Nonlinear vibration of symmetrically laminated composite skew plates by finite element method

Transcription

1 International Journal of Non-Linear Mechanics 42 (27) Nonlinear vibration of symmetrically laminated composite skew plates by finite element method M.K. Singha, Rupesh Daripa Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi - 6, India Received 7 November 26; received in revised form 24 July 27; accepted 28 August 27 Abstract Here, the large amplitude free flexural vibration behavior of symmetrically laminated composite skew plates is investigated using the finite element method. The formulation includes the effects of shear deformation, in-plane and rotary inertia. The geometric non-linearity based on von Kármán s assumptions is introduced. The nonlinear matrix amplitude equation obtained by employing Galerkin s method is solved by direct iteration technique. Time history for the nonlinear free vibration of composite skew plate is also obtained using Newmark s time integration technique to examine the accuracy of matrix amplitude equation. The variation of nonlinear frequency ratios with amplitudes is brought out considering different parameters such as skew angle, fiber orientation and boundary condition. 27 Elsevier Ltd. All rights reserved. Keywords: Composite skew plate; Nonlinear vibration; Finite element; Frequency ratio; Amplitude. Introduction The vibration characteristics of isotropic and composite plates have received considerable attention of many researchers [ 6]. In the case of linear vibration, the approximate solution of the equation of motion is readily obtainable and the problem of determination of frequencies and mode shapes is reduced to a generalized eigenvalue problem. However, for the case of non-linear free vibration of plates, the equation of motion is complex, and obtaining an approximate solution is much more difficult. The first approximate study of the equation of motion for the large amplitude free flexural vibration behavior of simply supported isotropic plates is reported by Chu and Herrmann [7]. The equation of motion for the non-linear vibration of isotropic plates was reduced to a single non-linear differential equation in time by assuming single space mode, which was solved by classical elliptic function approach. Subsequently, assumed single space-mode approximate non-linear differential equations were solved by different investigators using elliptic function [8,9], perturbation theory [,], Corresponding author. Tel.: ; fax: address: maloy@am.iitd.ernet.in (M.K. Singha). Runge Kutta integration [,2], Rayleigh Ritz method [3] and Galerkin Method[4] to study the nonlinear flexural vibration frequencies of isotropic plates. Sheikh and Mukhopadhyay [5] employed spline finite strip method, while many others [6 22] employed finite element method to study the nonlinear vibration characteristics of isotropic and composite plates. It is observed from the existing literature that the nonlinear frequencies of rectangular plates reported by different researchers vary widely. Fundamental frequency ratio ω NL /ω L (subscripts NL and L correspond to the non-linear and linear, respectively) at w max /h=. for a simply supported isotropic thin square plate obtained by different investigators varies from a low of.2967 [6] to the high of.626 [4] with the analytical solution at.424 [7]. Benamar et al. [24] calculated the average and standard deviation of nonlinear frequencies reported by different investigators for a clamped square plate at different amplitude of vibration. Hence, the question arises whether the assumed single space-mode approximate solutions are accurate or the finite element solutions are more accurate. There are two flaws in the finite element studies of Refs. [6 23]. First one is related to the erroneous formulation of non-linear stiffness matrix in Refs. [6 8] and the second one is the selection of time function in Refs. [9 23]. The sources of error in nonlinear finite element formulation using linearization technique /$ - see front matter 27 Elsevier Ltd. All rights reserved. doi:.6/j.ijnonlinmec.27.8.

