Shell theory

saklviTüal½y GnþrCatiINTERNATIONAL UNIVERSITY

Master of Civil Engineering (Structural Engineering)

Shell Theory

Phnom Penh 2003

By Seun Sambath, Ph.D, Civil Eng.

Shell TheoryShell TheoryShell = 3D thin walled structure.

Thin shell KWCaGgÁFatu EdlekagtamTismYy b¤BIr edayKμanrbt; nigkMBUlRsYc nigmankMras;tUcCagTMhMBIreTot ya:geRcIn .RbsinebIeKykkMritlMeGogRtwm 5% enaH shell esþIg manlkçx½NÐ 201≤Rh

Edl R CakaMkMeNagtUcCageK .plRbeyaCn_sMxan;rbs; shell KW multiple function of internal large space.

Modeling of shell:

• Three-dimensional elastic body• Using static-geometric hypothesis of Kirchhoff-Love

approximate theory (thin shell theory)

Shell Theory

Mathematical theory Engineering theory

CaRTwsþI sMrab;epÞógpÞat;PaBRtwmRtUvelIRTwsþIEdleRbIR)as;kñúgkarGnuvtþn_ Cak;Esþg

sMrab;eRbIR)as;kñúgGnuvtþn_Cak;Esþg/KuNvibtþi³ EdnkMNt;eRbIR)as;RTwsþIenH

Page 1

Elements ofElements ofDifferential Geometry of SurfaceDifferential Geometry of Surface

Equation of surface in vector notation

( ) ( ) ( ) ( )kjirr βα+βα+βα=βα= ,,,, zyx

In parametric form( ) ( ) ( )βα=βα=βα= ,;,;, zzyyxx

where α, β = independent parameters.

Equation of surface in Cartesian coordinates:

( )yxzz ,=

or ( ) 0,, =zyxFAs a function z of coordinates x, y.

Coordinate lines α, β = curvilinear coordinates.

Ellipsoid:

12

2

2

2

2

2

=++cz

by

ax

Hyperboloid of one sheet:

12

2

2

2

2

2

=−+cz

by

ax

⎪⎭

⎪⎬

⎫

ϕ=θϕ=θϕ=

cos,cossin,sinsin

czbyaxor

⎪⎪⎭

⎪⎪⎬

⎫

=+⋅=

+⋅=

cvzvuby

vuax

,1cos

,1sin2

2or

Page 2

Hyperboloid of two sheets:

12

2

2

2

2

2

−=−+cz

by

ax

⎪⎭

⎪⎬

⎫

+±=

⋅=⋅=

1

,cos,sin

2vcz

vubyvuaxor

Cone:

02

2

2

2

2

2

=−+cz

by

ax

⎪⎭

⎪⎬

⎫

=⋅=⋅=

cvzvubyvuax

,cos,sinor

Elliptical paraboloid:

qy

pxz

22

22

+=

⎪⎪⎭

⎪⎪⎬

⎫

=

⋅=

⋅=

vzvuqy

vupx

,cos2

,sin2or

Hyperbolic paraboloid:

qy

pxz

22

22

−=

Page 3

Elliptical cylinder:

12

2

2

2

=+by

ax

⎪⎭

⎪⎬

⎫

===

vzubyuax

,sin,sinor

Hyperbolic cylinder:

12

2

2

2

=−by

ax

⎪⎪⎭

⎪⎪⎬

⎫

==

+±=

vzbuy

uax,

,1 2or

Parabolic cylinder:

pxy 22 =

( )

( )( )

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

==

=

vvuzuvuy

puvux

,,,

,2

,2

or

Page 4

z

x y

α β

O

r

r+dr

drχ

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; .

;rdds = ;ββ∂

∂+α

α∂∂

= ddd rrr

,2 22222 β+βα+α=⋅= dBdFddAddds rr

FirstQuadratic Form

Where coefficients of first quadratic form are

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛β∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛β∂

∂=

β∂∂

β∂∂

==

β∂∂

α∂∂

+β∂

∂α∂

∂+

β∂∂

α∂∂

=β∂

∂α∂

∂=

⎟⎠⎞

⎜⎝⎛

α∂∂

+⎟⎠⎞

⎜⎝⎛

α∂∂

+⎟⎠⎞

⎜⎝⎛

α∂∂

=α∂

∂α∂

∂==

.

;

;

2222

2222

zyxGB

zzyyxxF

zyxEA

rr

rr

rr

,cos;; χ⋅β∂

∂⋅

α∂∂

=β∂

∂==

α∂∂

==rrrr FGBEASo,

χ = angle between coordinate lines α and β.

For orthogonal network: ,90°=χ 0=F22222 β+α= dBdAds

Page 5

Area of surface:

∫∫∫∫ βα−=βα×=σ βα ddFBAdd 222rr

α∂∂

=αrr

Normal unit vector:

222 FBA −

×=

×

×= βα

βα

βα rrrrrr

n

β∂∂

=βrrtangential to α-line, tangential to β−line

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:

,212

22

dsNddMdLd

Rk

nn

β+βα+α=−= Rn = radius of curve

SecondQuadratic Form ,2 22

22

β+βα+α=

⋅=⋅−=ϕ

NddMdLd

ddd nrnr

,1222

βββ

ααα

αααααα

βα

βααααα

−=

×

×⋅=⋅=

zyxzyxzyx

FBAL

rrrrr

nr

,1222

βββ

ααα

αβαβαβ

βα

βααβαβ

−=

×

×⋅=⋅=

zyxzyxzyx

FBAM

rrrrr

nr

,1222

βββ

ααα

ββββββ

βα

βαββββ

−=

×

×⋅=⋅=

zyxzyxzyx

FBAN

rrrrr

nr

Page 6

;,,222

22

rrrrrrβ∂

∂=

β∂α∂∂

=α∂

∂= ββαβαα

L, M, N = coefficients of second quadratic form

x y

z r

r+dr

drrα rβh

n

ds1 ds2

h22 =ϕ

Principal curvatures:

⎪⎪⎭

⎪⎪⎬

⎫

=−==

=−==

22

max2

21

min1

1

,1

BN

Rkk

AL

Rkk

2222

221β+α

β+α=−

dBdANdLd

R

Gaussian curvature of the surface:

222

2

2121

1FBA

MLNRR

kkk−

−===

Mean curvature of the surface:

221 kkH +

=

•Elliptical surface: 0>k (surface of positive curvature)

•Hyperbolic surface: 0<k (surface of negative curvature)

•Parabolic surface: 0=k (surface of zero curvature)

•Minimal surface: 0=H

Page 7

Ellipsoid x2

a2

y2

b2+

z2

c2+ 1=

a

b

c

⎛⎜⎜⎝

⎞⎟⎟⎠

1

1.5

0.5

⎛⎜⎜⎝

⎞⎟⎟⎠

:= N 20:=

i 0 N..:= ϕi iπ

N⋅:=

j 0 N..:= θj j2 π⋅

N⋅:=

Xi j, a sin ϕi( )⋅ sin θj( )⋅:= Yi j, b sin ϕi( )⋅ cos θj( )⋅:= Zi j, c cos ϕi( )⋅:=

Ellipsoid

X Y, Z, ( )

Page 8

Hyperpoloid x2

a2

y2

b2+

z2

c2− 1=

a

b

c

⎛⎜⎜⎝

⎞⎟⎟⎠

1

1

1.5

⎛⎜⎜⎝

⎞⎟⎟⎠

:= f z( ) 1z2

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

y2

b2+

z2

c2− 1−=

Page 9

Cone x2

a2

y2

b2+

z2

c2− 0=

a

b

c

⎛⎜⎜⎝

⎞⎟⎟⎠

1

1

1.5

⎛⎜⎜⎝

⎞⎟⎟⎠

:= f z( )zc

:= 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⎛⎜⎝

⎞⎟⎠

:= α15

:= F u v, ( )

a2

v u+( )⋅

b2

v u−( )⋅

α12

⋅ 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 1y2

b2+⋅

y

z

⎛⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎠

:= F2 y z, ( )

a− 1y2

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( )15

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 tanu2

⎛⎜⎝

⎞⎟⎠

⎛⎜⎝

⎞⎟⎠

+⎛⎜⎝

⎞⎟⎠

⋅:=

mesh 30:= S CreateMesh x y, z, π

22 π⋅

5−,

π

23 π⋅

7+, 0, 2 π⋅, mesh, ⎛⎜

⎝⎞⎟⎠

:=

Caternary surface

S

Page 18

H 3:= R 1:=

N 20:=

i 0 N..:= ρiRN

i⋅:=

j 0 N..:= ϕj2 π⋅

Nj⋅:=

Xi j, ρi cos ϕj( )⋅:= Yi j, ρi sin ϕj( )⋅:=

Z1i j, HR

ρi⋅:= Z2i j, H R2ρi( )2−+:=

X stack X X, ( ):= Y stack Y Y, ( ):= Z stack Z1 Z2, ( ):=

X Y, Z, ( )

Page 19

R 1:=

N 20:=

i 0 N..:= φi2 π⋅

Ni⋅:=

j 0 N..:= ρjRN

j⋅:=

Xi j, ρj cos φi( )⋅:= Yi j, ρj sin φi( )⋅:=

Zi j, ρj( )2:=

Page 20

Moment Theory of ShellsMoment Theory of ShellsSymbols

h thicknessNα, Nβ normal forcesSα, Sβ tangential shearsQα, Qβ shearsMα, Mβ bending momentsMαβ, Mβα torsion momentsX, Y, Z external forces

C (α,β)D (α+dα,β+dβ)C1 (α+dα,β)D1 (α,β +dβ)CD = dsCC1 = AdαCD1 = Bdβ

β∂⎟⎠⎞

⎜⎝⎛ α

α∂∂

+= dBBDC1

α∂⎟⎟⎠

⎞⎜⎜⎝

⎛β

β∂∂

+= dAADD1

xy

z

C

D

D1C1

Nα

Qα

Sα

Mαβ

Qβ

Nβ Sβ

MβαZ

YX

n

M

βα

MαMβ

Page 21

z

x XZ

C1

CAdαdϕα

dϕα

Nα

Qα

αα∂

∂+ α

α dNN

αα∂

∂+ α

α dQQ R1

dϕβ

R2

Nβ

Qβz

YZ

dϕβBdβ

ββ∂

∂+ β

β dN

Nβ

β∂∂

+ ββ d

QQ

C

D1

y

α=ϕα dRAd

1

β=ϕβ dRBd

2

C

C1D1

D

YXy, β

x, α

Nβ NαSαSβ

MβαMβ

Mα

Mαβ

dψαdψβ

αβ∂

∂=

−=ψα dA

BCDCCDDd 1

1

11 βα∂

∂=

−=ψβ dB

ACCCDDCd 1

1

11

Page 22

Equilibrium Equations

0sincos

sinsin

sincoscos

cossin

cos

1

1

11

11

11

=βα+⋅ϕψ⎟⎠⎞

⎜⎝⎛ α

α∂∂

+−

−⋅ϕψ⎟⎟⎠

⎞⎜⎜⎝

⎛β

β∂∂

++

+⋅ψ⎟⎠⎞

⎜⎝⎛ α

α∂∂

++⋅ϕψ⎟⎠⎞

⎜⎝⎛ α

α∂∂

++

+⋅−⋅ϕψ⎟⎟⎠

⎞⎜⎜⎝

⎛β

β∂∂

+−

+⋅ψ⎟⎟⎠

⎞⎜⎜⎝

⎛β

β∂∂

++⋅−=

ααα

α

βββ

β

αα

αααα

α

αβββ

β

ββ

ββ∑

dXABdDCdddQQ

DDdddQ

Q

DCddSSDCdddNN

CDNDDdddN

N

DDddS

SCCSX

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

=+α∂

∂+

α∂∂

−β∂∂

=

=+β∂

∂+

β∂∂

−α∂∂

=

=−α∂∂

+β∂∂

++=

=+−α∂∂

+β∂

∂−

β∂∂

=

=+−β∂∂

+α∂

∂−

α∂∂

=

αβα

βαβ

αββα

β2

αβ

αβα

∑

∑

∑

∑

∑

,01:0

,01:0

,0:0

,01:0

,01:0

2

2

21

2

1

2

ABQBMBMHAA

M

ABQAMAMHBB

M

ABZBQAQNRABN

RABZ

ABYQRABSB

BANANY

ABXQRABSA

ABNBNX

y

x

0:012

≡+−−= αββαβα∑ R

MR

MSSM z

HMMSSS ==== βααββα ,Because

HMMQQSNN ,,,,,,, βαβαβα8 unknowns and 5 equations.

