This paper presents the extension of a novel superposition method, put forth for static analysis earlier, for carrying out free vibration and bifurcation buckling studies of orthotropic rectangular plates with any combination of classical boundary conditions viz. clamped, simply supported and free edges. It is shown that use of infinite series counterparts of conventional Levy-type closed-form expressions result in a tremendous simplification of the problem without any compromise on accuracy. The complicating effects of an elastic foundation and initial stresses are also shown to be easily accounted for. Results, useful as benchmarks in future, are presented for a chosen set of cases. © 2008 Elsevier Ltd. All rights reserved.

K. Bhaskar

Department of Aerospace Engineering

S. Kshirsagar

Boundary conditions

Buckling

Problem solving

Static analysis

stresses

CLASSICAL BOUNDARY CONDITIONS

ELEGANT FREE VIBRATIONS

ORTHOTROPIC RECTANGULAR PLATES

Superposition method

Vibrations (mechanical)

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It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Journal of Sound and Vibration

The purpose of this paper is to illustrate the application of a new superposition method for the title problem and to explain its advantages with respect to the traditional superposition method. It is shown that the analytical manipulations are significantly less in the new approach, and this is achieved without any loss of rigour or accuracy. Further, the new method is also shown to facilitate incorporation of complicating effects like material orthotropy, in-plane forces or an elastic foundation without any change in the analytical methodology or the nature of the final equations. © 2008 Elsevier Ltd. All rights reserved.

Thin-Walled Structures

Free vibration and stability analysis of orthotropic shear-deformable plates using untruncated infinite series superposition method

The title problems, considered difficult hitherto, are solved here using a rigorous superposition method comprising of elegant and exact infinite series counterparts of the conventional Levy-type solutions. The infinite series are summed without truncation to avoid any loss of accuracy. After due validation studies, numerical results are tabulated for some typical plate problems for use as baseline solutions in the future. © SAGE Publications 2008.

Journal of Reinforced Plastics and Composites

Accurate free vibration analysis of clamped unsymmetric cross-ply/antisymmetric angle-ply/functionally graded rectangular plates by superposition of untruncated infinite series

A new elegant, powerful and accurate superposition method is presented for isotropic/specially orthotropic rectangular plates with arbitrary transverse load and arbitrary combination of free/simply-supported/clamped/guided/elastically supported edges. All the component solutions used here are infinite series equivalents of the complicated closed-form Levy-type solutions employed in the conventional superposition method; it is shown that these series equivalents are easily derived. The mathematical equations pertaining to the various component solutions required for the application of this new method to any plate problem are clearly presented. A number of validation studies are carried out to verify the accuracy of the method. The method can be directly extended to the analysis of more complicated plates made of multifunctional or functionally graded materials. © 2007 Elsevier Ltd. All rights reserved.

Composite Structures

Untruncated infinite series superposition method for accurate flexural analysis of isotropic/orthotropic rectangular plates with arbitrary edge conditions

A superposition approach based on exact untruncated infinite series solutions, developed recently for cross-ply plates, is extended herein to obtain a fast-convergent analytical solution for anti-symmetric angle-ply plates with any combination of S1, S3, C1, and C3 type edges under arbitrary loading. Numerical results are presented to highlight the differences between the various edge conditions. It is finally concluded that the results are significantly influenced by the in-plane conditions when the edges are simply supported, but not when they are clamped. © 2006 SAGE Publications.

An elegant and rigorous analytical solution for anti-symmetric angle-ply rectangular plates with any combination of simply supported and clamped edges

Clamped cross-ply plates are analyzed here using a higher-order theory based on a realistic thickness-wise displacement field that accounts for a zig-zag variation of the in-plane displacements. Rigorous analytical solutions are obtained for this difficult problem using a methodology developed recently wherein simple Fourier series solutions of a corresponding simply-supported plate are superposed, each such component solution being summed without truncation using MATHEMATICA. Numerical results, for square homogeneous and three-layer plates under uniform and central point loads, are compared with those based on the classical and first-order theories. © 2005 Elsevier Ltd. All rights reserved.

Analytical solutions for flexure of clamped rectangular cross-ply plates using an accurate zig-zag type higher-order theory

Based on the idea developed in an earlier work on orthotropic rectangular plates [Compos. Struct. 63 (2004) 63], it is shown here that exact double Fourier series solutions can be easily obtained for simply supported rectangular unsymmetrically laminated cross-ply plates subjected to various edge forces and moments, and that these can be superposed to yield a simple solution for the title problem considered difficult hitherto. Numerical results, based on the summation of the infinite series without truncation using MATLAB, are presented for a set of symmetric and antisymmetric cross-ply laminates under uniform and central concentrated loads. © 2004 Elsevier Ltd. All rights reserved.

Analysis of clamped unsymmetric cross-ply rectangular plates by superposition of simple exact double Fourier series solutions

A rigorous analytical solution for the title problem is presented here using the method of superposition. The derivation of tedious Levy-type solutions, commonly used in such an approach, is obviated by the ingenious use of their infinite series counterparts, which are summed without truncation so as to retain accuracy. The methodology is fast convergent and yields accurate results without excessive computation. Tabulated results are presented for two cross-ply plates for future comparisons. © 2005 SAGE Publications.

Journal	Journal of Sound and Vibration
ISSN	0022460X
Open Access	No