Báo cáo " NON-LINEAR ANALYSIS OF MULTILAYERED REINFORCED COMPOSITE PLATES " ppt

VNU. JOURNAL OF SCIENCE, Mathematics - Physics. T.XXI, N
0
1 - 2005
NON-LINEAR ANALYSIS OF MULTILAYERED
REINFORCED COMPOSITE PLATES
KhucVanPhu,NguyenTienDat
Military Technical Academ y
Abstract.
This paper deals with the analysis of non-linaer multilayered reinforced com-
p osite plates with simply supported along its four edges by Bubnov - Galerkin and Finite
Element Methods. Numerical results are presented for illustrating theoretical analysis of
reinforced and unreinforced laminated composite plates.
Keywords: Stiffened laminated composite plate, multilayered reinforced composite plates
1. Introduction
Multilayered reinforced composite plates are used extensively in Naval, Aerospace,
Automobile applications and in Civil engineering.v.v Today, analysis of linear laminated
composite plates has been studied by many authors. However, the analysis of non-linear
laminated composite plates has received comparatively little attention [3, 4, 5, 6, ] spe-
cially for a nalysis of non-linear stiffen ed laminated composite plates and shells subjected
to distributed transverse loads. This problem is studied in the present paper.
Typeset by A
M
S-T
E
X
16
Non-linear analysis of multilayered reinforced composite plates 17
2. Governing equations of laminated plates
Let’s consider a rectangular multila yered reinforced composite plate, in which each
lay er is made of unidirectional composite material and stiffeners are made by composite
material. This plate is subjected to distributed transverse loads (Figure 1).

For multilayered reinforced composite plates working in the elastic state the relation
bet ween internal force and deformation is of the form

σ

=

D

{ε} (1)
where

σ

=

N
x
N
y
N
xy
M
x
M
y
M
xy

T


D

-Matrixofstiffness constants of multilay ered reinforced composite plates

D

=

[A][B]
[B][D]

(2)
in which
(A
ij
,B
ij
,D
ij
=
N

k=1
h
k

h
k−1


Q
ij

k
(1,z,z
2
)dz (i, j =1, 2, 6),
{ε} - the deformation of point of the middle surface.
The strain - displacement relations in the non-linear theory are of the form
ε
x
=
∂u
∂x
+
1
2

∂w
∂x

2
, ε
y
=
∂v
∂y
+
1
2


∂w
∂y

2
, γ
xy
=

∂u
∂y
+
∂v
∂x

+
∂w
∂x
∂w
∂y
, (3)
k
x
= −
∂
2
w
∂x
2
,k

y
= −
∂
2
w
∂y
2
,k
xy
= −2
∂
2
w
∂x∂y
,
where u, v, w are the middle displacements along the x, y and z axis respectively.
For a plate simply supported on all edges, the following boundary condition are in
posed
+Atedgesx =0,x = a: w =0; v =0; M
x
=0; (4)
+Atedgesy =0,y = b: w =0; u =0; M
y
=0; (5)
3. Bubno v - Galerkin methods
According to Lekhnistki theory when expanding internal forces - deformations (1),
we obtain the expression for stress resultants and flexion moments of multilayered rein-
forced composite plates
N
x

=(A
11
+ E
1
A
1
/s
1
)ε
x
+ A
12
ε
y
+(E
1
A
1
/s
1
)z
1
k
x
,
N
y
=(A
22
+ E

2
A
2
/s
2
)ε
y
+ A
12
ε
x
+(E
2
A
2
/s
2
)z
2
k
y
,
N
xy
= A
66
γ
xy
, (6)
M

x
=(D
11
+ E
1
I
1
/s
1
)k
x
+ D
12
k
y
+(E
1
A
1
/s
1
)z
1
ε
x
,
M
y
=(D
22

+ E
2
I
2
/s
2
)k
y
+ D
12
k
x
+(E
2
A
2
/s
2
)z
2
ε
y
,
M
xy
= D
66
k
xy
,

18 KhucVanPhu,NguyenTienDat
where
- A
ij
, D
ij
(i, j =1, 2 and 6) are extending and bending stiffnesses of the plate
without stiffeners,
- E
1
, E
2
are the Young modulus of the longitudinal and transversal stiffeners,
respectively,
- A
1
, A
2
are the section areas of the longitudinal and transversal stiffeners, respec-
tively,
- I
1
, I
2
are the inertial moments of cross-section of the longitudinal and transversal
stiffeners, respectively,
- s
1
, s
2

