Bending , Vibration and Buckling of Laminated Composite Plates Using a Simple Four Variable Plate Theory

In the present study, a simple trigonometric shear deformation theory is applied for the bending, buckling and free vibration of crossply laminated composite plates. The theory involves four unknown variables which are five in first order shear deformation theory or any other higher order theories. The in-plane displacement field uses sinusoidal function in terms of thickness co-ordinate to include the shear deformation effect. The transverse displacement includes bending and shear components. The present theory satisfies the zero shear stress conditions at top and bottom surfaces of plates without using shear correction factor. Equations of motion associated with the present theory are obtained using the dynamic version of virtual work principle. A closed form solution is obtained using double trigonometric series suggested by Navier. The displacements, stresses, critical buckling loads and natural frequencies obtained using present theory are compared with previously published results and found to agree well with those.

Since the composite materials are increasingly being used in many engineering applications due to their attractive properties, such as stiffness, strength, weight reduction, corrosion resistance, thermal properties, fatigue life, and wear resistance.The plates made up of such materials are required accurate structural analysis to predict the correct bending behaviour.The effect of transverse shear deformation is more significant in thick plates than in thin plates.Therefore, various plate theories have been developed by researchers to predict correct bending behaviour of thick plates.The classical plate theory (CPT) of Kirchhoff (1850) is not suitable for thick plate due to neglect of transverse shear deformation.The first order shear deformation theory (FSDT) developed by Mindlin (1951) is also not suitable for the analysis since it does not satisfy the zero stress conditions at top and bottom surfaces of the plate and required problem dependent shear correction factors.Therefore, many five variable and six variable plate theories have been developed for the analysis of plates (Reddy, 1984;Touratier, 1991;Soldatos, 1992;Karama et al., 2009;Sayyad Latin American Journal of Solids and Structures 13 (2016) 516-535 and Ghugal, 2012;Sayyad, 2013;Zenkour, 2006;Sayyad and Ghugal, 2013;2014a;2014b;Ghugal and Sayyad, 2010;2013a;2013b;Metin, 2009).
In the last decade, a new class of plate theories has been developed by researchers in which displacement field involves only four unknowns.Shimpi and Patel (2006) were the first to present a plate theory involving two unknowns for bending and free vibration analysis of orthotropic plates.This theory is further extended by Thai and Kim (2010) for the free vibration analysis of cross-ply and angle-ply laminated plates considering four and five unknowns.Kim et al. (2009) also used this theory for the buckling analysis of orthotropic plates using the Navier solution technique.Thai and Kim (2011;2012) employed Levy type solution for the bending and buckling analysis of orthotropic plates.After this, a lot of research is reported in the literature on different four variable plate theories.However, these theories are applied for bending, buckling and free vibration analysis of functionally graded plates only (Ameur, et al. 2011;Thai and Vo, 2013;Meiche et al., 2011;Daouadji et al., 2012;2013;Zenkour, 2013).Recently Sayyad and Ghugal (2015) have presented a critical review of literature on refined shear deformation theories for the free vibration analysis of laminated composite and sandwich plates.Wherein, theories involving four or more than four are reviewed and discussed.
In the present study, an attempt is made to check the efficiency of four variable refined trigonometric shear deformation theory for the bending, buckling and free vibration analysis of cross-ply laminated composite plates.A trigonometric function in terms of thickness co-ordinate is used in the kinematics of the theory to account for shear deformation effects.The theory enforces cosine distribution of transverse shear stresses and satisfies zero shear stress conditions at top and bottom surfaces of the plates.The theory does not need problem dependent shear correction factor.Governing equations and boundary conditions are obtained using the virtual work principle.A closed form solution is obtained by employing a double trigonometric series technique developed by Navier.Finally, the numerical results obtained by using present theory are compared with exact elasticity solutions given by Pagano (1970) for bending, Noor (1973) for free vibration and Noor (1975) for buckling analysis of laminated composite plates.

Laminated Plate Under Consideration
A rectangular plate of the sides 'a' and 'b' and total thickness 'h' as shown in Figure 1 is considered.The plate consists of N number of homogenous layers.All the layers are perfectly bounded together and made up of linearly elastic and orthotropic material.The plate occupies the region 0 ≤ x ≤ a, 0 ≤ y ≤ b, -h/2 ≤ z ≤ h/2 in Cartesian coordinate system.The downward z-direction is taken as positive.The plate is subjected to transverse load q(x, y) on the upper surface of the plate (i.e.z = -h/2) or subjected to in-plane compressive forces (uniaxial/biaxial).Structures 13 (2016) 516-535 Figure 1: Geometry and co-ordinate system of laminated plate.

