Structural Design for Cold Region Engineering

Structural Design Structural Design forfor

Cold Region EngineeringCold Region Engineering

Lecture 14 Lecture 14 Thory of PlatesThory of PlatesShunji KanieShunji Kanie

Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate

Pure Bending Pure Bending Kirchhoff PlateKirchhoff Plate

Such as Bernoulli EulerSuch as Bernoulli Euler

Isotropic and homogeneousIsotropic and homogeneous

The thickness of the plate is thinThe thickness of the plate is thin(Comparatively to the length and width )(Comparatively to the length and width )

Linear filaments of the plateLinear filaments of the plate(Even after the deformation)(Even after the deformation) Kirchhoff hypothesisKirchhoff hypothesis

AssumptionsAssumptions


Kirchhoff PlateKirchhoff Plate

y

x

z

0

w

w

zh

x

w

Length : Length : aa in in xx direction directionWidth : Width : bb in in yy direction direction

Thickness : Thickness : hh in in zz direction direction

xx

wx

yy

wy

dxdw

DeflectionDeflection ),( yxw

Rotation angleRotation angle



y

x

z

0

w

w

zh

x

w

xdx

dwx

ydy

dwy

Displacement due to deflectionDisplacement due to deflection

Rotation angleRotation angle

x xwzu

y ywzv



Stress-Strain RelationStress-Strain Relation

yxx E

1

xyy E

1

xyxy G 1

Plane Stress !Plane Stress !

yxz E

xy

y

x

xy

y

xE

2

100

01

01

)1( 2

yxx

E

21

xyy

E

21

yx

wGzG xyxy

2

2

2

2

2

2

21 y

w

x

wEzx

2

2

2

2

21 x

w

y

wEzy

yx

wEzxy

2

1


Bending Moments and Torsional Moment are calculated at leastBending Moments and Torsional Moment are calculated at least

Sectional ForceSectional Force

2

2

h

h xx zdzM 2

2

h

h yy zdzM

2

2

2

2

h

h yxyx

h

h xyxy zdzMzdzM

2

2

h

h xzx dzQ 2

2

h

h yzy dzQ

2

2

2

2

21 y

w

x

wEzx

2

2

2

2

21 x

w

y

wEzy

yx

wEzxy

2

1

12

32

2

2 hdzz

h

h 2

3

112

EhD


02

2

2

2

2

2

h

h

zxh

h

yxh

h

x zdzz

zdzy

zdzx

y

M

x

MQ

yxxx

0

zyxyzyxy

y

M

x

MQ

yxyy

X directionX direction

y directiony direction

z directionz direction

0

zyxzyzzx 02

2

dzzyx

h

hzyzzx


z directionz direction

0

zyxzyzzx 02

2

dzzyx

h

hzyzzx

x

zx Qx

dzx

y

yzQ

ydz

y

),()2()2(2

2yxqhzhzdz

z zz

h

hzz

0),(

yxqy

Q

x

Q yx


y

M

x

MQ

yxxx

y

M

x

MQ

yxyy

Governing EquationGoverning Equation

0),(

yxqy

Q

x

Q yx

0),(22

22

2

2

yxq

y

M

yx

M

x

M yxyx

2

2

2

2

y

w

x

wDM x

2

2

2

2

x

w

y

wDM y

yx

wDMM yxxy

21



0),(22

22

2

2

yxq

y

M

yx

M

x

M yxyx

2

2

2

2

y

w

x

wDM x

2

2

2

2

x

w

y

wDM y

yx

wDMM yxxy

21

),(24

4

22

2

4

4yxq

y

w

yx

w

x

wD


Introducing LaplacianIntroducing Laplacian

),(24

4

22

2

4

4yxq

y

w

yx

w

x

wD

2

2

2

22

yx

D

yxqw

),(4

wx

DQx2

wy

DQy2


Boundary ConditionsBoundary Conditions

Simple supportSimple support

Fixed supportFixed support

Free supportFree support

0w 0xM

0w 0x

w

0xM 0xV

xV Effective transverse shearEffective transverse shearKirchhoff forceKirchhoff force

2

3

3

322

yx

w

x

wD

y

M

x

M

y

MQV

xyxxyxx

y

M

x

MQ

yxxx

Theory of PlatesTheory of PlatesSolutionSolution

Simply Supported PlateSimply Supported Plate

y

x

z

0

a

b

Assuming DeformationAssuming Deformation

b

yn

a

xmCw

sinsin

Boundary Condition?Boundary Condition?

