FINITE DEFORMATIONS, FINITE ROTATIONS, AND STABILITY OF PLATES:
A COMPLEMENTARY ENERGY - FINITE ELEMENT ANALYSIS
H. Murakawa* and S.N. A t l u r i * *
C e n t e r f o r t h e Development of Computational Mechanics
School of C i v i l E n g i n e e r i n g
Georgia I n s t i t u t e of Technology, A t l a n t a , G e o r g i a 30332
Abstract
A f i n i t e element method, b a s e d on complement a r y e n e r g y , f o r t h e a n a l y s i s of f i n i t e deformat i o n s , f i n i t e r o t a t i o n s , b u c k l i n g and p o s t - b u c k l i n g
b e h a v i o r of p l a t e s i s p r e s e n t e d . R e p r e s e n t a t i v e num e r i c a l r e s u l t s , based on t h i s p r o c e d u r e , a r e p r e s e n t e d and compared w i t h a v a i l a b l e s o l u t i o n s b a s e d
on o t h e r methods i n c l u d i n g t h o s e of c l a s s i c a l pot e n t i a l energy.
t i c m a t e r i a l s [2-71, and i n f i n i t e d e f o r m a t i o n
a n a l y s i s of e l a s t i c beams and r o d s [ a ] . A l s o , a s
d i s c u s s e d by Masur and P o p e l a r [ 9 ] , a n d K o i t e r [ l o ] ,
t h e complementary e n e r g y approach h o l d s c o n s i d e r a b l e
promise i n t h e a n a l y s i s of s t a b i l i t y problems.
I n t h i s p a p e r , we e x p l o r e t h e c o n c e p t s a s o u t l i n e d i n [2-71 a s t h e y may b e a p p l i e d i n t h e a n a l y s i s of f i n i t e d e f o r m a t i o n s and s t a b i l i t y of t h i n
p l a t e s w h e r e i n c e r t a i n p l a u s i b l e d e f o r m a t i o n hy=
p o t h e s e s o f t h e well-known "Kirchhoff'Love'' t y p e
a r e invoked. We c o n s i d e r t h e c l a s s of problems of
p l a t e s e h i c h a r e c h a r a c t e r i z e d by l a r g e r o t a t i o n s
b u t y o n l y moderate s t r e t c h i n g ; a s w e l l a s t h o s e
which may b e c h a r a c t e r i z e d by t h e well-known Von
Karman p l a t e t h e o r y .
The m a t e r i a l i s c o n s i d e r e d
t o b e i s o t o p i c and s e m i l i n e a r , i . e . , e x h i b i t i n g a
l i n e a r r e l a t i o n between t h e s t r e t c h ( o r e n g i n e e r i n g
s t r a i n ) t e n s o r and t h e Jauman s t r e s s ( o r e q u i v a l e n t l y ,
i n t h e c a s e of i s o t r o p y , t h e L u r e ' o r B i o t s t r e s s )
tensor.
Introduction
U n t i l r e c e n t l y i t was f e l t t h a t a c o n s i s t e n t
complementary e n e r g y p r i n c i p l e f o r f i n i t e d e f o r mation problems, a n a l o g o u s t o t h a t i n t h e c a s e of
i n f i n i t e s i m a l d e f o r m a t i o n s , c a n n o t be d e r i v e d . T h i s
was due t o t h e f a c t t h a t a l l s u c h f o r m u l a t i o n s were
b3sed on t h e u s e of a symmetric second Piola-Kirchh o f f (P-K) s t r e s s t e n s o r (which i n h e r e n t l y o b e y s 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 s ) , i n which c a s e
t h e complementary e n e r g y f u n c t i o n a l becomes a funct i o n o f b o t h t h e second P-K s t r e s s t e n s o r a s w e l l
a s t h e d i s p l a c e m e n t g r a d i e n t t e n s o r . Moreover, f o r
f i n i t e d e f o r m a t i o n s , t h e l i n e a r momentum b a l a n c e
c o n d i t i o n s i n t e r m s of t h e second P-K s t r e s s t e n s o r
a r e n o n l i n e a r , and i n v o l v e c o u p l i n g w i t h t h e def o r m a t i o n g r a d i e n t t e n s o r , and hence a r e n e a r l y i m p o s s i b l e t o be s a t i s f i e d a p r i o r i ; thus rendering
t h e a s s o c i a t e d complementary e n r g y p r i n c i p l e d i f f i c u l t for practical application.