are the distances between two longitudinal stiffeners and between two
transversal stiffeners, respectively,
- z
1
, z
2
are the distances from the mid-plane to the centroids of the longitudinal
and transversal stiffeners, respectively,
The equilibrium equations of a plate according to [3] are
∂N
x
∂x
+
∂N
xy
∂y
=0,
∂N
xy
∂x
+
∂N
y
∂y
=0, (7)
∂
2
M
x
∂x

2
+2
∂
2
M
xy
∂x∂y
+
∂
2
M
y
∂y
2
+ N
x
∂
2
w
∂x
2
+2N
xy
∂
2
w
∂x∂y
+ N
y
∂

2
w
∂y
2
− q(x, y)=0.
Substituting (3) and (6) into (7) after some operations we obtain the equilibrium
equations of the multilayered reinforced composite plates
(A
11
+ E
1
A
1
/s
1
)
∂
2
u
∂x
2
+ A
66
∂
2
u
∂y
2
+(A
12

+ A
66
)
∂
2
v
∂x∂y
− (E
1
A
1
/s
1
)z
1
∂
3
w
∂x
3
+
+(A
11
+ E
1
A
1
/s
1
)

∂w
∂x
∂
2
w
∂x
2
+(A
12
+ A
66
)
∂w
∂y
∂
2
w
∂x∂y
+ A
66
∂w
∂x
∂
2
w
∂y
2
=0,
(A
22

+ E
2
A
2
/s
2
)
∂
2
v
∂y
2
+ A
66
∂
2
v
∂x
2
+(A
12
+ A
66
)
∂
2
u
∂x∂y
− (E
2

A
2
/s
2
)z
2
∂
3
w
∂y
3
+
+(A
22
+ E
2
A
2
/s
2
)
∂w
∂y
∂
2
w
∂y
2
+(A
12

+ A
66
)
∂w
∂x
∂
2
w
∂x∂y
+ A
66
∂w
∂y
∂
2
w
∂x
2
=0, (8)
(D
11
+ E
1
I
1
/s
1
)
∂
4

w
∂x
4
+2(D
12
+2D
66
)
∂
4
w
∂x
2
∂y
2
+(D
22
+ E
2
I
2
/s
2
)
∂
4
w
∂y
4
− (E

1
A
1
/s
1
)z
1
∂
3
u
∂x
3
− (E
2
A
2
/s
2
)z
2
∂
3
v
∂y
3
− (E
1
A
1
/s

1
)z
1
∂w
∂x
∂
3
w
∂x
3
− (E
2
A
2
/s
2
)z
2
∂w
∂y
∂
3
w
∂y
3
−
1
2
(A
11

+ E
1
A
1
/s
1
)
∂
2
w
∂x
2

∂w
∂x

2
−
1
2
A
12
∂
2
w
∂y
2

∂w
∂x


2
−
1
2
A
12
∂
2
w
∂x
2

∂w
∂y

2
−
1
2
(A
22
+ E
2
A
2
/s
2
)
∂

2
w
∂y
2

∂w
∂y

2
− 2A
66
∂w
∂x
∂w
∂y
∂
2
w
∂x∂y
− (A
11
+ E
1
A
1
/s
1
)
∂u
∂x

∂
2
w
∂x
2
− 2A
66
∂u
∂y
∂
2
w
∂x∂y
− A
12
∂u
∂x
∂
2
w
∂y
2
− A
12
∂v
∂y
∂
2
w
∂x

2
− 2A
66
∂v
∂x
∂
2
w
∂x∂y
− (A
22
+ E
2
A
2
/s
2
)
∂v
∂y
∂
2
w
∂y
2
− q(x, y)=0,
Non-linear analysis of multilayered reinforced composite plates 19
in which q(x, y) is the lateral load, which can be expanded in a double Fourier series
q(x, y)=
∞


m=1
∞

n=1
q
mn
sin
mπx
a
sin
nπy
b
. (9)
For uniformly distributed load of intensity q
0
,thecoefficients q
mn
are given by
q
mn
=
16q
0
mnπ
2

