Mechanics Mechanics of of

Thin StructureThin Structure

Lecture 15 Lecture 15 Wrapping Up the CourseWrapping Up the CourseShunji KanieShunji Kanie

Mechanics of Thin StructureMechanics of Thin Structure

What you learned are;What you learned are;

Introduction for Linear ElasticityIntroduction for Linear ElasticityStress and Strain with 3D General Stress and Strain with 3D General ExpressionsExpressionsPlane Stress and Plane StrainPlane Stress and Plane StrainPrinciple of EnergyPrinciple of EnergyPrinciple of Virtual WorkPrinciple of Virtual WorkCalculus of VariationsCalculus of VariationsTheory of BeamsTheory of BeamsTheory of PlatesTheory of Plates

ExampleExamplePure bending problemPure bending problem

v

x

Solve the Solve the problemproblem

EquilibriumEquilibrium

CompatibilityCompatibility

EnergyEnergy

Governing Governing EquationEquation

ExampleExamplePure bending problemPure bending problem

v

x

Solve the Solve the problemproblem

EquilibriumEquilibrium

CompatibilityCompatibility

0)( xqdx

dQ

02

)(

dxx

Mdxdxxqdxdx

x

QQ

Qdx

dM )(

2

2

xqdx

Md

VerticalVertical

MomentMoment

EquilibriumEquilibrium

ExampleExamplePure bending problemPure bending problem

v

x

Solve the Solve the problemproblem

EquilibriumEquilibrium

CompatibilityCompatibility

02

2

2

2

2

2

h

h

zxh

h

yxh

h

x zdzz

zdzy

zdzx

dx

dMQ xx

X directionX direction

0

zyxzxyxx

2

2

2

2

2

2

h

h zx

h

hzx

h

hzx dzzzdzz

EquilibriumEquilibrium

ExampleExamplePure bending problemPure bending problem

v

x

Solve the Solve the problemproblem

EquilibriumEquilibrium

CompatibilityCompatibility

dx

dvyyyu )(

2

2)(

dx

vdy

dx

duyx

2

2

2

2

dx

vdEIydA

dx

vdyEydAEM

AA

xx

CompatibilityCompatibilityu

y

dx

dv

ExampleExamplePure bending problemPure bending problem

v

x

Solve the Solve the problemproblem

EquilibriumEquilibrium

CompatibilityCompatibility

2

2

2

2

dx

vdEIydA

dx

vdyEydAEM

AAxx

)(

2

2

xqdx

Md

0)(4

4

xqdx

vdEIGoverning Governing

EquationEquation

ExampleExamplePure bending problemPure bending problem

v

x

u

y

dx

dv

dx

dvyyyu )(

2

2)(

dx

vdy

dx

duyx

Energy should be Minimum, Energy should be Minimum, so that Energy is calcualted such as;so that Energy is calcualted such as;

CompatibilityCompatibility

EnergyEnergy

ExampleExamplePure bending problemPure bending problem

v

x

ll

lx

lxx

dxdx

vdEIdxbdy

dx

vdEy

dxbdyEdxbdyU

0

2

2

2

0

2

2

22

02

0

22

1

2

1

2

1

bdyxx21

Energy for sectionEnergy for section

2

2)(

dx

vdy

dx

duyx

ExampleExamplePure bending problemPure bending problem

v

x

Potential EnergyPotential Energy

l

vdxxqW0

)(

l

dxvxqdx

vdEIWU

0

2

2

2)(

2

You have the Functional You have the Functional !!!!

l

dxvvxFvJ0

)'',,(

ExampleExamplePure bending problemPure bending problem

v

x

l

dxvxqvEI

WU0

2 )(''2

You have the Functional You have the Functional !!!!

l

dxvvxFvJ0

)'',,(

Apply the Euler’s Apply the Euler’s EquationEquation

0''' 2

2

v

F

dx

d

v

F

dx

d

v

F

ExampleExamplePure bending problemPure bending problem

v

x

0''' 2

2

v

F

dx

d

v

F

dx

d

v

F

qv

F

0'

v

F''

''EIv

v

F

0''4

4

2

2 q

dx

vdEIEIv

dx

dq

Governing Equation for pure bending Governing Equation for pure bending beambeam from the Euler’s from the Euler’s

EquationEquation

l

dxvxqvEI

0

2 )(''2

ExampleExamplePure bending problemPure bending problem

v

x

Shearing forceShearing force

Boundary ConditionBoundary ConditionSee eq. (6-31)See eq. (6-31)

xQEIvdx

d

v

F

dx

d

v

F

'''''

xMEIvv

F

''''

Bending Bending MomentMoment

v

dx

dvv'

ExampleExamplePure bending problemPure bending problem

v

x

vdxqEIvdx

dMvQJ

x

x

x

xx

x

xx

2

1

2

1

2

1''

2

2

Mechanical Mechanical Boundary Boundary ConditionCondition

Geometrical Geometrical Boundary Boundary ConditionCondition

or

or

xQ

vxM

dx

dvv'

