Free vibration of laminated composite plates using two variable refined plate theory

https://doi.org/10.1016/j.ijmecsci.2010.01.002Get rights and content

Abstract

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.

Introduction

Laminated composite plates are widely used in the aerospace, automotive, marine and other structural applications because of advantageous features such as high ratio of stiffness and strength to weight and low maintenance cost. In company with the increase in the application of laminates in engineering structures, a variety of laminated theories have been developed. The classical laminated plate theory (CLPT), which neglects the transverse normal and shear stresses, provides reasonable results for thin laminates. However, it underpredicts deflections and overpredicts frequencies as well as buckling loads with moderately thick laminates [1]. In order to overcome the limitations of CLPT, the shear deformation theories accounted for the effect of transverse shear deformation have been recommended. The first-order shear deformation theory (FSDT) assumes linear variation of in-plane displacements through the thickness. Many studies of the free vibration of laminates have been carried out using FSDT [2], [3], [4]. Since FSDT violates equilibrium conditions at the top and bottom faces of the plate, shear correction factors are required to correct the unrealistic variation of the shear strain/stress through the thickness. The value of shear correction factor depends not only on the lamination and geometric parameters, but also on the loading and boundary conditions. To avoid the use of shear correction factors, the higher-order shear deformation theories (HSDT) based on power series expansion of displacements with respect to the thickness coordinate have been developed. The HSDT has been widely used to investigate the free vibration of laminated plates [5], [6], [7], [8], [9], [10]. A review of various shear deformation theories for the analysis of laminated composite plates is available in Refs. [11], [12], [13]. Recently, a two variable refined plate theory (RPT) was first developed for isotropic plates by Shimpi [14], and was extended to orthotropic plates by Shimpi and Patel [15], [16] and Kim et al. [17]. The most interesting feature of this theory is that it does not require shear correction factor, and has strong similarities with the CLPT in some aspects such as governing equation, boundary conditions and moment expressions. Kim et al. [18] has developed this theory for the laminated composite plates. The accuracy of this theory has been demonstrated for static bending and buckling analyses of laminates by Kim et al. [18], therefore, it seems to be important to extend this theory to the free vibration analysis of laminates.

The purpose of this paper is to extend the RPT developed by Kim et al. [18] to the free vibration of laminated composite plates. Equations of motion are derived from the Hamilton's principle. The closed-form solutions for simply supported antisymmetric cross-ply and angle-ply laminates are obtained using Navier solution. The effects of parameters such as the aspect ratio, thickness ratio, modulus ratio and number of layers on the natural frequencies of the laminates are investigated. Numerical examples are presented to illustrate the accuracy and efficiency of the present theory in predicting the natural frequencies of laminates by comparing the predictions with those computed using various theories and the exact solutions of three-dimensional elasticity theory.

Section snippets

Basic assumptions

Consider a rectangular plate of total thickness h composed of n orthotropic layers with the coordinate system as shown in Fig. 1. Assumptions of the RPT are as follows:

  • (1)

    The displacements are small in comparison with the plate thickness and, therefore, strains involved are infinitesimal.

  • (2)

    The transverse displacement W includes three components of extension wa, bending wb, and shear ws. Both these components are functions of coordinates x, y, and time t only.

    W(x,y,z,t)=wa(x,y,t)+wb(x,y,t)+ws(x,y,t)

  • (3)

Analytical solutions for the simply supported rectangular laminates

The Navier approach is employed to obtain the closed-form solutions of the partial differential equations in Eq. (21) for simply supported rectangular plates. Two sets of simply supported boundary conditions are considered.

  • The boundary conditions for antisymmetric cross-ply laminates:

    At edges x=0 and x=a

    v=wa=wb=ws=wa/y=wb/y=ws/y=Nx=Mxb=Mxs=0

    At edges y=0 and y=b

    u=wa=wb=ws=wa/x=wb/x=ws/x=Ny=Myb=Mys=0

  • The boundary conditions for antisymmetric angle-ply laminates:

    At edges x=0 and x=a

    u=wa=

Numerical results and discussion

In this section, various numerical examples are presented and discussed for verifying the accuracy and efficiency of the present theory in predicting the natural frequencies of simply supported antisymmetric cross-ply and angle-ply laminated plates. The effects of aspect ratio, thickness ratio, modulus ratio and number of layers on the natural frequencies of the laminates are investigated. For the verification purpose, the results obtained using present theory are compared with those predicted

Concluding remarks

A two variable refined theory is extended to the free vibration of laminates. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The accuracy and efficiency of the present theory has been demonstrated for free vibration behaviors of simply supported antisymmetric cross-ply and angle-ply laminates. The conclusions of

Acknowledgements

This research was supported by Korea Ministry of Land, Transport and Maritime Affairs through grant B01 of Construction Technology Innovation Program.

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