Mechanics of Thin Structure Lecture 15 Wrapping Up the Course Shunji Kanie.

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


v

x




0)( xqdx

dQ

02

)(

dxx

Mdxdxxqdxdx

x

QQ

Qdx

dM )(

2

2

xqdx

Md

VerticalVertical

MomentMoment



v

x




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



v

x




dx

dvyyyu )(

2

2)(

dx

vdy

dx

duyx

2

2

2

2

dx

vdEIydA

dx

vdyEydAEM

AA

xx

CompatibilityCompatibilityu

y

dx

dv


v

x




2

2

2

2

dx

vdEIydA

dx

vdyEydAEM

AAxx

)(

2

2

xqdx

Md

0)(4

4

xqdx

vdEIGoverning Governing

EquationEquation


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;


EnergyEnergy


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


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

)'',,(


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


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


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'


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


AssumptionAssumption

xL

av

sin~

0sin4

xx

vdxqvdxxL

aL

EI

xL

v sin


v

x



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


v

x



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



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


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


xL

av

sin~


x

LLEIa

dx

vdEI

sin

~ 4

4

4 EstimatedEstimated


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


v

x

x

dxqxLL

EIaaa

24

sin

0sinsin244

x

dxxLL

EIqxLL

EIaa

0sinsin4

x

dxxL

qxLL

EIaa


v

x



xL

av

sin~

0sin4

xx

vdxqvdxxL

aL

EI

xL

v sin


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


xL

av

sin~

xL

v sin


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


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~


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


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


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


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


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


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


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

Mechanics of Thin Structure Lecture 15 Wrapping Up the Course Shunji Kanie.

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Transcript of Mechanics of Thin Structure Lecture 15 Wrapping Up the Course Shunji Kanie.