Shell theory
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This document provides information on shell theory and the modeling of various shell structures. It begins by defining a shell as a thin-walled three-dimensional structure. It then discusses shell thickness and functions. The document also covers modeling shells using the static-geometric hypothesis of Kirchhoff-Love and thin shell theory. It provides the mathematical foundations of differential geometry as applied to shell surfaces. Finally, it gives examples of generating surfaces for different shell types like ellipsoids, hyperboloids, cones, paraboloids, cylinders, and applies meshing techniques.
![In the distance z form midplane:
⎪⎬ ⎫
R = R +
z
,
( )
( ) ⎪⎭
ds A d
= α
z ( ) ( )
1 1
R = R +
z
,
z
2 2
⎪⎬ ⎫
,
,
z z
1
ds B d
= β
( ) ( ) ⎪⎭
2
z z
A A z
( )
⎫
⎪ ⎪
⎬
⎞
1 ,
⎞
⎛
⎛
R
1
z
B B z
( ) ⎪ ⎪
⎭
⎟ ⎟⎠
⎜ ⎜⎝
= +
⎟ ⎟⎠
⎜ ⎜⎝
= +
1 .
R
2
z
z
ε = ε + κ
( )
( )
( )
⎫
u u zV
z ( )
⎪⎭
⎪⎬
α α α
z
ε = ε + κ
z
β β β
,
,
z
2 .
ε = ε + κ
z
αβ αβ αβ
z
α α
u u zV
( )
( )
⎫
⎪⎭
⎪⎬
= +
= +
z
β β
u =
u
.
,
1
,
2
z z z
( )
u
1 1 ,
( )
( )
( ) ( )
( )
( )
( )
( )
1
u
1 1 ,
( )
( )
( ) ( )
( )
( )
( )
( )
B
( )
( )
( )
( )
∂
∂
A
( )
( )
⎫
⎪ ⎪ ⎪ ⎪
⎬
( )
( ) ⎪ ⎪ ⎪ ⎪
⎭
⎞
⎟ ⎟
⎠
⎛
∂β
⎜ ⎜
⎝
∂
⎞
+ ⎟ ⎟
⎠
⎛
⎜ ⎜
⎝
∂
∂α
ε =
+
∂α
+
∂β
∂
ε =
+
∂β
+
∂α
∂
ε =
z
z
u
u
β α
αβ
α
β
β
β
α
α
.
2
z
z
z
z
z
z
z
z z
z
z
z z
z
z
z
z z
z
z
z z
z
z
z
A
B
B
A
R
u
B
A B
u
B
R
u
A
A B
u
A
⎫
⎪ ⎪ ⎪
⎬
⎪ ⎪ ⎪
⎭
⎞
⎟⎠
⎛
⎜⎝
1 1 ,
1 1 ,
∂
1
∂α
∂
⎞
∂
+ ⎟⎠
⎛
∂β
⎜⎝
∂
α
κ =
∂α
+
∂
∂
∂β
κ =
∂β
+
∂α
κ =
αβ
β
V
A
V
B
2 2 1
.
2
2
1
A
B
B
A
B V
AB
V
B
AV
AB
V
A
⎫
⎪ ⎪
⎬
⎪ ⎪
⎭
1 ∂
,
∂
z
∂β
α
V u
u
= −
∂α
= −
β
1 .
2
2
1
1
z
u
R B
V
u
R A
Hooke’s law
E E z
( ) ( ) ( ) [ ( )]
z z
α α β α β α β
2 2
E E z
( ) ( ) ( ) [ ( )]
⎫
⎪ ⎪ ⎪
⎬
z z
β β α β α β α
( ) ( ) ( )( ) ⎪ ⎪ ⎪
⎭
2 .
ε + κ
+ ν
ε =
+ ν
τ = τ =
ε + νε + κ + νκ
− ν
ε + νε =
− ν
σ =
ε + νε + κ + νκ
− ν
ε + νε =
− ν
σ =
αβ βα αβ αβ αβ
2 1 2 1
,
1 1
,
1 1
2 2
E E z
z
Page 27](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-28-320.jpg)
![sikSakrN I X=0: ecjBIsmIkarlMnwgTI 1 eyIg)an
∫ ν
u C N
x = = → = + −
u dx
ν
+
x
x z
x
z
R
u C C x
C
du
dx R
0
6 5 6
CMnYscUleTAkñúgsmIkarTI 2 eyIgTTYl)an
ν
( )3 z + γ = − N
1 .
d u x
u Z
4 4 ,
4
4
RD
D
dx
z
− ν
2 2
2
4
R h
γ =
Common solution:
u 0
= e −γ x (C cos γ x + C sin γ x) + e γ x (C cos γ x + C sin γ x)
z 1 2 3 4
~
Particular solution: u (x) z
sMrab;krNI]TahrNxagmux KWecjBIlkçx½NÐ)atxagelITMenr eyIgrkeXIj
0 0 6 N = → C = x
( ) ( )
u q l x
D
u q l x
D
d u
dx
z z
z
4
4
4
4
4
4 ~
γ
−
→ =
−
+ γ =
( ) ( ) ( )
−
u = e −γ x C cos γ x + C sin γ x + e γ x C cos γ x + C sin
γ x +
q l x
z 1 2 3 4 4
γ
4 D
Boundary conditions:
x u u duz
= 0 : = 0, = 0, = 0.
dx
x z
3
x l M D d u Q D d u
z
: 0, 0. 2
3
2
= = = = =
d x
dx
x
z
x
∫ ν
= −
x
x zu dx
R
u C
0
5
Page 36](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-37-320.jpg)
![E.Meissner’s unknowns:
⎞
duz
1 ,
χ = − R Q
α α = ψ ⎟⎠
⎛ +
⎜⎝
α
R
u
d
2
1
ecjBIsmIkarbMErbMrYlragxagelI b¤ecjBIlkçx½NÐCab; edayeyagelIc,ab;
Hooke eyIg)an ³
1 ( cotg )
,
2
du
1 ⎛ +
,
N N
N N
α β α
( )
( )
⎞
M M
12
1 ⎡
1 ⎛
, M M
u du
d
cotg . 12
1 2
3
1 1
3
1
⎞
⎟⎠
⎛
⎜⎝
u du
α
−
α
=
− ν
−
⎤
⎥⎦
⎢⎣
⎞
⎟⎠
⎜⎝
α
−
α
=
− ν
−
⎟⎠
⎜⎝
α
=
− ν
= + α = ε
− ν
α
β α
α
α β
α β
β α
d
Eh R R
d
d R
Eh R
u
d
Eh R
u u
Eh R
z
z
z
z
ecjBIsmIkarBIrmun eyIgTTYl)an ³
[( R R ) N ( R R ) N
] α α β
du
α − α = + ν − + ν
α
Eh
u
d
1 2 2 1
cotg 1
eFVIDIepr:g;EsülelIsmIkarTImYy eyIgnwgman ³
⎤
R
u u d
⎡ − ν
α
( ) ( )
cotg ,
⎤
α β α
R
d
⎡ − ν
α
( ) .
u du
sin
cotg
2
2
2
⎥⎦
⎢⎣
=
α
+
α
α −
d
du
α
⎥⎦
⎢⎣
+ α =
α
β α
α α
N N
Eh
d
d
d
N N
Eh
d
d
z
z
ecjBIsmIkarBIrxagelIenH eyIgTTYl)an ³
1
R
R R N R R N d
u duz
cotg [( ) ( ) ] ( ) . 2
1 2 2 1
⎤
⎥⎦
⎡ − ν
α
⎢⎣
+ ν − + ν −
α
=
= χ =
α
−
α β β α
α
N N
Eh
d
Eh
R
d
Page 57](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-58-320.jpg)
![müa:geTot eyIgGacsresr)anfa
d
R
cotg 0
, 1 ,
α α β β
d
R
1 χ
, cotg ,
α β
d R
1 2
0
1
⎞
⎛
⎞
ψ
d
d
R
1 χ
cotg , cotg 1 ,
1 2 2 1
2
⎟ ⎟⎠
⎜ ⎜⎝
χ
α
χ + ν
α
− = ⎟ ⎟⎠
⎛
⎜ ⎜⎝
χ
α
+ ν
α
= −
χ
α
κ =
α
κ =
+
α
ψ + = −
α
= −
α β
d
R R
M D
d R
M D
N
d
N N
R
N
Edl 0 , 0 α β N N CakMlaMgEkg tamRTwsþIKμanm:Um:g; (zero moment
theory of shells) Ed;lmanragdUcteTA ³
⎤
sin ( cos sin ) ,
= ∫ α
1
α C R R Z X d
sin
⎡
2 1 2
1
2
0
⎥ ⎥⎦
⎢ ⎢⎣
+ α α − α α
α
α
R
N
⎞
.
1
N R Z N
2
0
⎟ ⎟⎠
⎛
⎜ ⎜⎝
= − α
β R
bnÞab;BICMnYstMélkMlaMgEkg cUleTAkñúgsmIkarlMnwgBIrdMbUg eyIgeXIjfa
vaRtUv)anepÞógpÞat; . rIÉsmIkarTIbI rYmCamYynwglkçx½NÐCab; begáIt)anCa
cotg 3
R
2
R
2
⎤
⎤
⎡
⎞
⎛
α
χ
dh
R
R
dh
d
R
d
3 cotg cotg ,
cotg
R
2
R
1
R
2
1
⎤
χ
d
ψ
⎤
⎡
⎞
⎛
α
ψ
dh
dh
d
cotg cotg ( ),
1
2
d
R
d
R
R
1
2
1
2
1
2
2
2
⎡
2
1
2 1
2
1
2
1
1
2
2
1
α Φ + χ = ψ ⎥⎦
⎡
⎢⎣
α − ν
α
ν
− α −
−
α
⎥⎦
⎢⎣
α
− α + ⎟ ⎟⎠
⎜ ⎜⎝
+
α
ψ − = χ ⎥⎦
⎢⎣
+ α
α
ν α
− ν −
−
α
⎥⎦
⎢⎣
α
+ α + ⎟ ⎟⎠
⎜ ⎜⎝
+
α
EhR
d
R h
d
d
R h
R
R
d
d
R
D
R
d
h
d
d
R h
R
R
d
d
R
where
( ) h d
R
( ) cotg [( ) ( ) 0 ].
Φ α = N N R R N R R N
2 1
0
1 2
⎡ − ν
α
2 0 0
⎤
β α α β ν + − ν + α − ⎥⎦
⎢⎣
h
d
Page 58](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-59-320.jpg)
![Case h=const:
( ) ν
ν
χ = − 1 ψ , ( ψ ) +
ψ = χ + 1 Φ ( α
).
1 1 1
χ −
R
Eh
R
L
R D
L
where
d
R
R
⎛
α
d
d
L R
⎤
⎡
⎞
1 cotg
2 L cotg L
2
( L
) (L)
2
2
1
2
1
1
2
2
2
1
d R
R
R
d
d R
R
α
−
α ⎥⎦
⎢⎣
α + ⎟ ⎟⎠
⎜ ⎜⎝
+
α
=
ecjBIsmIkarxagelI eyIgGacTaj)anfa
( ) ( ) ( )
( ) ( ) ( )⎪ ⎪
⎫
⎪ ⎪
⎬
⎭
Φ α
⎞
ν
− ⎟ ⎟⎠
Eh
⎛
⎜ ⎜⎝
−
⎞
ν
2
ψ = ψ
⎞
ν
− ⎟ ⎟⎠
⎞
⎛ χ
⎛ Φ
⎜ ⎜⎝
ψ −
ν
ψ −
Φ α
− χ ⎟ ⎟⎠
⎛
⎜ ⎜⎝
−
ν
χ =
ν
+ ⎟ ⎟⎠
⎜ ⎜⎝
χ − ν
,
,
2
1
2
2
1
1 1 1
1
2
1
1 1
Eh
D R
R
L
R R R
LL L
D R D
R
L
R R
LL L
ebI]bmafa
L ( )
, 1
ϕ
ϕ χ = −
ν
ψ = ϕ −
R D
1
enaHsmIkarTI1 nwgepÞógpÞat; ehIysmIkarTI2 nwgTTYl)anragCa
( ) ⎞
( ) ⎛ ν
⎞
( )
ν
+ ⎟ ⎟⎠
⎛ ϕ
LL L Φ α
1
2
2
1
L Eh
R R
= ϕ ⎟ ⎟⎠
⎜ ⎜⎝
ϕ + −
1 1 D R R
⎜ ⎜⎝
ϕ − ν
For spherical, toroidal, conical, cylindrical shells: R1=const. So,
( ) ( )
LL Φ α
1
2
R
ϕ + μ ϕ =
where
( ) 2 2
( ) μ = − 1 12 1 b
,
12 1 1
.
2
1
2
Eh =
2
1
2
2
1
2
R
h
R
D R R
− ν
≈
ν
− ν
b R
2
2
1
2
2
h
=
Page 59](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-60-320.jpg)
![smIkarcugeRkayenH Gacsresr)aneTACa
[ ( ) ][ ( ) ] ( ),
1 R
L i L i
Φ α
L + μ ϕ − μ =
b¤k¾ [ ( ) ] [ ( ) ] ( ),
1 R
L L i i L i
Φ α
ϕ + μ − μ ϕ + μ =
dMeNaHRsayrYmrbs;smIkarTaMgenH GacTTYl)anCaragkMpøic .
krNIEs‘Vr R1=R2=R smIkaredImrbs;smIkarDIepr:g;EsülxagelI manragCa
( ∇ 2
+ μ ) ϕ = 0, ( ∇ + μ ) ϕ =
0, 1 2 2
22
1 1
where
RL d
d
μ = 1 + bi = ζ ( ζ + 1), μ = 1 − bi = ζ ( ζ +
1), 1 1 1 2 2 2 ∇ = − = L L L
( )( ) ,
1 cotg 2 2
sin
2
2
1 L L
+ α
−
α
α
α
d
d
dMeNaHRsayBiessrbs;smIkarDIepr:g;EsülxagelI Gacrk)anecjBIsmIkar
( ) ( ).
L i i Φ α
b
ϕ + μϕ =
smIkaredImk¾Gacsresr)anCarag
⎫
⎤
d
cotg 1 1
( )
( ) ⎪ ⎪
⎪ ⎪
⎬
⎭
⎤
= ϕ ⎥⎦
d
cotg ⎡
1 1
⎢⎣
α
ϕ
ϕ
+ ζ ζ + −
α
+ α
ϕ
1
ϕ
α
= ϕ ⎥⎦
⎢⎣⎡
α
+ ζ ζ + −
α
+ α
α
0.
sin
0,
sin
2 2 2 2
2
2
2
2
1 1 2 1
1
2
2
d
d
d
d
d
d
smIkarDIepr:g;EsülxagelIenH GacGaMgetRkal)an edayeRbIGnuKmn_
Legendre .
Page 60](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-61-320.jpg)
![Example. Spherical Cupola
, const 1 2 R = R = R h =
Equations:
⎫
L R
( )
( ) ( )⎪⎭
⎪⎬
χ − νχ = − ψ
1
L EhR
ψ + νψ = χ +Φ α
,
,
1
D
d
L RL d
L L
( ) ( ) ( )
( ) (1 ) .
cotg cotg ,
= = L
R dZ
2 2
2
2
2
1
R X
d
d
d
+ + ν
α
Φ α =
α − α
α
+
α
L L
where
Common solutions:
C X C X C Y C Y
ψ = + + +
0 1 1 2 2 3 1 4 2
1
,
[ X ( C C ) X ( C C
)
( ) ( )] 1 1 3 2 2 4
χ = λ + ν + λ + ν +
0 1 3 1 2 4 2
EhR
Y C C Y C C
+ − λ + ν + − λ + ν
Legendre functions:
,
⎛ π
2 8
sin
⎛ π
2 8
cos
1 3 2 cotg
8
λ
4 sin
2
⎤
⎡
⎞
⎞
⎞
⎛
1 ⎥ ⎥⎦
⎢ ⎢⎣
⎟ ⎟
⎠
⎜ ⎜
⎝
−
λ
α − ⎟ ⎟
⎠
⎜ ⎜
⎝
−
λ
α ⎟ ⎟
⎠
⎜ ⎜
⎝
λ
α
−
π α
≈
λ
α
X e
,
⎛ π
2 8
sin
1 3 2 cotg
8
⎛ π
2 8
cos
λ
4 sin
2
⎤
⎡
⎞
⎞
⎛
⎞
1 ⎥ ⎥⎦
⎢ ⎢⎣
⎟ ⎟
⎠
⎜ ⎜
⎝
−
λ
α ⎟ ⎟
⎠
⎜ ⎜
⎝
λ
α
− + ⎟ ⎟
⎠
⎜ ⎜
⎝
−
λ
α
π α
≈
λ
α
Y e
,
⎛ π
2 8
sin
1 3 2 cotg
8
⎛ π
2 8
cos
λ
sin
2
⎤
⎡
⎞
⎞
⎛
⎞
2 ⎥ ⎥⎦
⎢ ⎢⎣
⎟ ⎟
⎠
⎜ ⎜
⎝
+
λ
α ⎟ ⎟
⎠
⎜ ⎜
⎝
λ
α
+ − ⎟ ⎟
⎠
⎜ ⎜
⎝
+
λ
α
π α
≈
λ
−α
X e
⎤
.
⎛ π
2 8
sin
⎛ π
2 8
cos
1 3 2 cotg
8
λ
sin
2
⎡
⎞
⎞
⎞
⎛
2 ⎥ ⎥⎦
⎢ ⎢⎣
⎟ ⎟
⎠
⎜ ⎜
⎝
+
λ
α + ⎟ ⎟
⎠
⎜ ⎜
⎝
+
λ
α ⎟ ⎟
⎠
⎜ ⎜
⎝
λ
α
+
π α
≈
λ
−α
Y e
Solution of differential equations:
( ) ( )
( ) ( ) ⎭ ⎬ ⎫
ψ = ψ α +ψ α
0 1 ψ (α) χ (α) 1 1 , = particular solutions
χ = χ α + χ α
.
,
0 1
Page 61](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-62-320.jpg)
![dMeNaHRsayBiess eKrkCarag
A A
sin sin cos ,
χ = α + α α
1 1 2
ψ = α + α α
B B
sin sin cos .
1 1 2
bnÞab;BICMnYstMélTaMgenH cUleTAkñúgsmIkarxagedIm eKrkeXIj
A R
g A + ν
R
D
( )
( ) + ν
(3 ).
25
2 , 5
+ ν
1
1
,
25
2 , 3
1
1
p
D
2
2 2
2
1 2
3
2 2
3
1 2
+ ν
λ +
+ ν = −
λ +
= −
λ +
+ ν = −
λ +
= −
B gR B pR
dMeNaHRsaysrubrbs;smIkarDIepr:g;Esül Gacsresr)anfa
( )
⎫
⎪⎬
C X C Y
ψ = + +ψ α
,
1 1 3 1 1
1 .
[ X ( ⎪⎭
C C ) Y ( C C
) ] ( )χ = λ + ν + − λ + ν + χ α
1 3 1 1 1 3 1
EhR
)a:ra:Em:Rtefr C1, C3 kMNt;ecjBIlkçx½NÐRBMEdn α=45° dUcteTA
X d
cotg 0
⎛ + νχ α
α
M D
45 = ⎟⎠
45
⎞
⎜⎝
χ
= −
α= °
α= ° R
d
C C dY
C C dX
1 ( ) ( )
cotg
+ ν
d
C C X C C Y R
[( ) ( ) ]}
( ) + ν
(cos 2 cos ) 0
25
g R
1 cos 3
1
2
45
2
2
3
2
3
3 1 1 1 3 1
1
1 3
1
3 1
α + ν α =
λ +
⋅ + ν α −
⋅
λ +
+ λ + ν + − λ + ν −
+
⎩ ⎨ ⎧
+ ν α
α
+ − λ + ν
α
λ + ν
α= °
p
D
D
d
EhR
Y Case of simple support
α α= ° z α= ° β α= ° u u
0 0 45 45 45 = = → ε =
Page 63](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-64-320.jpg)
![smIkarlMnwgsMrab;EpñkxagelIénBuH α manragdUcteTA ³
q
z
rNq Q
− 2π sin α − = 0, α
where r = Rsin α,
Q q
z = resultant of self weight,
α α
= ∫ 2 π α = 2 π ∫ sin α α = 2 π 2 ( 1 − cos
α)
Qq q rRd qR d qR
z
0
2
0
So,
.
1 cos
= −
− α
N Q
2 sin sin 2 α
1 + cos
α
= −
π α
= − α
qR qR
r
q
q z
eday Z = −q α R = R = R 1 2 cos , eyIgnwg)an
⎛
[ − α( + α)]
⎞
N R Z N
+ α
⎛
=
⎞
⎟⎠
⎜⎝
+ α
+ α − = ⎟ ⎟⎠
⎜ ⎜⎝
= −
β
α
β
1 cos 1 cos
1 cos
1 cos
cos
1
2
N qR
R q q
R
q
q
q
Analysis on vertical live load
eyagtamRTwsþIbT 1 eyIgGacsresrsmIkarlMnwg)andUcteTA
− 2π sin α − π 2 = 0, α rN p p r
where r = Rsin α.
N p = − pR α
.
2
eday Z = − p cosα⋅cosα eyIgnwgrkeXIj
⎞
⎟⎠
= ⎛− α + ⎟ ⎟⎠
⎜⎝
⎞
⎛
N R Z N
⎜ ⎜⎝
= − α
cos2
β 2
1
2
R p p
R
p
p
N p pR
= − α β cos 2
2
Page 72](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-73-320.jpg)
![Cylindrical and conical shells are shells with zero Gaussian
curvatures:
1 1 0
1 2
1 2 = = =
R R
k k k
For cylindrical shells:
A B y z
⎞
⎛
∂β
⎞
⎛
∂β
1, ;
[( ′ ) 2 + ( 3
′
) 2 ] 2
, .
R R y z
1 2
2 2
′ ′′ − ′ ′′
y z z y
= ∞ =
⎟ ⎟⎠
⎜ ⎜⎝
∂
+ ⎟ ⎟⎠
⎜ ⎜⎝
∂
= =
For conical shells:
⎞
⎛
∂β
A B
1, sin ;
[ ( ) ]
2
2 3 2
α θ + θ′
( ) .
, sin
2 2
cos sin 2 cos sin
1 2
2
2
θ θ + θ′ θ − θ′′ θ
= ∞ = −
⎟ ⎟⎠
⎜ ⎜⎝
∂θ
= = α θ +
R R
edayyk A=1 nig R1=∞ smIkarlMnwgsþaTic TTYl)anragdUcteTA ³
( )
∂
1 0,
( )
⎫
⎪⎪ ⎪ ⎪
⎬
⎪ ⎪ ⎪ ⎪
⎭
− =
∂
+
+ =
∂
∂α
+
∂
∂
∂β
+ =
∂β
∂α
−
∂α
β
β
α β
0.
0,
N
2
2
Z
R
B S BY
B
N
BN N B S BX
edaHRsaysmIkarenH eyIgTTYl)an ³
; 2 N = R Z = RZ β
( ) ( ) ∫ α
1 1 B RZ B Y d
α
⎤
α ⎥⎦
⎡
⎢⎣
+
∂
∂β
= β −
0
2
2 1 2
B
f
B
S
Page 74](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-75-320.jpg)
![( ) ( )
( β
) − α β
S f Tangential
force
=
⎤
⎥⎦
⎡
⎢⎣
α α
− β α + β α
∂
∂β
−
β
2
R
= ∫ ∫
2 sin
cos sin
2
f
1
0 0
2
2
1
q
R
q d q d
R
R
Normal
force
[ ( )] ( ) ∂
( )
[ ( )] ( )
1 1 2 sin
R
− = ∫ α
q
R
f f
R
q d
R R
f f
R
N
2
α β
−
β
β α +
∂
∂
∂β
= −
− α β α
∂β
+
β
β α +
∂β
α
1 cos
2
2 1
0
2
2 1
Boundary conditions:
0, 0 ( ) 0; 2 α = = → β = α N f
, 0 ( ) 2 sin .
1 α = l N = → f β = qR l β +C α
)a:ra:Em:Rtefr C/R2 KWCakMlaMgkat;BRgayesμI elIEKmrbs;bMBg; . dUecñH
RbsinebI bMBg;minrgkarrmYreT KWmann½yfa )a:ra:Em:RtefrenHesμIsUnü ³
0, ( ) 2 sin .
1 C = f β = R ql β
srubmk eyIgTTYl)an
( )
cos ,
N q l
α −α
R
cos ,
=
N qR
= − β
β
(2 )sin .
