FINITE DEFORMATION ANALYSIS OF SHELLS:
A HYBRID FINITE ELEMENT METHOD BASED
ON ASSUMED STRESS-FUNCTION VECTOR
AND ROTATION TENSOR
*
**
N. Fukuchi and S. N. A t l u r i
Georgia I n s t i t u t e of Technology
A t l a n t a , Georgia
Abstract
A new s h e l l t h e o r y based on polar-decomposition
of s h e l l deformation i n t o r i g i d r o t a t i o n and pure
s t r e t c h i s presented. Mixed v a r i a t i o n a l p r i n c i p l e s
f o r s h e l l s , undergoing l a r g e mid-plane s t r e t c h e s and
l a r g e r o t a t i o n s , i n terms of a s t r e s s - f u n c t i o n vect o r and r o t a t i o n t e n s o r , a r e given. A h y b r i d f i n i t e
element method, wherein t h e unsymmetric nominal
( f i r s t Piola-Kirchoff) s t r e s s t e n s o r and r i g i d r o t a t i o n a r e t r e a t e d a s v a r i a b l e s , i s developed. Solut i o n s f o r s e v e r a l example c a s e s , t h e i r comparison
w i t h o t h e r a v a i l a b l e r e s u l t s , and a d i s c u s s i o n of
noted advantages of t h e p r e s e n t method a r e included.
Introduction
L i n e a r and n o n l i n e a r s h e l l t h e o r i e s have been
s u b j e c t s of f r u i t f u l s c i e n t i f i c preoccupation of
many a d i s t i n g u i s h e d mechanician, such a s Sanders
[ I ] , K o i t e r [ 2 ] , Budiansky [ 3 ] , Reissner [ 4 ] , Sirnmonds [ 5 ] , P i e t r a s z k i e w i c z [ 6 ] , and many o t h e r s . The
works of Sanders [ l ] and Budiansky [ 3 ] d e a l w i t h
n o n l i n e a r s h e l l t h e o r i e s wherein t h e f i e l d e q u a t i o n s
and boundary c o n d i t i o n s a r e w r i t t e n i n terms of mids u r f a c e displacement components. The r e s u l t i n g s e t s
of e q u a t i o n s a r e w e l l known t o b e q u i t e complicated.
,
Important c o n t r i b u t i o n s i n t h e d i r e c t i o n of
n o v e l ways of w r i t i n g s h e l l e q u a t i o n s , w i t h a view
toward t h e i r s i m p l i f i c a t i o n , have been made by Reiss n e r [7-101, Simmonds and Danielson [11,12], and
P i e t r a s z k i e w i c z [ 6 ] . R e i s s n e r showed t h a t t h e equat i o n s of n o n l i n e a r axisymmetric s h e l l s of r e v o l u t i o n
can b e w r i t t e n i n a much s i m p l e r form i n terms of
r o t a t i o n and s t r e s s f u n c t i o n than i n terms of d i s placements.
L a t e r , Simmonds and Danielson [ I 2 ] a t tempted t o o b t a i n s i m i l a r s i m p l i f i c a t i o n s f o r a r b i t r a r y s h e l l s , u s i n g a f i n i t e r o t a t i o n v e c t o r and a
s t r e s s - f u n c t i o n v e c t o r . The work i n [12] may now b e
recognized t o b e based on a polar-decomposition of
s h e l l midsurface deformation g r a d i e n t i n t o a r i g i d
r o t a t i o n followed by pure s t r e t c h . I n t h i s p r o c e s s ,
they [I21 d e f i n e a bending s t r a i n measure t h a t i s
dependent s o l e l y on r i g i d r o t a t i o n s . P i e t r a s z k i e wicz, on t h e o t h e r hand, w h i l e n o t n e c e s s a r i l y havi n g t h e same o b j e c t i v e s a s i n [12], p r e s e n t s a cornprehensive s t u d y of t h e f o r m u l a t i o n of b a s i c r e l a t i o n s of t h e n o n l i n e a r s h e l l t h e o r y i n t h e Lagrangen
d e s c r i p t i o n , t h e t h e o r y of f i n i t e r o t a t i o n s i n
s h e l l s , and o t h e r a s s o c i a t e d problems.
