Document Type : Research Paper
Authors
^{1} University of Applied Science and Technology
^{2} University of Semnan
Abstract
Keywords

Mechanics of Advanced Composite Structures 4 (2017) 7587 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
A Numerical and Analytical Solution for the Free Vibration of Laminated Composites Using Different Plate Theories
M.A. Torabizadeh ^{a}^{*}, A. Fereidoon ^{b}
^{a }Department of Mechanical Engineering, University of Applied Science and Technology, Mashhad, Iran
^{b} Department of Mechanical Engineering, University of Semnan, Semnan, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 20161127 Revised 20170121 Accepted 20170209 
An analytical and numerical solution for the free vibration of laminated polymeric composite plates with different layups is studied in this paper. The governing equations of the laminated composite plates are derived from the classical laminated plate theory (CLPT) and the firstorder shear deformation plate theory (FSDT). General layups are evaluated by the assumption of crossply and angleply laminated plates. The solver is coded in MATLAB. As a verification method, a finite element code using ANSYS is also developed. The effects of lamination angle, plate aspect ratio and modulus ratio on the fundamental natural frequencies of a laminated composite are also investigated and good agreement is found between the results evaluated and those available in the open literature. The results show that the fundamental frequency increases with the modular ratio and the bendingstretching coupling lowers the vibration frequencies for both crossply and angleply laminates with the CLPT. Also it is found that the effect of bendingstretching coupling, transverse shear deformation and rotary inertia is to lower the fundamental frequencies.




Keywords: Free vibration Laminated composites Plate theories Numerical method Analytical method 

