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 Total Downloads : 1729
 Authors : Ayisha P. S. E, M. C. Narasimhan
 Paper ID : IJERTV2IS50756
 Volume & Issue : Volume 02, Issue 05 (May 2013)
 Published (First Online): 18052013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Static Analysis Of Laminated Composite Circular And Annular Plates
Ayisha Powmya.S.E. Department of Built Environment,
Muscat College, Oman
Mattur C. Narasimhan Department of Civil Engineering, National Institute of Technology Karnataka, India
Abstract
Analytical solutions are presented for the static deflection analysis of laminated polar orthotropic circular and annular plates. The analysis is based on the application of the first order shear deformation theory employed. Three linear partial differential equations for axisymmetric deformations are written in terms of displacements u, and w. Chebyshev collocation method is employed for the solution of the evaluation of static deflection problem. Numerical results are presented to show the validity and accuracy of the proposed method. Results of parametric studies conducted to evaluate the effect of parameters like orthotropic ratios, number of layers, lamination sequences and boundary conditions, on the response of laminated polar orthotropic circular and annular plates are also presented.

Introduction
Fiber reinforced laminated composites are being increasingly used in modern engineering applications due to their high specific strength and high specific modulus. The increased application of laminated composites in the primary components in structures like spacecrafts, high speed aircrafts, missiles, gas turbines, etc. are due to the number of advantages they offer in structural, operational, production and/or maintenance aspects.
The use of advanced composite materials for structural elements brings in the need to develop new analytical and design techniques. With the present level of their application being what it is, it becomes a necessity to develop better mathematical models to predict the mechanical behavior of structural elements made up of such materials, under service loads.
In the present work, it is proposed to study, the static deflection analysis of laminated polar orthotropic circular and annular plates by Chebyshev collocation
method. A first order shear deformation theory is used in terms of u, and w. These field variables are expanded in polynomials and then orthogonal point collocation method is used to discretise the governing equations. To demonstrate the convergence of the method, numerical results are presented for clamped and simply supported isotropic and polar orthotropic circular and annular plates. The validity of the analytical solution is confirmed by comparing with data obtained from open literature.

Mathematical formulation
The laminated plate of constant thickness h is composed of polar orthotropic laminae stacking symmetrically or antisymmetrically about the middle surface of plate. Plate coordinates (r, , z) used are as shown in figure 1, where u, v, w denote the displacements of any point of the plate in the corresponding r, , z directions.
Figure 2.1 Geometry of a nlayered laminate
In this study, first order shear deformation theory is employed and the general displacement field is assumed in the form
u (r, , z) = uÂº(r, ) + z 1(r, ),
v (r, , z) = vÂº(r, ) + z 2(r, ),
w (r, , z) = w(r, ) (1)
where uÂº, vÂº, w denote the displacements of any point
Nr
Mr
N M
N M
h/2
r
on the middle surface and 1, 2 are the rotations of the normal to the midplane about , r axes respectively.
Strain displacement relations are of the following form
N r Mr
(1, z) dz ,
h/2 r
in polar coordinates
Qr h/2 rz
z , z ,
h/2
dz
(6)
r
r
r r
r
r
r
r
z ,
rz
rz
rz
,
z z
(2)
Q
z
where the reference surface strains and curvatures are given by,
where z is the distance of the lamina from the middle plane.
Substituting the stress strain relations, we have the
=
u
,
=
1 v
u ,
constitutive matrix as
r r
r
Nr
A11
A12
0 B11
B12 0 0
0 r
1 u
v
N
N
A
A
12
A 22
0 B12
B22 0 0
0
Nr 0
0 A 66 0
0 B66 0
0
r =
r
v ,
r
r
Mr B11
B12
0 D11
D12 0 0
0 r
(7)
M
M

