 Open Access
 Total Downloads : 306
 Authors : V. Subrahmanyam, K. S. B. S. V. S. Sastry, V. V. Ramakrishna, Y. Dhanasekhar
 Paper ID : IJERTV3IS11001
 Volume & Issue : Volume 03, Issue 01 (January 2014)
 Published (First Online): 28012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Analysis of Composite Laminate Plates under UD Load using CLPT

Subrahmanyam K. S. B. S. V. S. Sastry V. V. RamaKrishna Y. DhanaSekhar
Assoc. Professor, Kakinada Institute of Tech. & Science,
Tirupathi (V), Divili533433.
Assoc. Professor, Sri VasaviEngg.
College, Pedatadepalli, Tadepalligudem.
Asst. Professor, KITS, Divi.li A.P., India.
Asst. Professor, KITS,Divili. A.P., India.
Abstract
In this paper the strength of a composite material configuration is obtained from the properties of the constituent laminate by using Classical Laminate Plate Theory. For the analysis simply supported 4 Ply Orthotropic laminate plate of Boron/Epoxy with uniformly distributed load is considered. CProgramme was developed to calculate the deflection, stress, strain and failure load of the laminate of different configurations. The strengths are calculated for Uniformly Distributed Loads by varying the dimensions and configurations of laminate. From the analysis it is found that as the number of layers of a laminate increases in a given thickness, the strength of the laminate increases. It is also found that the location and angles of the principle material direction affect the strength of rectangular laminate, but it is not affecting the strength of a square orthotropic laminate plate.
Keywords:Composite, Laminate, Classical Laminate Plate Theory.
boat bodies, propellershaft and hulls for small boats,civilian and military aircraft components, rocket components, heat shields for satellites and hypersonic aeroplanes, sports goods, musical instruments, safety protections etc.
Composite materials can be broadly classified into phase composites and layered composites. Layered composites are called laminates. Depending on the number of layers they are classified as Unilayer composites and multilayer composites. A composite laminate is two or more laminae bonded together to act as an integral structural element. Laminae principal material directions are oriented to produce a structural element capable of resisting load in several directions.
Each layer of a unidirectional composite is known as layer, ply or lamina. The behaviourof aLaminae is the basic building block for laminate construction. [1]
Z
Loading Axis

Introduction:
A composite is a combination of two or more dissimilar materials on a macroscopic scale, with the aim to get properties better than the individual constituentmaterialslike Strength,
stiffness, corrosion resistance, wear resistance,
Y
Transverse Axis
X
Longitudinal Axis
fatigue life thermal insulation, thermal conductivity, acoustical insulation, temperature dependent behavior etc.The Properties of a composite material depend on the properties of the constituents, geometry, and distribution of the phases. One of the most important parameters is the volume (or weight) fraction of reinforcement, or fiber volume ratio. We can find the applications of composite in automobiles as vehicle bodies, engine components, racing
Fig.1 Schematic diagram of unidirectional composite.
It consists of parallel fibers embedded in a matrix. The direction parallel to fibers is known as longitudinal direction (xaxis). The direction perpendicular to the fibers is called the transverse direction (along yaxis).
A lamina has the strongest properties in the longitudinal direction.
Nomenclature
:Area of plate, m2
:Extensionalstiffnesses
:Length of plate in longitudinal direction, m
:Couplingstiffnesses
Volume fraction of the matrix
A
Aij A
Bij b
(Vm) =
m
vm f
vm
c
—(1)
Similarly, volme fraction of the fibres
(Vf)
v f
—(2)
in transverse
plate
of
:Length
c
direction,m
Dij :Bendingstiffnesses , KNm E11

Theory:

