CE 595 Section 1

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School of Civil Engineering Spring 2007

CE 595:CE 595:

Finite Elements in ElasticityFinite Elements in Elasticity

Instructors:Instructors: Amit Varma, Ph.D.Amit Varma, Ph.D.

Timothy M. Whalen, Ph.D.Timothy M. Whalen, Ph.D.

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Review of Elasticity -2-

Section 1: Review of ElasticitySection 1: Review of Elasticity

1.1. Stress & StrainStress & Strain

2.2. Constitutive TheoryConstitutive Theory

3.3. Energy MethodsEnergy Methods

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Section 1.1: Stress and StrainSection 1.1: Stress and Strain

Stress at a pointStress at a point QQ ::

0 0 0lim ; lim ; lim .

yx z x xy xz

A A A

FF F

A A A( ( (

(( (W X X

( ( (p p p

! ! !

? A Stress matrix ( ) ; Stress vector ( ) .

x

y

x xy xz

z

xy y yz

xy

xz yz z

yz

xz

Q Q

W

W

W X X WX W X

XX X W

X

X

-

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1.1: Stress and Strain (cont.)1.1: Stress and Strain (cont.)

Stresses must satisfy equilibrium equations inStresses must satisfy equilibrium equations in pointwisepointwisemanner:manner:

Strong Form

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Stresses act on inclined surfaces as follows:Stresses act on inclined surfaces as follows:

? A

2 2

( ) .

; .

x xy xz x

xy y yz y

xz yz z z

n

n

n

W X XX W X

X X W

W X W

!

-

!

! !

n

n n

S

n

S n Sg

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Strain at a pt.Strain at a pt. QQ related torelated to displacementsdisplacements ::

: , , : , ,

Displacement functions, , , , , , , ,

defined by:

, , ;

, , ;

, , .

Q x y z Q x y z

u x y z v x y z w x y z

x x u x y z

y y v x y z

z z w x y z

d d dd

d!

d!

d!

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Normal strainNormal strain relates to changes inrelates to changes in sizesize ::

;

, , = , , .

, , . Also, ; .

x

D Q

x y z

Q D QD Q D dx

Q D dxQ D x x x dx u x dx y x u x y dx u x dx y u x y

u x dx y u x y u v wQ Q Q

dx x y z

I

I I I

d d

dd dd | !

dd! ! -

x x x@ ! } ! !

x x x

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Shearing strainShearing strain relates to changes inrelates to changes in angleangle ::

, ,

. . . xy xz yzv x dx y u x y dy v u u v

Q Q Q Q Q Qdx dy x y x z y z

K E F K K x x x x x x

! ! ! x x x x x x

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Sometimes FEA programs useSometimes FEA programs use elasticityelasticityshearing strainsshearing strains ::

Strains must satisfy 6Strains must satisfy 6 compatibility equationscompatibility equations::

(usually automatic for most formulations)(usually automatic for most formulations)

1 1 1

2 2 2. . . xy xy xz xz yz yzI K I K I K ! ! !

2 22

2 2E.g.: .

xy yx

x y y x

K IIx xx

! x x x x

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Section 1.2Section 1.2 : Constitutive Theory: Constitutive Theory

ForFor linear elasticlinear elastic materials, stresses and strains arematerials, stresses and strains arerelated by therelated by the Generalized Hookes LawGeneralized Hookes Law ::

? A _ a .o o! C

? A

11 12 13 14 15 16

12 22 23 24 25 26

13 23 33 34 35 36

14 24 34 44 45 46

15 25 35 45 55 56

16 26 36 46 56 66

; ;

x x

y y

z z

xy xy

yz yz

xz xz

c c c c c c

c c c c c c

c c c c c c

c c c c c c

c c c c c cc c c c c c

W I

W I

W I

X K

X KX K

! ! ! -

;

; .

o o

Elasticity matrix

residual stresses residualstrains

! !

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1.21.2 : Constitutive Theory (cont.): Constitutive Theory (cont.)

ForFor isotropic linear elasticisotropic linear elastic materials, elasticity matrixmaterials, elasticity matrixtakes special form:takes special form:

? A

1

2

1

2

1

2

1 0 0 0

1 0 0 0

1 0 0 0.

0 0 0 1 2 0 01 2 1

0 0 0 0 1 2 0

0 0 0 0 0 1 2

= Young's modulus, = Poisson's ratio.

E

E

R R R

R R R

R R R

RR R

R

R

R

!

-

C

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Special cases of GHL:Special cases of GHL: Plane StressPlane Stress : allout: allout--ofof--plane stresses assumed zero.plane stressesassumed zero.

Plane StrainPlane Strain: allout: allout--ofof--plane strainsassumed zero.plane strainsassumed zero.