2 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) has been discussed by Prathap [25], Sharma and Varadan [26], Singh et al. [27] and Dumir and Bhaskar [28]. The non-linear vibrations of plate like structures involve the periodic solution of the differential equations [K L + 2 N (δ) + 3 N 2(δ, δ)]{δ} +[M]{ δ} = with quadratic and cubic nonlinearity. The non-linear matrix formulation of Refs. [6 8] appears to be erroneous [28]. Ganapathi et al. [9], Sivakumar et al. [2], Singha and Ganapathi [2], Sundararajana et al. [22] and Rao et al. [23] have solved the governing equation at the instant of maximum displacement {δ max } and obtained the nonlinear vibration frequencies of isotropic, composite and FGM plates. However, their solutions do not satisfy the equilibrium equation at all times over the period and hence do not match with the analytical results. Moreover, the above free vibration results differ from that of the forced vibration results [28] obtained either by Runge Kutta integration or finite element time integration using Newmark s technique. The notable contributions to correct the differences between free and forced vibration results are the use of Harmonic balance method [3,3], Multi-mode approach [32], Incremental Hamilton s principle [33,34] and energy balance method [24,35]. Several aspects of the solution of matrix amplitude equations for the free vibration of nonlinear systems have been discussed by Lewandowski [29]. It is further observed from the existing literature that the large amplitude flexural vibration of rectangular and circular plates has received considerable attention of the researchers. The plates with non-rectangular plan-forms like skew plates find wide application in the aerospace industry. Though a huge body of literature exists on the linear free vibration of isotropic and composite skew plates [2], such analysis concerning the non-linear free vibration of skew plates is sparsely treated in the literature. Ray et al. [36] have analytically studied the nonlinear vibration behavior of isotropic skew plates. Sathyamoorthy and Pandalai [37] analyzed single layer orthotropic skew plates for movable in-plane edges and Berger approximation for immovable in-plane edge conditions. Prathap and Varadan [38], and Sathyamoorthy and Chia [39] have investigated large amplitude vibration of anisotropic skew plates using the Galerkin method on the basis of a single term assumed vibration mode. All these investigations introduce assumed mode while solving the problem. Numerical methods such as finite element procedure is preferable over the analytical methods, as there is no need for an a priori assumption of the mode shapes, and the solution itself predict the mode shapes [2]. Such analysis for skew plates appears to be limited in the literature. Most recently, Singha and Ganapathi [2] have employed high precision plate bending element to solve the nonlinear free vibration problem of composite skew plates. However, the authors have solved the governing equation at the point of maximum displacement, which was responsible for getting higher values of frequencies. In the present paper, a four-noded shear flexible quadrilateral high precision plate bending element is developed to analyze the large amplitude free vibration characteristics of thin laminated composite skew plates. As the element is free from locking phenomenon, all the energy terms are evaluated using full numerical integration scheme. The formulation includes inplane and rotary inertia effects. The weighted residual of the non-linear governing equations over a period is set to zero. The nonlinear matrix amplitude equation thus obtained is solved by direct iteration technique using necessary convergence criteria [2,29]. The formulation developed here is validated with the available analytical solutions and finite element results. A limited parametric study is carried out to examine the influences of the skew angle, lay-up sequence and boundary conditions on the non-linear free vibration behavior of laminated composite skew plates. 