− 1

m+n

2
,m,n=1, 3, 5, (10)
If the boundary conditions discussed here can be satified, the displacements are
represented by
u = U
mn
cos
mπx
a
sin
nπy
b
,
v = V
mn
sin
nπx
a
cos
nπy
b
, (11)
w = W
mn
sin
mπx
a
sin
nπy
b

,
where
- a, b: edges of plate i n x and y axial directions respectively,
- m, n: the numbers of halfwave in the x and y axial directions respectively.
Substituting expressions (11) into the equilibrium equations (8) and applying the
Galerkin procedure yield the set of three algebraic equations with respect to the amplitudes
U
mn
, V
mn
, W
mn
,wherethefirst two equations of this system are linear algebraic equations
for U
mn
, V
mn
:
a
1
U
mn
+ a
2
V
mn
= a
3
W
mn

+ a
4
W
2
mn
,
a
5
U
mn
+ a
6
V
mn
= a
7
W
mn
+ a
8
W
2
mn
. (12)
Getting from (12) expression U
mn
, V
mn
with respect to W
mn

and substituting into
the third equation we obtain a non-linear equation with respect to W
mn
a
9
W
3
mn
+ A
10
W
2
mn
+ a
11
W
mn
= q
mn
, (13)
where a
i
are coefficients which depend on the material, geometry and the half wave,
a
1
=(A
11
+ E
1
A

1
/s
1
)
m
2
b
a
+ A
66
n
2
a
b
,
a
2
= a
5
=(A
12
+ A
66
)mn,
a
3
=(E
1
A
1

/s
1
)z
1
m
3
πb
a
2
,
a
4
= −
16
9

2(A
11
+ E
1
A
1
/s
1
)

m
a

2

b
nπ
− (A
12
− A
66
)
n
bπ

,
a
6
=(A
22
+ E
2
A
2
/s
2
)
n
2
a
b
+ A
66
m
2

b
a
,
a
7
=(E
2
A
2
/s
2
)z
2
n
3
aπ
b
2
,
20 KhucVanPhu,NguyenTienDat
a
8
= −
16
9

2(A
22
+ E
2

A
2
/s
2
)

n
b

2
a
mπ
− (A
12
− A
66
)
m
aπ

,
a
9
=
3
128

(A
11
+ E

1
A
1
/s
1
)
m
4
b
a
3
+2

A
12
+
2
3
A
66

(mn)
2
ab
+(A
22
+ E
2
A
2

/s
2
)
n
4
a
b
3

+
+
H
1
(a
6
a
4
− a
2
a
3
)+H
2
(a
1
a
8
− a
5
a

4
)
a
1
a
6
− a
2
a
5
,
a
10
=
8
9

(E
1
A
1
/s
1
)z
1

m
a

3

b
nπ
2
+(E
2
A
2
/s
2
)z
2

n
b

3
a
mπ
2

+
H
1
(a
3
a
6
− a
2
a

7
)+H
2
(a
1
a
7
− a
3
a
5
)+H
3
(a
6
a
4
− a
2
a
8
)+H
4
(a
1
a
8
− a
5
a

4
)
a
1
a
6
− a
2
a
5
,
a
11
=
1
4

(D
11
+ E
1
I
1
/s
1
)
m
4
b
a

3
+2(D
12
+2D
66
)
(mn)
2
ab
+(D
22
+ E
2
I
2
/s
2
)
n
4
a
b
3

+
H
3
(a
3
a

6
− a
2
a
7
)+H
4
(a
1
a
7
− a
3
a
5
)
a
1
a
6
− a
2
a
5
,
H
1
= −
16
9


m
a

2
b
nπ
3
(A
11
+ E
1
A
1
/s
1
)+(A
12
+2A
66
)
n
bπ
3

,
H
2
= −
16

9

n
b

2
a
mπ
3
(A
22
+ E
2
A
2
/s
2
)+(A
12
+2A
66
)
m
aπ
3

,
H
3
= −

1
4
(E
1
A
1
/s
1
)z
1
m
3
b
a
2
π
,H
4
= −
1
4
(E
2
A
2
/s
2
)z
2
n

3
a
b
2
π
·
4. Finite element method
Based on strain energy principle, the finite element method has built equilibrium
equation of the plate [7]

