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Non-linear integro-differential equations used in orthotropic shallow spherical shell analysis
Copyright (c) 1991
093~6413191 $3.00 + .oo
Pergamon Press plc
'ON-LINEM INTEGRO-DIFFERENTIAL EQUATIONS USED IN ORTHOTROPIC HALLOW SPHERICAL SIiELLANALYSIS
;. G. Ladopoulos jepartment of Applied Mathematics :ational Technical University of Athens, Athens GR-157 73, Greet deceived 24 May 1990; accepted for print 71 December 1990)
.- Introduction Shallow spherical shells made of modern materials with nisotropic, or at least orthotropic character, have been of inreasing importance during the last years, because of their aplication in offshore, underwater and buried structures, like allistic missile bulkheads, submarine hulls, storage tanks and pace structures. Hence, several mathematical models have been proposed by esearchers in order to analyze the spherical shell structures, specially made by isotropic solids, considering the interaction ith the surroundings. Among them we shall mention the following udiansky and Roth [l] , Roth and Klosner [2] , Huang 131 , tephens [4] , Ganapathi and Varadan [5] . In the present communication the non-linear static delection analysis of an orthotropic shallow spherical shell esting on a linkler - Pasternak elastic foundation [6] , c73 , 111
12
E.G. LADOPOULOS
s presented. Thus, the problem is reduced to the solution of non-linear integro-differential equation, while several other ypes of non-linear integral equations used in many other applie echanics problems, have been proposed in the past by Ladopoulos nd Zisis ~8] and Ladopoulos
E9] • Therefore,
on-linear integro-differential
the above mentione
equation is numerically evaluate
y using a Chebyshev polynomials technique.
.- Oovernin~ equations ,for orthotropic spherical s h e l l s
analysi~
Consider an orthotropic spherical shell resting on a inkler - Pasternak elastic foundation o~ndaoion medium is homogeneous and
[6~ , [7] • Moreover,
the
isotropic, while there
xists a frictionless bonding between the shell and the foundaion (Fi~. I).
g(T,t)
q ]H
0 Figure
I
An orthotropic spherical shell on a Wi~kler.~Pasternak elastic foun@ati on.
SHALLOWORTHOTROPICSHELLS
Then, -
Pasternak
the foundation
foundation
interface
is equal
t @2v p = Kv - G ~ % r - ~ Beyond ~trains,
the above,
~hell vith constant
s8
=
radius
u, w
u
+
(2.1)
1 r
)
and radial
Sr
spherica]
as:
(2.2)
v R
surface
a8
of a shallow
are defined
R 1 + ~
are corresponding
Moreover,
~v ~
to:
v
r
Qents at the middle
p , for Winkle3
deformation R
~u ~r = - ~ where
pressurs
the circumferential
for the axisymmetric
113
2
~v (--~)
(2.3)
the in-plane
and normal
of the described
for a cylindrically
shell
orthotropic
displace°
(Fi~.
I).
solid we have:
(2.4)
L 'd vhere
aij,
(i,j = 1,2)
are the elastic
On the other hand, ~rthotropic
parameter
~
d3
12all a22
v
=
the flexural
and Poisson's
D =
=
consider
constants.
ratio (2.5)
( ~ - v 2) (2.6)
al I a12 all
(2.7)
v
rigidity :
D ,
14
E.G. LADOPOULOS
The equations ~irections
Nr
for the in-plane
and norma:
are as following:
~N~ + (Nr - Nc~) = 0 ~r
r
E
of equilibrium
~v
r
(-.~
+--H- ) +
~(rMr)- M~
+
(2.8) ~
~r
0
I $2v 6dr ~ 7
+ c
6v ~t ) dr
are the in-plane
forces,
r(q - p) dr = (2.9)
r = 0 ~here moments
N r , N@ and
6,
c
constants
(Fig.
~r
' Me
the bending
2).
P Figure Stress-resultants
2
and moments
for the spherical
I
shell
of Fi~.
