Shell theory

saklviTüal½y GnþrCati
INTERNATIONAL UNIVERSITY
Master of Civil Engineering (Structural Engineering)
Shell Theory
By Seun Sambath, Ph.D, Civil Eng.
Phnom Penh 2003

Shell Theory
Shell = 3D thin walled structure.
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nigmankMras;tUcCagTMhMBIreTot ya:geRcIn .
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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
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elIRTwsþIEdleRbIR)as;kñúgkarGnuvtþn_
Cak;Esþg
sMrab;eRbIR)as;kñúgGnuvtþnCak;Esþg/
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Page 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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
Parabolic Helicoid
S
Page 16

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
:= +
Torse
S
Page 17

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

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

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

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

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

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

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

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

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

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 =
− 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

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

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

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

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

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

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

∫ 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

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 = =
⎫
⎪ ⎪
⎬
⎪ ⎪
⎭
⎞
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

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

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
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

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

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

⎫
⎪ ⎪ ⎪ ⎪
⎬
⎪ ⎪ ⎪ ⎪
⎭
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

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

⎞
⎛
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

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
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

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

⎫
⎪ ⎪
⎬
⎪ ⎪
⎭
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

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

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

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
− 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

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

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

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

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

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

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

Bending moment Mx
−Mx1
Bending moment My
−My1
Page 55

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
( )
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

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

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

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

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

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

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

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

( ) ( ) ( )
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

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

Zero Moment (Membrane)
Theory of Shells
= = = 0, = = 0 α β α β M M H Q Q
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

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
( )
sin ,
,
A R
= α
B r R
= = α
cos .
B R
1
2
1
= α
∂
∂α
( )
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

⎞
⎛
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

( 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

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

.
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

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

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

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

( ) ( )
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

( ) ( )
( 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
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

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

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

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

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

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

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

Diagram N1p
Diagram N2p
Page 83

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

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

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

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

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

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

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

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

⎞
⎟ ⎟⎠
⎛
⎜ ⎜⎝
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

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

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

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

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

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

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
3.3. Cylindrical and conical shells
4. Examples of shell analysis
Page 98

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

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

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

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 =
− ν
σ =
− ν
σ =
αβ βα αβ αβ
2 1
,
1
,
1
2
E z
Page 102

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

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

⎫
⎪ ⎪ ⎪ ⎪
⎬
⎪ ⎪ ⎪ ⎪
⎭
⎞
⎟ ⎟⎠
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:
⎞
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

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

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
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
( )
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

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
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

Shell theory

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