Structural composites:
E1:Longitudinal youngs modulus of a laminae, Gpa
E2 :Transverse youngs modulus of a laminae, Gpa
:Youngs modulus of fiber, Gpa
:Youngs Modulus of matrix,Gpa
:Stress in longitudinal direction,
:Stress in transverse direction,
:Shear stress,
:Uniformly distributed load, N/m2
:Reducedstiffnesses , GPa
:In plane shear strength , Gpa
:thickness of each ply of the laminate, mm
:Longitudinal tensile strength, Gpa
:Longitudinal compressive strength, Gpa
:Transverse compressive strengths, Gpa
:Transverse tense strengths, Gpa
:Fiber volume ratio in longitudinal direction
:Fiber volume ratio in transverse direction
:Matrix volume ratio in longitudinal direction
:Matrix volume ratio in transverse direction
:Majorpoisson Ratio
:Minorpoisson Ratio
:Deflection, mm
:Density, kg/m3
E22 or
Ef Em fx fy fxy P
Qij S
t
X4
Xc
Yc Yt
f1
f2
m1
m2
12
21
or
Structural Composites are used for structural applications. These are classified into Laminar andSandwichcomposites.Laminar Composites are twodimensional sheets layered one on another.[1]
Fig.2Laminar composites.

Orthotropy or Anisotropy :
The materials which of their properties at a point vary with direction.The composite materials are Orthotropic in nature i.e., they exhibit different properties in three different directions. These three mutually perpendicular axes, called principal axes of material symmetry.

Laminate designations:
Unidirectional 6ply : [0/0/0/0/0/0] = [06] Crossplysymmetric : [0/90/90/0] = [0/90]
s

Deflection of simply supported laminated plates under uniformly distributed lateral load:
Consider the general Class of laminated rectangular Plates that are simply supported along edges x = 0, x = a, y = 0 and y = b and subjected to a distributed lateral load p(x, y).
The lateral load can be expanded in a double Fourier series
P(x,y)=
Pmn sin
mx
.sin
ny
—(3)
P(x,y)=
m1 n1 a b
16po .a 4
mn
(1) 2 1
Z 6
mnD m4 2D

2D
m2 n2 D
n4
m1,3.. n1,3..
11 12 66
—(7)
22
The maximum deflection occurs at center of plate i.e, at x = a/2, y = b/2 ThereforeCenterDeflection () =
X 16p
1 sin
mx
.sin
ny
b Y o
mn a b
a 6
m1,3.. n1,3..
m 4
m 2 n 2
4
n
D 2D
2D D
11 a
12 66 a b
—(8)
22 b
Fig.3 showing simply supported rectangular plate subjected to U.D. Load
2.4 Specially orthotropic laminates :
2.6 Stress Strain Relations :[2]
The stressstrain relations for an anistropic body can be written as in contracted nations as
A specially orthotropic laminate has either a single layer of a specially orthotropic
1
2
c11
c12 c22
c13 c23
c14 c24
c15 c25
c16 1
c
26 2
material/multiple specially orthotropic layers that
are symmetrically arranged about the laminate middle surface.
3
4
c33
c34
c44
c35
c45
c36 3
c
46 4
5
sym
c55
c56 5
Laminate stiff nesses consist solely of A11, A12, A22, A66 , D11, D12, D22, D66
There is neither shear/twist coupling nor bending extension, coupling exists. Thus, for plate problems, the lateral deflections are described by only one D.E. of e.g.
P(x, y)=D11W,xxxx+ 2(D12 + 2D66)W,xxyy+ D22W,yyyy
6
s11
s12
(Or)
s13
—(9)
s14
s15
c66 6
s16
—(4)
Subject to Boundary Conditions of simply supported edge
s
22
s23
s33
s24
s34
s25
s35
s26
s36
X =0, a : w = 0 m
= D w,
– D w, = 0
s44
s45
s46
x 11 xx
12 yy
sym
s s
Y= 0, b :w = 0 my = D12 w, xx – D22 w, yy = 0
2.5 NaviersSolution :
55 56
s66
—(10)
Deflection equation is give by,
= C.sin mx .sin ny —(5)