Furthermore, using
the in-plane
I.
let us define forces
Nr = --5--r
Nr
'
the stress-function
and
Ne = ~
N~
:
(2.10)
g
, b3
SHALLOWORTHOTROPICSHELLS
Beyond the above, for orthotropic :oments are equal to: $2v M r = -D ( ~
v r
+
-D ( IBr $r~)V +
M8 =
115
solids the bending
Sv %r )
(2.11)
~2 V V T~-- )
(2.12)
~here the flexural rigidity D , orthotropic parameter ~ and ?oisson's ratio v are corresponding given by (2.5), (2.6) and [2.7). Thus, by using eqs (2.1) to (2.12) we obtain the follow .ng equations in terms of the stresE function g and normal teforn~tlon v : ( 02g
1, r
+
%r 2
Og
- - .~g -
Or
r
g)
d
+
[i
( Ov
all ~
2 )
r +
--=
R
Or
(
~rJ
(2.13) 1
$3v
co ' ; 7
~2v
* - -r
~v
%7r
+
~r ) - g (
r
f
$v Dr
r
) +
R
r
r
(Kv
%r ~%2v G ,gv) d r r ~
G----~--
-
f
=
~
0
r
E
-
8d ( ~ ~t L
+
C
~v
7 ~t..I
d;
0
(2.14)
We further introduce
the following non-dimensional
;ities : r =
, '"'
~
V
v =
b
,
~
~
g
all
b
b2 Jll
Rd
K*=
( 13- v 2)
=
db
Kb4
,
D
Gb2
* ~
C
~-
D
quan
16
E.G.LADOPOULOS 1/2
p = qb3 D
Ed~qb4 , p * =---'-~
,
C* = 16db41 D
~~]1/2 C , ~ =
D
- t (2.15)
Hence, by using (2.15), then eqns (2.13) and (2.14) are equal to the following non-linear differential and integrodifferential
g2
equations:
+ g ,~.~_ _
*
)g
2
~v* 2
..g.~-)
(2d)
=
--b ~(
+r.
0
(2.16) *
~2 63v*
;-Dj
~2v*
+
7
•-
~v*
-
12(~-)
~
)
(g
* ~v
) + C ~ g]
+
1
+r
,(K*
v - c-
~-'j~z-J
- a
0 =~
0
~
~.- Numerical
- ( .~ ~2 +
evaluation
,~.~
(2.17)
d,~
by usin~ Cheb~shev
pol~rnomial s
technique. In order to evaluate
numerically
the non-linear
differ-
ential and integro-differential eqs (2.16) and (2.17) we shall use a general function F! i) representing either the deflec, J . tion v , stress function g , load p or their i derivatiw with respect to ~ at step j . This function can be approximated by Chebyshev polynomials in the non-dimensional radius
~ :
SHALLOW ORTHOTROPICSHELLS
117
N - i Fj (i) -- >
Fj (i),r r
~ere
(i)
(~ )
Tr (~)
,
~e [ 0 , 1 ]
(3.1)
= 0
denotes
the order of differentiation
and
( , )
~notes that the first term of the series is to be halved. ~.~oreover, the integral
terms in (2.17) are approximated
following: N-i+ I .,
v *(i) d~ = >
(3.2)
Vr* T r ( ~ ) r=O
0 th :
*(i) . VN_i+ I
VN-i =
4(N-i+1) (3.3)
*(i) * VN_ i
.
1
Vr
=
V0
= 2
4r
VN_i_ I =
4(N-i)
(.*(i) Vr-1
-
v.~ij)t % r+1
, r
=
1,2, .... ,(N-i-l)
~d: - .... + (-I) (N-i)
-V2+
Then, the non-linear differential ~s (2.16) and (2.17) may be approximated
-j
VN_i e
and integro-differenti~ as following:
2N-4 .
.
L (v r ' Vr+1
.
' .... ' gr
~=O [th
, gr+_1
, .... , pr ) T r
(~)
=
0
43.4) (2N + 2)
unknowns.