ExampleExampleSolution 1 : Direct IntegrationSolution 1 : Direct Integration

v

x

04

4

qdx

vdEI

x

Cdxxqdx

vdEI 13

3

)(

212

2

)( CxCdxdxxqdx

vdEI

x x

01302.0384

5 4

EI

qL

ExampleExampleSolution 2 : Virtual WorkSolution 2 : Virtual Work

v

x

04

4

qdx

vdEI

EquilibriuEquilibriumm

No Energy produced if the Equilibrium is No Energy produced if the Equilibrium is satisfiedsatisfied

04

4

x

vdxqdx

vdEI

Virtual DisplacementVirtual Displacement

Virtual WorkVirtual Work

What is the requirement for the virtual What is the requirement for the virtual displacement?displacement?

ExampleExample Solution 2 : Virtual Work Solution 2 : Virtual Work

v

x

04

4

x

vdxqdx

vdEI

Virtual DisplacementVirtual Displacement

AssumptionAssumption

xL

av

sin~

0sin4

xx

vdxqvdxxL

aL

EI

xL

v sin

ExampleExample Solution 2 : Virtual Work Solution 2 : Virtual Work

v

x

Virtual DisplacementVirtual Displacement

AssumptionAssumption

xL

av

sin~

0sin4

xx

vdxqvdxxL

aL

EI

xL

v sin

0sinsin24

xx

dxxL

qdxxL

aL

EI 0sin2cos1

2

14

xx

dxxL

qdxxL

aL

EI

ExampleExample Solution 2 : Virtual Work Solution 2 : Virtual Work

v

x

Virtual DisplacementVirtual Displacement

AssumptionAssumption

xL

av

sin~

xL

v sin

0sin2cos12

14

xx

dxxL

qdxxL

aL

EI

022

14

L

qLL

aEI

EI

qL

EI

qLa

44

01307.0306

4

5

4

4L

EI

qa

EI

qL

EI

qL 44

01302.0384

5

ExampleExampleSolution 2 modifiedSolution 2 modified

v

x

04

4

x

vdxqdx

vdEI

Virtual DisplacementVirtual Displacement

AssumptionAssumption

mm x

L

mav

sin~

0sin4

xx

vdxqvdxxL

aL

EI

xL

v sin

0sin4

xm x

m vdxqvdxxL

ma

L

mEI

xL

nvn

sin

ExampleExampleSolution 2 modifiedSolution 2 modified

0sin4

xm x

m vdxqvdxxL

ma

L

mEI

xL

nvn

sin

0sinsinsin4

xm x

m dxxL

nqdxx

L

nx

L

ma

L

mEI

x

dxxL

nq

sin

It’s same as the solution introduced for Plate It’s same as the solution introduced for Plate Theory Theory

Fourier TransformFourier Transform

ExampleExampleSolution 3 :Point CollocationSolution 3 :Point Collocation

v

x0

4

4

x

vdxqdx

vdEI

Dirac Delta FunctionDirac Delta Function

Assume you would like to have the value at Assume you would like to have the value at the centerthe center

2Lv

2021

2 Lx

LxL

2

4

4

4

4

Lxx

qdx

vdEIdxq

dx

vdEI

ExampleExampleSolution 3 :Point CollocationSolution 3 :Point Collocation

v

x

0

2

4

4

4

4

Lxx

qdx

vdEIdxq

dx

vdEI

AssumptionAssumption

xL

av

sin~

0sin

2

4

2

4

4

LxLx

qxLL

aEIqdx

vdEI

EI

qL

EI

qLa

44

01027.0

EI

qL

EI

qL 44

01302.0384

5

ExampleExampleSolution 4 :Least Square MethodSolution 4 :Least Square Method

v

x

04

4

qdx

vdEI

Error should be Error should be minimumminimum

AssumptionAssumption

xL

av

sin~

EquilibriumEquilibrium

x

LLEIa

dx

vdEI

sin

~ 4

4

4 EstimatedEstimated

ExampleExampleSolution 4 :Least Square MethodSolution 4 :Least Square Method

qxLL

EIaqdx

vdEI

sin

~ 4

4

4

Squared ErrorSquared Error

ErrorError

x

dxqxLL

EIa

24

sin

In order to expect the Error In order to expect the Error minimized, an appropriate value for minimized, an appropriate value for a must bea must be

