INTRODUCTION
Laminated composite plates on elastic foundation are widely used in engineering
applications such as construction of the plane, spacecraft, ship, sports equipment
and chemical industry. These plates are very attractive for researchers and
research is continuing on. In severe conditions of environments, the buckling
and vibration of these structures are occurred. It is important to know the
mechanical behavior of the composite plates under the different load.
The number of parameters associated with the manufacturing and fabrication
of composites and modeling of the foundation are many. The composite structures
supported by elastic foundation display a considerable amount of uncertainties
in their material and foundation properties. Therefore for accurate analysis
of composite laminates supported on elastic foundation, it is necessary to estimate
the values of the eigen solutions of the structures for a reliable design.
Plates are generally placed on an elastic foundation in many applications of
engineering. The modeling and analysis of such structures under large deflections
are important study area. For this reason, takes a lot of research on the subject.
Many publications have appeared in literature on the free vibration analysis
of laminated composite plates. The vibration analysis of elastic boundary of
composite plates, very little in the literature are discussed.
Lal et al. (2008) analyzed nonlinear free vibration
of laminated composite plates on elastic foundation. In analysis was used random
system properties.
Setoodeh and Karami (2004) studied on thick composite
plates. The analysis of static, free vibration and buckling was performed. The
plates supported elastic and a 3D layerwise FEM was used for analysis.
Sobhy (2013) investigated buckling and free vibration
of sandwich plates on elastic foundations. The plates were under various boundary
conditions.
Malekzadeh et al. (2010) employed vibration
analysis of composite plates. Composite plates were elastic foundation and simply
supported.
Baltacioglu et al. (2011) researched static
analysis of composite plates. The boundary conditions of plate was elastic installation.
In analysis was used the method of discrete singular convolution.
Quintana and Nallim (2013) studied free vibrations
of composite plates. Triangular and trapezoidaltype plates were elastic supported.
Wun and Lu (2011) presented vibration analysis of
composite plates. Boundary conditions of plates were inner side columns and
edges of the elastic support. The powerful pb2 Ritz method was used for analyzing.
Ismail et al. (2013) investigated effect of
the material properties on vibration of composite plate. Boundary conditions
of plates were elastic. In analysis were used the Inverse method and Fourier
series.
Li et al. (2009) determined vibrations analysis
of plates. An analytical method used for the analyzes and edges of plates were
supported elastic.
Hao and Kam (2009) performed modal analysis of composite
plates. Boundary conditions of composite plates were elastic supported at different
places.
Vosoughi et al. (2013) analyzed forced vibration
of composite plates on elastic installation. These plates were subjected to
moving load.
Ashour (2006) studied vibration of angleply symmetric
laminated composite plates. Elastically supported edges of the plate.
Karami et al. (2006) investigated vibrations
of laminated composite plates. In analysis were used the differential quadrature
method (DQM).
Civalek (2013) investigated a numerical model for
geometrically nonlinear dynamic analysis of thicklaminated plates based on the
firstorder shear deformation theory.
Thai and Choi (2012) studied free vibration of functionally
graded plates on elastic foundation. A refined shear deformation theory is developed
for analysis.
Shen (2011) analyzed nonlinear vibrations of thin
films. The films supported elastic in thermal environmental conditions.
In this study, free vibration analysis of composite Plates are investigated
using symmetrical modes considering three different elastic boundary. Free vibration
frequencies of the different boundary conditions are expressed as numerical
values. This paper is contribution to vibration analysis of composite plates
of elastic boundary.
MATERIALS AND METHODS
Main equations for the finite element method: A rectangular element,
which is under the effect of bending vibrations, is shown at Fig.
1. There are three degrees of freedom at each node, at each corner. There
are three degrees of freedom at each node, respectively, deflection of z direction
and the two rotations, w, θ_{x} = ∂w/∂y and θ_{y}
= ∂w/∂x. In terms of then ondimensional (ξ, η) coordinates,
these become
Since the element has twelve degrees of freedom, the displacement function
can be represented by a polynomial having twelve terms due to simplicity.