w
1 w
B12
B22
0 D12
D 22 0 0
0
= ,
= ,
rz 1 r
z 2
r
Mr 0
0 B66 0
0 D66 0
0 r
Q
Q
0 0
0 0 0
0 K 2 A
0
1
1 2
Qr
44 rz
r =
r ,
=
r
1 ,
0 0
0 0 0 0
0 K 2 A55
z
= 1 1
2
(3)
where, A =
n
(C ) (z z ) ,
r
r
2
r
ij
k1
ij k k
k1
i, j = 1, 2,6,4,5
According to the shear deformation theory, the
1 n 2 2
constitutive equations for the kth layer of a polar orthotropic laminated plate can be written in the
Bij =
(Cij ) k (z k zk1 ) ,
2
2
k1
following form in polar coordinates.
i, j = 1, 2,6
D = 1 n (C
) (z 3 z3 ) ,
3
3
r
C11
C12
0 r
ij
k1
ij k k
k1
C12
C22
0 ,
where i, j = 1, 2,6 (8)
r
0
0 C66 r
Aij are the extensional stiffnesses, Bij are the bending
rz
C44
0 rz
extension coupling stiffnesses, Dij are the bending
0
C
(4)
stiffnesses and K2 is the Shear correction factor
z
55 z
introduced to account for nonuniform distribution of the transverse shear strains through the thickness of the
where C11 =
E
Er
1 r
, C12 =
r E
1 r
= Er ,
1 r
plate, which is taken as Â²/12.
n is the total number of layers in the laminate.
The stress resultants and stress couples defined in (6)
C22 =
1 r
and (7) must satisfy the following equilibrium equations (Ravichandran, 1989)
Nr 1 Nr (N r N ) 0 ,
C44 = Grz , C55 = Gz, C66 = Gr (5)
Where E and E are Youngs moduli of elasticity in r
r r r
Nr N 2 Nr 0
and d
r
irec
tions.
r and
are Poissons ratios in r and
r r r
directions. Gr, Gz and Grz are the shear moduli in the respective planes.
The stress resultants acting on a laminate are obtained as:
Qr 1 Q Qr q(r, ) 0 ,
r r r
Mr 1 Mr (M r M ) Q
r r r r
Mr
1 M
Mr
1 2 U
1 U U
2
Q
(9)
b11 (a b)2 2 ( a (1 )b) (a b) b22 ( a (1 )b) 2
r r r
1 2
1
d11 (a b)2 2 ( a (1 )b) (a b) d 22 ( a (1 )b) 2
If the analysis is now restricted to only axisymmetric case, the deformations are symmetrical about zaxis,
a 44 W
(12)
thus the stresses and strains are independent of and
h 2
r
z
0 , vÂº = 0, 2 = 0 and also (
) = 0
2.1 Polynomial series solution by collocation at
The governing equations for axisymmetric case then can be written in terms of midplane displacements as,
Chebyshev zeroes
The Chebyshev polynomials Tn (x) are a class of
2 u u u 2
orthogonal polynomials, which are defined as,
A11
A22 B11 1 1 B22 1 0
r 2
r r r 2 r 2
r r r 2
Tn (x) cosn
K2A
1

1
2w
w
q(r) 0
Where cos x
, 1 x 1
(13)
44 r
2 u
r
u
r2
u
r r
2
These provide a solution of the second order differential equation,
B11
B22 D11 1 1
2 " ' 2
r2
r r r2 r2
r r
(1 x
) Tn x Tn n
Tn 0
(14)
1 2
w
(10)
where ( ) '
d ( )
D22 r 2
K A44 1
r
dx
In many applications, it is advantageous to redefine the
r
r
For convenience, the following dimensionless
polynomials in the range
0 1 . The shifted
parameters are introduced
Chebyshev polynomial
T* ()
of degree r in the range
u (a b) w
1 (a b)
0 1
is derived from the polynomial
Tr (x)
U ,
h 2
W ,
h
,
h
{1 x 1} using a linear transformation,
r b ,
a b
q (a b) 4
p ,
ET h 4
2
K A
K A
a44 44 ,
ET h
1 x 1
2
i.e., Tr () cosrt
where cost (2 1)
(15)
(16)