K

{q} = {F }. (14)
For building equation (14), we need to build matrix [K], which are built from
stiffness matrix of element [K
e
].
Accordingto[4]forbuilding[K
e
], we can see multilayered reinforced composite
plates, which are a system of unreinforced plates and beams. From this opinion, the
building stiffness matrix [K
e
] of reinforced plates is difined
[K
e
]=[K
t
e
]+K

d
e
], (15)
where: [K
t
e
], [K
d
e
]arestiffness matrices of the plate and beam elements.
* Stiffness matrix of the plate elements

K
t
e

The relation between deformation and node displacement is of the form
{ε
t
} =

B
t

{q
e
}, (16)
where

B

t

=

B
t
0

+

B
t
L

. (17)
Non-linear analysis of multilayered reinforced composite plates 21
in which is the same matrix as in linear infinitesimal strain analysis,

B
t
L

is the large
strain matrix depending on {q
e
}.
Thus
d{ε
t
} = d


B
t

{q
e
}

=

B
t

d{q
e
} + {q
e
}d

B
t

. (18)
Because

B
t
L

depends on {q

e
}, d

B
t

= d

B
t
L

and {q
e
}d

B
t

=

B
∗
L

d{q
e
},then
(18) become
d{ε} =


B
t

+

B
∗
L

d{q
e
}, (19)
where

B
∗
L

has the same form as d

B
t
L

but instead of dq
i
we put q
i
d


B
t
L

=

[0] d

B
t
Lu

[0] [0]


B
∗
L

=

B
L

. (20)
According to [7], the sum of internal and external forces is difined as follows

Q


=

S

B
t

T

σ
t

dS − {F } (21)
in which {F } - external forces, from (21) we receive
d

Q

=

S
d

B
t

T

σ
t


dS +

S

B
t

T
d

σ
t

dS. (22)
Otherwise, from (1) we obtain
d

σ
t

=

D
t

d

ε
t


−

D
t


B
t

+

B
∗
L


f{q
e
}. (23)
Substituting (23) into (22) yields
d

Q

=

S
Dd


B
t

T

σ
t

dS +

S

B
t


D
t


B
t

+

B
∗
L



dSd{q
e
}. (24)
Because d

B
t

T
= d

B
t
Lu

T
and

B
∗
L

=

B
t
L

,onecanget
d


Q

=

S
d

B
t
Lu

T

σ
t

dS +

K

, (25)
in which

K

=

S



B
t

T

D
t


B
t

+

B
t

T

D
t


B
T
L


dS. (26)

Substituting

B
t

from (17) into (26) and a fter some operations we obtain

K

=

K
t
0e

+

K
t
Le

, (27)
22 KhucVanPhu,NguyenTienDat
where

K
t
0e

isthesamestiffness matrix as in linear infinitesimal s train analysis. For

elements of the plate

K
t
0e

=

S

B
t
0

T

D
t


B
t
0

dS. (28)
Matrix

K
t
Le


is the large displacement matrix, which can be defined a s follows

K
t
Le

=

S

2

B
t
0

T

DF
t


B
t
L

+

B

t
L

T

D
t


B
t
0

+2

B
t
L

T

D
t


B
t
L



dS. (29)
The first term of equation (25) can generally be written as:

S
d

B
t

σ
t

dS =

K
t
σe

d{q
e
} (30)
where

K
t
σe

is a symme tric matrix which dependens on the stress level. This matrix is
known as initial stress matrix or geometric matrix.
According to [7] we have


K
t
σe

=

[0] [0]
[0]

K
u
σe


,
with

K
u
σe

=

S

G
t

T


N
t
x
N
t
xy
N
t
xy
N
t
y


G
t

dS, (31)
in which

G
t

=
⎡
⎢
⎢
⎣
∂N

t
u
1
∂x
∂N
t
u
2
∂x
··· ···
∂N
t
u
11
∂x
∂N
t
u
12
∂x
∂N
t
u
1
∂y
∂N
t
u
2
∂y

··· ···
∂N
t
u
11
∂y
∂N
t
u
12
∂y
⎤
⎥
⎥
⎦
(32)
Thus, for element of the plate we obtain
d

