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194

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64.

65.

66.

67.

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199

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78. Navaratna, D.R., Pian, T.H.H. and Witmer, E.A. "Analysis of Elastic Stability of Shells of Revolution by the Finite Element Method" Proceedings, AIAAjASMA 8th Structures, Structural Dynamics & Material Conf. Palm Spring, California, March, 1967.

79. Stricklin, J.A., Haisler, W.E., MacDougall, H.R. and Stebbins, F.J. "Non-Linear Analysis of Shells of Revolution by Matrix Displacement Method" AIAA J., Vol. 6, No. 12, pp. 2306-2312, Dec. 1968.

80. Stricklin, J.A. "Geometrically Non-Linear Static and Dynamic Analysis of Shells of Revolution" Proc. of the Symp. of Int. Union of Theoretical & Applied Mechanics Lieg, Aug. 1970, pp. 23-29.

81. Tottenham, H. and Barony, S.Y. "Mixed Finite Element Formulation for Geometrically Non­Linear Analysis of Shells of Revolution" Int. Journal Numerical Methods in Eng. Vol. 12, No.2 1978, pp. 195-201.

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200

85. Schaeffer, H.G. Ball, R.E. "Non-linear Deflections of Asymmetrically Loaded Shells of Revolution" Structure, Structural Dynamics and Material Conference 9th 1968, AIAA AS ME Technical Paper No. 68-292.

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87. Moy, S.S.J., and Niku, S.M., "The Equivalent load Method for the Analysis of Axisymmetric Shells with Imperfections" under preparation.

88. Gould, P.L., Han, K.J. and Tong, G.S. "Analysis of Hyperbolic Cooling Towers with Local Imperfections" Procedings of the 2nd Int. Symp. of Natural Draught Cooling Towers Sept. 5-7, 1984. Ruhr-Universitat, Bochum, Germany.

APPENDIX 1

.ELASTICITY MATRIX

The Elasticity Matrix [0] relates the stress

resultants to the strains, as shown in equations (3.5), in

which orthotropic material and different properties in

membrane and bending behaviour are permitted.

Cep Cepe 0 0 0 0 m m·

Ceep Ce 0 0 0 0 m m

0 0 C 0 0 0 m [0] (A1.1 )

0 0 0 Cep Cepe b 0

b

0 0 0 Ceepb Ce 0 b

0 0 0 0 0 Cb

where; Eep tm

Cep m

1-Vepe veep m m m

Cepe m

veep Cep m m

Ee tm Ce

m 1-Vepe veep m m m

Ceep m

Vepe Ce m m

Cm G t m

Cep Eepbtb3

b 12 (1-'VepebVeepb)

Cepe b VeepbCepb

Ee tb3

Ce b

b 12 (1-VepebVeepb)

202

C =Gt 3 /12 b b

Since [D] is a symmetric matrix for orthotropic

materials the following relations exist between the material

properties in different directions,

For an isotropic material, with the same properties

for membrane and bending action

E<jl = Ee E<jl = Ee E m m b b

ve<jl v<jle = Ve<jlb = v<jle v m m b

G Gb E/2 (1 +v) m

APPENDIX 2

5 0 0 0 0 0 0

0 0 s 0 0 0 0 f(5)

~X8 0 0 0 0 5 52 53

-K cp

-K 5 cp

0 a 0 1/L 25/L 35 2/L

(A2.1 )

5 (1-5) 5 2 (1-5) 0 0

0 0 S (1-5) 52 (1-5) g(5) (A2.2)

~x~ 0 0 0 0

-K 5(1-5) cp

-K 5 2 (1-5) cp

a 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

B -Kcp(1) 0 0 0

_(l 0 1/L 0 0 (A2.3)

8x 8 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

-Kcp(2) -Kcp(2) 0 0 0 1/L 2/L 3/L

o l/

L

o

Cos

cp/r

sCos

cp/r

j/

r

-j/r

-j

s/r

-C

oscp

/r

Cj

=1 _C

Y. 6

X8

0 -1

/ (LR

cp)

0

-cos

cp/ (

rRcp

) -s

Cos

cp/ (

rRcp

) -j

Sin

cp/r

2

j/rR

cp

js/r

Rcp

Si

n<pc

oscp

/r2

o l/R

cp

js/r

Si

ncp/

r

[l/L

j

0

-sC

oscp

/r

0 0

-jsS

incp

/r2

_j2

/ r2

[SSi

n<PC

OSC

P/r2

j jC

oscp

/r2

-Sin

cp/L

r

(A2.