Aij ,
T
T

Bij ,
T
T
d Dij
(11)
The shifted polynomials satisfy the recurrence relations
E
E
T
T
ij h
ij E h 2
ij E h 3
*
T
T
r1
()
*

T

T
r1
() 2 (2 1) T* ()
(17)
r
r
and the orthogonality conditions
where i, j = 1, 2,6
Here, ET is the reference Youngs modulus. In case of
1 T () T ()
0
= /2
for
for
m n
m n 0
(18)
laminated composites with layers of same material, ET is taken to be the Youngs modulus in the direction
0 (1 )
for
m n d
m n d
m n 0
transverse to fiber direction.
Thus the governing differential equations of the plate can now be expressed in dimensionless form as,

., the shifted Chebyshev polynomials are orthogonal with respect to the weighing function
1
1 2 U 1 U U
W()
1/2
(19)
a11 (a b)2 2 ( a (1 )b) (a b) a 22 ( a (1 )b) 2
{(1 )}
1 2
1
Any continuous function g() in the interval 0 1 ,
b11 (a b)2 2 ( a (1 )b) (a b) b22 ( a (1 )b) 2 0
1
1 2 W
1 W
can be represented by a infinite series of the form
a 44 (a b) ( a (1 )b) (a b) 2
( a (1 )b)
g * *
h 2
g()
T () grTr () 2 r1
(20)
p 0
(a b)3
where gi
(i=0,1,2,.) are the coefficients to be
determined so as to obtain a best possible fit. The series in eqn.(20) is a fast converging one and good
approximation is obtained by taking a finite number of
N1 (n 1)(n 2) n3
(n 1) n2
terms in the above series. i.e.,
d11
n1
n
(a b)2
n
( a (1 )b) (a b)
g * N *
N1

d

n1
n
g()
T () gr Tr ()
N
N
2
2
r1
(21)
22
n1
( a (1 )b)2
gr r0
T* ()
a44
p
N1
n
n1
n1 (n 1)Wnn2 0
(26)
r
r
Where, the plus sign indicates that the first term of the series is to be halved.
For a known function g(), using the orthogonality conditions (eqn.18), the coefficients gr can be calculated as
There are 3(N+1) constants to be determined. The stipulation of the three boundary conditions at each edge provide six equations. 3(N1) additional equations are obtained by forcing the satisfaction of each of the
2 1 g() T* ()
, 0 r N
(22)
three differential equations at the (N1) zeroes of
gr
r d
T () ,
0 1
– the (N1)th degree shifted
0
(1 )
(N1)
r
r
The shifted Chebyshev polynomial T*() of degree r
Chebyshev Polynomial.
The Nth degree Chebyshev Polynomial T
has N
has r zeroes in the range 0 1 , which are used as
N
zeroes at
the collocation points in the present study.
1 (2 i 1)
The (N1) collocation points are taken at the zeroes of the Chebyshev polynomial. For static loads, the radial displacement, deflection and rotation are expanded as
i 2 1 cos 2N
i = 1, 2, N (27)
polynomials in . Thus forcing the satisfaction of each of the three
U(), (), W()
Un , n , Wnn1
(23)
differential equations at the (N1) zeroes of
T
T
(N1)
() ,
N1
N1
n1
0 1
along with the stipulation of the three
Substituting eqn. (23) in eqn. (12), the following equations are obtained.
boundary conditions at each edge results in a set of simultaneous equations of the form,
[L11] {U} [L12 ] {} [L13] {W} {0}N1 (n 1)(n 2)Unn3 (n 1)Unn2
[L } {U} [L ] {} [L ] {W} {p}11
11

n1
N1
(a b)2
Unn1
( a (1 )b) (a b)
21
[L31] {U} [L22
32 ] {} [L
23
33] {W} {0}
(28)
a 22
n1 ( a (1 )b) 2
Where { U}T
{U , U
,……………..U }
N 1 (n 1)(n 2)
n 3
(n 1)
n 2
1 2 n1