α
β
S q l
= − α − β
Page 76](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-77-320.jpg)
![α β=0 N
α=0 S
-
+
-
ql
ql
ql2 4R
+ ql
+
-
β N
qR
qR
π
2
β= S
Diagrams
Analysis on Fluid Weight
Components of
fluid weight: 0, cos . 0 X = Y = Z = p − γR β
dUecñH eyIgrkeXIj
( cos ), 0 = = − γ β β N RZ R p R
( ) 1 R ∂
( p R cos ) f
( ) d sin ,
= ∫ α
S f
1 − γ α β
2
1
0
0
2
2 2
β
− γ β α =
∂β
−
β
R
R
R R
[ ( )] ( ) ( )
[ ( )] ( ) β
− = ∫ α
1 1 sin
γα
+
β
β α +
∂
∂
∂β
= −
γ α β α
∂
∂β
+
β
β α +
∂β
α
cos
2
1
2
2
3 1
0
2
3 1
R
f f
R
R d
R R
f f
R
N
p0 = fluid pressure in a plane zOx.
Page 77](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-78-320.jpg)
![Example 3. Analysis of Cylindrical Tank on Wind Load
y
x
α
l
p
Wind
direction
β
R
Components of wind load:
X Y
= =
0,
= ( − β − β)
0.7 0.5cos 1.2cos 2
Z p
where p = max. wind pressure.
]bmafa sMBaFxül;minERbRbUltamkMBs;
suILaMg KWminGaRs½ynwgkUGredaen x=α .
kMlaMgxül;elIsuILaMg
dUecñH eyIg)an
= = (0.7 − 0.5cosβ −1.2cos 2β), β N RZ pR
( ) ( ) ( β
) − α( β + β)
R RZ d f
S f
1 α =
p
∂
∂β
∫ α
β
= −
1 0.5sin 2.4sin 2
2
1
0
2 2
R
B R
( )
⎤
∂
α α
1 1
= ∫ ∫
0 0
( ) ( β
) α
β +
+
( β + β)
∂
∂
∂β
=
α
⎭ ⎬ ⎫
⎩ ⎨ ⎧
⎥⎦
⎢⎣ ⎡
α
∂β
∂β
α
0.5cos 4.8cos 2
2
1
2
2
3 1
2
R
p
R
f f
R
B RZ d d
B B
N
ecjBIlkçx½NÐRBMEdn α = 0, = = 0 α S N eyIgkMNt;)an
( ) ( ) 0 1 2 f β = f β =
Page 79](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-80-320.jpg)
![Zero-Moment Spherical Cupola
Radius R := 10
Self weight q := 0.100 ⋅ 25.00 ⋅ 1.1 q = 2.75
Vertical live load p := 0.50 ⋅ 1.3 p = 0.65
Normal forces:
Nαq(α)
q ⋅ R
1 + cos(α)
:= − Nβq(α)
q ⋅ R
1 + cos(α)
:= ⋅ [1 − cos(α) ⋅ (1 + cos(α))]
Nαp(α)
p ⋅ R
2
:= − Nβp(α)
p ⋅ R
2
:= − ⋅ cos(2 ⋅ α)
Equations of section:
x(α) := R ⋅ sin(α) y(α) := R ⋅ cos(α)
α1 π
:= − α2 π
2
2
:=
n := 50 Δα
α2 − α1
n
:=
i := 0 .. n αi := α1 + i ⋅ Δα
⎯→⎯
⎯→⎯
X := x(α)
Y :=
y(α)
Diagrams:
Nx(α, N, scale) := x(α) + scale ⋅ N ⋅ sin(α) Ny(α, N, scale) := y(α) + scale ⋅ N ⋅ cos(α)
⎯⎯⎯⎯⎯⎯⎯→⎯
Nαqx Nx(α, Nαq(α) , 0.1)
⎯⎯⎯⎯⎯⎯⎯→⎯
:= Nαqy :=
Ny(α, Nαq(α) , 0.1)
⎯⎯⎯⎯⎯⎯⎯→⎯
Nβqx Nx(α, Nβq(α) , 0.1)
⎯⎯⎯⎯⎯⎯⎯→⎯
:= Nβqy :=
Ny(α, Nβq(α) , 0.1)
⎯⎯⎯⎯⎯⎯⎯→⎯
Nαpx Nx(α, Nαp(α) , 0.5)
⎯⎯⎯⎯⎯⎯⎯→⎯
:= Nαpy :=
Ny(α, Nαp(α) , 0.5)
⎯⎯⎯⎯⎯⎯⎯→⎯
Nβpx Nx(α, Nβp(α) , 0.5)
⎯⎯⎯⎯⎯⎯⎯→⎯
:= Nβpy :=
Ny(α, Nβp(α) , 0.5)
Page 81](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-82-320.jpg)
![Example 4. Spherical Tank under Fluid
R
α0
α
A A
r
z
Nα
Nα
2α
p
TMrragrgVg; AA CaRbePT simple
kaMmuxkat; ³ r = Rsin α
sMBaFGgÁFaturav ³
p = γR(1− cosα)
ecjBIlkçx½NÐlMnwgtamG½kS
bBaÄr eKrkeXIj ³
N Qz z
= α 2 sin 2 Rsin2
π α
=
π α
Q
r
r
z
dQz ( )
ϕ
dϕ
R α
dP
dP p r Rd R rRd
2 1 cos 2
3
= ⋅ π ⋅ ϕ = γ − ϕ π ϕ
= π γ ϕ( − ϕ) ϕ
R d
2 sin 1 cos
dQ dP z
cos
3
= ϕ
= π γ ϕ ϕ( − ϕ) ϕ
R d
2 sin cos 1 cos
( )
= ∫ = ∫ π γ ϕ ϕ − ϕ ϕ
Q dQ R d z z
⎤
⎥⎦
⎡
3
cos 1 2
2
0
2 1
= π γ − α⎛ − α
⎢⎣
⎞
⎟⎠
⎜⎝
α α
cos
3
1
6
2 sin cos 1 cos
3 2
0
R
⎞
⎛
[ ( )] ⎟ ⎟⎠
⎜ ⎜⎝
α
1 2cos
+ α
−
N
R2 γ
− α − α =
R α
γ
= α 6
1 cos
1 cos 3 2cos
6sin
2 2
2
2
Page 91](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-92-320.jpg)
![PROBLEMS OF SHELL THEORY
1. Differential Geometry Of Surface
1.1. eKeGayépÞmYyCarag z = z(x, y) . cUrrk first nig second quadratic forms RBmTaMg
Gaussian nig mean curvatures .
1.2. eKeGayépÞrgVilmYyCarag
r(u,ϕ) = x(u) i + ρ(u)cosϕ j+ ρ(u)sin ϕ k, ρ(u) > 0
cUrkMNt; first nig second quadratic forms .
1.3. Translation surface KWCaépÞ EdlekIteLIgedayclnarMkilExSekagmYy z f (x) 1 1 =
tambeNþayExSekagmYyeTot z f (y) 2 2 = . ExSekagrag nigExSekagTis GacepSg²Kña b:uEnþCaTUeTA eK
eRCIserIsykragEtmYy dUcCa )ara:bUl/ FñÚrgVg; .l.
smIkarrbs;épÞrMkil manrag
z f (x) f (y) 1 2 = +
]TahrN_ ³
2
2
z f x R x a − R − a ⎟⎠
= = − ⎛ −
( ) ,
2
1
2 4
2
1 1 1
⎞
⎜⎝
2
2
z f y R y b − R − b ⎟⎠
= = − ⎛ −
( ) .
2
2
2 4
2
2 2 2
⎞
⎜⎝
sMrab;épÞxagelIenH cUrrk first nig second quadratic forms RBmTaMg curvatures .
1.4. ]bmafa mankUGredaensuILaMg (z = α,β) Edl β KWCamMucab;BIG½kS Ox dl;cMeNalénvicT½rkaM
r . dUecñH épÞrgVilGacmansmIkardUcxageRkam
r(z,β) = r(z)cosβ i + r(z)sinβ j + z k
cUrrk first nig second quadratic forms RBmTaMg curvatures rbs;épÞxagelIenH .
1.5. cUrkMNt; first nig second quadratic forms RBmTaMg curvatures rbs;épÞCak;EsþgmYy
cMnYnxageRkam ³
a) Ellipsoid
x = a cosu cos v, y = a cosu sin v, z = c sin v
b) Sphere
x = Rcosαcosβ, y = Rcosαsinβ, z = Rsinα
c) Cylinder of revolution
x = α, y = Rcosβ, z = Rsinβ
Page 95](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-96-320.jpg)
![2.2. Internal forces:
N C ( )
( )
( )
( ) ⎪⎭
⎫
⎪⎬
= ε + νε
α α β
N C
= ε + νε
β β α
S 1 C
1 ,
= − ν ε
αβ
,
,
2
⎫
⎪⎬
M D
= − κ + νκ
α α β
,
( )
( ) ⎪⎭
M D
= − κ + νκ
β β α
1 .
= − − ν κ
αβ
,
H D
C = Eh ( 2 )
Strains:
B u u
D = Eh
∂
1 1 .
2 R
β
ε = α
AB
u
B
+ z
∂
∂α
+
∂β
A u u
∂
, β 1 1
1 R
ε = β
AB
u
A
+ z
∂
∂β
+
α
∂α
α
⎞
⎟⎠
⎛
∂β
A
u
B
ε = β α
αβ A
⎜⎝
∂
⎞
+ ⎟ ⎟⎠
⎛
⎜ ⎜⎝
∂
∂α
u
B
B
A
1− ν2
3
12 1− ν
⎫
⎪ ⎪ ⎪
⎬
⎪ ⎪ ⎪
⎭
⎞
⎟⎠
⎛
⎜⎝
1 1 ,
1 1 ,
∂
1
∂α
∂
⎞
∂
+ ⎟⎠
⎛
∂β
⎜⎝
∂
α
κ =
∂α
+
∂
∂
∂β
κ =
∂β
+
∂α
κ =
αβ
β
V
A
V
B
2 2 1
.
2
2
1
A
B
B
A
B V
AB
V
B
AV
AB
V
A
⎫
⎪ ⎪
⎬
⎪ ⎪
⎭
1 ∂
,
∂
z
∂β
α
V u
u
= −
∂α
= −
β
1 .
2
2
1
1
z
u
R B
V
u
R A
Changes of curvatures:
Hooke’s law:
E z
[ ( )]
[ ( )]
⎫
⎪ ⎪ ⎪
⎬
α α β α β
2
E z
β β α β α
( )( ) ⎪ ⎪ ⎪
⎭
2 .
ε + κ
+ ν
τ = τ =
ε + νε + κ + νκ
− ν
σ =
ε + νε + κ + νκ
− ν
σ =
αβ βα αβ αβ
2 1
,
1
,
1
2
E z
Page 102](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-103-320.jpg)
![3.2. Cylindrical and Conical Shell
For cylindrical shells:
A B y z
⎞
⎛
∂β
⎞
⎛
∂β
1, ;
[( ′ ) 2 + ( 3
′
) 2 ] 2
, .
R R y z
1 2
2 2
′ ′′ − ′ ′′
y z z y
= ∞ =
⎟ ⎟⎠
⎜ ⎜⎝
∂
+ ⎟ ⎟⎠
⎜ ⎜⎝
∂
= =
⎫
⎪⎬
= α
( )
( ) ⎪⎭
x
y y
= β
= β
,
.
,
z z
For conical shells:
x = αcosθ, y = αsin θsinβ, z = αsin θcosβ, θ = θ(β).
⎞
⎛
∂β
A B
1, sin ;
[ ( ) ]
2
2 3 2
α θ + θ′
( ) .
, sin
2 2
cos sin 2 cos sin
1 2
2
2
θ θ + θ′ θ − θ′′ θ
= ∞ = −
⎟ ⎟⎠
⎜ ⎜⎝
∂θ
= = α θ +
R R
( )
∂
1 0,
( )
⎫
⎪⎪ ⎪ ⎪
⎬
⎪ ⎪ ⎪ ⎪
⎭
− =
∂
+
+ =
∂
∂α
+
∂
∂
∂β
+ =
∂β
∂α
−
∂α
β
β
α β
0.
0,
N
2
2
Z
R
B S BY
B
N
BN N B S BX
; 2 N = R Z = RZ β ( ) ( ) ∫ α
1 1 B RZ B Y d
α
⎤
α ⎥⎦
⎡
⎢⎣
+
∂
∂β
= β −
0
2
2 1 2
B
f
B
S
Equilibrium equations:
Solutions:
( ) ( )
1 ⎤
1
⎡ β
∂
∫ 1 2
∫
0 0
α
⎡
∂
∂
1 ∂
1
∫ ∫ ( )
α
α
α
α
α
α
α
α
α
⎛ −
∂α
⎪⎭
⎪⎬ ⎫
⎪⎩
⎪⎨ ⎧
⎤
α ⎥⎦
⎢⎣
+
∂β
∂β
∂β
+
⎞
+ α ⎟⎠
⎜⎝
∂
+
β
+ α ⎥⎦
⎢⎣
∂β
= −
0 0
2
2
B RZ B Y d d
B B
B RZ BX d
B B
d f
B
f
B
N
Page 110](https://image.slidesharecdn.com/shelltheory-141124042024-conversion-gate02/85/Shell-theory-111-320.jpg)
Shell theory
- 1. saklviTüal½y GnþrCati INTERNATIONAL UNIVERSITY Master of Civil Engineering (Structural Engineering) Shell Theory By Seun Sambath, Ph.D, Civil Eng. Phnom Penh 2003
- 2. Shell Theory Shell = 3D thin walled structure. Thin shell KWCaGgÁFatu EdlekagtamTismYy b¤BIr edayKμanrbt; nigkMBUlRsYc nigmankMras;tUcCagTMhMBIreTot ya:geRcIn . RbsinebIeKykkMritlMeGogRtwm 5% enaH shell esþIg manlkçx½NÐ h R ≤1 20 Edl R CakaMkMeNagtUcCageK . plRbeyaCnsMxan;rbs; shell KW multiple function of internal large space. Modeling of shell: • Three-dimensional elastic body • Using static-geometric hypothesis of Kirchhoff-Love Æ approximate theory (thin shell theory) Shell Theory Mathematical theory Engineering theory CaRTwsþI sMrab;epÞógpÞat;PaBRtwmRtUv elIRTwsþIEdleRbIR)as;kñúgkarGnuvtþn_ Cak;Esþg sMrab;eRbIR)as;kñúgGnuvtþnCak;Esþg/ KuNvibtþi³ EdnkMNt;eRbIR)as;RTwsþIenH Page 1
- 3. Elements of Differential Geometry of Surface Equation of surface in vector notation r = r(α,β) = x(α,β)i + y(α,β)j+ z(α,β)k In parametric form x = x(α,β); y = y(α,β); z = z(α,β) where α, β = independent parameters. Coordinate lines α, β = curvilinear coordinates. Equation of surface in Cartesian coordinates: z = z(x, y) or F(x, y, z) = 0 As a function z of coordinates x, y. Ellipsoid: 2 2 + + = 1 2 2 2 2 z c y b x a ⎫ sin sin , or x a = ϕ θ sin cos , y b = ϕ θ cos z c Hyperboloid of one sheet: 2 2 + − = 1 2 2 2 2 z c y b x a ⎪⎭ ⎪⎬ = ϕ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ or 2 x a u v sin 1 , = ⋅ + y = b u ⋅ + v z = cv 2 cos 1 , Page 2
- 4. Hyperboloid of two sheets: 2 2 + − = − 1 2 2 2 2 z c y b x a ⎫ ⎪⎭ ⎪⎬ or x a u v sin , = ⋅ y b u v cos , = ⋅ = ± + 1 z c v2 Cone: 2 2 + − = 0 2 2 2 2 z c y b x a ⎫ ⎪⎭ ⎪⎬ or x = a sin u ⋅ v , y = b u ⋅ v z = cv cos , Elliptical paraboloid: 2 2 y q = + p z x 2 2 ⎫ ⎪ ⎪⎭ ⎪ ⎪⎬ or x = 2 p sin u ⋅ v , y = q u ⋅ v z = v 2 cos , Hyperbolic paraboloid: 2 2 y q = − p z x 2 2 Page 3
- 5. Elliptical cylinder: 2 2 + = 1 2 2 y b x a ⎫ ⎪⎭ ⎪⎬ x a u or = sin , y = b u z = v sin , Hyperbolic cylinder: 2 2 − = 1 2 2 y b x a ⎫ ⎪ ⎪⎭ ⎪⎪⎬ or x = ± a 1 + u 2 , y = bu z = v , Parabolic cylinder: y2 = 2 px ( ) ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎬ = y u v u , = , z u v = v p x u v u , , 2 , 2 or Page 4
- 6. z O r r+dr χ dr α β sMNaj;kUGredaen (coodinate network) manlkçN³dUcxageRkam ³ 1- kat;tamcMNucmYyénépÞ manExS α nig β EtmYyKt; . 2- ral;ExS α nig β nImYy² kat;ExS β nig α EtmYydgKt; . x y dr ∂ = r ∂ d r d ds = dr ; β; ∂β α + ∂α ds2 = dr ⋅ dr = A2dα2 + 2Fdαdβ + B2dβ2 , First Quadratic Form Where coefficients of first quadratic form are ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ r r A E x y z ⎛ ∂α ⎞ ⎟ ⎟⎠ r r F x x y y z z ⎛ ∂β ⎜ ⎜⎝ ∂ ⎛ ∂α ∂ + ⎞ ∂ + ⎟ ⎟⎠ ⎛ ∂β ⎜ ⎜⎝ ∂ ⎞ ⎛ ∂α ∂ + ⎞ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ⎛ ∂ ∂β ∂ = ∂ ∂ = r ∂ r ∂β ∂ ∂ ∂β ∂ ∂ = = = ∂β ∂α ∂β ∂α ∂β ∂α ∂β ∂α ⎞ ⎟⎠ ⎜⎝ ∂ + ⎟⎠⎞ ⎜⎝ ∂ + ⎟⎠ ⎜⎝ ∂ = ∂α ∂α = = . ; ; 2 2 2 2 2 2 2 2 B G x y z So, A E r B G ∂ ∂ ∂ = = r F = r ⋅ r ; ; ⋅ cosχ, ∂β ∂α ∂β = = ∂ ∂α χ = angle between coordinate lines α and β. For orthogonal network: χ = 90°, F = 0 ds2 = A2dα2 + B2dβ2 Page 5
- 7. Area of surface: σ = ∫∫ × α β = ∫∫ − α β α β r r d d A2B2 F2 d d r r ∂ = α ∂α tangential to α-line, r r tangential to β−line Normal unit vector: r r r × r α β r r = α β A2B2 − F2 = × × α β n ∂ = β ∂β Normal section of the surface through a point C is its section by a plane containing the surface normal in this point. Curvature of normal section: 2 2 Ld Md d Nd 1 2 , 2 ds R k n n α + α β + β = − = Rn = radius of curve Second 2 dr dn d r n ϕ = − ⋅ = ⋅ 2 = α + α β + β Quadratic Form Ld 2 2 Md d Nd 2 , x y z αα αα αα 1 , r ⋅ r × r αα − 2 2 2 x y z α α α β β β αα α β α β = × = ⋅ = x y z A B F L r r r n x y z αβ αβ αβ 1 , r ⋅ r × r αβ − 2 2 2 x y z α α α β β β αβ α β α β = × = ⋅ = x y z A B F M r r r n x y z ββ ββ ββ 1 , r ⋅ r × r ββ − 2 2 2 x y z α α α β β β ββ α β α β = × = ⋅ = x y z A B F N r r r n Page 6
- 8. 2 2 2 2 2 r r r r r r ∂ ∂ ∂ = , = , = ; αα ∂α αβ ∂α∂β ββ ∂β L, M, N = coefficients of second quadratic form rα dr rβ h z r r+dr x y n ds1 ds2 2h 2 ϕ = Principal curvatures: ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ L 1 , = = − = = = − = N 2 2 2 max 2 1 1 min 1 B R k k A R k k 1 Ld α 2 + Nd β 2 2 2 2 2 α + β − = A d B d R Gaussian curvature of the surface: 2 LN − M 2 2 2 1 2 1 2 1 A B F R R k k k − = = = Mean curvature of the surface: k + H k 1 2 2 = •Elliptical surface: k > 0 (surface of positive curvature) •Hyperbolic surface: k < 0 (surface of negative curvature) •Parabolic surface: k = 0 (surface of zero curvature) •Minimal surface: H = 0 Page 7
- 9. Ellipsoid x2 a2 + z2 y2 b2 + = 1 c2 a b c ⎛⎜⎜⎝ ⎞⎟⎟⎠ 1 1.5 0.5 ⎛⎜⎜⎝ ⎞⎟⎟⎠ := N := 20 i := 0 .. N ϕi i π N := ⋅ j := 0 .. N θj j 2 ⋅ π N := ⋅ Xi, j a sin ϕ⋅ ( i) sin θ:= ⋅ ( j) Yi, j b sin ϕ⋅ ( i) cos θ:= ⋅ ( j) Zi, j c cos ϕ:= ⋅ ( i) Ellipsoid (X, Y, Z) Page 8
- 10. Hyperpoloid x2 a2 + z2 y2 b2 − = 1 c2 a b c ⎛⎜⎜⎝ ⎞⎟⎟⎠ 1 1 1.5 ⎛⎜⎜⎝ ⎞⎟⎟⎠ := f (z) 1 z2 c2 := + F(ϕ, z) a ⋅ cos(ϕ) ⋅ f (z) b ⋅ sin(ϕ) ⋅ f (z) z ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Hyperboloid F a b c ⎛⎜⎜⎝ ⎞⎟⎟⎠ 1 1 1.5 ⎛⎜⎜⎝ ⎞⎟⎟⎠ := F1(u, v) a ⋅ cos(u) ⋅ v b ⋅ sin(u) ⋅ v c ⋅ v2 + 1 ⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠ := F2(u, v) a ⋅ cos(u) ⋅ v b ⋅ sin(u) ⋅ v −c ⋅ v2 + 1 ⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠ := Hyperboloid F1, F2 x2 a2 + z2 y2 b2 − = −1 c2 Page 9
- 11. Cone x2 a2 + z2 y2 b2 − = 0 c2 a b c ⎛⎜⎜⎝ ⎞⎟⎟⎠ 1 1 1.5 ⎛⎜⎜⎝ ⎞⎟⎟⎠ := f (z) z c := F(ϕ, z) a ⋅ cos(ϕ) ⋅ f (z) b ⋅ sin(ϕ) ⋅ f (z) z ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Cone F Page 10
- 12. Elliptical paraboloid p q ⎛⎜⎝ ⎞⎟⎠ 4 4 ⎛⎜⎝ ⎞⎟⎠ := z(x, y) x2 2 ⋅ p y2 2 ⋅ q := + Elliptical Paraboloid z w(z, ϕ) := z u(z, ϕ) := 2 ⋅ p ⋅ sin(ϕ) ⋅ z v(z, ϕ) := 2 ⋅ q ⋅ cos(ϕ) ⋅ z H := 6 mesh := 20 S := CreateMesh(u, v, w, 0, H, 0, 2 ⋅ π, mesh) Elliptical Paraboloid S Page 11
- 13. Hyperboloic paraboloid p q ⎛⎜⎝ ⎞⎟⎠ 3 1 ⎛⎜⎝ ⎞⎟⎠ := z(x, y) x2 2 ⋅ p y2 2 ⋅ q := − Hyperbolic Paraboloid z a b ⎛⎜⎝ ⎞⎟⎠ 1 1 ⎛⎜⎝ ⎞⎟⎠ := α 1 5 := F(u, v) a 2 ⋅ (v + u) b ⋅ (v − u) 2 α 1 2 ⋅ ⋅ u ⋅ v ⎡⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎦ := F Page 12
- 14. Elliptical Cylinder x2 a2 y2 b2 + = 1 a b ⎛⎜⎝ ⎞⎟⎠ 5 6 ⎛⎜⎝ ⎞⎟⎠ := F(ϕ, z) a ⋅ sin(ϕ) b ⋅ cos(ϕ) z ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Elliptical Cylinder F Hypobolic Cylinder a b ⎛⎜⎝ ⎞⎟⎠ 0.8 1 ⎛⎜⎝ ⎞⎟⎠ := F1(y, z) a 1 y2 b2 ⋅ + y z ⎛⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎠ := F2(y, z) −a 1 y2 b2 ⋅ + y z ⎛⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎠ := Hyperbolic Cylinder F1, F2 Page 13
