Materials are having greater impact on the mankind due to their effective use, since from ancient times. Considering the extent of composites for the use of mankind from the last forty year, has remarkable applications of the composites in various fields. Nowadays it has been found that composite materials are having the tremendous applications compared to the other materials. Such an impact made us to look through the composites for the betterment of the mankind. Composite materials are made up of two or more materials that will give desired properties which cannot be achieved by using any one of the constituents alone itself. For example fiber reinforced composite materials consists of more than two materials which will give desired properties like high strength to weight ratio and also gives high modulus. The fiber reinforced composite materials are idle for structural applications where high strength to weight and stiffness to weight ratios are required. A variety of structural components made of composite materials such as aircraft wing, helicopter blade, vehicle axles and turbine blades. The requirement to study the bending and free vibration behavior in structural components plays an important role

A number of researchers have been developed numerous solution methods in recent 20-30 years. Cawley and Adams (1978) investigated the natural modes of square aluminium plates and square composite plates with different ply orientations for free-free boundary conditions, both theoretically as well as experimentally. Cawley and Adams (1979) also used dynamic analysis to detect, locate and roughly quantify damage to components fabricated from fiber reinforced plastic. Maiti and Sinha (1994) used the first order shear deformation theory to develop FEM methods to study the bending and free vibration behavior composite beam. The free vibration frequencies of cross ply laminated square plates for twelve different boundary conditions were determined using Ritz method by Aydogdu and Timarci (2003). G.Rajeshkumar et al.(2014) presented the free vibration characteristics of newly identified Phoenix Sp fiber reinforced polymer matrix composite beams, moreover the physical, chemical and mechanical properties of fiber was determined by using standard experimental methods. The modal analysis is carried out on composite beams having different fiber gauge

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G.Abdollahzadeh and M. Ahmadi (2015) made an attempt to study the non linear behavior of composite beams undergoing prebuckling snd postbuckling. Analytical solutions are obtained to analyze the free non linear vibrations. The numerical results are obtained which have good agreement. The influences of the parameters of laminated composite beams on the free vibrations are studied. J. Alexander and B. S. Augustine (2015) presented the free vibration and damping characteristics of GFRP and BFRP laminated composites at various boundary conditions. Basically this study made on aircraft wing because of severe vibration during the motion of aircraft. Vibration analysis carried out by using modal analysis set up for various end conditions. Natural frequencies and damping coefficients were determined by for all materials, end conditions and for fiber orientations. Fabrication of GFRP and BFRP composites is done by using compression moulding technique. The present work is mainly focused on the numerical study of fiber reinforced composite plate subjected to different boundary conditions and loading conditions first order shear deformation theory The first order shear deformation theory (FSDT), also called Mindlin‘s plate theory is based on the assumptions that the transverse shear strains are constant which leads to a particular kinematics of deformation. The transverse normals do not remain perpendicular to the mid surface after deformation. The displacement of this theory is in the form,

0 0 0

(,,,)(,,)(,,) (,,,)(,,)(,,) (,,,)(,,)

   

 (1)

Where000,,,,xyuvware unknown functions to be calculated. As before 000,,uvw denote the displacement of a mid surface i.e. z=0. Note that which indicates that xdudz and ydvdz are the rotations of a transverse normal relating to y and x- axes respectively. The notation x denotes the rotation about y-axis and ydenotes the rotation about x-axis. Shear correction factor used is 5/6. finite element formulation and methodology The laminated composite plate consists of number of thin elastic substrate laminates. The fibers are oriented with different angles and they are embedded in matrix material. A lamina is assumed as elastically orthotropic. Consider a fiber reinforced laminated composite plate having length a, width b and thickness h, shown figure

To determine the reduced stiffness matrix for a lamina, longitudinal elastic modulus, transverse elastic modulus, major poisons ratio, minor poisons ratio and shear modulus are required for the given composite. Formulas used for determining reduced matrix are as follows.

11111221 1222121221 22221221

1 1 1

     

 (2)

6612;QG 4413;QG 5523;QG

Transformed reduced stiffness matrix is calculated by following formula.

_44221122126611_22441166121222_44221122126622_3311126612226616_111226

cossin2(2)sincos (4)sincos(cossin) sincos2(2)sincos (2)cossin(2)cossin (2

       

3366122266_2244112212666666

)cossin(2)cossin (22)cossin(cossin)

  

 (3)

The generalized [ABD] matrix is written as follows



111216111216 122226122226 162666162666 111216111216 122226122226 162666162666

 (4)

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stiffness matrix, [D]= Bending stiffness matrix and are calculated using following formula

  

11

22111 33111

()

1()2 1()3 ,1,2,6.