Special orthotropic material:
In the case of an orthotropic material
a
Lateral load
P (x, y) = fm sin
m1 n1
b
mx .sin ny a b
—(6)
(which has mutually perpendicular planes of material symmetry) the stressstrain relations in general have the same for as equation9. However, the number of independent elastic constants is reduced to nine, as various stiffness and compliance terms are interrelated. When the reference system of coordinates is selected along principal planes of material symmetry, i.e., in the case of a specially orthotropic material, then
1
c11
0 1
WhereU
1 (3Q 3Q 2Q 4Q )
2
c
12
22 23
0
2
1 8 11 22 12 66
3 c13
0 3
U 1 (Q Q )
0
4
0
44 4
2 2 11 22
5 0
0 5
U 1 (Q Q 2Q 4Q )
3 8 11 22 12 66
6 0
c12
c13
0
0
c
c
0
0
c23
c33
0
0
0
0
c
0
0
0
0
c55
0
0
0
0
—(11)
c66 6
U 1 (Q Q 6Q 4Q )
(Or)
4 8 11
1
22 12 66
—(16)
s11 s12
s13 0
0 0
U5 2 (U1 U 4 )
s22
s23 0
0 0

Laminate Geometry:
sym
s33
0
s44
0
0
s55
0
0
0
The geometry of an N layer laminate is shown in Fig 4. The z coordinate is measured positive downward from the mid plane of the
s66
—(12)
for = 0, the shear coupling coefficients are zero and the orthotropic lamina becomes specially orthotropic lamina.
0
1 12
laminate. The total thickness of the laminate is
h. the first lamina, of thickness t1 is located at the top of the stacking. The jth lamina of thickness tj is at a distance hj1. The laminate itself is considered thin as in the case of thin place theory and a perfect inter laminar bond exists between all the laminae. Each lamina is also assumed to be macroscopically homogeneous and behaves in a linear elastic
xx
11
E11
E22
xx
manner.
1 0
yy
22
E yy
22
xy
12
sym
1 xy
—(13)
G12
The stress strain relations for a general orthotropic lamina are
xx
Q11
Q12
Q16 xx
Q
Q
—(14)
yy
Q12
22 26
yy
xy
Q16
Q26
Q66
xy
Fig.4 Distances of lamina from middle plane
Q11 = U1 + U2 cos2 + U3 cos4
Q12 = U4 U3 cos4
Depending on the individual lamina properties, the stresses are determined from equation, [3]
1 xx
Q11
Q12
Q16 xx
Q16
U 2 sin 2 U3 sin 4
Q
Q
—(17)
2 yy Q12 22 26 yy
Q Q Q
Q22
U1
U2
cos 2 U3
cos 4
xy
16
26 66
xy
Q Q
Q 0
Z
Q 1 U
sin 2 U
sin 4
xx
11 12
16 xx x
26 2 2 3
yy
Q12
Q22
Q26
yy
0

Z y

Q Q
Q 0
Z
Q66 U5 U3 cos 4
—(15)
xy
16
26 66 xy
xy
—(18)
Q
Stressdisplacement relations for the Kth layer can be written as:
1
Q11
Q12
0
1
2
2 12
Q22
0 —(24)12
2
x
x2
2
—(19)
6
0
0 Q66 6
y
z Qij k
y 2
=[Q]12 []12
xy k
2
2
xy
WhereQij = Cij –
ci3c j 3
c33
(i, j = 1, 2, 6)
= Q
x
—(20)The
The inverse relation can be written as,
[]1,2 = [s]1,2 []1,2ij k y
xy
Reduced stiffnesses in terms of elastic constants,
stiffness matrices are given by, Extensional stiffness matrix:
Q11 =
E1
1 12 21
A A A
Q22 = E2
11 12 16
[A] =1 12 21
A12 A22 A26
E E
A A A
Q12 =
21 1 12 2
16
26 66
1
12 21
1
12 21
ij k
k
Aij= N Q Z