- 15. Parabolic Cylinder y2 = 2 ⋅ p ⋅ x p := 2 F(y, z) y2 2 ⋅ p y z ⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠ := Parabolic Cylinder F R r ⎛⎜⎝ ⎞⎟⎠ 5 2 ⎛⎜⎝ ⎞⎟⎠ := F(ϕ, θ) (R + r ⋅ cos(ϕ)) ⋅ cos(θ) (R + r ⋅ cos(ϕ)) ⋅ sin(θ) r ⋅ sin(ϕ) ⎡⎢⎢⎣ ⎤⎥⎥⎦ := F Page 14
- 16. Helicoid c := 1 f (u) := 0 F(u, v) u ⋅ cos(v) u ⋅ sin(v) c ⋅ v + f (u) ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Straight Helicoid F c := 1 f (u) := 1.5 ⋅ u x(u, v) := u ⋅ cos(v) y(u, v) := u ⋅ sin(v) z(u, v) := c ⋅ v + f (u) r := 2 R := 5 N := 4 H := N ⋅ π mesh := 20 S := CreateMesh(x, y, z, 2, 5, 0, 4 ⋅ π, mesh) Parabolic Helicoid S Page 15
- 17. c := 1 f (u) 1 5 := ⋅ u2 x(u, v) := u ⋅ cos(v) y(u, v) := u ⋅ sin(v) z(u, v) := c ⋅ v + f (u) r := 2 R := 5 N := 4 H := N ⋅ π mesh := 20 S := CreateMesh(x, y, z, 2, 5, 0, 4 ⋅ π, mesh) Parabolic Helicoid S Page 16
- 18. Torse a b ⎛⎜⎝ ⎞⎟⎠ 1 0.5 ⎛⎜⎝ ⎞⎟⎠ := x(u, v) a ⋅ cos(v) a ⋅ u ⋅ sin(v) a2 + b2 := − y(u, v) a ⋅ sin(v) a ⋅ u ⋅ cos(v) a2 + b2 := + z(u, v) b ⋅ v b ⋅ u a2 + b2 := + mesh := 20 S := CreateMesh(x, y, z, 1, 5, 0, 4 ⋅ π, mesh) Torse S Page 17
- 19. Catenary surface x(u, v) := cosh(u) ⋅ cos(v) y(u, v) := cosh(u) ⋅ sin(v) z(u, v) := u mesh := 30 S := CreateMesh(x, y, z, −1, 1, 0, 2 ⋅ π, mesh) Caternary surface S Pseudosphere a := 1 x(u, v) := a ⋅ sin(u) ⋅ cos(v) y(u, v) := a ⋅ sin(u) ⋅ sin(v) z(u, v) a cos(u) ln tan u 2 ⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ + ⎛⎜⎝ ⎞⎟⎠ := ⋅ + , 0 , 2 π ⋅ , mesh , ⎛⎜⎝ mesh := 30 S CreateMesh x, y, z π 2 2 ⋅ π 5 , − π 2 3 ⋅ π 7 ⎞⎟⎠ := Caternary surface S Page 18
- 20. H := 3 R := 1 N := 20 i := 0 .. N ρi R N := ⋅ i j := 0 .. N ϕj 2 ⋅ π N := ⋅ j Xi, j ρi cos ϕ:= ⋅ ( j) Yi, j ρi sin ϕ:= ⋅ ( j) Z1i, j H R ρ( i):= + − 2 ρ:= ⋅ i Z2i, j H R2 X := stack(X, X) Y:= stack(Y, Y) Z := stack(Z1, Z2) (X, Y, Z) Page 19
- 21. R := 1 N := 20 i := 0 .. N φi 2 ⋅ π N := ⋅ i j := 0 .. N ρj R N := ⋅ j Xi, j ρj cos φ:= ⋅ ( i) Yi, j ρj sin φ:= ⋅ ( i) Zi, j ρ( j):= 2 Page 20
- 22. Moment Theory of Shells Symbols h thickness Nα, Nβ normal forces Sα, Sβ tangential shears Qα, Qβ shears Mα, Mβ bending moments Mαβ, Mβα torsion moments X, Y, Z external forces C Æ (α,β) D Æ (α+dα,β+dβ) C1 Æ (α+dα,β) D1 Æ (α,β +dβ) CD = ds CC1 = Adα CD1 = Bdβ ⎞ β ∂ ⎟⎠ ∂ C D = B + B d 1 ⎛ α ⎜⎝ ∂α ⎞ α ∂ ⎟ ⎟⎠ ⎛ D D = A+ A d 1 ⎜ ⎜⎝ β ∂ ∂β x y z C n Mα Mβ C1 D1 D Nα Qα Sα Mαβ Qβ Nβ Sβ Mβα Z X Y M α β Page 21
- 23. z x X Z C1 Qα C Adα dϕα dϕα Nα α ∂ + α α N N d ∂α ∂ Q + Q α d α α R1 ∂α Q ∂ + β β d C dϕβ R2 Qβ Nβ z Y Z dϕβ Bdβ y N ∂ + β β d β ∂β N β ∂β Q D1 d A ϕ = α α d R 1 d B ϕ = β β d R 2 C C1 D1 D X Y y, β x, α Nβ Sα Nα Sβ Mβα Mβ Mα Mαβ dψα dψβ d D D CC 1 ψ = α A d d C D CD 1 ∂ ψ = B d β 1 1 α β ∂ ∂β = − CD B 1 ∂α = − 1 1 CC A 1 Page 22
- 24. Equilibrium Equations ⎞ ⎛ S ∂ d d D D X S CC S 1 1 d sin d cos d D D N CD cos 1 1 N N N N d d d C D S S d ⎞ d C D ⎞ ∂ ∂ + + ⋅ ϕ ψ ⎟⎠ ⎛ α cos cos sin 1 1 d sin d sin d D D ⎞ Q cos sin 0 1 1 ∂ ⎞ = β α + ⋅ ϕ ψ ⎟⎠ ∂ ⎛ ∂ ⎛ ⎛ α Q ⎛ α ⎜⎝ ∂α − + − ⋅ ϕ ψ ⎟ ⎟⎠ ⎜ ⎜⎝ β ∂β + + + ⋅ ψ ⎟⎠ ⎜⎝ ∂α ⎞ ⎜⎝ ∂α + + + ⋅ − ⋅ ϕ ψ ⎟ ⎟⎠ ⎜ ⎜⎝ β ∂β − + + ⋅ ψ ⎟ ⎟⎠ ⎜ ⎜⎝ β ∂β = − ⋅ + + α α α α β β β β α α α α α α α β β α β β β β β β Σ Q Q d d d C D XABd d A S AB ∂ ∂ 0 : 1 0, ( ) ( ) ( ) ( ) α β α B S AB 0 : 1 0, ∂ + ∂ β 2 ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ∂ ∂ N AB Z AB 0 : 0, α β β α ( ) ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 1 2 0 : 1 0, β α β + = ∂ ∂ ∂α + ∂ − ∂ − ∂α ∂ ∂ ∂β = + = ∂β + ∂β ∂α = − = ∂α ∂β ∂ ∂ = + + − + = ∂α + ∂β − ∂β = − + = ∂β + ∂α − ∂α = α β α β α Σ Σ Σ Σ Σ 2 0 : 1 2 0, 2 1 2 A H BM M B ABQ A M B H AM M A ABQ B M N AQ BQ ABZ R R Q ABY R B Y AN N A Q ABX R A X BN N B x y M α β Σ R = − − βα + αβ ≡ 0 : 0 R 2 1 M M S S z S = S = S M = M = H α β αβ βα Because , N , N , S,Q ,Q ,M ,M ,H α β α β α β 8 unknowns and 5 equations. Page 23
- 25. Internal Forces Αdα ⎛ + d A z dϕα α ⎞ α ⎟ ⎟⎠ ⎜ ⎜⎝ R 1 1 h 2 h 2 dz z z σβ τβα τβz ⎞ N z 1 , ⎞ S z 1 , ⎞ Q z 1 , h β β − h β βα 2 ⎛ ⎛ β β 2 1 ∫ 2 2 1 2 2 1 ∫ ∫ − − ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ = − τ + ⎟ ⎟⎠ ⎜ ⎜⎝ = τ + ⎟ ⎟⎠ ⎜ ⎜⎝ = σ + h h z h h dz R dz R dz R ⎛ + τ = ⎟ ⎟⎠ ⎛ zdz M z M z ∫ ∫ 1 , 1 β ⎟⎠ ⎟ β βα βα − − ⎞ ⎜ ⎜⎝ ⎞ ⎜ ⎜⎝ = − σ + 2 2 1 2 2 1 h h h h zdz R R R1 ⎛ + τ − = ⎟ ⎟⎠ ⎛ + τ = ⎟ ⎟⎠ ⎛ dz Q z dz S z N z ∫ ∫ ∫ 1 , 1 ⎟ ⎟⎠ , 1 α α α α − α αβ − − ⎞ ⎜ ⎜⎝ ⎞ ⎜ ⎜⎝ ⎞ ⎜ ⎜⎝ = σ + 2 2 2 2 2 2 2 2 2 h h z h h h h dz R R R ⎛ + τ = ⎟ ⎟⎠ ⎛ zdz M z M z ∫ ∫ 1 , 1 α ⎟⎠ ⎟ α αβ αβ − − ⎞ ⎜ ⎜⎝ ⎞ ⎜ ⎜⎝ = − σ + 2 2 2 2 2 2 h h h h zdz R R ⎛ + ≈ ⎟ ⎟⎠ ⎛ z z R z 1 ≈ ⎟ ⎟⎠ 1 1, 1 1 1 2 ⎞ ⎜ ⎜⎝ ⎞ ⎜ ⎜⎝ << → + R R So, S = S = S M = M = H α β αβ αβ , Page 24
- 26. Strain Determination. Hooke’’s Law. Boundary Conditions n eβ M M’ u uz eα uβ uα eα, eβ, n = unit vectors e = r e r α A β B 1 , 1 ∂ , ∂β = ∂ ∂α r × r n = α β A2B2 − F2 α β u = resultant displacements; uα, uβ, uz = displacement components in α-, β- and z-direction Position of M: r, Position of M’: r r u r e e n z = + = + u + u + u α α β β ' For a point M’: ⎞ ⎛ ⎛ ∂ ∂ ′ ′ u u ⎟ e ⎟⎠ ′ = r e e α n ⎜ ⎜⎝ − ∂ ∂α + ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ∂ ∂β − ∂α ≈ + ∂α α β β α α 1 1 1 1 1 R A A u AB u A A z ⎞ ⎛ ∂ ′ ′ u u ⎟ e ⎟⎠ ′ = r e e β n ⎜ ⎜⎝ − ∂ ∂β ⎞ + + ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ ∂ ∂α − ∂ ∂β ≈ ∂β β α β α β 2 1 1 1 1 R B B u AB u B B z ⎞ ⎞ ⎟ ⎟⎠ A r A A u u z ′ = β ⎛ ∂ ∂ ∂ ′ B r B B u u z ′ = α ⎜ ⎜⎝ + ∂ ∂α + ∂ ∂β ≈ + ∂ ′ ∂β β 2 1 1 1 R AB u B ⎟ ⎟⎠ ⎜ ⎜⎝ ⎛ + ∂β + ∂α ≈ + ∂α α 1 1 1 1 R AB u A Normal strains: ds ds ′ − ds ds , 2 2 , 2 ′ − ε = 1 1 α β ds 1 ds ε = , , 1 2 ds = Adα ds = Bdβ , . 1 2 ds′ = A′dα ds′ = B′dβ Page 25
- 27. B u u ∂ 1 1 . 2 R β ε = α AB u B + z ∂ ∂α + ∂β A u u ∂ β 1 1 , 1 R ε = β AB u A + z ∂ ∂β + α ∂α α Shear strain: ′ ′ = ′ ′ ⎛ π − ε ⎞ ⎛ − ε ⎞ sin α β α β αβ αβ αβ αβ ε ≈ ε = ⎟⎠ ⎜⎝ π = ⎟⎠ ⎜⎝ 2 cos 2 e e e e cos ⎞ ⎟⎠ ⎛ ∂β A u B ε = β α αβ A ⎜⎝ ∂ ⎞ + ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ ∂ ∂α u B B A Kichhoff-Love’s Assumptions: 1. About normal to middle surface: ε = ε = ε = 0 βz zα z 2. About normal stress: σ = 0 z After deformation: ds ds ( ) ( ) ( )⎭ ⎬ ⎫ ′ = + ε 1 , α 1 1 ds ds ′ = + ε 1 , β 2 2 ′ = + ε 1 , A A ( )⎭ ⎬ ⎫ ′ = + ε α 1 . β B B ′ = ′ ′ π ⎞ 1 1 ( )( ) αβ α β αβ ε ε + ε + = ⎟⎠ ⎛ − ε F A B cos AB ⎜⎝ 2 Love’s formulas: M ′ ε ε 1 1 , 1 1 β , . A B 1 1 1 2 2 κ = αβ κ = − − + ′ κ = − + ′ R R R R R R ′ ′ 2 β α α κα, κβ = changes of bending curvatures ¬pldkkMeNagBt;¦, καβ = change of twisting curvatures ¬pldkkMeNagrmYr¦. Page 26
- 28. In the distance z form midplane: ⎪⎬ ⎫ R = R + z , ( ) ( ) ⎪⎭ ds A d = α z ( ) ( ) 1 1 R = R + z , z 2 2 ⎪⎬ ⎫ , , z z 1 ds B d = β ( ) ( ) ⎪⎭ 2 z z A A z ( ) ⎫ ⎪ ⎪ ⎬ ⎞ 1 , ⎞ ⎛ ⎛ R 1 z B B z ( ) ⎪ ⎪ ⎭ ⎟ ⎟⎠ ⎜ ⎜⎝ = + ⎟ ⎟⎠ ⎜ ⎜⎝ = + 1 . R 2 z z ε = ε + κ ( ) ( ) ( ) ⎫ u u zV z ( ) ⎪⎭ ⎪⎬ α α α z ε = ε + κ z β β β , , z 2 . ε = ε + κ z αβ αβ αβ z α α u u zV ( ) ( ) ⎫ ⎪⎭ ⎪⎬ = + = + z β β u = u . , 1 , 2 z z z ( ) u 1 1 , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 u 1 1 , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) B ( ) ( ) ( ) ( ) ∂ ∂ A ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟ ⎠ ⎛ ∂β ⎜ ⎜ ⎝ ∂ ⎞ + ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ ∂ ∂α ε = + ∂α + ∂β ∂ ε = + ∂β + ∂α ∂ ε = z z u u β α αβ α β β β α α . 2 z z z z z z z z z z z z z z z z z z z z z z z z z A B B A R u B A B u B R u A A B u A ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ⎞ ⎟⎠ ⎛ ⎜⎝ 1 1 , 1 1 , ∂ 1 ∂α ∂ ⎞ ∂ + ⎟⎠ ⎛ ∂β ⎜⎝ ∂ α κ = ∂α + ∂ ∂ ∂β κ = ∂β + ∂α κ = αβ β V A V B 2 2 1 . 2 2 1 A B B A B V AB V B AV AB V A ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 1 ∂ , ∂ z ∂β α V u u = − ∂α = − β 1 . 2 2 1 1 z u R B V u R A Hooke’s law E E z ( ) ( ) ( ) [ ( )] z z α α β α β α β 2 2 E E z ( ) ( ) ( ) [ ( )] ⎫ ⎪ ⎪ ⎪ ⎬ z z β β α β α β α ( ) ( ) ( )( ) ⎪ ⎪ ⎪ ⎭ 2 . ε + κ + ν ε = + ν τ = τ = ε + νε + κ + νκ − ν ε + νε = − ν σ = ε + νε + κ + νκ − ν ε + νε = − ν σ = αβ βα αβ αβ αβ 2 1 2 1 , 1 1 , 1 1 2 2 E E z z Page 27
- 29. Internal forces: N C ( ) ( ) ( ) ( ) ⎪⎭ ⎫ ⎪⎬ = ε + νε α α β N C = ε + νε β β α S 1 C 1 , = − ν ε αβ , , 2 ⎫ ⎪⎬ M D = − κ + νκ α α β , ( ) ( ) ⎪⎭ M D = − κ + νκ β β α 1 . = − − ν κ αβ , H D C = Eh shell stiffness (rigidity) for tension, 1− ν2 3 D = Eh shell stiffness (cylindrical rigidity) for bending, ( 2 ) 12 1− ν Boundary Conditions Equations (17) • 5 equations of statics, • 6 strain components, • 6 physical equations. Unknowns (17) • 8 internal forces: α β α β α β N , N , S,M ,M ,H,Q ,Q • 3 displacements: z u ,u ,u α β • 6 strains: α β αβ α β αβ ε ,ε , ε , κ , κ , κ Generalized shears and tangential shears (β=const): H S S H ~ 1 , ~ . 1 R ∂ Q Q = − A ∂α = + β β enAelIRCugnImYy² RtUvman 4 lkçx½NÐRBMEdn Page 28
- 30. Rim β=const is free: H N S H 0, 1 0, 0, 0. = + β β β R = = − = 1 ∂ ∂α A M Q Rim β=const is built-in: ∂ 0, 1 = 0. 2 ∂β = = = = − α β z z u B u u u V Rim β=const is hinge supported: = 0, = = = 0. β α β z M u u u Rim β=const is simple supported with normal movement: = + β β α β H u u ∂ 0, 1 = 0, = = 0. ∂α A M Q Rim β=const is simple supported with tangential movement: M N S H = = − = = β β z u 0, 0, 0, 0. R 1 Page 29
- 31. Analysis of Cylindrical Shells z y x f l β=s α=x dx ds a x=l x=0 C Z X Y z x,α y,β D C1 D1 Qx S Nx S H H Mx Qs Ns Ms Equations of cylindrical shell: x = α, y = y(β), z = z(β) Coordinate lines: α = x, β = s, s = arc length. A B F d dx d ds = ∞ = ϕ = ϕ = 1, 0, , ,cos 0, = = = α = β = χ = R R R s d d ds , ( ), 0, . 1 2 α β R Equilibrium equations: ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 0, − = Q s s Q ∂ + ∂ S ∂ + N x ∂ + Q ∂ + ∂ − + = ∂ N ∂ S ∂ ∂ + = ∂ ∂ 0, 0, N 2 2 Z x s R Y R s x X s x s s x ⎫ ⎪ ⎪⎭ ⎪ ⎪⎬ + = M ∂ − H ∂ − ∂ H ∂ − M ∂ − ∂ + = ∂ ∂ 0, 0, x x s s Q s x Q s x ⎫ ⎪ ⎪⎭ ⎪ ⎪⎬ H ∂ + M ∂ + ∂ ∂ Q = M Q ∂ = H ∂ ∂ ∂ , . s x s x s s x x Page 30
- 32. CMnYs Qx nig Qs cUleTAkñúgsmIkarbIxagmux eyIgTTYl)an ³ ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 0, M H 1 1 0, − = ∂ s x ∂ + M ∂ ∂ ∂ H ∂ ∂ + S ∂ + N x ∂ + ∂ + M ∂ + = ∂ − ∂ − ∂ N ∂ S ∂ ∂ + = ∂ ∂ 2 0. 2 2 2 2 2 N 2 Z s x s x R Y s x R s R x X s x s x s Strain components: u u u u ∂ + ∂ ∂ , , , 2 u u u u , , 2 1 2 . u 2 2 ∂ x s x ⎞ s R R ∂ x s s x R s u x s z xs ∂ − s z y z x s x xs s z y x x ∂ ∂ − ∂ ∂ = κ ⎟⎠ ⎛ ⎜⎝ ∂ ∂ κ = ∂ ∂ ∂ κ = − ∂ ∂ + ε = ∂ ε = ∂ ε = Internal forces: ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ u ⎛ + ∂ u u ∂ s z x ⎞ ⎟⎠ ⎡ N C u N C ⎡ u ∂ ∂ u S − ν C ⎛ ∂ u ⎜⎝ u ∂ + ∂ ∂ = ⎤ ⎥⎦ ⎢⎣ ∂ + + ν ∂ = ⎤ ⎥⎦ ⎢⎣ ⎞ ⎟⎠ ⎜⎝ ∂ + ν ∂ = , 2 1 , , s x x R s R s x s x s x s z x M D u ⎡ u s ∂ s z z 1 1 ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ u ∂ − u ⎞ ⎟ ⎟⎠ ∂ ⎛ ⎛ ⎜ ⎜⎝ ⎛ u ∂ ν − ⎟⎠ ∂ − u ∂ ∂ u ∂ − u ∂ ∂ = − − ν ⎞ ⎤ ⎥⎦ ⎢⎣ ∂ ⎞ ⎜⎝ ∂ ∂ = − ⎤ ⎥⎦ ⎢⎣ ⎡ ⎟⎠ ⎜⎝ ∂ ∂ ∂ + ν ∂ = − − . 2 , , 2 2 2 2 2 x s x R H D x s R M D s R x s s z s z s z x CMnYstMélkMlaMgkñúgxagelIcUleTAkñúgsmIkarlMnwg eyIgnwg)an Page 31
- 33. X − ν ∂ 0, ⎞ 1 2 u u 2 + ν ∂ ⎛ + ν ∂ 2 2 2 12 1 2 ⎡ u ν ∂ + 2 2 2 12 − ν ∂ 1 ⎛ + ν ∂ 2 1 ∂ 2 2 2 2 2 2 2 2 ∂ 2 2 2 ⎛ ⎤ ⎞ u Y = + ⎥⎦ ⎡ ⎢⎣ ⎞ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ ∂ ∂ ∂ ∂ ⎞ − ⎟⎠ ∂ ⎛ ∂ ⎜⎝ ∂ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎤ ⎥⎦ ⎢⎣ ∂ + ⎟⎠ ⎜⎝ ∂ + ∂ + ∂ + u ∂ ∂ C R s x s h s R u R s R R x h x s s x z s x ∂ + ⎟⎠ 4 ⎡ 2 2 ∂ 4 2 ⎤ ⎞ ∂ + ∂ ⎤ ⎞ ⎛ ⎛ ∂ u Z 2 0. ∂ ⎛ 2 4 12 1 12 1 4 2 2 4 2 2 2 = − ⎥⎦ u ν ∂ ⎡ ⎢⎣ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ ∂ ∂ + ∂ ∂ + + + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥⎦ ⎢⎣ ⎟⎠ ⎜⎝ ∂ ⎞ ⎜⎝ ∂ ∂ − ∂ + ∂ C x x s s h R u s x R s R h x R s R z s x 0, 2 1 2 2 2 + = ∂ ∂ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ C x x s R x s s z x For circular cylindrical shell: R = r = const ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ L u L u L u X + + + = x s z 11 12 13 C L u L u L u Y + + + = x s z 21 22 23 C L u L u L u Z + + − = 0, 0, 0. 31 32 33 C x s z Equilibrium equations , 1 + ν ∂ 2 12 21 x s 2 L L ∂ ∂ , = = − ν ∂ ∂ = 11 x s 2 1 2 2 2 2 L ∂ + ∂ , ∂ + + ν ∂ 22 x s 2 1 2 2 2 2 L ∂ ∂ = L L ν ∂ = = , 13 31 r ∂ x , ⎛ 2 3 ⎡ ⎞ ∂ + ∂ ∂ 23 32 12 ⎥⎦ 1 3 3 2 ⎤ ⎢⎣ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ ∂ ∂ − ∂ = = x s s h r s L L 4 4 ⎛ ∂ + ∂ ∂ 2 33 ⎟ ⎟⎠ 2 . 2 4 12 1 4 2 2 4 ⎞ ⎜ ⎜⎝ ∂ ∂ ∂ + ∂ = + x x s s h r L Page 32
- 34. Case X=Y=0: u L u L s 1 4 L L 11 12 ∇ = , = , , x L L s x 2 21 22 − ν = = L L L L − L u − = . , L u L z 13 12 L u L 23 22 L z x − 11 13 22 23 z z L s L − L u = ⎞ , r u uz 1 z z z ⎛ 2 5 + ν h u 1 12 4 5 3 2 2 3 3 3 4 ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + u ∂ ∂ ∂ ∂ ∂ − ν − ∂ + u ∂ ∂ ∂ ∂ ∇ = −ν x s x s x s x x 5 ⎡ 2 5 3 r u ∂ uz ∂ − u h ∂ u ∂ u ∂ u z z z z ( ) ( ) 2 (3 ) (1 ) . 4 ⎥⎦ 2 5 s 12 1 12(1 ) 2 (3 ) (2 ) 1 4 , 5 2 3 4 3 3 2 ⎤ ⎢⎣ ∂ + − ν ∂ ∂ + − ν ∂ ∂ − ν + ∂ ∂ ∂ ∇ = − + ν s x s x s s x s 6 u ⎡ ∂ ∂ ∂ − ν ∂ ⎥⎦ u z z z z = ∇ z 4 2 6 u 2 4 u 6 6 u 4 2 4 2 2 2 8 Z x s D x s s x r r h ⎤ ⎢⎣ ∂ ∂ + + ν ∂ ∂ + + ν ∂ + ∂ ∇ + 4 4 4 ∂ + ∂ ∂ Where 4 2 , 4 2 2 4 x ∂ x ∂ s ∂s + ∂ ∇ = 8 8 8 ∂ 2 6 8 ∂ ∂ ∂ 8 4 4 4 6 4 4 4 8 6 2 8 8 ∂ + x x s x s ∂ x ∂ s ∂s + ∂ + ∂ ∂ + ∂ ∇ = ∇ ∇ = L.N.Donnel’s equations: u 1 , ν ∂ 3 u u ∇ = − 2 1 , u z z x ∂ ∂ 2 3 u 3 3 4 x s x r r ∂ + ∂ + ν ∂ u z z s ∂ 3 3 2 4 s x s r r ∂ − ∂ ∂ ∇ = − 12 ( 1 − ν ) ∂ u 1 4 . u z z = ∇ 4 4 2 2 2 8 Z x D r h ∂ ∇ + For closed shell: Σ Σ∞ ( )cos , ( )cos , z zm u u x m Z Z x m = ϕ = ϕ 0 = 0 ∞ = m m m Page 33
