International Journal of Solids and Structures 38 (2001) 5807±5817
www.elsevier.com/locate/ijsolstr
Hamiltonian system based Saint Venant solutions for multilayered composite plane anisotropic plates
Weian Yao *, Haitian Yang
State Key Lab of Structural Analysis of Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology,
Dalian 116023, People's Republic of China
Received 8 September 1999; in revised form 9 October 2000
Abstract
This paper presents Hamiltonian system based Saint Venant solutions for the problem of multi-layered composite
plane anisotropic plates. A mixed energy variational principle is proposed, and dual equations are derived in the
symplectic space. The schemes of separation of variables and eigenfunction expansion, instead of the traditional semiinverse method, are implemented, and compatibility conditions at interfaces are formulated by dual variables. By
expending eigenfunctions in the subspace with zero eigenvalue, an analytical solution of a cantilever composite plate is
presented to illustrate the proposed approach. Ó 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Multi-layer; Anisotropic plate; Hamiltonian system; Saint Venant problem; Symplectic space
1. Introduction
Airy stress function (see, e.g., Timoshenko and Goodier, 1970) and other semi-inverse techniques are
often used to solve the problem of elasticity, however for the case of multi-layered plate, the compatibility
condition of displacement seems an obstacle for them to formulate, although compatibility condition of
stress could be relatively easy to be described since each layer has an individual stress function.
Zhong and Zhong (1990) presented an analogy theory between computational structural mechanics and
optimal control, the theory of Hamiltonian system can therefore be utilized for the solution of elasticity
(see, e.g., Zhong, 1991), and a new solution system was established by using a translation from Euclidean
space to symplectic space, in which, the schemes of separation of variables and eigenfunction expansion,
instead of traditional semi-inverse methods are implemented (see, e.g., Zhong and Yao, 1997; Zhong and
Yang, 1992). For the problem of elastic composite plate, compatibility conditions of displacements and
stress at interfaces are easy to be described by dual variables in the symplectic space.
Variational method is a useful tool solving the problem of elasticity. Steele and Kim (1992) presented
a Fourier transform based modi®ed mixed variational principle from the Hellinger±Reissner mixed
*
Corresponding author. Fax: +86-411-4708400.
E-mail address: zwoce@dlut.edu.cn (W. Yao).
0020-7683/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 0 - 7 6 8 3 ( 0 0 ) 0 0 3 7 1 - 1
5808
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
variational principle, and derived state-vector equations for both elastic bodies and shells of revolution.
Further application of this principle can be found in the solution of plate bending (see, e.g., Kang et al.,
1995). In this paper, a Hamiltonian mixed energy variational principle is established, and a group of
corresponding dual equations are derived.
Saint Venant principle based solution, in the sense of static equivalence, is often employed to solve the
problem with complex stress boundary conditions. Zhong and Yao (1997) presented a Saint Venant solution for multi-layered composite orthotropic plates via an eigenfunction expansion in the eigensubspace.
This paper is a further extension of the above work, a solution of Saint Venant problem is given for multilayered composite plane anisotropic plates.
2. Variational principles and Hamiltonian operator matrix
A composite plate with n layers is shown in Fig. 1 where xj L …j ˆ 0; 1; . . . ; n† is longitudinal coordinate of the upper surface of jth layer. The boundary conditions at two side surfaces are speci®ed by
rx ˆ X 1 …z† and
sxz ˆ Z 1 …z†
xˆ0
1a†
rx ˆ X 2 …z† and
sxz ˆ Z 2 …z†
x ˆ xn
1b†
The relationship between stress and strain for ith layer can be described as
38
9 2
9
9 2
9
8
8
38
b
b
b
e
e
r
c
c
c
>
>
>
>
>
>
>
x;i
x;i
1;i
2;i
3;i
x;i
1;i
2;i
3;i
=
=
= 6
=
<
<
<
< rx;i >
7
6
7
ez;i
rz;i ˆ 4 c2;i c4;i c5;i 5 ez;i
rz;i
or
ˆ6
b2;i b4;i b5;i 7
4
5
>
>
>
>
>
>
>
>
;
;
;
;
:
:
:
:
cxz;i
cxz;i
sxz;i
c3;i c5;i c6;i
sxz;i
b3;i b5;i b6;i
i ˆ 0; 1; . . . ; n
1†
2†
Following symbols are de®ned to simplify the derivation
d i ˆ det cj;i ;
di ˆ
1
;
b4;i
cj;i ˆ
cj;i
;
d i b4;i
bj;i ˆ
bj;i
b4;i
j ˆ 1; 2; . . . ; 6; i ˆ 0; 1; . . . ; n
1†
3†
where subscript i denotes ith layer, it will not appear in the derivation afterward except the case where it
may cause confusion.