Page 23

Internal Forces

dϕα α

Αdα

α⎟⎟⎠

⎞⎜⎜⎝

⎛+ d

RzA

1

1

2h2h

zdz

z

σβ

τβα

τβz

,1

,1

,1

2

21

2

21

2

21

∫

∫

∫

−

ββ

−

βαβ

−

ββ

⎟⎟⎠

⎞⎜⎜⎝

⎛+τ−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+τ=

⎟⎟⎠

⎞⎜⎜⎝

⎛+σ=

h

hz

h

h

h

h

dzRzQ

dzRzS

dzRzN

∫∫−

βαβα

−

ββ ⎟⎟⎠

⎞⎜⎜⎝

⎛+τ=⎟⎟

⎠

⎞⎜⎜⎝

⎛+σ−=

2

21

2

21

1,1

h

h

h

h

zdzRzMzdz

RzM

R1

∫∫∫−

αα

−

αβα

−

αα ⎟⎟⎠

⎞⎜⎜⎝

⎛+τ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛+τ=⎟⎟

⎠

⎞⎜⎜⎝

⎛+σ=

2

22

2

22

2

22

1,1,1

h

hz

h

h

h

h

dzRzQdz

RzSdz

RzN

∫∫−

αβαβ

−

αα ⎟⎟⎠

⎞⎜⎜⎝

⎛+τ=⎟⎟

⎠

⎞⎜⎜⎝

⎛+σ−=

2

22

2

22

1,1

h

h

h

h

zdzRzMzdz

RzM

11,1121

1 ≈⎟⎟⎠

⎞⎜⎜⎝

⎛+≈⎟⎟

⎠

⎞⎜⎜⎝

⎛+→<<

Rz

RzRz

So, HMMSSS ==== αβαββα ,

Page 24

Strain Determination.Strain Determination.HookeHooke’’s Law. Boundary Conditionss Law. Boundary Conditions

MM’

u

uz

uβuα

eα

n

eβ

α β u = resultant displacements;uα, uβ, uz = displacement components in α-, β- and z-direction

eα, eβ, n = unit vectors

,1,1β∂

∂=

α∂∂

=rere βα BA

222 FBA −

×= βα rr

n

Position of M: r,

Position of M’: neerurr zuuu +++=+= ββαα'

For a point M’:

neere ⎟⎟⎠

⎞⎜⎜⎝

⎛−

α∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂−

α∂∂

+≈α∂′∂

′=′ α

βαβ

αα1

1111Ruu

AuA

ABu

AAz

neere ⎟⎟⎠

⎞⎜⎜⎝

⎛−

β∂∂

++⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂−

β∂∂

≈β∂′∂

′=′ β

βαβα

β2

1111Ruu

BuB

ABu

BBz

⎟⎟⎠

⎞⎜⎜⎝

⎛+

α∂∂

+β∂

∂+≈

β∂′∂

=′ αβ

2

111RuuB

ABu

BBB zr

⎟⎟⎠

⎞⎜⎜⎝

⎛+

β∂∂

+α∂

∂+≈

α∂′∂

=′ βα

1

111RuuA

ABu

AAA zr

Normal strains:

,, 21 β=α= BddsAdds ., 21 β′=′α′=′ dBsddAsd

,,2

22

1

11

dsdssd

dsdssd −′

=ε−′

=ε βα

Page 25

.11

2RuuB

ABu

Bz+

α∂∂

+β∂

∂=ε α

ββ,11

1RuuA

ABu

Az+

β∂∂

+α∂

∂=ε β

αα

Shear strain:

αβαβαβαββαβα ε≈ε=⎟⎠⎞

⎜⎝⎛ ε−

π=⎟

⎠⎞

⎜⎝⎛ ε−

π′′=′′ sin2

cos2

coseeee

⎟⎠⎞

⎜⎝⎛

β∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛α∂∂

=ε αβαβ A

uBA

Bu

AB

Kichhoff-Love’s Assumptions:

1. About normal to middle surface: 0=ε=ε=ε αβ zzz

2. About normal stress: 0=σ z

After deformation:

( )( ) ⎭

⎬⎫

ε+=′ε+=′

β

α

,1,1

22

11

dssddssd ( )

( ) ⎭⎬⎫

ε+=′ε+=′

β

α

.1,1

BBAA

( )( ) αββααβ εε+ε+=⎟⎠⎞

⎜⎝⎛ ε−

π′′=′ 112

cos ABBAF

Love’s formulas:

.,11,11

22111 BAM

RRRRRR ′′′

−=κε

+−′

=κε

+−′

=κ αβ2

ββ

αα

κα, κβ = changes of bending curvatures ¬pldkkMeNagBt;¦,καβ = change of twisting curvatures ¬pldkkMeNagrmYr¦.

Page 26

In the distance z form midplane:

( )

( ) ⎪⎭

⎪⎬⎫

+=

+=

,,

22

11

zRRzRR

z

z ( ) ( )

( ) ( ) ⎪⎭

⎪⎬⎫

β=

α=

,,

2

1

dBdsdAds

zz

zz( )

( ) ⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

.1

,1

2

1

RzBB

RzAA

z

z

( )

( )

( )⎪⎭

⎪⎬

⎫

κ+ε=ε

κ+ε=ε

κ+ε=ε

αβαβαβ

βββ

ααα

.2,,

zzz

z

z

z ( )

( )

( )⎪⎭

⎪⎬

⎫

=

+=

+=

ββ

αα

.,,

2

1

zzz

z

z

uuzVuuzVuu

( )( )

( )

( ) ( )

( )( )

( )

( )( )

( )

( ) ( )

( )( )

( )

( )( )

( )

( )

( )

( )

( )

( )

( ) ⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛

β∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂

=ε

+α∂

∂+

β∂∂

=ε

+β∂

∂+

α∂∂

=ε

αβαβ

αβ

β

βα

α

.

,11

,11

2

1

z

z

z

z

z

z

z

zz

zzz

z

zz

z

zz

zzz

z

zz

z

zz

Au

BA

Bu

AB

Ru

uB

BAu

B

Ru

uA

BAu

A

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎟⎠⎞

⎜⎝⎛

α∂∂

+⎟⎠⎞

⎜⎝⎛

β∂∂

=κ

α∂∂

+β∂

∂=κ

β∂∂

+α∂

∂=κ

αβ

β

α

.2

,11

,11

12

12

21

AV

BA

BV

AB

VBAB

VB

VAAB

VA

⎪⎪⎭

⎪⎪⎬

⎫

β∂∂

−=

α∂∂

−=

β

α

.1

,1

22

11

z

z

uBR

uV

uAR

uV

Hooke’s law

( ) ( )( ) ( )[ ]

( ) ( )( ) ( )[ ]

( ) ( ) ( ) ( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

κ+εν+

=εν+

=τ=τ

νκ+κ+νε+εν−

=νε+εν−

=σ


=νε+εν−

=σ

αβαβαββααβ

αβαβαββ

βαβαβαα

.21212

,11

,11

22

22

zEE

zEE

zEE

z

zz

zz

Page 27

Internal forces:

( )( )

( ) ⎪⎭

⎪⎬

⎫

εν−=

νε+ε=

νε+ε=

αβ

αββ

βαα

,1

,,

21 CS

CNCN ( )

( )( ) ⎪

⎭

⎪⎬

⎫

κν−−=

νκ+κ−=

νκ+κ−=

αβ

αββ

βαα

.1,,

DHDMDM

21 ν−=

EhC shell stiffness (rigidity) for tension,

( )2

3

112 ν−=

EhD shell stiffness (cylindrical rigidity) for bending,

Boundary ConditionsBoundary ConditionsEquations (17)• 5 equations of statics,• 6 strain components,• 6 physical equations.

Unknowns (17)• 8 internal forces: βαβαβα QQHMMSNN ,,,,,,,• 3 displacements: zuuu ,, βα

• 6 strains: αββααββα κκκεεε ,,,,,

Generalized shears and tangential shears (β=const):

.~,1~

1RHSSH

AQQ −=

α∂∂

+= ββ

enAelIRCugnImYy² RtUvman 4 lkçx½NÐRBMEdn

Page 28

Rim β=const is free:

.0,0,01,01

=−==α∂

∂+= βββ R

HSNHA

QM

Rim β=const is built-in:

.01,0 2 =β∂

∂−==== βα

zz

uB

Vuuu

Rim β=const is hinge supported:

.0,0 ==== βαβ zuuuM

Rim β=const is simple supported with normal movement:

.0,01,0 ===α∂

∂+= βαββ uuH

AQM

Rim β=const is simple supported with tangential movement:

.0,0,0,01

==−== ββ zuRHSNM

Page 29

Analysis of Cylindrical ShellsAnalysis of Cylindrical Shells

z

y

xβ=sα=

x

dx ds

a

lf

x=l

x=0

ZYX

C

x,α y,β

z

D

C1

D1

Qx

NxSSMxHH

Qs

Ns

Ms

Equations of cylindrical shell: ( ) ( )β=β=α= zzyyx ,,

Coordinate lines: ,, sx =β=α s = arc length.

( ) .,0,,

,0cos,,,0,1

21 RdsddsRRR

dsddxdFBA

=ϕ=ϕ=∞=

=χ=β=α===

βα

Equilibrium equations:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−∂∂

+∂∂

+

=+−∂∂

+∂∂

=+∂∂

+∂∂

,0

,0

,0

2

2

Zx

Qs

QRN

YRQ

sN

xS

XsS

xN

xss

ss

x

⎪⎪⎭

⎪⎪⎬

⎫

=+∂∂

−∂∂

−

=+∂∂

−∂∂

−

,0

,0

xx

ss

QsH

xM

Qs

MxH

⎪⎪⎭

⎪⎪⎬

⎫

∂∂

+∂∂

=

∂∂

+∂∂

=

.

,

sM

xHQ

sH

xMQ

ss

xx

Page 30

CMnYs Qx nig Qs cUleTAkñúgsmIkarbIxagmux eyIgTTYl)an ³

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−∂∂

+∂∂

∂+

∂∂

+

=+∂∂

−∂∂

−∂∂

+∂∂

=+∂∂

+∂∂

.02

,011

,0

2

22

2

2

2

ZsM

sxH

xM

RN

Ys

MRx

HRs

NxS

XsS

xN

sxs

xs

x

Strain components:

.212,,

,,,

2

2

2

sxu

xu

Rsu

Ru

sxu

su

xu

Ru

su

xu

zsxs

zsy

zx

xsxs

zsy

xx

∂∂∂

−∂∂

=κ⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=κ∂∂

−=κ

∂∂

+∂∂

=ε+∂∂

=ε∂∂

=ε

Internal forces:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ν−

=

⎥⎦⎤

⎢⎣⎡

∂∂

ν++∂∂

=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +∂∂

ν+∂∂

=

,2

1

,

,

su

xuCS

xu

Ru

suCN

Ru

su

xuCN

xs

xzss

zsxx

( )⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂−

∂∂

ν−−=

⎥⎦

⎤⎢⎣

⎡∂∂

ν−⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

−=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

ν+∂∂

−−=

.211

,

,

2

2

2

2

2

sxu

xu

RDH

xu

su

Ru

sDM

su

Ru

sxuDM

zs

zzss

zszx

CMnYstMélkMlaMgkñúgxagelIcUleTAkñúgsmIkarlMnwg eyIgnwg)an

Page 31

,012

21

1221

21

2

2

2

22

2

2

2

22

2

2

2

22

=+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡∂∂ν−

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ν+

+∂∂

+∂∂

∂ν+

CYu

sxsRh

Rs

uxRRsR

hxssx

u

z

sx

.0212

1

121

4

4

22

4

4

42

2

2

2

2

22

=−⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂∂

+∂∂

++

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

−∂∂

+∂∂ν

CZu

ssxxh

R

uRsRxs

hsRx

uR

z

sx

,02

12

1 2

2

2

2

2

=+∂∂ν

+∂∂

∂ν++⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂ν−

+∂∂

CX

xu

Rsxuu

sxzs

x

For circular cylindrical shell: const== rR

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−++

=+++

=+++

.0

,0

,0

333231

232221

131211

CZuLuLuL

CYuLuLuL

CXuLuLuL

zsx

zsx

zsx

Equilibrium equations

,2

1 2

2112 sxLL

∂∂∂ν+

==,2

12

2

2

2

11 sxL

∂∂ν−

+∂∂

=

,2

12

2

2

2

22 sxL

∂∂

+∂∂ν+

=

,3113 xrLL

∂∂ν

==

,12

13

3

2

32

3223 ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂−

∂∂

==ssx

hsr

LL

.212

14

4

22

4

4

42

233 ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂∂

+∂∂

+=ssxx

hr

L

Page 32

Case X=Y=0:

,,LLu

LLu s

sx

x == ,2

1 4

2221

1211 ∇ν−

==LLLL

L

,2223

1213

LuLLuL

Lz

zx −

−= .

2322

1311

z

zs uLL

uLLL

−−

=

,121

14

5

23

52

2

3

3

34

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂+

∂∂∂

ν−ν+

−∂∂

∂+

∂∂

ν−=∇sx

usx

uhsx

uxuur zzzz

x

( ) ( ) ( ) ( ) .132112

2 5

5

32

5

4

52

3

3

2

34

⎥⎦

⎤⎢⎣

⎡∂∂

ν−+∂∂

∂ν−+

∂∂∂

ν−+

∂∂

−∂∂

∂ν+−=∇

su

sxu

sxuh

su

sxuur zzzzz

s

( ) ( ) ( ) ,1232112 424

6

42

6

6

6

24

4

22

28 Z

Dsxu

sxu

su

rxu

hru zzzz

z ∇=⎥⎦

⎤⎢⎣

⎡∂∂

∂ν++

∂∂∂

ν++∂∂

+∂∂ν−

+∇

Where ,2 4

4

22

4

4

44

ssxx ∂∂

+∂∂∂

+∂∂

=∇

8

8

62

8

44

8

26

8

8

8448 464

ssxsxsxx ∂∂

+∂∂∂

+∂∂

+∂∂∂

+∂∂

=∇∇=∇

L.N.Donnel’s equations:

,12

3

3

34

sxu

rxu

ru zz

x ∂∂∂

+∂∂ν

−=∇ ,123

3

2

34

su

rsxu

ru zz

s ∂∂

−∂∂

∂ν+−=∇

( ) .1112 44

4

22

28 Z

Dxu

hru z

z ∇=∂∂ν−

+∇

For closed shell:

( ) ( ) ,cos,cos00∑∑∞

=

∞

=

ϕ=ϕ=m

mm

zmz mxZZmxuu

Page 33

where ( ) .cos, ∫π

π−

ϕϕ==ϕ dmZxZrs

m

,2

2

2

2

2

2

2

22

rm

xsx−

∂∂

=∂∂

+∂∂

=∇ ,2 4

4

2

2

2

2

4

44

rm

xrm

x+

∂∂

−∂∂

=∇

,464 8

8

2

2

6

6

4

4

4

4

6

6

2

2

8

88

rm

xrm

xrm

xrm

x+

∂∂

−∂∂

+∂∂

−∂∂

=∇

smIkar Donnel TI3 manragCa( )

( ) 0cos21

411264

4

4

2

2

2

2

4

4

08

8

2

2

6

6

4

4

22

2

4

4

6

6

2

2

8

8

=⎭⎬⎫

⎥⎦

⎤⎢⎣

⎡+−−

⎪⎩

⎪⎨⎧

−⎥⎦

⎤⎢⎣

⎡+−⎟⎟

⎠

⎞⎜⎜⎝

⎛ ν−++−∑

∞

=

rmsxZ

rm

dxd

rm

dxd

D

xurm

dxd

rm

dxd

hrrm

dxd

rm

dxd

m

mzm

( ) ,cos0∑∞

=

ϕ=m

xmx mxuu

Tangential displacements:

