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Structural Design for Cold Region Engineering. Lecture 14 Thory of Plates Shunji Kanie. Theory of Plates Kirchhoff Plate. Kirchhoff Plate. Pure Bending. Such as Bernoulli Euler. Assumptions. Isotropic and homogeneous The thickness of the plate is thin - PowerPoint PPT Presentation
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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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
Kirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
Kirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
Governing EquationGoverning Equation
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesKirchhoff PlateKirchhoff Plate
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
Theory of PlatesTheory of PlatesSolutionSolution
Simply Supported PlateSimply Supported Plate
y
x
z
0
a
b
Governing EquationGoverning Equation
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
Theory of PlatesTheory of PlatesSolutionSolution
Simply Supported PlateSimply Supported Plate
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
Theory of PlatesTheory of PlatesSolutionSolution
Simply Supported PlateSimply Supported Plate
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
Theory of PlatesTheory of PlatesSolutionSolution
Simply Supported PlateSimply Supported Plate
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
Theory of PlatesTheory of PlatesSolutionSolution
Simply Supported PlateSimply Supported Plate
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
Theory of PlatesTheory of PlatesSolutionSolution
Simply Supported PlateSimply Supported Plate
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
Theory of PlatesTheory of PlatesSolutionSolution
Simply Supported PlateSimply Supported Plate
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
Theory of PlatesTheory of PlatesSolutionSolution
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''''
Theory of PlatesTheory of PlatesSolutionSolution
Plate supported likePlate supported like
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,…….
Theory of PlatesTheory of PlatesSolutionSolution
Plate supported likePlate supported like
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
Theory of PlatesTheory of PlatesSolutionSolution
Plate supported likePlate supported like
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,…….
Theory of PlatesTheory of PlatesSolutionSolution
Plate supported likePlate supported like
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
Theory of PlatesTheory of PlatesSolutionSolution
Plate supported likePlate supported like
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
Theory of PlatesTheory of PlatesSolutionSolution
Plate supported likePlate supported like
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
Theory of PlatesTheory of PlatesSolutionSolution
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
Theory of PlatesTheory of PlatesSolutionSolution
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
Theory of PlatesTheory of PlatesSolutionSolution
Galerkin MethodGalerkin Method
y
x
z
0
a
b
simple support
arbitrary
Governing EquationGoverning Equation
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
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