The Hellinger±Reissner variational principle for the composite plates can be written as (see, e.g., Hu,
1981)
Fig. 1. A n-layer composite plate.
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
(Z
d
n 1 Z xi‡1
LX
0
xi
iˆ0
rx
ou
ow
ou ow
‡ rz
‡ sxz
‡
ox
oz
oz ox
U …rx ; rz ; sxz †
Zw dx dz ‡ Ue1 ‡ Ue2
Xu
5809
)
ˆ0
4†
where
U …rx ; rz ; sxz † ˆ 12 b1 r2x ‡ b4 r2z ‡ b6 s2xz ‡ b2 rx rz ‡ b3 rx sxz ‡ b5 rz sxz
Ue1 ˆ
Z
0
L
n
X 1 u ‡ Z 1 w xˆ0
(
Ue2
o
X 2 u ‡ Z 2 w xˆxn dz
L
R xn Z w ‡ X 3 u 0 dx
R0xn 3
L
‰…w w†rz ‡ …u u†sxz Š0 dx
0
ˆ
5†
6a†
for prescribed tractions
for prescribed displacements
6b†
Z 3 , X 3 and w, u are prescribed vectors of traction and displacement at the end surfaces of z ˆ 0 and z ˆ L,
respectively.
Coordinate z here is employed to simulate the time variable in the Hamiltonian system, x is a spatial
coordinate. A symbol Ô.Õ in the following derivation will be used denoting the di€erential with respect to z,
i.e. …† ˆ o=oz.
By making stationary for Eq. (4) with respect to rx , rx can be expressed in terms of r and s as follows
1
ou
rx ˆ
d
7†
b 2 r b3 s
b1
ox
where r and s represent rz and sxz , respectively. Substitution Eq. (7) for Eq. (4) can lead to a Hamiltonian
mixed energy variational principle
(Z "
#
)
n 1 Z xi‡1 L X
1
2
d
rw_ ‡ su_ H…w; u; r; s† Zw Xu dx dz ‡ Ue ‡ Ue ˆ 0
8†
0
iˆ0
xi
where H represents the Hamiltonian function (or density of mixed energy) having the form
#
"
2
1
ou
ou
ou
ow
d
H…w; u; r; s† ˆ
c6 r2 ‡ c4 s2 2c5 rs ‡ 2b2 r ‡ 2b3 s
2b1 s
2b1
ox
ox
ox
ox
9†
The state function vector can be described by v ˆ fw; u; r; sgT , where r and s are dual variables of w and
u in the symplectic space, respectively. The stationary requirement of Eq. (8) can yield a group of dual
equations as follows: v_ ˆ Hv ‡ Q; where
9
8
2
3
0
b2 oxo
c6
c5
0 >
>
>
>
=
<
b1 oxo
b3 oxo
c5
c4 7
16
0
6
7
10†
Hˆ 4 0
Q
ˆ
o 5;
0
0
b1 ox
Z>
>
b1
>
>
;
:
o2
o
o
X
0
d ox2 b2 ox
b3 ox
where H is a Hamiltonian operator matrix.