Zk 1
—(21)
Q66 = G12 —(25)
K 1
Coupling stiffness matrix:
2.10 Tsai – Hill criterion or Deviatoric Strain
B11 B12 B16
Energy Theory :[4] [B] = B B B
2 f 2 f f f 2
12
22 26
f x y
x y xy
=1 —(26)
B16
B26
B66
X 2 Y 2 X 2 S 2
Bij= 1 N Q
Z 2 Z 2
—(22)
The TsaiHill failure theory is expressed
2 K 1
ij k k
k 1
in terms of a single criterion instead of the three sub criteria required in the maximum stress and
Bending stiffness matrix:
D11 D12 D16
D D D
[D] =
12 22 26
D16 D26 D66
maximum strain theories. The Tsai Hill theory allows for considerable interaction among the stress components. It is based on Hills theory for ductile anisotropy and adopted to more brittle heterogeneous composites.
Dij= 1 N Q
Z 3 Z 3
—(23)
3 K 1
ij k k
k 1



Results & Discussion:
2.9 Orthotropic mateial under plane stress:
In most structural applications composite materials are used in the form of thin laminates loaded in the plane of the laminated. Thus composite laminae or laminates can be considered to be under a condition of plane stress with all stress components in the out of plane direction ( 3 direction) being Zero i.e.,
3 = 0,4 = 0,5 = 0
The orthotropic stressstrain relations reduced to
In this work, the strength of Composite Laminated plates of varying number of layers, configurations and dimensions are calculated using Classical Laminate Plate Theory. Numerical results are generated for the Boron Epoxy laminated composite plate and the properties are listed in Table 1.
Table 1 Material properties of Boron/Epoxy composite:
Longitudinal Modulus E11
206.84 Gpa
Transverse Modulus E22
20.68 Gpa
Major Poissons ratio v12
0.30
Minor Poissons ratio v21
0.30
Planar shear modulus G12
6.89 Gpa
Ultimate longitudinal tensile
strength X
1.378 Gpa
Ultimate transverse tensile
strength Y
0827 Gpa
Ultimate shear strength S
0.124 Gpa
Details of the plates : Thickness of laminate : 10mm
Plate Dimensions : Rectangular plates of 2m along principal plane direction, 1m across principal plane direction .
Square plate 1.414m 1.414m Laminate Material : Boron / Epoxy
Ply angle : 00 and 900 only (i.e., crossply) Numbers of layers : 2,4,6,8 with equal thickness of laminae.
Edge conditions : Simply supported on all sides.
Loading : Uniformly Distributed Lateral Load.
An efficient Cprogram [7] is developed to evaluate the response of the laminate, initial failure load and ultimate failure load under different load and boundary conditions. The developed program is run for the problem given in the book mechanics of laminated composites by J.N.Reddy in the chapter 5 Analysis of specially orthotropic laminates using CLPT. And the results are compared here under.
Given data: [3]
Simply supported Square plate under distributed mechanical loading laterally applied With Material properties
E11 = 25E22; v12 = 0.25;
and G12 = 0.5E22;
Table(2) comparison of the results with published one
Published result
Out put of the
program
Non dimensional
deformation
.666
.699
Non dimensional
Longitudinal stress
.8075
.8549
Non dimensional
Transverse stress
.0306
.0324
The results are found satisfactory.
The program is run for the material properties and lamina dimensions under consideration and the following results are obtained.
Reduced stiffness matrices for the given material(Boron/Epoxy):
Q[ij] Matrix for 0Degrees
———————————– 208.72 6.26 0.00
6.26 20.872 0.00 in GPa
0.00 0.00 6.89
———————————–
Q[ij] Matrix for 90Degrees
———————————– 20.87 6.26 0.00
6.26 208.72 0.00 in Gpa
0.00 0.00 6.89
———————————–
4Layer Symmetric Laminate[900090]:
Coupling Matrix (D) – Matrix in KNm:
———————————– 3.70 0.52 0.00
0.52 15.44 0.00
0.00 0.00 0.57
———————————–
The maximum allowable load : 294.0000 KPa and deflection : 0.297 m
4Layer antisymmetriclaminate(090090):
D – Matrix in KNm:
———————————– 9.57 0.52 0.00
0.52 9.57 0.00
0.00 0.00 0.57
———————————–
The maximum allowable load : 263. KPa and deflection : 0.377 m
Fig.( 5 ) Failure Load of Laminates Vs Deflection curve for 4Ply laminate of various configurations
6PLY LAMINATE
Fig.(6) Failure Load of 6plyLaminates Vs Deflection curve for 6Ply laminate
ORTHROTOPIC SQUARE PLATE:
Dimensions: a=1 m; b=1 m; t=.01 m
Failure load
,
Kpa
Deflection , mm
2Ply laminate (090)
(900)
76
.224
4Ply laminate
152
.441
(090090)
(090900)
(900090)
6Ply Laminate
228
.665
(090090090)
(09000900)
(9009090090)
Table 3 Failure loads and Deflections of square laminated plates
Fig 7 Number of layers Vs Ultimate failure load of square laminate plates of
varying layers
Fig. (8) Longitudinal stress Vs Strain in outer layer of a four ply laminate (900090)
Fig. (9) Lateral stress Vs strain in outer layers of a four ply laminate (900090)
Fig. (10) Longitudinal stress Vs Strain in inner layers of a four plylaminate (900090)
Fig. (11) Lateral stress Vs Strain in inner layers of a four plylaminate (900090)