measures, some b e l i e v e d t o b e new, of s t r e s s r e s u l t a n t s and s t r e s s - c o u p l e s i n a f i n i t e l y deformed s h e l l were d e f i n e d n a t u r a l l y from t h e i r counterp a r t s i n 3-D continuum mechanics; ( i i ) t h e equat i o n s of f o r c e and moment b a l a n c e f o r a f i n i t e l y
deformed s h e l l were w r i t t e n down c o n c i s e l y i n terms
of t h z s e a l t e r n a t e s t r e s s measures. These equations exhibit t h e i r essential similarity t o t h e i r
3-D c o u n t e r p a r t s ; ( i i i ) mixed v a r i a t i o n a l p r i n c i p l e s
i n v o l v i n g t h e r i g i d r o t a t i o n t e n s o r and s t r e s s funct i o n v e c t o r were developed f o r a r b i t r a r y f i n i t e deformations ( a r b i t r a r y mid-plane s t r e t c h a s w e l l a s
a r b i t r a r y r i g i d r o t a t i o n ) of an a r b i t r a r y shaped
s h e l l . I n doing s o , b o t h t y p e s of polar-decomposit i o n , ( a ) pure midplane s t r e t c h followed by r i g i d
r o t a t i o n a s w e l l a s (b) r i g i d r o t a t i o n followed by
a pure midplane s t r e t c h , a r e considered. I n t h e
c a s e of (b) t h e p r e s e n t r e s u l t s were compared w i t h
t h o s e of [ 1 2 ] , and t h e d i f f e r e n c e s were c r i t i c a l l y
examined; ( i v ) two new bending s t r a i n measures, b o t h
of which depend s o l e l y on r i g i d r o t a t i o n s , and two
a l t e r n a t e s t r e t c h i n g s t r a i n measures were i n t r o duced; and (v) even though t h e developed theory i s
v a l i d f o r a r b i t r a r y midplane s t r e t c h e s , and a r b i t r a r y r o t a t i o n s , no ad hoc d e f i n i t i o n s of "modified
s t r e s s - r e s u l t a n t s " and "modified bending s t r a i n s ' '
a r e employed, a s appears t o b e t h e c a s e i n t h e
c e l e b r a t e d works of K o i t e r [ 2 ] , Sanders [ I ] , and
Budiansky [ 3 I
.
The main o b j e c t i v e of t h e p r e s e n t paper i s t o
p r e s e n t a numerical development based on a mixed
v a r i a t i o n a l p r i n c i p l e i n [13]. The numerical development i s based on a hybridlmixed f i n i t e element
method wherein t h e s t r e s s - f u n c t i o n v e c t o r and r i g i d r o t a t i o n t e n s o r a r e approximated.
The c o n t e n t s of t h e p r e s e n t paper, i n t h e o r d e r
(i) a s y n o p s i s of t h e
of t h e i r appearance, a r e :
new s h e l l t h e o r y , ( i i ) an o u t l i n e of t h e h y b r i d /
mixed f i n i t e element development, and ( i i i ) numeric a l r e s u l t s f o r s e v e r a l t e s t problems and t h e i r
comparison w i t h e x i s t i n g s o l u t i o n s . The paper ends
w i t h a d i s c u s s i o n of t h e advantages of t h e p r e s e n t
method.
Nomenclature
ca
:
Recently one of t h e a u t h o r s undertook a compre( i ) several alternate
hensive s t u d y [13] wherein:
c3
:
c o o r d i n a t e along t h e normal t o midsurface
(-)
:
under a symbol denotes a v e c t o r
*
(-) :
V i s i t o r , C u r r e n t l y A s s o c i a t e P r o f e s s o r , Nagasaki
U n i v e r s i t y , Japan
**~ e g e n t ' sP r o f e s s o r
of Mechanics, Member AIAA
Copyrighl O American lnslitute of Aeronautics and
Astronautics. h e . , 1983. All rights reserved.
4
CY = 1 , 2 ; c u r v i l i n e a r c o o r d i n a t e s on t h e s h e l l
midsurface
under a symbol denotes a second-order
: A?;&
tation
a second-order
tensor
t e n s o r i n dyadic no-
where
undeformed midsurface of s h e l l
i s t h e c u r v a t u r e t e n s o r of
b a s e v e c t o r s on s
u n i t normals t o S and s r e s p e c t i v e l y
midsurface deformation g r a d i e n t
p o l a r decomposition of F
-0
Midplane s t r e t c h i n a Kirchhoff-Love t h e o r y
a
a n
-ar
Cauchy s t r e s s - r e s u l t a n t t e n s o r f o r a
shell
(TLa)
x
and i s s y m e -
c3.
b a s e v e c t o r s on S
n
-
s1
A s noted e a r l i e r , we w i l l use convected coordinates
and
A f t e r deformation, l e t s1 b e mapped i n t o s , and l e t Po and P b e mapped t o po and p,
r e s p e c t i v e l y . To d e f i n e p, we s h a l l invoke t h e
well-known Kirchhoff-Love hypotheses, v i z . , ( i ) t h e
m a t e r i a l f i b e r s o r i g i n a l l y normal t o s1 a r e mapped
i n t o f i b e r s normal t o s , and ( i i ) t h e r e i s no t h i c k n e s s s t r e t c h . I f t h e p o s i t i o n v e c t o r s of po and p
a r e _ro and 5, r e s p e c t i v e l y , we have:
deformed midsurface of s h e l l
E(.vo
5
Thus,
tric.
gradient operator
6
= AaB-Aa A
- = A-6-A 6
I.