Compatibility conditions of displacement and stress at interfaces are speci®ed by
1
oui
di
b1;i
ox
b2;i ri
si ˆ si 1 wi ˆ wi
b3;i si
1
and
ˆ
1
b1;i
di
1
ui ˆ ui
1
oui 1
1
ox
x ˆ xi
b2;i 1 ri
1
b3;i 1 si
i ˆ 1; 2; . . . ; n
1
;
1†
11†
5810
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
The boundary conditions at two side surfaces can be rewritten as
1
ou0
d0
b2;0 r0 b3;0 s0 ˆ X 1 s0 ˆ Z 1 x ˆ 0
b1;0
ox
1
b1;n
1
dn
oun 1
1
ox
12a†
b2;n 1 rn
1
b3;n 1 sn
1
ˆ X2
sn
1
ˆ Z 2 x ˆ xn
12b†
3. Hamiltonian eigenvalue problem and symplectic adjoint orthogonal relationship
The homogeneous equation corresponding to Eq. (10) can be written as
v_ ˆ Hv
13†
For the free boundary conditions
1
ou
d
b2 r b3 s ˆ 0 and
b1
ox
sˆ0
x ˆ 0 or xn
14†
Eq. (13) can be solved by using a scheme of separation of variables, i.e.
v ˆ exp …lz†W…x†
and
HW ˆ lW
15†
where l is an eigenvalue, and W, satisfying the requirements of Eqs. (11) and (14), is an eigenfunction vector
only related with x.
To describe the behavior of Hamiltonian operator, following unit symplectic matrix is introduced
0 I
1 0
Jˆ
; Iˆ
16†
I 0
0 1
and a symplectic inner product is de®ned as
n 1 Z xi‡1
X
hv1 ; v2 i …vT1 Jv2 † dx
iˆ0
xi
17†
If v1 and v2 meet the requirement of Eqs. (11) and (14), it can be proved that
hv1 ; Hv2 i hv2 ; Hv1 i
18†
H is therefore a Hamiltonian operator matrix in the symplectic space. The Hamiltonian eigenvalue problem
is not self-adjoint; however, it can be termed as an symplectic adjoint eigenvalue problem, that is to say, if l
is an eigenvalue, then l must also be one. Thus the eigenvalues can be divided into two groups, i.e. group
A and group B
lAj …j ˆ 1; 2; . . .†;
Re…lAj † P 0
lBj …j ˆ 1; 2; . . .†;
lBj ˆ
lAj
in the group A
in the group B
19a†
19b†
In addition, there must exist two groups of eigenfunction vectors WAj and WBj corresponding to eigenvalue
lAj and lBj respectively, in which there must exist some of WAj and WBj satisfying an symplectic adjoint
orthogonal relationship
hWAi ; WBj i ˆ dij ;
hWAi ; WAj i ˆ hWBi ; WBj i ˆ 0
where dij is Kronecker symbol.
20†
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
5811
So the solution of Eq. (10) can be obtained by eigenfunction expansion (Yao et al., 1999) with following
combination
vˆ
1
X
Aj …z†WAj ‡ Bj …z†WBj
21†
jˆ1
4. Eigenfunction vectors for eigenvalue zero
In the Hamiltonian eigenvalue problem, zero, if it is an eigenvalue, must be multiple with even number,
and there often exist subsidiary eigenfunction vectors with various orders in Jordan normal form (Zhong
and Yao, 1997; Xu et al., 1997; Van Loan, 1984). A combination of eigenfunction vectors for zero eigenvalue with relevant eigenfunction vectors in Jordan form can be exploited to describe Saint Venant
solutions.
Two fundamental solutions for equation HW ˆ 0 with conditions (11) and (14) can be written as
0†
u ˆ 0;
r ˆ 0;
s ˆ 0 gT
and
v1 ˆ W 1
0†
u ˆ 1;
r ˆ 0;
s ˆ 0 gT
and
v2 ˆ W 2
W1 ˆ f w ˆ 1;
W2 ˆ f w ˆ 0;
0†
W1
0†
0†
22†
0†
0†
23†
0†
W2
and
describe rigid body translations along z and x axes, respectively, and are located at the heads
of two Jordan chains.
The governing equation of Jordan normal form eigenfunction for eigenvalue zero can be written as
HW…k† ˆ W…k
1†
24†
where k refers to kth order Jordan normal form (k ˆ 1; 2; . . .).
