Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses

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Abstract

Natural frequencies and buckling stresses of angle-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Numerical results are compared with those of the published existing theories. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. The present global higher-order approximate theories can predict the natural frequencies, buckling stresses and modal stresses of thick multilayered angle-ply composite laminates accurately within small number of unknowns which is not dependent on the number of layers.

Introduction

While a considerable effort has been expended in the analysis of vibration and buckling problems of cross-ply laminated composite plates, very few research investigations on angle-ply laminated composite plates can be found in the survey papers [1], [2], [3]. A very flexible design of the structure at the lamina level can be developed in multilayered angle-ply plates by changing the lamination angle. The most appropriate plate stiffness may be designed by selecting suitable values of angles of laminate reinforcements. The mechanical behaviors of angle-ply laminated composite plates are strongly dependent on the degree of orthotropy of individual layers with arbitrary angles of reinforcements, the low ratio of transverse shear modulus to the in-plane modulus and the stacking sequence of laminates. However, in spite of the importance for technical applications, only a few investigations on vibration and stability problems can be found for the case of multilayered angle-ply plates. Within the limitation of the classical linear theory, several papers dealing with the vibration and buckling problems of angle-ply laminated composite plates has been published [4], [5], [6].

The classical two-dimensional laminated plate theory based on the Kirchhoff hypothesis yields sufficiently accurate results only for thin laminated composite plates. For moderately thick laminated plates with relatively soft transverse shear modulus and for highly anisotropic composites, classical plate theory leads to a significant over-prediction of the natural frequencies and buckling stresses. This inaccuracy is due to neglecting the effects of transverse shear strains and transverse normal strain in the laminate (for example, Ref. [7]). In order to take into account the effects of low ratio of transverse shear modulus to the in-plane modulus, a number of first-order shear deformation theories have been developed. However, the inherent deficiency of Mindlin-type shear deformation plate theory is the presence of a correction coefficient, which is introduced to correct the contradictory shear stress distribution over the thickness of plates and cannnot be found from within the assumptions of the theory itself. The shear correction coefficient should be adjusted when studying higher mode vibrational behavior of plates, because the dynamic shear strain distribution may differ significantly from the parabolic form of the static shear strain distribution. It has been shown that the classical and Mindlin-type first-order shear deformation theories are inadquate to predict the accurate solutions of laminated composite plates.

In order to obtain the accurate predictions of the gross response characteristics, such as natural frequencies and buckling stresses, and also the distributions of displacements and stresses at the ply level, a number of contributions based on the three-dimensional elasticity theory have been made to analyze angle-ply laminated composite plates [8], [9], [10]. However, accurate solutions based on the three-dimensional elasticity theory are often computationally expensive. For one of the best alternatives to the three-dimensional elasticity solutions, the layer-wise theories and individual layer theories have been presented to obtain more accurate information of cross-ply laminated composite plates [11], [12], [13], [14]. These theories require numerous unknowns for multilayered plates and are often computationally expensive to obtain the accurate solutions. The total number of unknowns depends on the number of layers in a laminate and will increase dramatically as the number of layers increases.

Without requiring the specification of a shear correction coefficient in the Mindlin-type first-order shear deformation plate theory, various higher-order plate theories have been developed. A number of single-layer (global) higher-order plate theories that include the effects of transverse shear deformations have been published in the literature. Although various models of higher-order displacement fields have been considered (for example, [15], [16]), most of these theories are the third-order theories in which the in-plane displacements are assumed to be a cubic expression of the thickness coordinate and the out-of-plane displacement to be a quadratic expression at most. It has been pointed out that, in general, single-layer second- and third-order theories are adequate in representing global responses, such as natural frequencies and buckling stresses, but inadequate in representing the local responses, such as stress distributions in each layer of laminates (e.g., Ref. [13]). This seems to be a hasty conclusion to analyze the response characteristics of laminated composite plates by using single-layer higher-order theories. For a thick isotropic plates, a two-dimensional higher-order theory has been developed and has been applied to the statics and dymamics of a very thick plate by Matsunaga [17], [18], [19]. Natural frequencies and buckling stresses of thick isotropic plates subjected to in-plane stresses have been analyzed by using the approximate two-dimensional higher-order theories. Remarkable effects of transverse shear deformations and thickness changes have been predicted in the results. A general nonlinear higher-order theory of isotropic shells for large deformations and finite strains in reference to a certain natural state has been presented by Yokoo and Matsunaga [20]. Recently, Matsunaga [21] proposed a globa higher-order theory for the vibration and buckling problems of cross-ply laminated composite plates. The modal transverse stresses were obtained by integrating the three-dimensional equations of motion in the thickness direction starting from the bottom surface of the laminates. However, general higher-order theories of plates which take into account the complete effects of shear deformations, thickness changes and rotatory inertia have not been investigated in the vibration and stability problems of angle-ply multilayered composite plates.

