school of civil engineeringspring 2007 ce 595: finite elements in elasticity instructors: amit...

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.

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


Section 1.1: Stress and StrainSection 1.1: Stress and Strain

Stress at a point Stress at a point QQ : :

0 0 0lim ; lim ; lim .yx z

x xy xzA A A

FF F

A A A

Stress matrix ( ) ; Stress vector ( ) .

x

yx xy xz

zxy y yz

xyxz yz z

yz

xz

Q Q

σ σ


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

Stresses must satisfy equilibrium equations in Stresses must satisfy equilibrium equations in pointwisepointwise manner: manner:

“Strong Form”



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

ˆ

2 2ˆ ˆ

ˆ ( ) .

ˆ ; .

x xy xz x

xy y yz y

xz yz z z

n

Q n

n

Q

Q Q

n

n n

S

σ n

S n S



Strain at a pt. Strain at a pt. QQ related to related 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



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

;

, , = , , .

, , . Also, ; .

x

D Q

x y z

Q D QD Q D dx

QD dx

Q 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



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

, ,= . . .xy xz yz

v x dx y u x y dy v u w u w vQ Q Q Q Q Q

dx dy x y x z y z



Sometimes FEA programs use Sometimes FEA programs use elasticity elasticity shearing strainsshearing strains : :

Strains must satisfy 6 Strains must satisfy 6 compatibility compatibility equationsequations::

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

1 1 12 2 2. . .xy xy xz xz yz yz

2 22

2 2E.g.: .xy yx

x y y x


Section 1.2Section 1.2 : Constitutive Theory : Constitutive Theory

For For linear elasticlinear elastic materials, stresses and materials, stresses and strains are related by the strains are related by the Generalized Hooke’s Generalized Hooke’s LawLaw : : .

o o σ C ε ε σ

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 c

c c c c c c

σ ε C

;

; .

o o

Elasticity matrix

residual stresses residual strains

σ ε


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

For For isotropic linear elasticisotropic linear elastic materials, elasticity materials, elasticity matrix takes special form:matrix takes special form:

12

12

12

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

C



Special cases of GHL:Special cases of GHL:– Plane Stress Plane Stress : all “out-of-plane” stresses assumed : all “out-of-plane” stresses assumed

zero.zero.

– Plane Strain Plane Strain : all “out-of-plane” strains assumed : all “out-of-plane” strains assumed zero.zero.

212

1 0

1 0 . 1

0 0 1

; ;

require Note: d.1

x x

y y

xy xy

z x y

E

Cσ ε

21

1 01

11 0 .

12

0 01

Note

; ;

requir: ed .

x x

y y

xy xy

z x y

E

σ ε C



Other constitutive relations:Other constitutive relations:– Orthotropic Orthotropic : material has “less” symmetry than isotropic case.: material has “less” symmetry than isotropic case.

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

– ViscoelasticViscoelastic : stresses in material depend on both strain and : stresses in material depend on both strain and strain rate.strain rate.

Asphalt, soils, concrete (creep), …Asphalt, soils, concrete (creep), …

– NonlinearNonlinear : stresses not proportional to strains. : stresses not proportional to strains.

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



Strain EnergyStrain Energy– Energy stored in an elastic material during deformation; Energy stored in an elastic material during deformation;

can be recovered can be recovered completelycompletely..

Work done during 1 1 :

.

; .

.

.

If all external work is 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



Strain Energy Density Strain Energy Density : strain energy : strain energy per per unit volumeunit volume..

In general, In general,

.final

o

o x x

Volume

U U V d

U UdV

.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


Section 1.3 Section 1.3 : Energy Methods: Energy Methods

Energy methods are techniques for satisfying Energy methods are techniques for satisfying equilibrium or compatibility on a equilibrium or compatibility on a globalglobal level rather level rather than pointwise.than pointwise.

Two general types can be identified:Two general types can be identified:– Methods that assume equilibrium and enforce Methods that assume equilibrium and enforce

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

– Methods that assume displacement compatibility and Methods that assume displacement compatibility and enforce equilibrium.enforce equilibrium.(Virtual displacement principle, Castigliano’s 1(Virtual displacement principle, Castigliano’s 1stst theorem, theorem, …)…)

Most important for FEA!