 

   

  

(5)

An eight node quadrilateral isoparametric element is used in this analysis. Let {u, v, w,x,y} be the displacements field variables. The plate is discretized into finite elements. The analysis is based on first order shear deformation theory. Shape functions for 8 node quadrilateral element are as follows

(6)

Elemental stiffness matrix for the plate element is given by, [][][][]TeKBDBJdd (7) Elemental load vector is given by TeFNqJdd (8) Where, []eK=Element stiffness matrix eF=Element load vector J =Determinant of Jocobian Jacobian matrix,

Jxy

 (9)



0000 0000 000 0000 0000 000 000 000 1,2....8

 (10)

For free vibration analysis equations used are 0MuKu (11) Where, u= Displacement vector u =Velocity vector u=Acceleration vector [K]= Global stiffness and [M]=Global mass matrix. The element mass matrix is given by

/2

/2

ThAeehMNNdzdA (12) For equations (7) and (11), [N] is given by



0000 0000 0000 0000 0000 1,2.......8



 (13)

The above equations are used to find the bending deflection and fundamental frequency for a fiber reinforced composite plate.

In the present work, MATLAB software is used to perform the objective. The methodology described above is used to study the bending and free vibration

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are given below 1225,EE121320.5*,GGE2320.2*GE,Density

Plate length (a) and width (b) = 100, thickness (h)=1

In the present work, the validation study is carried out by comparing the present results with the Reddy (2004). It is concluded from the table1 that the present results are good agreement with the Reddy for both loading conditions (CPL and UDL).

Present results Ref. Present results Ref.

0/90 1.6985 1.6955 4.6886 4.6664 90/0 1.6985 1.6955 4.6886 4.6664 (0/90)2 0.8116 0.8085 2.2372 2.2105 (0/90)4 0.7181 0.7150 1.9806 1.9536

(a) Bending analysis of fiber reinforced composite square plate A study on bending behavior of composite square plate with different fiber orientations, loadings and boundary conditions are carried out and the results are computed by considering the following load types and boundary conditions.

Uniformly distributed load (UDL) and central point load (CPL). Boundary conditions (BCs): SSSS: plate with all edges simply supported i.e. edges along x=0 and a, and also for y=0 and b, simply supported. CCCC: plate with all edges clamped. SCSC: plate with edges along x=0 and a, simply supported and edges along y=0 and b, clamped. reinforced composite square plate is studied based on following figures,

The variation of non dimensional central deflections (w*) for a single layer fiber reinforced composite square plate having different fiber orientation (0, 15, 30, 45, 60, 75, 90) with different load and boundary conditions are presented in Fig. 2. The non dimensional central deflections of CCCC type boundary conditions are smaller than that of both SSSS and SCSC boundary conditions (both UDL and CPL), this is because stiffness of the square plate is more under CCCC boundary condition and for SCSC boundary condition non dimensional central

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orientation(both UDL and CPL).

The variation of non dimensional central deflections (w*) for different layers (2, 4, 6, 8, 10, 20) of anti- different fiber orientation; =15, 30, 45, 60, and 75, with different load and boundary conditions are presented e.g. Fig.3 shows non dimensional central deflection for two layer composite square plate. The non dimensional central deflections of CCCC type boundary conditions are smaller than that of both SSSS and SCSC boundary conditions (both UDL and CPL) and for SCSC boundary condition non dimensional central deflection decreases with increase in the fiber orientation(both UDL and CPL). For SSSS boundary conditions the non dimensional central deflection decreases with increase in the fiber orientation up to (45/-45) and increases, further increase in the orientation(for both CPL and UDL). The deformation behavior of angle ply plates are influenced by presence of extension-twisting and

B26.

