Elasticity2.ppt

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Formulation of Two-Dimensional

Elastici ty Problems

Professor M. H. Sadd



Simplif ied Elastici ty Formulations

Displacement Formulation

Eliminate the stresses and strains

from the general system of equations.

This generates a system of three

equations for the three unknown

displacement components.

Stress Formulation

Eliminate the displacements and

strains from the general system of

equations. This generates a system of

six equations and for the six unknown

stress components.

The General System of Elasticity F ield Equations

of 15 Equations for 15 Unknowns I s Very Diff icul t

to Solve for Most Meaningful Problems, and So

Modif ied Formulations Have Been Developed.



Solution to Elastici ty Problems

F(z)

G(x,y)

z

x

y

Even Using Displacement and Stress FormulationsThree-Dimensional Problems Are Diff icult to Solve!

So Most Solutions Are Developed for Two-Dimensional Problems



Two and Three Dimensional Problems

x

y

z

x

y

z

Three-Dimensional Two-Dimensional

x

y

z

Spherical Cavity





Examples of Plane Strain Problems

x

y

z

x

y

z

P

Long Cylinders

Under Unif orm Loading

Semi-I nf ini te Regions

Under Uni form Loadings



Examples of Plane Stress Problems

Thin Plate WithCentral Hole

Circular Plate Under

Edge Loadings



Plane Strain Formulation

0

2

1,,

yz xz z

xy y x

eee

x

v

y

ue

y

ve

x

ue

Strain-Displacement

0,2

)()(

2)(

2)(

yz xz xy xy

y x y x z

y y x y

x y x x

e

ee

eee

eee

Hooke’s Law

0,),(,),( w y xvv y xuu



Plane Strain Formulation

0)(

0)(

2

2

y

x

F y

v

x

u

yv

F y

v

x

u

xu


0

0

y

y xy

x

xy x

F y x

F y x

y

F

x

F y x y x

1

1)(

2

Stress Formulation

R

S o

S i

S = S i + S o

x

y



Plane Strain Example

0

)()(

02)(

1

)1(22

)21)(1(

)21)(1(

2)(

:

0,)1(

,1

:

0,)1(

,1

,

2

2

2

yz xz xy

o y x y x z

y y x y

o

oo

x y x x

z zx yz xyo yo x

oo

ee

eee

E

E

E

E

eee

eeee

E y

ve

E x

ue

w y E

v x E

u

Stresses

Strains

StresseandStrainstheDeterminentsDisplacemeFollowingtheGiven



Plane Stress Formulation

Hooke’s Law

0,1

)(1

)(

)(1

)(1

yz xz xy xy

y x y x z

x y y

y x x

ee E

e

ee E

e

E e

E e

Strain-Displacement

02

1

02

1

2

1

,,

x

w

z

u

e

y

w

z

ve

x

v

y

ue

z

we

y

ve

x

ue

xz

yz

xy

z y x

0,),(,),(,),( yz xz z xy xy y y x x y x y x y x

Note plane stress theory normally neglects some of the

strain-displacement and compatibility equations.



Plane Stress Formulation

R

S o

S i

S = S i + S o

x

y


0

)1(2

0)1(2

2

2

y

x

F

y

v

x

u

y

E v

F y

v

x

u

x

E u

0

0

y

y xy

x

xy x

F

y x

F y x

Stress Formulation

y

F

x

F y x y x )1()(

2



Correspondence Between Plane Problems

Plane Strain Plane Stress

0)(

0)(

2

2

y

x

F y

v

x

u

yv

F y

v

x

u

xu

0

0

y

y xy

x

xy x

F y x

F y x

y

F

x

F y x y x

1

1)(

2

0

)1(2

0)1(2

2

2

y

x

F

y

v

x

u

y

E v

F y

v

x

u

x

E u

0

0

y

y xy

x

xy x

F y x

F y x

y

F

x

F y x y x )1()(

2



Elastic Modul i Transformation Relations for Conversion

Between Plane Stress and Plane Strain Problems

21 E

1

2)1(

)21(

E

1

E v

Plane Stress to Plane Strain

Plane Strain to Plane Stress

Plane Strain Plane Stress

Therefore the solution to one plane problem also yields the solution

to the other plane problem through this simple transformation



Airy Stress Function Method

Plane Problems with No Body Forces

0

0

y x

y x

y xy

xy x

0)(2 y x

Stress Formulation

y x x y xy y x

2

2

2

2

2

,,

Airy Representation

02 4

4

4

22

4

4

4

y y x x

Biharmonic Governing Equation

(Single Equation with Single Unknown)





Solutions to Plane Problems

Cartesian Coordinates

y x x y xy y x

2

2

2

2

2

,,

Airy Representation

02 4

4

4

22

4

4

4

y y x x


),(,),( y x f T y x f T y y x x

Traction Boundary Conditions

R

S

x

y



Solutions to Plane Problems

Polar Coordinates

R

S),(,),(

r f T r f T

r r

Traction Boundary Conditions

Airy Representation

r r r r r r r r

1,,

112

2

2

2

2


01111 2

2

22

2

2

2

22

2

4

r r r r r r r r

x

y

r



Cartesian Coordinate Solutions

Using Polynomial Stress Functions

024

4

22

4

4

4

y y x x

2

0211

2

20011000

0 0

),( y A xy A x A y A x A A y x A y xm n

nm

mn

y x x y xy y x

2

2

2

2

2

,,

terms do not contribute to the stresses and are therefore dropped1nm

terms will automatically satisfy the biharmonic equation 3nm

terms require constants Amn to be related in order to satisfy biharmonic equation3nm

Solution method limited to problems where boundary traction conditions

can be represented by polynomials or where more complicated boundaryconditions can be replaced by a statically equivalent loading



Stress Function Example

)(6

,0

)(6

:

)23(

:

3

2

2

2

32

2

2

3

yd yd

F

y x x

yd xd

F

y

yd xyd

F

xy y

x

StressestheDetermine

FunctionStressFollowingtheConsider

Appears to Solve the Beam Problem:

x

y

dF

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