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

Principle of Virtual DisplacementsPrinciple of Virtual Displacements (Elastic (Elastic case): 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 force b 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 admissible is equivalent to static equilibrium.

u

e iW W u



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

ˆ .

.

e

volume surface volume surface

i

volume

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

volume

W dV dA dV dA

W U U dV

dV

b δu T δu b δu σ n δu

δu

ˆ .volume surface volume

dV dA dV b δu σ n δu σ δε



Recall: Recall: Integration by Parts Integration 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 dx

, , , , , , , , , , , , .x

volume 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


+

+

yz yzyz yz yz y yz z

volume surface volume surface volume

xz xzxz xz xz x xz z

volume surface volume surface

dV w n dA w dV v n dA v dVy z

dV w n dA w dV u n dA ux z

volume

dV

+ .

xy

xy xyxy xy xy x xy y

volume surface volume surface volume

v u

x y

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


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

x x x x x


udV u n dA u dV

x x

zz z z z z


wdV w n dA w dV

z z

yy y y y y


vdV v n dA v dV

y y



ˆsurface

x

xyx xz

x xy x

xz yz zyzxz z

xy y yz

xy y yi y

surfacez

d

z

A

z n

W n d

x y z

vx y z

A

n w

x z

u

y

σ n δu

ˆ ˆ

for an

volume

i e

surface volume volume surface

volume

dV

W W dA dV dV dA

dV arbit y

u

r

v

ra

w

A

σ n δu A δu b δu σ n δu

A b δu 0 δu

A b 0

• By reversing the steps, can show that the equilibrium equations imply

• is called the weak form of static equilibrium.

i eW W

i eW W



Rayleigh-Ritz MethodRayleigh-Ritz Method : a specific way of implementing : a specific way of implementing the Principle of Virtual Displacements.the Principle of Virtual Displacements.– Define Define total potential energytotal potential energy ; PVD is then stated ; PVD is then stated

as as – Assume you can approximate the displacement functions as Assume you can approximate the displacement functions as

a sum of a sum of knownknown functions with functions with unknownunknown coefficients. coefficients.– Write everything in PVD in terms of virtual displacements Write everything in PVD in terms of virtual displacements andand

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

i eW W 0i eW W

1

unknown virtual coefficient * equation involving real coefficients 0.n

i ii



Rayleigh-Ritz Method: ExampleRayleigh-Ritz Method: Example

GivenGiven:: An axial bar has a length An axial bar has a length LL, constant modulus of elasticity , constant modulus of elasticity EE, , and a variable cross-sectional area given by the function and a variable cross-sectional area given by the function , where , where ββ is a known parameter. Axial forces is a known parameter. Axial forces FF11 and and FF22 act at act at x = 0x = 0 and and x= Lx= L, respectively, and the corresponding , respectively, and the corresponding displacements are displacements are uu1 1 and and uu22 . .

RequiredRequired: : Using the Rayleigh-Ritz method and the assumed Using the Rayleigh-Ritz method and the assumed displacement function , determine the displacement function , determine the equation that relates the axial forces to the axial displacements equation that relates the axial forces to the axial displacements for this element. for this element.

( ) 1 sin xLoA x A

1 2( ) 1 x xL Lu x u u



Solution :Solution :1)1) Treat Treat uu1 1 and and uu22 as unknown parameters. Thus, the virtual as unknown parameters. Thus, the virtual

displacement is given bydisplacement is given by

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

1 2( ) 1 x xL Lu x u u

2 1

2 1 2 1

1 1 2 2

1 11 2

(no body force terms).

( ) .

* * .

and * .

e

i x x x xbar bar

u uL L Lx

u u u uL Lx x

W F u F u

W dV A x dx

uu u

x

E



(Cont) :(Cont) :2)2)

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

2 1 2 1

2 1 2 1

2 1 2 1

0

2

2 22 1

* * * 1 sin

* * * *

* 1 * 1 .

x Lu u u u x

L L Li ox

u u u u LL L o

u u u uL Li o o

W E A dx

E A L

W u EA u EA

2 1 2 1

1 2

2 1

2 21 1 2 2 2 1

21 1 1 12

22 22 2

* 1 * 1 .

For : 1 1 11 .

1 1For : 1

u u u uL Lo o

u uLo

o

u uLo

F u F u u EA u EA

u F EA u FEAu FLu F EA

school of civil engineeringspring 2007 ce 595: finite elements in elasticity instructors: amit...

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