I n t h e f o l l o w i n g we p r e s e n t , a s a s t a r t i n g
point, a general v a r i a t i o n a l principle f o r f i n i t e
e l a s t i c i t y , a n a l o g o u s t o t h e well-known Hu-Washizu
p r i n c i p l e of l i n e a r e l a s t i c i t y , which i n v o l v e s t h e
d i s p l a c e m e n t s , s t r e t c h e s , r o t a t i o n s , and t h e f i r s t
Piola-Kirchhoff s t r e s s e s a s v a r i a b l e s .
By i n c o r p o r a t i n g t h e a p p r o p r i a t e " p l a u s i b l e " assumptions
f o r t h e c a s e of a t h i n p l a t e , t h e above p r i n c i p l e
is specialized t o t h e case of f i n i t e deformations
of a p l a t e .
From t h i s g e n e r a l p r i n c i p l e , a n app r o p r i a t e complementary e n e r g y p r i n c i p l e and a n a s s o c i a t e d f i n i t e element method, a r e developed f o r
plate analysis.
I n t h e l a s t s e c t i o n of t h e p a p e r ,
we p r e s e n t d e t a i l e d r e s u l t s f o r b u c k l i n g (from a n
i r r o t a t i o n a l fundamental s t a t e ) , l a r g e d e f o r m a t i o n s ,
and p o s t - b u c k l i n g of i n i t i a l l y f l a t , s i m p l y s u p p o r t e d
a s w e l l a s clamped r e c t a n g u l a r p l a t e s . Comparisons
of t h e s e r e s u l t s w i t h e x i s t i n g o n e s a r e made, and
t h e p o s s i b l e a d v a n t a g e s o f t h e p r e s e n t method a r e
discussed.
S e v e r a l a t t e m p t s , a s reviewed by t h e a u t h o r s
[ I ] , were made r e c e n t l y by Zubov, K o i t e r , F. d e
Veubeke, C h r i s t o f f e r s e n , and t h e a u t h o r s i n formul a t i n g complementary e n e r g y p r i n c i p l e s f o r f i n i t e
d e f o r m a t i o n problems. As concluded i n [ I ] , t h e m o s t
p r o m i s i n g i d e a , which may b e a t t r i b u t e d t o F. d e
Veubeke, i s t o e x p r e s s t h e complementary e n e r g y
f o r m u l a t i o n i n t e r m s of b o t h t h e f i r s t P i o l a - K i r c h h o f f s t r e s s t e n s o r a s w e l l a s t h e t e n s o r of p o i n t w i s e r i g i d r o t a t i o n of m a t e r i a l e l e m e n t s .
In t h i s
f a s h i o n , t h e complementary e n e r g y becomesa f u n c t i o n
of t h e symmetrized Biot-Lure' s t r e s s t e n s o r o r t h e
s o - c a l l e d Jaumann s t r e s s t e n s o r . The a d v a n t a g e s
stemming from t h i s c o n c e p t a r e :
( i ) the linear
momentum b a l a n c e c o n d i t i o n i n t e r m s of t h e uasymm e t r i c f i r s t P i o l a - K i r c h h o f f stress t e n s o r c a n b e
s a t i s f i e d q u i t e e a s i l y , a p r i o r i , ( i i ) t h e angular
momentum b a l a n c e c o n d i t i o n i s a n unambiguous E u l e r
e q u a t i o n o f t h e v a r i a t i o n a l p r i n c i p l e , and ( i i i ) t h e
c o n s t r a i n t condition of orthogonality of r i g i d rot a t i o n c a n b e met e a s i l y . The a u t h o r s have exp l o i t e d t h e above c o n c e p t s i n t h e " r a t e " a n a l y s i s
of f i n i t e s t r a i n s i n n o n l i n e a r e l a s t i c and i n e l a s -
I.
Preliminaries
We u s e a f i x e d r e c t a n g u l a r C a r t e s i a n c o o r d i n a t e
system. We a d o p t t h e n o t a t i o n :
(-) under a symb o l d e n o t e s a v e c t o r ; ( - ) under a symbol d e n o t e s
a second-order t e n s o r ; +=A.b i m p l i e s t h a t a i=Ai kb k ;
+.Ej
- -
d e n o t e s a p r o d u c t s u c h t h a t (4.;)
.
ij
=A
.
B
ik kj'
(A:B)=A i j ' and u . t = u i t i'
*
The p o s i t i o n v e c t o r of a p a r t i c l e i n t h e undeformed body i s fi'(x e ) where e a r e u n i t Car-
**
t e s i a n b a r e s , and t h e g r a d i e n t o p e r a t o r
Formerly P o s t - D o c t o r a l Fellow, now w i t h H i t a c h i
Co., J a p a n .
Regents' Professor.
Released to AlAA to publish in all forms.
