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REFERENCES
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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).