1 18
E.G. LADOPOULOS
4.- A p p l i c a t i o n deflection
to the d e t e r m i n a t i o n
As an a p p l i c a t i o n determine
of the s h a l l o w
spherical
the s p h e r i c a l
shell
cases
i) Clamped
,
~v*
The following
~2v*
of
the
s~atic
= 0
,
+ v
~v
static
by u s i n g
aforementioned
K the
-
behaviour
in S e c t i o n the
_v g
conditions
2. Hence,
following
two
= 0
(4.1)
(S.E.C.) T @ =
. g. u . . . . . shell
50
, G
central
spherical
and
*
= 0
=
we shall
(C.E.C.)r ~ = I :
u = ~g
*
method,
deflection
conditions:
spherical
3 shows
p
described
edge
parameters:
load
shell
conditions
shallow
.e._ceU ~ , Figure
central
is studied
ii I Simol,y-supported = 0
static
of b o u n d a r y
edge
v = O,
v
of the p r o p o s e d
the n o n - l i n e a r
different
of the n o n - l i n e a r
anal~sis.
:
0
(4.2)
1~nder c o n s i d e r a t i o n =
100
,
~=
deflection shell
orthotropic
I
in
5
,
response
comparison
parameter
~
has v= V
I/3 (O)/d
with
.
1.6 --
C.E.G.
--- g.E.G.
¢
1.2
\
, "~:2D /
,.Y.,513--1.5
,,.f
0.8
o :>
O.4
0.0 0
10
2O
3O
40
5O
p'=cLbYEd* Figure St|t a t i c - l o at d - d e f l e c t i o n shallow
s o| h e r i c a l
shellI
response of Fi~.
of the o r t h o t r o o|i c 1.
the
the
.
SHALLOWORTHOTROPICSHELLS
119
Finally, from Fig. 3 it is shown that as the ortho'opy ( ~ ) of the shell decreases, then deflection decreases, ~o, while it becomes almost linear for ~ = 0.5 . When ~ = 0 m d the solid becomes iso~ropic), zed.
then deflections are mini-
- Conclusions The results presented gave an insight for the more reastic design of orthotropic shallow spherical shells considerir~ eir interaction with the surroundings. It has been shown that e cylindrical orthotropy and the foundation interaction
play
basic role in improving the load carrying capacity of shallow ,herical shells.
ferences B. Bu~iansky and R. S. Roth, Axisymmetric dynamic buckling of clamped shallow spherical shells, NASA TN-D-1510, 597 606 (1962). R. S. Roth ~ud J. H. Klosner, Non-linear response of cylindrically shells subjeete~ to dynamic axial loads, AL~A J, 2, 1788-1794 (1964). N. C. Huang, Axisy~metric dynamic snap through of elastic clamped spherical ~hells, AIAA J., 7, 215-220 (1969). W. B. Stephens, Computer program for static and dynamic axisymmetric non-linear response of symetrically loaded orthotropic shells of revolution, NASA TN-D-6158, (1970). ~. Ganapathi and T. K. Varadan, Dynamic buckling of orthotropic shmllow spherical shells, Comp. Struct., 15, 517-520 (!982). P. L. Pasternak, On a New Method of Analysis of an Elastic Foundation by means of two Foundation Constants, St. Publ. Cons Lit., Moscow (1954). A. D. Kerr, Elastic and Viscoelastic Foundation Models, ASM~ J. Appl. Mech., 31, 491-498 (1964). E. G. Ladopoulos and V. A. Zisis, Existence and uniqueness for non-linear eingular integral equations used in fluid mechanics, Int. J. Math. Math Scien., to be published (1990). E. G. Ladopoulos, Non-linear parabolic integral equations used in stationary and dynamic visccplasticity, Non-lin. Anal. Theor. Meth. Appl., (submitted i or publication). E. G. Ladopoulos, On the solution of the two-dimensional prob] of a plane crack or arbitrary shape in an anisotropic material, J. Engng Fract. Mech., 28, 187-195 (1987).