Cauchy s t r e s s - c o u p l e t e n s o r f o r a
shell
A ( t-na) F i r s t Piola-Kirchhoff
t a n t tensor
-2
k(r2a)
A ( ra)
-a t-
s t r e s s resul-
Biot-Luri? s t r e s s - r e s u l t a n t
F i r s t Piola-Kirchhoff
tensor
stress-couple
tensor
% (r,fa)
-
where _u i s t h e displacement Popo, and 2 i s a u n i t
normal t o s. The b a s e v e c t o r s a t po and p a r e respectively,
where 1, i s t h e symmetric c u r v a t u r e t e n s o r of s . and
Biot-LurL s t r e s s - c o u p l e t e n s o r
second fundamental form of t h e deformed s h e l l
midsurface s
bending s t r a i n measure, e q u a l t o
A d i f f e r e n t i a l v e c t o r a t Po i s denoted by
w h i l e t h a t a t P by 2, such t h a t
a,
b*.$l
stress-function vector
:
f u n c t i o n a l s defined i n t e x t
w h i l e i n t h e deformed c o n f i g u r a t i o n s t h e i r maps a r e
denoted by CIJo and CIJ, r e s p e c t i v e l y , where
A Synopsis of t h e New S h e l l Theory
We d e f i n e t h e r e f e r e n c e s u r f a c e of t h e undeformed s h e l l t o b e s1 on which a g e n e r i c p o i n t i s denoted a s Po. The s u r f a c e s1 i s defined by two conv e c t i n g c u r v i l i n e a r ~ o o r d i n a t e sca(a=1,2). The u n i t
and
i s a coordinate
normal t o s1 i s denoted by
along
An a r b i t r a r y p o i n t P i n t h e undeformed
s h e l l i s measured by
such t h a t
N,
N.
where &E
t o r s on
c3
R
i s t h e p o s i t i o n v e c t o r of Po. The b a s e veca r e d e f i n e d by:
s1
= a dca
+ ndc3
(12)
d~ = g dca
+ ndc3
-
(13)
-a
and
-a
Thus, i n t h e p r e s e n t converted c o o r d i n a t e system,
t h e deformation g r a d i e n t s Fo and F , which a r e def i n e d by:
and
9=
a r e given by:
Fo
=
faia
+ nN
where aa6 = eaB/&, where ea6 i s t h e permutation
t e n s o r and A = d e t h B= d e t ( & a $ ) .
The b a s e vect o r s a t P are:
-
We w i l l now c o n s i d e r t h e polar-decomposition of F
0
i n t o s t r e t c h and r o t a t i o n a s follows:
course, t h e same i s n o t t r u e of c o n t r a v a r i a n t o r
mixed components.
where Llo is t h e midplane s t r e t c h and R t h e r i g i d
I n t h e p r e s e n t convected c o o r d i n a t e sysrotation.
tem, b e a r i n g i n mind t h e Kirchhoff-Love hypotheses,
we d e f i n e :
and
$
7~
= a a*B
- B-
We now c o n s i d e r t h e f r a m e - i n d i f f e r e n t form
i n 3-D continuum mechanics, is c a l l e d
t h e Cauchy deformation t e n s o r and is r e l a t e d t o t h e
concept of a change i n t h e s q u a r e of a d i f f e r e n t i a l
l e n g t h element. For t h e Kirchhoff-Love s h e l l , usi n g Eqs. (25 and 29) we s e e t h a t :
F ~ . F , which,
+ p rJ
a*'
whereby:
=
;
= &1.~6
aa.a*-6 = & a B; and l i k e w i s e , a g =
-
,
such t h a t
&.sT
G
and
From Eqs. (16,
aa = 5.aw such t h a t _aB-_aa = gag
19, and 20) i t i s seen, under t h e p r e s e n t KirchhoffLove hypotheses, t h a t :
.
P r a c t i c a l ways of r e p r e s e n t i n g
Appendix I.
R
a r e given i n t h e
Now we d e f i n e a new bending s t r a i n measure,
such t h a t
<*,
Using (21) i n ( 9 ) , i t i s seen t h a t
%
= a
-a
+
(a2/aca)c3
I n terms of t h i s new bending s t r a i n measure, we may
r e w r i t e Eq. (30) a s :
E q u i v a l e n t l y , i n terms of t h e second fundamental
form b of s l , we have:
The v a l i d i t y of Eq. (32) can b e immediately v e r i f i e d by d i r e c t expansion and t h e u s e of Eq. (31).
Thus we reach t h e i n t e r e s t i n g conclusion t h a t t h e
quantity
Thus t h e deformation g r a d i e n t a t any p o i n t i n t h e
s h e l l can b e w r i t t e n a s :
may b e considered a s a measure of t h e three-dimens i o n a l s t r a i n s t a t e i n t h e s h e l l . We s h a l l consid e r Uo a s t h e midplane s t r e t c h i n g s t r a i n measure
t h e bending s t r a i n measure. Note t h a t go i s
and
symmetric, w h i l e , a s seen from Eq. ( 3 1 ) ,
is
unsyrmnetric i n g e n e r a l .