4)

S/Rc

p

sSin

cp/r

0 0

[Cos

cp/rL

j

_j2

s/r2

S2/R

cp

S3/

Rcp

s2Si

ncp/

r s3

Sinc

p/r

0 0

2/L

2 6s

/L2

[2sco

scp/

rLj

[3s2

CO

SCP/

rLj

.2

2 2

.2

3 2

-] s

/r

-] s

/r

[jSCO

SCP/

r2 j

[js2C

OSC

P/r2

j [js3C~SCP/r2j

-2j/

Lr

-4js

/Lr

-6js

/L

r

I\)

0 .,..

(1-2

s/L

) (2

s-3

s2)/L

0

0

s(1

-s)C

ose

j>/r

S

2 (1

-s)

Co

sej>

/r

js(1

-s)/r

j

s2

(1

-s)/r

-js

(1

-s)/r

-j

s2

(1

-s)/r

f(

1-2

S)/

L

r (2

s-3

s2)/L

_S

(1-S

)co

se

j>1

l-

s2

(1

-s)C

ose

j>/r

cj

I r

-8

6x

8 I -(

1-2

s)/

Re

j>L

-(

2s-3

s2

)/R

ej>

L

0 0

..., 0 U1

-s(1

-s)C

ose

j>/r

Re

j>

-S 2

(1

-s)

CO

Sej

>/rR

ej>

js(1

-s)S

ine

j>/r

2 jS

2 (

1-s

)Sin

ej>

/r2

jS(1

-S)/

rRe

j>

jS2

(1

-s)/

rRe

j>

s(1

-s)S

ine

j>C

ose

j>

S2

(1-s

) S

inej

>Cos

ej>

r2

r2

(1-2

s)S

ine

j>

_ (2

s-3

s2

)S!n

ej>

L

r L

r

(A2

.5)

APPENDIX 3

A3.a: Shell under Self Weight

Since the physical properties have been assumed

to be symmetric about the axis of revolution, this is an

axisymmetric problem, and only zero harmonic is required.

Dead load can be reduced to two components

given by:

and T (A3. 1)

where;

Nand T; are the normal and meridional components

of dead load as in Fig. (A3.1).

Ws; force density of dead load per unit area of

the middle surface, obtained by

where;

(A3.2)

P is the mass density of the material of the shell

g is the acceleration due to gravity

t thickness of the shell

Therefore the force density vector of (3.31.b)

which in this case is independent of e, is given by;

Po T

Pv 0 P (s) (A3. 2)

Pw N

Ps 0 ¢

207

A3.b: Shell under Wind Load

Since the wind load acts normal to the shell's

surface (3.31.c) becomes

~ 0

pj(S) ~ 0 (A3.4)

~ pj

~ 0 ¢

where Pj is the jth Fourier component of the wind load.

APPENDIX 4

DERIVATION OF STIFFNESS MATRIX OF THE

SUPPORT COLUMNS AT INTERFACE

Consider a pair of columns, made of the two inclined

members AB and AC, fixed at Band C as shown in Fig. (A4.1)

A

Fig. (A4.1)

B C

B

Fig. (A4.2.a) Fig. (A4.2.b)

If an axial displacement 01 (shown in Fig. A4.2.a)

is applied at A of member AB then the corresponding total

strain energy becomes

(A4.1 )

where;

is the axial stiffness of member AB, equal

to

209

But 01 can be expressed in terms of meridional and

circumferential displacements (ub,vb ) at the interface

(see Fig. A4.2.a)

01 = ubSina-vbcosa

or in matrix form

(Sina - Cosa)

Vb

Substituting 01 from (A4.2) into (A4.1) gives

u b m ~AB

2x2

where;

(A4 . 2)

(M.3)

~~B is the membrane stiffness matrix of member

AB at interface defined in coordinate

(ub,vb ) and given by

m ~AB

2x2

Following the same calculation for member AC leads to

m ~Jl.C

2xT

EA Cos2a/L c c

(M.4)

(A4.5)

The total membrane stiffness at A can be obtained

by combining (A4.4) and (A4.5)

2EA Sin2a/L c c o

m m ~AB + ~AC (M.6)

o 2EA COS2a/L c c

210

B (A4.3.a) (A4.3.b)

Similarly the total flexural strain energy of

member AB, due to normal displacements Wand rotation of Sr

and tortion of St (see Fig. A4.3.a), at A can be expressed

by

where;

given by

f ~ (W Sr St)x~1 x Sr (A4.7)

St

Kf is the flexural stiffness at A of member AB, _1

(12EI /L 3 -6Elc /Lc 2 0

c c

Kf -6Elc /Lc 2 4E1c/Lc 0 (A4.8) _1

0 0 GJ /L c c

c

As Fig. (A4.3.a) shows, the rotational and tortional

displacements at A of member AB can be expressed in terms of

the meridional and circumferential rotations of S~ and Se

at interface

13~Sina + 13ecosa

-13~cosa + SeSina

expressing this in matrix form and including W

(A4 .9)