DOI: 10.22075/MACS.2017.1768.1090 
© 2017 Published by Semnan University Press. All rights reserved. 
A composite material can be defined as a combination of two or more materials that results in better properties than those of the individual components used alone. In contrast to metallic alloys, each material retains its separate chemical, physical and mechanical properties. The two constituents are reinforcement and a matrix. When composites are compared to bulk materials, the main advantages of composite materials are their high strength and stiffness, combined with low density, allowing for a weight reduction in the finished part. The reinforcing phase provides the strength and stiffness. In most cases, the reinforcement is harder, stronger and stiffer than the matrix. The reinforcement is usually a fiber or a particulate. Particulate composites have dimensions that are approximately equal in all directions. They may be spherical, platelets, or any other regular or irregular geometry. Particulate composites tend to be much weaker and less stiff than continuousfiber composites, but they are usually much less expensive. Particulate reinforced composites usually contain less reinforcement (up to 40 volume percent to 50 volume percent) due to processing difficulties and brittleness [1].
A fiber’s length is much greater than its diameter. The lengthtodiameter (l/d) ratio is known as the aspect ratio and can vary greatly. Continuous fibers have long aspect ratios, whereas discontinuous fibers have short ones. Continuousfiber composites normally have a preferred orientation, whereas discontinuous fibers generally have a random orientation. Examples of continuous reinforcements include unidirectional, woven cloth and helical winding, whereas examples of discontinuous reinforcements are chopped fibers and random material. Continuousfiber composites are often made into laminates by stacking single sheets of continuous fibers in different orientations to obtain the desired strength and stiffness properties with fiber volumes as high as 60 percent to 70 percent. Fibers produce highstrength composites because of their small diameter; they contain far fewer defects (normally surface defects) compared to those in the material produced in bulk. As a general rule, the smaller the diameter of the fiber, the higher its strength, but often the cost increases as the diameter becomes smaller. In addition, smallerdiameter/highstrength fibers have greater flexibility and are more amenable to fabrication processes, such as weaving or forming over radius. Typical fibers include glass, aramid and carbon, which may be continuous or discontinuous. The continuous phase is the matrix, which is a polymer, metal or ceramic. Polymers have low strength and stiffness, metals have intermediate strength and stiffness but high ductility, and ceramics have high strength and stiffness but are brittle. The matrix (continuous phase) performs several critical functions, including maintaining the fibers in the proper orientation and spacing and protecting them from abrasion and the environment. In polymer and metal matrix composites that form a strong bond between the fiber and the matrix, the matrix transmits loads from the matrix to the fibers through shear loading at the interface. In ceramicsmatrix composites, the objective is often to increase the toughness rather than the strength and stiffness; therefore, a low interfacial strength bond is desirable [1].
Tan and Nie [2] studied free and forced vibration of variable stiffness composite annular thin plates with elastically restrained edges based on the classical plate theory. They found that the transverse mode shapes of the plates with inplane variable stiffness are different from those with constant stiffness. Zhang et al. [3] analyzed free vibration analysis of triangular CNTreinforced composite plates subjected to inplane stresses using the FSDT elementfree method. Chakraborty et al. [4] presented a novel approach, referred to as polynomial correlated function expansion (PCFE), for a stochastic freevibration analysis of a composite laminate. Finally, based on the numerical results, new physical insights had been created on the dynamic behavior of composite laminates. Ganesh et al. [5] studied the free vibration analysis of delaminated composite plates using a finite element method. Mantari and Ore [6] presented a simplified firstorder shear deformation theory (FSDT) for a laminated composite and sandwich plates. Their approach had a novel displacement field that includes undetermined integral terms and contains only four unknowns. Su et al. [7] illustrated a modified Fourier series to study the free vibration of a laminated composite and fourparameter functionally graded sector plates with general boundary conditions. Zhang et al. [8] studied the freevibration analysis of functionally graded carbon nanotubereinforced composite triangular plates using the FSDT and the elementfree IMLSRitz method. They also examined the influence of a carbon nanotube volume fraction, plate thicknesstowidth ratio, plateaspect ratio and a boundary condition on the plate’s vibration behavior. Marjanović and Vuksanović [9] illustrated a layerwise solution to free vibrations and the buckling of a laminated composite and sandwich plates with embedded delamination. The effects of plate geometry, lamination scheme, degree of orthotropy and delamination size or position on the dynamic characteristics of the plate were presented. Boscolo [10] presented an analytical closedform solution for a freevibration analysis of multilayered plates by using a layerwise displacement assumption based on Carrera’s Unified Formulation. A wide range of boundary conditions were analyzed by using a Levytype solution. Ou et al. [11] presented an efficient method for predicting the free and transient vibrations of multilayered composite structures with parallelepiped shapes, including beams, plates and solids. Rafiee et al. [12] analyzed the geometrically nonlinear free vibration of shear deformable piezoelectric carbon nanotube/fiber/polymer multiscale laminated composite plates. Akhras and Li [13] used a spline finite strip with higherorder shear deformation for stability and a freevibration analysis of piezoelectric composite plates. Grover et al. [14] assessed a new shear deformation theory for freevibrationresponse laminated composite and sandwich plates. They compared the results with finite element and analytical solutions. Jafari et al. [15] presented a freevibration analysis of a generally laminated composite beam (LCB) based on the Timoshenko beam theory using the method of Lagrange multipliers. They examined some parameters, such as the slenderness ratio, the rotary inertia, the shear deformation, material anisotropy, ply configuration and boundary conditions on the natural frequency and mode shape. Tai and Kim [16] illustrated the free vibration of laminated composite plates using two variable refined plate theories. They applied the Navier technique to obtain the closedform solutions of antisymmetric crossply and angleply laminates. Srinivasa et al. [17] and Ramu and Mohanty [18] used finite element results as a verification method with those obtained from experimental tests on the free vibration of composite plates. Chandrashekhara [19] presented an exact solution for the free vibration of symmetrically laminated composite beams. Ke et al. [20] investigated the nonlinear free vibration of functionally graded nanocomposite beams reinforced by singlewalled carbon nanotubes (SWCNTs) based on the Timoshenko beam theory and von Kármán geometric nonlinearity. Also, the free vibration of anisotropic thinwalled composite beams and delaminated composite beams were performed by Song [21] and Lee [22], respectively.
Based on papers reviewed in the literature, few investigations were found that compared the analytical and numerical analyses of different theories and lamination layups. Therefore, in this paper, the analytical and numerical solutions for the free vibration of laminated polymeric composite plates with different layups are compared. Two different theories and layups are selected. Also, finiteelement analysis is performed using ANSYS to validate results obtained by analytical methods. The solver is coded in MATLAB. Also investigated are the effects of different parameters, such as the lamination angle, the plate aspect ratio and the modulus ratio on the fundamental natural frequencies of laminated composite. The main objective of this paper is to compare different theories and lamination schemes on the vibration response of laminated composites.
2.1. Classical lamination plate theory (CLPT)
2.1.1 Displacement and strains
A rectangular plate of sides a and b with thickness h is shown in Fig. 1. Based on the classical lamination plate theory, the following displacement field can be assumed [23]:
(1) 