6"
<*
I n t h e undeformed c o n f i g u r a t i o n , we have:
A t t h i s p o i n t , i t i s worth n o t i n g some propert i e s of !, which h a s t h e form:
F u r t h e r , i n t h e undef ormed c o n f i g u r a t i o n ,
F u r t h e r , i t can b e e a s i l y v e r i f i e d , a s i n [ 1 3 ] , t h a t
*
i n v a r i a n t under a superposed r i g i d -
*
v0
and b a r e
body motion.
Thus, even though b and b* a r e e v i d e n t l y d i f f e r e n t
t e n s o r s , they have t h e i n t e r e s t i n g p r o p e r t y t h a t :
t h e c o v a r i a n t components of b i n t h e b a s i s system
ga a t po a r e numerically e q u a l t h e c o v a r i a n t components of 12" i n t h e b a s i s system 4, a t Po. of
From (33), ( 3 4 ) , and ( 3 5 ) , i t i s s e e n t h a t one
mzy p o s t u l a t e "semi-linear i s o t r o p i c e l a s t i c behavi o u r " [ s e e 1 3 f o r f u r t h e r d e t a i l s ] and an a t t e n d a n t
s t r a i n - e n e r g y d e n s i t y f u n c t i o n Wo p e r u n i t i n i t i a l
volume, i n terms of t h e " s t r a i n measure",
A s shown i n [13], f o r "plane-stress" t h e e x p r e s s i o n
f o r Wo f o r a s h e l l of s e m i l i n e a r i s o t r o p i c m a t e r i a l
can b e w r i t t e n as:
It i s s e e n t h a t rg i s symmetric, w h i l e -,*:
is unsymmetric. As d i s c u s s e d i n [13], i n analogy w i t h
3-D continuum problems,
w i l l be called here the
symmetrized Biot-Lure s t r e s s t e n s o r , and .*r i s t h e
unsymmetric Biot-LurB s t r e s s couple t e n s o r .
K i n e t i c s of a F i n i t e l y Deformed S h e l l
where K
=
v(l
-
v)/(l
-
2v)
.
(37)
We now c o n s i d e r t h e t h i c k n e s s of t h e s h e l l t o b e h
and t h a t t h e midsurface of t h e s h e l l i s t h e r e f e r ence s u r f a c e . We now d e f i n e Wo t o b e t h e s t r a i n energy d e n s i t y l u n i t a r e a of t h e undeformed midsurface. Thus, we d e f i n e :
I n t h e deformed s h e l l l e t us c o n s i d e r an e l e ment w i t h "lengths" (aldS i ) and ( _ a 2 d ~ 2i)n t h e r e f e r e n c e p l a n e s and "height" (&) i n t h e t h i c k n e s s
d i r e c t i o n . L e t t h e Cauchy s t r e s s i n t h e s h e l l b e
The t r a c t i o n on a s t r i p of a r e a spanned by d<B and
from t h e r e f e r e n c e p l a n e , i s
dc3, a t a h e i g h t
given by:
x.
c3
where g i s t h e determinent of g
aB
The d i f f e r e n t i a l f o r c e p e r u n i t of
.
cB i s :
Following K o i t e r 1141, we s e e t h a t
A s shown i n [ 1 3 ] , we d e f i n e a Cauchy s t r e s s - r e s u l t a n t t e n s o r Tft such t h a t
where H and K a r e t h e mean and Guassian c u r v a t u r e s
of t h e undeformed midsurface.
Following K o i t e r ' s
[14] argument f o r a c o n s i s t e n t f i r s t approximation
t o e v a l u a t e t h e i n t e g r a l i n (39), i t i s seen t h a t :
Likewise, we d e f i n e a Cauchy s t r e s s - c o u p l e
Tg such t h a t
tensor
where
such t h a t
I n t h e above, a i s t h e determinant of t h e s u r f a c e
m e t r i c aaB of s.
