imsc002_2delasticproblems

IMSC002 by W.M.Quach, 2006

Plane Stress & Plane Strain Problems

IMSC002 Theory of Elasticity and Plasticity IMSC002 Theory of Elasticity and Plasticity


Validities of Stresses and Strains• A valid stress field τij must satisfy the Equilibrium Equations [i.e. Eq. (1.11) ], and the traction boundary conditions.

• A valid strain field εij must be related to the displacements through the Strain-Displacement Relation [i.e. Compatibility Equations, Eq. (2.9) ].

• A stress field τij should be related to the corresponding strain field εij by a valid Stress-Strain Relationship [i.e. Constitutive Law, Eq. (2.8) for Elasticity & Eqs for Plasticity]


Problem Solving for a 3-D Elastic Stress Analysis

For a 3-D elastic structure subjected to a given load & boundary conditions, we want to determine the following quantities (15 unknowns) at each material point within the structure:

• Stresses – 6 components (σx, σy, σz, τxy, τyz, τxz)

• Strains – 6 components (εx, εy, εz, γxy, γyz, γxz)• Displacements – 3 components (u, v, w)

The solution of these 15 unknowns, should make the following conditions (totally 15 independent equations) satisfied:• The Boundary Conditions and the Stress Equilibrium Equations are satisfied.• The strains are compatible with the displacements (i.e. Compatibility Conditions).• The stresses are related to the strains through the elastic stress-strain relation (i.e. Generalized Hooke’s Law).



These 15 independent equations for a 3-D elastic solid are resulted from:

c. Generalized Hooke’s Law (6 eqs.):

a. Stress Equilibrium Equations (3 eqs.):

b. Compatibility Equations (6 eqs.):

( )[ ]

( )[ ]

( )[ ]GE

GE

GE

xzxzyxzz

yzyzzxyy

xyxyzyxx

τγσσνσε

τγσσνσε

τγσσνσε

=+−=

=+−=

=+−=

,1

,1

,1

∂

∂−

∂∂

+∂

∂

∂∂

=∂∂

∂∂∂

+∂∂

=∂∂

∂

∂

∂+

∂∂

−∂

∂

∂∂

=∂∂

∂

∂∂

+∂

∂=

∂∂

∂

∂

∂+

∂∂

+∂

∂−

∂∂

=∂∂

∂∂

∂+

∂∂

=∂∂

∂

zyxzyxzxxz

zyxyxzyzzy

zyxxzyxyyx

xyxzyzzxzxz

xyxzyzyzyyz

xyxzyzxyxxy

γγγεεεγ

γγγεεεγ

γγγεεεγ

21,

21,

21,

2

2

2

2

22

2

2

2

2

22

2

2

2

2

22

0

0

0

=+∂

∂+

∂∂

+∂∂

=+∂

∂+

∂

∂+

∂

∂

=+∂∂

+∂

∂+

∂∂

zyzxzz

yyzxyy

xxzxyx

Fyxz

Fzxy

Fzyx

ττσ

ττσ

ττσ

.. (2.9)

.. (2.28).. (1.11)

wuvuuuzyxjiiu

ju

zyx

jiij

====

∂

∂+

∂∂

=

,,,,,

,21ε

Note:



For a 2-D elastic structure subjected to a given load & boundary conditions (i.e. Plane Stress or Plane Strain), there are only 8 unknownsfor each material point within the structure:

• Stresses – 3 components (σx, σy, τxy)

• Strains – 3 components (εx, εy, γxy)• Displacements – 2 components (u, v)

The solution of these 8 unknowns, should make the following conditions (totally 8 independent equations) satisfied:• The Boundary Conditions and the Stress Equilibrium Equations are satisfied.• The strains are compatible with the displacements (i.e. Compatibility Conditions).• The stresses are related to the strains through the elastic stress-strain relation (i.e. Generalized Hooke’s Law).


Plane Strain Problems

Strain-Displacement Relations

Plane strain in a cylindrical body under in-planar loading (i.e. internal &

external pressures)

Frictionless nature of the fixed end constraint permits x, y deformation, but precludes z displacement (w = 0, at z = EL/2).