211

W

~ 0

c:s~ W

Sr Sina S¢

St -Cosa Sina Se

(M.10)

~AB

Substituting (A4.10) into (M.7) leads to;

W f UAB ~ (W ]" ¢ ]" e) ~~ ]"<jJ (M .11)

]"e

where; f

~AB is the flexural stiffness at pOint A of member

AB in local coordinate of the shell at interface, given by

,'EIdLe' -6El Sma/L 2 -6El Cosa/L 2 c c C c

4El Sin2a/L 4ElcSinaCosa/Lc c c I -6El Sina/L 2 +

Kf Tt Kf c c

:.:AB :AB _1 :As GJ Cos 2 a/L -GJ cSinaCosa/L c 3X3 3X3 3X3 3X3 C c

4ElcSinacosa/Lc 4El Cos 2a/L c c -6El Cosa/L 2 + C c

-GJcSinaCosa/Lc GJ Sin2a/L c c

(M .12)

For member AC, transformation ~AC becomes (see Fig. A4.3.b)

o Sina

Cosa

and by a similar process as before

(M .13)

f ~AC

3 x 3

12EI /L 3 c c

-6EI Sina/L 2 c c

-6EI Cosa/L 2 c c

212

-6EI Sina/L 2 c c

[-4EI SinaCosa/L c c

+ GJ cSinaCosa/L c

6EI Cosa/L 2 c c

[4EI SinaCosa/L c c

+ GJ cSinaCosa/L c

(A4.14)

The total flexural stiffness at interface A in local

coordinates of the shell is obtained by

f f f ~=~+~C

3X3 3X3 3X3

24EI~C3

o

-12EI SinajL 2 c c

o

o

o

(A4 .15)

Therefore the total stiffness at A can be derived by combining

the membrane and flexural stiffness of (A4.6) and (A4.15)

2EA

Si

n2o.

j~

0 C

C

0

0 0

0 -2

EA

C

Os

2 o.j

L

0 0

0 c

-c

0 0

24

EI

jL

3 -1

2E

I S

ino

.jL

2

0 c

C

C

c

~b

14

EI

Sin

2 o.j

L

c c

SX

S 0

0 -1

2E

I S

ino

.jL

2

2x

J 0

c c

GJ

CO

S2

o.j

'" -

C

C

Co)

4E1

co

s'.

/L 1

c

c 0

0 0

0 2

xl G

J S

in2 o

.jL

c

c

(A4

.16

)

214

Since the circumferential rotation has been

disregarded in the analysis, ~b from (A4.16) reduces to

a (4 x4) matrix.

2EA Sin2ajL 0 0 c c o

0 2EA Cos 2 a/L 0 c c o

0 0 24EI~c 3

K = _b -12EI Sina/L 2 c c

0 0 -12EI Sina/L 2 c c

4EI Sin2 a./L 2x c + c

GJ Cos 2a/L c c

(A4.17)

APPENDIX 5

DERIVATION OF EQUIVALENT NODAL FORCES

OF A DISTRIBUTED EDGE LOADING

The derivation of the equivalent nodal forces of

self equilibrated edge loading at the base is represented

here. The same approach is applicable to any other case of

distributed edge loading applied at the top or the base

of the shell.

As was obtained in eqn. (4.15), section (4.3.1),

the self equilibrated edge loading N defined in local _s coordinates can be expressed by

N (e) _s

Nhs L

i=O 4X4

Ni _s

4Xl

(A5. 1 )

Therefore corresponding displacements at the base

can be expressed by the similar Fourier Series

where;

Nhs - \' ej - j ~b(e) = j~O _ ~b

~b( e) is the displacement vector in local

(A5. 2)

coordinates of the nodal circle at the base, due to edge loading ~s(e).

The contribution of the potential energy of the

applied edge loading N (e) becomes -s

(A5. 3)

where;

Rb ; is the horizontal radius at the base.

Substituting ~b(e) and ~s(e) from (A5.1) and (A5.2)

into (A5.3) gives

Nhs Nhs f2TI Vp = ilo jlo 0 Rb ~~,T e j e i ~! de (A5.4)

216

Rb , ~~ and ~! are independent of e, and therefore from

(3.31.e)

(AS.S)

where;

(AS.S.l)

Ni is defined as the ith Fourier component of the _se equivalent nodal force, since it is equal to the first

derivative of the potential energy w.r.t. the nodal displacements.

becomes

where;

The equivalent nodal force Ni in global coordinate _se

~b (AS.6) 4X4

~b is the transformation matrix at the base

given by (4.6.a).