- 35. ∫ π where ϕ = s , Z ( x ) = Z cos mϕdϕ . −π r m 2 m , 2 2 2 2 4 m m ∂ ∂ ∇ = 2 , 2 4 2 2 2 2 2 r ∂ = ∂ + ∂ x s x − ∂ ∂ ∂ 4 2 2 4 4 r r x x + ∂ − ∂ ∇ = 8 2 6 4 4 6 2 m m m m ∂ ∂ ∂ 4 6 4 , 2 6 4 4 6 2 8 8 8 8 r r x r x r x ∂ x + ∂ − ∂ + ∂ − ∂ ∇ = smIkar Donnel TI3 manragCa ( ) 4 2 ⎛ 4 − ν 6 2 ⎞ d m d m 4 6 121 4 4 2 d + + − Σ∞ = 2 4 2 6 ⎤ ⎡ d m Z x ms m d m d 1 2 ( ) cos 0 4 − − + 2 2 4 0 8 m 8 2 6 4 2 2 4 6 2 8 8 = ⎭ ⎬ ⎫ ⎥⎦ ⎢⎣ ⎪⎩ ⎪⎨ ⎧ ⎤ − ⎥⎦ ⎡ ⎢⎣ + − ⎟ ⎟⎠ ⎜ ⎜⎝ r r dx r dx D u x r dx r dx r r h dx r dx m m zm Tangential displacements: ( )cos , Σ∞ x xm u u x m = ϕ 0 = m ( )sin , Σ∞ s sm u u x m = ϕ 0 = m For open shell: u u s m x ⎫ ( ) ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎭ cos , l u u s m x sin , l u u s m π x x xm m s sm = π = π = ∞ Σ = ∞ 0 Σ = m ∞ 0 Σ z zm = sin , 0 m l ⎫ ⎪ ⎪ ⎪ X X s m x cos , Y Y s m x ( ) ( ) ⎪ ⎪ ⎪ ⎬ ⎭ sin , l π Z = Z s m x π = π = ∞ Σ = ∞ 0 Σ = 0 ∞ Σ = sin . 0 m m m m m m l l Z m x l Y m x l X m x l 2 cos , 2 sin , 2 sin . ∫ π ∫ π π = = ∫ 0 0 0 = l m l m l m dx l dx Z l dx Y l X Boundary conditions = 0 and = : = = = = 0. s z x x x x l u u N M where (Simple-supported on the rigid diaphragm) Page 34
- 36. Example Axis-symmetrical Cylindrical Shell z x l 2R h x Z External forces: X = Y = 0, Z = q(l − x) Data: R = 1 m , h = 5 mm , l = 5 m = 0.001 3 q kgf cm Steel: 2 106 2 , E = ⋅ kgf cm 0.3 ν = In a case of axis symmetry (Y = 0): ∂ 0, = 0. ∂ = = = = s u Q S H s s L Internal forces: du N C u ⎞ ⎞ = ⎛ + ν ⎟⎠ , , D d u ⎟⎠ Q dM u ⎜⎝ M D d u = ⎛ + ν , , . 3 3 2 2 N C du ⎜⎝ M D d u 2 2 dx dx dx dx dx R R dx x z x z s z x z x s x z x = = ν = = Equilibrium equations: ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⎞ u ZR = − ⎟ ⎟⎠ x z ⎛ ⎜ ⎜⎝ d u du du ν + + X + = ν + 0. 2 4 Rh d 12 1 0, 4 2 2 C dx dx R C dx dx R z x Page 35
- 37. sikSakrN I X=0: ecjBIsmIkarlMnwgTI 1 eyIg)an ∫ ν u C N x = = → = + − u dx ν + x x z x z R u C C x C du dx R 0 6 5 6 CMnYscUleTAkñúgsmIkarTI 2 eyIgTTYl)an ν ( )3 z + γ = − N 1 . d u x u Z 4 4 , 4 4 RD D dx z − ν 2 2 2 4 R h γ = Common solution: u 0 = e −γ x (C cos γ x + C sin γ x) + e γ x (C cos γ x + C sin γ x) z 1 2 3 4 ~ Particular solution: u (x) z sMrab;krNI]TahrNxagmux KWecjBIlkçx½NÐ)atxagelITMenr eyIgrkeXIj 0 0 6 N = → C = x ( ) ( ) u q l x D u q l x D d u dx z z z 4 4 4 4 4 4 ~ γ − → = − + γ = ( ) ( ) ( ) − u = e −γ x C cos γ x + C sin γ x + e γ x C cos γ x + C sin γ x + q l x z 1 2 3 4 4 γ 4 D Boundary conditions: x u u duz = 0 : = 0, = 0, = 0. dx x z 3 x l M D d u Q D d u z : 0, 0. 2 3 2 = = = = = d x dx x z x ∫ ν = − x x zu dx R u C 0 5 Page 36
- 38. Circular Tank Radius R := 1 Heigth L := 3 Thickness h := 0.1 Fluid density q := 10 Modulus of elasticity E 2 10 4 ⋅ 10− 3 := ⋅ Poisson ratio ν := 0.2 10− 6 Cylindrical stiffness D E ⋅ h3 ⋅ ( − 2) 12 1 ν := γ4 ⋅ ( − 2) R2 ⋅ h2 3 1 ν := γ 4 := γ4 Particular solution u1z(x) q ⋅ (L − x) 4 ⋅ γ4 ⋅ D := u01z(x) q L ⋅ x x2 2 − ⎛⎜⎝ ⎞⎟⎠ ⋅ 4 ⋅ γ4 ⋅ D := F(x) e − γ⋅x ⋅ cos(γ ⋅ x) e − γ⋅x ⋅ sin(γ ⋅ x) eγ⋅x ⋅ cos(γ ⋅ x) eγx ⋅ ⋅ sin(γ ⋅ x) ⎛⎜⎜⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎟⎟⎠ := K −γ γ 0 0 −γ −γ 0 0 0 0 γ γ 0 0 −γ γ ⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠ := K2 := K ⋅ K K3 := K2 ⋅ K K01 K− 1 := F1(x) := K ⋅ F(x) F2(x) := K2 ⋅ F(x) F3(x) := K3 ⋅ F(x) D1 −q := D2 := 0 D3 := 0 4 ⋅ γ4 ⋅ D Boundary conditions: A 0 〈 〉 := F(0) A 1 〈 〉 := F1(0) A 2 〈 〉 := F2(L) A 3 〈 〉 := F3(L) B0 := −u1z(0) B1 := −D1 B2 := −D2 B3 := −D3 Integration constants: C (AT) − 1 := ⋅ B Page 37
- 39. Normal displacement uz(x) := C ⋅ F(x) + u1z(x) u1x(x) ν R K01T C ⋅ ( ) F x ( ) F 0 ( ) − ( ) ⋅ u01z x ( ) u01z 0 ( ) − ( ) + ⎡⎣ ⎤⎦ := ⋅ c5 := u1x(0) c5 = 0 Longitudinal displacement ux(x) := c5 − u1x(x) C0 E ⋅ h 1 ν − 2 := ⋅ ( − 2) uz(x) Normal force Ns(x) C0 1 ν R := ⋅ Bending moment Mx(x) := D ⋅ (C ⋅ F2(x) + D2) Ms(x) := ν ⋅ Mx(x) Shear Qx(x) := D ⋅ (C ⋅ F3(x) + D3) ξ := 0, 0.02 ⋅ L .. L 0 1 2 3 30 20 10 0 − 10 Normal forces Ns(x) Ns(ξ) x, ξ L1 := 0.2 ⋅ L ξ := 0, 0.02 ⋅ L1 .. L1 0 0.2 0.4 0.6 1 0.8 0.6 0.4 0.2 0 − 0.2 Bending moments Mx(x) Mx(ξ) x, ξ Page 38
- 40. Analysis of Shallow Shells Shallow shell: 20, 5, min min R h ≥ l f ≥ where lmin = least dimension in plane, f = rise. x y z α ≡ x, β ≡ y cosϕ =1, sin ϕ = 0, ϕ− slope angle Tangential stresses = their projectives Assumptions: 1. In rectangular coordinate: z = z(x, y) 2 2 2 ds = dx + dy 2. Zero Gauss’s curvature k = k k = 0 1 2 3. Q Q α = 0, β = 0 R R 1 2 1 , 2 2 2 2 2 → = = ⎪⎭ ⎪⎬ ⎫ = α + β A B ds A d B d In polar coordinates (r, β): ds2 = dr2 + r2dβ2 → A =1, B = r(z) u 4. α = 0, β = 0. R R 1 2 u So, Page 39
- 41. ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ A u u 1 1 , ⎞ ⎟⎠ B u u ⎛ ∂β ⎜⎝ ∂ ⎞ ∂ ∂ + ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ ∂ ∂ ∂ ∂α ε = + ∂α + ∂β ε = + ∂β + ∂α ε = z 1 u A u β α αβ α β β β α α , 1 1 , 2 A B B B A R AB u B R AB u A z ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟⎠ 1 ⎛ 1 1 , ⎛ z z 2 1 1 1 , ⎛ ⎜ ⎜⎝ A ∂ u ∂α ∂ ∂β ∂ ∂ z z − ∂ ∂ ∂ ∂α ∂ 2 ∂α ⎞ ⎞ − ∂ ∂ ∂ 2 ∂α∂β α β κ = − ∂α ∂α − ⎟ ⎟⎠ ⎜ ⎜⎝ ∂β ∂ ∂ ∂β κ = − ∂β ∂β − ⎟⎠ ⎜⎝ ∂α ∂α κ = − αβ 1 1 1 z z z . A B u B u AB B u A B u B B A u AB u A A 1 ∂ 0, Equilibrium Equations: ∂ ( ) ( 2 ) ( ) ( ) ∂ + ∂ ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ∂ ∂ α β N AB α β β α ( ) ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ AB 1 2 1 0, β α β + = ∂ ∂ ∂α + ∂ − ∂ − ∂α ∂ ∂ ∂β + = ∂β + ∂β ∂α − = ∂α ∂β + + + = ∂α + ∂β − ∂ ∂ ∂β + = ∂β + ∂α − ∂α α β α β α 2 1 0, 0, 1 0, 2 2 A H BM M B ABQ A B H AM M A ABQ B N AQ BQ ABZ R R B S ABY B AN N A A S ABX A BN N B Page 40
- 42. Integration of equilibrium equations ecjBIsmIkarBIrxageRkay eyIgTTYl)an³ ⎤ ⎡ 1 1 ∂ , ( ) ( 2 ) ⎤ α α β ⎡ ∂ ∂ B H M A 1 ( ) 1 ( 2 ) . ⎥⎦ ⎢⎣ ∂ ∂β − ∂ ∂α − ∂ ∂β = ⎥⎦ ⎢⎣ ∂α − ∂β − ∂α = β β α B AM AB Q A H M B A BM AB Q edayeyageTAelIlkçx½NÐCab; (compatibility conditions) 1 ∂ 0, ∂ ( ) ( 2 ) α α αβ ( ) 1 ∂ ( 2 κ ) = 0, ∂α − ∂ ∂β κ − κ ∂ ∂ ∂β κ = ∂β − ∂α κ − κ ∂α B β β αβ B A A A A B B eyIgnwgTTYl)an ³ D ∂ ∂ ( ) ( ) ( ) ( ) . 1 α α β 12 1 1 , 12 1 2 2 3 2 2 3 z z u D A B Q Eh u A A Q Eh ∂ ∇ ∂β κ + κ = ∂ ∂β − ν = − ∇ ∂α κ + κ = ∂α − ν = − β α β CMnYstMélxagelI eTAkñúgsmIkarTIbI eyIgTTYl)an³ ⎡ 1 1 1 ∂ ( ) ( ) β α ⎤ ∂ ∂ ∂ ( ) 1 ( 2 ) 0 2 N 1 2 − = ⎭ ⎬ ⎫ ⎥⎦ ⎢⎣ ⎡ ∂ ∂α − ∂ ∂β − ∂ ∂α ∂ + ∂α + ⎩ ⎨ ⎧ ⎤ ⎥⎦ ⎢⎣ ∂β − ∂α − ∂β ∂β + + α β α β A H M B Z A BM B H M A B AM R AB B N R tagGnuKmnsMBaFkñúg (stress function) ϕ tamrUbmnþxageRkam ³ Page 41
- 43. ⎞ ⎛ 1 1 1 , 2 ∂ϕ ∂ ∂ϕ ∂ 1 1 1 , 2 2 ⎞ ∂ ⎛ ∂ϕ ∂ ⎞ ⎛ ∂ϕ 1 1 1 . ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ϕ ∂α ∂ ∂β − ∂ϕ ∂β ∂ ∂α − ∂ ϕ ∂α∂β α = − ∂β ∂β + ⎟⎠ ⎜⎝ ∂α ∂α = ∂α ∂α + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂β ∂β = β A A B AB B S A A A AB N B B B A B N bnÞab;BICMnYscUleTAkñúgsmIkarlMnwgsþaTic eyIgsegáteXIjfa smIkarbYn RtUv)anepÞógpÞat; KWBIrxagmux cMeBaHkrNI X=Y=0 nigBIrxageRkay rIÉsmIkarTIbI nwgTTYl)anrag ³ N ⎛ − α + β Eh u Z ( ) 0 12 1 2 2 2 3 1 2 ∇ ∇ + = − ν ⎞ − ⎟ ⎟⎠ ⎜ ⎜⎝ R N R z eyIgman N + N = ∇2ϕ , k N + k N = ∇ 2 ϕ , α β 1 α 2 β k edayEp¥kelIsmIkar Kodazzi ∂ ∂ B ∂ ∂ k B k k A k A ( ) , ( ) = , 2 1 1 2 ∂β ∂β ∂α = ∂α Edl ⎤ ⎡ ⎞ ⎛ A B 1 , ⎤ ⎡ ⎞ ∂ ⎛ k A A ∂ + ⎟⎠ ∂ ∂ 1 . ∂ + ⎟⎠ ∂ 2 1 2 2 ⎥⎦ ⎢⎣ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ ∂β ∂β ⎞ ⎛ ⎜⎝ ∂α ∂ ∂α ∇ = ⎥⎦ ⎢⎣ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂β ∂β ⎞ ⎜⎝⎛ ∂α ∂α ∇ = L L L L L L k B B AB B A AB k dUecñH lkçx½NÐCab;TIbI nigsmIkarlMnwg manragdUcteTA ³ 1 ∇2∇2ϕ −∇2u = 0, ∇2ϕ + D∇2∇2u − Z = 0. Eh k z k z Page 42
- 44. Analysis of Rectangular Shallow Shells ∂ ∂ Strain components: u y u x u u u ∂ + , , , y R x y 1 2 u R x xy y z y x z ∂ x ∂ ∂ + ε = ∂ + ε = ∂ ε = 2 2 u ∂ u ∂ u z z κ = − αβ , , . 2 2 x y y x y z κ = − ∂ x ∂ ∂ ∂ κ = − ∂ Internal forces: ( ) ( ) ⎡ M D u ⎡ ∂ M D ∂ u ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ 1 2 ⎞ ⎛ u u u ∂ + N C u u ∂ S C u ⎤ , ⎤ ⎡ ⎡ ∂ ∂ ∂ ∂ z z H D u ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ∂ ∂ ∂ = − − ν ⎤ , ⎤ ⎥⎦ ⎢⎣ ∂ ∂ ∂ + ν ∂ = − ⎥⎦ ⎢⎣ ∂ + ν ∂ = − ⎪ ⎪ ⎪ ⎪ ⎭ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ ∂ = − ν ⎥⎦ ⎢⎣ + + ν ∂ + ν ∂ = ⎥⎦ ⎢⎣ + + ν ∂ + ν ∂ = 1 . , 1 , 2 , 2 u u 2 2 2 2 2 2 2 2 2 1 x y x y y x x y k k u x y N C k k u y x z y z z x x y z y x y z x y x 2 2 3 k z C Eh D Eh k z ∂ = ∂ = where = , = 1 − ν 2 12 ( 1 − ν 2 ), , . 1 ∂ x 2 2 ∂ y 2 Equilibrium equations: k k u ( ) ( ) + ν ∂ 2 2 − ν ∂ ⎛ ⎞ 2 2 2 ⎤ ∂ u u k k ∂ u ⎞ X Y ⎡ + ∇ + + ν + − ν ∂ u ∂ u k k ∂ u ∂ ∂ + ν ∂ ⎛ h k k k k u Z ( ) ( ) ( 2 ) 0, 12 0, 2 1 2 1 0, 2 1 2 1 2 1 2 2 2 1 4 2 1 2 2 1 2 2 1 2 1 2 2 2 2 = − ⎥⎦ ⎢⎣ ∂ + + ν ∂ + ν + = ∂ ν + + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ + ∂ ∂ + = ∂ + + ν ∂ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ C y k k x C y x y y x C x x y u x y z x y z y x y z x ∂ ( ) ( ) ⎪ ⎪⎭ ⎫ ⎪ ⎪⎬ x x y z ∂ 2 ∇ ∂ κ + κ = ∂ ∂ ∂ = − ∇ ∂ κ + κ = ∂ = − , . 2 u y x y z y D y Q D u x D x Q D Page 43
- 45. Stress function ϕ = ϕ(x, y): 2 ∂ ϕ = ∂ ϕ = N x y ∂ ∂ , , . 2 2 2 2 x y S y N x ∂ ϕ = − ∂ ∂ Mixed differential equations of shallow shells: ⎪⎭ ⎪⎬ ⎫ 2 2 2 D ∇ ∇ u +∇ ϕ = Z z k Eh u ∇ ∇ ϕ− ∇ = , 0, 2 2 2 k z where ∂ ∂ , , 4 ∂ + ∂ 2 . 4 4 2 2 4 4 4 2 2 2 1 2 2 2 2 2 2 2 2 x x y y y k x k ∂ + x y k ∂ ∂ ∂ + ∂ ∂ ∇ = ∂ + ∂ ∇ = ∂ ∂ ∂ ∇ = L L L L L L L L L L Example 1. Mixed Method Equation of shallow shell: ( ) ( ) . ⎞ 2 4 z z x z y − − , ⎟⎠ = −⎛ − 2 4 , 2 2 2 2 2 2 2 2 2 1 2 = 2 −⎛ − 1 1 1 2 z R x a R a z R y b R b − − ⎟⎠ ⎜⎝ ⎞ ⎜⎝ = + 1 , 1 . k k x y Curvatures: ≈ = ≈ = Assume that all rims are simple supported: 1 R 2 2 1 k k R x x a u u M N 0, 0, = = → = = = = = = → = = = = z y x x y y b u u M N 0, 0. z x y y dUecH eyIgGaceRCIserIsykGnuKmnbMlas;TI nigGnuKmnsMBaFkñúg dUcmanrag Page 44
- 46. ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ n y u C m x sin sin , n y D m x π π z mn ϕ = π π Σ Σ = Σ ∞ Σ m = n 1,3 1,3 ∞ = ∞ = ∞ = sin sin , 1,3 1,3 mn m n b a b a where Cmn, Dmn = const. Surface distributed forces in double Fourier’s series: n y Z q m x sin sin , Σ ∞ Σ ∞ = 1,3 = 1,3 π π = m n mn a b where n y Z m x 4 sin sin ∫ ∫ π π = a b mn dxdy b a ab q 0 0 2 n y dxdy q b m x Z q q q 4 sin sin 16 0 0 π = − π π = − → = − mn ∫ ∫ a mn a b mn smIkarDIepr:g;Esülrbs;sMbk nwgmanragCasmIkarBICKNit ³ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⎤ = − ⎤ ⎥ ⎥⎦ EhC k n ⎛ π ⎡ DC m ⎛ π ⎢ ⎢⎣ k m ⎛ π ⎞ ⎟⎠ n ⎛ π + ⎟⎠ ⎜⎝ ⎞ ⎜⎝ n D m mn mn − ⎤ ⎥ ⎥⎦ ⎡ ⎛ π ⎡ D k n ⎢ ⎢⎣ ⎞ ⎟⎠ ⎤ ⎞ ⎛ π + ⎟⎠ k m ⎛ π ⎜⎝ ⎞ + ⎟⎠ ⎛ π ⎜⎝ = ⎥ ⎥⎦ ⎡ ⎢ ⎢⎣ ⎞ ⎟⎠ ⎜⎝ + ⎟⎠⎞ ⎜⎝ + ⎥ ⎥⎦ ⎢ ⎢⎣ ⎟⎠ ⎜⎝ ⎞ ⎜⎝ . 0, 2 2 2 2 2 1 2 2 2 1 2 2 q mn mn mn b a a b a b b a edaHRsaysmIkarenH eyIgTTYl)an ³ D Ehl q , , 2 mn mn D ⎛ k + Eh ⎞ mn mn D k Eh 4 2 4 2 ⎞ ⎟⎠ ⎜⎝ = − ⎟⎠ ⎜⎝⎛ + = mn mn mn mn mn mn l D l D C k q 2 ⎞ k m ⎛ π ⎞ l k n ⎛ π k m mn mn , . 2 2 1 2 2 ⎟⎠ ⎜⎝ + ⎟⎠ ⎜⎝ ⎞ = ⎟⎠ n ⎛ π + ⎟⎠ ⎜⎝ ⎞ ⎛ π = ⎜⎝ a b b a Page 45
- 47. Example 2. Method of Displacements For rectangular shallow shell of simple-supported rims: ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ n y u A m x cos sin , n y u B m x sin cos , n y u C m x π π x mn m n y mn = π π = π π = ΣΣ 0 1 ΣΣ m n ∞ 1 0 ΣΣ z mn m = n ∞ = ∞ = ∞ = ∞ = ∞ = sin sin . 1 1 b a b a b a External distributed forces: ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ n y X a m x cos sin , n y Y b m x sin cos , n y Z c m x π π = π π = π π = ΣΣ 0 1 ΣΣ 1 0 ∞ ΣΣ m = n ∞ = ∞ = ∞ = ∞ = ∞ = sin sin , 1 1 mn m n mn m n mn b a b a b a where n y X m x 4 cos sin , ∫ ∫ 0 0 n y Y m x 4 sin cos , ∫ ∫ 0 0 n y Z m x 4 sin sin . ∫ ∫ 0 0 π π = π π = π π = a b mn a b mn a b mn dxdy b a ab c dxdy b a ab b dxdy b a ab a a b c q mn mn mn If 16 X = Y = 0, Z = −q, then = = 0, = − . π2 mn dUecñH smIkarlMnwgsþaTic manragCasmIkarBICKNitdUcteTA ³ B k k m ( ) 0, A mn 2 1 n − ν ⎛ π 2 1 1 2 ⎤ 2 2 2 = π − + ν + ν π + ⎥ ⎥⎦ ⎡ m ⎛ π ⎢ ⎢⎣ ⎞ ⎟⎠ ⎜⎝ ⎞ + ⎟⎠ ⎜⎝ mn mn mn C a ab b a B k k n ( ) 0, m − ν ⎛ π 2 1 2 1 2 1 ⎡ 2 2 2 = π − + ν ⎤ ⎥ ⎥⎦ A n ⎛ π + ⎢ ⎢⎣ ⎞ ⎟⎠ ⎜⎝ ⎞ + ⎟⎠ ⎜⎝ + ν π mn mn mn C b a b mn ab Page 46
- 48. A k k n a ( ) ( ) mn mn 1 2 2 1 ⎤ k k k k C c 2 . ⎛ 4 2 2 h m 12 2 1 2 2 2 1 2 2 2 2 C n b a B b k k m mn mn = − ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ ⎞ + ν + + ⎟ ⎟⎠ ⎜ ⎜⎝ + π − − π + + ν π + ν edaHRsaysmIkarxagelI eyIgTTYl)an ( ) ( ) ( ) ( ) m A k k k l 1 , mn mn n B k k k l 1 , mn mn , 1 2 2 1 2 2 2 mn mn D ⎛ k + Eh 4 2 ⎞ ⎟⎠ ⎜⎝ = − + + ν π = − + + ν π = mn mn mn mn mn mn mn mn mn l D C c k C b k C a k ⎫ ⎪ ⎪⎭ ⎪ ⎪⎬ 2 2 π n + 2 2 2 2 π where, k m mn l = k m 2 2 π + π = . , 2 k n 2 1 2 2 2 b a b a mn Page 47