01-
7
--a
'
i n the
.
initial configuration is V=e a/ax
The position
---CL
a
vector of the same particle in the deformed body is
y and the corresponding gradient operator! N =e.a/ay
-1
i'
The defromation gradient tensor F is given by
F=(&) T ; Fia=yi.a=ayi/a~a. For nonsingular the
F
polardecomposition, F=a.(I+Q) exists, where (I+h)
is a symmetric positive definite tensor called the
stretch tensor (with Q often being called the engineering strain tensor), I an identity tensor; and
T -1
5 an orthogonal rotation tensor such that
=?
T
2
The deformation tensor G is defined by G=F .F(I+Q)
The Green-Lagrange strain tensor is defined by
2
g=1/2(5-_I)=1/2 (h + ~ ~ ) = 1 / 2 ( e + e ~ + ~ ~
where
. ~ ) e is
the gradient of-the displaceknt vector
a
.
u(-F-z),
-
i.e.,
.
.
such that e. =u.
For our present
la l,a
purposes we introduce the stress measures: (i) the
true (Cauchy) stress tensor, r, (ii) the PiolaLagrange or the first Piola-Kirchhoff stress tensor,
5 ; (iii) and the Jaumann stress tensor (or what is
also at times referred to as the symmetrized Lure'
stress tensor or the symmetrized Biot stress tensor)
r . As discussed in [I], and elsewhere, the relations
between L , 5 , and r are:
$=(z)l
The vanishing of the above first variation
leads to the following Euler-Lagrange equations
from the usual arguments of calculus of variations:
(i) the constitutive equation (corresponding to Ah);
(ii) the linear momentum balance condition for t
(corresponding to 6%); (iii) the angular moment;m
balance condition for 5 , viz., that (E+Q).t.?=
symmetric (due to the skewy-symmetric nature of
-
-
aL.6a,
since a is required to be orthogonal a
rt.
priori, such that %'.?=I)
; (iv) the compatibility
condition between g , a , and
(corresponding to 65);
, viz., t=c.t
( V ) the tractions conaition at S
h
0
(corresponding to 6g at Su ) ; and (vi) the displace0
ment boundary condition at S
where F-I is the inverse of F, and J is the determinant-of the Jacobian y.
As also discussed in
1,a
[I], and elsewhere, and r are symmetric, while 5
is in general unsymmetric tensor. In the case of
isotropic elastic materials, h, g , and r become
coaxial, and the relation (2)-becomes:
corresponding to 6t
.
As discussed in detail in [I], a functional
n(u,h,a,~)
- - . whose stationary conditions are the field
equations governing the finite deformation of a nonlinear elastic body can be stated as:
In the technical theory of beams, or plates
and shells, certain "plausible" approximations are
introduced to reduce these problems, respectively,
to one or two dimensional in nature from what may
rigorously be classified as three-dimensional problems. It is well-known that variational principles
often provide a convenient way of deriving the field
equations and consistent boundary conditions for
these problems. One may systematically introduce
"plausible" approximations for the field variables
in a functional, then the stationary conditions of
the functional yield the relevant field equations
and boundary conditions for structural memeber.
Thus, the "modus operandi" of the present procedure
is to introduce certain approximations to 2, h, 2,
and 5 appearing in E q . (4) so that the relevant
field equations for the structural problemsofplates
may be derived from the stationary condition of the
thus approximated functional n
HW'
11.
The stretch h is required to be symmetric a
priori, the rotation ? is required to be orthogonal,
a priori, and the first Piola-Kirchhoff stresses t
must be allowed to be unsymmetric, a priori. In
Eq. (3), V is the volume of the space occupied by
the undeformed body; S
and S
0
are, respectively,
0
the surfaces where displacements and traction are
prescribed; W ( h ) is the strain energy density per
unit initial volume, as a function of the symmetric
engineering strain tensor h; g are body forceslunit
mass; p is the mass densitylunit initial volume:
t=n.f
are surface tractions, and superposed bar
implies a prescribed quantity. The first variation
of the functional in Eq. (4) can be shown [I] to be:
Finite Deformation Analysis of Plates
We consider, without loss of generality, an
initially flat plate, with (x1,x2) being the Cartesian coordinates in the mid-plane, and x3 normal
to the mid-plane of the plate. The position vector
of an arbitrary material point in the undeformed
plate is x=x.e. (i=1,
3).
In the present work
1-1
...,
we invoke the well-known "plausible deformation
hypotheses" of Kirchhoff and Love, viz., the -"3
lines of the undeformed plate remain normal to the
deformed midplane and, moreover, remain unstretched.
