A new shear deformation theory for laminated composite plates

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Abstract

In the present study, a new higher order shear deformable laminated composite plate theory is proposed. It is constructed from 3-D elasticity bending solutions by using an inverse method. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. It was observed that this theory gives most accurate results with respect to 3-D elasticity solutions for bending and stress analysis when compared with existing five degree of freedom shear deformation theories [Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51:745–52; Touratier M. An efficient standard plate theory. Int J Eng Sci 1991;29(8):901–16; Karama M, Afaq KS, Mistou S. Mechanical behaviour of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct 2003;40:1525–46]. All shear deformation theories predict the vibration and buckling results with reasonable accuracy, generally within %2 for investigated problems. Previous exponential shear deformation theory of Karama et al. (2003) can be found as a special case.

Introduction

Laminated composite structures are used in many engineering applications such as aerospace, automotive, submarines, sport and health instrument applications due to low specific density and low specific modulus. Composite plates are one of the most important structural elements that were studied by many researchers in the last 6 or 7 decades.

In the open literature, basically two different approaches were used in order to study laminated composite structures: single layer theories and discrete layer theories. In the single layer theories laminated structures are assumed to be composed from one layer whereas in latter case each layer is considered in the analysis. Other important point in the static and dynamic analysis of composite plates is deformation assumptions used in the dynamic and static analysis. Plate deformation theories can be divided in to two groups: stress based and displacement based theories. Since in the present study a new displacement based theory will be analyzed, a brief review of displacement based theories is given below:

Displacement based theories can be divided in to two parts: The classical laminated plate theory and shear deformation plate theories. Transverse shear stress components are neglected in the classical plate theory where it is included in the shear deformation theories. The classical plate theory can be used only for thin plates since it gives erroneous results for thick plates especially made from advanced composites. The first shear deformation theory is uniform or the first order shear deformation theory proposed by Mindlin [4] and Reissner [5]. According to this theory, transverse lines before deformation will be line after deformation but they are not normal to the mid-plane. This theory assumes constant transverse shear stress and it needs a shear correction factor in order to satisfy the plate boundary conditions on the lower and upper surface. Different higher order theories were proposed in order to satisfy the plate boundary conditions. Ambartsumian [6], proposed a transverse shear stress function in order to explain plate deformation. A similar method was used later by Soldatos and Timarci [7], for dynamic analysis of laminated shells. Later some new functions were proposed by Reddy [1], Touratier [2], Karama et al. [3] and Soldatos [8]. Different shear deformation theories were compared for dynamic and static analysis of laminated composites (Aydogdu [9]).

Swaminathan and Patil [10] used a higher order computational model for the free vibration analysis of antisymmetric angle-ply plates. Liu et al. [11] studied static, free vibration and buckling of shear deformable composite laminates using mesh-free radial basis function method.

In recent years, the layer-wise theories and individual layer theories have been presented to obtain more accurate information on the ply level by Wu and Chen, [12] and Cho et al. [13], Plagianakos and Saravanos [14], Fares and Elmarghany [15]. These theories require numerous unknowns for multilayered plates and are often computationally expensive to obtain accurate results.

It was shown in the previous studies that conventional plate theories are not effective when dealing with stress and bending problems. In the present study a new set transverse shear deformation function is proposed. This theory satisfies plate boundary conditions. The shear deformation function was chosen according to 3-D results by using inverse method. Validity of present shear deformation model was checked by comparing some example problems for bending, vibration and buckling analysis of composite laminates with existing 3-D and various 2-D results.

Section snippets

Laminated composite plates

In this section, derivation of the governing equations for the laminated composite plates is briefly explained. Consider a rectangular plate having the length a, the width b and a uniform thickness h. The plate is assumed to be constructed of arbitrary number, N, of linearly elastic orthotropic layers.

Laminated composite beams

In this section of present study, analysis given in Section 2 was modified for laminated composite beams. Consider a straight uniform composite beam having length L with rectangular cross section h and b. The beam is assumed to be constructed of arbitrary number, N, of linearly elastic orthotropic layers. Therefore, the state of stress in each layer is given by using Eq. (2):σx(k)=Q11(k)εx,τxz(k)=Q55(k)γxz,where Qij(k) are well-known reduced stiffnesses ([21]) and k is the number of layers.

Exact solutions for symmetric cross-ply plates

Exact solutions of Eq. (10) for simply supported, symmetric cross-ply rectangular plate were obtained by using the Navier approach. For symmetric cross-ply plates, the following plate stiffness components are identically zero.A16=A26=D16=D26=F16=F26=H16=H26=0,Bij=0andEij=0,i,j=1,2.Thus, the coupling between stretching and bending is zero. The following simply supported boundary conditions are assumed:Nx=v=w=Mxc=Mxa=v1=0atx=0anda,Nyc=u=w=Myc=Mya=v1=0aty=0andb.These boundary conditions and the

Exact solutions for symmetric laminated composite beams

A similar analysis was conducted for laminated composite beam vibration and buckling that given in Section 4 for composite plates. The following simply supported boundary conditions are assumed:Nxc=w=Mx=Mxa=0atx=0andL,These boundary conditions and the governing equations are satisfied by the following middle surface displacement functions:w=m,n=1Wmnsin(αx)sin(ωt),(Lu1)=m,n=1Xmncos(αx)sin(ωt),where α = mπ/a, and m is the half-wave numbers along the x direction. Time dependent part of above

Selection of α parameter in f(z) function for AYDOGDU model

As mentioned in Section 2, various f(z) functions were used in the previous studies in order to characterize static and dynamic behaviour of laminate composite structures. In the selection of f(z) function following points should be taken into account.

1 – Apporoximatelly parabolic transverse shear deformation distribution should be satisfied by selecting suitable f(z) function.

2 – Boundary conditions should be satisfied on the plate upper and lower surfaces.

Derivative of present model given in

Conclusions

A new shear deformation theory was proposed to analyse laminated composite static and dynamic behaviour. Bending and stress analysis under transverse load, free vibration and buckling and cross ply simply supported composites were analysed and results were compared with previous studies. From the numerical examples worked out the following observations are made: Dynamic and static behaviours are best predicted by present theory when compared with three-dimensional elasticity solutions. Accuracy

References (33)

  • A.K. Noor

    Stability of multilayered composite plates

    Fibre Sci Technol

    (1975)
  • A.A. Khdeir et al.

    Free vibration of cross-ply laminated beams with arbitrary boundary conditions

    Int J Eng Sci

    (1994)
  • A.A. Khdeir et al.

    Buckling of cross-ply laminated beams with arbitrary boundary conditions

    Compos Struct

    (1997)
  • M. Aydogdu

    Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method

    Int J Mech Sci

    (2005)
  • M. Aydogdu

    Buckling analysis of cross-ply laminated beams with general boundary conditions by Ritz method

    Compos Sci Technol

    (2006)
  • J.N. Reddy

    A simple higher-order theory for laminated composite plates

    J Appl Mech

    (1984)
  • Cited by (0)

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