(2) 

(3) 
where are the displacements along the coordinate lines of a material point on xyplane.
The von Karman strains associated with the displacement field in static loading can be computed using the straindisplacement relations for small strains:
(4) 
Note that the transverse strains are identically zero in classical plate theory. The first three strains have the form
(5) 
2.1.2 Equilibrium equations
By using Eqs. (4) and (5), the constitutive equations are obtained. Equations of equilibrium can be derived using the variational principle, which is not explained in detail here (see [23]). The EulerLagrange equations of the theory are obtained as follows,
(6) 

(7) 

(8) 

(9) 
where, the quantities N_{ij} are called the inplane force resultants and M_{ij} are called the moment resultants and (I_{0}, I_{1}, I_{2}) are the mass moments of inertia.
2.1.3 Navier solution methodology
The displacement fields are assumed by the following form:
(10) 

(11) 

(12) 
where U_{mn}, V_{mn} and W_{mn} are the coefficients that should be determined and and
The consideration of Eqs. (10) – (12), shows that the mechanical transverse load q should also be expanded in a double sine series. Thus,
(13) 

(14) 
Figure 1. The geometry of simply supported rectangular laminated plates used in the analytical solutions.
Substituting expansions (10–12) into expressions given in Eqs. (68) without thermal loads yields
(15) 
where A_{ij}, D_{ij} and B_{ij} are called extensional, bending and bendingextensional coupling stiffness, respectively [23]. Also, and . Note that the edge shear force is necessarily zero.
Substituting the expansion (13) into (15), we obtain expressions of the form

(16) 
where a_{mn}, b_{mn} and c_{mn} are coefficients whose explicit form will be given shortly. Since Eq. (16) must hold for any m, n, x and y, it follows that a_{mn}=0, b_{mn}=0 and c_{mn}=0 for every m and n. The explicit forms of these coefficients are given by:
(17) 
or in matrix form
(18) 
where is
(19) 
Eqs. (18) provide three secondorder differential equations among the three variables U_{mn}, V_{mn} and W_{mn} for any fixed values of m and n.
For free vibration, all applied loads and the inplane forces are set to zero, and we assume a periodic solution of the form:
(20) 
where and is the frequency of natural vibration. Then Eq. (18) reduces to the eigenvalue problem:
( 
(21) 
For a nontrivial solution, the determinant of the coefficient matrix in (21) should be zero, which yields the characteristic polynomial
, 
(22) 
where is the eigenvalue and
(23) 
The real positive roots of this cubic equation give the square of the natural frequency associated with mode (m,n). The smallest of the frequencies is called the fundamental frequency. In general, is not the fundamental frequency; the smallest frequency might occur for values other than m = n = 1.
If the inplane inertias are neglected (i.e., ), and irrespective of whether the rotary inertia is zero, Eq. (22) will be
(24) 
Note that if the inplane inertias are not neglected, the eigenvalue problem cannot be simplified to a single equation, even if the rotary inertia is zero.
2.2. Firstorder shear deformation theory (FSDT)
2.2.1 Displacement and strains
Under the same assumptions and restrictions as in the classical laminate theory, the displacement field of the firstorder theory is of the form:

(25) 
Where
(26) 
which indicate that and are the rotations of a transverse normal about the y and x axes, respectively.
The nonlinear strains associated with the displacement field (25) are obtained as
(27) 
Note that the strains are linear through the laminate thickness, whereas the transverse shear strains are constant through the thickness of the laminate in the firstorder laminated theory. These strains have the fo
(28) 
2.2.2 Equilibrium equations
The governing equations of the firstorder theory will be derived using the dynamic version of the principle of virtual displacements. The EulerLagrange equations are obtained as follows
, 
(29) 
where Q_{x} and Q_{y} are called transverse force resultants and
(30) 
Parameter K is called a shear correction coefficient and is used because of a discrepancy between the actual stress state and the constant stress state predicted by the firstorder theory.
2.2.2 Boundary condition
The natural boundary conditions are obtained by setting the coefficients of and to zero separately:
(31) 
where
(32) 
Thus, the primary and secondary variables of the theory are
primary variables: secondary variables : 
(33) 
The initial conditions of the theory involve specifying the values of the displacements and their first derivatives with respect to time at t = 0.
2.2.3 Equations of motion
The boundary conditions are satisfied by the following expansions
(34) 
The Navier solution can be calculated from
(35) 
where

(36) 
For free vibration, all thermal and mechanical loads are set to zero and substitute to Eq. (29) and obtain
(37) 
where
,

(38) 
and . When rotary inertia is omitted, Eq. (37) can be simplified by eliminating Xmn and Ymn (i.e., using the static condensation method) as follows
(39) 
where
(40) 
If the inplane and rotary inertias are omitted (i.e., ), we have
(41) 
3. Finite Element Method
The finite element method (FEM), known as a powerful tool for many engineering problems, has been used to compute such matters as elasticplastic, residual and thermal stresses, and buckling and vibration analysis. Because of this, ANSYS software that is a commercial FEM program was preferred for the vibration analysis of the laminated composite plates. The Shell 99 element type was selected for the 2D modeling of solid structures in ANSYS. Initially, the plates are to get an initial estimate of the undamped natural frequencies ωn and mode shape n. The element type of Shell 99 may be used for layered applications of a structural shell model. The element has six degrees of freedom at each node; and translations in the nodal x and y directions and rotations about the nodal zaxis. This element is constituted by layers designated by numbers (LNlayer number), increasing from the bottom to the top of the laminate; the last number quantifies the existing total number of layers in the laminate (NLtotal number of layers). The boundary conditions have been applied to the nodes, i.e., the dimensions in the x and y are 400 mm for 2D, and the displacements and rotations of all nodes about the y–z plane are also taken as zero. The model of the laminated plate is generated with a different number of layers (based on different sidetothickness ratio). The boundary conditions and mesh shape are shown in Fig. 2. It is mentioned that for a free vibration analysis, the subspace method is applied. The subspace iteration method was described in detail by Bathe [24].
Figure 2: View of a laminated composite plate with boundary conditions and mesh shape.
After the mesh generation process, a laminated composite plate with six layers has 500 elements and 1,488 nodes. By increasing the number of layers, the numbers of elements and the nodes of the plates increase. The normal penalty stiffnesses of the contact element are chosen between 10^{4} and 10^{9}.
4. Results and Discussions
The nondimensionalized frequencies , of specially orthotropic and antisymmetric crossply square laminates are presented in Table 1 for modulus ratios and 20 (G_{12 }= G_{13 }= 0.5E_{2}, G_{23 }= 0.2E_{2}, ʋ_{12 }= 0.25). All layers are of equal thickness. Results are presented for m,n = 1,2,3 and for when the rotary inertia is neglected.
The fundamental frequency increases with the modular ratio. The effect of including rotary inertia is to decrease the frequency of vibration. Note that the first four frequencies for an antisymmetric crossply plates are (m,n) = (1,1), (1,2), (2,1) and (2,2) and for antisymmetric laminates. Also, good agreement was found between the analytical solution and the FEM analysis.
Fig. 3 shows a plot of fundamental frequency versus aspect ratio a/b for symmetric (0/90)_{s} crossply and antisymmetric (0/90)_{2} crossply laminates. The material properties used are , G_{12 }= G_{13}= 0.6E_{2}, G_{23 }= 0.2E_{2}, ʋ_{12 }= 0.25). Fig. 4 shows the effect of coupling between bending and extension on the fundamental frequencies of antisymmetric crossply laminates. The material properties used are , G_{12 }= G_{13 }= 0.5E_{2}, G_{23 }= 0.2E_{2}, ʋ_{12 }= 0.25). With an increase in the number of layers, the frequencies approach those of the orthotropic plate. The bendingstretching coupling lowers the vibration frequencies. For example, the twolayer plate has vibration frequencies about 40 percent lower than those of an eightlayer antisymmetric laminate or orthotropic plate with the same total thickness.