A s shown i n [131, we may now d e f i n e t h e s o - c a l l ed [13] f i r s t Piola-Kirchhoff s t r e s s r e s u l t a n t and
s t r e s s - c o u p l e t e n s o r s , tg and t r , r e s p e c t i v e l y , as:
From t h e g e n e r a l theory of conjugate s t r e s s and
s t r a i n measures p r e s e n t e d i n [13], i t i s s e e n t h a t ,
by d e f i n i t i o n ,
A s a l s o shown i n [ 1 3 ] , we may d e f i n e t h e s o - c a l l e d
[13] Biot-Lurd s t r e s s r e s u l t a n t and s t r e s s - c o u p l e
tensors, respectively, as:
-
c*
Thus t h e p h y s i c a l meaning of t h e t e n s o r
i s evid e n t : i t i s a t e n s o r whose c o v a r i a n t components i n
numerically e q u a l t o t h e
t h e mixed b a s i s
components bau, of t h e second fundamental form b of
t h e deformed midsurface i n t h e b a s i s
where
are
and
(v)
-
We may d e f i n e a symmetrized Biot-Lure s t r e s s tensor,
as
rn,
(at2)
,a
+ fi
T
Jli tg T
(vi)
=
=
-
0 ( l i n e a r momentum b a l a n c e )
*
( a
Symmetric
(64)
T
(644
The above can b e shown [13] t o b e e q u i v a l e n t t o t h e
equations:
Making u s e of ( 5 4 ) , (55) i n ( 5 2 ) , (53) and compari n g w i t h Eqs. ( 5 0 , 5 l ) , we have:
nu'
and
+ b:(,ra')
= ( no')
+ b:
(Trau)
(64~)
where drLa i s t h e C h r i s t o f f e l symbol of t h e deformed
midsurface, ( ) ; a i m p l i e s c o v a r i a n t d e r i v a t i v e w.r.t
P on t h e deformed midsurface. Thus (64a) o r equiv a l e n t l y (64b and c ) r e p r e s e n t t h e moment b a l a n c e
conditions.
and
-
Mixed V a r i a t i o n a l P r i n c i p l e s
(vii) y = g ;
(g.5)
=
(E)a t
C
(65)
uo
It was shown [13] t h a t t h e f i e l d e q u a t i o n s and
boundary c o n d i t i o n s f o r a f i n i t e l y deformed s h e l l
can b e derived a s t h e Euler-Lagrange c o n d i t i o n s
a r i s i n g o u t of t h e s t a t i o n a r i t y of t h e following
general functional:
Eqs. (60-66) r e p r e s e n t t h e complete s e t of f i e l d
equations f o r t h e s h e l l .
We now c o n s i d e r c e r t a i n s i m p l i f i c a t i o n s t o t h e
g e n e r a l v a r i a t i o n a l p r i n c i p l e s t a t e d i n Eq. (59).
F i r s t we n o t e t h a t t h e s t r a i n - e n e r g y
(41) can b e w r i t t e n a s :
W
of Eq.
where
6"
wherein JE i s a symmetric t e n s o r ;
i s an a r b i t r a r y unsgmmetric t e n s o r , g i s an o r t h o g o n a l t e n s o r ,
t g and
a r e unsymmetric t e n s o r s . For such admiss i b l e f i e l d s , when 6F1 = 0 f o r a d m i s s i b l e 6v0, 66*,
6 5 6t"
and 6tg, i t can b e shown [13] t h a t t h e
corresponding E u l e r Lagrange e q u a t i o n s a r e :
tz
(i)
an
9=
a uo
&( t-n-R-
+ RT.
- t-nT)
5
5
awe = - r-5 =
7
t
at!
-
( i i i ) (A
-a
+ U7a)&a E
-
(61)
r*E
Gob
=
~h~
12 (1-v2)
n
r(60)
-
(ii)
+ ,AnT)
and
wherein t h e s u b s c r i p t s ( s ) and (b) s t a n d f o r
" s t r e t c h i n g " and *'bending8*,r e s p e c t i v e l y , such t h a t
awoS =
a Aa =
-a-
E-tJo
(62)
avo
awob
n ; -=
r-
at;;*
-
.*E
Now we s t a t e a mixed v a r i a t i o n a l p r i n c i p l e i n v o l v i n g
only tg and R a s v a r i a b l e s a s f o l l o w s :
= b
Aa (_auF.$
a 0-
= b
Aaa*'
au- -
it follows from (77
-
80) t h a t
(82)
where
wherein t h e following a p r i o r i c o n s t r a i n t s and def i n i t i o n s apply:
r@
Xu
a r e C h r i s t o f f e l symbols of t h e undeformed
midsurf ace.
We now p r e s e n t a second s i m p l i f i e d v a r i a t i o n a l
p r i n c i p l e i n v o l v i n g tg; R; and tr a s v a r i a b l e s , a s
t h e c o n d i t i o n of s t a t i o n a r i t y of t h e f u n c t i o n a l :
Thus Eq. (73) i m p l i e s t h a t a c o n t a c t t r a n s f o r mation has been e s t a b l i s h e d t o e x p r e s s-t h e complementary energy d e n s i t y i n s t r e t c h i n g , Wcs,
as a
f u n c t i o n of t h e symmetized Biot-Lurb s t r e s s r e s u l t a n t t e n s o r , g. Eqs. (74,75) imply t h a t t h e f o r c e
b a l a n c e e q u a t r o n s a s w e l l a s t h e f o r c e boundary
conditions a r e s a t i s f i e d a p r i o r i .