1†
0†
Solving equation HW1 ˆ W1 with conditions (11) and (14) can give an eigenfunction in the ®rst order
Jordan form in the chain 1
T
1†
W1;i ˆ b5;i x ‡ fi ‡ D0 ; b2;i x ‡ gi ‡ D1 ; di 0
i ˆ 0; 1; . . . ; n 1†
25†
where
f0 ˆ 0;
fi ˆ fi
g0 ˆ 0;
gi ˆ gi
1
1
xi …b5;i
b5;i 1 †
i ˆ 1; 2; . . . ; n
1†
26†
xi …b2;i
b2;i 1 †
i ˆ 1; 2; . . . ; n
1†
27†
1†
0†
W1 itself is not the solution of the original Eq. (13); however by combining with W1 a solution can be
obtain to describe a simple extension in the form
1†
1†
0†
v1 ˆ W1 ‡ zW1
28†
1†
W1
0†
W1 ,
Because
is symplectic adjoint with
eigenfunction in the second order Jordan form in the chain 1
will not exist.
1†
1†
0†
Similarly, v2 , a solution for rigid body rotation can be obtained by solving equation HW2 ˆ W2 and
1†
0†
combining W2 with W2 , where
1†
W2 ˆ f D2
1†
W2
x;
D3 ;
0;
0g
T
and
1†
1†
0†
v2 ˆ W2 ‡ zW2
29†
1†
W2
is an eigenfunction in the ®rst order Jordan form in the chain 2. Because
is symplectic orthogonal
0†
0†
with the eigenfunctions W1 and W2 , eigenfunction in the second order Jordan form in the chain 2 will
exist.
5812
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
The eigenfunction in second order Jordan form in the chain 2 is
9
8
2†
9
8
>
>
w
> > 1 b5;i … x D2 †2 D3 …x D2 † ‡ pi ‡ D4 >
>
2;i
>
> <
>
> …2† >
> 2
=
=
<
2
1
u2;i
2†
b … x D 2 † ‡ qi ‡ D 5
2 2;i
ˆ
i ˆ 0; 1; . . . ; n
W2;i ˆ
> >
>
> r…2†
>
di … x D2 †
>
>
2;i >
>
;
>
:
>
;
: s…2† >
0
1†
30†
2;i
where
p0 ˆ 0;
pi ˆ pi
1
‡ 12…xi
D2 † …b5;i
2
b5;i 1 †
i ˆ 1; 2; . . . ; n
1†
31†
q0 ˆ 0;
qi ˆ qi
1
‡ 12…xi
D2 † …b2;i
2
b2;i 1 †
i ˆ 1; 2; . . . ; n
1†
32†
A solution of pure bending can be given by
2†
2†
1†
0†
v2 ˆ W2 ‡ zW2 ‡ 12z2 W2
33†
2†
W2
0†
W1
0†
W2
is also symplectic orthogonal with the eigenfunctions
and
by choosing appropriate constant D2 (see Appendix A), eigenfunction in the third order Jordan form in the chain 2 will exist.