ANALYSIS:
Based on the results the graphs are drawn :
Fig. (6)indicates the failure of 4ply laminates with different configurations. The first point on each of the lines indicates the First ply failure which occurred in outer layers and the second point indicates the last ply failure which is in inner layers. Out off the three configurations (900090) is the best configuration that an with stand a higher ultimate load of 294 KPa with small deflection of .297m.
Table (3) presents ultimate loads with varying number of layers of aOrthotropic Square laminate with different configurations. It is observed that the ultimate loads forall the configurations is giving same failure loads and deflections for a given number of layers.
Fig. (7) presents the ultimate loads for varying number of layers in a Square laminate plate. It is observed that as the number of layers increases the strength of the plate also increases.
Fig. (8) to (11) presents the stress and strain relations in both longitudinal and transverse in a 4ply laminate (900090).It is observed that there is a linear relation ship in stress and strain and the stresses are equal in magnitude at equal distance from mid plane on both sides but in opposite nature i.e., tensile in lower laminae and compressive at upper laminae.

CONCLUSIONS
From the study of the work carried out for simply supported orthotropic laminated plates under uniformly distributed load, the following conclusions are made.
As the number of layers increases in a laminated composite plate for a given thickness, its strength also increases.
The strength of the laminated composite plate depends on the number of layers and also the arrangement of the layers.
The strength of an orthotropic square laminated plate is independent of the arrangement of the layers.

References:


Robert M.Jones, Mechanics of Composite Materials, First Edition,McGrawHill Koga Kusha Ltd.

Isaac N .Daniel &OriIshai, Engineering Mechanics of Composite Materials, Oxford University press,1994.

J.N.Reddy, Mechanics of laminated composite plates & shells, First Edition,McGrawHill.

Timoshenko &WoinowskyKrieger , Theory of plates & vessels, 2nd Edition,McGrawHill.

Ansel c Ugural, Stresses in plates & shells, 2ndEdition, McGrawHill.

Bryan Haris, Engineering Composite Materials, 2nd Edition, McGrawHill.

E.Bala Guru Samy, C and Data Structures,2nd Edition, Tata McGrawHill.

Autar K. Kaw, Mechanics of Composite Materials, 2nd Edition, Taylor & Francis, 2006.