Eq. (76) i m p l i e s
t h a t t h e s t r a i n energy d e n s i t y i n bending, Web, i s
expressed a s a f u n c t i o n of g, through s a t i s f y i n g t h e
c o m p a t i b i l i t y r e l a t i o n (63) between $* and R.
where t h e following a p r i o r i c o n s t r a i n t s and d e f i n i t i o n s apply:
It can b e e a s i l y v e r i f i e d t h a t t h e a p o s t e r i o r i
= 0 are:
c o n d i t i o n s t h a t f o l l o w from 6F2(6t"6R)
( i ) c o m p a t i b i l i t y of midplane s t r a i n s , Eq. (62) ;
( i i ) t h e a n g u l a r momentum b a l a n c e c o n d i t i o n , Eq.
(64) ; ( i i i ) t h e moment boundary c o n d i t i o n (66b) ;
and ( i v ) t h e deformation b - c , Eq. (65).
.
ax
ax
where F i s a s t r e s s - f u n c t i o n v e c t o r , E
= e /JZ;
e l 2 = z-21 = 1, e l 1 = e22 = 0; d
,
pa is a p a r t i c u l a r s o l u t i o n given by:
We can e x p r e s s F and pa i n t h e form of components
along t h e b a s i s v e c t o r s of t h e undeformed midsurface, a s
Thus, i f one e x p r e s s e s t$
i n component form a s :
-
v
n = N
a t t-
As shown i n [ 1 3 ] , t h e a p r i o r i s a t i s f a c t i o n of
f o r c e b a l a n c e , Eq. ( 7 4 ) , i s p o s s i b l e i f one s e t s
a t Coo
The a p o s t e r i o r i c o n d i t i o n s t h a t follow from
6F3 = 0 a r e : ( i ) c o m p a t i b i l i t y of midplane s t r e t c h i n g s t r a i n s , Eq. ( 6 2 ) ; ( i i ) t h e c o m p a t i b i l i t y cond i t i o n f o r c u r v a t u r e s t r a i n s , Eq. (63) ; ( i i i ) t h e
moment b a l a n c e c o n d i t i o n , Eq. ( 6 4 ) ; and ( i v ) t h e
moment and deformation b - c , Eqs. (66b, 6 5 ) , respectively.
I n t h e following, we p r e s e n t a f i n i t e element
a p p l i c a t i o n of t h e v a r i a t i o n a l p r i n c i p l e of Eq.(71).
E a r l i e r , some p r e l i m i n a r y r e s u l t s of a p p l i c a t i o n of
Eq. (84) were p r e s e n t e d [15].
Hybrid F i n i t e Element Method
R e c a l l t h a t i n t h e a p p l i c a t i o n of Eq. (71), t h e
a p r i o r i c o n s t r a i n t s and d e f i n i t i o n s (72-76) apply.
I n t h e f i n i t e element approach, one s t a r t s by assuming tg such t h a t Eq. (74) is s a t i s f i e d a p r i o r i .
This i s done w i t h t h e a i d of t h e s t r e s s - f u n c t i o n
I n t h e p r e s e n t apv e c t o r F a s i n Eqs. (82,83).
proach, a l l t h e 3 components of F a r e assumed t o b e
cubic polynomials i n
c2. However, t h e assumed
do n o t s a t i s f y t h e i n t e r e l e m e n t t r a c t i o n r e c i p r o c i t y c o n d i t i o n , namely,
tn
cl,
-.
[17,18], which a r e s t r i c t l y assumed displacement
methods.
where pm i s t h e i n t e r e l e m e n t boundary, and v i s a
w
u n i t normal t o pm, and (+) and (-) denote a r b i t r a i l y t h e two s i d e s of pm. Condition (91) i s i n t r o duced a s an a u x i l i a r y c o n s t r a i n t through a Lagrange
multiplier
( a t t h e i n t e r e l e m e n t boundary) i n t o
t h e f u n c t i o n a l F2 of Eq. (71). Thus, f o r a f i n i t e
element assembly, t h e a s s o c i a t e d f u n c t i o n a l becomes :
I n Fig. 2, t h e p r e s e n t l y computed v a r i a t i o n s of
t a n g e n t i a l displacements, a s a f u n c t i o n of l o a d , a r e
given along w i t h comparison r e s u l t s of [18].
An E l l i p t i c a l P a r a b o l o i d a l S h e l l
An e l l i p t i c a l s h e l l w i t h c u r v a t u r e s (118) and
(114) i s analyzed, and t h e s o l u t i o n s a r e p r e s e n t e d
i n Figs. 3-6.
I n Fig. 3, t h e r e l a t i o n between t h e
c e n t r a l d e f l e c t i o n (w/a) and t h e t r a n s v e r s e l o a d
(PIE) i s shown along w i t h t h e comparison s o l u t i o n
of [18]. The v a r i a t i o n of t a n g e n t i a l displacements
of an e l l i p t i c a l p a r a b o l o i d a l s h e l l w i t h t h e a p p l i e d
l o a d i s shown i n Fig. 4. I n F i g s . 5 and 6, t h e cont o u r curves of r o t a t i o n a n g l e s and t h e d i r e c t i o n of
r o t a t i o n a x i s a r e given.