- 49. Analysis of Rectangular Shallow Shell (method of displacements) ORIGIN := 1 a b ⎛⎜⎝ ⎞⎟⎠ 8 6 ⎛⎜⎝ ⎞⎟⎠ := R1 R2 ⎛⎜⎝ ⎞⎟⎠ 20 2000 ⎛⎜⎝ ⎞⎟⎠ := E ν ⎛⎜⎝ ⎞⎟⎠ 2 10 8 ⋅ 0.2 ⎛⎜⎝ ⎞⎟⎠ := h := 0.15 n1 n2 ⎛⎜⎝ ⎞⎟⎠ 3 3 ⎛⎜⎝ ⎞⎟⎠ := q := 1.1 External force: Z(x, y) := −q Equation of shallow shell: − R12 a2 z1 x ( ) R1 2 x a 2 − ⎛⎜⎝ ⎞⎟⎠ 2 − R22 b2 − − := z2 y ( ) R2 2 y 4 b 2 − ⎛⎜⎝ ⎞⎟⎠ 2 4 := − − z(x, y) := z1(x) + z2(y) z a 2 b 2 , ⎛⎜⎝ ⎞⎟⎠ = 0.406 Axial stiffness C1 E ⋅ h 1 ν − 2 := Flexural stiffness D E ⋅ h3 ⋅ ( − 2) 12 1 ν := Curvatures k1 1 R1 := k2 1 R2 := m := 1 .. max(n1, n2) Im := 2 ⋅ m − 1 I 1 3 5 ⎛⎜⎜⎝ ⎞⎟⎟⎠ = m := 1 .. n1 αm Im ⋅ π a := n := 1 .. n2 βn In ⋅ π b := Coefficients of external forces: m := 1 .. n1 n := 1 .. n2 cm, n 4 a ⋅ b b 0 y a Z(x, y) sin α x ⋅ ( m ⋅ x) sin β⋅ ( n ⋅ y) ⌠⎮⌡ 0 d ⌠⎮⌡ := ⋅ d c ⎛⎜⎜⎝ ⎞⎟⎟⎠ −1.783 −0.594 −0.357 −0.594 −0.198 −0.119 −0.357 −0.119 −0.071 = Page 48
- 50. i := 0 .. 10 j := 0 .. 10 z0i+1, j+1 z a i 10 ⋅ b j 10 ⋅ , ⎛⎜⎝ ⎞⎟⎠ := Rectangular Shallow Shell z0 ⋅ 10 Coefficients of system: A11(m, n) α( m)2 1 − ν β( n):= + ⋅ 2 A12(m, n) 2 1 + ν 2 αm ⋅ βn := ⋅ A13(m, n) −(k1 + ν ⋅ k2) αm := ⋅ A21(m, n) 1 + ν 2 αm ⋅ βn := ⋅ A22(m, n) β( n)2 1 − ν α( m):= + ⋅ 2 A23(m, n) k2 −( + ν ⋅ k1) βn := ⋅ 2 A31(m, n) (k1 + ν ⋅ k2) αm := ⋅ A32(m, n) (k2 + ν ⋅ k1) βn := ⋅ Page 49
- 51. A33(m, n) h2 12 ⎡⎣ ( 2 αm)+ ( 2 βn)⎤⎦ 2 ⋅ + k12 + 2 ⋅ ν ⋅ k1 ⋅ k2 + k22 ⎡⎢⎣ ⎤⎥⎦ := − B1(m, n) 0 0 cm, n − C1 ⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠ := A1(m, n) A11(m, n) A21(m, n) A31(m, n) A12(m, n) A22(m, n) A32(m, n) A13(m, n) A23(m, n) A33(m, n) ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Coefficients of displacement: m := 1 .. n1 n := 1 .. n2 Am, n Bm, n Cm, n ⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠ A1(m, n)− 1 := ⋅ B1(m, n) A −3.445 10− 6 × −1.538 10− 7 × −9.168 10− 9 × −2.012 10− 8 × −5.915 10− 9 × −1.149 10− 9 × −7.331 10− 10 × −4.193 10− 10 × −1.707 10− 10 × ⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠ = B 5.673 10− 7 × 4.462 10− 8 × 2.034 10− 9 × −5.318 10− 9 × 9.742 10− 10 × 3.227 10− 10 × −4.153 10− 10 × −4.655 10− 12 × 2.811 10− 11 × ⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠ = C −4.264 10− 5 × −3.622 10− 6 × −3.568 10− 7 × −1.267 10− 6 × −2.197 10− 7 × −5.05 10− 8 × −1.209 10− 7 × −2.948 10− 8 × −1.057 10− 8 × ⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠ = Displacements ux(x, y) 1 n1 n2 Am, n cos α⋅ ( m ⋅ x) sin βn y ⋅ ( ) ⋅ ( ) Σ= m n 1 Σ= := uy(x, y) Bm, n sin α( m ⋅ x) cos βn y ⋅ ( ) ⋅ ( ) Σ=Σ= 1 n1 n2 m n 1 := uz(x, y) 1 n1 n2 Cm, n sin α⋅ ( m ⋅ x) sin βn y ⋅ ( ) ⋅ ( ) Σ= m n 1 Σ= := Page 50
- 52. Internal forces: Nx(x, y) C1 1 n1 n2 m n 1 (k1 + ν ⋅ k2) ⋅ Cm, − α⋅ Am, − ν ⋅ β⋅ Bm, ⋅ n m n n n sin ( α⋅ m x) ⋅ sin ( β⋅ n y ⎡⎣ ⎡⎣ ) ⎤⎦ ⎤⎦ Σ= Σ= := ⋅ Ny(x, y) C1 1 n1 n2 m n 1 (k2 + ν ⋅ k1) ⋅ Cm, − ⋅ n ν α⋅ m Am, − β⋅ Bm, ⋅ sin ( α⋅ n n n m x) ⋅ sin ( β⋅ n y ⎡⎣ ⎡⎣ ) ⎤⎦ ⎤⎦ Σ= Σ= := ⋅ S(x, y) 1 − ν 2 ⋅ C1 1 n1 n2 m n 1 αm Am, n ⋅ βn Bm, n ( + ⋅ ) cos α⋅ ( m ⋅ x) cos βn y ⋅ ( ) ⋅ ⎡⎣ ⎤⎦ Σ= Σ= := ⋅ Mx(x, y) −D 1 n1 n2 m n 1 ⎡⎣ Cm, ⋅ ( 2 αm)+ 2 n ν ⋅ ( βn)⋅ sin ( α⋅ x) ⋅ sin ( β⋅ m n ⎡⎣ y) ⎤⎦ ⎤⎦ Σ= Σ= := ⋅ My(x, y) −D 1 n1 n2 m n 1 ⎡⎣ Cm, ⋅ ( 2 βn)+ 2 n ν ⋅ ( αm)⋅ sin ( α⋅ x) ⋅ sin ( β⋅ m n ⎡⎣ y) ⎤⎦ ⎤⎦ Σ= Σ= := ⋅ H(x, y) (1 − ν) ⋅ D 1 n1 n2 Cm, n αm ⋅ βn ⋅ cos α⋅ ( m ⋅ x) cos βn y ⋅ ( ) ⋅ ( ) Σ= m n 1 Σ= := ⋅ Qx(x, y) D 1 n1 n2 m n 1 ⎡⎣ Cm, ⋅ α⋅ ( 2 αm)+ n m ( βn)2 ⋅ cos ( α⋅ x) ⋅ m sin ( β⋅ n y) ⎡⎣ ⎤⎦ ⎤⎦ Σ= Σ= := ⋅ Qy(x, y) D 1 n1 n2 m n 1 ⎡⎣ Cm, ⋅ β⋅ ( 2 αm)+ n n ( βn)2 ⋅ sin ( α⋅ x) ⋅ m cos ( β⋅ n y) ⎡⎣ ⎤⎦ ⎤⎦ Σ= Σ= := ⋅ Rx(y) D 1 n1 n2 m n 1 ⎤⎦⋅ sin β⋅ ( n ⋅ y) ⎡⎣ Cm, n αm ⋅ α( m)2 (2 − ν) β( n)2 ⋅ + ⎡⎣ ⎤⎦ Σ= Σ= := ⋅ Ry(x) D 1 n1 n2 m n 1 Cm, ⋅ ⋅ n βn ⎡⎣ ( 2 βn)+ (2 − ν) ⋅ ( m)2 α⋅ sin ( α⋅ m x) ⎡⎣ ⎤⎦ ⎤⎦ Σ= Σ= := ⋅ R0 2 ⋅ (1 − ν) ⋅ D 1 n1 n2 Cm, n αm ⋅ βn ⋅ ( ) Σ= m n 1 Σ= := ⋅ Page 51
- 53. At the section y b 2 := x := 0, 0.01 ⋅ a .. a 0 2 4 6 8 0 − 1 10− 5 × − 2 10− 5 × − 3 10− 5 × − 4 10− 5 × Deflection uz at section y=b/2 uz(x, y) x 0 2 4 6 8 0 − 10 − 20 − 30 Normal force diagrams at y=b/2 Nx(x, y) Ny(x, y) x 0 2 4 6 8 0 − 0.2 − 0.4 − 0.6 Bending moment diagrams at y=b/2 − Mx(x, y) − My(x, y) x 0 2 4 6 8 1 0.5 0 − 0.5 − 1 Shearing force diagrams at y=b/2 Qx(x, y) Qy(x, y) x Page 52
- 54. At the section x a 2 := y := 0, 0.01 ⋅ b .. b 0 2 4 6 0 − 1 10− 5 × − 2 10− 5 × − 3 10− 5 × − 4 10− 5 × Deflection uz at section x=a/2 uz(x, y) y 0 2 4 6 0 − 10 − 20 − 30 Normal force diagrams at x=a/2 Nx(x, y) Ny(x, y) y 0 2 4 6 0 − 0.2 − 0.4 − 0.6 Bending moment diagrams at x=a/2 − Mx(x, y) − My(x, y) y 0 2 4 6 1 0.5 0 − 0.5 − 1 Shearing force diagrams at x=a/2 Qx(x, y) Qy(x, y) y Page 53
- 55. m := 0 .. 20 x1m+1 a m 20 := ⋅ n := 0 .. 20 y1n+1 b n 20 := ⋅ uz1m+1, n+1 uz x1m+1 y1n+1 := ( , ) Mx1m+1, n+1 Mx x1m+1 y1n+1 := ( , ) My1m+1, n+1 My x1m+1 y1n+1 := ( , ) Deflection uz uz1 ⋅ 105 Page 54
- 56. Bending moment Mx −Mx1 Bending moment My −My1 Page 55
- 57. Shells of Revolution r O α dα α Nα r ds1 dr Nα+d Nα α z dz C C1 R2 R1 z = sin α 2 r R ds CC R d Ad = = α = α 1 1 1 A R ⇒ = (α) 1 ( α, β = meridian and parallel. r(α) – meridian equation. ds rd R sin d = β = α β 2 2 B R sin ⇒ = α 2 dr CC cos R cos d 1 1 B r R = α ∂ = ∂α ∂ ∂α ⇒ = α = α α cos 1 ( Case of Axis-Symmetrical Shell: Y = 0 = = = 0, = ε = κ = 0 β β αβ αβ S Q H u = 0 ∂ k L ∂β k Equilibrium equations: ( ) R N R N R Q R R X sin α − cos α − sin α + sin α = 0, 2 α 1 β 2 α 1 2 ( ) ⎫ ⎪ ⎪ ⎪ ⎬ d R N R N d sin cos sin sin 0, 2 α 1 β 2 α 1 2 ( ) ⎪ ⎪ ⎪ ⎭ R M R M R R Q sin α + cos α + sin α = 0. d α − α − α = α α + α + α 2 1 1 2 α β α d R Q R R Z d d Strains: du 1 ⎛ + ⎞ , 1 ( cotg ) , z β α z α 1 2 ⎤ ⎡ u du u du d 1 1 ⎛ − ⎞ , cotg . 1 1 1 2 ⎞ ⎟⎠ ⎛ − ⎜⎝ α = κ ⎥⎦ ⎢⎣ ⎟⎠ ⎜⎝ α α κ = + α = ε ⎟⎠ ⎜⎝ α ε = α α β α dz dz R R d R R u u R u d R z z Page 56
- 58. E.Meissner’s unknowns: ⎞ duz 1 , χ = − R Q α α = ψ ⎟⎠ ⎛ + ⎜⎝ α R u d 2 1 ecjBIsmIkarbMErbMrYlragxagelI b¤ecjBIlkçx½NÐCab; edayeyagelIc,ab; Hooke eyIg)an ³ 1 ( cotg ) , 2 du 1 ⎛ + , N N N N α β α ( ) ( ) ⎞ M M 12 1 ⎡ 1 ⎛ , M M u du d cotg . 12 1 2 3 1 1 3 1 ⎞ ⎟⎠ ⎛ ⎜⎝ u du α − α = − ν − ⎤ ⎥⎦ ⎢⎣ ⎞ ⎟⎠ ⎜⎝ α − α = − ν − ⎟⎠ ⎜⎝ α = − ν = + α = ε − ν α β α α α β α β β α d Eh R R d d R Eh R u d Eh R u u Eh R z z z z ecjBIsmIkarBIrmun eyIgTTYl)an ³ [( R R ) N ( R R ) N ] α α β du α − α = + ν − + ν α Eh u d 1 2 2 1 cotg 1 eFVIDIepr:g;EsülelIsmIkarTImYy eyIgnwgman ³ ⎤ R u u d ⎡ − ν α ( ) ( ) cotg , ⎤ α β α R d ⎡ − ν α ( ) . u du sin cotg 2 2 2 ⎥⎦ ⎢⎣ = α + α α − d du α ⎥⎦ ⎢⎣ + α = α β α α α N N Eh d d d N N Eh d d z z ecjBIsmIkarBIrxagelIenH eyIgTTYl)an ³ 1 R R R N R R N d u duz cotg [( ) ( ) ] ( ) . 2 1 2 2 1 ⎤ ⎥⎦ ⎡ − ν α ⎢⎣ + ν − + ν − α = = χ = α − α β β α α N N Eh d Eh R d Page 57
- 59. müa:geTot eyIgGacsresr)anfa d R cotg 0 , 1 , α α β β d R 1 χ , cotg , α β d R 1 2 0 1 ⎞ ⎛ ⎞ ψ d d R 1 χ cotg , cotg 1 , 1 2 2 1 2 ⎟ ⎟⎠ ⎜ ⎜⎝ χ α χ + ν α − = ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ χ α + ν α = − χ α κ = α κ = + α ψ + = − α = − α β d R R M D d R M D N d N N R N Edl 0 , 0 α β N N CakMlaMgEkg tamRTwsþIKμanm:Um:g; (zero moment theory of shells) Ed;lmanragdUcteTA ³ ⎤ sin ( cos sin ) , = ∫ α 1 α C R R Z X d sin ⎡ 2 1 2 1 2 0 ⎥ ⎥⎦ ⎢ ⎢⎣ + α α − α α α α R N ⎞ . 1 N R Z N 2 0 ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ = − α β R bnÞab;BICMnYstMélkMlaMgEkg cUleTAkñúgsmIkarlMnwgBIrdMbUg eyIgeXIjfa vaRtUv)anepÞógpÞat; . rIÉsmIkarTIbI rYmCamYynwglkçx½NÐCab; begáIt)anCa cotg 3 R 2 R 2 ⎤ ⎤ ⎡ ⎞ ⎛ α χ dh R R dh d R d 3 cotg cotg , cotg R 2 R 1 R 2 1 ⎤ χ d ψ ⎤ ⎡ ⎞ ⎛ α ψ dh dh d cotg cotg ( ), 1 2 d R d R R 1 2 1 2 1 2 2 2 ⎡ 2 1 2 1 2 1 2 1 1 2 2 1 α Φ + χ = ψ ⎥⎦ ⎡ ⎢⎣ α − ν α ν − α − − α ⎥⎦ ⎢⎣ α − α + ⎟ ⎟⎠ ⎜ ⎜⎝ + α ψ − = χ ⎥⎦ ⎢⎣ + α α ν α − ν − − α ⎥⎦ ⎢⎣ α + α + ⎟ ⎟⎠ ⎜ ⎜⎝ + α EhR d R h d d R h R R d d R D R d h d d R h R R d d R where ( ) h d R ( ) cotg [( ) ( ) 0 ]. Φ α = N N R R N R R N 2 1 0 1 2 ⎡ − ν α 2 0 0 ⎤ β α α β ν + − ν + α − ⎥⎦ ⎢⎣ h d Page 58
- 60. Case h=const: ( ) ν ν χ = − 1 ψ , ( ψ ) + ψ = χ + 1 Φ ( α ). 1 1 1 χ − R Eh R L R D L where d R R ⎛ α d d L R ⎤ ⎡ ⎞ 1 cotg 2 L cotg L 2 ( L ) (L) 2 2 1 2 1 1 2 2 2 1 d R R R d d R R α − α ⎥⎦ ⎢⎣ α + ⎟ ⎟⎠ ⎜ ⎜⎝ + α = ecjBIsmIkarxagelI eyIgGacTaj)anfa ( ) ( ) ( ) ( ) ( ) ( )⎪ ⎪ ⎫ ⎪ ⎪ ⎬ ⎭ Φ α ⎞ ν − ⎟ ⎟⎠ Eh ⎛ ⎜ ⎜⎝ − ⎞ ν 2 ψ = ψ ⎞ ν − ⎟ ⎟⎠ ⎞ ⎛ χ ⎛ Φ ⎜ ⎜⎝ ψ − ν ψ − Φ α − χ ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ − ν χ = ν + ⎟ ⎟⎠ ⎜ ⎜⎝ χ − ν , , 2 1 2 2 1 1 1 1 1 2 1 1 1 Eh D R R L R R R LL L D R D R L R R LL L ebI]bmafa L ( ) , 1 ϕ ϕ χ = − ν ψ = ϕ − R D 1 enaHsmIkarTI1 nwgepÞógpÞat; ehIysmIkarTI2 nwgTTYl)anragCa ( ) ⎞ ( ) ⎛ ν ⎞ ( ) ν + ⎟ ⎟⎠ ⎛ ϕ LL L Φ α 1 2 2 1 L Eh R R = ϕ ⎟ ⎟⎠ ⎜ ⎜⎝ ϕ + − 1 1 D R R ⎜ ⎜⎝ ϕ − ν For spherical, toroidal, conical, cylindrical shells: R1=const. So, ( ) ( ) LL Φ α 1 2 R ϕ + μ ϕ = where ( ) 2 2 ( ) μ = − 1 12 1 b , 12 1 1 . 2 1 2 Eh = 2 1 2 2 1 2 R h R D R R − ν ≈ ν − ν b R 2 2 1 2 2 h = Page 59
- 61. smIkarcugeRkayenH Gacsresr)aneTACa [ ( ) ][ ( ) ] ( ), 1 R L i L i Φ α L + μ ϕ − μ = b¤k¾ [ ( ) ] [ ( ) ] ( ), 1 R L L i i L i Φ α ϕ + μ − μ ϕ + μ = dMeNaHRsayrYmrbs;smIkarTaMgenH GacTTYl)anCaragkMpøic . krNIEs‘Vr R1=R2=R smIkaredImrbs;smIkarDIepr:g;EsülxagelI manragCa ( ∇ 2 + μ ) ϕ = 0, ( ∇ + μ ) ϕ = 0, 1 2 2 22 1 1 where RL d d μ = 1 + bi = ζ ( ζ + 1), μ = 1 − bi = ζ ( ζ + 1), 1 1 1 2 2 2 ∇ = − = L L L ( )( ) , 1 cotg 2 2 sin 2 2 1 L L + α − α α α d d dMeNaHRsayBiessrbs;smIkarDIepr:g;EsülxagelI Gacrk)anecjBIsmIkar ( ) ( ). L i i Φ α b ϕ + μϕ = smIkaredImk¾Gacsresr)anCarag ⎫ ⎤ d cotg 1 1 ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎬ ⎭ ⎤ = ϕ ⎥⎦ d cotg ⎡ 1 1 ⎢⎣ α ϕ ϕ + ζ ζ + − α + α ϕ 1 ϕ α = ϕ ⎥⎦ ⎢⎣⎡ α + ζ ζ + − α + α α 0. sin 0, sin 2 2 2 2 2 2 2 2 1 1 2 1 1 2 2 d d d d d d smIkarDIepr:g;EsülxagelIenH GacGaMgetRkal)an edayeRbIGnuKmn_ Legendre . Page 60
- 62. Example. Spherical Cupola , const 1 2 R = R = R h = Equations: ⎫ L R ( ) ( ) ( )⎪⎭ ⎪⎬ χ − νχ = − ψ 1 L EhR ψ + νψ = χ +Φ α , , 1 D d L RL d L L ( ) ( ) ( ) ( ) (1 ) . cotg cotg , = = L R dZ 2 2 2 2 2 1 R X d d d + + ν α Φ α = α − α α + α L L where Common solutions: C X C X C Y C Y ψ = + + + 0 1 1 2 2 3 1 4 2 1 , [ X ( C C ) X ( C C ) ( ) ( )] 1 1 3 2 2 4 χ = λ + ν + λ + ν + 0 1 3 1 2 4 2 EhR Y C C Y C C + − λ + ν + − λ + ν Legendre functions: , ⎛ π 2 8 sin ⎛ π 2 8 cos 1 3 2 cotg 8 λ 4 sin 2 ⎤ ⎡ ⎞ ⎞ ⎞ ⎛ 1 ⎥ ⎥⎦ ⎢ ⎢⎣ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − λ α − ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − λ α ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ λ α − π α ≈ λ α X e , ⎛ π 2 8 sin 1 3 2 cotg 8 ⎛ π 2 8 cos λ 4 sin 2 ⎤ ⎡ ⎞ ⎞ ⎛ ⎞ 1 ⎥ ⎥⎦ ⎢ ⎢⎣ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − λ α ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ λ α − + ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − λ α π α ≈ λ α Y e , ⎛ π 2 8 sin 1 3 2 cotg 8 ⎛ π 2 8 cos λ sin 2 ⎤ ⎡ ⎞ ⎞ ⎛ ⎞ 2 ⎥ ⎥⎦ ⎢ ⎢⎣ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ + λ α ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ λ α + − ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ + λ α π α ≈ λ −α X e ⎤ . ⎛ π 2 8 sin ⎛ π 2 8 cos 1 3 2 cotg 8 λ sin 2 ⎡ ⎞ ⎞ ⎞ ⎛ 2 ⎥ ⎥⎦ ⎢ ⎢⎣ ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ + λ α + ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ + λ α ⎟ ⎟ ⎠ ⎜ ⎜ ⎝ λ α + π α ≈ λ −α Y e Solution of differential equations: ( ) ( ) ( ) ( ) ⎭ ⎬ ⎫ ψ = ψ α +ψ α 0 1 ψ (α) χ (α) 1 1 , = particular solutions χ = χ α + χ α . , 0 1 Page 61
- 63. h X 45° 45° 20m α z R Z q p R = 14.4 m , h = 1 cm E kgf = ⋅ 6 ν = Self weight: Support 2 2 10 , 0.3 2 cm g = kgf 2 0.008 cm Live load: p = kgf 2 0.02 cm Support 1 2 2 R EhR λ2 = 2μ2 = − ν D enARtg;kMBUlEs‘Vr α=0 GnuKmn_ X2, Y2 mantMél infinity . RbkarenHxusBI karBitCak;Esþg dUecñHRtUvlubbM)at;va edaydak;eGay C2 = C2 = 0 . rIÉ )a:ra:Em:Rtefr C1, C3 kMNt;ecjBIlkçx½NÐRBMEdn . Vertical load on 1m2 of cupola surface: q = g + p cosα Components of the vertical load: X q g p sin sin sin cos , = α = α + α α Z = q α = g α + p 2 α cos cos cos . Load function: R dZ ( ) ( ) R 1 X 2 2 + + ν α Φ α = = 2 ( + ν) α α + 2 ( + ν) α pR gR 3 sin cos 2 sin d Page 62
- 64. dMeNaHRsayBiess eKrkCarag A A sin sin cos , χ = α + α α 1 1 2 ψ = α + α α B B sin sin cos . 1 1 2 bnÞab;BICMnYstMélTaMgenH cUleTAkñúgsmIkarxagedIm eKrkeXIj A R g A + ν R D ( ) ( ) + ν (3 ). 25 2 , 5 + ν 1 1 , 25 2 , 3 1 1 p D 2 2 2 2 1 2 3 2 2 3 1 2 + ν λ + + ν = − λ + = − λ + + ν = − λ + = − B gR B pR dMeNaHRsaysrubrbs;smIkarDIepr:g;Esül Gacsresr)anfa ( ) ⎫ ⎪⎬ C X C Y ψ = + +ψ α , 1 1 3 1 1 1 . [ X ( ⎪⎭ C C ) Y ( C C ) ] ( )χ = λ + ν + − λ + ν + χ α 1 3 1 1 1 3 1 EhR )a:ra:Em:Rtefr C1, C3 kMNt;ecjBIlkçx½NÐRBMEdn α=45° dUcteTA X d cotg 0 ⎛ + νχ α α M D 45 = ⎟⎠ 45 ⎞ ⎜⎝ χ = − α= ° α= ° R d C C dY C C dX 1 ( ) ( ) cotg + ν d C C X C C Y R [( ) ( ) ]} ( ) + ν (cos 2 cos ) 0 25 g R 1 cos 3 1 2 45 2 2 3 2 3 3 1 1 1 3 1 1 1 3 1 3 1 α + ν α = λ + ⋅ + ν α − ⋅ λ + + λ + ν + − λ + ν − + ⎩ ⎨ ⎧ + ν α α + − λ + ν α λ + ν α= ° p D D d EhR Y Case of simple support α α= ° z α= ° β α= ° u u 0 0 45 45 45 = = → ε = Page 63