This paper presents a global higher-order theory for analyzing natural frequencies and buckling stresses of angle-ply laminated composite plates. The complete effects of shear deformations, thickness changes and rotatory inertia can be taken into account within the approximate two-dimensional theory. Several sets of the governing equations of truncated approximate theories are applied to the analysis of vibration and stability problems of a simply supported multilayered elastic plate subjected to in-plane stresses. Based on the power series expansions of displacement components (for example, [15], [22]), a fundamental set of equations of a two-dimensional higher-order plate theory is derived through Hamilton's principle. Natural frequencies and buckling stresses of an angle-ply laminated composite plate subjected to in-plane stresses are obtained by solving the eigenvalue problem numerically. Convergence properties of the present numerical solutions are shown to be accurate for the natural frequencies and buckling stresses with respect to the order of approximate theories. A comparison of the present results is also made with previously published results. In-plane and transverse stress components of thick isotropic pates were determined accurately by integration of equilibrium equations of three-dimensional elastic continuum with satisfying the stress boundary conditions on the upper and lower surfaces of a plate [17]. For multilayered angle-ply plates the distribution of modal displacements and modal stresses in the thickness direction has also been obtained accurately in the ply level. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction starting from the top or bottom surface of the laminates. The present results obtained by various sets of approximate theories are considered to be accurate enough for general angle-ply laminated composite plates with small length-to-thickness ratio. Two-dimensional global higher-order theory in the present paper can predict the natural frequencies, buckling stresses and modal stress distributions of simply supported thick multilayered angle-ply plates accurately within small number of unknowns when compared with the three-dimensional elasticity theory and layer-wise higher-order plate theories.

Section snippets

Fundamental equations of angle-ply laminated composite plates

Consider an angle-ply laminated composite plate of uniform thickness h, having a rectangular plan a×b as shown in Fig. 1. Introducing the Cartesian reference coordinate xi(i=1,2,3) on the middle plane of a plate of uniform thickness h, the dynamic displacement components in a plate are expressed asvα≡vα(xi;t),v3≡v3(xi;t),where t denotes time. The displacement components may be expanded into power series of the thickness coordinate x3 as follows:vα=n=0vα(n)x3n,v3=n=0v3(n)x3n,where n

Fourier series solution for angle-ply laminated plates

In the following analysis, the Cartesian reference coordinate x=x1,y=x2 and z=x3 and displacement components u=v1,v=v2 and w=v3 are followed. The in-plane stresses sαβ0 are assumed to distribute uniformly in the thickness direction. The following combination of the uniaxial (κ=0) and biaxial (κ≠0) in-plane stresses is taken into consideration:syy0=κsxx0andsxy0=syx0=0,where κ is ratio of in-plane stresses in y- and x-direction.

The Navier approach can be used to find the solution for general

Eigenvalue problem for vibration and stability problems

The equations of motion , can be rewritten by collecting the coefficients for the generalized displacements of any fixed values r and s. The generalized displacement vector {U} for the Mth-order approximate theory is expressed as{U}T={ũrs(1),…,ũrs(2M−1);ṽrs(1),…,ṽrs(2M−1);w̃rs(0),…,w̃rs(2M−2);ūrs(0),…,ūrs(2M−2);v̄rs(0),…,v̄rs(2M−2);w̄rs(1),…,w̄rs(2M−3)}.

For free vibration problems, the equations of motion can be expressed as the following eigenvalue problem:([K]−Ω2[M]){U}={0},where

Determination of modal stress distributions

Although the stress components can be calculated from the constitutive relations, these stresses may not satisfy the stress boundary conditions on the top and bottom surfaces of a laminated plate. In-plane stress components have no reference to the boundary conditions. With the use of the in-plane stress components, therefore, transverse stress components are determined by integration of equations of motion of three-dimensional elastic continuum with satisfying the stress boundary conditions on

Numerical examples

Natural frequency and buckling stress of angle-ply multilayered composite plates with simply supported edges are analyzed. The orthotropic material constants of each layer are E1(k), E2(k), E3(k), G13(k), G23(k) and G12(k). Poisson's ratio are given by ν13(k), ν23(k) and ν12(k) and other Poisson's ratio can be obtained by the reciprocal theorem. Although various material properties have been used for parametric studies of numerical examples in the literature, the material properties of the

Conclusions

Natural frequencies and buckling stresses of simply supported angle-ply laminated composite plates have been obtained by using a global higher-order plate theory. In order to analyze the complete effects of higher-order deformations on the natural frequencies and buckling stresses of angle-ply thick laminates, various orders of the expanded approximate laminate theories have been presented. It is shown through the numerical examples that the present global higher-order theories can provide

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