n
oT
3†
3†
3†
3†
3†
W2;i ˆ w2;i
i ˆ 0; 1; . . . ; n 1†
34†
; u2;i ; r2;i ; s2;i
where
3†
w2;i ˆ 16 b2;i ‡ b6;i
3
2b25;i … x D2 †
qi D5 ‡ ti ‡ D6
‡ c1;i ri
3†
1
6
u2;i ˆ
b2;i b5;i ‡ c3;i di … x
D2 †
3
2
1
b D …x
2 5;i 3
D2 † ‡ … x
D2 † b5;i …pi ‡ D4 †
35a†
1
b D …x
2 2;i 3
2
D2 † ‡ … x
D2 † b2;i …pi ‡ D4 †
c3;i ri ‡ si ‡ D7
35b†
3†
r2;i ˆ
b5;i di … x
3†
D2 †
2
di D3 … x
D2 † ‡ di …pi ‡ D4 †
b5;i ri
35c†
2
s2;i ˆ 12di … x
D 2 † ‡ ri
35d†
where
r0 ˆ
1
d D2 ;
2 0 2
s0 ˆ 0;
si ˆ si
b2;i
b2;i
ri ˆ ri
‡ 1…xi
6
xi
1
1
i ˆ 1; 2; . . . ; n
t0 ˆ 0;
ti ˆ ti
1
b5;i
1
b5;i
1
x
2 i
1
2
D2 † …di
di 1 †
i ˆ 1; 2; . . . ; n
1†
36†
3
2
D2 † b2;i b5;i ‡ c3;i di b2;i 1 b5;i 1 c3;i 1 di 1 ‡ 12D3 …xi D2 †
D2 † b2;i …pi ‡ D4 † c3;i ri b2;i 1 …pi 1 ‡ D4 † ‡ c3;i 1 ri 1
1†
i ˆ 1; 2; . . . ; n
1
x
6 i
xi
37†
3
D2 † b2;i ‡ b6;i 2b25;i b2;i 1 b6;i 1 ‡ 2b25;i 1 ‡ 12D3 …xi
D2 † b5;i …pi ‡ D4 † ‡ c1;i ri qi b5;i 1 …pi 1 ‡ D4 † c1;i 1 ri
1†
D2 †
1
2
‡ qi
1
38†
A solution of a bending problem with constant shear force can be given by
3†
3†
2†
1†
0†
v2 ˆ W2 ‡ zW2 ‡ 12z2 W2 ‡ 16z3 W2
39†
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
5813
Table 1
Adjoint symplectic orthogonal relationship of eigenfunctions for eigenvalue zero
0†
0†
W1
1†
0†
1†
2†
3†
W1
W1
W2
W2
W2
W2
0
0
0
D2
D4
0
0
D2
D4
D1 , D6
0
0
0
0
0
0
D5
1†
W1
0†
W2
1†
W2
2†
W2
3†
W2
0
3†
0†
Because W2 is symplectic adjoint with the W2 , eigenfunction in the fourth order Jordan form in the chain
2 will not exist.
The above six eigenfunctions constitute symplectic adjoint and orthogonal relationships, as shown in
Table 1 where 0 represents a symplectic orthogonal relationship, represents a symplectic adjoint relationship, and Di …i ˆ 1; 2; 4; 5; 6† represent a set of parameters. Proper choice of Di …i ˆ 1; 2; 4; 5; 6† can
make two eigenfunction vector symplectic orthogonal. The expressions of Di …i ˆ 1; 2; 4; 5; 6† are listed in
Appendix A, where k1 and k2 denote extensional and bending sti€ness of a cross-section, respectively, and
D2 represents the position of centroidal axis of a cross-section in the pure bending problem.
5. The Saint Venant solutions
Combining eigenfunction vectors for zero eigenvalue with eigenfunction vectors in Jordan form, an
analytic solution of Saint Venant problem can be obtained via expanding eigenfunction in the subspace for
zero eigenvalue, having the form
0†
0†
1†
1†
2†
3†
v ˆ a1 …z†W1 ‡ a2 …z†W2 ‡ a3 …z†W1 ‡ a4 …z†W2 ‡ a5 …z†W2 ‡ a6 …z†W2
40†
Substituting Eq. (40) for Eq. (8) and choosing proper Di …i ˆ 0; 3; 7†, listed in Appendix A, then yields
Z L k1 a3 a_ 1 ‡ k2 a5 a_ 4 k2 a6 a_ 2 12k1 a23 12k2 a25 ‡ k2 a4 a6 N …z†a1 Q…z†a2 W …z†a3
d
0
L
M…z†a4 h…z†a5 V …z†a6 dz ‡ …k3 a3 a5 ‡ k4 a5 a6 †jzˆ0 ‡ Ue2 ˆ 0
41†
The expressions of ki …1; 2; 3; 4†, N …z†, and Q…z† etc. are listed in Appendix A.