Figure 3: The nondimensionalized fundamental frequency versus the plate aspect ratio (a/b) for crossply laminates. 
Table 1. The nondimensionalized frequencies of crossply laminates, according to the classical plate theory 
m 
n 
Layup 

(0/90) 

(0/90)_{2} 

(0/90)_{3} 

Method 







1 
1 
Analytical 

1.066 

0.977 

1.359 

1.257 

1.445 

1.321 
FEM 

1.254 

1.098 

1.536 

1.478 

1.613 

1.528 

Reddy [23] 

1.183 

0.990 

1.479 

1.386 

1.545 

1.469 
















1 
2 
Analytical 

3.090 

2.697 

3.987 

3.851 

4.136 

4.087 
FEM 

3.265 

2.861 

4.182 

4.023 

4.351 

4.295 

Reddy [23] 

3.174 

2.719 

4.077 

3.913 

4.274 

4.158 
















2 
1 
Analytical 

3.090 

2.697 

3.987 

3.851 

4.136 

4.087 
FEM 

3.265 

2.861 

4.182 

4.023 

4.351 

4.295 

Reddy [23] 

3.174 

2.719 

4.077 

3.913 

4.274 

4.158 
















2 
2 
Analytical 

4.266 

3.911 

5.812 

5.449 

5.985 

5.774 
FEM 

4.882 

4.078 

6.081 

5.631 

6.227 

5.937 

Reddy [23] 

4.733 

3.959 

5.918 

5.547 

6.179 

5.877 
















3 
1 
Analytical 

6.542 

5.747 

8.537 

8.329 

8.989 

8.778 
FEM 

6.741 

5.829 

8.773 

8.546 

9.231 

9.097 

Reddy [23] 

6.666 

5.789 

8.698 

8.456 

9.136 

8.998 
















3 
2 
Analytical 

7.386 

6.014 

9.894 

9.423 

10.398 

9.997 
FEM 

8.011 

6.271 

10.125 

9.696 

10.553 

10.201 

Reddy [23] 

7.927 

6.193 

10.034 

9.507 

10.494 

10.088 
Figure 4: The nondimensionalized fundamental frequency versus the plate aspect ratio (a/b) for antisymmetric crossply laminates. 
Fig. 5 shows a plot of fundamental frequency versus the modulus ratio E_{1}/E_{2} for antisymmetric (0/90) crossply laminates for various values of plate aspect ratios. The plate aspect ratio lowers the vibration frequencies. The rectangle plate has vibration frequencies about 50 percent lower than those of a square plate with the same total thickness.
Figure 5: The nondimensionalized fundamental frequency versus the modulus ratio for antisymmetric crossply laminates for various plate aspect ratios. 
The nondimensionalized fundamental frequencies of graphiteepoxy composites with , G_{12}/E_{2 }= 0.5, ʋ_{12 }= 0.25 and a/b = 1 are shown as a function of the lamination angle in Fig. 6. The bendingstretching coupling due to the presence of B_{16} and B_{26} lowers the frequencies. The coupling is the maximum for twolayer plates, and it rapidly decreases with increasing number of layers. At θ = 45^{°}, the fundamental frequency of the twolayer plate is about 40 percent lower than that of the eightlayer laminate.
The nondimensionalized fundamental frequencies of graphiteepoxy composites with , G_{12}/E_{2 }= 0.5, ʋ_{12 }= 0.25 are shown as a function of plate aspect ratios in Fig. 7. With an increase in the number of layers, the frequencies approach those of the orthotropic plate. The bendingstretching coupling lowers the vibration frequencies. For example, the twolayer plate has vibration frequencies about 40 percent lower than those of a fourlayer antisymmetric angleply laminate or orthotropic plate with the same total thickness and aspect ratio. Also, effects of an aspect ratio on the fundamental frequencies of a laminated composite with same total thickness are more significant for values less than 1.
Figure 6: The nondimensionalized fundamental frequency versus the lamination angle of antisymmetric angleply square laminates.