( ) ,sin0∑∞

=

ϕ=m

sms mxuu

For open shell:

( )

( )

( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

π=

π=

π=

∑

∑

∑

∞

=

∞

=

∞

=

,sin

,sin

,cos

0

0

0

mzmz

msms

mxmx

lxmsuu

lxmsuu

lxmsuu ( )

( )

( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

π=

π=

π=

∑

∑

∑

∞

=

∞

=

∞

=

.sin

,sin

,cos

0

0

0

mm

mm

mm

lxmsZZ

lxmsYY

lxmsXX

.sin2,sin2,cos2

000∫∫∫

π=

π=

π=

l

m

l

m

l

m dxl

xmZl

Zdxl

xmYl

Ydxl

xmXl

X

Boundary conditions

.0: and 0 ====== xxzs MNuulxx

where

(Simple-supported on the rigid diaphragm)

Page 34

AxisAxis--symmetrical Cylindrical Shellsymmetrical Cylindrical ShellExampleExample

z

x

l2R

h

x

Z

External forces:

( )xlqZYX −=== ,0

Data:

3001.05,5,1

cmkgfqmlmmhmR

=

===

Steel:

3.0,102 26

=ν⋅= cmkgfE

In a case of axis symmetry (Y = 0):

.0,0 =∂∂

====s

HSQu ssL

Internal forces:

.,,

,,

3

3

2

2

2

2

dxudD

dxdMQ

dxudDM

dxudDM

dxdu

RuCN

Ru

dxduCN

zxx

zs

zx

xzs

zxx

==ν==

⎟⎠⎞

⎜⎝⎛ ν+=⎟

⎠⎞

⎜⎝⎛ ν+=


⎪⎪⎭

⎪⎪⎬

⎫

=−⎟⎟⎠

⎞⎜⎜⎝

⎛++ν

=+ν

+

.012

1

,0

4

42

2

2

CZRu

dxdRh

Rdxdu

CX

dxdu

Rdxud

zx

zx

Page 35

sikSakrNI X=0: ecjBIsmIkarlMnwgTI 1 eyIg)an

∫ν

−+=→==ν

+x

zxx

zx dxu

RxCCu

CNCu

Rdxdu

0656

CMnYscUleTAkñúgsmIkarTI 2 eyIgTTYl)an,4 4

4

4

RDN

DZu

dxud x

zz ν

−=γ+( ).13

22

24

hRν−

=γ

Common solution:

( ) ( )xCxCexCxCeu xxz γ+γ+γ+γ= γγ− sincossincos 43210

Particular solution: ( )xuz~

sMrab;krNI]TahrN_xagmux KWecjBIlkçx½NÐ)atxagelITMenr eyIgrkeXIj00 6 =→= CNx

( ) ( )Dxlqu

Dxlqu

dxud

zzz

44

4

4

4~4

γ−

=→−

=γ+

( ) ( ) ( )DxlqxCxCexCxCeu xx

z 44321 4sincossincos

γ−

+γ+γ+γ+γ= γγ−

Boundary conditions:

.0,0,0:0 ====dxduuux z

zx

.0,0: 3

3

2

2

=====xd

udDQdx

udDMlx zx

zx

∫ν

−=x

zx dxuR

Cu0

5

Page 36

Circular Tank

Radius R 1:=

Heigth L 3:=

Thickness h 0.1:=

Fluid density q 10:=

Modulus of elasticity E 2 104⋅

10 3−

10 6−⋅:= Poisson ratio ν 0.2:=

Cylindrical stiffness DE h3

⋅

12 1 ν2

−( )⋅:=

γ43 1 ν

2−( )⋅

R2 h2⋅

:= γ4

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⋅( )⋅

eγ x⋅ 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( )⋅:=

D1q−

4 γ4⋅ D⋅:= D2 0:= D3 0:=


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

RK01T C⋅( ) F x( ) F 0( )−( )⋅ u01z x( ) u01z 0( )−( )+⎡⎣ ⎤⎦⋅:=

c5 u1x 0( ):= c5 0=

Longitudinal displacement ux x( ) c5 u1x x( )−:=

C0E h⋅

1 ν2

−:=

Normal force Ns x( ) C0 1 ν2

−( )⋅uz x( )

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 2 310−

0

10

20

30Normal forces

Ns x( )

Ns ξ( )

x ξ,

L1 0.2 L⋅:=

ξ 0 0.02 L1⋅, L1..:=

0 0.2 0.4 0.60.2−

0

0.2

0.4

0.6

0.8

1Bending moments

Mx x( )

Mx ξ( )

x ξ,

Page 38

xy

z

Analysis of Shallow ShellsAnalysis of Shallow ShellsShallow shell: ,5,20 minmin ≥≥ flhR

where lmin = least dimension in plane, f = rise.

yx ≡β≡α ,

angle slope ,0sin,1cos −ϕ=ϕ=ϕ

Tangential stresses = their projectives

Assumptions:

2. Zero Gauss’s curvature 021 == kkk

3. 0,021

== βα

RQ

RQ

1. In rectangular coordinate: ( )yxzz ,=

1,

22222

222

==→⎪⎭

⎪⎬⎫

β+α=

+=BA

dBdAdsdydxds

In polar coordinates (r, β):

( )zrBAdrdrds ==→β+= ,12222

4. .0,021

== βα

Ru

Ru

So,

Page 39

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎠⎞

⎜⎝⎛

β∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛α∂∂

=ε

+α∂

∂+

β∂∂

=ε

+β∂

∂+

α∂∂

=ε

αβαβ

αβ

β

βα

α

,

,11

,11

2

1

Au

BA

Bu

AB

RuuB

ABu

B

RuuA

ABu

A

z

z

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂β∂

∂−

α∂∂

α∂∂

−β∂α∂

∂−=κ

α∂∂

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂β∂∂

−=κ

β∂∂

β∂∂

−⎟⎠⎞

⎜⎝⎛

α∂∂

α∂∂

−=κ

αβ

β

α

.111

,111

,111

2

2

2

zzz

zz

zz

uAA

uBB

uAB

uBBA

uBB

uAAB

uAA

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

=+α∂

∂+

α∂∂

−β∂∂

=+β∂

∂+

β∂∂

−α∂∂

=−α∂∂

+β∂∂

++

=+α∂∂

+β∂

∂−

β∂∂

=+β∂∂

+α∂

∂−

α∂∂

αβα

βαβ

αββα

αβ

βα

,01

,01

,0

,01

,01

2

2

21

2

2

ABQBMBMHAA

ABQAMAMHBB

ABZBQAQNRABN

RAB

ABYSBB

ANAN

ABXSAA

BNBN

Equilibrium Equations:

Page 40

Integration of equilibrium equations

ecjBIsmIkarBIrxageRkay eyIgTTYl)an³

( ) ( )

( ) ( ) .11

,11

2

2

⎥⎦

⎤⎢⎣

⎡β∂

∂−

α∂∂

−β∂∂

=

⎥⎦

⎤⎢⎣

⎡α∂

∂−

β∂∂

−α∂∂

=

αββ

βαα

AMHBB

AMAB

Q

BMHAA

BMAB

Q

edayeyageTAelIlkçx½NÐCab; (compatibility conditions)

( ) ( )

( ) ( ) ,01

,01

2

2

=κα∂∂

−β∂

∂κ−κ

β∂∂

=κβ∂∂

−α∂

∂κ−κ

α∂∂

αβββ

αβαα

BB

AA

AA

BB

eyIgnwgTTYl)an ³

( ) ( )

( ) ( ) .1112

,1112

22

3

22

3

z

z

uAD

BEhQ

uAD

AEhQ

∇β∂∂

=κ+κβ∂∂

ν−−=

∇α∂∂

=κ+κα∂∂

ν−−=

βαβ

βαα

CMnYstMélxagelI eTAkñúgsmIkarTIbI eyIgTTYl)an³

( ) ( )

( ) ( ) 01

111

2

2

21

=−⎭⎬⎫

⎥⎦

⎤⎢⎣

⎡α∂

∂−

β∂∂

−α∂∂

α∂∂

+

+⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡β∂

∂−

α∂∂

−β∂∂

β∂∂

++

βα

αββα

ZBMHAA

BM

AMHBB

AMBABR

NRN

tagGnuKmn_sMBaFkñúg (stress function) ϕ tamrUbmnþxageRkam ³

Page 41

.111

,111

,111

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛α∂ϕ∂

β∂∂

−β∂ϕ∂

α∂∂

−β∂α∂

ϕ∂−=

β∂ϕ∂

β∂∂

+⎟⎠⎞

⎜⎝⎛

α∂ϕ∂

α∂∂

=

α∂ϕ∂

α∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛β∂ϕ∂

β∂∂

=

β

α

AA

BBAB

S

AABAA

N

BBABB

N

bnÞab;BICMnYscUleTAkñúgsmIkarlMnwgsþaTic eyIgsegáteXIjfa smIkarbYn RtUv)anepÞógpÞat; KWBIrxagmux cMeBaHkrNI X=Y=0 nigBIrxageRkay rIÉsmIkarTIbI nwgTTYl)anrag ³

( ) 0112

222

3

21

=+∇∇ν−

−⎟⎟⎠

⎞⎜⎜⎝

⎛+− βα ZuEh

RN

RN

z

eyIgman ,, 221

2 ϕ∇=+ϕ∇=+ βαβα kNkNkNN

edayEp¥kelIsmIkar Kodazzi

( ) ( ) ,, 2112 β∂∂

=β∂∂

α∂∂

=α∂∂ AkAkBkBk

Edl

.1

,1

122

2

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂β∂∂

+⎟⎠⎞

⎜⎝⎛

α∂∂

α∂∂

=∇

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂β∂∂

+⎟⎠⎞

⎜⎝⎛

α∂∂

α∂∂

=∇

LLL

LLL

kBAk

AB

AB

BA

AB

AB

k

dUecñH lkçx½NÐCab;TIbI nigsmIkarlMnwg manragdUcteTA ³

.0,01 222222 =−∇∇+ϕ∇=∇−ϕ∇∇ ZuDuEh zkzk

Page 42

Analysis of Rectangular Shallow ShellsAnalysis of Rectangular Shallow Shells

,,,21 y

ux

uRu

yu

Ru

xu xy

xyzy

yzx

x ∂∂

+∂∂

=ε+∂∂

=ε+∂∂

=ε

.,,22

2

2

yxu

yu

xu zz

yz

x ∂∂∂

−=κ∂

∂−=κ

∂∂

−=κ αβ

Strain components:

Internal forces:

( )

( )

( ) ( )⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

∂∂∂

ν−−=

⎥⎦

⎤⎢⎣

⎡∂∂

ν+∂∂

−=

⎥⎦

⎤⎢⎣

⎡∂∂

ν+∂∂

−=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

ν−=

⎥⎦

⎤⎢⎣

⎡ν++

∂∂

ν+∂

∂=

⎥⎦

⎤⎢⎣

⎡ν++

∂∂

ν+∂∂

=

.1

,

,

,12

,

,

2

2

2

2

2

2

2

2

2

12

21

yxuDH

xu

yuDM

yu

xuDM

xu

yuCS

ukkxu

yu

CN

ukkyu

xuCN

z

zzy

zzx

yx

zxy

y

zyx

x

where ( ) .,,112

,1 2

2

22

2

12

3

2 yzk

xzkEhDEhC

∂∂

=∂∂

=ν−

=ν−

=


( )

( )

( ) ( ) ( ) ,0212

,02

12

1

,02

12

1

2221

21

42

1221

122

2

2

22

21

2

2

2

2

2

=−⎥⎦

⎤⎢⎣

⎡+ν++∇+

∂

∂ν++

∂∂

ν+

=+∂∂

ν++⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ν−

+∂∂

+∂∂

∂ν+

=+∂∂

ν++∂∂

∂ν++⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂ν−

+∂∂

CZukkkkh

yu

kkxukk

CY

yukku

xyyxu

CX

xukk

yxu

uyx

zyx

zy

x

zyx

( )

( )⎪⎪⎭

⎪⎪⎬

⎫

∇∂∂

=κ+κ∂∂

−=

∇∂∂

=κ+κ∂∂

−=

.

,

2

2

zyxy

zyxx

uy

Dy

DQ

ux

Dx

DQ

Page 43

Stress function ( ):, yxϕ=ϕ

.,,2

2

2

2

2

yxS

yN

xN yx ∂∂

ϕ∂−=

∂ϕ∂

=∂

ϕ∂=

Mixed differential equations of shallow shells:

⎪⎭

⎪⎬⎫

=∇−ϕ∇∇

=ϕ∇+∇∇

,0

,222

222

zk

kz

uEh

ZuD

where

.2

,,

4

4

22

4

4

44

2

2

12

2

22

2

2

2

22

yyxx

yk

xk

yx k

∂∂

+∂∂

∂+

∂∂

=∇

∂∂

+∂

∂=∇

∂∂

+∂

∂=∇

LLLL

LLL

LLL

Example 1. Mixed Method

Equation of shallow shell:( ) ( )

.42

,42

,2

22

2222

22

1

22

11

21

bRbyRzaRaxRz

yzxzz

−−⎟⎠⎞

⎜⎝⎛ −−=−−⎟

⎠⎞

⎜⎝⎛ −−=

+=

.1,1

22

11 R

kkR

kk yx =≈=≈Curvatures:

Assume that all rims are simple supported:

.0,0

,0,0

====→==

====→==

yyxz

xxyz

NMuubyy

NMuuaxx

dUecH eyIgGaceRCIserIsykGnuKmn_bMlas;TI nigGnuKmn_sMBaFkñúg dUcmanrag

Page 44

⎪⎪⎭

⎪⎪⎬

⎫

ππ=ϕ

ππ=

∑ ∑

∑ ∑∞

=

∞

=

∞

=

∞

=

,sinsin

,sinsin

3,1 3,1

3,1 3,1

m nmn

m nmnz

byn

axmD

byn

axmCu

where Cmn, Dmn = const.