The implement of variations for Eq. (41) with respect to ai …i ˆ 1; 2; . . . ; 6† leads to following di€erential
equations
a_ 3 ˆ
N …z†=k1
with respect to da1
a_ 1 ˆ a3 ‡ W …z†=k1
a_ 6 ˆ Q…z†=k2
a_ 5 ˆ a6
with respect to da3
with respect to da2
M…z†=k2
with respect to da4
42a†
42b†
42c†
42d†
a_ 4 ˆ a5 ‡ h…z†=k2
with respect to da5
42e†
a_ 2 ˆ a4
with respect to da6
42f†
V …z†=k2
5814
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
Usually it is dicult to describe the required boundary conditions on two ends exactly with these six ai , a
group of variational principles (Eq. (41)) based boundary conditions of tractions are therefore presented,
having the form
a3 ˆ N =k1 ;
where
Nˆ
Z
xn
0
a5 ˆ …M ‡ D3 Q†=k2 ;
Z
Z 3 dx;
Qˆ
xn
0
a6 ˆ
Q=k2
Z
X 3 dx;
Mˆ
xn
0
when z ˆ 0 or L
D2
43†
x†Z 3 dx
44†
On the other hand, on the basis of Eq. (41), the boundary conditions of displacement can be rewritten as
k 1 a1 ‡ k 3 a5 ˆ W ;
where
W ˆ
n 1 Z
X
iˆ0
xi
xi‡1
wdi dx;
k2 a4 ‡ k3 a3 ‡ k4 a6 ˆ h;
hˆ
n 1 Z
X
iˆ0
xi‡1
xi
2†
k2 a2
wr2;i dx;
k 4 a5 ˆ
V
n 1 Z
X
xi‡1
V ˆ
iˆ0
xi
45†
h
i
3†
3†
wr2;i ‡ us2;i dx
46†
W , h and V represent equivalent displacements.
For clamped end, boundary conditions can be described by Eq. (45) or
w ˆ u ˆ ow=ox ˆ 0
when z ˆ 0 or L
and
x ˆ x …0 6 x 6 xn †
47†
Assuming x ˆ 0, analytic solutions of Saint Venant problem can be given by integrating Eq. (42) with
boundary conditions (43) and (45) or Eq. (47).
As a example, a cantilever plate is considered, which is clamped at the end z ˆ 0, and subjected to a load
P in the direction of x axis at the another end z ˆ L. By integrating Eq. (42) with conditions (43) and (45)
(or Eq. (47)), the distribution of stress can be described as
h
i.
2†
3†
3†
ri ˆ P …L ‡ D3 z†r2;i r2;i k2 ; si ˆ P s2;i =k2
i ˆ 0; 1; . . . ; n 1†
48†
If the distribution of load P is described as Eq. (48), can be considered as an elastic analytic solution
since the requirements of both Eq. (10) and two side boundary conditions (1) are satis®ed. The di€erence
between two solutions of taking boundary condition (45) or (47) can be proved to be only a minute rigid
body displacement.
6. Conclusions
The major objective of this paper is to present a Saint Venant solution for elastic multi-layered composite plane anisotropic plates in the proposed Hamiltonian system. The merits of using the proposed
approach lie in
1. being convenient for the application of conventional schemes, such as separation of variables, and eigenfunction expansion etc.,
2. facilitating to describe compatibility conditions at interfaces for displacements and stress.
In some cases, such as interlaminates stresses analysis (see, e.g., Pipes and Pagano, 1970), the end e€ects
must be taken into account, it is de®nitely necessary to consider non-zero eigenvalues and their eigenfunctions to give more exact description. Due to the limited capacity of a paper, this issue is not discussed
here. In fact, the major e€ect of the addition of non-zero eigenfunctions will appear at the neighbor areas of
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
5815
the ends, these non-zero eigenfunctions in Eq. (21) will decay from two ends quickly, the solution with zero
eigenvalue is still a fairly good description of stress distribution in the region far enough from ends.
Acknowledgements
The research leading to this paper is funded by NSF (19732020) of People's Republic of China.