Thus, under this hypothesis, the position vector of
a material particle after deformation is given by:
where t i j = t i j (xl ,x2 ,x3) and uij=aij
where u* ( i = 1 , 2 , 3 ) a r e t h e d i s p l a c e m e n t s of a p a r 1
t i c l e i n t h e midplane o f t h e undeformed p l a t e , and
N i s a u n i t v e c t o r normal t o t h e deformed midplane.
F u r t h e r , u t ( i = 1 , 2 , 3 ) a r e f u n c t i o n s o n l y of x
( a = 1 , 2 ) . Thus t h e d i s p l a c e m e n t o f a n a r i b t r a r y p a r +
= u*e
t i c l e i n t h e p l a t e i s g i v e n by: 2 =
1-1
u*e
u*e + x (N-e ) . The b a s e v e c t o r s a t a n a r 2-2
3 3
3 - -3
b i r a r y p o i n t i n t h e deformed p l a t e a r e g i v e n by:
=
+
and
G
-3
ax
= -- =
ax 3
(x1,x2).
(16)
F i n a l l y , t h e e x t e r n a l f o r c e s d i s t r i b u t e d on t h e
p l a t e a r e assumed t o b e s p e c i f i e d p e r u n i t a r e a on
t h e mid p l a n e of t h e p l a t e t o b e g . = g . ( x ) ( i = 1 , 2 , 3 ;
l l a
a = 1 , 2 ) . When t h e a s s u m p t i o n s i n Eqs. (11-16) a r e
s u b s t i t u t e d i n Eq. ( 4 ) , we f i n d t h r o u g h s t r a i g h t forward a l g e b r a , t h a t :
N
-
.
where u*
E au*/ax
The d e f o r m a t i o n g r a d i e n t
1,a
1
CL
t e n s o r , F , d e f i n e d s u c h t h a t dy=F.dx, i s g i v e n by:
-
/c [l.2*+B.
(g-;) . e l d c
-3
t
i n t h e well-known d y a d i c r e p r e s e n t a t i o n f o r secondorder tensors.
For t h e p r e s e n t d e f o r m a t i o n h y p o t h e s e s , we
assume t h e s t r e t c h t e n s o r t o b e :
and W* =
As p e r d i s c u s s i o n i n t h e p r e v i o u s s e c t i o n , t h e nons i n g u l a r F w i l l b e decomposed i n t o r o t a t i o n and
From Eqs. (8 and
s t r e t c h according a s F = a.(I+h).
9) i t i s s e e n t h a t : -
-
Wdx3.
L 3
I n Eq.
( 1 7 ) , So i s t h e a r e a of t h e undeformed mid-
p l a n e , and C
U
and C
t
a r e respectively the displace-
ment- and t r a c t i o n - p r e s c r i b e d
A c c o r d i n g l y , t h e d i s p l a c e m e n t v e c t o r can b e w r i t t e n
a s:
Note t h a t
F
i n Eq.
(8) i s l i n e a r i n x3 under t h e
( C o n s t i t i t u v e Law)
r- = t- . a-.
(14)
The f i r s t P i o l a - K i r c h h o f f s t r e s s t e n s o r t and t h e
r o t a t i o n t e n s o r a a r e assumed t o b e :
-
t = tijeiej and a =
e . . ;
iJ-1-J
i , j = , 2 , 3
(15)
It
aW*/ah*
=
I A
TAT -(T.e+?
.I
) = R(21)
2 -
when Eqs. (21, 22 and 23) s a t i s f i e d a p r i o r i , one
can e l i m i n a t e from Eq. ( 1 7 ) :
(i) h* and x t h r o u g h
t h e u s u a l c o n t a c t t r a n s f o r m a t i o n s and by & t a b l i s h i n g a complementary e n e r g y d e n s i t y W* s o t h a t
(ii u* i n t h e i n f e r i o r o f
aW*/aR=h* and aw*/aN=x;
C
where f i s t h e s o - c a l l e d Jaumann s t r e s s . S i n c e ,
f o r i s o t r o p y , h , g , and r a r e c o a x i a l , we have:
.
e x p r e s s i o n due t o t h e p r e s e n t l y invoked d e f o r m a t i o n
hypotheses.
The c o n s t i t u t i v e e q u a t i o n s and l i n e a r
momentum b a l a n c e (LMB) c o n d i t i o n s o b t a i n a l l e from
Eq. (17) a r e :
p r e s e n t h y p o t h e s e s . A c c o r d i n g l y , we assume t h a t
h
c a n b e approximated a s :
a6
For t h e 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 we assume t h e
c o n s t i t u t i v e law:
b o u n d a r i e s of S
i s s e e n t h a t o n l y tai e n t e r i n t o t h e above energy
S
- -
C
- -
t h r o u g h s a t i s f y i n g Eq.
(23), a p r i o r i .