Surface distributed forces in double Fourier’s series:

,sinsin3,1 3,1

∑ ∑∞

=

∞

=

ππ=

m nmn b

yna

xmqZ

where

∫ ∫ππ

=a b

mn dydxb

yna

xmZab

q0 0

sinsin4

20 0

16sinsin4π

−=ππ

−=→−= ∫ ∫ mnqdxdy

byn

axm

mnqqqZ

a b

mn

smIkarDIepr:g;Esülrbs;sMbk nwgmanragCasmIkarBICKNit ³

⎪⎪

⎭

⎪⎪

⎬

⎫

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ π

+⎟⎠⎞

⎜⎝⎛ π

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ π

+⎟⎠⎞

⎜⎝⎛ π

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ π

+⎟⎠⎞

⎜⎝⎛ π

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ π

+⎟⎠⎞

⎜⎝⎛ π

.

,0

222

2

2

1

2

2

2

1

22

mnmnmn

mnmn

qbn

amDC

amk

bnkD

amk

bnkEhC

bn

amD

edaHRsaysmIkarenH eyIgTTYl)an ³

,,2424

2

⎟⎠⎞

⎜⎝⎛ +

−=⎟⎠⎞

⎜⎝⎛ +

=

mnmn

mnmnmn

mnmn

mnmnmn

lDEhkD

qEhlDl

DEhkD

qkC

.,2

2

2

1

22

⎟⎠⎞

⎜⎝⎛ π

+⎟⎠⎞

⎜⎝⎛ π

=⎟⎠⎞

⎜⎝⎛ π

+⎟⎠⎞

⎜⎝⎛ π

=a

mkb

nklbn

amk mnmn

Page 45

Example 2. Method of Displacements

For rectangular shallow shell of simple-supported rims:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

ππ=

ππ=

ππ=

∑∑

∑∑

∑∑

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

.sinsin

,cossin

,sincos

1 1

1 0

0 1

m nmnz

m nmny

m nmnx

byn

axmCu

byn

axmBu

byn

axmAu

External distributed forces:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

ππ=

ππ=

ππ=

∑∑

∑∑

∑∑

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

,sinsin

,cossin

,sincos

1 1

1 0

0 1

m nmn

m nmn

m nmn

byn

axmcZ

byn

axmbY

byn

axmaX

where

.sinsin4

,cossin4

,sincos4

0 0

0 0

0 0

∫ ∫

∫ ∫

∫ ∫

ππ=

ππ=

ππ=

a b

mn

a b

mn

a b

mn

dydxb

yna

xmZab

c

dydxb

yna

xmYab

b

dydxb

yna

xmXab

a

If ,,0 qZYX −=== then .16,0 2π−===

mnqcba mnmnmn

dUecñH smIkarlMnwgsþaTic manragCasmIkarBICKNitdUcteTA ³

( ) ,02

12

121

222

=π

ν+−πν+

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ πν−

+⎟⎠⎞

⎜⎝⎛ π

mnmnmn Ca

mkkBabmnA

bn

am

( ) ,02

12

112

222

=π

ν+−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ πν−

+⎟⎠⎞

⎜⎝⎛ π

+πν+

mnmnmn CbnkkB

am

bnA

abmn

Page 46

( ) ( )

.212

2221

21

2

2

2

2

224

1221

CcCkkkk

bn

amh

BbnkkA

amkk

mnmn

mnmn

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡+ν++⎟⎟

⎠

⎞⎜⎜⎝

⎛+

π−

−π

ν++π

ν+

edaHRsaysmIkarxagelI eyIgTTYl)an

( ) ( )

( ) ( )

,

,1

,1

24

2

212

221

⎟⎠⎞

⎜⎝⎛ +

=

πν++−=

πν++−=

mnmn

mnmnmn

mnmn

mnmnmn

mnmn

mnmnmn

lDEhkD

kcC

Cbn

klkkkB

Ca

mk

lkkkA

⎪⎪⎭

⎪⎪⎬

⎫

π+

π=

π+

π=

.

,

2

22

12

22

2

2

22

2

22

bnk

amkl

bn

amk

mn

mn

where,

Page 47

Analysis of Rectangular Shallow Shell(method of displacements)

ORIGIN 1:=

a

b⎛⎜⎝

⎞⎟⎠

8

6⎛⎜⎝

⎞⎟⎠

:=R1

R2⎛⎜⎝

⎞⎟⎠

20

2000⎛⎜⎝

⎞⎟⎠

:=E

ν

⎛⎜⎝

⎞⎟⎠

2 108⋅

0.2

⎛⎜⎝

⎞⎟⎠

:=

h 0.15:=n1

n2⎛⎜⎝

⎞⎟⎠

3

3⎛⎜⎝

⎞⎟⎠

:= q 1.1:=

External force: Z x y, ( ) q−:=

Equation of shallow shell:

z1 x( ) R12 xa2

−⎛⎜⎝

⎞⎟⎠

2− R12 a2

4−−:= z2 y( ) R22 y

b2

−⎛⎜⎝

⎞⎟⎠

2− R22 b2

4−−:=

z x y, ( ) z1 x( ) z2 y( )+:= za2

b2

, ⎛⎜⎝

⎞⎟⎠

0.406=

Axial stiffness C1E h⋅

1 ν2

−:=

Flexural stiffness DE h3

⋅

12 1 ν2

−( )⋅:=

Curvatures k11

R1:= k2

1R2

:=

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⋅0

by

0

axZ x y, ( ) sin αm x⋅( )⋅ sin βn y⋅( )⋅

⌠⎮⌡

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 ai

10⋅ b

j10

⋅, ⎛⎜⎝

⎞⎟⎠

:=

Rectangular Shallow Shell

z0 10⋅

Coefficients of system:

A11 m n, ( ) αm( )2 1 ν−

2βn( )2

⋅+:= A12 m n, ( )1 ν+

2αm⋅ βn⋅:=

A13 m n, ( ) k1 ν k2⋅+( )− αm⋅:= A21 m n, ( )1 ν+

2αm⋅ βn⋅:=

A22 m n, ( ) βn( )2 1 ν−

2αm( )2

⋅+:= A23 m n, ( ) k2 ν k1⋅+( )− βn⋅:=

A31 m n, ( ) k1 ν k2⋅+( ) αm⋅:= A32 m n, ( ) k2 ν k1⋅+( ) βn⋅:=

Page 49

A33 m n, ( )h2

12αm( )2

βn( )2+⎡

⎣⎤⎦

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

m 1

n2

n

Am n, cos αm x⋅( )⋅ sin βn y⋅( )⋅( )∑=

∑=

:=

uy x y, ( )

1

n1

m 1

n2

n

Bm n, sin αm x⋅( ) cos βn y⋅( )⋅( )∑=

∑=

:=

uz x y, ( )

1

n1

m 1

n2

n

Cm n, sin αm x⋅( )⋅ sin βn y⋅( )⋅( )∑=

∑=

:=

Page 50

Internal forces:

Nx x y, ( ) C1

1

n1

m 1

n2

n

k1 ν k2⋅+( ) Cm n, ⋅ αm Am n, ⋅− ν βn⋅ Bm n, ⋅−⎡⎣ ⎤⎦ sin αm x⋅( )⋅ sin βn y⋅( )⋅⎡⎣ ⎤⎦∑=

∑=

⋅:=

Ny x y, ( ) C1

1

n1

m 1

n2

n

k2 ν k1⋅+( ) Cm n, ⋅ ν αm⋅ Am n, ⋅− βn Bm n, ⋅−⎡⎣ ⎤⎦ sin αm x⋅( )⋅ sin βn y⋅( )⋅⎡⎣ ⎤⎦∑=

∑=

⋅:=

S x y, ( )1 ν−

2C1⋅

1

n1

m 1

n2

n

αm Am n, ⋅ βn Bm n, ⋅+( ) cos αm x⋅( )⋅ cos βn y⋅( )⋅⎡⎣ ⎤⎦∑=

∑=

⋅:=

Mx x y, ( ) D−

1

n1

m 1

n2

n

Cm n, αm( )2ν βn( )2

⋅+⎡⎣

⎤⎦⋅ sin αm x⋅( )⋅ sin βn y⋅( )⋅⎡

⎣⎤⎦∑

=∑=

⋅:=

My x y, ( ) D−

1

n1

m 1

n2

n

Cm n, βn( )2ν αm( )2

⋅+⎡⎣

⎤⎦⋅ sin αm x⋅( )⋅ sin βn y⋅( )⋅⎡

⎣⎤⎦∑

=∑=

⋅:=

H x y, ( ) 1 ν−( ) D⋅

1

n1

m 1

n2

n

Cm n, αm⋅ βn⋅ cos αm x⋅( )⋅ cos βn y⋅( )⋅( )∑=

∑=

⋅:=

Qx x y, ( ) D

1

n1

m 1

n2

n

Cm n, αm⋅ αm( )2βn( )2

+⎡⎣

⎤⎦⋅ cos αm x⋅( )⋅ sin βn y⋅( )⋅⎡

⎣⎤⎦∑

=∑=

⋅:=

Qy x y, ( ) D

1

n1

m 1

n2

n

Cm n, βn⋅ αm( )2βn( )2

+⎡⎣

⎤⎦⋅ sin αm x⋅( )⋅ cos βn y⋅( )⋅⎡

⎣⎤⎦∑

=∑=

⋅:=

Rx y( ) D

1

n1

m 1

n2

n

Cm n, αm⋅ αm( )2 2 ν−( ) βn( )2⋅+⎡

⎣⎤⎦⋅ sin βn y⋅( )⋅⎡

⎣⎤⎦∑

=∑=

⋅:=

Ry x( ) D

1

n1

m 1

n2

n

Cm n, βn⋅ βn( )2 2 ν−( ) αm( )2⋅+⎡

⎣⎤⎦⋅ sin αm x⋅( )⋅⎡

⎣⎤⎦∑

=∑=

⋅:=

R0 2 1 ν−( )⋅ D⋅

1

n1

m 1

n2

n

Cm n, αm⋅ βn⋅( )∑=

∑=

⋅:=

Page 51

At the section yb2

:=

x 0 0.01 a⋅, a..:=

0 2 4 6 84− 10 5−×

3− 10 5−×

2− 10 5−×

1− 10 5−×

0Deflection uz at section y=b/2

uz x y, ( )

x

0 2 4 6 830−

20−

10−

0Normal force diagrams at y=b/2

Nx x y, ( )

Ny x y, ( )

x

0 2 4 6 80.6−

0.4−

0.2−

0Bending moment diagrams at y=b/2

Mx x y, ( )−

My x y, ( )−

x

0 2 4 6 81−

0.5−

0

0.5

1Shearing force diagrams at y=b/2

Qx x y, ( )

Qy x y, ( )

x

Page 52

At the section xa2

:=

y 0 0.01 b⋅, b..:=

0 2 4 64− 10 5−×

3− 10 5−×

2− 10 5−×

1− 10 5−×

0Deflection uz at section x=a/2

uz x y, ( )

y

0 2 4 630−

20−

10−

0Normal force diagrams at x=a/2

Nx x y, ( )

Ny x y, ( )

y

0 2 4 60.6−

0.4−

0.2−

0Bending moment diagrams at x=a/2

Mx x y, ( )−

My x y, ( )−

y

0 2 4 61−

0.5−

0

0.5

1Shearing force diagrams at x=a/2

Qx x y, ( )

Qy x y, ( )

y

Page 53

m 0 20..:= x1m 1+ am20

⋅:=

n 0 20..:= y1n 1+ bn20

⋅:=

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 RevolutionShells of Revolution

α

dα

αNα

r

Or

ds1

drα

Nα+d Nα

z

dz

C

C1

R1

R2

z

α= sin2Rr

( )α=⇒α=α==

1

111

RAAddRCCds

(

α, β = meridian and parallel.r(α) – meridian equation.

α=⇒βα=β=

sinsin

2

22

RBdRrdds

α=α∂

∂=

α∂∂

⇒

αα=α=

cos

coscos

1

11

RrBdRCCdr

(

Case of Axis-Symmetrical Shell: 0=Y

0,0 =κ=ε==== αβαβββ uHQS

0=β∂

∂k

k L


( )

( )

( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=α+α+αα

−

=α−αα

+α+α

=α+α−α−αα

αβα

αβα

αβα

.0sincossin

,0sinsincossin

,0sinsincossin

2112

21212

21212

QRRMRMRdd

ZRRQRddNRNR

XRRQRNRNRdd

Strains:

( )

.cotg,11

,cotg1,1

2111

21

⎟⎠⎞

⎜⎝⎛ −

α=κ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

α=κ

+α=ε⎟⎠⎞

⎜⎝⎛ +

α=ε

αβαα

αβα

α

dzduu

RRdzduu

Rdd

R

uuR

uddu

R

zz

zz

Page 56

E.Meissner’s unknowns:

αα =ψ⎟⎠⎞

⎜⎝⎛ +

α−=χ QR

Ru

dduz

21

,1

( )

( )

( ).cotg12

,1112

,1

,cotg1

213

113

1

2

⎟⎠⎞

⎜⎝⎛

α−

α=

ν−−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

α−

α=

ν−−

⎟⎠⎞

⎜⎝⎛ +

α=

ν−

ε=α+=ν−

ααβ

αβα

αβα

βααβ

dduu

RREhMM

dduu

Rdd

REhMM

uddu

REhNN

uuREh

NN

z

z

z

z

ecjBIsmIkarbMErbMrYlragxagelI b¤ecjBIlkçx½NÐCab; edayeyagelIc,ab; Hooke eyIg)an ³

ecjBIsmIkarBIrmun eyIgTTYl)an ³( ) ( )[ ]βαα

α ν+−ν+=α−α

NRRNRREh

uddu

12211cotg

eFVIDIepr:g;EsülelIsmIkarTImYy eyIgnwgman ³

( ) ( )