n n
{}T {1 , 2 ,……………… n1}
11 n 1
N 1
(a b)2
n 1
( a (1 )b) (a b)
{ W}T {W1, W2 ,………..Wn1}
(29)

b
n 0
(24)
It can be seen that [L13] = [0] and [L21] = [0].
22 n 1 ( a (1 )b) 2
(n 1) n2
n1
These algebraic equations are then solved using LU
n n
Decomposition method. (William H. Press et al, 2002)
a N1
(a b)
( a (1 )b)
The coefficients are then substituted in eqn. (23) to get
n1 n n
n1 n n
44 (n 1)(n 2)W n3 (n 1)W n2
radial displacement, transverse deflection and rotation at any point inside the plate.
(a b)
p
p 0
(a b)3
( a (1 )b)
(25)
N1 (n 1)(n 2)Unn3
(n 1)Unn2



Convergence and comparison studies
b11
2
n1
N1
(a b)
Unn1
( a (1 )b) (a b)
In order to validate the procedure implemented, static analysis has been conducted on isotropic plates. The
b22
n1 ( a (1 )b) 2
convergence analysis of the Chebyshev collocation method for static deflection of circular and annular
plates is first carried out. Table 3.1 shows the results of analysis of an isotropic circular plate subjected to axisymmetric pressure q, with increasing number of terms used in each of the series in eqn. (23). It can be observed that sufficiently accurate, converged solution is obtained taking 10 12 terms used in each of the series.
Isotropic circular and annular plates subjected to various types of pressure loads are now analyzed. Table 3.2 and Table 3.3 shows the results of static deflection and rotation of the transverse normal in radial direction, obtained using the present solution. It can be observed that the present solution provides results comparing reasonably well with the results obtained by CPT [Warren C. Young (1989)]. The deviations between the results can be attributed in fact to effect of shear deformations as the plates considered herein are moderately thick plates. (a/h = 10)
Table 3.1 Static deflection of isotropic circular plates – convergence study
a = 100 mm, h=10 mm, = 0.3, q = 100 N/mm2, E = 2.10 kN/mm2
Boundary condition: clamped edges
2 w
W 10
h
Number of terms in Chebyshev series
N
Maximum deflection W
4
0.8438
6
0.8437
8
0.8438
10
0.8437
12
0.8437
Table 3.2 Static solution for isotropic circular plates
a/h = 10 , = 0.3
Type of load
Boundary Condition
Present (FSDT)
Warren C. Young (1989) (CPT)
Uniformly Distributed Load
Clamped
Kymax
0.01622
0.01563
Simply Supported
Kymax Kmax
0.06430
0.09615
0.06370
0.09615
Uniformly Varying Load
Clamped
Kymax
0.00693
0.00667
Simply Supported
Kymax Kmax
0.03257
0.05128
0.03231
0.05128
Parabolically Varying Load
Clamped
Kymax
0.00362
0.00347
Simply Supported
Kymax Kmax
0.01964
0.03205
0.01949
0.03205
Wmax = Kymax
q a 4 D
q a3
max = Kmax
D
Table 3.3 Static solution for isotropic annular plates
a/h = 10 , b/a = 0.1, = 0.3
Type of load
Boundary Condition
Present (FSDT)
Reference (CPT)*
Uniformly Distributed Load
C C
Kymax
0.0020
0.0018
Sa C
Kymax
0.0025
0.0025
Kb
0.0100
0.0135
Ca S
Kymax
0.0044
0.0040
Ka
0.0140
0.0147
S S
Kymax Kb
0.0061
0.0196
0.0060
0.0264
Ka
0.0176
0.0198
Uniformly Varying Load
C C
Kymax
0.0010
0.0009
Sa C
Kymax
0.0013
0.0013
Kb
0.0043
0.0059
Ca S
Kymax
0.0026
0.0024
Ka
0.0087
0.0093
S S
Kymax
0.0035
0.0034
Kb Ka
0.0114
0.0108
0.0137
0.0119
Parabolically Varying Load
C C
Kymax
0.0006
0.0005
Sa C
Kymax
0.0007
0.0007
Kb
0.0025
0.0031
Ca S
Kymax Ka
0.0017
0.0060
0.0016
0.0064
S S
Kymax
0.0022
0.0022
Kb
0.0074
0.0083
Ka
0.0073
0.0080
Wmax = Kymax
q a 4 D
, = K
q a3 D
* Warren C. Young (1989)
a Boundary condition at inner edge of the plate