0,0,0

,,,

=∂∂

+∂∂

==∂∂

+∂∂

==∂∂

=

∂∂

+∂∂

=∂∂

=∂∂

=

yw

zv

zu

xw

zw

xv

yu

yv

xu

yzxzz

xyyx

γγε

γεε

wuvuuuzyxjiiu

ju

zyx

jiij

====

∂

∂+

∂∂

=

,,,,,

,21ε

General 3-D

Plane Strain

.. (3.1)

.. (3.2)

0=== zxyzz γγε (Plane Strain Condition)

.. (2.3)


( )( )( ) xzxzzyxzz

yzyzzyxyy

xyxyzyxxx

GG

GG

GG

γτεεελεσ

γτεεελεσ

γτεεελεσ

=+++=

=+++=

=+++=

,2

,2

,2

Plane Strain ProblemsStress-Strain Relationships

0=== zxyzz γγε

General 3-D

( )( )

( ) ( )

( )( )νννλ

σσνεελσττ

γτ

εελεσ

εελεσ

211

,0&

2

2

−+=

+=+===

=

++=

++=

Ewhere

GGG

yxyxzyzxz

xyxy

yxyy

yxxx

Plane Strain

.. (3.3)

.. (3.4)

(Plane Strain Condition)

.. (2.30)

A cylindrical body under in-planar loading


Plane Strain ProblemsStrain-Stress Relations

General 3-D

Plane Strain .. (3.5)

( )[ ]

( )[ ]

( )[ ]GE

GE

GE

xzxzyxzz

yzyzzxyy

xyxyzyxx

τγσσνσε

τγσσνσε

τγσσνσε

=+−=

=+−=

=+−=

,1

,1

,1

(Plane Strain Condition) ( ) ( )yxyxzyzxz

zxyzz

σσνεελσττ

γγε

+=+===

===

,0

0

.. (2.28)

G

E

E

xyxy

xyy

yxx

τγ

σν

νσνε

σν

νσνε

=

−−

−=

−−

−=

11

11

2

2



Plane Strain ProblemsEquilibrium Equations

General 3-D



.. (1.11)

0

0

0

=+∂

∂+

∂∂

+∂∂

=+∂

∂+

∂

∂+

∂

∂

=+∂∂

+∂

∂+

∂∂

zyzxzz

yyzxyy

xxzxyx

Fyxz

Fzxy

Fzyx

ττσ

ττσ

ττσ

0

0

=+∂

∂+

∂

∂

=+∂

∂+

∂∂

yxyy

xxyx

Fxy

Fyxτσ

τσ

( )0&

,0

=

+===

z

yxzyzxz

Fσσνσττ



Plane Strain Problems

Boundary Conditions



.. (1.22)

mlpmlp

yxyy

xyxx

στ

τσ

+=

+=

0

,0

=

==

z

yzxz

pττ

General 3-D

nmlpnmlpnmlp

zyzxzz

yzyxyy

xzxyxx

σττ

τστ

ττσ

++=

++=

++=

Boundary condition -- The conditions of equilibrium with respect to external forces or stresses acting on the boundary of a body.


Surface forces acting on the boundary of a body


Plane Strain ProblemsCompatibility Conditions

Plane Strain

.. (3.8)yxxyxyyx

∂∂

∂=

∂

∂+

∂∂ γεε 2

2

2

2

2

..(3.5)

( )[ ] ( )[ ]yxxyxy

xyyx ∂∂

∂=−−

∂∂

+−−∂∂ τ

νσσννσσν2

2

2

2

2

211

Strain-Stress Relations

( )

∂

∂+

∂∂

−−=+

∂∂

+∂∂

yF

xF

yxyx

yx νσσ

11

2

2

2

2

0

0

=+∂

∂+

∂

∂

=+∂

∂+

∂∂

yxyy

xxyx

Fxy

Fyxτσ

τσ

Equilibrium Eqs.

.. (3.9)(3.6)

Differentiation of Eq.(3.6)

Equation of Compatibility in terms of Stresses

G

E

E

xyxy

xyy

yxx

τγ

σν

νσνε

σν

νσνε

=

−−

−=

−−

−=

11

11

2

2


Plane Strain Problems:Solutions of Stresses

.. (3.4)

3 Governing Equations

Equilibrium Eqs.