When
t h i s i s done, we o b t a i n t h e complementary e n e r g y
functional :
+ sinwcos8e
e + coswe e
-3-2
-+3 .
(31)
The other constraint on the variables appearing in
Eq. (24), namely, the linear momentum balance condition (23), can be satisfied by setting
In Eq. (24), 9 is required to be orthogonal and
further a is, as in Eq. (16) a function only of
x1 and xi. Also, the variables R and N in Eq. (24)
are assumed to be defined in terms of 4, M, and ?
as in Eqs. (21 and 22). Further, we assume, for
simplicity, that the edges of the plate are normal
to the midplane; thus a unit normal to the boundary
of So is given by:
Thus,
A
A
T
and M appearing
in the intergrals on c
and
c are defined as:
t
As noted aboae, the rotation 2 appearing in
Eq. (24) must be such that a = a(x x ) anf further
-T1'2
a should be orthogonal, i.e., a.9 =I. To satisfy
this condition, the concept of-a "f inite-rotationvector" [ ] is useful. Let w be the finite angle
of rotation around an arbitrarily oriented unit vector g in the midplane of the plate. The finite
rotation vector 2 is defined to be:
-
where, $i are once-differentialbe stress functions,
are any particular solutions such that:
and P
ai
With the admissible variables chosen as per Eqs.
(31 and 32) and the complementary energy density
defined as in Eqs. (21, 22), the complementary
energy principle based on the stationary condition
of the functional in Eq. (24) can be used (either
as is, or in an appropriate incremental form, as
discussed in detail in [ I ] * to solve arbitrary
finite deformations of the plate.
Now we consider a some-what restricted class
of problems, characterized by the well-known VonKarman plate theory, in which the rotations are
assumed to be only moderatly large. This approximation can be invoked in Eq. (31) by setting:
(1-cosw)
2
2
w2 and sinw
However, 8 is arbitrary in Eq. (31).
fines :
wsin8 5 $2 and wcose
The action of the finite rotation 2 on a vector
can be written [ ] as the transformation of
to
as,
-
where a
=
I
-
+ (GI) + [ (2~:).
(2~:)1 /2
v*
W.
$ 1'
(33)
If one rede-
(34)
8 and 8 are rotations around xl and x2 axes,
1
2
respectively. The rotation tensor would then have
the form:
cosL(w/2) (29)
It can be verified that 9 of Eq. (29) is orthogonal.
The vector g in Eq. (27) can be written as:
e
-
=
e cos 8
-1
+ -e2
sin 8.
(30)
Thus, the rotation tensor a of Eq. (29) is a function of two parameters: 8Xx ,x ) and w(x x ) .
1 2
1' 2
The explicit expression for a can be shown to be:
a
= <
1-(1-cosw)
+ < (1-cosw)
+
sinL8>e e
-1-1
sinecose e e
-1-2
(sinwsin8)e e
-1-3
+<l-(1-cosw)
+
(1-cosw) sinecosee e
-2-1
2
cos 8>gle2 - sinwcos8e e
-2-3
Thus, for a von-Karman type plate theory, when Eq.
(35) is used, the functional of Eq. (24) becomes:
where R
aB
=
1
+ T a
[Taiai~
Bi ial?
It is seen that Eq. (36) can be written alternately
as :
=
js
1
{-W:(RaB9~aB)
+
2
1
2
T2241
-
T ~,,4, - 7 ~
~
~
0
1
-5
+ T13'2
-
M11,142
-
Lt
+
1
T~14142+
T
M12 ,141 - M21,242
+
T2341
nal tractions and displacement, respectively, are
prescribed.
M22,241 dS
(M2-naMa~)'l'~~.
We now define W*.
First we note:
1
=-(M
a +M a )
N a ~ 2 ai iB Bi ia
(Ti-naTai) U; + (i1-naMal) 42
-
where aS =p +cmu+cmt; asmo is the boundary of
mo m
~ S m pm ~is the interelement boundary,
the4 element
mo'
and cmu' cmt are the segments of aSmo where exter-
(41)
where 9 for the von-Karman type theory is given in
(38)
Eq. (35).
From the constitutive law, Eq. (13), we have
We now consider a finite element method based on
Eq. (38). First we use Tai (a=1,2; i=1,2,3) in
each element such that they satisfy Eq. ( 2 3 ) ,
through using stress-functions as in (32a,b).