Parametric study
In order to bring out the effect of shear deformation in case of plates undergoing axisymmetric deformations, different shear moduli are considered.
0.5 E
assumed to be appropriate even in the case of composite plates and the same is followed here. It can be seen that, the increase in orthotropy ratio decreases the displacements for all the lamination schemes considered herein. Also, in case of annular plates, among the different lamination sequences, the least deflection is observed when the laminate is made up of
(1) Grz
0.75 E
(4) Grz

0.375 E
laminae in which fibers are arranged in 0Âº only. Surprisingly, in case of circular plates with simply supported boundary condition, the least deflection is

Grz (5) Grz
1
0.625 E
1
0.25 E
observed when the laminate is made up of laminae in which fibers are arranged in 90Âº only. In case of unsymmetric and antisymmetric laminates, since the

Grz
1
(6) Grz
1
coupling rigidity [Bij] exists, radial dsplacements are
Table 4.1 shows the results of analysis of an isotropic plate undergoing axisymmetric deformations. For very large value of shear modulus, the results of the present work are comparable with the results given by Warren
C. Young (1989). It can also be observed from this table that the results with different values of shear moduli considered (in the narrow range) are very close to each other and to the value due to CPT, which neglects shear deformation.
Table 4.2 and Table 4.3 show the results of a study conducted to know the effect of orthotropy ratio on the transverse deflection and radial displacement of laminated polar orthotropic circular and annular plates respectively. The need to select proper shear correction factor K2 in the analysis makes the first order shear deformation theory of the present work somewhat
also observed.
Table 4.1 Effect of shear deformation on the deflection of plate
a/h = 10, Boundary condition: clamped edges
G (1 ) E
Kymax
Present (FSDT)
(CPT)*
0.75
0.625
0.5
0.375
0.25
0.01562
0.01602
0.01610
0.01622
0.01642
0.01682
0.01563
q a 4
empirical. There are different recommendations for the selection of shear correction factors. Quite often the value of 2/12 recommended for isotropic plates is
Wmax = Kymax
D
* Warren C. Young (1989)
Table 4.2 Effect of orthotropy ratio on displacements of laminated polar orthotropic circular plates
a = 100 mm, h = 10 mm, r = 0.31, Grz/E = 0.253, q = 100 N/mm2 ,
BC
Displacement
Er/E
5
10
15
30
Lamination scheme
90Âº
0.0582
0.0408
0.0327
0.0223
90Âº/0Âº/90Âº
0.0559
0.0378
0.0297
0.0195
90Âº/0Âº/0Âº/90Âº
0.0516
0.0333
0.0256
0.0166
W
0Âº/90Âº
0.0469
0.0304
0.0234
0.0153
Clamped
90Âº/0Âº/90Âº/0Âº
0.0438
0.0271
0.0206
0.0136
0Âº
0.0428
0.0250
0.0191
0.0125
0Âº/90Âº/0Âº
0.0417
0.0248
0.0190
0.0125
0Âº/90Âº/90Âº/0Âº
0.0403
0.0232
0.0186
0.0123
U
0Âº/90Âº
0.0172
0.0056
0.0040
0.0027
90Âº/0Âº/90Âº/0Âº
0.0026
0.0018