( ) .sin

cotg

,cotg

22

2

⎥⎦⎤

⎢⎣⎡ ν−

α=

α+

α−α

α

⎥⎦⎤

⎢⎣⎡ ν−

α=α+

α

αβαα

αβα

NNEhR

dd

dduu

ddu

NNEhR

dduu

dd

z

z

ecjBIsmIkarBIrxagelIenH eyIgTTYl)an ³

( ) ( )[ ] ( ) .cotg 21221

1

⎥⎦⎤

⎢⎣⎡ ν−

α−ν+−ν+

α=

=χ=α

−

αββα

α

NNEhR

ddNRRNRR

Eh

Rdduu z

Page 57

müa:geTot eyIgGacsresr)anfa

,1cotg,cotg1

,cotg,1

,1,cotg

1221

21

0

1

0

2

⎟⎟⎠

⎞⎜⎜⎝

⎛αχ

ν+χα

−=⎟⎟⎠

⎞⎜⎜⎝

⎛χ

αν+

αχ

−=

χα

=καχ

=κ

+αψ

−=+ψα

−=

βα

βα

ββαα

dd

RRDM

Rdd

RDM

Rdd

R

Ndd

RNN

RN

Edl 00 , βα NN CakMlaMgEkg tamRTwsþIKμanm:Um:g; (zero moment

theory of shells) Ed;lmanragdUcteTA ³

( ) ,sincossinsin1

1

2122

0

⎥⎥⎦

⎤

⎢⎢⎣

⎡αα−αα+

α= ∫

α

αα dXZRRC

RN

.1

20

⎟⎟⎠

⎞⎜⎜⎝

⎛−= α

β RNZRN

bnÞab;BICMnYstMélkMlaMgEkg cUleTAkñúgsmIkarlMnwgBIrdMbUg eyIgeXIjfa vaRtUv)anepÞógpÞat; . rIÉsmIkarTIbI rYmCamYynwglkçx½NÐCab; begáIt)anCa

( ),cotgcotg

cotg

,cotgcotg3

3cotg

12

2

1

1

2

2

1

1

22

2

1

2

12

2

1

1

2

1

2

1

22

2

1

2

αΦ+χ=ψ⎥⎦

⎤⎢⎣

⎡ν−α

αν

−α−

−αψ

⎥⎦

⎤⎢⎣

⎡

α−α+⎟⎟

⎠

⎞⎜⎜⎝

⎛α

+αψ

ψ−=χ⎥⎦

⎤⎢⎣

⎡α+

ααν

−ν−

−αχ

⎥⎦

⎤⎢⎣

⎡

α+α+⎟⎟

⎠

⎞⎜⎜⎝

⎛α

+α

χ

EhRddh

hRR

dd

ddh

hRR

RR

RR

dd

dd

RR

DR

RR

ddh

h

dd

ddh

hRR

RR

RR

dd

dd

RR

where

( ) ( ) ( ) ( )[ ].cotg 012

021

002βααβ ν+−ν+α−⎥⎦

⎤⎢⎣⎡ ν−

α=αΦ NRRNRRNN

hR

ddh

Page 58

Case h=const:

( ) ( ) ( ).1,1

111

αΦ+χ=ψν

+ψψ−=χν

−χR

EhR

LDR

L

where

( ) ( )LLL

L2

2

1

2

1

2

12

2

21

2 cotgcotg1Rd

dRR

RR

dd

Rdd

RRL α

−α⎥

⎦

⎤⎢⎣

⎡α+⎟⎟

⎠

⎞⎜⎜⎝

⎛α

+α

=

ecjBIsmIkarxagelI eyIgGacTaj)anfa

( ) ( ) ( )

( ) ( ) ( ) ⎪⎪

⎭

⎪⎪

⎬

⎫

αΦν

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

νψ=ψ

ν−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Φ−ψ

ν−ψ

αΦ−χ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

ν=χ

ν+⎟⎟

⎠

⎞⎜⎜⎝

⎛ χν−χ

,

,

21

21

2

111

12

1

2

11

RDEh

RL

RRRLLL

DRDEh

RL

RRLLL

ebI]bmafa( )

ϕ−=χϕ

ν−ϕ=ψ

DRL 1,

1

enaHsmIkarTI1 nwgepÞógpÞat; ehIysmIkarTI2 nwgTTYl)anragCa

( ) ( ) ( )1

21

2

11 RRDEhL

RRLLL αΦ

=ϕ⎟⎟⎠

⎞⎜⎜⎝

⎛ ν−+ϕ

ν+⎟⎟

⎠

⎞⎜⎜⎝

⎛ ϕν−ϕ

For spherical, toroidal, conical, cylindrical shells: R1=const. So,

( ) ( )1

2

RLL αΦ

=ϕμ+ϕ

where

( ) ,11212

1

221

2

21

21

22

Rb

hR

RRDEh

=ν−

≈ν

−=μ( ) .112

2

21

22

hRb ν−

=

Page 59

smIkarcugeRkayenH Gacsresr)aneTACa( )[ ] ( )[ ] ( ) ,

1RiLiL αΦ

=μ−ϕμ+L

b¤k¾ ( )[ ] ( )[ ] ( ) ,1R

iLiiLL αΦ=μ+ϕμ−μ+ϕ

dMeNaHRsayrYmrbs;smIkarTaMgenH GacTTYl)anCaragkMpøic .

krNIEs‘Vr R1=R2=R smIkaredImrbs;smIkarDIepr:g;EsülxagelI manragCa( ) ( ) ,0,0 22

2211

21 =ϕμ+∇=ϕμ+∇

where( )( ) ,

sincotg1 22

221 α

−α

α+α

=−=∇LLL

LLdd

ddRL

( ) ( ),11,11 222111 +ζζ=−=μ+ζζ=+=μ bibi

dMeNaHRsayBiessrbs;smIkarDIepr:g;EsülxagelI Gacrk)anecjBIsmIkar( ) ( ) .

biiL αΦ

=ϕμ+ϕ

smIkaredImk¾Gacsresr)anCarag

( )

( ) ⎪⎪⎭

⎪⎪⎬

⎫

=ϕ⎥⎦⎤

⎢⎣⎡

α−+ζζ+

αϕ

α+αϕ

=ϕ⎥⎦⎤

⎢⎣⎡

α−+ζζ+

αϕ

α+αϕ

.0sin

11cotg

,0sin

11cotg

22222

22

2

12111

21

2

dd

dd

dd

dd

smIkarDIepr:g;EsülxagelIenH GacGaMgetRkal)an edayeRbIGnuKmn_ Legendre .

Page 60

Example. Spherical CupolaExample. Spherical Cupolaconst,21 === hRRR

Equations:( )

( ) ( ) ⎪⎭

⎪⎬⎫

αΦ+χ=νψ+ψ

ψ−=νχ−χ

,

,

1

1

EhRLDRL

( ) ( ) ( )

( ) ( ) .1

,cotgcotg

22

22

2

1

XRddZR

dd

ddRLL

ν++α

=αΦ

α−αα

+α

== LLL

LL

where

Common solutions:

( ) ( )[( ) ( )]422311

2421310

241322110

1,

CCYCCY

CCXCCXEhR

YCYCXCXC

ν+λ−+ν+λ−+

+ν+λ+ν+λ=χ

+++=ψ

Legendre functions:

,82

sin82

cos8

cotg231sin4

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ π−

λα−⎟⎟

⎠

⎞⎜⎜⎝

⎛ π−

λα⎟⎟

⎠

⎞⎜⎜⎝

⎛

λα

−απ

λ≈

λα

eX

,82

sin8

cotg23182

cossin4

21

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ π−

λα⎟⎟

⎠

⎞⎜⎜⎝

⎛

λα

−+⎟⎟⎠

⎞⎜⎜⎝

⎛ π−

λα

απλ

≈λ

αeY

,82

sin8

cotg23182

cossin

22

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ π+

λα⎟⎟

⎠

⎞⎜⎜⎝

⎛

λα

+−⎟⎟⎠

⎞⎜⎜⎝

⎛ π+

λα

απλ

≈λ

α−eX

.82

sin82

cos8

cotg231sin

22

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ π+

λα+⎟⎟

⎠

⎞⎜⎜⎝

⎛ π+

λα⎟⎟

⎠

⎞⎜⎜⎝

⎛

λα

+απ

λ≈

λα−

eY

Solution of differential equations:

( ) ( )( ) ( ) ⎭

⎬⎫

αχ+αχ=χαψ+αψ=ψ.

,

10

10 ( ) ( )αχαψ 11 , = particular solutions

Page 61

20m

45°45°

α

z

R

h X

Zq

p

Self weight:

Support 2

3.0,102

1,4.14

26 =ν⋅=

==

cmkgfE

cmhmR

2008.0cmkgfg =

Live load:

202.0cmkgfp =

Support 1

22

222 ν−=μ=λD

EhRR

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:

α+= cospgq

Components of the vertical load:

.coscoscos,cossinsinsin

2 α+α=α=

αα+α=α=

pgqZpgqX

Load function:

( ) ( )

( ) ( ) αν++ααν+=

ν++α

=αΦ

sin2cossin3

1

22

22

gRpR

XRddZR

Page 62

dMeNaHRsayBiess eKrkCarag

.cossinsin,cossinsin

211

211

αα+α=ψαα+α=χ

BBAA

bnÞab;BICMnYstMélTaMgenH cUleTAkñúgsmIkarxagedIm eKrkeXIj

( )

( ) ( ).325

5,21

1

,25

3,21

1

222

221

3

22

3

21

ν++λ

ν+−=ν+

+λν+

−=

+λν+

−=ν++λ

−=

pRBgRB

pDRAg

DRA

dMeNaHRsaysrubrbs;smIkarDIepr:g;Esül Gacsresr)anfa( )

( ) ( )[ ] ( )⎪⎭

⎪⎬⎫

αχ+ν+λ−+ν+λ=χ

αψ++=ψ

.1,

1311131

11311

CCYCCXEhR

YCXC

)a:ra:Em:Rtefr C1, C3 kMNt;ecjBIlkçx½NÐRBMEdn α=45° dUcteTA

0cotg45

45 =⎟⎠⎞

⎜⎝⎛ ανχ+

αχ

−=°=α

°=α dd

RDM

( ) ( )

( ) ( )[ ]}

( ) ( ) 0cos2cos25

3cos1

12

cotg1

45

22

3

2

3

131113

131

113

=αν+α+λ

ν+−αν+⋅

⋅+λν+

−ν+λ−+ν+λ+

+⎩⎨⎧ αν+

αν+λ−+

αν+λ

°=α

pDRg

DRYCCXCC

ddYCC

ddXCC

EhR

Case of simple support

00454545

=ε→==°=αβ°=α°=αα zuu

Page 63

( ) ( ) ( )

01

cotgsin

11

45

113

11

113112

=⎥⎦

⎤⎟⎠⎞

⎜⎝⎛

αψ

+α

+α

+

⎢⎣⎡ +ψ++α

ν+

αα

ν+−→

°=αdd

ddYC

ddXC

R

YCXCRR

FZREh

enARtg;enH( ) ( )

( )α−−α−=

=αα−αα=α ∫α

cos1sin

sincossin

22221

0

2

gRpR

dXZRF

Case of roller support

0,0cossin4545

==α−α°=αα°=ααα uNQ

Internal forces:

( ) ( ) ;1sin

;cotgsin

;

;cotg;cotg1

12

12

22

2

2

1221

αψ

−α

α−=α

ψ−

αα

=

ψ=

⎟⎟⎠

⎞⎜⎜⎝

⎛αχν

+χα

−=⎟⎟⎠

⎞⎜⎜⎝

⎛χ

αν+

αχ

−=

βα

α

βα

dd

RRFZRN

RRFN

RQ

dd

RRDM

Rdd

RDM

Strains:( )

( ) .cotg11sin

1

;1cotg1sin

1

22

2122

22

2122

⎥⎦

⎤⎢⎣

⎡α

νψ++

αψ

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ν+

αα

−=ε

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛αψ

−ν−αψ

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ν+

αα

=ε

β

α

RZR

dd

RRRF

Eh

dd

RZR

RRRF

Eh

Page 64

Displacements:edaHRsaysmIkar

( ),cotg1,1

21zz uu

Ru

ddu

R+α=ε⎟

⎠⎞

⎜⎝⎛ +

α=ε αβ

αα

eyIgTTYl)an( )

( ) ( ) .1sin

cotg1cotg

,sinsin

2sin11sin

2

22

1

2

⎥⎦⎤

⎢⎣⎡ α+

αα

−+⎟⎠⎞

⎜⎝⎛ ανψ−

αψ

−α−=

αα

⎥⎦⎤

⎢⎣⎡ −

αα

αν+

+ψν+

+α=

α

α

αα ∫

RFRZ

EhR

dd

Ehuu

dRZR

FREhEh

Au

z

Edl A2 Ca)a:ra:Em:Rtefr nigkMNt;)anecjBIlkçx½NÐRBMEdn .

Page 65

Zero Moment Zero Moment (Membrane)(Membrane)Theory of ShellsTheory of Shells

0,0 ===== βαβα QQHMM


( ) ( )

( ) ( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

=−+

=+α∂∂

+β∂

∂−

β∂∂

=+β∂∂

+α∂

∂−

α∂∂

βα

αβ

βα

.0

,01

,01

21

2

2

ZRN

RN

ABYSBB

ANAN

ABXSAA

BNBN

The problem is statically determinate.

eKaledAénkar KNnaK μanm:Um:g; KWkMNt;rksPaB sMBaFkñúgem (principalstress state)

mYyEdledIr tYnaTIsMxan; .

lkçx½NÐ zero-moment stress-strain state:

Shell RtUvEtmankMras;efr b¤ERbRbYledaysnSwm² ehIydUcKñaEdr cMeBaH kaMkMeNag minRtUvERbRbYlya:gxøaMgenaHeT .

kMlaMgeRkA RtUvEtCab;Kña nigERbRbYledaysnSwm². Zero-moment shell minGaceFVIkarnwgkMlaMgeTal)aneT .

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 RevolutionAnalysis of Shells of Revolution

α

dα

αNα

r

Or

ds1

drα

Nα+d Nα

z

dz

C

C1

R1

R2

z

α, β = meridian and parallel.

