

m

b

p

s

Strain energy

V T

V T

### { } { } e s 2 { } [ ]{ } e e

Potential energy of body force

b

V

T

p

T

### P

Potential energy of distributed load (or surface force)

S

S

T

(18)

## zzz zz

p

V

T

V

T

T

S

T T

T T

V

T V

T

T T

S T

T T

V

T

where

V

T

S

### = zzz + + zz

T (7.2.46)

Condition having pole values

p T

V

®

T

## zzz

V

### =

(19)

So, the element stiffness matrix is

T

V

### = zzz

Case of an element having constant thickness

:

A T

T

## zz

Matrix

### [ ] k

is a 6×6 matrix, and the element equation is as below

x y x y x y 1

1

2

2

3

3

11 12 16

21 22 26

61 62 66

1

1

2

2

3

3

### u

(20)

a global coordinate system of equation.

( )e

e N

=

1

and

( )e

e N

=

1

### { } F = [ ]{ } K d

Step 6: Calculation of nodal displacement

Step 7: Calculation of force(stress) in an element

Transformation from the global coordinate system to the local coordinate system: (See Ch. 3)

### d = Tdf = T fk = T k T

T

Constant-strain triangle(CST) has 6 degrees of freedom.

(21)

where

### q

A triangular element with local coordinates system not along to the global coordinate system

(22)

### Finite Element Method in a plane stress problem

Find nodal displacements and element stresses in the case of the thin plate(see below figure) under surface force.

thickness

6

### psi , n = 0 30 .

(23)

(1) Discretization: Surface tension force is replaced by the following nodal loads.

### ( psi )( in . in. ) lb

The global system of the governing equation is

or

x y x y x y x y

x y x y

x y x y x y x y

x y x y 1

1

2

2

3

3

4

4

1

1

2

2

1

1

2

2

3

3

4

4

3

3

4

4

where

### [ ] K

is a 8×8 matrix.

(24)

(2) A combination of stiffness matrix:

### [ ] k = tA B [ ] [ ][ ] D B

· Element 1

- Calculation of matrix

i j m

i j m

i i j j m m

where

i j m

j m i

m i j

i m j

j i m

m j i

and

2

(25)

Then is

- Matrix

(plane stress)

2

6

### n n

- Calculation of stiffness matrix

T

### .

(26)

- Calculation of matrix

i j m

i j m

i i j j m m

where

i j m

j m i

m i j

i m j

j i m

m j i

and

2

(27)

Then matrix is

- Matrix

(plane stress)

### 1 0 3 0 0 3 1 0 0 0 0 35

6

- Calculation of stiffness matrix

(28)

Element 1:

Element 2:

### Q PP PP PP PP PP P

(29)

(3) Calculation of displacement: Superpositioning element stiffness matrix, global system of stiffness matrix is obtained as below.

(30)

Substituting

to

, R

R R R

d d d

x y x y

x y x 1

1 2 2

3 3 4

d4y

### |

Applying given boundary conditions with elimination of columns and rows.

3

3

4

4

### dddd

x y x y

(31)

Transposing the displacement matrix to the left side

x y x y 3

3

4

4

1

6

-

-

### in.

The solution of 1-D beam under tension force is

-

6

6

### in.

Therefore, x-component of the displacement at nodes in the equation (7.5.27) of 2-D plane is quite accurate when the grid is considered as coarse grid.

(32)

(4) Stresses at each node:

- Element 1

E

A

d d d d d d

x y x y x y

2

1 3 2

1 3 2

1 1 3 3 2 2

1 1 3 3 2 2

Calculating,

x y xy

(33)

- Element 2

E

A

d d d d d d

x y x y x y

2

1 4 3

1 4 4

1 1 4 4 3 3

1 1 4 4 3 3

Calculating,

x y xy

Updating...

## References

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