Figure 7: The nondimensionalized fundamental frequency versus the plate aspect ratio of antisymmetric angleply laminates. 
Figure 8: The nondimensionalized fundamental frequency versus the modulus ratio of antisymmetric angleply square laminates. 
Fig. 8 shows a nondimensionalized fundamental frequency versus a modulus ratio of antisymmetric angleply square laminates. The effect of coupling is significant for all modulus ratios and the difference between the twolayer solution and orthotropic solution increases with the modulus ratio.
Table 2 and 3 contain the nondimensionalized fundamental frequencies for symmetric crossply laminates using the FSDT. The effect of the shear correction factor is to decrease the frequencies. The smaller the K, the smaller the frequencies are. The rotary inertia (RI) also decreases the frequencies.
Fig. 9 shows the effect of transverse shear deformation and rotary inertia on the fundamental natural frequencies of orthotropic and symmetric crossply (0/90/90/0) square plates with the following lamina properties:
, G_{12 }= G_{13 }= 0.5E_{2}, G_{23 }= 0.2E_{2}, ʋ_{12 }= 0.25.
The symmetric crossply plate behaves much like an orthotropic plate. The effect of rotary inertia is negligible in the FSDT and, therefore, is not shown in the figure.
Fig. 10 shows the effect of transverse shear deformation, bendingextensional coupling and rotary inertia on the fundamental natural frequencies of twolayer and eightlayer antisymmetric crossply laminates ( , G_{12} = G_{13} = 0.5E_{2}, G_{23} = 0.2E_{2}, ʋ_{12} = 0.25).
Table 2. The effect of shear deformation on the dimensionless natural frequencies of simply supported symmetric crossply plates. 
a/h 