093~6413191 $3.00 + .oo
Pergamon Press plc
'ON-LINEM INTEGRO-DIFFERENTIAL EQUATIONS USED IN ORTHOTROPIC HALLOW SPHERICAL SIiELLANALYSIS
;. G. Ladopoulos jepartment of Applied Mathematics :ational Technical University of Athens, Athens GR-157 73, Greet deceived 24 May 1990; accepted for print 71 December 1990)
.- Introduction Shallow spherical shells made of modern materials with nisotropic, or at least orthotropic character, have been of inreasing importance during the last years, because of their aplication in offshore, underwater and buried structures, like allistic missile bulkheads, submarine hulls, storage tanks and pace structures. Hence, several mathematical models have been proposed by esearchers in order to analyze the spherical shell structures, specially made by isotropic solids, considering the interaction ith the surroundings. Among them we shall mention the following udiansky and Roth [l] , Roth and Klosner [2] , Huang 131 , tephens [4] , Ganapathi and Varadan [5] . In the present communication the non-linear static delection analysis of an orthotropic shallow spherical shell esting on a linkler - Pasternak elastic foundation [6] , c73 , 111
12
E.G. LADOPOULOS
s presented. Thus, the problem is reduced to the solution of non-linear integro-differential equation, while several other ypes of non-linear integral equations used in many other applie echanics problems, have been proposed in the past by Ladopoulos nd Zisis ~8] and Ladopoulos
E9] • Therefore,
on-linear integro-differential
the above mentione
equation is numerically evaluate
y using a Chebyshev polynomials technique.
.- Oovernin~ equations ,for orthotropic spherical s h e l l s
analysi~
Consider an orthotropic spherical shell resting on a inkler - Pasternak elastic foundation o~ndaoion medium is homogeneous and
[6~ , [7] • Moreover,
the
isotropic, while there
xists a frictionless bonding between the shell and the foundaion (Fi~. I).
g(T,t)
q ]H
0 Figure
I
An orthotropic spherical shell on a Wi~kler.~Pasternak elastic foun@ati on.
SHALLOWORTHOTROPICSHELLS
Then, -
Pasternak
the foundation
foundation
interface
is equal
t @2v p = Kv - G ~ % r - ~ Beyond ~trains,
the above,
~hell vith constant
s8
=
radius
u, w
u
+
(2.1)
1 r
)
and radial
Sr
spherica]
as:
(2.2)
v R
surface
a8
of a shallow
are defined
R 1 + ~
are corresponding
Moreover,
~v ~
to:
v
r
Qents at the middle
p , for Winkle3
deformation R
~u ~r = - ~ where
pressurs
the circumferential
for the axisymmetric
113
2
~v (--~)
(2.3)
the in-plane
and normal
of the described
for a cylindrically
shell
orthotropic
displace°
(Fi~.
I).
solid we have:
(2.4)
L 'd vhere
aij,
(i,j = 1,2)
are the elastic
On the other hand, ~rthotropic
parameter
~
d3
12all a22
v
=
the flexural
and Poisson's
D =
=
consider
constants.
ratio (2.5)
( ~ - v 2) (2.6)
al I a12 all
(2.7)
v
rigidity :
D ,
14
E.G. LADOPOULOS
The equations ~irections
Nr
for the in-plane
and norma:
are as following:
~N~ + (Nr - Nc~) = 0 ~r
r
E
of equilibrium
~v
r
(-.~
+--H- ) +
~(rMr)- M~
+
(2.8) ~
~r
0
I $2v 6dr ~ 7
+ c
6v ~t ) dr
are the in-plane
forces,
r(q - p) dr = (2.9)
r = 0 ~here moments
N r , N@ and
6,
c
constants
(Fig.
~r
' Me
the bending
2).
P Figure Stress-resultants
2
and moments
for the spherical
I
shell
of Fi~.