x

dxqxLL

EIaaa

24

sin

ExampleExampleSolution 4 :Least Square MethodSolution 4 :Least Square Method

v

x

x

dxqxLL

EIaaa

24

sin

0sinsin244

x

dxxLL

EIqxLL

EIaa

0sinsin4

x

dxxL

qxLL

EIaa

ExampleExample Solution 2 : Virtual Work Solution 2 : Virtual Work

v

x

Virtual DisplacementVirtual Displacement

AssumptionAssumption

xL

av

sin~

0sin4

xx

vdxqvdxxL

aL

EI

xL

v sin

ExampleExampleSolution 4 :Least Square MethodSolution 4 :Least Square Method

v

x

0sinsin4

x

dxxL

qxLL

EIaa

0sinsin24

xx

dxxL

qdxxL

aL

EI

ExampleExampleSolution 5 :Galerkin Method 1Solution 5 :Galerkin Method 1

v

x0

4

4

x

vdxqdx

vdEI

Starting Equation is Starting Equation is SameSame

Weighting FunctionWeighting Function

AssumptionAssumption

xL

av

sin~

xL

v sin

ExampleExampleSolution 5 :Galerkin Method 2Solution 5 :Galerkin Method 2

v

x0

4

4

x

vdxqdx

vdEI

Starting Equation is Starting Equation is SameSame

xx

L

x

vdxqdxx

v

dx

vdEIv

dx

vdEIvdxq

dx

vdEI

3

3

03

3

4

4

xx

LL

vdxqdxx

v

dx

vdEI

x

v

dx

vdEIv

dx

vdEI

2

2

2

2

02

2

2

2

03

3

ExampleExampleSolution 5 :Galerkin Method 2Solution 5 :Galerkin Method 2

xx

LL

vdxqdxx

v

dx

vdEI

x

v

dx

vdEIv

dx

vdEI

2

2

2

2

02

2

2

2

03

3

02

2

2

2

xx

vdxqdxx

v

dx

vdEI

dcxbxaxv 23~ cxbxaxv 23~

0~ 23 cLbLaLv bLaLc 2

LxbxLxaxv 22~

ExampleExampleSolution 5 :Galerkin Method 2Solution 5 :Galerkin Method 2

02

2

2

2

xx

vdxqdxx

v

dx

vdEI

LxbxLxaxv 22~

baxdx

vd26

2

2

221 Lxxv Lxxv 2

xdx

vd6

21

2

2

22

2

dx

vd

cbxaxdx

vd 23

~2

ExampleExampleSolution 5 :Galerkin Method 2Solution 5 :Galerkin Method 2

02

2

2

2

xx

vdxqdxx

v

dx

vdEI

baxdx

vd26

2

2

22

1 Lxxv

Lxxv 2

xdx

vd6

21

2

222

2

dx

vd

0626 22 xx

dxLxqxdxxbaxEI

0226 xx

dxLxxqdxbaxEI

ExampleExampleSolution 5 :Galerkin Method 2Solution 5 :Galerkin Method 2

0626 22 xx

dxLxqxdxxbaxEI

0226 xx

dxLxxqdxbaxEI

024

61244

23

LL

qbLaLEI

023

4633

2

LL

qbLaLEI

qL

bLaLEI4

6124

23

qL

bLaLEI6

463

2 EI

qLb

24

2

0a

LxbxLxaxv 22~

EI

qL

EI

qLv

44

0104166.096

EI

qL

EI

qL 44

01302.0384

5

LxxEI

qLv

24~

2

ExampleExampleSolution 5 :Galerkin Method 3Solution 5 :Galerkin Method 3

dcxbxaxv 23~

223 ~0~ vcLbLaLv

12 bLaLc

cbxaxdx

vd 23

~2

222 223 bLaLcbLaL

221

La

L

b 212

xxL

xL

v 12213

221 2~

2

2

2

3

1

2

2

3

2~

L

x

L

xx

L

x

L

xv

1~~ vdv

ExampleExampleSolution 5 :Galerkin Method 3Solution 5 :Galerkin Method 3

2

2

2

3

1

2

2

3

2~

L

x

L

xx

L

x

L

xv

22122

2 126

146

~

LL

x

LL

x

dx

vd

02

2

2

2

xx

vdxqdxx

v

dx

vdEI

ExampleExampleSolution 5 :Galerkin Method 3Solution 5 :Galerkin Method 3

012

24 2

21

q

L

LLEI

012

42 2

21

q

L

LLEI

qL

LEI

12

36 2

1 EI

qL

24

3

1 EI

qL

24

3

2

ExampleExampleSolution 5 :Galerkin Method 3Solution 5 :Galerkin Method 3

EI

qL

24

3

1 EI

qL

24

3

2

2

2

2

3

1

2

2

3

2~

L

x

L

xx

L

x

L

xv

EI

qLLLLLLv

9648228~

4

21

EI

qL

EI

qLv

44

0104166.096

EI

qL

EI

qL 44

01302.0384

5

Mechanics of Thin StructureMechanics of Thin Structure

What you learned are;What you learned are;

Introduction for Linear ElasticityIntroduction for Linear ElasticityStress and Strain with 3D General Stress and Strain with 3D General ExpressionsExpressionsPlane Stress and Plane StrainPlane Stress and Plane StrainPrinciple of EnergyPrinciple of EnergyPrinciple of Virtual WorkPrinciple of Virtual WorkCalculus of VariationsCalculus of VariationsTheory of BeamsTheory of BeamsTheory of PlatesTheory of Plates