( )

.cos

,sin,

1

2

1

α=α∂

∂α==

α=

RBRrB

RA


( )

( )

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−+

=α+αα∂∂

α+

β∂∂

=α+β∂

∂+α−α

α∂∂

βα

β

βα

.0

,0sinsinsin1

,0sincossin

2112

2122

22

1

21112

ZRRNRNR

YRRSRR

NR

XRRSRRNNR

Case of axis symmetrical problem: 0,0 =β∂

∂= k

k

Y L

0=== β SQH

( )⎪⎭

⎪⎬⎫

=−+

=α+α−αα

βα

βα

.0

,0sincossin

2112

2112

ZRRNRNR

XRRRNNRdd

Page 67

ecjBIsmIkarTI 2 eyIgTTYl)an ³ ⎟⎟⎠

⎞⎜⎜⎝

⎛−= α

β1

2 RNZRN

CMnYscUleTAkñúgsmIkarTI 1 eyIgnwgman ³

( ) ( ) 0cossinsin 1 =α−α+αα α ZXrRrN

dd

ecjBIenH ( ) CdXZrRrN +αα−α=α ∫

α

αα

1

sincossin 1

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡αα−αα+

α= ∫

α

αα

1

sincossinsin1

2122

dXZRRCR

N

Edl C Ca)a:ra:Em:Rt nigrk)anecjBIlkçx½NÐRBMEdn .

RbsinebI smIkaremrIdüanRtUv)aneKeGayCarag ( )zrr = enaHsmIkarrbs;épÞrgVil KitenAkUGredaenEkg Gacsresr)anfa

zzryrx =β=β= ,cos,sin

dUecñH eyIg)an ,cotg α==′dzdrr

( )( ) ⎪⎭

⎪⎬⎫

=

′+=.

,1 212

zrBrA

⎪⎭

⎪⎬⎫

β==

′+==

.

,1

12

211

rdDCds

rdzdsCC(

(

Curvatures:

( ) ( ).

1

1,1 2

122

212

1rr

kr

rk′+

=′+

′′−=

Page 68


( )

( )

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=′+−+′+′′

−

=′++∂∂

+β∂

∂′+

=′++β∂

∂′++′−∂∂

βα

β

βα

.011

,0111

,011

22

222

22

ZrrNNrrr

YrrSrzr

Nr

XrrSrNrrNz

For homogeneous problem: 0=== ZYX

eKtag stress function:

⎟⎠⎞

⎜⎝⎛ ϕ

∂∂

−=β∂ϕ∂′′

=β∂ϕ∂

= βα rzS

rArN

rAN ,,2

enAkñúgkrNIenH smIkarTI 1 nigTI 3 epÞógpÞat; rIÉsmIkarTTYl)anrag ³

02

2

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛β∂ϕ∂

+ϕ′′

−∂

ϕ∂rr

z

For axis symmetrical problem: 0=Y

( )

⎪⎪⎭

⎪⎪⎬

⎫

=′+−+′+′′

−

=′++′−

βα

βα

.011

,01

22

2

ZrrNNrrr

XrrNrrNdzd

Equilibrium equations

( )

.11

;1

22

2

0

ZrrNr

rrN

dzXZrrCr

rNz

z

′++′+′′

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−′+

′+=

αβ

α ∫Solution

Page 69

z

α0 r0

q qQz

R2

R1 α

dα

r

k2 k1XNα

Nαsinα

Z

rUbmnþ Nα Gacsresr)anfa³( ) CdXZRRRN π+αα−ααπ=π⋅α⋅α ∫

α

αα 2sincossin22sinsin

0

212

Integration Technique

( ) qrdRrXZNr ⋅π+α⋅π⋅α−α=α⋅π ∫α

αα 01 22sincossin2

0

or

tYeqVgénsmPaBxagelI KWCacMeNalelIG½kS z énpÁÜbrbs;kMlaMgEkg tamrgVg; EdlmankaM r . edayehtufa 2πrR1dα KWCaépÞénvgStUcminkMNt;mYy EdlRtUvnwgmMu dα/ rIÉ Zcosα nig Xsinα KWCacMeNalelIG½kS z énkMlaMgeRkA dUecñH

( ) zQdRrXZ =α⋅π⋅α−α∫α

α0

12sincos

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

.sin2 απ

=α rQN z

kñúgkarkMNt; Qz eKGaceRbIR)as;RTwsþIbT dUcxageRkam .RTwsþIbT 1> RbsinebI elIépÞNamYy eFVIGMeBIsMBaFBRgayesμ I p enaHminGaRs½y nwgrUbragépÞ cMeNalénkMlaMgpÁÜbrbs;sMBaFelIG½kSNamYy esμ IplKuNsMBaF p enaH nwgRkLaépÞrbs;cMeNalénépÞelIbøg; EdlEkgnwgG½kSenaH .

RTwsþIbT 2> RbsinebI elIépÞNamYy eFVIGMeBIsMBaFGgÁFaturav enaHkMlaMgpÁMúbBaÄrrbs;sMBaFenaH esμ ITMgn;GgÁFaturavkñúgmaD EdlenAelIépÞ .

Example 1.Example 1.

Rα

p

ααdα

p

R

qNα

qNα

q

Spherical cupola:

Thickness h,

Self weight q,

Vertical live load p,

Simple support at α = 90°

Page 71

smIkarlMnwgsMrab;EpñkxagelIénBuH α manragdUcteTA ³,0sin2 =−απ− α

qz

q QrN

where ,sin α= RrqzQ = resultant of self weight,

( )α−π=ααπ=απ= ∫∫αα

cos12sin22 2

0

2

0

qRdqRrRdqQqz

So,.

cos1sincos1

sin2 2 α+−=

αα−

−=απ

−=αqRqR

rQN

qzq

eday RRRqZ ==α−= 21,cos eyIgnwg)an

( )[ ]α+α−α+

=

⎟⎠⎞

⎜⎝⎛

α++α−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

β

αβ

cos1cos1cos1

cos1cos

12

qRN

qqRRNZRN

q

qq

Analysis on vertical live load

eyagtamRTwsþIbT 1 eyIgGacsresrsmIkarlMnwg)andUcteTA,0sin2 2 =π−απ− α rprN p

where .sin α= Rr

.2

pRN p −=α

eday α⋅α−= coscospZ eyIgnwgrkeXIj

⎟⎠⎞

⎜⎝⎛ +α−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−= α

β 2cos2

12

ppRRNZRN

pp

α−=β 2cos2

pRN p

Page 72

qNαDiagram qNβDiagram

pNαDiagram pNβDiagram

C

Cylindrical Cylindrical andand Conical ShellsConical Shellsx

y

z

αβ

xy

z

Cα β

θ

( )( ) ⎪

⎭

⎪⎬

⎫

β=β=

α=

.,

,

zzyy

x( ).

,cossin,sinsin

,cosβθ=θ

⎪⎭

⎪⎬

⎫

βθα=βθα=

θα=

zyx

Page 73

Cylindrical and conical shells are shells with zero Gaussian curvatures:

011

2121 ===

RRkkk

For cylindrical shells:

( ) ( )[ ] .,

;,1

2322

21

22

yzzyzyRR

zyBA

′′′−′′′′+′

=∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛β∂

∂==

For conical shells:

( )[ ]( )

.sincos2sincos

sin,

;sin,1

22

2322

21

22

θθ′′−θθ′+θθθ′+θα

−=∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛β∂θ∂

+θα==

RR

BA

edayyk A=1 nig R1=∞ smIkarlMnwgsþaTic TTYl)anragdUcteTA ³

( )

( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

=−

=+α∂∂

+β∂

∂

=+β∂

∂+

α∂∂

−α∂∂

β

β

βα

.0

,01

,0

2

2

ZRN

BYSBB

N

BXSBNBN

edaHRsaysmIkarenH eyIgTTYl)an ³;2 RZZRN ==β

( ) ( )∫α

α

α⎥⎦

⎤⎢⎣

⎡+

β∂∂

−β=0

2212

11 dYBRZBB

fB

S

Page 74

( ) ( )

( )∫ ∫

∫∫α

α

α

α

α

α

α

αα

α⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

α⎥⎦

⎤⎢⎣

⎡+

β∂∂

β∂∂

β∂∂

+

+α⎟⎠⎞

⎜⎝⎛ −

α∂∂

+β

+α⎥⎦⎤

⎢⎣⎡ β

β∂∂

−=

0 0

00

22

21

11

11

ddYBRZBBB

dBXRZBBB

fdB

fB

N

enARtg;enH f1(β), f2(β) CaGnuKmn_GaRs½ynwgGefr β .

Example 2.Example 2. Horizontal Pipeline of Circular SectionHorizontal Pipeline of Circular Section

α (x)

y y

β

z

l

qYZR

O

For cylindrical shell:

RBRR == ,2

Analysis on Self Weight

Components of self weight: β=β== cos,sin,0 qZqYX

Normal forces: β−==β cosqRRZN

Rims are rigidly in plane and free out plane.

Page 75

( ) ( )

( )βα−

β=

⎥⎦

⎤⎢⎣

⎡αβ+αβ−

β∂∂

−β

= ∫∫αα

sin2

sincos

21

002

2

21

qRf

dqdqRR

RfSTangential

force

Normal force

( )[ ] ( ) ( )

( )[ ] ( )R

qR

ffR

dqRR

ffR

N

βα−

β+αβ

β∂∂

−=

αβα−β∂∂

+β

+αββ∂∂

−= ∫α

α

cos1

sin211

22

12

0

212


( ) ;00,0 2 =β→==α α fN

( ) .sin0, 21 ClqRfNl +β=β→==α α

)a:ra:Em:Rtefr C/R2 KWCakMlaMgkat;BRgayesμ I elIEKmrbs;bMBg; . dUecñH RbsinebI bMBg;minrgkarrmYreT KWmann½yfa )a:ra:Em:RtefrenHesμ IsUnü ³

( ) .sin,0 21 β=β= qlRfC

srubmk eyIgTTYl)an( )

( ) .sin2,cos

,cos

β−α−=

β−=

βα−α

=

β

α

lqSqRN

RlqN

Page 76

0=βαN

0=αS

-+

-

qlql

Rql 42

+ ql+

-βN

qR

qR

2π

=βS

Diagrams

dUecñH eyIgrkeXIj ( ),cos0 βγ−==β RpRRZN

( ) ( ) ( ) ,sincos12

1

00

222

1 βαγ−β

=αβγ−β∂∂

−β

= ∫α

RRfdRpR

RRfS

( )[ ] ( ) ( )

( )[ ] ( )β

γα+

β+αβ

β∂∂

−=

αβαγβ∂∂

+β

+αββ∂∂

−= ∫α

α

cos2

1

sin11

22

13

0

213

Rff

R

dRRR

ffR

N

Analysis on Fluid WeightAnalysis on Fluid Weight

Components of fluid weight: .cos,0 0 βγ−=== RpZYX

p0 = fluid pressure in a plane zOx.

Page 77

srubmk eyIgnwgmanlTæpl

( )

( )

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

β⎟⎠⎞

⎜⎝⎛ α−γ=

βγ−=

βα−αγ

−=

β

α

.sin2

,cos

,cos2

0

lRS

RpRN

lN

edayeRbIR)as;lkçx½NÐRBMEdn dUcbgðajxagmux eyIgGackMNt;)an ³

( ) ( ) .sin2

,02

12 βγ

=β=βlRff

π=βαN

0=αS

-+

2Rlγ

+2Rlγ

+

-

βN

( )RpR γ+0

2π

=βS

Diagrams

+

82lγ

2Rlγ

( )RpR γ−0

Page 78

Example 3.Example 3. Analysis of Cylindrical Tank on Wind LoadAnalysis of Cylindrical Tank on Wind Load

y

xl

α

Wind direction

pβ

R

Components of wind load:

( )β−β−===

2cos2.1cos5.07.0,0

pZYX

where p = max. wind pressure.

]bmafa sMBaFxül;minERbRbUltamkMBs; suILaMg KWminGaRs½ynwgkUGredaen x=α .

kMlaMgxül;elIsuILaMg

dUecñH eyIg)an

( ),2cos2.1cos5.07.0 β−β−==β pRRZN

( ) ( ) ( ) ( )β+βα−β

=αβ∂∂

−β

= ∫α

2sin4.2sin5.012

1

022

1 pRfdRZR

RBfS

( )

( ) ( ) ( )β+βα

+β

+ββ∂∂

=

α⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡α

β∂∂

β∂∂

= ∫ ∫α α

α

2cos8.4cos5.02

1

11

22

13

0 02

Rp

Rff

R

ddRZBBB

N

ecjBIlkçx½NÐRBMEdn 0,0 ===α αNS eyIgkMNt;)an ( ) ( ) 021 =β=β ff

Page 79

srubmk eyIgnwgman

( ).2sin4.2sin5.0 β+βα−= pS

( ),2cos8.4cos5.02

2

β+βα

=α RpN

( ),2cos2.1cos5.07.0 β−β−=β pRN

Diagramsl

N=αα

lS

=αβN

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( )⋅:=

α1π

2−:= α2

π

2:=

n 50:= Δαα2 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

Nαqxi

⎛⎜⎜⎝

⎞⎟⎟⎠

:= Y1 i⟨ ⟩Yi

Nαqyi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

X2 i⟨ ⟩Xi

Nβqxi

⎛⎜⎜⎝

⎞⎟⎟⎠

:= Y2 i⟨ ⟩Yi

Nβqyi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

X3 i⟨ ⟩Xi

Nαpxi

⎛⎜⎜⎝

⎞⎟⎟⎠

:= Y3 i⟨ ⟩Yi

Nαpyi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

X4 i⟨ ⟩Xi

Nβpxi

⎛⎜⎜⎝

⎞⎟⎟⎠

:= Y4 i⟨ ⟩Yi

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−( )⋅

Rcos β( )⋅:=

N 50:= Δαα1 α0−

N:=

α α0 α0 Δα+, α1..:=

0 0.2 0.4 0.6 0.80

0.01

0.02

0.03Diagram Nx

Nα α π, ( )

Nα α π, ( )