S h e l l s w i t h D i f f e r e n t Curvatures
- I ,uomG - [ ~ ~ . B +t -r-(g-iii-g] i i i d c
Now, t h e T+i,
which i s p h y s i c a l l y t h e i n t e r e l e m e n t
boundary displacement f i e l d , must b e continuous a
p r i o r i a t pm. I n t h e p r e s e n t approach, t h e inp l a n e displacements u l , u2, a s w e l l as t h e t r a n s v e r s e displacement w a r e assumed i n t h e form of &
noded i s o p a r a m e t r i c q u a d r a t i c approximation.
Now, s i n c e W0b depends on t h e f i r s t d e r i v a t i v e
of g, it i s s e e n t h a t g should b e C0 continuous,
i. e. continuous a t i n t e r e l e m e n t boundary. This
w i l l a s s u r e t h a t t h e i n t e r e l e m e n t moment r e c i p r o c i t y w i l l b e preserved. I n t h e p r e s e n t work, t h e
t e n s o r g i s expressed i n terms of a f i n i t e r o t a t i o n
v e c t o r 2 a s i n d i c a t e d i n t h e Appendix. The r o t a t i o n a n g l e w and t h e d i r e c t i o n of t h e a x i s of r o t a t i o n a r e assumed i n an &node i s o p a r a m e t r i c f a s h i o n
i n t h e p r e s e n t work.
The s o l u t i o n procedure i s t h e well-documented
[16] i n c r e m e n t a l procedure and w i l l n o t b e r e p e a t e d
h e r e due t o space reasons. A key s t e p t o b e mentioned is t h a t s i n c e t
o
i
sn o t s u b j e c t t o nodal
c o n n e c t i v i t y , t h e parameters i n
a r e eliminated
a t t h e element l e v e l and expressed i n terms of noda l displacements and r o t a t i o n s . The p r e s e n t h y b r i d
method thus l e a d s t o a " s t i f f n e s s matrix" approach
[161.
tz
Four d i f f e r e n t types of s h e l l s w i t h d i f f e r e n t
geometries ( c u r v a t u r e s ) a r e analyzed, and t h e r e l a t i o n s between t h e maximum d e f l e c t i o n (wmax/a) and
t h e l a t e r a l p r e s s u r e (PIE) a r e shown i n Fig. 7. I n
Figs. 8 and 9 , l a t e r a l and t a n g e n t i a l displacements
a r e shown, r e s p e c t i v e l y , a s t h e l o a d i n c r e a s e s .
Closure
A new s h e l l theory based on a p o l a r decomposit i o n of t h e s h e l l deformation i n t o r i g i d r o t a t i o n
and pure s t r e t c h i s d e t a i l e d . Preliminary r e s u l t s
based on t h i s t h e o r y a r e presented. While t h e s e a r e
found t o b e encouraging, much remains t o b e done t o
b r i n g t h e f e a t u r e s of t h e s h e l l theory i n t o a more
p r a c t i c a l analysis tool.
Acknowledgements
T h i s work was supported by AFOSR under a g r a n t
t o Georgia Tech. The a u t h o r s g r a t e f u l l y acknowledge
t h i s s u p p o r t a s w e l l a s t h e encouragement of D r . A.
Amos. It i s a p l e a s u r e t o acknowledge t h e a s s i s t a n c e of M s . J. Webb i n t h e p r e p a r a t i o n of t h i s
manuscript
.
Appendix I
A s noted i n 1121, it i s convenient t o e x p r e s s
t h e r i g i d body r o t a t i o n by a s i n g l e r o t a t i o n of
magnitude o about an a x i s p a r a l l e l t o some u n i t
v e c t o r g on t h e r e f e r e n c e s u r f a c e of t h e s h e l l .
We r e p r e s e n t t h e f i n i t e r o t a t i o n v e c t o r as:
Some Numerical R e s u l t s
I n a l l t h e 3 examples considered below, t h e
following assumptions a r e made:
( i ) a l l t h e edges
of t h e s h e l l a r e clamped, ( i i ) t h e l o a d i n g i s t h a t
of uniform t r a n s v e r s e p r e s s u r e , and ( i i i ) geometric a l and m a t e r i a l d a t a a r e shown i n Figs. 1, 3, and
7 r e s p e c t i v e l y . The r e s u l t s a r e b r i e f l y d i s c u s s e d
below.