Q

=


K
t
0e

+


K
t
Le

+

K
t
σe


d{q
e
} =

K
t
e

d{q
e
}
and stiffness matrix of the element of the plate

K
t
e

=


K
t
0e

+

K
t
Le

+

K
t
σe

. (33)
*Stiffness matrix of element of the beam

K
t
σe

Using two-noded element of the beam with three degree of freedom at each node

u
d
1
,w
d

1
, ϕ
d
1
,u
d
2
,w
d
2
, ϕ
d
2

T
Non-linear analysis of multilayered reinforced composite plates 23
Acording to [8] for a element of multilayered composite beam, which works in the
elastic state the relation between internal force and deformation are of the form

σ
d

=

D
d

{ε
d
}, (34)

in which
{σ
d
} =

N
d
x
M
d
y

T
;

D
d

=

[A
d
][B
d
]
[B
d
][D
d
]


(35)
The matrices [A
d
], [B
d
], [D
d
]aredefinedin[8],{ε
d
} the deformation of point of
the middle surface

ε
d

=

du
dx
+
1
2

dw
dx

2
−
d

2
w
dx
2

T
(36)
or we can be rewritten in the form

ε
d

=


ε
d
0m


ε
d
0u


+


ε
d

L

{0}

(37)
Expressing the defomation with noded diplacement as follows

ε
d

=

B
d

q
d
e

, (38)
where

B
d

=

B
d
0


+

B
d
L

. (39)
Similar to the multilayered composite plate, we obtain stiffness matrix of element
of beam as follow

K
d
e

=

K
d
0e

+

K
d
Le

+

K

d
σe

, (40)
where

K
d
0e

=

L

B
d
0

T

D
d


B
d
0

dx


K
d
Le

=

L

2

B
d
0

T

D
d


B
d
L

+

B
d
L



D
d


B
d
0

+2

B
d
L


D
d


B
d
L


dx (41)

K
d
σe


=

L

G
d

T

N
d

G
d

dx.
5. Numerical examples
Let’s consider a simply supported stiffened rectangular symmetrical composite plate:
a =0.8m; b =0.5 m. The materials of the plate are composed by Thornel 300 graphite
fibers and Narmco 5205 Thermosetting Epoxy resin [5], the properties of these materials
are: E
1
= 127.4GPa; E
2
=13GPa;G
12
=6.4GPa; ν
12
=0.38; The plate has six layers:

[45/ − 45/90/90/ − 45/45]; thickness of each la yer: t =0.5 mm; The laminate plate is re-
inforced by longitudinal and transversal stiffeners, which were made of CPS material, the
stiffeners have the same sizes, as follows: b
g
× h
g
=4mm × 6 mm; Spacing of longitudinal
and transverse stiffeners is: s
1
= s
2
=0.1m.
The results according to two methods are presented on t he Figs 2, 3, 4.
24 KhucVanPhu,NguyenTienDat
.
Fig. 2. Displacement of cut trace, going over the center of plate and paralled with x axis
Fig. 3, Relation between displacement and external force
Fig. 4. Effecty of thickness of the plate
Non-linear analysis of multilayered reinforced composite plates 25
Conclusions
The results by the Bupnov - Galerkin method agree qualitatively with those by the
Finite element method, but the results by the Bupnov - Galerkin method are smaller than
that by the FEM. This difference can be reduced, if we take more number of terms in the
double Fourier series of the displacemen ts.
Displacement of the non-linear analysis of multilaye red reinforced CPS plates are
smaller than that of multilayered unreinforced CPS plates. This means, the hardness of
multilayered reinforced CPS plates is bigger than that of multilaye red unreinforced CPS
plates.
Displacement and stress of the linear analysis of multilayered CPS plate are directly
proportional to external force, but displacement and stress of the non-linear analysis of

multilayered CPS plate aren’t direct by proportional to external force. If external forces are
small, displacement in non-linear problem is approximately equal with linear displacement.
When external force increases, the difference between linear and non-linear analysis also
gets increased. This means non-linear analysis is exacter than linear analysis.
If the thickness of the plate is increased, the difference between reinforced and
unreinforced plate also gets reduced, so the stiffener takes effect for thin plates.
Acknowledgements. The author would like to thank Professor Dao Huy Bich for helping
him to complete this work. This publication is p artly supported by the National Council
for Natural Sciences.
References
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