A new shear deformation theory for laminated composite plates
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):where 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.Thus, the coupling between stretching and bending is zero. The following simply supported boundary conditions are assumed: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:These boundary conditions and the governing equations are satisfied by the following middle surface displacement functions: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)
An efficient standard plate theory
Int J Eng Sci
(1991)- et al.
Mechanical behaviour of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity
Int J Solids Struct
(2003) - et al.
A unified formulation of laminated composite, shear deformable, five-degrees-of-freedom cylindrical shell theories
Compos Struct
(1993) - et al.
Analytical solutions using a higher order refined computational model with 12 degrees of freedom for the free vibration analysis of antisymmetric angle-ply plates
Compos Struct
(2008) - et al.
Mesh-free radial basis function method for static, free vibration and buckling analysis of shear deformable composite laminates
Compos Struct
(2007) - et al.
Vibration and stability of laminated plates based on a local high order plate theory
J Sound Vibr
(1994) - et al.
Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory
J Sound Vibr
(1991) - et al.
Higher-order layerwise laminate theory for the prediction of interlaminar shear stresses in thick composite and sandwich composite plates
Compos Struct
(2009) - et al.
A refined zigzag nonlinear first-order shear deformation theory of composite laminated plates
Compos Struct
(2008) An accurate simple theory of statics and dynamics of elastic plates
Mech Res Commun
(1980)