- 65. ( ) ( ) ( ) 1 0 ZR F ⎡ α + +ψ + cotg sin 1 1 45 d ψ C dY 1 1 3 1 C dX R 1 2 1 1 3 1 1 ⎤ = ⎥⎦ ⎞ ⎟⎠ ⎛ ⎜⎝ α + α + α + ⎢⎣ ν + α α → − + ν α= ° d d d C X C Y R R Eh enARtg;enH = α α − α α = α ∫ α ( ) ( ) F R sin Z cos X sin d = − 2 2 α − 2 ( − α) sin 1 cos 1 2 0 2 pR gR Z Case of roller support sin cos 0, 0 45 45 α − α = = α α α= ° α α= ° Q N u Internal forces: ⎞ ⎛ d ν χ ⎞ χ d R 1 cotg ; cotg ; ⎛ α β ( ) N ZR F ( ) 1 ; sin cotg ; ; R N F sin 1 2 1 2 2 2 ψ = 2 2 1 2 2 1 ψ α − α α α = − ψ − α α = ⎟ ⎟⎠ ⎜ ⎜⎝ α χ + α − = ⎟ ⎟⎠ ⎜ ⎜⎝ χ ν α + α = − α α β d d R R R R Q d R R M D d R M D Strains: ( ) ⎤ ⎞ ⎛ ψ ⎞ ⎛ ν α d R ( ) 1 1 cotg . sin 1 1 cotg 1 ; sin 1 2 2 ⎞ ⎛ ν 2 1 2 α 2 2 2 2 1 2 2 ⎤ ⎥⎦ ⎡ ⎢⎣ α νψ ψ + + α − ⎟ ⎟⎠ ⎜ ⎜⎝ + α α ε = − ⎥⎦ ⎢⎣ ⎡ ⎟ ⎟⎠ ⎜ ⎜⎝ α α − ν − ψ − ⎟ ⎟⎠ ⎜ ⎜⎝ + α ε = β R ZR d d R R R F Eh d ZR R R R F Eh Page 64
- 66. Displacements: edaHRsaysmIkar du 1 ⎞ , 1 ( u cotg u ), z z 1 2 ε = β α R u ⎛ + d R + α = ε ⎟⎠ ⎜⎝ α α α eyIgTTYl)an ( ) α ⎤ ( ) (1 ) . sin R F sin 1 1 sin 2 α ∫ d cotg 1 cotg , ⎡ − sin sin RZ F ⎡ + α 2 1 2 2 2 ⎤ ⎥⎦ ⎢⎣ α α ⎞ R − + ⎟⎠ ⎛ − νψ α α ⎜⎝ ψ = − α − α ⎥⎦ ⎢⎣ α α α + ν ψ + + ν = α + α α α R Eh d Eh u u RZ d R Eh Eh u A z Edl A2 Ca)a:ra:Em:Rtefr nigkMNt;)anecjBIlkçx½NÐRBMEdn . Page 65
- 67. Zero Moment (Membrane) Theory of Shells = = = 0, = = 0 α β α β M M H Q Q Equilibrium equations: 1 ∂ 0, ∂ ( ) ( ) ( ) ( ) ⎫ ⎪⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ∂ α β β α N + − = + = ∂ ∂α + ∂β − ∂ ∂ ∂β + = ∂β + ∂α − ∂α α β 0. 2 1 0, 1 2 2 Z R N R B S ABY B AN N A A S ABX A BN N B The problem is statically determinate. eKaledAénkar KNnaKμanm:Um:g; KWkMNt;rksPaB sMBaFkñúgem (principal stress state) mYyEdledIr tYnaTIsMxan; . lkçx½NÐ zero-moment stress-strain state: X Shell RtUvEtmankMras;efr b¤ERbRbYledaysnSwm² ehIydUcKñaEdr cMeBaH kaMkMeNag minRtUvERbRbYlya:gxøaMgenaHeT . Y kMlaMgeRkA RtUvEtCab;Kña nigERbRbYledaysnSwm². Zero-moment shell minGaceFVIkarnwgkMlaMgeTal)aneT . Z Shell RtUvmanTMrya:gNa Edlpþl;lTæPaBeFVIclnatamTisEkg edayesrI KWenAelIEKmrbs; shelltamTisEkg minRtUvTb;sáat;mMurgVil nigbMlas;TIeT . edIm,IeGayeBjelj TMrkñúgbøg;b:H k¾minRtUvnaMeGaymankarBt;esaHeLIy . [ kMlaMg Edlsgát;elIEKmrbs; shell RtUvsßitenAkñúgbøg;b:Hnwg shell enaH. Page 66
- 68. Analysis of Shells of Revolution r O α dα α Nα r ds1 dr Nα+d Nα α z dz C C1 R2 R1 z α, β = meridian and parallel. ( ) sin , , A R = α B r R = = α cos . B R 1 2 1 = α ∂ ∂α Equilibrium equations: ( ) sin cos sin 0, ( ) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ∂ β 1 + − = ∂ sin α + sin α = 0, ∂α α + ∂ ∂β + α = ∂β α − α + ∂ ∂α α β α β 0. sin 2 1 1 2 1 2 2 2 2 2 1 2 1 1 1 2 R N R N R R Z R S R R Y R N R R N N R R S R R X ∂ k Y L Case of axis symmetrical problem: 0, = 0 = k ∂β = = = 0 β H Q S ( ) ⎫ ⎪⎭ ⎪⎬ R N N R R R X sin α − cos α + sin α = 0, 2 1 1 2 d R N R N R R Z + − = α α β α β 0. 2 1 1 2 d Page 67
- 69. ⎞ ⎛ ecjBIsmIkarTI 2 eyIgTTYl)an ³ ⎟ ⎟⎠ ⎜ ⎜⎝ = − α β 2 R 1 N R Z N CMnYscUleTAkñúgsmIkarTI 1 eyIgnwgman ³ d ( sin ) ( sin cos ) 0 1 α + α − α = α α rN rR X Z d ecjBIenH rN sin α = rR ( Z cos α − X sin α ) d α + C 1 ∫ α α α 1 ( ) ⎤ ⎥ ⎥⎦ ⎡ = ∫ α ⎢ ⎢⎣ + α α − α α α α α 1 sin cos sin 1 sin 2 1 2 2 C R R Z X d R N Edl C Ca)a:ra:Em:Rt nigrk)anecjBIlkçx½NÐRBMEdn . RbsinebI smIkaremrIdüanRtUv)aneKeGayCarag r = r(z) enaHsmIkarrbs; épÞrgVil KitenAkUGredaenEkg Gacsresr)anfa x = r sinβ, y = r cosβ, z = z r dr dUecñH eyIg)an ′ = = cotgα, dz ( ) ( ) ⎪⎭ ⎪⎬ ⎫ = + ′ A r = 1 2 , . 2 1 B r z ⎪⎭ ⎪⎬ ⎫ = = + ′ CC ds dz r 1 , . = = β 2 1 2 1 1 ( ds CD rd ( Curvatures: , 1 k r ′′ ( ) . 1 2 1 2 2 ( 1 2 ) 1 2 1 r r k r + ′ = + ′ = − Page 68
- 70. Equilibrium equations: ( rN ) ∂ − r ′ N + + r ′ S r r X z 1 1 1 0, ( ) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ∂ α β β 2 2 2 + − + ′ = ′′ rr + ′ ∂ − + + ′ = ∂ + ∂ ∂β + ′ + + ′ = ∂β ∂ α β 1 0. 1 1 1 0, 2 2 2 2 N N r r Z r r S r r Y r z N r For homogeneous problem: X = Y = Z = 0 eKtag stress function: ⎞ ⎟⎠ ⎛ ϕ ∂ N A , N r , 2 = α β z r ⎜⎝ ∂ = − ′′ ∂ϕ ∂β = ∂ϕ ∂β S rA r enAkñúgkrNIenH smIkarTI 1 nigTI 3 epÞógpÞat; rIÉsmIkarTTYl)anrag ³ ⎞ ∂ ϕ 0 2 2 ∂ ϕ 2 2 = ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ ∂β ϕ+ ′′ − ∂ r r z For axis symmetrical problem: Y = 0 ( ) ⎫ ⎪ ⎪⎭ ⎪ ⎪⎬ rN r N r r X α β + − + ′ = ′′ rr + ′ d − − ′ + + ′ = α β 1 0. 1 1 0, 2 2 2 N N r r Z r dz Equilibrium equations ( ) α ∫ 1 . N r N rr 1 ⎤ ⎡ 1 ; 2 2 2 0 N r r Z ′′ r C r r Z X dz r z z + + ′ + ′ = ⎥ ⎥⎦ ⎢ ⎢⎣ + ′ − + ′ = β α Solution Page 69
- 71. z q Q q z α0 r0 R2 R1 α dα r k2 k1 X Nα Nαsinα Z rUbmnþ Nα Gacsresr)anfa³ N sin α ⋅ R sin α ⋅ 2 π = 2 π R R sin α ( Z cos α − X sin α ) d α + 2 π C ∫ α 2 1 2 α α 0 Integration Technique ( ) q r d R r X Z N r ⋅ π + α ⋅ π ⋅ α − α = α ⋅ π ∫ α α 1 0 2 sin cos sin 2 2 or α 0 tYeqVgénsmPaBxagelI KWCacMeNalelIG½kS z énpÁÜbrbs;kMlaMgEkg tamrgVg; EdlmankaM r . edayehtufa 2πrR1dα KWCaépÞénvgStUcminkMNt;mYy EdlRtUvnwgmMu dα/ rIÉ Zcosα nig Xsinα KWCacMeNalelIG½kS z énkMlaMgeRkA dUecñH ( Z cos α − X sin α ) ⋅ 2 π r ⋅ R d α = Q 1 z ∫ α α0 Edl Qz CacMeNalénpÁÜbrbs;kMlaMgeRkA EdleFVIGMeBIelIépÞrbs; shell enA EpñkxagelIénmuxkat; α . )a:ra:Em:Rtefr C GacsresrCarag C=r0q/ Edl q CaGaMgtg;suIeténkMlaMg tamTisG½kS z Edlsgát;tamrgVg;kaM r0 . sMrab;krNIGvtþmankMlaMgenH KW C=0 ehIy Page 70
- 72. . N Qz = α r 2π sin α kñúgkarkMNt; Qz eKGaceRbIR)as;RTwsþIbT dUcxageRkam . RTwsþIbT 1> RbsinebI elIépÞNamYy eFVIGMeBIsMBaFBRgayesμI p enaHminGaRs½y nwgrUbragépÞ cMeNalénkMlaMgpÁÜbrbs;sMBaFelIG½kSNamYy esμIplKuNsMBaF p enaH nwgRkLaépÞrbs;cMeNalénépÞelIbøg; EdlEkgnwgG½kSenaH . RTwsþIbT 2> RbsinebI elIépÞNamYy eFVIGMeBIsMBaFGgÁFaturav enaHkMlaMgpÁMú bBaÄrrbs;sMBaFenaH esμITMgn;GgÁFaturavkñúgmaD EdlenAelIépÞ . Example 1. R α p dα α α p R Nq α Nq α q Spherical cupola: Thickness h, Self weight q, Vertical live load p, Simple support at α = 90° Page 71
- 73. smIkarlMnwgsMrab;EpñkxagelIénBuH α manragdUcteTA ³ q z rNq Q − 2π sin α − = 0, α where r = Rsin α, Q q z = resultant of self weight, α α = ∫ 2 π α = 2 π ∫ sin α α = 2 π 2 ( 1 − cos α) Qq q rRd qR d qR z 0 2 0 So, . 1 cos = − − α N Q 2 sin sin 2 α 1 + cos α = − π α = − α qR qR r q q z eday Z = −q α R = R = R 1 2 cos , eyIgnwg)an ⎛ [ − α( + α)] ⎞ N R Z N + α ⎛ = ⎞ ⎟⎠ ⎜⎝ + α + α − = ⎟ ⎟⎠ ⎜ ⎜⎝ = − β α β 1 cos 1 cos 1 cos 1 cos cos 1 2 N qR R q q R q q q Analysis on vertical live load eyagtamRTwsþIbT 1 eyIgGacsresrsmIkarlMnwg)andUcteTA − 2π sin α − π 2 = 0, α rN p p r where r = Rsin α. N p = − pR α . 2 eday Z = − p cosα⋅cosα eyIgnwgrkeXIj ⎞ ⎟⎠ = ⎛− α + ⎟ ⎟⎠ ⎜⎝ ⎞ ⎛ N R Z N ⎜ ⎜⎝ = − α cos2 β 2 1 2 R p p R p p N p pR = − α β cos 2 2 Page 72
- 74. Nq α Diagram Nq β Diagram N p α Diagram N p β Diagram Cylindrical and Conical Shells C x y z α β x y z C α β θ ⎫ ⎪⎬ = α ( ) ( ) ⎪⎭ x y y = β = β , . , z z ( ). = α θ cos , = α θ β sin sin , sin cos , θ = θ β ⎫ ⎪⎭ ⎪⎬ = α θ β x y z Page 73
- 75. Cylindrical and conical shells are shells with zero Gaussian curvatures: 1 1 0 1 2 1 2 = = = R R k k k For cylindrical shells: A B y z ⎞ ⎛ ∂β ⎞ ⎛ ∂β 1, ; [( ′ ) 2 + ( 3 ′ ) 2 ] 2 , . R R y z 1 2 2 2 ′ ′′ − ′ ′′ y z z y = ∞ = ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ = = For conical shells: ⎞ ⎛ ∂β A B 1, sin ; [ ( ) ] 2 2 3 2 α θ + θ′ ( ) . , sin 2 2 cos sin 2 cos sin 1 2 2 2 θ θ + θ′ θ − θ′′ θ = ∞ = − ⎟ ⎟⎠ ⎜ ⎜⎝ ∂θ = = α θ + R R edayyk A=1 nig R1=∞ smIkarlMnwgsþaTic TTYl)anragdUcteTA ³ ( ) ∂ 1 0, ( ) ⎫ ⎪⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ − = ∂ + + = ∂ ∂α + ∂ ∂ ∂β + = ∂β ∂α − ∂α β β α β 0. 0, N 2 2 Z R B S BY B N BN N B S BX edaHRsaysmIkarenH eyIgTTYl)an ³ ; 2 N = R Z = RZ β ( ) ( ) ∫ α 1 1 B RZ B Y d α ⎤ α ⎥⎦ ⎡ ⎢⎣ + ∂ ∂β = β − 0 2 2 1 2 B f B S Page 74
- 76. ( ) ( ) 1 ⎤ 1 ⎡ β ∂ ∫ 1 2 ∫ 0 0 α ⎡ ∂ ∂ 1 ∂ 1 ∫ ∫ ( ) α α α α α α α α α ⎛ − ∂α ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ⎤ α ⎥⎦ ⎢⎣ + ∂β ∂β ∂β + ⎞ + α ⎟⎠ ⎜⎝ ∂ + β + α ⎥⎦ ⎢⎣ ∂β = − 0 0 2 2 B RZ B Y d d B B B RZ BX d B B d f B f B N enARtg;enH f1(β), f2(β) CaGnuKmnGaRs½ynwgGefr β . Example 2. Horizontal Pipeline of Circular Section α (x) y y Y R β Z O z l q Rims are rigidly in plane and free out plane. For cylindrical shell: R = R, B = R 2 Analysis on Self Weight Components of self weight: X = 0, Y = qsinβ, Z = q cosβ Normal forces: = = − β β N RZ qRcos Page 75
- 77. ( ) ( ) ( β ) − α β S f Tangential force = ⎤ ⎥⎦ ⎡ ⎢⎣ α α − β α + β α ∂ ∂β − β 2 R = ∫ ∫ 2 sin cos sin 2 f 1 0 0 2 2 1 q R q d q d R R Normal force [ ( )] ( ) ∂ ( ) [ ( )] ( ) 1 1 2 sin R − = ∫ α q R f f R q d R R f f R N 2 α β − β β α + ∂ ∂ ∂β = − − α β α ∂β + β β α + ∂β α 1 cos 2 2 1 0 2 2 1 Boundary conditions: 0, 0 ( ) 0; 2 α = = → β = α N f , 0 ( ) 2 sin . 1 α = l N = → f β = qR l β +C α )a:ra:Em:Rtefr C/R2 KWCakMlaMgkat;BRgayesμI elIEKmrbs;bMBg; . dUecñH RbsinebI bMBg;minrgkarrmYreT KWmann½yfa )a:ra:Em:RtefrenHesμIsUnü ³ 0, ( ) 2 sin . 1 C = f β = R ql β srubmk eyIgTTYl)an ( ) cos , N q l α −α R cos , = N qR = − β β (2 )sin . α β S q l = − α − β Page 76
- 78. α β=0 N α=0 S - + - ql ql ql2 4R + ql + - β N qR qR π 2 β= S Diagrams Analysis on Fluid Weight Components of fluid weight: 0, cos . 0 X = Y = Z = p − γR β dUecñH eyIgrkeXIj ( cos ), 0 = = − γ β β N RZ R p R ( ) 1 R ∂ ( p R cos ) f ( ) d sin , = ∫ α S f 1 − γ α β 2 1 0 0 2 2 2 β − γ β α = ∂β − β R R R R [ ( )] ( ) ( ) [ ( )] ( ) β − = ∫ α 1 1 sin γα + β β α + ∂ ∂ ∂β = − γ α β α ∂ ∂β + β β α + ∂β α cos 2 1 2 2 3 1 0 2 3 1 R f f R R d R R f f R N p0 = fluid pressure in a plane zOx. Page 77
- 79. edayeRbIR)as;lkçx½NÐRBMEdn dUcbgðajxagmux eyIgGackMNt;)an ³ f f R γ β = β = l ( ) ( ) sin . srubmk eyIgnwgmanlTæpl ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ cos , N l N R p R = − γ β ⎞ β ⎟⎠ α = γ ⎛ −α ⎜⎝ α −α β γ = − β sin . 2 cos , 2 0 S R l 2 0, 2 2 1 β α β=π N α=0 S - + γRl 2 + γRl 2 + - β N R(p + γR) 0 γl2 8 π 2 β= S Diagrams + γRl 2 R(p − γR) 0 Page 78
- 80. Example 3. Analysis of Cylindrical Tank on Wind Load y x α l p Wind direction β R Components of wind load: X Y = = 0, = ( − β − β) 0.7 0.5cos 1.2cos 2 Z p where p = max. wind pressure. ]bmafa sMBaFxül;minERbRbUltamkMBs; suILaMg KWminGaRs½ynwgkUGredaen x=α . kMlaMgxül;elIsuILaMg dUecñH eyIg)an = = (0.7 − 0.5cosβ −1.2cos 2β), β N RZ pR ( ) ( ) ( β ) − α( β + β) R RZ d f S f 1 α = p ∂ ∂β ∫ α β = − 1 0.5sin 2.4sin 2 2 1 0 2 2 R B R ( ) ⎤ ∂ α α 1 1 = ∫ ∫ 0 0 ( ) ( β ) α β + + ( β + β) ∂ ∂ ∂β = α ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥⎦ ⎢⎣ ⎡ α ∂β ∂β α 0.5cos 4.8cos 2 2 1 2 2 3 1 2 R p R f f R B RZ d d B B N ecjBIlkçx½NÐRBMEdn α = 0, = = 0 α S N eyIgkMNt;)an ( ) ( ) 0 1 2 f β = f β = Page 79
- 81. srubmk eyIgnwgman (0.5cos 4.8cos 2 ), N p α = α R 2 2 β + β = (0.7 − 0.5cosβ −1.2cos 2β), β N pR S = − pα(0.5sinβ + 2.4sin 2β). Diagrams l N α α= l S β α= N Page 80
- 82. Zero-Moment Spherical Cupola Radius R := 10 Self weight q := 0.100 ⋅ 25.00 ⋅ 1.1 q = 2.75 Vertical live load p := 0.50 ⋅ 1.3 p = 0.65 Normal forces: Nαq(α) q ⋅ R 1 + cos(α) := − Nβq(α) q ⋅ R 1 + cos(α) := ⋅ [1 − cos(α) ⋅ (1 + cos(α))] Nαp(α) p ⋅ R 2 := − Nβp(α) p ⋅ R 2 := − ⋅ cos(2 ⋅ α) Equations of section: x(α) := R ⋅ sin(α) y(α) := R ⋅ cos(α) α1 π := − α2 π 2 2 := n := 50 Δα α2 − α1 n := i := 0 .. n αi := α1 + i ⋅ Δα ⎯→⎯ ⎯→⎯ X := x(α) Y := y(α) Diagrams: Nx(α, N, scale) := x(α) + scale ⋅ N ⋅ sin(α) Ny(α, N, scale) := y(α) + scale ⋅ N ⋅ cos(α) ⎯⎯⎯⎯⎯⎯⎯→⎯ Nαqx Nx(α, Nαq(α) , 0.1) ⎯⎯⎯⎯⎯⎯⎯→⎯ := Nαqy := Ny(α, Nαq(α) , 0.1) ⎯⎯⎯⎯⎯⎯⎯→⎯ Nβqx Nx(α, Nβq(α) , 0.1) ⎯⎯⎯⎯⎯⎯⎯→⎯ := Nβqy := Ny(α, Nβq(α) , 0.1) ⎯⎯⎯⎯⎯⎯⎯→⎯ Nαpx Nx(α, Nαp(α) , 0.5) ⎯⎯⎯⎯⎯⎯⎯→⎯ := Nαpy := Ny(α, Nαp(α) , 0.5) ⎯⎯⎯⎯⎯⎯⎯→⎯ Nβpx Nx(α, Nβp(α) , 0.5) ⎯⎯⎯⎯⎯⎯⎯→⎯ := Nβpy := Ny(α, Nβp(α) , 0.5) Page 81
- 83. i := 0 .. n X1 i 〈 〉 Xi := Y1 i 〈 〉 Yi Nαqxi ⎛⎜⎜⎝ ⎞⎟⎟⎠ Nαqyi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := X2 i 〈 〉 Xi := Y2 i 〈 〉 Yi Nβqxi ⎛⎜⎜⎝ ⎞⎟⎟⎠ Nβqyi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := X3 i 〈 〉 Xi := Y3 i 〈 〉 Yi Nαpxi ⎛⎜⎜⎝ ⎞⎟⎟⎠ Nαpyi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := X4 i 〈 〉 Xi := Y4 i 〈 〉 Yi Nβpxi ⎛⎜⎜⎝ ⎞⎟⎟⎠ Nβpyi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Diagram N1q Diagram N2q Page 82
- 84. Diagram N1p Diagram N2p Page 83
- 85. Analysis of Horizontal Pipeline Radius: R := 10 Length: L := 1 Self weight: q := 1 Components of self weight: X(β) := 0 Y(β) := q ⋅ sin(β) Z(β) := −q ⋅ cos(β) Coefficients of first quadratic form: A := 1 B := R Range: α0 := 0 α1 := L Normal forces: Nβ(β) := −q ⋅ R ⋅ cos(β) S(α, β) := −q ⋅ (2 ⋅ α − L) ⋅ sin(β) Nα(α, β) q ⋅ α ⋅ (α − L) := ⋅ cos(β) R N := 50 Δα α1 − α0 N := α := α0, α0 + Δα .. α1 0 0.2 0.4 0.6 0.8 0.03 0.02 0.01 0 Diagram Nx Nα(α, π) Nα(α, π) α Page 84
- 86. 0 0.2 0.4 0.6 0.8 1 0.5 0 − 0.5 − 1 Diagram S S α π 2 , ⎛⎜⎝ ⎞⎟⎠ S α π 2 , ⎛⎜⎝ ⎞⎟⎠ α N := 50 Δβ π N := i := 0 .. N βi := i ⋅ Δβ := Sy i 〈 〉 βi S1i S 0 βi := ( , ) Sx i 〈 〉 0 S1i ⎛⎜⎝ ⎞⎟⎠ βi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := := Ny i 〈 〉 βi N2i Nβ β:= ( i) Nx 〈i〉 0 N2i ⎛⎜⎝ ⎞⎟⎠ βi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Diagram S 3 2 1 − 0.5 0 0.5 1 β Sy S1, Sx Diagram N2 3 2 1 − 10 − 5 0 5 10 β Ny N2, Nx Page 85
- 87. Fluid density γ := 1 Fluid pressure p := 0.5 ⋅ γ ⋅ R Normal and tangential forces: Να(α, β) γ := − ⋅ α ⋅ (L − α) ⋅ cos(β) 2 Nβ(β) := R ⋅ (p − γ ⋅ R ⋅ cos(β)) S(α, β) γ ⋅ R L − α 2 ⎛⎜⎝ ⎞⎟⎠ := ⋅ ⋅ sin(β) N := 50 Δα α1 − α0 N := α := α0, α0 + Δα .. α1 0 0.2 0.4 0.6 0.8 0.03 0.02 0.01 0 Diagram Nx Nα(α, π) Nα(α, π) α 0 0.2 0.4 0.6 0.8 6 4 2 0 − 2 − 4 − 6 Diagram S S α π 2 , ⎛⎜⎝ ⎞⎟⎠ S α π 2 , ⎛⎜⎝ ⎞⎟⎠ α Page 86