2. Finite element formulation The displacement components at an arbitrary point (x,y,z) of a quadrilateral plate can be expressed as u(x,y,z)= u (x, y) + z{ w,x + γ x (x, y)}, v(x,y,z) = v (x, y) + z{ w,y + γ y (x, y)}, w(x,y,z) = w (x, y). () Here, u, v, w are the mid-surface displacements; γ x and γ y are the rotations due to shear; (),x and (),y represent the partial differentiation with respect to x and y; and x = w,x +γ x (x, y) and y = w,y + γ y (x, y) are the nodal rotations. Following von Kármán strain displacement relation, the inplane and shear strains can be written as { } { } εx u,x w,x 2 /2 ε y = v,y + w,y ε xy v,x + u,y 2 /2 w,x w,y { } w,xx + γ x,x + z w,yy + γ y,y 2w,xy + γ y,x + γ x,y ={ε }+z{κ} (2a) and { } { } γxz γx =. (2b) γ yz γ y In the present work, oblique coordinate system (Fig. ) is used. ψ is the skew angle measured from Y -axis. The relationships between global coordinate system (x G,y G,z G ) and oblique coordinate system (x,y,z) are as follows: x G = x + y sin ψ, y G = y cos ψ, z G = z. (3) The first and second derivatives of a function f(x,y) may be expressed in global Cartesian coordinate as [4] f x G f y g [ = tan ψ sec ψ ] f x f y, (4)

3 46 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) Y G Y Substituting Eqs. (6) and (8) in Lagrange s equation of motion, one obtains the governing equation as [K L + 2 N (δ) + 3 N 2(δ, δ)]{δ}+[m]{ δ}={f }, (9) Ψ 2 f x 2 G 2 f y 2 G 2 2 f x G y G a b X G Fig.. Oblique coordinate system for the skew plate. [ = tan 2 ψ sec 2 ψ sec ψ tan ψ 2 tan ψ sec ψ 2 f x 2 2 f y f x y. (5) The internal strain energy (U), and the work done (W) by the external load q(x,y) acting over the surface of the plate can be written as U(δ) = [{ε} T [A]{ε}+{ε} T [B]{κ}+{κ} T [B] T {ε} 2 A +{κ} T [D]{κ}+{γ} T [S]{γ}] da or U(δ) ={δ} T [ 2 K L + 6 N (δ) + 2 N 2(δ, δ)]{δ} (6) and W(δ) = q(x,y)da, (7) A where [A], [B], [D], and [S] are extensional, bending extensional, bending, and shear stiffness coefficients, respectively. For a composite laminate of thickness h, comprising of N layers with stacking angles θ i (i =, 2,...,N)and layer thicknesses h i (i =, 2,...,N), the necessary expressions to compute the stiffness coefficients are available in the literature [4]. The kinetic energy of the plate is given by T(δ) = [(u v2 + w2 )ρ + ( 2 x + 2 y )ρz2 ] dv. (8) V ] X where [M] is mass matrix, [K L ] is the linear stiffness matrix, [N ] and [N 2 ] are non-linear stiffness matrices and {δ} is the vector of nodal displacements and {F } is the vector of nodal forces. The governing Eq. (9) is solved using finite element approach based on C continuous four-noded quadrilateral plate bending element with 4 degrees of freedom per node, namely u, u,x, u,y, v, v,x, v,y, w, w,x, w,y, w,xx, w,xy, w,yy, γ x, and γ y.. The cubic polynomial shape functions are employed to describe the field variables corresponding to in-plane displacements (u,v ), quintic polynomial function is considered for the lateral displacement (w), whereas, linear polynomial shape functions are used for the rotations due to shear of the middle surfaces (γ x, γ y ), and are expressed as follows: u =[,x,y,x 2,xy,y 2,x 3,x 2 y,xy 2,y 3,x 3 y,xy 3 ]{c i }, i =, 2, v =[,x,y,x 2,xy,y 2,x 3,x 2 y,xy 2,y 3,x 3 y,xy 3 ]{c i }, i = 3, 24, w =[,x,y,x 2,xy,y 2,x 3,x 2 y,xy 2,y 3,x 4,x 3 y, x 2 y 2,xy 3,y 4,x 5,x 4 y,x 3 y 2,x 2 y 3,xy 4,y 5,x 5 y, x 3 y 3,xy 5 ]{c i }, i = 25, 48, γ x =[,x,y,xy]{c i }, i = 49, 52, γ y =[,x,y,xy]{c i }, i = 53, 56, () where c k are constants and are expressed in terms of nodal displacements in the finite element discretization. It is to be noted here that the present element is an improvement over the element used in Ref. [2]. The linear interpolation functions of Ref. [2] are changed to cubic interpolation polynomials to describe the in-plane displacements to achieve better convergence. The shear correction factor is taken as 5 6. The full integration scheme with 6 6 Gaussian integration rule is adopted for computing the element mass matrix M, whereas 4 4 Gaussian) integration rule is used to calculate the stiffness matrices K and N and N Solution procedure Introducing a harmonic solution of the form {δ} = {δ max } sin ωt (ω is the natural frequency) for the large amplitude free flexural vibration problem (F = ) of composite plates, Eq. (9) reduces into the following expression: [K L + 2 N (δ max ) sin ωt + 3 N 2(δ max, δ max )sin 2 ωt] {δ max } sin ωt ω 2 [M]{δ max } sin ωt ={R}, () where {R} is the residual vector.