However, at the interelement boundaries, the
generalized traction-reciprocity condition implies:
= 2uhaB
r
aB
=
+ XhYY6a5
2~ (hEB+x3xaB)
+ X (h*YY+x 3xY Y) 6aB'
(42)
We thus obtain:
where p
is the interelement boundary between two
m
neighboring elements. We choose T
and M
arai
ai
bitrarily in each element and enforce Eqs. (39a,b)
through a Lagrange-multiplier technique. The
Lagrange multiplier to enforce Eq. (39a) can be
seen to be the inter-element-boundary displacement
field ; while the one to enforce Eq. (39b) can
ip'
be seen to be the inter-element boundary rotation
Further, on an element boundary where
field 4
aP
displacemen~s/rotationsare prescribed, we can
such that they meet the prescribed
choose ti
ipl'ap
conditions a priori. With these concepts in mind,
the complementary energy functional for an assemblage of finite elements can be written (see [l]
for more details) as:
Thus, Eqs. (43a,b) can be solved for h
and x
a5
af3
in terms of R
and N
The complementary energy
a5
aB'
density then becomes,
.
With the assumptions of field variables being made
as discussed above, the functional of Eq. (38) can
be used as it is, or in an incremental form, to
construct the finite element algebraic equations.
Before doing so however, we consider the case of
bifurcation buckling problems characterized by
irrotational fundamental states and wherein T
aB
are specified and constant. Thus, in bifurcation
buckling probl.ems the modified complementary energy
functional of Eq. (40) reduces to:
111.
Finite Element Development:
In the development based on Eq. (47), we note
the the variables to be assumed are: (i) Tai and
Tai as per Eq. (32a,b);
(ii) arbitrary AMa6:
(iii)
arbitrary A$ within the element, and (iv) A@
oB
and
Aii
It can be seen that the stationary condition of the
functional in Eq. (45) leads to a linear eigenvalue problem for the buckling loads T
We now
aB'
present some details of incremental finite element
analysis of finite deformations and post buckling
analysis of plates based on Eq. (40), and linear
buckling analysis based on Eq. (45).
In using Eq. (40) in an incremental analysis,
we use the now well-known total Lagrangean approach
Ill. Suppose
we consider a generic increment from
.
-state CN to CN+l during which the relevant vari.-.
N
N
N
N - N
ables change from [T
Mai, ma, ma6, and c.10 1 to
-
N+l M~+',
$N+l, iN+',
iiN+l] respectively. We
a
a0
10
[Tai ' ai
then write, formally, from Eq. (40):
where Ii (cN) is the functional in Eq. (40) evalumc
ated at cN, Il is a constant, Anrnc involves first
at the element boundary such that they are
io
inherently compatible with those of the neighbouring elements. On the other hand, in a linear
buckling analysis based on Eq. (45) the variables
to be assumed are: (i) arbitrary M
(ii) arbitaB'
rary ba, (iii) Ta3 that satisfy the equation
=O [or T3=~3,2;T23=-$3,1], and (iv) 1 and
Ta3,a
ip
at the element boundary such that inter-element
@UP
compatibility is auto~aticallymet.
In the present series of analyses, the following
assumptions were used: First of all, the element,
of a general quadrilateral shape, has 8 nodes at
the boundary, 4 at the corners and 4 at mid-sides.
Tai (or ATai, u=1,2, i=1,3) satisfying Eq. (32a,b)
were chosen from 3 stress functions $
i
x2.
each of
Each M
1.'
a6
(orAM ) (a,B=1,2) was chosen to be a linear funca6
tion. The rotations $ (a=1,2) in the interior of
which is a cubic polynomial in x
the element were chosen to be linear polynomials.
On the boundary of the element, the inplane dis(a=1,2) were chosen to be linear in
placements c
Pa
terms of the displacements at the corner nodes. The
transverse displacement 1 were chosen to be
P3
quadratic in terms of the transverse displacements
at the 8 boundary nodes. The boundary rotations
(a=1,2) were chosen to be linear functions in
order terms in the incremental variables [AT =
ai
2
T~?' - T
etc.], and A nmc involves second order
a1
ai'
terms in the incremental variables. As shown in
[I], anmc can be used to check the equilibrium
terms of rotations at the 4 comer nodes. In
choosing these fields it was verified that the
necessary rang conditions [ ] for hybrid methods
were satisfied.
while the condition
and compatibility in C
N'
2
6A nmc=O gives the solution from CN to CN+l.
Carrying out the necessary integrations as the
functional in Eq. (47) is expressed in terms of the
undetermined parameters in (ATai' AMaB,Ama,Ailip, A & ~ ~ )
Through straight-forward algebra, it is seen that:
in the case of Eq. (47), while the functional in
Eq. (45) is expressed in terms-of the undetermined
parameters in (MaB ,Ta3,$a,fiiP,$ap).
In the case of Eq. (45), first the functional
is varied w.r.t. the parameters in ATai, AMaB, and
A$ which are independent for each element.