- 88. N := 50 Δβ π N := i := 0 .. N βi := i ⋅ Δβ := Sy i 〈 〉 βi S1i S 0 π βi := ( , − ) Sx i 〈 〉 0 S1i ⎛⎜⎝ ⎞⎟⎠ βi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := := Ny i 〈 〉 βi := ( − ) Nx i 〈 〉 0 N2i Nβ π βi N2i ⎛⎜⎝ ⎞⎟⎠ βi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := 3 2 1 Diagram S − 2 0 2 4 6 β Sy S1, Sx 3 2 1 Diagram N2 − 50 0 50 100 150 β Ny N2, Nx Page 87
- 89. Analysis of Cylindrical Tank on Wind Load Radius R := 1 Heigth L := 3 ⋅ R Wind load p := 0.50 Z(β) := p ⋅ (0.7 − 0.5 ⋅ cos(β) − 1.2 ⋅ cos(2 ⋅ β)) Section: y(β) := R ⋅ cos(β) z(β) := R ⋅ sin(β) Diagram of wind load: Sz := 0.5 Zx(β) := −(y(β) − Z(β) ⋅ cos(β) ⋅ Sz) Zy(β) := z(β) − Z(β) ⋅ sin(β) ⋅ Sz N := 50 i := 0 .. N βi i 2 ⋅ π N := ⋅ vxi y β:= − ( i) vyi z β:= ( i) Z1i Zx β:= ( i) Z2i Zy β:= ( i) L1 i 〈 〉 vxi Z1i ⎛⎜⎜⎝ ⎞⎟⎟⎠ ⎞⎟⎟⎠:= := L2 i 〈 〉 vyi Z2i ⎛⎜⎜⎝ vy Z2 L2 vx, Z1, L1 Page 88
- 90. Normal and tangential forces: Nα(α, β) ⋅ 2 2 ⋅ R p α := ⋅ (0.5 ⋅ cos(β) + 4.8 ⋅ cos(2 ⋅ β)) Nβ(β) := p ⋅ R ⋅ (0.7 − 0.5 ⋅ cos(β) − 1.2 ⋅ cos(2 ⋅ β)) S(α, β) := −p ⋅ α ⋅ (0.5 ⋅ sin(β) + 2.4 ⋅ sin(2 ⋅ β)) Diagram scales: s1 1 25 := s2 1 2 := s3 1 20 := Nαx(α, β) := −(y(β) + Nα(α, β) ⋅ cos(β) ⋅ s1) Nαy(α, β) := z(β) + Nα(α, β) ⋅ sin(β) ⋅ s1 Nβx(β) := −(y(β) + Nβ(β) ⋅ cos(β) ⋅ s2) Nβy(β) := z(β) + Nβ(β) ⋅ sin(β) ⋅ s2 Sx(α, β) := −(y(β) + S(α, β) ⋅ cos(β) ⋅ s3) Sy(α, β) := z(β) + S(α, β) ⋅ sin(β) ⋅ s3 i := 0 .. N N1xi Nαx L βi := ( , ) N1yi Nαy L βi := ( , ) L1x i 〈 〉 vxi := L1y i 〈 〉 vyi N1xi ⎛⎜⎜⎝ ⎞⎟⎟⎠ N1yi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := N2xi Nβx β:= ( i) N2yi Nβy β:= ( i) L2x i 〈 〉 vxi := L2y i 〈 〉 vyi N2xi ⎛⎜⎜⎝ ⎞⎟⎟⎠ N2yi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Diagram N1 vy N1y L1y vx, N1x, L1x Page 89
- 91. Sxi := Sx(L, βi) Syi Sy L β:= ( , i) L3x i 〈 〉 vxi Sxi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := L3y i 〈 〉 vyi Syi ⎛⎜⎜⎝ ⎞⎟⎟⎠ := Diagram N2 vy N2y L2y vx, N2x, L2x Diagram S vy Sy L3y vx, Sx, L3x Page 90
- 92. Example 4. Spherical Tank under Fluid R α0 α A A r z Nα Nα 2α p TMrragrgVg; AA CaRbePT simple kaMmuxkat; ³ r = Rsin α sMBaFGgÁFaturav ³ p = γR(1− cosα) ecjBIlkçx½NÐlMnwgtamG½kS bBaÄr eKrkeXIj ³ N Qz z = α 2 sin 2 Rsin2 π α = π α Q r r z dQz ( ) ϕ dϕ R α dP dP p r Rd R rRd 2 1 cos 2 3 = ⋅ π ⋅ ϕ = γ − ϕ π ϕ = π γ ϕ( − ϕ) ϕ R d 2 sin 1 cos dQ dP z cos 3 = ϕ = π γ ϕ ϕ( − ϕ) ϕ R d 2 sin cos 1 cos ( ) = ∫ = ∫ π γ ϕ ϕ − ϕ ϕ Q dQ R d z z ⎤ ⎥⎦ ⎡ 3 cos 1 2 2 0 2 1 = π γ − α⎛ − α ⎢⎣ ⎞ ⎟⎠ ⎜⎝ α α cos 3 1 6 2 sin cos 1 cos 3 2 0 R ⎞ ⎛ [ ( )] ⎟ ⎟⎠ ⎜ ⎜⎝ α 1 2cos + α − N R2 γ − α − α = R α γ = α 6 1 cos 1 cos 3 2cos 6sin 2 2 2 2 Page 91
- 93. ⎞ ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ α 5 6cos 2cos + α − α + Normal component of external force: ⎞ γ = − = ⎟ ⎟⎠ ⎛ N R Z N ⎜ ⎜⎝ = − α α β 6 1 cos 2 2 1 2 RZ N R R rUbmnþ Nα nig Nβ xagelIenH eRbI)ansMrab;EtkrNI . 0 0 ≤ α ≤ α edIm,IkMNt;kMlaMgpÁÜb Qα sMrab;EpñkxageRkamTMr eRkABIsMBaFkñúg eKRtUv KitRbtikmμbBaÄrrbs;TMrcUlbEnßmeTot EdlesμITMgn;GgÁFaturavTaMgmUl ³ R 4 R A = π 3γ 3 dUecñH ⎤ ⎥⎦ ⎡ Q 4 R3 R3 2 z cos 1 2 2 2 1 = π γ + π γ − α⎛ − cosα ⎢⎣ ⎞ ⎟⎠ ⎜⎝ 3 1 6 3 Z = p = γR(1− cosα) ecjBIenH eyIgnwgTTYl)an ⎞ . ⎛ 2 2 5 2cos ⎞ α N R 6 2 2 1 6cos 2cos 1 cos N R 6 , 1 cos ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ α − α − α − γ = ⎟ ⎟⎠ ⎜ ⎜⎝ − α + γ = α β enARtg;cMNuc α=α0 tMélkMlaMg Nα nig Nβ minCab;Kña . enHmann½yfa RTwsþIKμan m:Um:g; minGacbMeBjlkçx½NÐCab;enARtg;TMrxagelI)aneT . ehtudUecñH enAEk,rTMr nwgekItman local bending Edl stresses rbs;va GackMNt;)antamRTwsþIm:Um:g;. Page 92
- 94. Example 5. Ellipsoid of Revolution r z p a b α α r z Nα Nα p α α r p CasMBaFBRgayesμIelI shell. kMlaMgpÁÜbbBaÄr ³ = π = π 2 sin α Q r2 p R z 2 R1 ecjBIsmIkarlMnwgtamG½kSbBaÄr eyIg)an ³ N Qz = 2 pr pR r 2 sin 2sin α 2 = π α = α Equation of ellipse: 2 2 + = 1 2 2 z b r a ⎞ ⎟ ⎟⎠ ⎛ − = ⎟ ⎟⎠ pR R ⎜ ⎜⎝ ⎞ ⎛ N R Z N ⎜ ⎜⎝ = − α β 2 1 1 2 2 1 R R , 1 1 ′′ Curvatures: . 1 1 R r 1 r 1 2 2 2 2 1 k r r R k + ′ = = + ′ = = − Radius of curvature: 2 R a r b z = R R 3 b , . 4 4 2 4 2 2 b 2 1 2 a + = 2 3 b R = R = a . enARtg;kMBUl r = 0, z = b : , 1 2 b N = N = pa α β 2 2 R = a R = a , enAeGkVaT½r r = a, z = 0 : , , 2 2 b N = pa α 2 ⎞ , N pa a 1 2 2 2 ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ = − β b Page 93
- 95. Example 6. Conical Shell under Fluid z β β l z Qz V2 Nα Nα V1 β β z l R2 2α r γ r = z tgβ N Qz z ecjBIlkçx½NÐlMnwgtamG½kSbBaÄr eyIg)an π β = = α 2 π cos β 2 z sin Q r kMlaMgpÁÜbbBaÄr ³ ⎞ 1 2 2 2 Q = γ V +V = γ⎡ πr z + πr l − z r l z z 3 = γπ ⎛ − ⎥⎦ ( ) ( ) ⎟⎠ ⎜⎝ ⎤ ⎢⎣ 2 3 1 2 ⎞ β ⎟⎠ 2 2 z l z z β 2 γπ ⎛ − γ ⎛ − = = α 2cos ⎜⎝ r l z 3 π β ⎞ ⎟⎠ ⎜⎝ tg 3 2 sin N N N l z l ( ) tg 3 2 γ β β = = α max α = 3 4 16 cos Radius: β β R r z = β = tg cos cos 2 Normal component of force: Z = γ(l − z) ( ) N R Z l z z γ − β β = = β cos tg 2 N l γ β = β 4cos ( ) β 2 tg max z α N β N + l 2 + 3l 4 Page 94
- 96. PROBLEMS OF SHELL THEORY 1. Differential Geometry Of Surface 1.1. eKeGayépÞmYyCarag z = z(x, y) . cUrrk first nig second quadratic forms RBmTaMg Gaussian nig mean curvatures . 1.2. eKeGayépÞrgVilmYyCarag r(u,ϕ) = x(u) i + ρ(u)cosϕ j+ ρ(u)sin ϕ k, ρ(u) > 0 cUrkMNt; first nig second quadratic forms . 1.3. Translation surface KWCaépÞ EdlekIteLIgedayclnarMkilExSekagmYy z f (x) 1 1 = tambeNþayExSekagmYyeTot z f (y) 2 2 = . ExSekagrag nigExSekagTis GacepSg²Kña b:uEnþCaTUeTA eK eRCIserIsykragEtmYy dUcCa )ara:bUl/ FñÚrgVg; .l. smIkarrbs;épÞrMkil manrag z f (x) f (y) 1 2 = + ]TahrN_ ³ 2 2 z f x R x a − R − a ⎟⎠ = = − ⎛ − ( ) , 2 1 2 4 2 1 1 1 ⎞ ⎜⎝ 2 2 z f y R y b − R − b ⎟⎠ = = − ⎛ − ( ) . 2 2 2 4 2 2 2 2 ⎞ ⎜⎝ sMrab;épÞxagelIenH cUrrk first nig second quadratic forms RBmTaMg curvatures . 1.4. ]bmafa mankUGredaensuILaMg (z = α,β) Edl β KWCamMucab;BIG½kS Ox dl;cMeNalénvicT½rkaM r . dUecñH épÞrgVilGacmansmIkardUcxageRkam r(z,β) = r(z)cosβ i + r(z)sinβ j + z k cUrrk first nig second quadratic forms RBmTaMg curvatures rbs;épÞxagelIenH . 1.5. cUrkMNt; first nig second quadratic forms RBmTaMg curvatures rbs;épÞCak;EsþgmYy cMnYnxageRkam ³ a) Ellipsoid x = a cosu cos v, y = a cosu sin v, z = c sin v b) Sphere x = Rcosαcosβ, y = Rcosαsinβ, z = Rsinα c) Cylinder of revolution x = α, y = Rcosβ, z = Rsinβ Page 95
- 97. d) Shallow shell z ∂ = z z x y ∂ z ( , ), ≈ 0 ∂ ∂ = y x e) Conical surface of revolution x = α, y = Rcosβ⋅α, z = Rsinβ ⋅α 2. Shell Analysis 2.1. eFVIkarKNna circular cylindrical shallow shell nwgbnÞúkeRkAbBaÄrBRgayesμI q sMrab; krNIEdlTMrTaMgbYnRCugrbs;va CaRbePTsnøak; (simple supports) . a b h . 8m, 6m, 0 2m = = = R R f 40m, 1.2m = = = 2 = ⋅ ν = , 0.25 2 10 kg m 2 9 E y z f b a 2.2. eFVIkarKNnaEkvragekan EdlmanmMukMBUlesμI 2β nigpÞúk x edayGgÁFaturav Edlmanma:smaD γ . 2.3. cUreFVIkarKNna spherical tank EdlRTedayTMr kMNl;ragrgVg; AA nigpÞúkeBjedayGgÁFatu rav Edlmanma:smaD γ . β β l R α0 α A A Page 96
- 98. 3. Miscellaneous 3.1. dUcemþcEdlehAfa shallow shell ? etIkarKNna shallow shell RtUv)ansMrYlya:gdUc emþcxøH ? 3.2. cUreGayniymn½y cylindrical nig conical shell ? etIlkçN³Biessrbs; shells TaMgenH ya:gdUcemþcxøJH ? 3.3. etI shell RbePTNa GacTukCa zero moment )an ? 3.4. cUrerobrab;KuNsm,tþirbs;eRKOgpÁMúsMNg; shell ? Page 97
- 99. Content 1. Differential geometry of surface 1.1. Equation of surface 1.2. First and second quadratic forms, Gaussian and mean curvature 2. Moment theory of shells 2.1. Differential equations of equilibrium 2.2. Internal forces, strains, change of curvatures, Hooke’s law and boundary conditions 2.3. Analysis of cylindrical shells 2.4. Analysis of shallow shells 2.5. Shells of revolution 3. Zero moment (membrane) theory of shells 3.1. Equilibrium equations 3.2. Shells of revolution 3.3. Cylindrical and conical shells 4. Examples of shell analysis Page 98
- 100. Reference: 1. Krivoshapko C.N. Fundamentals of thin-walled structure design.- Moscow: PFU, 1986. 2. Krivoshapko C.N. Textbook: differential geometry of surface. – Moscow: PFUR, 1992. 3. Krivoshapko C.N. Textbook: analysis of shallow shells in rectangular coordinates using displacement method. – Moscow: PFU, 1987. 4. Kashin P.A. Textbook: moment theory analysis of shells. – Moscow: PFU, 1987. 5. Kashin P.A. Textbook: examples of shell analysis. – Moscow: PFU, 1986. 6. Philin A.P. Shell theory. – Leningrad: Construction Publishing, 1970. 7. Alexandrov A.V., Potapov V.D. Fundamentals of theory of elasticity and plasticity. – Moscow: High School, 1990. 8. Samul V.I. Fundamentals of theory of elasticity and plasticity. – Moscow: High School, 1970. 9. Timoshenko S., Woinowsky-Krieger S. Theory of plates and shells. - New York: McGraw-Hill, 1959. 10. Darkov A.V. Structural Mechanics. – Moscow: Mir Publishers, 1986. Page 99
- 101. Summary 1. Differential Geometry of Surface 1.1. Equation of surface: r = r(α,β) = x(α,β)i + y(α,β)j+ z(α,β)k In vector x x or ( ) ( ) ( ) ⎪⎭ ⎫ ⎪⎬ , , = α β , , y y = α β , . z z = α β In function z = z(x, y) or F(x, y, z) = 0 1.2. First quadratic form: ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ r r A E x y z ⎛ ∂α ⎞ ⎟ ⎟⎠ r r F x x y y z z ⎛ ∂β ⎜ ⎜⎝ ∂ ⎛ ∂α ∂ + ⎞ ∂ ⎞ + ⎟ ⎟⎠ ⎛ ∂β ⎜ ⎜⎝ ∂ ⎞ ⎛ ∂α ∂ + ⎞ ∂ + ⎟ ⎟⎠ ⎛ ∂β ⎜ ⎜⎝ ∂ ∂ = ∂ ∂ = r ∂ r ∂β ∂ ∂ ∂β ∂ ∂ = = = ∂β ∂α ∂β ∂α ∂β ∂α ∂β ∂α ⎞ ⎟⎠ ⎜⎝ ∂ + ⎟⎠ ⎜⎝ ∂ + ⎟⎠ ⎜⎝ ∂ = ∂α ∂α = = . ; ; 2 2 2 2 2 2 2 2 B G x y z Principal curvatures: ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ L 1 , = = − = = = − = N 2 2 2 max 2 1 1 min 1 B R k k A R k k 1 Ld α 2 + Nd β 2 2 2 2 2 α + β − = A d B d R 2 LN − M Gaussian curvature of the surface: 2 2 2 1 2 1 2 1 A B F R R k k k − = = = Page 100
- 102. 1 2 H k k x y z αα αα αα + 1 , Mean curvature of the surface: Second quadratic form: r ⋅ r × r αα − 2 2 2 x y z α α α β β β αα α β α β = × = ⋅ = x y z A B F L r r r n x y z αβ αβ αβ 1 , r ⋅ r × r αβ − 2 2 2 x y z α α α β β β αβ α β α β = × = ⋅ = x y z A B F M r r r n x y z ββ ββ ββ 1 , r ⋅ r × r ββ − 2 2 2 x y z α α α β β β ββ α β α β = × = ⋅ = x y z A B F N r r r n 2 = 2. Moment Theory of Shell 2.1. Differential equations of equilibrium A S AB ∂ ∂ 0 : 1 0, ( ) ( ) ( ) ( ) α β α B S AB 0 : 1 0, ∂ + ∂ β 2 ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ∂ ∂ N AB Z AB 0 : 0, α β β α ( ) ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 1 2 0 : 1 0, β α β + = ∂ ∂ ∂α + ∂ − ∂ − ∂α ∂ ∂ ∂β = + = ∂β + ∂β ∂α = − = ∂α ∂β ∂ ∂ = + + − + = ∂α + ∂β − ∂β = − + = ∂β + ∂α − ∂α = α β α β α Σ Σ Σ Σ Σ 2 0 : 1 2 0, 2 1 2 A H BM M B ABQ A M B H AM M A ABQ B M N AQ BQ ABZ R R Q ABY R B Y AN N A Q ABX R A X BN N B x y Page 101
- 103. 2.2. Internal forces: N C ( ) ( ) ( ) ( ) ⎪⎭ ⎫ ⎪⎬ = ε + νε α α β N C = ε + νε β β α S 1 C 1 , = − ν ε αβ , , 2 ⎫ ⎪⎬ M D = − κ + νκ α α β , ( ) ( ) ⎪⎭ M D = − κ + νκ β β α 1 . = − − ν κ αβ , H D C = Eh ( 2 ) Strains: B u u D = Eh ∂ 1 1 . 2 R β ε = α AB u B + z ∂ ∂α + ∂β A u u ∂ , β 1 1 1 R ε = β AB u A + z ∂ ∂β + α ∂α α ⎞ ⎟⎠ ⎛ ∂β A u B ε = β α αβ A ⎜⎝ ∂ ⎞ + ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ ∂ ∂α u B B A 1− ν2 3 12 1− ν ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ⎞ ⎟⎠ ⎛ ⎜⎝ 1 1 , 1 1 , ∂ 1 ∂α ∂ ⎞ ∂ + ⎟⎠ ⎛ ∂β ⎜⎝ ∂ α κ = ∂α + ∂ ∂ ∂β κ = ∂β + ∂α κ = αβ β V A V B 2 2 1 . 2 2 1 A B B A B V AB V B AV AB V A ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 1 ∂ , ∂ z ∂β α V u u = − ∂α = − β 1 . 2 2 1 1 z u R B V u R A Changes of curvatures: Hooke’s law: E z [ ( )] [ ( )] ⎫ ⎪ ⎪ ⎪ ⎬ α α β α β 2 E z β β α β α ( )( ) ⎪ ⎪ ⎪ ⎭ 2 . ε + κ + ν τ = τ = ε + νε + κ + νκ − ν σ = ε + νε + κ + νκ − ν σ = αβ βα αβ αβ 2 1 , 1 , 1 2 E z Page 102
- 104. 2.3. Cylindrical Shells Equations of cylindrical shell: x = α, y = y(β), z = z(β) A B F d dx d ds = ∞ = 1, 0, , , cos 0, = = = α = β = χ = , ( ). 1 2 R R R s Q M M s Q H ∂ + ∂ = , . s x H s x s x ∂ + ∂ x ∂ ∂ ∂ ∂ Shears: = ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ Equations of equilibrium: 0, M H 1 1 0, − = ∂ s x ∂ + M ∂ ∂ ∂ H ∂ ∂ + S ∂ + N x ∂ + ∂ + M ∂ + = ∂ − ∂ − ∂ N ∂ S ∂ ∂ + = ∂ ∂ 2 0. 2 2 2 2 2 Z s x s x N R Y s x R s R x X s x s x s Strain components: u u u u ∂ + ∂ ∂ , , , 2 u u u u , , 2 1 2 . u 2 2 ∂ x s x ⎞ s R R ∂ x s s x R s u x s z xs ∂ − s z y z x s x xs s z y x x ∂ ∂ − ∂ ∂ = κ ⎟⎠ ⎛ ⎜⎝ ∂ ∂ κ = ∂ ∂ ∂ κ = − ∂ ∂ + ε = ∂ ε = ∂ ε = Internal forces: ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ u ⎛ + ∂ u u ∂ s z x ⎞ ⎟⎠ ⎡ N C u N C ⎡ u ∂ ∂ u S − ν C ⎛ ∂ u ⎜⎝ u ∂ + ∂ ∂ = ⎤ ⎥⎦ ⎢⎣ ∂ + + ν ∂ = ⎤ ⎥⎦ ⎢⎣ ⎞ ⎟⎠ ⎜⎝ ∂ + ν ∂ = , 2 1 , , s x x R s R s x s x s x s z x ⎡ M D u ⎡ u s ∂ s z z 1 1 ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ u ∂ − u ⎞ ⎟ ⎟⎠ ∂ ⎛ ⎛ ⎜ ⎜⎝ ⎛ u ∂ ν − ⎟⎠ ∂ − u ∂ ∂ u ∂ − u ∂ ∂ = − − ν ⎞ ⎤ ⎥⎦ ⎢⎣ ∂ ⎞ ⎜⎝ ∂ ∂ = − ⎤ ⎥⎦ ⎢⎣ ⎟⎠ ⎜⎝ ∂ ∂ ∂ + ν ∂ = − − . 2 , , 2 2 2 2 2 x s x R H D x s R M D s R x s s z s z s z x Page 103