4 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) For the linear free vibration problem, the nonlinear matrices N and N 2 does not exist and Eq. () reduces to the following eigenvalue problem: [K L ]{δ} ω 2 [M]{δ}={}. (2) 3.. Nonlinear free vibration At the point of maximum displacement {δ}={δ max }, t =T/4 (T is the time period), ωt = π/2, sin ωt =, Eq. () reduces to [9 23] [K L + 2 N (δ max ) + 3 N 2(δ max, δ max )] {δ max } ω 2 [M]{δ max }={}. (3) It may be noted here that some of the earlier investigators [9 23] have solved Eq. (3) as an eigenvalue problem and predicted higher nonlinear frequencies. Since eigenvalue equation (3) does not satisfy the equilibrium equation (9) for all the times, the nonlinear frequencies so obtained are incorrect. Taking the weighted residual [29] along the path t = tot/4({δ}={} to {δ max }) we get T/4 {R} sin ωt dt =. (4) Evaluating the integral (4) with the relations T/4 sin 2 ωt dt= T/8; T/4 sin 3 ωt dt = T/3π and T/4 sin 4 ωt dt = 3T/32, the following eigenvalue problem is obtained: [ K L + 4 3π N (δ max ) + ] 4 N 2(δ max, δ max ) {δ max } ω 2 [M]{δ max }={}. (5) The nonlinear vibration problem is solved using the matrix amplitude equation (5) as an eigenvalue problem [9 23,29]. Firstly, the eigenvector (mode shape) is obtained from the linear vibration analysis (Eq. (2)), and normalized. Next, the normalized vector is amplified/scaled up so that the maximum displacement is equal to the desired amplitude, say w/h =.4 (w is the maximum lateral displacement, h is the thickness of the plate). This gives the initial vector, denoted by δ. The iterative solution procedure for the nonlinear analysis starts with initial vector, δ. Based on this initial mode shape ( δ), the nonlinear stiffness matrices N and N 2, that depends on displacement, is formed and subsequently, the updated eigenvalue and its corresponding eigenvector are obtained. This eigenvector is further normalized, and scaled up by the same amplitude (w/h), and the iterative procedure adopted here continues till the frequency values and mode shapes evaluated from the subsequent two iterations satisfy the prescribed convergence criteria [2,29]. For the time history analysis, Newmark s time integration technique is employed to solve Eq. (9) with the initial condition {δ} ={δ max } obtained from Eq. (5). A convergence study is carried out to select a time step, which yields a stable and accurate solution. 4. Results and discussion Large amplitude free flexural vibration characteristics of thin laminated composite skew plates are studied here. The material properties, unless specified otherwise, used in the present analysis are E L /E T = 4., G LT /E T =.6, G TT /E T =.5, ν LT =.25, E T =. and ρ =., where E, G, ν, and ρ are Young s modulus, shear modulus, and Poisson s ratio and density. Subscripts L and T represent the longitudinal and transverse directions, respectively, with respect to the fibers. All the layers are of equal thickness. Fiber orientation is measured from X-axis. The boundary conditions considered here are: Simply supported case: u = v = w = Clamped edge: u = v = w =, u = v = w =, at x =,a and y =,b. w,x = atx =,a, w,y = aty =,b. Before proceeding for the detailed study for the nonlinear free vibration behavior, the formulation developed herein is validated against the linear free vibration of laminated composite skew plate. The non-dimensional linear frequencies (ϖ = ωa 2 /π 2 h ρ/e T ; a and h are length and thickness of the plate) obtained for simply supported thin (a/h = ) and thick (a/h = ) cross-ply [9 / /9 / /9 ] composite skew plates are presented in Table along with the available