The a c t i o n of a f i n i t e r o t a t i o n v e c t o r 3 on a vect o r A can b e expressed a s t h e t r a n s f o r m a t i o n of A
t o A* such t h a t :
Thus, t h e r o t a t i o n t e n s o r _R can b e w r i t t e n a s :
Circular Cylindrical Shell
The computed r e l a t i o n between t h e c e n t r a l def l e c t i o n (w/a) and t r a n s v e r s e l o a d (PIE) i s shown
i n Fig. 1. The p r e s e n t r e s u l t i s compared t o t h a t
of Brebbia and Connor [17] and a s o l u t i o n by t h e
f i n i t e s t r i p method [18]. It i s n o t e d t h a t t h e
p r e s e n t s o l u t i o n i s more f l e x i b l e than t h o s e i n
We may now e x p r e s s 2 a s :
such t h a t
16.
A t l u r i , S. N. and Murakawa, H., i n F i n i t e Elements i n Nonlinear Mechanics, Vol. 1 (Eds. P.G.
Bergan, e t a l . ) pp. 1-43, 1977.
Using (A. 1 ) and (A. 4) i n (A. 3 ) , g can b e expressed
a s a f u n c t i o n of 66 alone. Thus, 5 involves only
t h e geometric v a r i a b l e s of t h e undeformed r e f e r ence s u r f a c e .
17.
Brebbia, C. and Connor, J . , J o u r n a l of ASCE
Engineering Mechanics D i v i s i o n , A p r i l 1969.
18.
Fukuchi, N . , T r a n s a c t i o n s of W. Japan S o c i e t y
of Naval A r c h i t e c t u r e , No. 50, 1975.
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S i m o n d s , J. G., i n Trends i n S o l i d Mechanics
(Eds. J. F. B e s s e l i n g and A. M. A. Van d e r
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0
04
Centrol
R e i s s n e r , E., J o u r n a l of Applied Mechanics, Vol.
36, pp. 267-270, 1969.
R e i s s n e r , E., S t u d i e s i n Applied Mathematics,
Vol. 48, pp. 171-175, 1969.
0.2
08
0.6
Deflection
(x idt)
(We )
Fig. 1 A c y l i n d r i c a l s h e l l s u b j e c t e d t o uniform
pressure
R e i s s n e r , E . , Proceeding of Symposia on Applied
Mathematics, Vol. 3, pp. 27-52, 1950.
R e i s s n e r , E., P r o g r e s s i n Applied Mechanics,
Prager Anniversary Volume, McMillan, pp. 171178, 1963.
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Konink Nederlandse Ada van Wetenschappen, Seri e s B , Vol 73, pp. 460-478, 1970.
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19 72.
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S t r a i n Measures, and Mixed V a r i a t i o n a l Formulat i o n s I n v o l v i n g Rigid R o t a t i o n s , f o r Computat i o n a l Analyses of F i n i t e l y Deformed S o l i d s ,
w i t h A p p l i c a t i o n t o P l a t e s and S h e l l s - P a r t I:
Theory", Report GIT-CACM-SNA-81-34,
October
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K o i t e r , W. T . , Proceedings of IUTAM Symposium
on t h e Theory of Thin E l a s t i c S h e l l s , NorthHolland P u b l i s h e r s , pp. 12-33, 1960.
Fukuchi, N. and A t l u r i , S. N . , i n Nonlinear
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ASME, pp. 233-248, 1981.
( a ) The sddion by present amlysh
( b ) The sosolutiar bj
Finite strip methad
Fig. 2
T a n g e n t i a l displacements of clamped cyl i n d r i c a l s h e l l s u b j e c t e d t o uniform
pressure
I
-
Resent analysis
F~nite strip method
a = 1.0
Fig. 5
0
o1
0.2
0.3
Central deflection
Fig.
3
04
The c o n t o u r c u r v e s o f r o t a t i o n a n g l e s on
an e l l i p t i c a l paraboloidal s h e l l
0.5 (x,0-2
(w/a)
An e l l i p t i c a l p a r a b o l o i d a l s h e l l s u b j e c t e d t o uniform p r e s s u r e
0
sco l e
w
scale
=05x102
Fig. 6
u = Iolloz
The d i r e c t i o n s o f r o t a t i o n a x i s on a n
e l l i p t i c a l paraboloidal s h e l l
v
~0.3
d
Elliptical
paraboloidal
Cylindrical
0 0
Fig. 4
T a n g e n t i a l d i s p l a c e m e n t o f clamped e l l i p t i c a l paraboloidal s h e l l
0.1
02
0.3
04
Mox~mum deflection ( w / a )
Fig. 7
shell
05
(x 10')
Shells with various curvature subiected
t o uniform p r e s s u r e
(XI
8)
F1
6-01
e
-
-
0
-0.2
0.2x1d6
04x16~
-
(Y
C
5
8 -03-
2
e
Ellipticol poroboloidal
?
-04
-
shell
Ellipticol porabdoidol
shell
J
Fig. 8 Lateral Displacements of clamped shells
Elliptical
paroboloidol shell
Fig. 9
Cylindrical shell
poraboloidol shell
Tangential displacements of clamped shells