Fig. 1: 
Geometry of a rectangular element 
It can be written as follows related to ξ = ±1 and η = ±1
coordinate, at the node points:
where, {w}_{e} is the displacement and rotations vector:
At (2.1), defined the N (ξ, η) is:
And (ξ_{i}, η_{j})are the coordinates of node j (Petyt,
1990).
Mass matrix for plate element: The kinetic energy expressions for thin
plate bending element is:
where, is ρ density, h is thick plate and A is area of plate. Substituting
Eq. 1 into 4 gives:
Where:
Equation 6 is element mass matrix. If NJ (ξ, η)
substitute from Eq. 3 and integrate Eq. 6,
the result will be as follows (Petyt, 1990):
Linear stiffness matrix for composite plate element: The strain energy
can be expressed (Petyt, 1990):
Where:
And:
where,
a matrix of reduced stiffness components for the kth layer whose surfaces are
at distances z_{k1}, z_{k }from the middle surface of the plate.
are the components transformed lamina stiffness matrix which are defined as
follows:
Terms of the
matrix are:
Substituting Eq. 8 into 1 gives:
[K]_{e }can be written as follows:
Equation 14 is the element stiffness matrix. And:
dA = dx dy, ,
dA = abd ξdη equalby is (2.14) new expression:
Element stiffness matrix terms for easiness are separated square underside
matrix and:
and stiffness matrix is symmetric (Morgul and Kucukrendeci,
2008).
Analysis of linear undumped free vibration: Free vibration analysis
of the laminated composite plates is made by Eq. 18:
where, [M] and [K] are system mass matrix and system stiffness matrix respectively.
System matrices are composed of [M]_{e} element mass matrix and [K]_{e}
element stiffness matrix:
In order to determine the frequencies, ω and modes {Φ}, of free vibration
of a structure, it is necessary to solve the linear eigenvector problem (Morgul
and Kucukrendeci, 2008).
In this study, the Eq. 18 [K] matrix by adding parameters
of the spring ([K^{s}]) has been rewritten in the form of (Eq.
2021). In addition, the computer program used for the
analyzes rearranged:
RESULTS AND DISCUSSION
Effect of different elastic boundary on free vibration: In this study,
Morgul and Kucukrendeci (2008) is used as a model of
the solution. The Eq. 22 is the natural frequency parameter
and this parameter is used in the analyzes:
In this study, the equations of boundary conditions were written as special
Eq. 21. Special software was developed to analyze of these
equations in Matlab program. The linear undumped free vibrations of five layers
symmetrically laminated rectangular plates were analyzed. The angleplyis (θ,
θ, θ, θ, θ) and as an example is (30^{0}, 30^{0},
30^{0}, 30^{0}, 30^{0}) for plates. The angles of orientations
of the layerare taken as 0^{0}, 15^{0}, 30^{0}, 45^{0}.
The dimensions of the rectangular plate are a = 0.45 m, b = 0.30 m, a/b = 1,
5. Each layer thickness of 0.2 mm, the total crosssection of plate thickness
of 1 mm. The model of three different boundary condition is designed for plates.
The boundary conditions of plates shown in Fig. 24.
In the Fig. 2 is fully clamped and supported with five spring.
In the Fig. 3 is four corner simply supports and supported
with five spring. In the Fig. 4 is four corner free and supported
with five spring. Stiffness of spring is selected k = 1500 N m^{1}.

Fig. 2: 
Rectangular composite plate of symmetrically laminated is
fully clamped and supported with five spring, (4x4) mesh element model (fully
clamped) 
In other studies, two or fourlayer composite plates were analyzed. The free
vibration analysis of angleplay, fivelayered, elastic restricted of composite
plates with finite element method are no other studies and is unique to this
study. Finite element method used in this study is unique and there is no other
studies.
The material of laminated composite plate is AS/3501 Graphite/epoxy. Its mechanical
properties are as follows: density “ρ = 1630 kg m^{3}”,
modulus of elasticity “E_{1} = 138 Gpa, and E_{2}= 9.0
Gpa”, shear module “G_{12} = 6.9 Gpa” and poisons ratios
“v_{12}= 0.3 and v_{21}= 0.019”.

Fig. 3: 
Rectangular composite plate of symmetrically laminated is
four corner simply supports and supported with five spring, (4x4) mesh element
model (simply support) 

Fig. 4: 
Rectangular composite plate of symmetrically laminated is
four corner free and supported with five spring and (4x4) mesh element model
(free) 

Fig. 5: 
AS/3501 Graphite/epoxy, Linear natural frequency parameters
λ of symmetrically fivelayer angleply (0^{0}) in different
elastic boundary 

Fig. 6: 
AS/3501 Graphite/epoxy, Linear natural frequency parameters
λ of symmetrically fivelayer angleply (15^{0}) in different
elastic boundary 
Plate type and material properties retrieved from (Schwartz,
1992).
Composite plates of AS/3501 Graphite/epoxy of fivelayer symmetrically analyzed
for different elastic boundary. Linear natural frequency parameters of the undamped
vibration of the plates are calculated. The linear frequency for the symmetric
mode 20 of these plates are displayed in the Fig. 58.
Figure 58, frequency parameters for the
first 20 modes are shown free, simply support, fully clamped of boundary conditions
of composite plate. Angles of plate layer (angleplay) are (0^{0}, 0^{0},
0^{0}, 0^{0}, 0^{0}), (15^{0},15^{0},
15^{0}, 15^{0}, 15^{0}), (30^{0},30^{0},
30^{0}, 30^{0}, 30^{0}), (45^{0}, 45^{0},
45^{0},45^{0}, 45^{0}).
In Figure 5, in all the boundary conditions, between 49
and 1316 modes are approximate the frequency values. In the first ten mode,
fully clamped and simply support conditions frequencies are consistent. After
the seventeenth mode differs values. Between 3 and 16 modes in all the boundary
conditions the characteristic of vibration can be said to same. In Figure
6, all the boundary conditions between 116 modes the frequency values are
approximate.

Fig. 7: 
AS/3501 Graphite/epoxy, Linear natural frequency parameters
λ of symmetrically fivelayer angleply (30^{0}) in different
elastic boundary 

Fig. 8: 
AS/3501 Graphite/epoxy, Linear natural frequency parameters
λ of symmetrically fivelayer angleply (45^{0}) in different
elastic boundary 
After seventeenth mode values differs. In Fig. 7, in all
modes conditions of simple support and free frequency values overlap. In fully
clamped condition, frequency values of the first three and 1016 modes are approximate
the other terms. After sixteenth mode has higher values. In Figure
8, in all the boundary conditions, the frequency values of the first five
modes and between 1216 modes are approximate. In other modes differs values.
CONCLUSIONS
Linear frequency parameters calculated according to the plate on elastic foundations.
The effect of boundary conditions on the plate vibrations was observed (Fig.
58). In Fig. 58
frequency values close to each other. Frequency values of Fig.
78 greater than Fig. 56.
Composite plates can be formed with different angle of arrangement of layers.
These plates can be found dynamic properties. Types of plates can be selected
according to the characteristics of user location. As a result, in usage area
of plates with different boundary conditions, is preferred over one another.
This are important to overcome the difficulties of construction, reduce costs
durability.