analytical solutions of Wang [42,43] and they match very well for both thin and thick cases. Numerical full integration scheme is used in evaluating the various element level matrices and the element does not show any locking as it can be seen from the thin plate results of Table. It is also observed that the element employed here has a good convergence property and thus, an 8 8 mesh (with a total degree of freedom 34) is found to be adequate to model the full plate. Further, the efficacy of the present finite element formulation is tested by studying the nonlinear free flexural vibrations of thin isotropic square plates (a/ h = ) with simply supported and clamped boundary conditions for which several analytical and numerical results are available in the literature. The variation of non-dimensional nonlinear frequency ratio (ω NL /ω L ; where subscripts NL and L correspond to the nonlinear and linear, respectively) with respect to non-dimensional maximum amplitude (w max /h; w max is the maximum amplitude of the plate) is evaluated for simply supported and clamped square plates and are shown in Tables 2 and 3, respectively, along with published results. It is observed that present results (obtained from Eq. (5)) are in close agreement with those of assumed space-mode analytical solutions [7 2] and finite element studies with solution methodologies of harmonic balance method [3], multi-mode approach [32], incremental Hamilton s principle [33,34] and energy balance

5 48 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) Table Convergence study of non-dimensional linear frequency (ϖ=ωa 2 /π 2 h ρ/e T ) of five-layered [9 / /9 / /9 ] simply supported skew laminates (a/b=) Skew angle Mesh size Modes Thin plate (a/h = ) Present study Ref. [42] Present study Ref. [42] Thick plate (a/h = ) Present study Ref. [43] Present study Ref. [43] Table 2 Comparison of nonlinear frequency ratio (ω NL /ω L ) of simply supported isotropic square plate (a = b, a/h = ) W/h Analytical integration [7] Direct integration [,] a Perturbation method [] Perturbation method [] Runge Kutta integration [] Runge Kutta integration [2] Rayleigh-Ritz method [3] Spline finite strip method [5] Finite element [6 8] a Chen et al. [4].626 FE eigenvalue equation (3) [9,2 23] a FE contd. method [33] Multi-mode approach [32] FE time history [Present] Matrix amplitude equation (5) [Present] a Ref.: References within % error. method [24,35]. The time history response for the large amplitude free vibration problem is also solved by using Newmark s time integration technique with the initial condition {δ}={δ max } obtained from Eq. (5) and the corresponding non-linear period of vibration are found to be in excellent agreement with the above results. Here, some published finite element results underestimates the non-linear frequencies [6 8] due to their erroneous element formulation, while some others [9,2 23] overestimates the nonlinear frequencies due to their solution methodology [5,25 28]. It is observed that the present

6 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) Table 3 Comparison of nonlinear frequency ratio (ω NL /ω L ) of clamped isotropic square plate (a = b, a/h = ) W/h Elliptic integral [8] FE eigenvalue equation (3) [23] Spline finite strip method [5] HBM [3] FE contd. method [34] Energy balance method [24] FE time history [Present] Matrix amplitude equation (5) [Present] Table 4 The non-linear frequency ratios (ω NL /ω L ) of simply supported cross-ply [ /9 / /9 / ] thin skew plates (a = b, a/h = ) w max /h Skew angle = Skew angle = 3 Eq. (3) Eq. (5) FE time history Eq. (3) Eq. (5) FE time history Linear response T L =.548 