0.0014
0.0008
0Âº
0.6379
0.4349
0.4204
0.4090
0Âº/90Âº/90Âº/0Âº
0.2270
0.0494
1.2854
0.3627
0Âº/90Âº/0Âº
0.1846
0.4104
0.3639
0.2728
W
0Âº/90Âº
0.1933
0.1820
0.1328
0.0796
Simply
90Âº/0Âº/90Âº/0Âº
0.1812
0.1063
0.0759
0.0425
Supported
90Âº/0Âº/0Âº/90Âº
0.1459
0.0835
0.0597
0.0340
90Âº/0Âº/90Âº
0.1418
0.0815
0.0585
0.0336
90Âº
0.1404
0.0810
0.0584
0.0337
U
0Âº/90Âº
0.0914
0.0498
0.0334
0.0193
90Âº/0Âº/90Âº/0Âº
0.0098
0.0074
0.0057
0.0033
w
w
W U
h
u a p
Table 4.3 Effect of orthotropy ratio on displacements of laminated polar orthotropic annular plates
a = 100 mm, h = 10 mm, b = 10 mm, r = 0.31, Grz/E = 0.253, q = 100 N/mm2
BC
Displacement
Er/E
5
10
15
30
Lamination scheme
90Âº
1.2031
1.0107
0.8956
0.7150
90Âº/0Âº/90Âº
1.1303
0.9063
0.7823
0.6038
90Âº/0Âº/0Âº/90Âº
0.9981
0.7600
0.6437
0.4938
W
0Âº/90Âº
0.8386
0.6513
0.5634
0.4489
C C
90Âº/0Âº/90Âº/0Âº
0.7408
0.5472
0.4704
0.3831
0Âº/90Âº/90Âº/0Âº
0.5934
0.4470
0.3941
0.3375
0Âº/90Âº/0Âº
0.5746
0.4354
0.3857
0.3327
0Âº
0.5673
0.4309
0.3825
0.3309
U
0Âº/90Âº
0.2743
0.2283
0.1888
0.1236
90Âº/0Âº/90Âº/0Âº
0.1108
0.0796
0.0610
0.0360
90Âº
3.9604
1.7014
1.4370
1.0204
90Âº/0Âº/90Âº
2.9180
1.6621
1.2933
0.9317
90Âº/0Âº/0Âº/90Âº
2.3716
1.5949
1.2860
0.8963
0Âº/90Âº
1.7369
1.1620
0.9406
0.6936
90Âº/0Âº/90Âº/0Âº
1.7061
1.1210
0.9014
0.6576
W
0Âº/90Âº/90Âº/0Âº
1.3923
0.9238
0.7560
0.5806
S – S
0Âº/90Âº/0Âº
1.3455
0.8932
0.7346
0.5722
0Âº
1.3255
0.8814
0.7265
0.5692
U
0Âº/90Âº
0.3120
0.3015
0.2637
0.2036
90Âº/0Âº/90Âº/0Âº
0.1333
0.1162
0.0954
0.0664
W 10


w h
U 10


Conclusions
Studies have been conducted to understand the static deflection characteristics of composite circular and annular plates by the Chebyshev collocation technique using first order shear deformation theory. Convergence tests were conducted for the Chebyshev collocation technique and it can be seen that there is excellent convergence even when we take four or six terms in the series for the problems considered. Further, numerical results have been worked out for isotropic and polar orthotropic laminated circular and annular plates with different combinations of clamped and simply supported boundary conditions.
From the results it can be seen that, the increase in orthotropy ratio decreases the displacements for all the lamination schemes considered. Also, in case of annular plates, among the different lamination sequences, the least deflection is observed when the laminate is made up of laminae in which fibers are arranged in 0Âº only. Surprisingly, in case of circular plates with simply supported boundary condition, the least deflection is observed when the laminate is made up of laminae in which fibers are arranged in 90Âº only. In case of unsymmetric and antisymmetric laminates, since the coupling rigidity [Bij] exists, radial displacements are also observed.

References

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