0x ax 0 xMw

0y by 0 yMw



y

x

z

0

a

b


b

yn

a

xmCw

sinsin

),(24

4

22

2

4

4

yxqy

w

yx

w

x

wD

Assumed DeformationAssumed Deformation

b

yn

a

xmq

b

yn

a

xm

b

n

a

mCDq

mn

sinsin

sinsin

2

2

2

2

24

2

2

2

2

24

b

n

a

mCDqmn



y

x

z

0

a

b

b

yn

a

xmCw

sinsin

Assumed DeformationAssumed Deformation

2

2

2

2

24

b

n

a

mCDqmn

DeformationDeformation

b

yn

a

xm

anbm

ba

D

qw mn

sinsin

22222

44

4



y

x

z

0

a

b

DeformationDeformation

b

yn

a

xm

anbm

ba

D

qw mn

sinsin

22222

44

4

2

2

2

2

y

w

x

wDM x

2

2

2

2

x

w

y

wDM y

yx

wDMM yxxy

21

Bending & Twisting MomentsBending & Twisting Moments



y

x

z

0

a

b

b

yn

a

xm

anbm

baanbmqM mn

x

sinsin

22222

222222

2

b

yn

a

xm

anbm

babmanqM mn

y

sinsin

22222

222222

2

b

yn

a

xm

anbm

bmnaqM mn

xy

coscos

122222

33

2

Bending & Twisting MomentsBending & Twisting Moments

If we are very LUCKY enoughIf we are very LUCKY enoughb

yn

a

xmqq mn

sinsin



b

y

a

xqyxq

sinsin),( 0

y

x

z

0

a

b

0q

IfIf

qmn is successfully calculated and we can have the solutionqmn is successfully calculated and we can have the solution

Is there any good idea if q is uniformly distributed load?Is there any good idea if q is uniformly distributed load?

b

yn

a

xmqq mn

sinsin

2

2

2

2

24

b

n

a

mCDqmn



y

x

z

0

a

b

Apply Double Fourier Apply Double Fourier Expansion for qExpansion for q

b

yj

a

xiqyxq

i jij

sinsin),(

1 1

dxdyb

yn

a

xmyxq

abq

a bmn

sinsin),(

40 0

1 122222

44

4sinsin

m n

mn

b

yn

a

xm

anbm

ba

D

qw



y

x

z

0

a

b

When q is constant as When q is constant as q0q0 .),( 0 constqyxq

2016

mn

qqmn

1 1

2

2

2

2

260 sinsin

116

m n b

yn

a

xm

b

n

a

mmn

D

qw

aa

m

a

a

xm

m

adx

a

xm0

0

2cossin

dxdyb

yn

a

xmyxq

abq

a bmn

sinsin),(

40 0

You can solve the problem for any shape of load distributionYou can solve the problem for any shape of load distribution


Plate supported likePlate supported like

Assuming DeformationAssuming Deformation

Single Fourier ExpansionSingle Fourier Expansion

y

x

z

0

a

b

simple support

arbitrary

1

sin)(),(m

m a

xmyYyxw

1

sin)(m

m a

xmqxq

),(2

4

4

22

2

4

4

yxqy

w

yx

w

x

wD

mmmm qYa

mY

a

mYD

42

''2''''