α

Page 84

0 0.2 0.4 0.6 0.81−

0.5−

0

0.5

1Diagram S

S απ

2, ⎛⎜

⎝⎞⎟⎠

S απ

2, ⎛⎜

⎝⎞⎟⎠

α

N 50:= Δβπ

N:=

i 0 N..:= βi i Δβ⋅:=

S1i S 0 βi, ( ):= Sx i⟨ ⟩ 0

S1i

⎛⎜⎝

⎞⎟⎠

:= Sy i⟨ ⟩βi

βi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

N2i Nβ βi( ):= Nx i⟨ ⟩ 0

N2i

⎛⎜⎝

⎞⎟⎠

:= Ny i⟨ ⟩βi

βi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

0.5− 0 0.5 1

1

2

3

Diagram S

β

Sy

S1 Sx,

10− 5− 0 5 10

1

2

3

Diagram N2

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

2− α⋅ L α−( )⋅ cos β( )⋅:=

Nβ β( ) R p γ R⋅ cos β( )⋅−( )⋅:=

S α β, ( ) γ R⋅L2

α−⎛⎜⎝

⎞⎟⎠

⋅ sin β( )⋅:=

N 50:= Δαα1 α0−

N:=

α α0 α0 Δα+, α1..:=

0 0.2 0.4 0.6 0.80

0.01

0.02

0.03Diagram Nx

Nα α π, ( )

Nα α π, ( )

α

0 0.2 0.4 0.6 0.86−

4−

2−

0

2

4

6Diagram S

S απ

2, ⎛⎜

⎝⎞⎟⎠

S απ

2, ⎛⎜

⎝⎞⎟⎠

α

Page 86

N 50:= Δβπ

N:=

i 0 N..:= βi i Δβ⋅:=

S1i S 0 π βi−, ( ):= Sx i⟨ ⟩ 0

S1i

⎛⎜⎝

⎞⎟⎠

:= Sy i⟨ ⟩βi

βi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

N2i Nβ π βi−( ):= Nx i⟨ ⟩ 0

N2i

⎛⎜⎝

⎞⎟⎠

:= Ny i⟨ ⟩βi

βi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

2− 0 2 4 6

1

2

3

Diagram S

β

Sy

S1 Sx,

50− 0 50 100 150

1

2

3

Diagram N2

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 i2 π⋅

N⋅:=

vxi y βi( )−:= vyi z βi( ):=

Z1i Zx βi( ):= Z2i Zy βi( ):=

L1 i⟨ ⟩vxi

Z1i

⎛⎜⎜⎝

⎞⎟⎟⎠

:= L2 i⟨ ⟩vyi

Z2i

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

vy

Z2

L2

vx Z1, L1,

Page 88


Nα α β, ( )p α

2⋅

2 R⋅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: s1125

:= s212

:= s3120

:=

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

N1xi

⎛⎜⎜⎝

⎞⎟⎟⎠

:= L1y i⟨ ⟩vyi

N1yi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

N2xi Nβx βi( ):= N2yi Nβy βi( ):=

L2x i⟨ ⟩vxi

N2xi

⎛⎜⎜⎝

⎞⎟⎟⎠

:= L2y i⟨ ⟩vyi

N2yi

⎛⎜⎜⎝

⎞⎟⎟⎠

:=

Diagram N1

vy

N1y

L1y

vx N1x, L1x,

Page 89

Sxi Sx L βi, ( ):= Syi Sy L βi, ( ):=

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.Example 4. Spherical Tank under FluidSpherical Tank under Fluid

R

α0

α

AA

r

z

Nα

Nα

2αp

TMrragrgVg; AA CaRbePT simple

sMBaFGgÁFaturav ³( )α−γ= cos1Rp

α= sinRrkaMmuxkat; ³

ecjBIlkçx½NÐlMnwgtamG½kSbBaÄr eKrkeXIj ³

απ=

απ=α 2sin2sin2 R

QrQN zz

r

z

R αϕ dϕ

dPdQz ( )

( ) ϕϕ−ϕγπ=

ϕπϕ−γ=ϕ⋅π⋅=

dRrRdRRdrpdP

cos1sin22cos12

3

( ) ϕϕ−ϕϕγπ=

ϕ=

dR

dPdQz

cos1cossin2

cos3

( )

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ α−α−γπ=

ϕϕ−ϕϕγπ== ∫∫αα

cos321cos

21

612

cos1cossin2

23

0

3

0

R

dRdQQ zz

( )[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛α+α

−γ

=α−α−α

γ=α cos1

cos216

cos23cos1sin6

222

2

2 RRN

Page 91

⎟⎟⎠

⎞⎜⎜⎝

⎛α+α

+α−γ

=−=⎟⎟⎠

⎞⎜⎜⎝

⎛−= α

αβ cos1

cos2cos656

22

12

RNRZRNZRN

Normal component of external force:

rUbmnþ Nα nig Nβ xagelIenH eRbI)ansMrab;EtkrNI .00 α≤α≤

edIm,IkMNt;kMlaMgpÁÜb Qα sMrab;EpñkxageRkamTMr eRkABIsMBaFkñúg eKRtUv KitRbtikmμbBaÄrrbs;TMrcUlbEnßmeTot Edlesμ ITMgn;GgÁFaturavTaMgmUl ³

γπ= 3

34 RRA

dUecñH

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ α−α−γπ+γπ= cos

321cos

21

612

34 233 RRQz

( )α−γ== cos1RpZ

ecjBIenH eyIgnwgTTYl)an

.cos1

cos2cos616

,cos1

cos256

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛α−α

−α−γ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛α−α

+γ

=

β

α

RN

RN

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.Example 5. Ellipsoid of RevolutionEllipsoid of Revolution

r

z

p

a

b

αα

r

z

p

αα

NαNα

r

p CasMBaFBRgayesμ IelI shell.

kMlaMgpÁÜbbBaÄr ³ απ=π= sin22

2 RprQz

R1

ecjBIsmIkarlMnwgtamG½kSbBaÄr eyIg)an ³

2sin2sin22pRpr

rQN z =

α=

απ=α

Equation of ellipse:

12

2

2

2

=+bz

ar

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−= α

β1

22

12 1

RRpR

RNZRN

Curvatures: .111,

11

22

221

1rrR

kr

rR

k′+

==′+

′′−==

Radius of curvature:

., 4

23212

2424

2 abRR

bzbraR =

+=

enARtg;kMBUl :,0 bzr == ,2

21 baRR == .

2

3

bpaNN == βα

enAeGkVaT½r :0, == zar ,,2

22 baRaR == ,

2paN =α

,2

1 2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−=β b

apaN

Page 93

Example 6.Example 6. Conical Shell under FluidConical Shell under Fluid

z

ββl

z

ββ

NαNα

zl

R22α

Qz

r

γ

β= tgzr

ecjBIlkçx½NÐlMnwgtamG½kSbBaÄr eyIg)anβπ

=βπ

=α sin2cos2 zQ

rQN zz

V1

V2

kMlaMgpÁÜbbBaÄr ³( ) ( ) ⎟

⎠⎞

⎜⎝⎛ −γπ=⎥⎦

⎤⎢⎣⎡ −π+πγ=+γ= zlrzlrzrVVQz 3

231 222

21

β

β⎟⎠⎞

⎜⎝⎛ −γ

=βπ

⎟⎠⎞

⎜⎝⎛ −γπ

=α cos2

tg32

sin2322 zlz

z

zlrN

( )ββγ

===αα cos

tg163 2

43max

lNNlz

Radius:ββ

=β

=cos

tgcos2

zrR

Normal component of force: ( )zlZ −γ=

( )β

β−γ==β cos

tg2

zzlZRN

( )ββγ

=β cos4tg2

max

lN

z

+

2l

+

43l

αNβN

Page 94

PROBLEMS OF SHELL THEORY

1. Differential Geometry Of Surface

1.1. eKeGayépÞmYyCarag ( )yxzz ,= . cUrrk first nig second quadratic forms RBmTaMg Gaussian nig mean curvatures .

1.2. eKeGayépÞrgVilmYyCarag ( ) ( ) ( ) ( ) ( ) 0,sincos >ρϕρ+ϕρ+=ϕ uuuuxu, kjir

cUrkMNt; first nig second quadratic forms . 1.3. Translation surface KWCaépÞ EdlekIteLIgedayclnarMkilExSekagmYy ( )xfz 11 =

tambeNþayExSekagmYyeTot ( )yfz 22 = . ExSekagrag nigExSekagTis GacepSg²Kña b:uEnþCaTUeTA eK eRCIserIsykragEtmYy dUcCa )ara:bUl/ FñÚrgVg; .l.

smIkarrbs;épÞrMkil manrag ( ) ( )yfxfz 21 +=

]TahrN_ ³

( ) ,42

22

1

22

111aRaxRxfz −−⎟

⎠⎞

⎜⎝⎛ −−==

( ) .42

222

22222

bRbyRyfz −−⎟⎠⎞

⎜⎝⎛ −−==

sMrab;épÞxagelIenH cUrrk first nig second quadratic forms RBmTaMg curvatures . 1.4. ]bmafa mankUGredaensuILaMg ( )βα= ,z Edl β KWCamMucab;BIG½kS Ox dl;cMeNalénvicT½rkaM

r . dUecñH épÞrgVilGacmansmIkardUcxageRkam ( ) ( ) ( ) kjir zzrzrz +β+β=β sincos,

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

vczvuayvuax sin,sincos,coscos ===

b) Sphere α=βα=βα= sin,sincos,coscos RzRyRx

c) Cylinder of revolution β=β=α= sin,cos, RzRyx

Page 95

d) Shallow shell

( ) 0,, ≈∂∂

=∂∂

=yz

xzyxzz

e) Conical surface of revolution α⋅β=α⋅β=α= sin,cos, RzRyx

2. Shell Analysis

2.1. eFVIkarKNna circular cylindrical shallow shell nwgbnÞúkeRkAbBaÄrBRgayes μ I q sMrab; krNIEdlTMrTaMgbYnRCugrbs;va CaRbePTsnøak; (simple supports) .

25.0,mkg102

m2.1m,40m20m,6m,8

29

2

=ν⋅=

======

E

fRR.hba

2.2. eFVIkarKNnaEkvragekan EdlmanmMukMBUlesμ I β2 nigpÞúk edayGgÁFaturav Edlmanma:smaD γ .

2.3. cUreFVIkarKNna spherical tank EdlRTedayTMr kMNl;ragrgVg; AA nigpÞúkeBjedayGgÁFatu rav Edlmanma:smaD γ .

x

y

z

f

b a

β β l

R α0

α

AA

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

Content1. Differential geometry of surface

1.1. Equation of surface1.2. First and second quadratic forms, Gaussian and mean

curvature

2. Moment theory of shells2.1. Differential equations of equilibrium2.2. Internal forces, strains, change of curvatures, Hooke’s

law and boundary conditions2.3. Analysis of cylindrical shells2.4. Analysis of shallow shells2.5. Shells of revolution

3. Zero moment (membrane) theory of shells3.1. Equilibrium equations3.2. Shells of revolution3.3. 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

SummarySummary1. Differential Geometry of Surface

1.1. Equation of surface:

( ) ( ) ( ) ( )kjirr βα+βα+βα=βα= ,,,, zyx

or( )( )( ) ⎪

⎭

⎪⎬

⎫

βα=βα=βα=

.,,,,,

zzyyxxIn vector

In function

( )yxzz ,= or ( ) 0,, =zyxF

1.2. First quadratic form:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛β∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛β∂

∂=

β∂∂

β∂∂

==

β∂∂

α∂∂

+β∂

∂α∂

∂+

β∂∂

α∂∂

=β∂

∂α∂

∂=

⎟⎠⎞

⎜⎝⎛

α∂∂

+⎟⎠⎞

⎜⎝⎛

α∂∂

+⎟⎠⎞

⎜⎝⎛

α∂∂

=α∂

∂α∂

∂==

.

;

;

2222

2222

zyxGB

zzyyxxF

zyxEA

rr

rr

rr

Principal curvatures:

⎪⎪⎭

⎪⎪⎬

⎫

=−==

=−==

22

max2

21

min1

1

,1

BN

Rkk

AL

Rkk

2222

221β+α

β+α=−

dBdANdLd

R

Gaussian curvature of the surface: 222

2

2121

1FBA

MLNRR

kkk−

−===

Page 100

,1222

βββ

ααα

αααααα

βα

βααααα

−=

×

×⋅=⋅=

zyxzyxzyx

FBAL

rrrrr

nr

,1222

βββ

ααα

αβαβαβ

βα

βααβαβ

−=

×

×⋅=⋅=

zyxzyxzyx

FBAM

rrrrr

nr

,1222

βββ

ααα

ββββββ

βα

βαββββ

−=

×

×⋅=⋅=

zyxzyxzyx

FBAN

rrrrr

nr

Second quadratic form:

Mean curvature of the surface:2

21 kkH +=

2. Moment Theory of Shell

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

=+α∂

∂+

α∂∂

−β∂∂

=

=+β∂

∂+

β∂∂

−α∂∂

=

=−α∂∂

+β∂∂

++=

=+−α∂∂

+β∂

∂−

β∂∂

=

=+−β∂∂

+α∂

∂−

α∂∂

=

αβα

βαβ

αββα

β2

αβ

αβα

∑

∑

∑

∑

∑

,01:0

,01:0

,0:0

,01:0

,01:0

2

2

21

2

1

2

ABQBMBMHAA

M

ABQAMAMHBB

M

ABZBQAQNRABN

RABZ

ABYQRABSB

BANANY

ABXQRABSA

ABNBNX

y

x

2.1. Differential equations of equilibrium

Page 101

2.2. Internal forces:

( )( )

( ) ⎪⎭

⎪⎬

⎫

εν−=

νε+ε=

νε+ε=

αβ

αββ

βαα

,1

,,

21 CS

CNCN ( )

( )( ) ⎪

⎭

⎪⎬

⎫

κν−−=

νκ+κ−=

νκ+κ−=

αβ

αββ

βαα

.1,,

DHDMDM

Strains:

.11

2RuuB

ABu

Bz+

α∂∂

+β∂

∂=ε α

ββ,11

1RuuA

ABu

Az+

β∂∂

+α∂

∂=ε β

αα

⎟⎠⎞

⎜⎝⎛

β∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛α∂∂

=ε αβαβ A

uBA

Bu

AB

21 ν−=

EhC ( )2

3

112 ν−=

EhD

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎟⎠⎞

⎜⎝⎛

α∂∂

+⎟⎠⎞

⎜⎝⎛

β∂∂

=κ

α∂∂

+β∂

∂=κ

β∂∂

+α∂

∂=κ

αβ

β

α

.2

,11

,11

12

12

21

AV

BA

BV

AB

VBAB

VB

VAAB

VA

⎪⎪⎭

⎪⎪⎬

⎫

β∂∂

−=

α∂∂

−=

β

α

.1

,1

22

11

z

z

uBR

uV

uAR

uV

Changes of curvatures:

Hooke’s law:

( )[ ]

( )[ ]

( ) ( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

κ+εν+

=τ=τ


=σ


=σ

αβαββααβ

αβαββ

βαβαα

.212

,1

,1

2

2

zE

zE

zE

Page 102

2.3. Cylindrical Shells

Equations of cylindrical shell: ( ) ( )β=β=α= zzyyx ,,

( ).,,0cos,,,0,1

21 sRRRdsddxdFBA

=∞==χ=β=α===

.,s

MxHQ

sH

xMQ s

sx

x ∂∂

+∂∂

=∂∂

+∂

∂=Shears:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−∂

∂+

∂∂∂

+∂

∂+

=+∂

∂−

∂∂

−∂

∂+

∂∂

=+∂∂

+∂

∂

.02

,011

,0

2

22

2

2

ZsM

sxH

xM

RN

Ys

MRx

HRs

NxS

XsS

xN

sxs

xs

x

Equations of equilibrium:

Strain components:

.212,,

,,,

2

2

2

sxu

xu

Rsu

Ru

sxu

su

xu

Ru

su

xu

zsxs

zsy

zx

xsxs

zsy

xx

∂∂∂

−∂∂

=κ⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=κ∂∂

−=κ

∂∂

+∂∂

=ε+∂∂

=ε∂∂

=ε

Internal forces:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ν−

=

⎥⎦⎤

⎢⎣⎡

∂∂

ν++∂∂

=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +

∂∂

ν+∂∂

=

,2

1

,

,

su

xuCS

xu

Ru

suCN

Ru

su

xuCN

xs

xzss

zsxx

( )⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂−

∂∂

ν−−=

⎥⎦

⎤⎢⎣

⎡∂∂

ν−⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

−=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

ν+∂∂

−−=

.211

,

,

2

2

2

2

2

sxu

xu

RDH

xu

su

Ru

sDM

su

Ru

sxuDM

zs

zzss

zszx

Page 103

,012

21

1221

21

2

2

2

22

2

2

2

22

2

2

2

22

=+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡∂∂ν−

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ν+

+∂∂

+∂∂

∂ν+

CYu

sxsRh

Rs

uxRRsR

hxssx

u

z

sx

.0212

1

121

4

4

22

4

4

42

2

2

2

2

22

=−⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂+

∂∂

++

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

−∂∂

+∂∂ν

CZu

ssxxh

R

uRsRxs

hsRx

uR

z

sx

,02

12

1 2

2

2

2

2

=+∂∂ν

+∂∂

∂ν++⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂ν−

+∂∂

CX

xu

Rsxuu

sxzs

x

Equilibrium equations in displacements:

2.4. Shallow Shells

.5,20 minmin ≥≥ flhR 0,0 ≈∂∂

≈∂∂

yz

xz

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

=+α∂

∂+

α∂∂

−β∂∂

=+β∂

∂+

β∂∂

−α∂∂

=−α∂∂

+β∂∂

++

=+α∂∂

+β∂

∂−

β∂∂

=+β∂∂

+α∂

∂−

α∂∂

αβα

βαβ

αββα

αβ

βα

,01

,01

,0

,01

,01

2

2

21

2

2

ABQBMBMHAA

ABQAMAMHBB

ABZBQAQNRABN

RAB

ABYSBB

ANAN

ABXSAA

BNBN

Equilibrium Equations:

Page 104

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂β∂

∂−

α∂∂

α∂∂

−β∂α∂

∂−=κ

α∂∂

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂β∂∂

−=κ

β∂∂

β∂∂

−⎟⎠⎞

⎜⎝⎛

α∂∂

α∂∂

−=κ

αβ

β

α

.111

,111

,111

2

2

2

zzz

zz

zz

uAA

uBB

uAB

uBBA

uBB

uAAB

uAA

Changes of curvature:

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎠⎞

⎜⎝⎛

β∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛α∂∂

=ε

+α∂

∂+

β∂∂

=ε

+β∂

∂+

α∂∂

=ε

αβαβ

αβ

β

βα

α

,

,11

,11

2

1

Au

BA

Bu

AB

RuuB

ABu

B

RuuA

ABu

A

z

zStrains:

( ) ( )

( ) ( ) .1112

,1112

22

3

22

3

z

z

uAD

BEhQ

uAD

AEhQ

∇β∂∂

=κ+κβ∂∂

ν−−=

∇α∂∂

=κ+κα∂∂

ν−−=

βαβ

βαα

Shears:

.111

,111

,111

2

2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛α∂ϕ∂

β∂∂

−β∂ϕ∂

α∂∂

−β∂α∂

ϕ∂−=

β∂ϕ∂

β∂∂

+⎟⎠⎞

⎜⎝⎛

α∂ϕ∂

α∂∂

=

α∂ϕ∂

α∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛β∂ϕ∂

β∂∂

=

β

α

AA

BBAB

S

AABAA

N

BBABB

N


Page 105

.0,01 222222 =−∇∇+ϕ∇=∇−ϕ∇∇ ZuDuEh zkzk

Equation of shallow shell:

,,,21 y

uxu

Ru

yu

Ru

xu xy

xyzy

yzx

x ∂∂

+∂∂

=ε+∂∂

=ε+∂∂

=ε

.,,22

2

2

yxu

yu

xu zz

yz

x ∂∂∂

−=κ∂

∂−=κ

∂∂

−=κ αβ

Strain components:

Rectangular Shallow Shell

Internal forces:

( )

( )

( ) ( )⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

∂∂∂

ν−−=

⎥⎦

⎤⎢⎣

⎡∂∂

ν+∂∂

−=

⎥⎦

⎤⎢⎣

⎡∂∂

ν+∂∂

−=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

ν−=

⎥⎦

⎤⎢⎣

⎡ν++

∂∂

ν+∂

∂=

⎥⎦

⎤⎢⎣

⎡ν++

∂∂

ν+∂∂

=

.1

,

,

,12

,

,

2

2

2

2

2

2

2

2

2

12

21

yxuDH

xu

yuDM

yu

xuDM

xu

yuCS

ukkxu

yu

CN

ukkyu

xuCN

z

zzy

zzx

yx

zxy

y

zyx

x

( )

( )⎪⎪⎭

⎪⎪⎬

⎫

∇∂∂

=κ+κ∂∂

−=

∇∂∂

=κ+κ∂∂

−=

.

,

2

2

zyxy

zyxx

uy

Dy

DQ

ux

Dx

DQ

Page 106


( )

( )

( ) ( ) ( ) ,0212

,02

12

1

,02

12

1

2221

21

42

1221

122

2

2

22

21

2

2

2

2

2

=−⎥⎦

⎤⎢⎣

⎡+ν++∇+

∂

∂ν++

∂∂

ν+

=+∂∂

ν++⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂ν−

+∂∂

+∂∂

∂ν+

=+∂∂

ν++∂∂

∂ν++⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂ν−

+∂∂

CZukkkkh

yu

kkxukk

CY

yukku

xyyxu

CX

xukk

yxu

uyx

zyx

zy

x

zyx

Stress function ( ):, yxϕ=ϕ

.,,2

2

2

2

2

yxS

yN

xN yx ∂∂

ϕ∂−=

∂ϕ∂

=∂

ϕ∂=

Mixed differential equations of shallow shells:

⎪⎭

⎪⎬⎫

=∇−ϕ∇∇

=ϕ∇+∇∇

,0

,222

222

zk

kz

uEh

ZuD

2.5. Shells of revolution

( ),1 α= RA

α, β = meridian and parallel.r(α) – meridian equation.

α= sin2RB

Case of Axis-Symmetrical Shell: 0=Y

0,0 =κ=ε==== αβαβββ uHQS

0=β∂

∂k

k L


( )

( )

( ) ⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=α+α+αα

−

=α−αα

+α+α

=α+α−α−αα

αβα

αβα

αβα

.0sincossin

,0sinsincossin

,0sinsincossin

2112

21212

21212

QRRMRMRdd

ZRRQRddNRNR

XRRQRNRNRdd

Page 107

Strains:

( )

.cotg,11

,cotg1,1

2111

21

⎟⎠⎞

⎜⎝⎛ −

α=κ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

α=κ

+α=ε⎟⎠⎞

⎜⎝⎛ +

α=ε

αβαα

αβα

α

dzduu

RRdzduu

Rdd

R

uuR

uddu

R

zz

zz

E.Meissner’s unknowns:

αα =ψ⎟⎠⎞

⎜⎝⎛ +

α−=χ QR

Ru

dduz

21

,1

Case h=const

( ) ( ) ( ),1,1

111

αΦ+χ=ψν

+ψψ−=χν

−χR

EhR

LDR

L

where

( ) ( )LLL

L2

2

1

2

1

2

12

2

21

2 cotgcotg1Rd

dRR

RR

dd

Rdd

RRL α

−α⎥

⎦

⎤⎢⎣

⎡α+⎟⎟

⎠

⎞⎜⎜⎝

⎛α

+α

=

0,0 ===== βαβα QQHMM


( ) ( )

( ) ( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

=−+

=+α∂∂

+β∂

∂−

β∂∂

=+β∂∂

+α∂

∂−

α∂∂

βα

αβ

βα

.0

,01

,01

21

2

2

ZRN

RN

ABYSBB

ANAN

ABXSAA

BNBN

3. Zero Moment (Membrane) Theory of Shell:

Page 108

3.1. Shell of revolution

α, β = meridian and parallel.

( ) .sin, 21 α==α= RrBRA


( )

( )

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=−+

=α+αα∂∂

α+

β∂∂

=α+β∂

∂+α−α

α∂∂

βα

β

βα

.0

,0sinsinsin1

,0sincossin

2112

2122

22

1

21112

ZRRNRNR

YRRSRR

NR

XRRSRRNNR

Case of axis symmetrical problem 0,0 =β∂

∂= k

k

Y L

0=== β SQH

( )⎪⎭

⎪⎬⎫

=−+

=α+α−αα

βα

βα

.0

,0sincossin

2112

2112

ZRRNRNR

XRRRNNRdd

⎟⎟⎠

⎞⎜⎜⎝

⎛−= α

β1

2 RNZRN

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡αα−αα+

α= ∫

α

αα

1

sincossinsin1

2122

dXZRRCR

N

Solution:

Page 109

3.2. Cylindrical and Conical Shell

For cylindrical shells:

( ) ( )[ ] .,

;,1

2322

21

22

yzzyzyRR

zyBA

′′′−′′′′+′

=∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛β∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛β∂

∂==

( )( ) ⎪

⎭

⎪⎬

⎫

β=β=

α=

.,

,

zzyy

x

For conical shells:

( )[ ]( )

.sincos2sincos

sin,

;sin,1

22

2322

21

22

θθ′′−θθ′+θθθ′+θα

−=∞=

⎟⎟⎠

⎞⎜⎜⎝

⎛β∂θ∂

+θα==

RR

BA

( ).,cossin,sinsin,cos βθ=θβθα=βθα=θα= zyx

( )

( )

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

=−

=+α∂∂

+β∂

∂

=+β∂

∂+

α∂∂

−α∂∂

β

β

βα

.0

,01

,0

2

2

ZRN

BYSBB

N

BXSBNBN

;2 RZZRN ==β ( ) ( )∫α

α

α⎥⎦

⎤⎢⎣

⎡+

β∂∂

−β=0

2212

11 dYBRZBB

fB

S


Solutions:

( ) ( )

( )∫ ∫

∫∫α

α

α

α

α

α

α

αα

α⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

α⎥⎦

⎤⎢⎣

⎡+

β∂∂

β∂∂

β∂∂

+

+α⎟⎠⎞

⎜⎝⎛ −

α∂∂

+β

+α⎥⎦⎤

⎢⎣⎡ β

β∂∂

−=

0 0

00

22

21

11

11

ddYBRZBBB

dBXRZBBB

fdB

fB

N

Page 110

saklviTüal½y GnþrCati Program: Master of Civil Engineering

saRsþacarüTTYbnÞúk ³ bNÐit esOn sm,tþi Page 1 of 4

sMNYreRtomRblgbBa©b;karsikSa fñak;GnubNÐitsMNg;suIvil

Epñk ³ Advanced Structural Analysis

muxviC¢a ³ Theory of Elasticity

1> cUrerobrab;sm μtikmμ (hypothesis) énRTwsþIeGLasÞic RBmTaMgBnül;xøwmsarrbs;va. 2> dUcemþcEdlehAfa sMBaFkñúg (stress)? enAelImuxkat;mYy etIsMBaFkñúgbMEbkecjCab:un μanbgÁMú

(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 nigGtßRbeyaCn_énkarrgVilG½kS kñúgkarsikSasMBaFkñúg. etITItaMgfμ Irbs;

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 nigGtßRbeyaCn_énlkçx½NÐCab; (compatibility conditions) RBmTaMg

bgðajBIviFITajrklkçx½NÐTaMgenaH. 9> dUcemþcEdlehAfa bMErbMrYlragem (principal strains)? etIeKkMNt;bMErbMrYlragemenH

ya:gdUcemþc? 10> etIGVIeTACac,ab; Hook? etIc,ab;enHmanGtßRbeyaCn_GVIxøH 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).



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 manRbeyaCn_GVIxøH 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) nigRTwsþIK μanm:Um:g; (zero moment

theory). cUrbgðajBIPaBxusKñarvagRTwsþITaMgBIrenH nigGtßRbeyaCn_énkarEbgEckenH? 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 tamRTwsþIK μanm:Um:g;? 26> cUrerobrab;smIkarlMnwg énRTwsþIPñas Kμanm:Um:g;? etIlMhat;énRTwsþIKμanm:Um:g;enH CaRbePTsþaTic

kMNt; b¤sþaTicminkMNt;? mUlehtuGVI?



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.



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.