- 105. X − ν ∂ 0, Equilibrium equations in displacements: ⎞ 1 2 u u 2 + ν ∂ ⎛ + ν ∂ 2 2 2 12 1 2 ⎡ u ν ∂ + 2 2 2 12 − ν ∂ 1 ⎛ + ν ∂ 2 1 ∂ 2 2 2 2 2 2 2 2 ∂ 2 2 2 ⎛ ⎤ ⎞ u Y = + ⎥⎦ ⎡ ⎢⎣ ⎞ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ ∂ ∂ ∂ ∂ ⎞ − ⎟⎠ ∂ ⎛ ∂ ⎜⎝ ∂ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎤ ⎥⎦ ⎢⎣ ∂ + ⎟⎠ ⎜⎝ ∂ + ∂ + ∂ + u ∂ ∂ C R s x s h s R u R s R R x h x s s x z s x ∂ + ⎟⎠ 4 ⎡ 2 2 ∂ 4 2 ⎤ ⎞ ∂ + ∂ ⎤ ⎞ ⎛ ⎛ ∂ u Z 2 0. ∂ ⎛ 2 4 12 1 12 1 4 2 2 4 2 2 2 = − ⎥⎦ u ν ∂ ⎡ ⎢⎣ ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ ∂ ∂ + ∂ ∂ + + + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥⎦ ⎢⎣ ⎟⎠ ⎜⎝ ∂ ⎞ ⎜⎝ ∂ ∂ − ∂ + ∂ C x x s s h R u s x R s R h x R s R z s x 0, 2 1 2 2 2 + = ∂ ∂ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ C x x s R x s s z x 2.4. Shallow Shells z ∂ z ∂ 20, 5. min min R h ≥ l f ≥ 0, ≈ 0 ∂ ≈ ∂ y x 1 ∂ 0, Equilibrium Equations: ∂ ( ) ( 2 ) ( ) ( ) ∂ + ∂ ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ∂ ∂ α β N AB α β β α ( ) ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ AB 1 2 1 0, β α β + = ∂ ∂ ∂α + ∂ − ∂ − ∂α ∂ ∂ ∂β + = ∂β + ∂β ∂α − = ∂α ∂β + + + = ∂α + ∂β − ∂ ∂ ∂β + = ∂β + ∂α − ∂α α β α β α 2 1 0, 0, 1 0, 2 2 A H BM M B ABQ A B H AM M A ABQ B N AQ BQ ABZ R R B S ABY B AN N A A S ABX A BN N B Page 104
- 106. ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ⎞ ⎟ ⎟⎠ A u u 1 1 , Strains: z B Changes of curvature: ⎞ B u u ⎛ ∂β ⎞ ⎛ ∂ ∂ ∂ ∂ ∂ 1 u A u 1 ⎛ 1 1 , ⎛ z z 2 1 1 1 , ⎛ ⎜ ⎜⎝ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ A ∂ u ∂α ∂ ∂β ∂ ∂ z z − ∂ ∂ ∂ ∂α ∂ 2 ∂α ⎞ ⎞ − ∂ ∂ ∂ 2 ∂α∂β α β κ = − ∂α ∂α − ⎟ ⎟⎠ ⎜ ⎜⎝ ∂β ∂ ∂ ∂β κ = − ∂β ∂β − ⎟⎠ ⎜⎝ ∂α ∂α κ = − αβ 1 1 1 z z z . A B u B u AB B u A B u B B A u AB u A A ⎪ ⎪ ⎪ ⎪ ⎭ ⎟⎠ ⎜⎝ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂α ε = + ∂α + ∂β ε = + ∂β + ∂α ε = β α αβ α β β β α α , 1 1 , 2 A B B A R AB u B R AB u A z D ∂ ∂ ( ) ( ) ( ) ( ) . 1 α α β 12 1 1 , 12 1 2 2 3 2 2 3 z z u D A B Q Eh u A A Q Eh ∂ ∇ ∂β κ + κ = ∂ ∂β − ν = − ∇ ∂α κ + κ = ∂α − ν = − β α β Shears: Normal and tangential forces: ⎞ 1 1 1 , 2 ∂ϕ ∂ ∂ϕ ∂ 1 1 1 , 2 2 ⎞ ∂ ⎛ ∂ϕ ∂ ⎞ ⎛ ∂ϕ 1 1 1 . ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ϕ ∂α ∂ ∂β − ∂ϕ ∂β ∂ ∂α − ∂ ϕ ∂α∂β α = − ∂β ∂β + ⎟⎠ ⎜⎝ ∂α ∂α = ∂α ∂α + ⎟ ⎟⎠ ⎜ ⎜⎝ ⎛ ∂β ∂β = β A A B AB B S A A A AB N B B B A B N Page 105
- 107. Equation of shallow shell: 1 ∇2∇2ϕ−∇2u = 0, ∇2ϕ+ D∇2∇2u − Z = 0. Eh k z k z ∂ ∂ Rectangular Shallow Shell Strain components: u y u x u u u ∂ + , , , y R x y 1 2 u R x xy y z y x z ∂ x ∂ ∂ + ε = ∂ + ε = ∂ ε = 2 2 u ∂ u ∂ u z z , , . κ = − αβ 2 2 x y y x y z κ = − ∂ x ∂ ∂ ∂ κ = − ∂ Internal forces: ( ) ( ) ⎡ M D u ⎡ ∂ M D ∂ u ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎛ u u u ∂ + N C u u ∂ S C u ⎡ ⎡ ∂ ∂ ∂ ∂ ⎤ , 1 2 z z H D u ( ) ( ) ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ∂ ∂ ∂ = − − ν ⎤ , ⎤ ⎥⎦ ⎢⎣ ∂ ∂ ∂ + ν ∂ = − ⎥⎦ ⎢⎣ ∂ + ν ∂ = − ⎪ ⎪ ⎪ ⎪ ⎭ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ∂ ∂ = − ν ⎥⎦ ⎤ ⎢⎣ + + ν ∂ + ν ∂ = ⎥⎦ ⎢⎣ + + ν ∂ + ν ∂ = 1 . , 1 , 2 , 2 u u 2 2 2 2 2 2 2 2 2 1 x y x y y x x y k k u x y N C k k u y x z y z z x x y z y x y z x y x ∂ ( ) ( ) ⎪ ⎪⎭ ⎫ ⎪ ⎪⎬ x x y z ∂ 2 ∇ ∂ κ + κ = ∂ ∂ ∂ = − ∇ ∂ κ + κ = ∂ = − , . 2 u y x y z y D y Q D u x D x Q D Page 106
- 108. Equilibrium equations: k k u ( ) ( ) + ν ∂ 2 2 − ν ∂ ⎛ ⎞ 2 2 2 ⎤ ∂ u u k k ∂ u ⎞ X Y ⎡ + ∇ + + ν + − ν ∂ u ∂ u ∂ k k u ∂ ∂ + ν ∂ ⎛ h k k k k u Z ( ) ( ) ( 2 ) 0, 12 0, 2 1 2 1 0, 2 1 2 1 2 1 2 2 2 1 4 2 1 2 2 1 2 2 1 2 1 2 2 2 2 = − ⎥⎦ ⎢⎣ ∂ + + ν ∂ + ν + = ∂ ν + + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ + ∂ ∂ + = ∂ + + ν ∂ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ∂ C y k k x C y x y y x C x x y u x y z x y z y x y z x Stress function ϕ = ϕ(x, y): 2 ∂ ϕ = ∂ ϕ = N x y ∂ ∂ , , . 2 2 2 2 x y S y N x ∂ ϕ = − ∂ ∂ Mixed differential equations of shallow shells: ⎪⎭ ⎪⎬ ⎫ 2 2 2 D ∇ ∇ u +∇ ϕ = Z z k Eh u ∇ ∇ ϕ− ∇ = , 0, 2 2 2 k z 2.5. Shells of revolution α, β = meridian and parallel. r(α) – meridian equation. ( ), 1 A = R α = sin α 2 B R Case of Axis-Symmetrical Shell: Y = 0 = = = 0, = ε = κ = 0 β β αβ αβ S Q H u = 0 ∂ k L ∂β k Equilibrium equations: ( ) R N R N R Q R R X sin α − cos α − sin α + sin α = 0, 2 α 1 β 2 α 1 2 ( ) ⎫ ⎪ ⎪ ⎪ ⎬ d R N R N d sin cos sin sin 0, 2 α 1 β 2 α 1 2 ( ) ⎪ ⎪ ⎪ ⎭ R M R M R R Q sin α + cos α + sin α = 0. d α − α − α = α α + α + α 2 1 1 2 α β α d R Q R R Z d d Page 107
- 109. Strains: du 1 ⎛ + ⎞ , 1 ( cotg ) , z β α z α 1 2 ⎤ ⎡ u du u du d 1 1 ⎛ − ⎞ , cotg . 1 1 1 2 ⎞ ⎟⎠ ⎛ − ⎜⎝ α = κ ⎥⎦ ⎢⎣ ⎟⎠ ⎜⎝ α α κ = + α = ε ⎟⎠ ⎜⎝ α ε = α α β α dz dz R R d R R u u R u d R z z E.Meissner’s unknowns: ⎞ duz 1 , χ = − R Q α α = ψ ⎟⎠ ⎛ + ⎜⎝ α R u d 2 1 Case h=const ( ) ν ν χ = − 1 ψ , ( ψ ) + ψ = χ + 1 Φ ( α ), 1 1 1 χ − R Eh R L R D L where d R R ⎛ α d d L R ⎤ ⎡ ⎞ 1 cotg 2 L cotg L 2 ( L ) (L) 2 2 1 2 1 1 2 2 2 1 d R R R d d R R α − α ⎥⎦ ⎢⎣ α + ⎟ ⎟⎠ ⎜ ⎜⎝ + α = 3. Zero Moment (Membrane) Theory of Shell: = = = 0, = = 0 α β α β M M H Q Q Equilibrium equations: 1 ∂ 0, ∂ ( ) ( ) ( ) ( ) ⎫ ⎪⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ∂ α β β α N + − = + = ∂ ∂α + ∂β − ∂ ∂ ∂β + = ∂β + ∂α − ∂α α β 0. 2 1 0, 1 2 2 Z R N R B S ABY B AN N A A S ABX A BN N B Page 108
- 110. 3.1. Shell of revolution α, β = meridian and parallel. ( ), sin . 1 2 A = R α B = r = R α Equilibrium equations: ( ) sin cos sin 0, ( ) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ∂ β 1 + − = ∂ sin α + sin α = 0, ∂α α + ∂ ∂β + α = ∂β α − α + ∂ ∂α α β α β 0. sin 2 1 1 2 1 2 2 2 2 2 1 2 1 1 1 2 R N R N R R Z R S R R Y R N R R N N R R S R R X ∂ k Y L Case of axis symmetrical problem 0, = 0 = k ∂β = = = 0 β H Q S ( ) ⎫ ⎪⎭ ⎪⎬ R N N R R R X sin α − cos α + sin α = 0, 2 1 1 2 d R N R N R R Z + − = α α β α β 0. 2 1 1 2 d ⎞ ⎟ ⎟⎠ ⎛ ⎜ ⎜⎝ Solution: = − α β 2 R 1 N R Z N ( ) ⎤ ⎥ ⎥⎦ ⎡ = ∫ α ⎢ ⎢⎣ + α α − α α α α α 1 sin cos sin 1 sin 2 1 2 2 C R R Z X d R N Page 109
- 111. 3.2. Cylindrical and Conical Shell For cylindrical shells: A B y z ⎞ ⎛ ∂β ⎞ ⎛ ∂β 1, ; [( ′ ) 2 + ( 3 ′ ) 2 ] 2 , . R R y z 1 2 2 2 ′ ′′ − ′ ′′ y z z y = ∞ = ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ + ⎟ ⎟⎠ ⎜ ⎜⎝ ∂ = = ⎫ ⎪⎬ = α ( ) ( ) ⎪⎭ x y y = β = β , . , z z For conical shells: x = αcosθ, y = αsin θsinβ, z = αsin θcosβ, θ = θ(β). ⎞ ⎛ ∂β A B 1, sin ; [ ( ) ] 2 2 3 2 α θ + θ′ ( ) . , sin 2 2 cos sin 2 cos sin 1 2 2 2 θ θ + θ′ θ − θ′′ θ = ∞ = − ⎟ ⎟⎠ ⎜ ⎜⎝ ∂θ = = α θ + R R ( ) ∂ 1 0, ( ) ⎫ ⎪⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ − = ∂ + + = ∂ ∂α + ∂ ∂ ∂β + = ∂β ∂α − ∂α β β α β 0. 0, N 2 2 Z R B S BY B N BN N B S BX ; 2 N = R Z = RZ β ( ) ( ) ∫ α 1 1 B RZ B Y d α ⎤ α ⎥⎦ ⎡ ⎢⎣ + ∂ ∂β = β − 0 2 2 1 2 B f B S Equilibrium equations: Solutions: ( ) ( ) 1 ⎤ 1 ⎡ β ∂ ∫ 1 2 ∫ 0 0 α ⎡ ∂ ∂ 1 ∂ 1 ∫ ∫ ( ) α α α α α α α α α ⎛ − ∂α ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ⎤ α ⎥⎦ ⎢⎣ + ∂β ∂β ∂β + ⎞ + α ⎟⎠ ⎜⎝ ∂ + β + α ⎥⎦ ⎢⎣ ∂β = − 0 0 2 2 B RZ B Y d d B B B RZ BX d B B d f B f B N Page 110
- 112. saklviTüal½y GnþrCati Program: Master of Civil Engineering sMNYreRtomRblgbBa©b;karsikSa fñak;GnubNÐitsMNg;suIvil Epñk ³ Advanced Structural Analysis muxviC¢a ³ Theory of Elasticity 1> cUrerobrab;smμtikmμ (hypothesis) énRTwsþIeGLasÞic RBmTaMgBnül;xøwmsarrbs;va. 2> dUcemþcEdlehAfa sMBaFkñúg (stress)? enAelImuxkat;mYy etIsMBaFkñúgbMEbkecjCab:unμanbgÁMú (components)? cUrbgðajrUbmnþkMNt;bgÁMúTaMgenaH? etIeKTajrUbmnþTaMgenH)anya:gdUcemþc? 3> etIbgÁMúrbs;sMBaFkñúg Rtg;cMNucmYy ya:gdUcemþcxøH? cUrerobrab;lkçx½NÐlMnwg. ehtuGVI)anCa lkçx½NÐlMnwgmYycMnYn minRtUv)aneKcat;TukCasmIkarlMnwg? 4> GVIeTACalkçx½NÐRBMEdn (boundary conditions)? cUrbgðajBIkarTajlkçx½NÐTaMgenH. 5> cUrBnül;BImUlehtu nigGtßRbeyaCnénkarrgVilG½kS kñúgkarsikSasMBaFkñúg. etITItaMgfμIrbs; G½kSkUGredaen kMNt;ecjBIlkçx½NÐNa nigya:gdUcemþc? 6> dUcemþcEdlehAfa sMBaFkñúgem (principal stresses)? etIeFVIya:gdUcemþc edIm,ITTYl)an sMBaFkñúgem? 7> GVIeTACa bMErbMrYlragEkg (normal strain) nigbMErbMrYlragb:H (shear strain)? etIbgÁMúénbMErbMrYl rag (strain components) TaMgenaH kMNt;ya:gdUcemþc? ehtuGVI)anCacaM)ac;sBaØaN strain enH? 8> cUrBnül;BImUlehtu nigGtßRbeyaCnénlkçx½NÐCab; (compatibility conditions) RBmTaMg bgðajBIviFITajrklkçx½NÐTaMgenaH. 9> dUcemþcEdlehAfa bMErbMrYlragem (principal strains)? etIeKkMNt;bMErbMrYlragemenH ya:gdUcemþc? 10> etIGVIeTACac,ab; Hook? etIc,ab;enHmanGtßRbeyaCnGVIxøH enAkñúgkarsikSaGMBI stresses nig strains? etIrUbFatu (material) mYyRbePTsMKal;eday)a:ra:Em:RtGVIxøH? cUrbBa¢ak;BIGtßn½yén )a:ra:Em:RtnImYy². 11> etIenAkñúgRTwsþIeGLasÞic mansmIkar nigGBaØtþiGVIxøH? cUrbBa¢ak;BIKMrUKNiténsmIkarnImYy² nig bgðajpøÚvsMrab; edaHRsaylMhat;eGLasÞic (elasticity problem). saRsþacarüTTYbnÞúk ³ bNÐit esOn sm,tþi Page 1 of 4
- 113. saklviTüal½y GnþrCati Program: Master of Civil Engineering 12> GVIeTACaviFIbMlas;TI (solution in displacements) sMrab;edaHRsaylMhat;eGLasÞic? cUrbgðaj BIkarTajrksmIkarénviFIenH? 13> GVIeTACaviFIkMlaMg (solution in stresses) sMrab;edaHRsaylMhat;eGLasÞic? cUrbgðajBIkarTaj rksmIkarénviFIenH? 14> enAeBleRKOgpÁMúeFVIkar etImanfamBleGLasÞicGVIxøH? cUrerobrab;. etIRTwsþIfamBleGLasÞic manGtßRbeyaCn_ eRbIR)as;enAeBlNaxøH? etIlkçx½NÐlMnwg nigfamBleGLasÞic manTMnak; TMngnwgKña ya:gdUcemþc? 15> etIeKsikSaRTwsþIeGLasÞic manRbeyaCnGVIxøH sMrab;RTwsþIeRKOgpÁMú (theory of structures)? cUrerobrab;. muxviC¢a ³ Shell Theory 16> cUrsresrTMrg;epSg² énsmIkarrbs;épÞ (equation of surface). 17> etIépÞmYy sMKal;eday)a:ra:Em:RtGVIxøH? cUrbgðajrUbmnþ sMrab;kMNt;)a:ra:Em:RtTaMgenaH. 18> cUrerobrab;KuNsm,tþi éneRKOgpÁMúsMNg;RbePTPñas (shells). 19> eKEckRTwsþIPñas CaBIr KWRTwsþImanm:Um:g; (moment theory) nigRTwsþIKμanm:Um:g; (zero moment theory). cUrbgðajBIPaBxusKñarvagRTwsþITaMgBIrenH nigGtßRbeyaCnénkarEbgEckenH? 20> cUrsresrlkçx½NÐlMnwg énsmIkarlMnwgrbs;Pñas; tamRTwsþImanm:Um:g;. enAkñúgPñas etIman kMlaMgkñúgGVIxøH EdltMrUveGayKNnark? 21> cUrerobrab;smIkar nigGBaØtþi énRTwsþImanm:Um:g;rbs;Pñas. 22> cUrsresrlkçx½NÐRBMEdn sMrab;krNIEKmmYycMnYn EdleKEtgEtCYbRbTH. ehtuGVI )anCacaM)ac; tMrUveGaybMeBjlkçx½NÐRBMEdnTaMgenH? 23> cUreGayniymn½y PñasragsuILaMg (cylindrical shells) nigPñasragekan (conical shells). etI lkçN³Biessrbs;Pñas;TaMgenH ya:gdUcemþcxøH? 24> dUcemþcEdlehAfa shallow shell ? etIkarKNna shallow shell RtUv)ansMrYlya:gdUcemþcxøH? 25> etIRtUvmanlkçx½NÐGVIxøH edIm,IKNnaPñas tamRTwsþIKμanm:Um:g;? 26> cUrerobrab;smIkarlMnwg énRTwsþIPñas Kμanm:Um:g;? etIlMhat;énRTwsþIKμanm:Um:g;enH CaRbePTsþaTic kMNt; b¤sþaTicminkMNt;? mUlehtuGVI? saRsþacarüTTYbnÞúk ³ bNÐit esOn sm,tþi Page 2 of 4
- 114. saklviTüal½y GnþrCati Program: Master of Civil Engineering 27> cUrbgðajBIbec©keTsmYycMnYn EdleKeRbIkñúgkaredaHRsaylMhat;KNnaPñas. muxviC¢a ³ Matrix Methods of Structural Analysis 28> enAkñúg Structural Analysis eKEckeRKOgpÁMúCaRbePTsþaTickMNt; (statically determinate) nig sþaTicminkMNt; (statically indeterminate). cUrerobrab;GMBIviFIsaRsþKNnaeRKOgpÁMúTaMgBIr RbePTenH? 29> edIm,IKNnaeRKOgpÁMúRbePTsþaTicminkMNt; eKGaceRbIviFIkMlaMg (method of forces) b¤viFI bMlas;TI (slope-deflection method). cUrGtßaFib,ayeRbobeFob GMBIviFITaMgBIrenH. 30> dUcemþcEdlehAfa ma:RTIs? etIsMenrma:RTIs sMrYlGVIxøH kñúgkaredaHRsaylMhat;? 31> etIeKeRbI singularity method sMrab;KNnaFñwmRbePTNa? a. Statically determinate beams b. Single span beams with uniform stiffness c. Uniform stiffness beams (statically determinate / indeterminate and single / multiple span) d. Beams with variable stiffness 32> erobrab;cMNucsMxan;²én singularity method. 33> etIeKeRbIviFItMN (method of joints) sMrab;KNnaeRKOgpÁMúRbePTNa? a. Statically determinate structures b. Statically indeterminate structures 34> etIGVIeTACaGBaØtiénviFItMN? 35> cUrGtßaFib,ayBIsmIkarsMxan;²énviFItMN dUcmanxageRkam³ - Static equations of member, - Geometric equations, - Equilibrium equations of joint. 36> etIeKeRbI stiffness method sMrab;KNnaeRKOgpÁMúRbePTNa? A. Statically determinate structures B. Statically indeterminate structures 37> enAkñúg stiffness method etIGVIxøHCaGBaØti? 38> cUrGtßaFib,ayBIsmIkarsMxan;²én stiffness method dUcmanxageRkam ³ - Static equations of member, - Physical equations, - Geometric equations, - Equilibrium equations of joint. saRsþacarüTTYbnÞúk ³ bNÐit esOn sm,tþi Page 3 of 4
- 115. saklviTüal½y GnþrCati Program: Master of Civil Engineering 39> etI duality principle niyayGMBIGVI? muxviC¢a ³ Finite Element Methods 40> cUrerobrab;BIsmIkarsMxan;²én theory of elasticity edaybBa¢ak;BIxøwmsar. 41> etIfamBleGLasÞic )anmkBIfamBlGVIxøH? etIfamBleGLasÞic niglkçx½NÐlMnwg man TMnak;TMngCamYyKña ya:gdUcemþc? 42> cUreFVIkareRbobeFob stiffness method nig finite element method (FEM) in displacements? 43> etIeKeRbI FEM sMrab;eFVIkarKNnaeRKOgpÁMúsMNg;RbePTNaxøH? etIeRKOgpÁMúsMNg;GVI EdlcaM)ac; FEM edIm,IeFVIkarKNna? 44> cUrerobrab;BIcMNuceKal²én FEM? 45> cUrTajrkTMnak;TMngsMxan;²én FEM sMrab; plane truss. 46> cUrTajrkTMnak;TMngsMxan;²én FEM sMrab; space truss. 47> cUrTajrkTMnak;TMngsMxan;²én FEM sMrab; bending beam. 48> cUrTajrkTMnak;TMngsMxan;²én FEM sMrab; plane frame. 49> enAkñúg bar structures etIGVIeTACa finite elements? cMeBaH plates nig shells vij etI finite elements manragGVIxøH? sMrab;karKNna etIeKKYreRCIserIsykmYyNa mkeRbICa model? 50> smIkarsMKal;kareFVIkarrbs;eRKOgpÁMú CaTUeTACasmIkarDIepr:g;Esül. bnÞab;BIeKGnuvtþviFI kMNat; (FEM) mk etIsmIkarxagelIenH enAEtCasmIkarDIepr:g;EsüldEdl b¤)anbMElgeTA CasmIkarRbePTepSgvij. cUrbBa¢ak;mUlehtu. saRsþacarüTTYbnÞúk ³ bNÐit esOn sm,tþi Page 4 of 4