Theory 

0^{o} 

Threeply 

Fiveply 

Sevenply 

Nineply 
5 

FSDT 

8.388 

8.094 

8.569 

8.673 

8.713 



9.019 

8.698 

9.197 

9.312 

9.357 




9.534 

9.196 

9.706 

9.829 

9.877 


FEM 

9.643 

9.234 

9.857 

9.997 

10.017 


CLPT 

14.750 

14.750 

14.750 

14.750 

14.750 

10 

FSDT 

12.067 

11.730 

12.167 

12.290 

12.342 



12.540 

12.223 

12.621 

12.735 

12.783 




12.890 

12.592 

12.956 

13.062 

13.107 


FEM 

12.901 

12.668 

13.078 

13.215 

13.295 


CLPT 

15.104 

15.104 

15.104 

15.104 

15.104 

20 

FSDT 

14.220 

14.042 

14.229 

14.288 

14.312 



14.411 

14.254 

14.412 

14.461 

14.461 




14.542 

14.402 

14.538 

14.580 

14.598 


FEM 

14.568 

14.523 

14.638 

14.705 

14.712 


CLPT 

15.197 

15.197 

15.197 

15.197 

15.197 

25 

FSDT 

14.569 

14.433 

14.563 

14.604 

14.621 



14.700 

14.582 

14.688 

14.722 

14.737 




14.789 

14.682 

14.774 

14.803 

14.815 


FEM 

14.812 

14.723 

14.835 

14.907 

14.918 


CLPT 

15.208 

15.208 

15.208 

15.208 

15.208 

50 

FSDT 

15.079 

15.015 

15.052 

15.063 

15.068 



15.115 

15.057 

15.086 

15.096 

15.100 




15.139 

15.085 

15.110 

15.117 

15.121 


FEM 

15.238 

15.128 

15.262 

15.236 

15.240 


CLPT 

15.223 

15.223 

15.223 

15.223 

15.223 

100 

FSDT 

15.215 

15.173 

15.183 

15.186 

15.187 



15.225 

15.184 

15.192 

15.194 

15.195 




15.231 

15.191 

15.198 

15.200 

15.200 




15.312 

15.284 

15.293 

15.301 

15.302 


CLPT 

15.227 

15.227 

15.227 

15.227 

15.227 
The first line corresponds to the shear correction coefficient of K=2/3 and the second and third lines correspond to the shear correction coefficient of K=5/6 and K=1.0, respectively.
Table 3. The effect of shear deformation, rotary inertia and the shear correction coefficient on the dimensionless natural frequencies of simply supported symmetric crossply (0/90/0) plates

a/h 

m 

n 

CLPT w/o RI 

CLPT with RI 

FSDT w/o RI 

FSDT with RI 
10 

1 

1 

15.228 

1.104 

12.593 

12.573 










12.223 

12.163 


1 

2 

22.877 

22.421 

19.440 

19.203 










18.942 

18.729 


1 

3 

40.229 

38.738 

32.496 

31.921 










31.421 

30.932 


2 

1 

56.885 

55.751 

33.097 

32.931 










31.131 

30.991 


2 

2 

60.911 

59.001 

36.786 

36.362 










34.794 

34.434 


1 

4 

66.754 

62.526 

48.837 

47.854 










46.714 

45.923 


2 

3 

71.522 

67.980 

45.484 

44.720 










43.212 

42.585 













100 

1 

1 

15.228 

15.227 

15.192 

15.191 










15.185 

15.183 


1 

2 

22.877 

22.873 

22.831 

22.827 










22.822 

22.817 


1 

3 

40.299 

40.283 

40.190 

40.147 










40.169 

40.153 


2 

1 

56.885 

56.874 

56.330 

56.319 










56.221 

56.210 


2 

2 

60.911 

60.891 

60.342 

60.322 










60.230 

60.211 


1 

4 

66.754 

66.708 

66.466 

66.421 










66.409 

66.364 


2 

3 

71.522 

71.484 

70.919 

70.882 










70.801 

70.764 
The first line corresponds to the shear correction coefficient of K=1.0 and the second line corresponds to the shear correction coefficient of K=5/6.
Figure 9: The nondimensionalized fundamental frequency versus the sidetothickness ratio for simply supported orthotropic and symmetric crossply (0/90/90/0) laminates.

. The eightlayer antisymmetric crossply plate behaves much like an orthotropic plate. The effect of rotary inertia is negligible in the FSDT and, therefore, is not shown in the figure.
Table 4 contains numerical values of the fundamental frequencies of antisymmetric crossply laminated plates for various modular ratios. The results for both twolayer and eightlayer laminated plates for square and rectangular geometries are presented.
Figure 10: The nondimensionalized fundamental frequency versus the sidetothickness ratio for simply supported orthotropic and antisymmetric crossply (0/90) laminates. 
Table 4. The effect of shear deformation on the nondimensionalized fundamental frequencies of simply supported antisymmetric crossply plates. 


5. Conclusions
Analytical and numerical solutions for the free vibration of laminated polymeric composite plates with different layups are compaired based on different plate theories. Also, the effects of some parameters on the fundamental frequencies of laminated plate were performed. As a verification method, an FEM was applied with ANSYS to compare the results with those obtained from a closedform solution. Based on the results observed, the following comments are as such:
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