T L = 8.5 W max /h Time, t (sec) W max /h Time, t (sec) ω NL ω L T = L TNL =.587 ω NL ω L T L = TNL =.545 Nonlinear response at w max / h =. T NL = T NL = 5.52 W max /h Time, t (sec) W max /h Time, t (sec) Fig. 2. Time history for the nonlinear free vibration of simply supported cross-ply [ /9 / /9 / ] skew plate studied in Table 4 (a = b, a/h = ): (a) square plate and (b) skew plate (skew angle =3 ). simplified eigenvalue formulation (Eq. (5)) based on weighted residual approach appears to be accurate for isotropic plates. Next, the nonlinear free vibration behaviors of simply supported cross-ply [ /9 / /9 / ] thin (a/h = ) square and skew plates (skew angle 3 ) are examined in Table 4. The results are obtained using three different solution methodologies: (a) eigenvalue equation (3), which satisfies the equilibrium equation only at the point of maximum displacement, (b) eigenvalue equation (5) based on weighted residual approach, and (c) time integration by Newmark s technique with the initial condition {δ} ={δ max }. The time history for the nonlinear free vibration problem of the thin (a/ h = ) square and skew composite plates at a vibration amplitude equal to thickness of the plate (w max /h =.) is presented in Fig. 2. The period of vibration (T NL ) is measured from the time history curves and the non-dimensional non-linear frequency ratio (ω NL /ω L = T L /T NL ) is calculated. It is observed that the results obtained from Eq. (5) closely matches with those from time history response for the case of isotropic plates in Tables 2, 3 and symmetrically laminated composite skew plates in Table 4. The influence of thickness-to-span ratio on the nonlinear vibration frequencies of composite skew plates is

7 5 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) Table 5 The effect of thickness on the nonlinear frequency ratios (ω NL /ω L ) of simply supported cross-ply [ /9 / /9 / ] skew plates (a = b) w max /h Skew angle = Skew angle = 3 a/h = a/h = 5 a/h = 2 a/h = a/h = a/h = 5 a/h = 2 a/h = Table 6 The nonlinear frequency ratios (ω NL /ω L ) of simply supported skew plate (a/b = ; a/h = ) w max /h Skew angle [ /9 / /9 / ] [45 / 45 /45 / 45 /45 ] Table 7 The nonlinear frequency ratios (ω NL /ω L ) of clamped skew plate (a/b = ; a/h = ) w max /h Skew angle [ /9 / /9 / ] [45 / 45 /45 / 45 /45 ] studied in Table 5. It is observed that the nonlinear frequency ratio (ω NL /ω L ) increases with the increase in thickness of the plate. However, for the case of thick plates with a/h ratio, there is a drop in the frequency values at w max /h =.8. This is possibly due to change in stiffness values and thus leading to the redistribution of mode shapes at higher amplitudes of vibration as demonstrated in Ref. [2]. Thereafter, a detailed study is carried out considering multilayered cross-ply [ /9 / /9 / ] and angle-ply [45 / 45 /45 / 45 /45 ] skew laminates for which results are

8 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) not available in the literature. The variation of nonlinear free vibration frequency ratio (ω NL /ω L ) of a simply supported skew plate (a/b =.) is presented in Table 6 for different values of skew angle. From this table, it is observed that the nonlinear frequency ratio increases with the increase in amplitude of vibration, indicating hardening type of nonlinear behavior. Furthermore, for the chosen amplitude, it can be noted that, with the increase in skew angle, the degree of nonlinearity is high. It is further observed from Table 6 that there is a sudden drop in the value of frequency ratio at certain amplitude of vibration at skew angle 45 and 6 indicating redistribution of mode shapes. The influence of clamped boundary condition on the nonlinear vibration behavior of skew laminates (a/b =, a/h = ) is studied in Table 7. The behavior, in general, is qualitatively similar to those of simply supported case. It can be further concluded that although the type of nonlinear behavior is hardening, the degree of nonlinearity is less compared to those of simply supported skew laminates. 