( )

∂

∂+

∂∂

−−=+

∂∂

+∂∂

=+∂

∂+

∂

∂

=+∂

∂+

∂∂

yF

xF

yx

Fxy

Fyx

yxyx

yxyy

xxyx

νσσ

τσ

τσ

11

0

0

2

2

2

2

( )xyyx τσσ ,,

Equation of Compatibility

.. (3.6)

.. (3.9)

3 Unknown Stresses

( )yxzyzxz σσνσττ +=== ,0Note: The direct stress in z–direction, is then given by Eq.(3.4):

Provide the boundary condition through Eq. (3.7)


Plane Strain Problems:Solutions of Strains & Displacements

( )xyyx τσσ ,, .. (3.5)Stresses

0,0,0

,,,

=∂∂

+∂∂

==∂∂

+∂∂

==∂∂

=

∂∂

+∂∂

=∂∂

=∂∂

=

yw

zv

zu

xw

zw

xv

yu

yv

xu

yzxzz

xyyx

γγε

γεε



( )xyyx γεε ,,Strains

Displacements .. (3.1)

.. (3.2)

G

E

E

xyxy

xyy

yxx

τγ

σν

νσνε

σν

νσνε

=

−−

−=

−−

−=

11

11

2

2

( )0&

,=wvu


Plane Stress Problems

Plane Stress

(Plane Stress Condition)

A thin plate under Plane Stress


General 3-D

( )[ ]

( )[ ]

( )[ ]GE

GE

GE

xzxzyxzz

yzyzzxyy

xyxyzyxx

τγσσνσε

τγσσνσε

τγσσνσε

=+−=

=+−=

=+−=

,1

,1

,1

.. (2.28)

0=== yzxzz ττσ

( )

( )

( ) ( )yxyxzyzxz

xyxyxyy

yxx

E

GE

E

εεννσσνεγγ

τγνσσε

νσσε

+−−

=+−

===

=−=

−=

1,0&

,1

1

.. (3.10)

.. (3.11)




wuvuuuzyxjiiu

ju

zyx

jiij

====

∂

∂+

∂∂

=

,,,,,

,21ε

General 3-D

Plane Stress


.. (2.3)

0,0,0

,,,

=∂∂

+∂∂

==∂∂

+∂∂

=≠∂∂

=

∂∂

+∂∂

=∂∂

=∂∂

=

yw

zv

zu

xw

zw

xv

yu

yv

xu

yzxzz

xyyx

γγε

γεε .. (3.1)

0

0

≠

==

z

zxyz

but εγγ

Its value can be given by Eq.(3.11)



Plane Stress ProblemsEquilibrium Equations

General 3-D

Plane Stress .. (3.6)


.. (1.11)

0

0

0

=+∂

∂+

∂∂

+∂∂

=+∂

∂+

∂

∂+

∂

∂

=+∂∂

+∂

∂+

∂∂

zyzxzz

yyzxyy

xxzxyx

Fyxz

Fzxy

Fzyx

ττσ

ττσ

ττσ

0

0

=+∂

∂+

∂

∂

=+∂

∂+

∂∂

yxyy

xxyx

Fxy

Fyxτσ

τσ

0&0

=

===

z

yzxzz

Fττσ

The same Equilibrium Equations used for “Plane Strain”




Boundary Conditions

.. (3.7)


.. (1.22)

mlpmlp

yxyy

xyxx

στ

τσ

+=

+=

nmlpnmlpnmlp

zyzxzz

yzyxyy

xzxyxx

σττ

τστ

ττσ

++=

++=

++=

Plane Stress

General 3-D

Boundary condition -- The conditions of equilibrium with respect to external forces or stresses acting on the boundary of a body.

0&

0

=

===

z

yzxzz

pττσ

The same equations of Boundary Conditions used for “Plane Strain”Surface forces acting on the

boundary of a body



Plane Stress ProblemsCompatibility Conditions

Plane Stress .. (3.8)yxxyxyyx

∂∂

∂=

∂

∂+

∂∂ γεε 2

2

2

2

2

.. (3.10)


0

0

=+∂

∂+

∂

∂

=+∂

∂+

∂∂

yxyy

xxyx

Fxy

Fyxτσ

τσ

Equilibrium Eqs.