D

qY

a

mY

a

mY m

mmm

42

''2''''



y

x

z

0

a

b

simple support

arbitrary

dxa

xi

a

xmqdx

a

xixq

mm

sinsinsin)(

1

im

imadx

a

xi

a

xm

02sinsin

a

m dxa

xmxq

aq

0sin)(

2

If q is constant in x directionIf q is constant in x direction

1cos2

cos2

sin)(2 0

000

mm

q

a

xm

m

aq

adx

a

xmxq

aq

aa

m

40

32

20

12

1cos

m

m

m

m

m

mq

qm

m02

1 41

m=1,3,5,…….m=1,3,5,…….



y

x

z

0

a

b

simple support

arbitraryIf q is linear in x directionIf q is linear in x direction

m=1,2,3,4,…….m=1,2,3,4,…….

xa

qxq 0)(

aa

m dxa

xmx

a

qdx

a

xmxq

aq

020

0sin

2sin)(

2

m

m

adx

a

xm

a

xmx

m

adx

a

xmx

aa

acoscoscossin

2

00

0

m

qm

m

qq mm

010 21cos

2



y

x

z

0

a

b

simple support

arbitrary

If q is linear in x directionIf q is linear in x direction

m=1,2,3,4,…….m=1,2,3,4,…….

m

qm

m

qq mm

010 21cos

2

D

qY

a

mY

a

mY m

mmm

42

''2''''SolveSolve

If q is constant in x directionIf q is constant in x direction

mq

qm

m02

1 41

m=1,3,5,…….m=1,3,5,…….



y

x

z

0

a

b

simple support

arbitrary

D

qY

a

mY

a

mY m

mmm

42

''2''''SolveSolve

General SolutionGeneral Solution

0''2''''42

mmm Y

a

mY

a

mY

pym eY

024

22

4

a

mp

a

mp

Characteristic EquationCharacteristic Equation

0

222

a

mp

a

mp



y

x

z

0

a

b

simple support

arbitraryGeneral SolutionGeneral Solution

ym

ym

ym

ymm yeCeCyeCeCY 4321

yyDyCyyByAY mmmmm coshsinhsinhcosh

Singular SolutionSingular Solution

Dm

qDm

q

Ya

mY

a

mY

m

m

mmm

01

02)1(

42

2)1(

4)1(

''2''''

mY mFshould be constant such asshould be constant such as



y

x

z

0

a

b

simple support

arbitraryGeneral SolutionGeneral Solution

ym

ym

ym

ymm yeCeCyeCeCY 4321

yyDyCyyByAY mmmmm coshsinhsinhcosh

Singular SolutionSingular Solution

Dm

qDm

q

Fa

m

m

m

m

01

02)1(

4

2)1(

4)1(

Dm

aq

Dm

aq

Fm

m

m

5

401

5

402

)1(

2)1(

4)1(

1

1

sincoshsinhsinhcosh

sin)(),(

mmmmmm

mm

a

xmFyyDyCyyByA

a

xmyYyxw

SolutionSolution


Difference MethodDifference Method

y

x

z

0

a

b

simple support

arbitrary

x

ww

x

w mm

211

211

2

2 2

x

www

x

w mmm

32112

3

3

2

22

x

wwww

x

w mmmm

42112

4

4 464

x

wwwww

x

w mmmmm

44

4 464

y

wwwww

y

w pnmlk

22

11111122

4 42

yx

wwwwwwwww

yx

w mnlmmnnll


Difference MethodDifference Method

y

x

z

0

a

b

simple support

arbitrary

Governing EquationGoverning Equation),(2

4

4

22

2

4

4

yxqy

w

yx

w

x

wD

22

2222

1111

211

222

2

11468

yxD

q

wwwwwwww

wwwww

mmpknnll

nlmmm

xy

Simple supportSimple support

Fixed supportFixed support

0sw 11 ss ww

0sw 11 ss ww


Galerkin MethodGalerkin Method

y

x

z

0

a

b

simple support

arbitrary


nmm n

mn YXCw

1 1

Assume ApproximationAssume Approximation

0),(

0 04

dxdyYX

D

yxqw nm

a b

0),(4

D

yxqw

Weighted ResidualWeighted Residual

Same with Double FourierSame with Double Fourier

Structural Design for Cold Region Engineering

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