5. Conclusions Large amplitude free vibration of composite skew plate has been investigated using a four-noded shear flexible quadrilateral high precision plate bending element. The present formulation has been verified with the available analytical and finite element results. It is observed that the matrix amplitude equation is quite accurate to predict the nonlinear frequency of symmetrically laminated composite skew plates. Limited numerical studies are conducted to examine the effect of skew angle, fiber orientation, and boundary conditions on the large-amplitude frequency of composite skew plates. Present study reveals that the nonlinear frequency ratio in general increases with the increase in thickness and skew angle. The degree of hardening type of nonlinearity is less for clamped boundary condition compared to those of simply supported boundary condition. References [] A.W. Leissa, Plate vibration research, : classical theory, Shock Vib. Dig. 3 (98) 22. [2] A.W. Leissa, Plate vibration research, : complicating factors, Shock Vib. Dig. 3 (98) [3] C.W. Bert, Research on dynamic behavior of composite and sandwich plates IV, Shock Vib. Dig. 7 (985) 3 5. [4] C.Y. Chia, Geometrically nonlinear behavior of composite plates: a review, ASME Appl. Mech. Rev. 4 (988) [5] R.K. Kapania, S. Raciti, Recent advances in analysis of laminated beams and plates, Part II: Vibration and wave propagation, Am. Inst. Aeronaut. Astronaut. J. 27 (989) [6] M. Sathyamoorthy, Nonlinear vibrations of plates: an update of recent research developments, ASME Appl. Mech. Rev. 49 (996) S55 S62. [7] H.N. Chu, G. Herrmann, Influence of large amplitudes on free flexural vibrations of rectangular plates, ASME J. Appl. Mech. 23 (956) [8] C.S. Hsu, On the application of elliptic functions in nonlinear forced oscillations, Q. J. Appl. Math. 7 (96) [9] G. Chandrasekharappa, H.R. Srirangarajan, Nonlinear free vibration of elastic plates, Am. Inst. Aeronaut. Astronaut. J. 27 (989) [] A. Bhimaraddi, Large amplitude vibrations of imperfect antisymmetric angle-ply laminated plates, J. Sound Vib. 62 (993) [] Y.S. Shih, P.T. Blotter, Non-linear vibration analysis of arbitrarily laminated thin rectangular plates on elastic foundations, J. Sound Vib. 67 (993) [2] B.N. Rao, S.R.R. Pillai, Non-linear vibrations of a simply supported rectangular antisymmetric cross-ply plate with immovable edges, J. Sound Vib. 52 (992) [3] I.S. Raju, G.V. Rao, K.K. Raju, Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates, J. Sound Vib. 49 (976) [4] C.-S. Chen, C.-P. Fung, R.-D. Chein, A further study on nonlinear vibration of initially stressed plates, Appl. Math. Comput. 72 (26) [5] A.H. Sheikh, M. Mukhopadhyay, Large amplitude free flexural vibration of stiffened plates, Am. Inst. Aeronaut. Astronaut. J. 34 (996) [6] C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates, Comput. Structures 3 (973) [7] C. Mei, K. Decha-Umphai, A finite element method for nonlinear forced vibrations of rectangular plates, Am. Inst. Aeronaut. Astronaut. J. 23 (985) 4. [8] G.V. Rao, I.S. Raju, K.K. Raju, A finite element formulation for large amplitude flexural vibrations of thin rectangular plates, Comput. Struct. 