Assumptions Made in Mathematical Formulation.
Mathematical formulation of the present theory is based on the following assumptions.
1.The displacements are small in comparison with the plate thickness and, therefore, strains involved are infinitesimal.2. The displacements u in x-direction and v in y-direction consist of extension (u0), bending (ub) and shear components (us).(2)

Kinematics and Constitutive Relations
Based on the before mentioned assumptions the following displacement field associated with present theory is obtained.
The non-zero normal and shear strain components are obtained using strain displacement relations given by Jones (1975).
Latin American Journal of Solids and Structures 13 (2016) 516-535 The constitutive relationships for the k th layer can be given as, where ij Q are the plane stress reduced elastic constants in the material axes of the plate, and are defined as:  ,  ,  ,  1 1 1 where E1, E2 are the Young's moduli along and transverse direction of the fibre, G12, G13, G23 are the in-plane and transverse shear moduli and 12 21 ,   are the Poisson's ratios.The force and moment resultants of a present theory can be obtained by integrating stresses given by Eq. ( 5) through the thickness and are as follows:   where hk is the thickness ordinate of k th layer.The terms  

EQUATIONS OF MOTION
The variationally consistent governing equations of motion and boundary conditions associated with the present theory can be derived using the principle of virtual work.The analytical form of the principle of virtual work can be written as: where  be the variational operator.Integrating Eq. ( 8) by parts and collecting the coefficients of : Latin American Journal of Solids and Structures 13 (2016) 516-535 Substituting the stress resultants in terms of displacement variables from Eq. ( 7) into the Eqs.( 9) -( 12), the governing equations of equilibrium can be rewritten as: A ,B , As ,D ,Bs , Ass , Acc are the laminate stiffness coefficients and I0, I1, I2, I3 are the inertia constants which are given as: Latin American Journal of Solids and Structures 13 (2016) 516-535 This completes the mathematical formulation of the present trigonometric shear deformation theory.

NAVIER SOLUTION TECHNIQUE
The Navier solution technique (Szilard, 2004) is used for the bending, buckling and free vibration analysis of laminated composite plates simply supported at all four edges (pinned edges) satisfying the following boundary conditions:

Bending Analysis of Laminated Composite Plates
Following the Navier solution technique, the governing equations of the simply supported laminated composite plates in case of bending analysis are obtained by discarding in-plane compressive loads ( 0 0 0 xx y y x y N ,N ,N ) and inertia terms (I0, I1, I2, I3) from Eqs. ( 13) -( 16).
The plate is subjected to transverse load q(x, y) at top surface i.e. z = -h/2.The transverse load is presented in double trigonometric series as given in Eq. ( 25).
  where and qmn is the coefficient of Fourier expansion.The coefficient of Fourier expansion (qmn) is: where q0 is the maximum intensity of load at the center of plate.The following solution form is assumed for the unknown displacement variables 0 u , 0 v , b w and s w satisfying the boundary conditions of simply supported plates exactly.
where and mn mn bmn smn u ,v ,w w are the unknown constants to be determined.In case of sinusoidally distributed load, positive integers are unity (m = 1, n = 1).Substitution of this form of solution and transverse load q(x, y) into the governing equations ( 21) -( 24) leads to the set of algebraic equations which can be written in matrix form as follows.
where elements of stiffness matrix [K] are as follows: Latin American Journal of Solids and Structures 13 (2016) 516-535 From the solution of Eq. ( 28), unknown constants and mn mn bmn smn u ,v ,w w can be obtained.Having obtained values of these unknown constants one can then calculate all the displacement and stress components within the plate using Eqs.( 3 The displacements and stresses are presented in the following non-dimensional form.

Buckling Analysis of Laminated Composite Plates
In this section, an analytical solution for the buckling analysis of plate is developed using Navier solution technique.The governing equations of the plate in case of static buckling are obtained by discarding transverse load (q) and inertia terms (I0, I1, I2, I3) from Eqs. ( 13) -( 16).The in-plane compressive ( 0 0 0 and xx y y x y N ,N N ) forces now represents loads instead of reaction forces, as there is no transverse load.The values of in-plane compressive forces are taken as 0 The governing equations for static buckling are as follows: where N0 is the intensity of in-plane compressive force.After substituting Eq. ( 27) into Eqs.( 32) -( 35), the following system of equations in matrix form is obtained.
where the element of stiffness matrix [Kij] are given in Eq. ( 29).For nontrivial solution, the determinant of the coefficient matrix in Eq. ( 36) must be zero.For each choice of m and n, there is a corresponding unique value of N0.The critical buckling load is the smallest value of N0(m, n).A simply supported laminated composite square and rectangular plates subjected to the uniaxial and biaxial loading conditions, as shown in Figure 5, is considered to illustrate the accuracy of the present theory in predicting the buckling behaviour.The following material properties are used in the numerical study.

Free Vibration Analysis of Laminated Composite Plates
According to Navier solution technique, the governing equations of the plate in case of free vibration analysis are obtained by discarding transverse load (q) and in-plane compressive forces ( 0 from Eq. ( 13) -( 16).These equations are as follows: The following solution form is assumed for unknown displacement variables 0 Substituting Eq. ( 43) into the Eqs.( 39) -( 42), the following system of equations is obtained.