From
the thus obtained equations, the parameters in
ATai, AMaB and A$ for each element are expressed
,
in terms of the "generalized" displacement parameters in ii. and ;d
of the respective element.
I n
ao
By resubsti;uting these relations in Eq. (45), the
complementary energy functional is expressed entirely in terms of element "generalized displacement" parameters; from this, the incremental element stiffness and loads can be identified in the
usual way. The incremental analysis follows the
usual procedures [I]. In the case of Eq. (47), by
following procedures similar to above, the undeand $ in each
termined parameters in M
aB7 Ta3'
a
element are first expressed in terms of generalized
nodal-displacement parameters of the respective
element. Again, Eq. (47) is expressed in terms of
nodal generalizated-displacements only; thus identifying the element stiffness (which now depends
on the given T,@) and loads. In the case of linear
buckling problem, the eigen value problem for T
a0
is solved to identify the values of T
which make
a0
the properly constrained global stiffness matrix
singular, following procedures that are well-known.
For reasons of space, we omit the finite
element equations and their development, which are
more or less analogous to those that are detailed
in [a].
In the following we present certain example
results.
IV.
Results:
(a)
Buckling from an irrotational fundamental
state:
We consider a simply supported square plate
under uniform biaxial compression. One quarter of
the plate was analyzed. Three meshes, (lxl),
(2x2), (3x3) respectively, were considered. The
analytical solution for the buckling load, as is
widely known in literature, is o*=(232~/a~t). The
convergence of the buckling load with number of
elements for this case is shwon in Fig. 1. For
each of the considered finite element meshes, the
convergence of the lowest eigen value (to zero) of
the stiffness matrix, with the applied axial stress,
is shown in Fog. 2. Finally, the case of a clamped
square plate under uniform biaxial compression is
considered. The lower limit to the buckling load
as given by G.I. Taylor[l2] is o* =5.30n2~/a2t.
cL
The convergence of the computed buckling load with
number of elements for this case is shown in Fig. 3.
Results similar to those in Fig. 2, but for the
case of clamped plate, are shown in Fig. 4. The
computed buckling modes for the two cases are
shown in Fog. 5.
f
Next, the problem of a simply supported
square plate that is subject to uniform uniaxial
compression at two edges is considered. Only a
2x2 mesh in a quarter of the plate was used. The
pertinent geometrical and material property data
are indicated in Fig. 6. In the lower half of
Fig. 6, the maximum transverse displacement
(wmax/t) is shown magnified for lower values of
2
(P a 2 / ~ t) where ? is the axial compressive stress.
lnXthe present anaTYsis a small uniform lateral
~x1
applied
0~
along with the
load of ( ~ a ~ / ~ t ~ ) =was
axial Px. The linear buckling load predicted by
Levy [13] is (P a2/~t~)=3.6. The post-buckling
displacement wX
as obtained by Levy [13] is also
max
shown in both parts of Fie. 6. It is seen that the
present results agree excellently with those in
[13] upto displacemnet levels of (w
It) = 3.5.
max
Finally the problem of a similar isotripic
thin elastic simply supported plate that is sybject
to uniform transverse load P is considered. The
pertinent problem-data is shown in Fig. 7. The
presently computed variation of the maximum transis seen to agree excellently
verse displacement w
with that of ~evy[l3ql?~
The results presented in this paper for finite
deformation analysis of plates via a complementary
energy method appear encouraging. We defer any
conclusive comments on the merits of the present
finite element methods until our studies underway,
which deal with a mathematical analysis of convergence of the method, are completed.
Acknowledgements:
The results presented here were obtained
during the course of an investigation supported
by AFOSR under Grant No. AFOSR-81-0057.
The authors gratefully acknowledge these supports.
The authors thank Ms. Margarete Eiteman for her
care in typing this manuscript.
References:
Atluri, S.N. and Murakawa, H., "On Hybrid
Finite Element Models in Nonlinear Solid
Mechanics" in Finite Elements in Nonlinear
Mechanics vol. 1 (Ed. P.G. Bergan, et. al.)
TAPIR, Norway, 1977 pp 3-41.
Murakawa, H. and Atluri, S.N., "Finite Elasticity Solutions Using Hybrid Finite Elements
Based on a Complementary Energy
Principle"
Journal of
~echanics,Trans AS&,
vol.
45, 1978, pp 539-948.
li lied
~
Murakawa, H. and Atluri, S.N., "Finite Elasticity Solutions Using Hybrid Finite Elmenets
Based on a Complementary Energy Principle - 11.
Incompressible Materials" Journal of Applied
Mechanics vol. 46, 1979, pp 71-78.