6 (976) [9] M. Ganapathi, T.K. Varadan, B.S. Sarma, Nonlinear flexural vibrations of laminated orthotropic plates, Comput. Struct. 39 (99) [2] K. Sivakumar, N.G.R. Iyengar, K. Deb, Free vibration of laminated composite plates with cutouts, J. Sound Vib. 22 (999) [2] M.K. Singha, M. Ganapathi, Large amplitude free flexural vibrations of laminated composite skew plates, Int. J. Non-linear Mech. 39 (24) [22] N. Sundararajana, T. Prakash, M. Ganapathi, Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments, Finite Element Anal. Des. 42 (25) [23] S.R. Rao, A.H. Sheikh, M. Mukhopadhyay, Large amplitude finite element flexural vibration of plates/stiffened plate, J. Acoust. Society Am. 93 (993) [24] R. Benamar, M.M.K. Bennouna, R.G. White, The effect of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, Part II: Fully clamped rectangular isotropic plates, J. Sound Vib. 64 (993) [25] G. Prathap, Comments on Large amplitude vibration of circular plates, J. Sound Vib. 54 (977) [26] B.S. Sharma, T.K. Varadan, Certain discussions in the finite element formulation of nonlinear beam vibrations, Comput. Struct. 5 (982) [27] G. Singh, K. Kanaka Raju, G.V. Rao, Nonlinear vibration of simply supported rectangular cross-ply plates, J. Sound Vib. 42 (99) [28] P.C. Dumir, A. Bhaskar, Some erroneous finite element formulations of non-linear vibrations of beams and plates, J. Sound Vib. 23 (988) [29] R. Lewandowski, Free vibration of structures with cubic nonlinearity remarks on amplitude equation and Rayleigh quotient, Comput. Methods Appl. Mech. Eng. 92 (23) [3] W. Han, M. Petyt, Geometrically nonlinear vibration analysis of thin, rectangular plates using the hierarchical finite element method I. The fundamental mode of isotropic plates, Comput. Struct. 63 (2) (997) [3] P. Ribeiro, Nonlinear vibrations of simply supported plates by the p-version finite element method, Finite Elem. Anal. Des. 4 (25) [32] Y. Shi, R.Y.Y. Lee, C. Mei, Finite element method for nonlinear free vibration of composite plates, Am. Inst. Aeronaut. Astronaut. 35 (997) [33] S.L. Lau, Y.K. Cheung, Amplitude incremental variational principle for nonlinear vibration of elastic systems, ASME J. Appl. Mech. 48 (98)

9 52 M.K. Singha, R. Daripa / International Journal of Non-Linear Mechanics 42 (27) [34] S.L. Lau, Y.K. Cheung, S.Y. Wu, Nonlinear vibration of thin-elastic plates, Part : Generalized incremental Hamilton s principle and element formulation, ASME J. Appl. Mech. 5 (984) [35] B. Harras, R. Benamar, R.G. White, Geometrically nonlinear free vibration of fully clamped symmetrically laminated rectangular composite plates, J. Sound. Vib. 25 (22) [36] A.K. Ray, B. Banerjee, B. Bhattacharjee, Large amplitude free vibrations of skew plates including transverse shear deformation and rotary inertia a new approach, J. Sound Vib. 8 (995) [37] M. Sathyamoorthy, K.A.V. Pandalai, Nonlinear flexural vibration of orthotropic skew plates, J. Sound Vib. 24 (972) 5 2. [38] G. Prathap, T.K. Varadan, Non-linear flexural vibration of anisotropic skew plate, J. Sound Vib. 63 (979) [39] M. Sathyamoorthy, C.Y. Chia, Effects of shear and rotary inertia on large amplitude vibrations of anisotropic skew plates, Parts I and II, ASME J. Appl. Mech. 47 (98) [4] C.L. Liao, Z.Y. Lee, Elastic stability of skew laminated composite plates subjected to biaxial follower forces, Int. J. Numer. Methods Eng. 36 (993) [4] R.M. Jones, Mechanics of Composite Materials, McGraw-Hill, New York, 975. [42] S. Wang, Vibration of thin skew fiber reinforced composite laminates, J. Sound Vib. 2 (997) [43] S. Wang, Free vibration analysis of skew fibre-reinforced composite laminates based on first-order shear deformation plate theory, Comput. Struct. 63 (997)