? A

2

12

1 0

1 0 .1

0 0 1

; ;

requireNote: d.1

x x

y y

xy xy

z x y

EW I

W I

X KR

I I IR

R

RR

R

! !

!

! -

C

? A

21

1 01

11 0 .

1

20 0

1

.

x x

y y

xy xy

z x y

E

W IW I

X K

RR

R RR

W R W W

R

! !

!

! -

C

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Other constitutive relations:Other constitutive relations: OrthotropicOrthotropic: material has less symmetry than isotropiccase.: material has less symmetry than isotropiccase.

FRP, wood, reinforced concrete, FRP, wood, reinforced concrete,

ViscoelasticViscoelastic: stresses in material depend on bothstrain and strain rate.: stresses in material depend on bothstrain and strain rate.Asphalt, soils, concrete (creep), Asphalt, soils, concrete (creep),

NonlinearNonlinear: stresses notproportional tostrains.: stresses notproportional tostrains.

Elastomers, ductile yielding, cracking, Elastomers, ductile yielding, cracking,

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Strain EnergyStrain Energy Energy stored in an elastic material during deformation; can beEnergy stored in an elastic material during deformation; can be

recoveredrecovered completelycompletely..

Work done during1 1 :

.

; .

.

.

Ifall external workis stored,

.

final

o

final

o

x o x o

x x o o

o o x x

o x x

dW F dF dL FdL

F A dL d L

dW d A L

W A L d

U W V d

I

I

I

I

W I

W I

W I

W I

dp

! }

! !

@ !

!

! !

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Strain Energy DensityStrain Energy Density : strain energy: strain energy per unitper unitvolumevolume..

In general,In general,

.

final

o

o x x

olu e

U U V d

U UdV

I

I

W I! !

!

.

final final final final final final

o o o o o o

x x y y z z xy xy yz yz xz xzU d d d d d d

I I I K K K

I I I K K K

W I W I W I X K X K X K!

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Section1.3Section1.3 : Energy Methods: Energy Methods

Energy methods are techniques for satisfying equilibriumEnergy methods are techniques for satisfying equilibriumor compatibility onaor compatibility ona globalglobal level rather thanpointwise.level rather thanpointwise.

Two general types canbe identified:Two general types canbe identified: Methods that assume equilibrium and enforce displacementMethods that assume equilibrium and enforce displacement

compatibility.compatibility.(Virtual force principle, complementary strain energy theorem, )(Virtual force principle, complementary strain energy theorem, )

Methods that assume displacement compatibility and enforceMethods that assume displacement compatibility and enforce

equilibrium.equilibrium.(Virtual displacement principle, Castiglianos 1(Virtual displacement principle, Castiglianos 1stst theorem, )theorem, )

Most important for FEA!

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1.31.3 : Energy Methods (cont.): Energy Methods (cont.)

Principle of Virtual DisplacementsPrinciple of Virtual Displacements (Elastic case):(Elastic case):(aka Principle of Virtual Work, Principle of Minimum Potential Energy)(aka Principle of Virtual Work, Principle of Minimum Potential Energy)

Elastic body under the action of body forceb

and surface stresses T.

Apply an admissible virtual displacement

Infinitesimal in size and speed

Consistent with constraints

Has appropriate continuity

Otherwise arbitrary

PVD states that for any admissibleis equivalent to static equilibrium.

Hu

e iW WH H! Hu

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External and Internal Work:External and Internal Work:

So, PVD for an elastic body takes the formSo, PVD for an elastic body takes the form

? A

_ a

.

.

e

volu e surface volu e surface

ivolu e

x x y y z z xy xy yz yz xz xz

volu e

W dV dA dV dA

W U U dV

dV

H

H H

W HI W HI W HI X HK X HK X HK

! !

! !

!

b u T u b u u

u

? A .volu e surface volu e

dV dA dV ! b u u

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Recall:Recall: Integration by PartsIntegration by Parts

In 3D, the corresponding rule is:In 3D, the corresponding rule is:

.b b

b

aa a

f x g x dx f x g x g x f x dxd d! -

, , , , , , , , , , , , .xvolume surface volume

g ff x y z x y z dV f x y z g x y z n dA g x y z x y z dV

x x

x x!

x x

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_ a

_ a

+

+

yz yz

yz yz yz y yz z

volume surface volume surface volume

xz xz xz xz xz x xz z

volume surface volume surface

dV w n dA w dV v n dA v dV y z

dV w n dA w dV u n dA ux z

X XX HK X H H X H H

X XX HK X H H X H H

x x !

x x

x x ! x x

volume

dV

_ a .

xy

xy xy

xy xy xy x xy yvolu e surface volu e surface volu e

v u

x y

dV v n dA v dV u n dA u dVx y

H HHK

X XX HK X H H X H H

x x!

x x

x x !

x x


Take a closer look at internal work:Take a closer look at internal work:

_ a .x x x x x xvolume surface volume

udV u n dA u dV

x x

H WHI W HI W H H

x x ! !

x x

_ a z z z z z zvolu e surface volu e

wdV w n dA w dV

z z

H WHI W HI W H H

x x ! ! x x

_ a y y y y y yvolume surface volume

vdV v n dA v dV

y y

WHHI W HI W H H

xx ! !

x x

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? A surface

x

xy x xz

x xy x

xz yz z

yzxz z

x

y y yz

xy y yi y

surface

z

d

z

A

z n

W n d

x y z

v x y z

A

n w

x z

u

y

X W XX

XW X

W X X H

X X W

W X H

X W

H

HX

xx

x x@ ! xx

-

x

x x x

x

x x

x

x x x

x

n u

1 4 4 4 4 4 44 2 4 4 4 4 4 4 43

? A ? A

_ a

foran

volume

i e

surface volume volume surface

volume

dV

W W dA dV dV dA

dV arbit y

u

r

v

ra

w

H

H

H

H

H

@ ! !

!

!

A

n u A u b u n u

A b u 0 u

A b 0

g

1 4 4 4 2 4 4 4 3

By reversing the steps, can show thatthe equilibrium equations imply

is called the weak form ofstatic equilibrium.

i eW WH H!

i eW WH H!

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RayleighRayleigh--Ritz MethodRitz Method : a specific way of implementing: a specific way of implementingthe Principle ofVirtual Displacements.the Principle ofVirtual Displacements.

DefineDefine total potential energytotal potential energy ; PVD is then stated; PVD is then statedasas

Assume you can approximate the displacement functions as aAssume you can approximate the displacement functions as asum ofsum ofknownknown functions withfunctions with unknownunknown coefficients.coefficients.

Write everything in PVD in terms of virtual displacementsWrite everything in PVD in terms of virtual displacements andandreal displacements. (Note: stresses are real, not virtual!)real displacements. (Note: stresses are real, not virtual!)

Using algebra, rewrite PVD in the formUsing algebra, rewrite PVD in the form

Each unknown virtual coefficient generates one equation toEach unknown virtual coefficient generates one equation tosolve for unknown real coefficients.solve for unknown real coefficients.

i eW W4 ! 0

i eW WH H H! !

1

unknown virtual coefficient * equation involving real coefficients 0.n

i ii!

!

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RayleighRayleigh--Ritz Method: ExampleRitz Method: Example

GivenGiven::An axial bar has a lengthAn axial bar has a length LL, constant modulus of elasticity, constant modulus of elasticity EE, and a, and avariable crossvariable cross--sectional area given by the function ,sectional area given by the function ,wherewhere is a known parameter. Axial forcesis a known parameter. Axial forces FF11 andand FF22 act atact at x = 0x = 0 andand x=x=LL, respectively, and the corresponding displacements are, respectively, and the corresponding displacements are uu11 andand uu22 ..

RequiredRequired:: Using the RayleighUsing the Rayleigh--Ritz method and the assumed displacementRitz method and the assumed displacementfunction , determine the equation that relates thefunction , determine the equation that relates theaxial forces to the axial displacements for this element.axial forces to the axial displacements for this element.

( ) 1 sin xLoA x A TF!

1 2

( ) 1 x xL L

u x u u!

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Solution :Solution :

1)1) TreatTreat uu11 andand uu22 as unknown parameters. Thus, the virtualas unknown parameters. Thus, the virtualdisplacement is given bydisplacement is given by

2)2) Calculate internal and external work:Calculate internal and external work:

1 2

( ) 1 x x u x u uH H H!

2 1

2 1 2 1

1 1 2 2

1 1

1 2

(no body orce terms).

( ) .

* * .

and * .

e

i x x x xbar bar

u u

x

u u u u

x x

W u u

W dV A x dx

u u ux

EH H

H H H

H W HI W HI

I

HI W

!

! !

x! ! !x

! !

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(Cont) :(Cont) :

2)2)

3)3) Equate internal and external work:Equate internal and external work:

_ a_ a _ a

_ a _ a

2 1 2 1

2 1 2 1

2 1 2 1

0

2

2 2

2 1

* * * 1 sin

* * * *

* 1 * 1 .

x

u u u u x

i ox

u u u u

o

u u u u

i o o

W E A dx

E A L

W u EA u EA

H H

H H FT

F FT T

H F

H H H

!

!

@ !

!

!

_ a _ a

2 1 2 1

1 2

2 1

2 2

1 1 2 2 2 1

2

1 11 12

2

2 22 2

* 1 * 1 .

For : 1 1 11 .

1 1For : 1

u u u u

L Lo o

u u

Loo

u u

Lo

F u F u u EA u EA

u F EA u FEA

u FLu F EA

F FT T

FT F

TFT

H H H H

H

H

!

! ! ! -

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CE 595 Section 1