Appendix A
(1) Expressions of constants D0 , D2 and k1
n 1 Z xi‡1
X
b0
b1
D0 ˆ
; D2 ˆ ;
k1 ˆ
di dx
k1
k1
iˆ0 xi
where
b0 ˆ
n 1 Z
X
xi‡1
xi
iˆ0
di b5;i x ‡ fi dx;
b1 ˆ
n 1 Z
X
A:1†
xi‡1
xi
iˆ0
di x dx
A:2†
(2) Expressions of constants D3 , D4 , k2 and k3
b3
k 3 b2
D3 ˆ
;
D4 ˆ
;
k2
k1
k2 ˆ
n 1 Z
X
iˆ0
where
b2 ˆ
xi‡1
xi
n 1 Z
X
iˆ0
D2 †2 dx;
di … x
k3 ˆ
n 1 Z
X
xi
1
b …x
2 5;i
di
2
xi‡1
xi
iˆ0
xi‡1
A:3†
di …D2
D2 † ‡ pi dx;
b3 ˆ
x† b5;i x ‡ fi dx
n 1 Z
X
iˆ0
xi
xi‡1
h
di 12b5;i … x
A:4†
D2 †3
pi … x
i
D2 † dx
A:5†
(3) Expressions of constants D1 , D5 , D6 , D7 and k4
b4
b6 ‡ b7
b5
b8
D1 ˆ ; D5 ˆ
; D6 ˆ
; D7 ˆ D3 D5 ‡ ;
k2
2k2
k1
k2
where
b4 ˆ
n 1 Z
X
iˆ0
b5 ˆ
xi
n 1 Z
X
iˆ0
xi
xi‡1
n
1
d
2 i
b2;i
2b25;i … x
3
D2 † ‡ 12di b2;i D2 ‡ gi
h ‡ ri b2;i
b25;i ‡ di b5;i …pi ‡ D4 D2 D3 †
o
‡ b5;i D2 ‡ fi di D4 ‡ di pi b5;i ri
dx
xi‡1
di
n
1
6
b2;i ‡ b6;i
‡ c1;i ri
qi … x
2b25;i … x
D2 †
o
D2 † ‡ ti dx
3
k4 ˆ 12…b7
2b5;i b5;i D2 ‡ D3 ‡ fi … x
i
di D3 fi … x
1
b D …x
2 5;i 3
b6 †
A:6†
D2 †
2
D2 † ‡ ri b2;i D2 ‡ gi
A:7†
2
D2 † ‡ b5;i …pi ‡ D4 †
A:8†
5816
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
b6 ˆ
n 1 Z
X
iˆ0
xi‡1
xi
di
n
1
b2;i ‡ b6;i
6
‡ c1;i ri
b7 ˆ
n 1 Z
X
iˆ0
xi‡1
xi
n
1
d
4 i
qi … x
2b25;i
‡ ri b25;i
2b25;i … x
2
D 2 † ‡ ti … x
b2;i … x
b2;i
i
x
‡ di pi …pi ‡ D4 † ‡ ri qi
b8 ˆ
n 1 Z
X
iˆ0
xi
D2 †
4
o
D2 † dx
4
D2 † ‡ 32di b5;i D3 … x
D2 †
2
3
D2 † ‡ b5;i …pi ‡ D4 †
1
b D …x
2 5;i 3
D3 2di pi
o
b5;i pi dx
A:9†
h
3
D2 † ‡ 12 di qi ‡ 2D23
b5;i ri … x
3b5;i pi
b5;i D4
D2 †
A:10†
i
1 h
1
5
di 2b5;i …2b2;i ‡ b6;i 2b25;i † b3;i … x D2 † ‡ di D3 10b25;i 5b2;i 2b6;i
12
12
h
1
4
x D2 † ‡ di …pi ‡ D4 † b6;i ‡ 4b2;i 8b25;i ‡ 3di b5;i D23 ‡ 2qi ‡ 2D5
6
i
1h
ri 2b2;i b5;i ‡ 4c3;i di 8b35;i ‡ 7b5;i b6;i … x D2 †3 ‡ ri D3 3b25;i b2;i 2b6;i
2
i
‡ di D3 2qi 3b5;i pi 3b5;i D4 ‡ si di 2b5;i di ti … x D2 †2
h
‡ di …pi ‡ D4 † b5;i pi ‡ b5;i D4 ‡ c1;i ri qi D5
di ti D3 ri2 c3;i ‡ c1;i b5;i
i
‡ ri b5;i …qi ‡ D5 † ‡ ri …pi ‡ D4 † b2;i b25;i … x D2 †
‡ si ri ‡ ti di …pi ‡ D4 † ti ri b5;i dx
A:11†
xi‡1
(4) Expressions of functions N …z†, Q…z†, M…z†, W …z†, h…z†, and V …z† in Eq. (41)
N …z† ˆ
n 1 Z
X
xi
iˆ0
Q…z† ˆ
n 1 Z
X
iˆ0
M…z† ˆ
xi‡1
xi
n 1 Z
X
iˆ0
W …z† ˆ
xi‡1
xi‡1
xi
n 1 Z
X
iˆ0
xi
xi‡1
Z…x; z†dx ‡ Z 2 …z†
Z 1 …z†
A:12†
X …x; z†dx ‡ X 2 …z†
X 1 …z†
A:13†
‰…D2
x†Z ‡ D3 X Š dx ‡ …D2
xn †Z 2 ‡ D3 X 2
h
i