Atluri, S.N., "On Rate Principles for Finite
Strain Analysis of Elastic and Inelastic Nonlinear Solids", in Recent RLesearch or Mechanical Behavior of Solids (Miyamoto Anniversary
Volume) Univ. o f Tokyo Press, Tokyo, 1979,
pp 79-io7.
Atluri, S.N., "On Some New General and Complementary Energy Principles for the Rate
Problems of Finite Strain, Classical Elastoplasticity" Journal of Structural Mechanics,
vol. 8, no. 1, 1980, pp 61-92.
Atluri, S.N., "Rate Complementary Energy Principles, Finite Strain Plasticity, and Finite
Elements" in Proc. IUTAM Symp. on Variational
Methods, (S. Nemat-Nasser and K. Washizu, Eds.)
Pergamon, 1980 (In Press).
Atluri, S.N. and Murakawn, H., "New General
and Complementary Energy Theorems, Finiie
Strain, Rate-Sensitive Inelasticity, and
Finite Elements: Some Computational Studies",
in Proc. U.S.-Europe Workshop on Finite Elements in Nonlinear Structural Mechanics, Ruhr
University, Bochum, W. Germany, July 1980.
LOWEST
E IGENVALUE
I
2.0-
[8] Murakawa, H., Reed, K.W., Atluri, S.N., and
Rubenstein, R., "Stability Analysis of
Structures Via a Complementary Energy Method"
Proc. Symp. on Computational Meth. in Nonlinear Structural and Solid Mechanics, (A.K.
Noor, et a1 Eds.) Wash. DC Oct. 1980 (Also
in Jnl of Computers and Structures).
0.883
1.0
1.02
1.04
I
1
x
l MESH
1.06 -7223 1.08
[9] Masur, E.F. and Popelar, C.H., "On the Use
of the Complementary Energy in the Solution
of Buckling Problems", Int. Jnl. of Solids
and Structures, vol. 12, p 203, 1976.
q,q= 1.005
2
x
2 MESH
[lo] Koiter, W.T., "Complementary Energy, Neutral
Equilibrium, and Buckling", Proc. Kon. Ned.
Akad Wetensch. Series B, p 183, 1977.
[ll] Lure, A.I., Analytical Mechanics, Mowscow,
1961.
[12] Taylor, G.I., Z. Angew. Math. u. Mech., vol.
13, p. 147, 1933.
q =zolm
[13] Levy, S., "Bending of Rectangular Plates with
Large Deflections", NACA Tech. Note 846 and
NACA Tech. Report 737, 1942, p. 139.
C
Z
b
*
=
1.004
3 1 3 MESH
-Q2
-0.255
qx: ANALYTICAL
d=
SOLUTION
z200.847
Fig. 2. Rate o f Convergence to Zero of Lowest
Eigenvalue of Stiffness matrix for
different meshes for SS Plate.
5 :EXACT SOLUTION
q, : G.I. Taylor
SIMPLY SUPORTED
1.0
;
Fig. 1.
4
7
9
NUMBER
AF
Convergence of Buckling Load with Number
of Elements SS Plate.
NUMBER
OF
ELEMENTS
Fig. 3.
Convergence of Buckling Load with
number of Elements, clamped plate.
LOWEST
EIMNVALUE
-
LEVY LNACA PER 7371
PRESENT F E M
YOUNG'S MODULUS ~ = 1 0 '
='b 0 316
POISON RATIO
WIDTH OF PLATE
a=1 0
THICKNESS
1 =0.01
AVERAGE LOAD
PER UNlT LENGTH Px
0 928
I
Yb
2 x 2 MESH
11
1.2
-0 435
uJdC'
Wmax
t
0
PRESENT F.E M
LEVY
0.2
3 x 3 MESH
Fig. 4.
Results s i m i l a r t o those i n Fig.
2, f o r a clamped p l a t e .
F i g . 6.
P o s t - B u c k l i n g Behavior o f a SS
Plate.
-
PRESENT F.E.M.
LEVY INACA
YOWCS MODULUS
POISSON S RATIO
WIDTH OF PLATE
THICKNESS
LATERAL PRESSURE
PER UNIT AREA
0
O E F O R U T O N MODE
OF A CLAMPED SOUARE
PLATE FOR THE LOWEST
EIGENVALUE AT
DEFORMATION MODE
OF A SIMPLY SUPORTED
1.2
REP 7371
E = 10'
0 318
= 1.0
I= a 0 1
I
p
BOUNDARY CONDITIONS,
SRUARE PLATE FOR THE
LOWEST EIGENVALUE AT
F i g . 7.
F i g . 5.
Computed B u c k l i n g Modes
Large D e f o r m a t i o n A n a l y s i s o f a
Transversley loaded SS P l a t e .