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Fig.4 and 5 presents plots of non dimensional central deflection (w*) for cross ply and angle ply composite square plate versus different number of layers under different loading conditions (both CPL and UDL) and for different boundary conditions. From fig.5 and 6 it is observed that, the increase in number of layers (n) decreases the non dimensional central deflection of the square plate. It is also observed that the decrease is small with the increase of the total number of layers beyond six. This is because of decrease in the bending stretching coupling effect with increase in the number of layers. It is also noticeable that the non dimensional central deflections of CCCC type boundary conditions are smaller than that of both SSSS and SCSC boundary conditions (both UDL and CPL). (b) Free vibration behavior of fiber reinforced composite square plate

SSSS: plate with all edges simply supported i.e. edges along x=0 and a, and also for y=0 and b, simply supported. CCCC: plate with all edges clamped. SCSC: plate with edges along x=0 and a, simply supported and edges along y=0 and b, clamped. SFSF: plate with edges along x=0 and a, simply supported and edges along y=0 and b, free. Non dimensional fundamental frequency is given by

2*

 (13)

0 15.1366 32.7245 16.7381 14.1596 15 15.0251 31.2471 16.6650 12.6348 30 15.0319 28.0235 17.6885 8.9435 45 15.2519 26.3951 21.3656 5.4592 60 15.1041 28.4814 26.6965 3.3797 75 15.0896 32.3307 31.4033 2.8755 90 15.1899 34.0732 33.2395 2.8486

15/-15 13.9354 24.3875 15.5944 8.9909 30/-30 14.4248 21.2378 16.3949 6.3033 45/-45 15.1493 20.8051 18.2006 4.4064 60/-60 14.4398 21.7122 20.1407 3.2367 75/-75 13.9672 25.3439 23.7065 2.8744

The tables2 and 3 shows the fundamental frequencies of single and two layers of composite square plate with different fiber orientation and different boundary conditions. From tables it is observed that for SSSS boundary conditions, the increase in fundamental frequency is marginal. For SFSF boundary condition the fundamental frequency decreases with increase in fiber orientation. For SCSC boundary conditions, the fundamental frequency increases with increase in the fiber orientation. It is noted that the fundamental frequencies of CCCC boundary conditions are maximum compared to other boundary conditions , this is because increase in the number of boundary conditions will increase the fundamental frequency.

The finite element modeling is done by MATLAB software to study the bending and free vibration behavior of fiber reinforced composite plate. A study on the effect of boundary conditions, ply orientation

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vibration analysis is carried out for fiber reinforced composite plate. Finite element model is formulated efficiently for linear bending and free vibration behavior of composite square plate. Bending deflections and fundamental frequencies for fiber reinforced composite plate with different ply orientation and boundary conditions are efficiently derived from the analysis using MATLAB software. From bending analysis, it is came to know that the central deflection of composite plate dependent on the ply orientation, number of layers used and type of loading. From free vibration analysis it is concluded that the increase in the number of boundary constraints will increase the fundamental frequency.

P. Cawley, and R.D. Adams, ―The predicted and experimental natural modes of free-free CFRP plates,‖ Journal of composite materials,12,336, 1978. G. Abdollahzadeh, and M. Ahmadi, ―Nonlinear vibration anlysis of prebuckling and post buckling in laminated composite beams,‖ Archives of civil engineering vol. LXI, issue-2, 2015. S.B. Kerur and A. Gosh, ―Active vibration control of composite plate using AFC actuator and PVDF sensor,‖ International journal of structural sability and dynamic, vol.11(2),p.237-255,2011. N.Aydogdu, and T. Timarci, ―Vibration analysis of cross ply laminated square plate with general boundary conditions,‖ Composite science and technology, 63,p.1061-1070, 2003. L. Jaehong, ―Free vibration analysis of delaminated composite beams,‖Computers and structures 74(2000), 121-129. D. Maiti, and P.K. Sinha, ―Bending and free vibration analysis of shear deformable laminated composite beams by finite element method,‖Composite structures 29(1994), p. 421 -431. M. Mukhopadhyay, Mechanics of composite materials and structures,Universities press, 2005. J.N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC press,2004. M. Samikshya, and N. Nishant, ―Experimental and numerical study on vibration and buckling characteristics of laminated composite plates,‖ Department of civil Engineering NIT Rourkela. damping characteristics of GFRP and BFRP laminated composites at various boundary conditions,‖ in Indian journal of science and technology, vol. 8(12). G. Rajeshkumar, and V. Hariharan, ―Free vibration charactristics of pheonix Sp fiber reinforced polymer matrix composite beams,‖ procedia Engineering 97(2014), p. 687-693. D. Nicoara, ―On the numerical analysis of laminated composite,‖ annals of the Oredea University issue-1(2013). T. Kant, and K. Swaminathan, ―Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher order refined theory,‖ Composite structures(2001). F.G. Ronald, ―Modal vibration response measurements for characterization of composite materials and structures,‖ Composites science and technology,60(2000), p.2769-2780.

PG Student, Dept. of mechanical Engg., BEC. Bagalkot, Karnataka, India