Furthermore, using
the in-plane
I.
let us define forces
Nr = --5--r
Nr
'
the stress-function
and
Ne = ~
N~
:
(2.10)
g
, b3
SHALLOWORTHOTROPICSHELLS
Beyond the above, for orthotropic :oments are equal to: $2v M r = -D ( ~
v r
+
-D ( IBr $r~)V +
M8 =
115
solids the bending
Sv %r )
(2.11)
~2 V V T~-- )
(2.12)
~here the flexural rigidity D , orthotropic parameter ~ and ?oisson's ratio v are corresponding given by (2.5), (2.6) and [2.7). Thus, by using eqs (2.1) to (2.12) we obtain the follow .ng equations in terms of the stresE function g and normal teforn~tlon v : ( 02g
1, r
+
%r 2
Og
- - .~g -
Or
r
g)
d
+
[i
( Ov
all ~
2 )
r +
--=
R
Or
(
~rJ
(2.13) 1
$3v
co ' ; 7
~2v
* - -r
~v
%7r
+
~r ) - g (
r
f
$v Dr
r
) +
R
r
r
(Kv
%r ~%2v G ,gv) d r r ~
G----~--
-
f
=
~
0
r
E
-
8d ( ~ ~t L
+
C
~v
7 ~t..I
d;
0
(2.14)
We further introduce
the following non-dimensional
;ities : r =
, '"'
~
V
v =
b
,
~
~
g
all
b
b2 Jll
Rd
K*=
( 13- v 2)
=
db
Kb4
,
D
Gb2
* ~
C
~-
D
quan
16
E.G.LADOPOULOS 1/2
p = qb3 D
Ed~qb4 , p * =---'-~
,
C* = 16db41 D
~~]1/2 C , ~ =
D
- t (2.15)
Hence, by using (2.15), then eqns (2.13) and (2.14) are equal to the following non-linear differential and integrodifferential
g2
equations:
+ g ,~.~_ _
*
)g
2
~v* 2
..g.~-)
(2d)
=
--b ~(
+r.
0
(2.16) *
~2 63v*
;-Dj
~2v*
+
7
•-
~v*
-
12(~-)
~
)
(g
* ~v
) + C ~ g]
+
1
+r
,(K*
v - c-
~-'j~z-J
- a
0 =~
0
~
~.- Numerical
- ( .~ ~2 +
evaluation
,~.~
(2.17)
d,~
by usin~ Cheb~shev
pol~rnomial s
technique. In order to evaluate
numerically
the non-linear
differ-
ential and integro-differential eqs (2.16) and (2.17) we shall use a general function F! i) representing either the deflec, J . tion v , stress function g , load p or their i derivatiw with respect to ~ at step j . This function can be approximated by Chebyshev polynomials in the non-dimensional radius
~ :
SHALLOW ORTHOTROPICSHELLS
117
N - i Fj (i) -- >
Fj (i),r r
~ere
(i)
(~ )
Tr (~)
,
~e [ 0 , 1 ]
(3.1)
= 0
denotes
the order of differentiation
and
( , )
~notes that the first term of the series is to be halved. ~.~oreover, the integral
terms in (2.17) are approximated
following: N-i+ I .,
v *(i) d~ = >
(3.2)
Vr* T r ( ~ ) r=O
0 th :
*(i) . VN_i+ I
VN-i =
4(N-i+1) (3.3)
*(i) * VN_ i
.
1
Vr
=
V0
= 2
4r
VN_i_ I =
4(N-i)
(.*(i) Vr-1
-
v.~ij)t % r+1
, r
=
1,2, .... ,(N-i-l)
~d: - .... + (-I) (N-i)
-V2+
Then, the non-linear differential ~s (2.16) and (2.17) may be approximated
-j
VN_i e
and integro-differenti~ as following:
2N-4 .
.
L (v r ' Vr+1
.
' .... ' gr
~=O [th
, gr+_1
, .... , pr ) T r
(~)
=
0
43.4) (2N + 2)
unknowns.