h
i
1†
1†
1†
1†
w1;i Z ‡ u1;i X dx ‡ w1;n 1 Z 2 ‡ u1;n 1 X 2
xˆxn
D2 Z 1
h
1†
D3 X 1
1†
w1;0 Z 1 ‡ u1;0 X 1
A:14†
i
xˆ0
A:15†
W. Yao, H. Yang / International Journal of Solids and Structures 38 (2001) 5807±5817
h…z† ˆ
n 1 Z
X
iˆ0
V …z† ˆ
xi‡1
iˆ0
xi
i
h
i
2†
2†
2†
2†
w2;i Z ‡ u2;i X dx ‡ w2;n 1 Z 2 ‡ u2;n 1 X 2
h
h
i
h
i
3†
3†
3†
3†
w2;i Z ‡ u2;i X dx ‡ w2;n 1 Z 2 ‡ u2;n 1 X 2
h
xˆxn
xi
n 1 Z
X
h
xi‡1
xˆxn
2†
2†
w2;0 Z 1 ‡ u2;0 X 1
3†
3†
i
w2;0 Z 1 ‡ u2;0 X 1
xˆ0
i
xˆ0
5817
A:16†
A:17†
References
Hu, H.C., 1981. Variational principle of elasticity and its application. Science Press, Beijing, China (in Chinese).
Kang, L.C., Wu, C.H., Steele, C.R., 1995. Fourier series for polygonal plate bending: a very large plate element. Applied Mathematics
and Computation 67, 157±225.
Pipes, R.B., Pagano, N.J., 1970. Interlaminar stresses in composite laminates under uniform axial extension. Journal of Composite
Materials 4, 538±548.
Steele, C.R., Kim, Y.Y., 1992. Modi®ed mixed variational principle and the state-vector equation for elastic bodies and shells of
revolution. Journal of Applied Mechanics 59 (3), 587±595.
Timoshenko, S.P., Goodier, J.N., 1970. Theory of Elasticity, third ed., McGraw-Hill, New York.
Van Loan, C.F., 1984. A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix. Linear Algebra and
Applications 61, 233±251.
Xu, X.S., Zhong, W.X., Zhang, H.W., 1997. The Saint-Venant problem and principle in elasticity. International Journal of Solids and
Structures 34 (22), 2815±2827.
Yao, W.A., Zhong, W.X., Su, B., 1999. New solution system for circular sector plate bending and its application. ACTA Mechanics
Solida Sinica 12 (4), 307±315.
Zhong, W.X., 1991. Plane elasticity problem in strip domain and Hamiltonian system. Journal of Dalian University of Technology 31
(4), 373±384 (in Chinese).
Zhong, W.X., Yang, Z.S., 1992. Partial di€erential equations and Hamiltonian system. In: Cheng, F.Y., Zizhi, Fu. (Eds.),
Computational Mechanics in Structural Engineering, Elsevier, Amsterdam, pp. 32±48.
Zhong, W.X., Yao W.A, 1997. The Saint Venant solutions of multi-layered composite plates. Advances in Structural Engineering 1 (2),
127±133.
Zhong, W.X., Zhong, X.X., 1990. Computational structural mechanics, optimal control and semi-analytical method for PDE.
Computers and Structures 37 (6), 993±1004.