Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory
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:
At edges y = 0 and y = b: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
References (30)
An accurate simple theory of the statics and dynamics of elastic plates
Mech. Res. Commun.
(1980)Numerical analysis of thick plates
Comput. Meth. Appl. Mech. Eng.
(1982)- et al.
A refined higher order C0 plate bending element
Comput. Struct.
(1982) - et al.
A consistent refined theory for flexure of a symmetric laminate
Mech. Res. Commun.
(1987) - et al.
Higher order shear deformable theories for flexure of sandwich plates – finite element evaluations
Int. J. Solids Struct.
(1988) - et al.
Flexure analysis of laminated composites using refined higher order C0 plate bending elements
Comput. Meth. Appl. Mech. Eng.
(1988) - et al.
A refined higher order generally orthotropic C0 plate bending element
Comput. Struct.
(1988) - et al.
Finite element stress analysis of laminated composite plates using higher order displacement model
Compos. Sci. Technol.
(1988) - et al.
On accurate estimation of transverse stresses in multilayer laminates
Comput. Struct.
(1994) Application of higher order theories to the bending analysis of layered composite plates
Int. J. Solids Struct.
(1992)
Bending and stretching of certain types of heterogeneous aelotropic elastic plates
ASME J. Appl. Mech.
Theory of plates and shells
Theory and analysis of plates (classical and numerical methods)
The effect of transverse shear deformation on the bending of elastic plates
ASME J. Appl. Mech.
Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates
ASME J. Appl. Mech.
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