Bending Analysis of Laminated Composite Plates
In this example the efficacy of present theory is proved for the bending analysis of simply supported two layered (0 0 /90 0 ) anti-symmetric laminated composite square plates subjected to sinusoidally distributed load.The non-dimensional displacement and stresses obtained using present theory are compared and discussed with those obtained by classical plate theory (CPT) of Kirchhoff (1850), first order shear deformation theory (FSDT) of Mindlin (1951), higher order shear deformation theory (HSDT) of Reddy (1984), sinusoidal shear and normal deformation theory (SSNDT) of Sayyad and Ghugal (2014a) and exact elasticity solution given by Pagano (1970).The non-dimensional numerical results are summarized in Table 1.It is observed that the in-plane displacement predicted by present theory is in good agreement with other theories.In-plane displacement is maximum in 90 0 layer whereas minimum in 0 0 layer (Figure 2).The present theory underestimates the value of transverse displacement for aspect ratio 4 but it is in good agreement with exact solution and other higher order


. Through thickness distribution of this stress is shown in Figure 4.

Buckling Analysis of Laminated Composite Plates
A comparison of the critical buckling load parameters obtained by the present theory for a three layered (0 0 /90 0 /0 0 ) and five layered (0 0 /90 0 /0 0 /90 0 /0 0 ) symmetric cross-ply laminated composite square plates subjected to uniaxial and biaxial compressions for various modular ratios (E1/E2) is presented in Table 2.All the layers are of equal thickness.The results of present theory are compared with HSDT of Reddy (1984), SSNDT of Sayyad and Ghugal (2014b) FSDT of Mindlin (1951) and CPT of Kirchhoff (1850) and exact elasticity solution given by Noor (1975).The material properties used for this example are shown in Eq. ( 37).From the examination of Table 2 it is observed that the present results are in excellent agreement with exact solution as well as HSDT of Reddy (1984).It is also observed that the buckling loads predicted by CPT are significantly higher than those obtained by the present theory.This is the consequence of neglecting the transverse shear deformation effect in the CPT.It can be seen from Table 2 that the critical buckling loads in case of biaxial compression are exactly half of those of uniaxial compression for square plates.Table 3 shows the critical buckling load parameter for four layered (0 0 /90 0 /90 0 /0 0 ) symmetric laminated composite rectangular plate.The numerical results are obtained for various values of b/a ratios and a/h ratios.From Table 3 it is observed that the critical buckling load increases with respect to increase in 'b/a' and 'a/h' ratios.It is also pointed out that the present theory is in excellent agreement while predicting the buckling behaviour of rectangular laminated composite plates.

Free Vibration Analysis of Laminated Composite Plates
In Table 4, non-dimensional natural frequencies of simply supported square laminated composite plates for various modular ratios (E1/E2) are presented and compared with those obtained by SSNDT of Sayyad and Ghugal (2015), HSDT of Reddy (1984), FSDT of Mindlin (1951) and CPT of Kirchhoff (1850).In all the lamination schemes, the layers are of equal thickness.The material properties are shown in Equation ( 37).The exact elasticity solution for free vibration analysis of laminated composite plates given by Noor (1973) is used for the purpose of comparison.From the Table 8 it is observed that the present theory is in excellent agreement while predicting the natural frequencies of laminated composite plates.The CPT overestimates the natural frequencies because of neglect of the transverse shear deformation effect.It is also observed that the natural frequencies of laminated composite plates increase with respect to increase in modular ratios (E1/E2).

CONCLUSIONS
In the present study, a refined trigonometric shear deformation theory is applied for the bending, buckling and free vibration analysis of laminated composite plates.The most important feature of the present theory is that it involves only four unknowns which are five in case of first order shear Latin American Journal of Solids and Structures 13 (2016) 516-535 deformation theory and other higher order theories.The present theory satisfies the traction free conditions at top and bottom surfaces of plates without using shear correction factor.From the mathematical formulation of present theory, it is observed that, due to four unknown variables, the present theory requires less computational efforts compared to five and six variable shear deformation theories.From the numerical results and discussion it is concluded that present theory is in good agreement while predicting the bending, buckling and free vibration behaviour of laminated composite plates.
moduli along and transverse directions of the fibre G12, G13, G23 In-plane and transverse shear moduli

) 3 .
The transverse displacement w includes two components, i.e. bending   are the in- plane force and moment resultants associated with the classical plate theory whereas,  , shear force and moment resultants associated with the transverse shear deformation.
governing equations of equilibrium and boundary conditions (Euler-Lagrange equations) associated with the present theory are obtained using fundamental lemma of calculus of variation.The governing equations of plate equilibrium are as follows:

Figure 5 :
Figure 5: A simply supported plate subjected to in-plane compressive forces.   ) -(5).Transverse shear stresses     to ascertain the continuity at layer interface.The following material properties are used for the bending analysis of simply supported anti-symmetric laminated composite square plates subjected to sinusoidally distributed load.

Table 1 :
Comparison of non-dimensional displacements and stresses for the two layered (0 0 /90 0 ) laminated composite square (b = a) plate subjected to sinusoidally distributed load.

Table 4 :
Comparison of non-dimensional natural frequencies of simply supported square laminated composite plates (b = a, a/h = 5).

Latin American Journal of Solids and Structures 13 (2016) 516-535 theories
for aspect ratio 10.Figure3shows that, in-plane normal stress  