Elsevier

Composite Structures

Volume 56, Issue 4, June 2002, Pages 329-344
Composite Structures

Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory

https://doi.org/10.1016/S0263-8223(02)00017-XGet rights and content

Abstract

Analytical formulations and solutions to the static analysis of simply supported composite and sandwich plates hitherto not reported in the literature based on a higher order refined theory developed by the first author and already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate – thus modelling the warping of transverse cross-sections more accurately and eliminating the need for shear correction coefficients. In addition, a few higher order theories and the first order theory developed by other investigators and already available in the literature are also considered for the evaluation. The equations of equilibrium are obtained using principle of minimum potential energy (PMPE). Solutions are obtained in closed form using Navier's technique by solving the boundary value problem. The comparison of the present results with the available elasticity solutions and the results computed independently using the first order and the other higher order theories available in the literature shows that this refined theory predicts the transverse displacement and the stresses more accurately than all other theories considered in this paper. After establishing the accuracy of present results for composite and sandwich plates, new results for the stretching–bending coupling behaviour of antisymmetric sandwich laminates using all the theories considered in this paper are presented which will serve as a benchmark for future investigations.

Introduction

Laminated composite plates are being increasingly used in the aeronautical and aerospace industry as well as in other fields of modern technology. To use them efficiently a good understanding of their structural and dynamical behaviour and also an accurate knowledge of the deformation characteristics, stress distribution, natural frequencies and buckling loads under various load conditions are needed. The classical laminate plate theory (CLPT) [1], which is an extension of classical plate theory (CPT) [2], [3], neglects the effects of out-of-plane strains. The greater differences in elastic properties between fibre filaments and matrix materials lead to a high ratio of in-plane young's modulus to transverse shear modulus for most of the composite laminates developed to date. Because of this reason the transverse shear deformations are much pronounced for laminated plates than for isotropic plates. Thus the CLPT which ignores the effect of transverse shear deformation becomes inadequate for the analysis of multilayer composites. In general the CLPT often underpredicts deflections and overpredicts natural frequencies and buckling loads. The first order theories (FSDTs) based on Reissner [4] and Mindlin [5] assume linear in-plane stresses and displacements, respectively through the laminate thickness. Since the FSDT accounts for layerwise constant states of transverse shear stress, shear correction coefficients are needed to rectify the unrealistic variation of the shear strain/stress through the thickness and which ultimately define the shear strain energy.

In order to overcome the limitations of FSDT, higher order shear deformation theories (HSDTs) that involve higher order terms in Taylor's expansions of the displacements in the thickness coordinate were developed. In these higher order theories with each additional power of the thickness coordinate an additional dependent unknown is introduced into the theory. Hildebrand et al. [6] were the first to introduce this approach to derive improved theories of plates and shells. Nelson and Lorch [7], Librescu [8] presented higher order displacement based shear deformation theories for the analysis of laminated plates. Lo et al. [9], [10] have presented a closed form solution for a laminated plate with higher order displacement model which also considers the effect of transverse normal deformation. Levinson [11] and Murthy [12] presented third order theories neglecting the extension/compression of transverse normal but used the equilibrium equations of the first order theory used by Whitney and Pagano [13] in the analysis which are variationally inconsistent. Kant [14] was the first to derive the complete set of variationally consistent governing equations for the flexure of a symmetrically laminated plate incorporating both distortion of transverse normals and effects of transverse normal stress/strain by utilizing the complete three-dimensional generalized Hooke's law and presented results for isotropic plate only. Reddy [15] derived a set of variationally consistent equilibrium equations for the kinematic models originally proposed by Levinson and Murthy. Using the theory of Reddy, Senthilnathan et al. [16] presented a simplified higher order theory by introducing a further reduction of the functional degrees of freedom by splitting up the transverse displacement into bending and shear contributions. Kant et al. [17] are the first to present a finite element formulation of a higher order flexure theory. This theory considers three-dimensional Hooke's law, incorporates the effect of transverse normal strain in addition to transverse shear deformations. Pandya and Kant [18], [19], [20], [21], [22], Kant and Manjunatha [23], [24] and Manjunatha and Kant [25] have extended this theory and presented C0 finite element formulations and solutions for the stress analysis of symmetric and unsymmetric laminated composite and sandwich plates. Rohwer [26] made a comparative study of various higher order theories for the bending analysis of multilayer composite plates. The advantages and disadvantages of the various theories were highlighted. Noor and Burton [27] presented a complete list of references of FSDTs and HSDTs for the static, free vibration and buckling analyses of laminated composites. Pagano [28] presented exact three dimensional elasticity solutions for the stress analysis of laminated composite and sandwich plates which serve as benchmark solutions for comparison by many researchers. The present paper deals with the analytical formulations and solutions hitherto not reported in the literature of the refined theory already proposed by the senior author as applied to static analysis of laminated composite and sandwich plate problems with simply supported edge conditions. Comparison of results with the three-dimensional elasticity solutions available in the literature shows that this theory predicts the transverse displacements and the in plane stresses more accurately than all other theories considered in this paper. After establishing the accuracy of the present results for composite and sandwich plates, benchmark results for stretching–bending coupling behaviour of antisymmetric sandwich plates are presented.

Section snippets

Displacement models

In order to approximate the three-dimensional elasticity problem to a two-dimensional plate problem, the displacement components u(x,y,z), v(x,y,z) and w(x,y,z) at any point in the plate space are expanded in a Taylor's series in terms of the thickness coordinate. The elasticity solution indicates that the transverse shear stresses vary parabolically through the plate thickness. This requires the use of a displacement field in which the in-plane displacements are expanded as cubic functions of

Analytical solutions

Here the exact solutions of , , , , , , , , for cross-ply rectangular plates are considered. Assuming that the plate is simply supported in such a manner that normal displacement is admissible, but the tangential displacement is not, the following boundary conditions are appropriate:

At edges x=0 and x=a:v0=0,w0=0,θy=0,θz=0,Mx=0,v0*=0,w0*=0,θy*=0,θz*=0,Mx*=0,Nx=0,Nx*=0.

At edges y = 0 and y = b:u0=0,w0=0,θx=0,θz=0,My=0,u0*=0,w0*=0,θx*=0,θz*=0,My*=0,Ny=0,Ny*=0.Following Navier's solution

Numerical results and discussion

In this section, various numerical examples solved are described and discussed for establishing the accuracy of the various theories for the stress analysis of laminated composite and sandwich plates. The description of the various displacement models compared is given in Table 1. A shear correction factor of 5/6 is used in computing results using Whitney–Pagano's theory. For all the problems a simply supported (diaphgram supported) plate is considered for the analysis. The transverse loading

Conclusion

Analytical formulations and solutions to the static analysis of simply supported composite and sandwich plates hitherto not reported in the literature based on a higher order refined theory developed by the first author and already reported in the literature are presented. The displacement field of this theory takes into account both the transverse shear and normal deformations thus making it more accurate than the first order and other higher order theories considered. For laminated composite

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