.. (3.12)

.. (3.6)

Differentiation of Eq.(3.6)

Equation of Compatibility in terms of Stresses

( )

( )

G

E

E

xyxy

xyy

yxx

τγ

νσσε

νσσε

=

−=

−=

1

1

( ) ( )

∂

∂+

∂∂

+−=+

∂∂

+∂∂

yF

xF

yxyx

yx νσσ 12

2

2

2

The same equation used for “Plane Strain”


( ) ( )

∂

∂+

∂∂

+−=+

∂∂

+∂∂

=+∂

∂+

∂

∂

=+∂

∂+

∂∂

yF

xF

yx

Fxy

Fyx

yxyx

yxyy

xxyx

νσσ

τσ

τσ

1

0

0

2

2

2

2

Plane Stress Problems:Solutions of Stresses

3 Governing Equations

Equilibrium Eqs.

( )xyyx τσσ ,,

Equation of Compatibility

.. (3.6)

.. (3.12)

3 Unknown Stresses

Provide the boundary condition through Eq. (3.7)


Plane Stress Problems:Solutions of Strains & Displacements

( )xyyx τσσ ,,Stresses



Strains

Displacements

( )

( )

( ) ( )0

,1

&

,,1

,1

==

+−−

=+−

=

=−=

−=

yzxz

yxyxz

xyxyxyy

yxx

E

GE

E

γγ

εεννσσνε

τγνσσε

νσσε

.. (3.11)

( )z

xyyx

ε

γεε

&

,,(3.10)

0,0,0

,,,

=∂∂

+∂∂

==∂∂

+∂∂

=≠∂∂

=

∂∂

+∂∂

=∂∂

=∂∂

=

yw

zv

zu

xw

zw

xv

yu

yv

xu

yzxzz

xyyx

γγε

γεε

( )vu,


Airy’s Stress Function3 Governing Equations(with body forces)

( )

( ) ( )

∂

∂+

∂∂

+−=+

∂∂

+∂∂

∂

∂+

∂∂

−−=+

∂∂

+∂∂

=+∂

∂+

∂

∂=+

∂

∂+

∂∂

yF

xF

yx

oryF

xF

yx

Fxy

Fyx

yxyx

yxyx

yxyy

xxyx

νσσ

νσσ

τστσ

1

11

0,0

2

2

2

2

2

2

2

2

( ) 0

0,0

2

2

2

2

=+

∂∂

+∂∂

=∂

∂+

∂

∂=

∂

∂+

∂∂

yx

xyyxyx

yx

xyyx

σσ

τστσ

3 Governing Equations(without body forces)

Equilibrium Eqs.

Plane Stress[Eq.(3.12)]

Plane Strain[Eq.(3.9)]

0=== zyx FFFwith

2

2

2

22

4

4

22

4

4

44 02

yxwhere

yyxx

∂∂

+∂∂

=∇

=∂Φ∂

+∂∂Φ∂

+∂Φ∂

=Φ∇( )

yxxy

whichforyxfunctionStress

xyyx ∂∂Φ∂

−=∂Φ∂

=∂Φ∂

=

Φ2

2

2

2

2

,,

:,,

τσσ

Compatibility Equation in terms of Φ

Introduce a stress function Φ

.. (3.13)

.. (3.14)


Airy’s Stress Function:Plane Stress Problems

For Plane Stress Problems, in addition to the “Governing Equation” (3.12) or (3.14), the following conditions should also be satisfied:

0,0,02

2

2

2

2

=∂∂

∂=

∂∂

=∂∂

yxxyzzz εεε

Plane Stress Conditions:

∂

∂−

∂∂

+∂

∂

∂∂

=∂∂

∂∂∂

+∂∂

=∂∂

∂

∂

∂+

∂∂

−∂

∂

∂∂

=∂∂

∂

∂∂

+∂

∂=

∂∂

∂

∂

∂+

∂∂

+∂

∂−

∂∂

=∂∂

∂∂

∂+

∂∂

=∂∂

∂

zyxzyxzxxz

zyxyxzyzzy

zyxxzyxyyx

xyxzyzzxzxz

xyxzyzyzyyz

xyxzyzxyxxy

γγγεεεγ

γγγεεεγ

γγγεεεγ

21,

21,

21,

2

2

2

2

22

2

2

2

2

22

2

2

2

2

22

3-D Compatibility Equation

0,0

,0

≠∂∂===

===

zwzyzxz

yzxzz

εγγ

ττσ ( )

( )yx

yxz E

εενν

σσνε

+−−

=

+−

=

1( )( ) .&,,,,,

.&,,,,

zoftindependenareyxfunctions

zoftindependenareyxfunctions

xyzyx

xyyx

=

=

γεεε

τσσ

The solution of the stress function Φ, obtained from Eq.(3.12) or (3.14), may provide the approximation only, since the above additional compatibility conditions for may not always be satisfied by a given stress function Φ.


Airy’s Stress Function:Polynomial Solutions

.. (3.14)

For Plane Stress or Plane Strain Problems in absence of body forces, the governing equations is given as a Biharmonic Equation as follows:

Compatibility Equation in terms of Φ, in absence of body forces

2

2

2

22

4

4

22

4

4

44 ,02

yxwhere

yyxx ∂∂

+∂∂

=∇=∂Φ∂

+∂∂Φ∂

+∂Φ∂

=Φ∇

( ).deg""

,0

reethitheofpolynomialtheiswhere

yx

i

ii

−Φ

Φ=Φ ∑∞

=

The solution Φ of the above biharmonic equation is thus:

LL

,2012661220

,1262612

,6226

,22

,,

554532523545555

443422434444

332323333

222

22211100

yf

xye

yxd

yxc

yxb

xa

yexydyxcyxbxa

ydxycyxbxaycxybxaybxaa

+++++=Φ

++++=Φ

+++=Φ++=Φ+=Φ=Φwith

The values of these constants (a’s, b’s, c’s, etc.)

are obtained for a given boundary condition.



( ).deg""

,0

reethitheofpolynomialtheiswhere

yx

i

ii

−Φ

Φ=Φ ∑∞

=

The solution Φ of the biharmonic equation is :

For examples:

The approximation for the solution Φ can be obtained by • a single polynomial Φi

• or the linear combination of several selected polynomials Φi’s.

( )222

222

222

,,

:22

,

bac

ycxybxayxFor

xyyx −===

++=Φ=Φ

τσσ

( )ycxbybxaydxc

yd

xyc

yxb

xa

yxFor

xyyx 333333

332323333

,,

:6226

,

−−=+=+=

+++=Φ=Φ

τσσ

Eq.(3.13)

yx

xy

Eq

xy

yx

∂∂Φ∂

−=

∂Φ∂

=∂Φ∂

=

2

2

2

2

2

,

:)13.3.(

τ

σσ

Note:

Eq.(3.13)



For examples:

yx

xy

Eq

xy

yx

∂∂Φ∂

−=

∂Φ∂

=∂Φ∂

=

2

2

2

2

2

,

:)13.3.(

τ

σσ

Note: Eq.(3.13)

( )( )

244

24

244

24

2444

24

443422434444

22

2

2

:1262612

,

ydxycxbycxybxa

yacxydxc

yexydyxcyxbxayxFor

xy

y

x

−−−=

++=

+−+=

++++=Φ=Φ

τ

σ

σEq.(3.13)

( )

( )

( )

( ) ( ) 355

25

25

355

3525

255

35

35

255

25

35

554532523545555

233123

31

323

233

:2012661220

,

ycdxydyxcxdf

ydxycyxdfxa

yfxycayxdxc

yfxyeyxdyxcyxbxayxFor

xy

y

x

++−−+=

+++−=

++−+=

+++++=Φ=Φ

τ

σ

σ

( )

.)14.3.(

,2 444

satisfiedisEqequationgoverningtheaceWith +−=


Airy’s Stress Function:Example

imsc002_2delasticproblems

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