1 18
E.G. LADOPOULOS
4.- A p p l i c a t i o n deflection
to the d e t e r m i n a t i o n
As an a p p l i c a t i o n determine
of the s h a l l o w
spherical
the s p h e r i c a l
shell
cases
i) Clamped
,
~v*
The following
~2v*
of
the
s~atic
= 0
,
+ v
~v
static
by u s i n g
aforementioned
K the
-
behaviour
in S e c t i o n the
_v g
conditions
2. Hence,
following
two
= 0
(4.1)
(S.E.C.) T @ =
. g. u . . . . . shell
50
, G
central
spherical
and
*
= 0
=
we shall
(C.E.C.)r ~ = I :
u = ~g
*
method,
deflection
conditions:
spherical
3 shows
p
described
edge
parameters:
load
shell
conditions
shallow
.e._ceU ~ , Figure
central
is studied
ii I Simol,y-supported = 0
static
of b o u n d a r y
edge
v = O,
v
of the p r o p o s e d
the n o n - l i n e a r
different
of the n o n - l i n e a r
anal~sis.
:
0
(4.2)
1~nder c o n s i d e r a t i o n =
100
,
~=
deflection shell
orthotropic
I
in
5
,
response
comparison
parameter
~
has v= V
I/3 (O)/d
with
.
1.6 --
C.E.G.
--- g.E.G.
¢
1.2
\
, "~:2D /
,.Y.,513--1.5
,,.f
0.8
o :>
O.4
0.0 0
10
2O
3O
40
5O
p'=cLbYEd* Figure St|t a t i c - l o at d - d e f l e c t i o n shallow
s o| h e r i c a l
shellI
response of Fi~.
of the o r t h o t r o o|i c 1.
the
the
.
SHALLOWORTHOTROPICSHELLS
119
Finally, from Fig. 3 it is shown that as the ortho'opy ( ~ ) of the shell decreases, then deflection decreases, ~o, while it becomes almost linear for ~ = 0.5 . When ~ = 0 m d the solid becomes iso~ropic), zed.
then deflections are mini-
- Conclusions The results presented gave an insight for the more reastic design of orthotropic shallow spherical shells considerir~ eir interaction with the surroundings. It has been shown that e cylindrical orthotropy and the foundation interaction
play
basic role in improving the load carrying capacity of shallow ,herical shells.
ferences B. Bu~iansky and R. S. Roth, Axisymmetric dynamic buckling of clamped shallow spherical shells, NASA TN-D-1510, 597 606 (1962). R. S. Roth ~ud J. H. Klosner, Non-linear response of cylindrically shells subjeete~ to dynamic axial loads, AL~A J, 2, 1788-1794 (1964). N. C. Huang, Axisy~metric dynamic snap through of elastic clamped spherical ~hells, AIAA J., 7, 215-220 (1969). W. B. Stephens, Computer program for static and dynamic axisymmetric non-linear response of symetrically loaded orthotropic shells of revolution, NASA TN-D-6158, (1970). ~. Ganapathi and T. K. Varadan, Dynamic buckling of orthotropic shmllow spherical shells, Comp. Struct., 15, 517-520 (!982). P. L. Pasternak, On a New Method of Analysis of an Elastic Foundation by means of two Foundation Constants, St. Publ. Cons Lit., Moscow (1954). A. D. Kerr, Elastic and Viscoelastic Foundation Models, ASM~ J. Appl. Mech., 31, 491-498 (1964). E. G. Ladopoulos and V. A. Zisis, Existence and uniqueness for non-linear eingular integral equations used in fluid mechanics, Int. J. Math. Math Scien., to be published (1990). E. G. Ladopoulos, Non-linear parabolic integral equations used in stationary and dynamic visccplasticity, Non-lin. Anal. Theor. Meth. Appl., (submitted i or publication). E. G. Ladopoulos, On the solution of the two-dimensional prob] of a plane crack or arbitrary shape in an anisotropic material, J. Engng Fract. Mech., 28, 187-195 (1987).