DQM free vibration analysis of moderately thick symmetric laminated plates with elastically restrained edges G. Karami a, * , P. Malekzadeh b , S.R. Mohebpour b a Department of Mechanical Engineering and Applied Mechanics, North Dakota State University, Fargo, ND 58105-5285, United States b Department of Mechanical Engineering, Persian Gulf University, Bushehr, Iran Abstract The application of differential quadrature method will be shown for free vibration analysis of moderately thick composite plates with edges elastically restrained against translation and rotation. The governing equations employed are based on the first order shear defor- mation theory including the effects of rotary inertia. Different combinations of constraints at edges are tested, which includes plates with at least a corner without a rigid support. Angle-ply and cross-ply laminates with different aspect ratios, thickness-to-length ratios are examined. Comparisons are made with results for thin as well as moderately thick angle-ply and cross-ply laminated plates. Highly accu- rate solutions can be achieved with only a few grid points. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Laminates; DQM; Vibration; Elastic supports 1. Introduction Composite laminated plates are widely used in different fields of engineering, ranging from more conventional civil engineering and mechanical engineering to aerospace engi- neering, due to their high strength-to-weight ratio and flex- ibility in design. The vibration analyses of laminated plates are essential for design proposes. It is well known that the actual restraints along the edges of plates often deviates from that provided by classical edge conditions traditionally denoted as simply supported, clamped and free. In many cases, although the computed results seem to be satisfactory for design proposes, but in many other situations, the simplified classical modeling of the plate edge conditions leads to significant errors and thus elastic restraints should be included for modeling pur- poses. Most of the previous works on plates with elastically restrained edges are limited to isotropic thin plates. An excellent review of the works on plates with elastic restrained edges can be found in a recent paper by Ashour [1]. Chung et al. [2] investigated the free vibration analysis of orthotropic Mindlin plates with edges elastically restrained against rotation using Rayleigh–Ritz method. Hung et al. [3] studied the effects of elastic support at edges on natural frequencies of thin symmetric cross-ply lami- nates. Bert and Malik [4] used a semi-analytical differential quadrature method to investigate the effects of rotationally restrained boundaries on natural frequencies of symmetric cross-ply laminated plates with two opposite edges simply supported. Liew et al. [5] studied the free vibration of symmetric cross-ply laminated plates being elastically restrained against rotation and translation. Due to complicated mathematical structures of the boundary conditions for anisotropic plates, closed form solution may be found for only a few special cases. There- fore, approximate methods must be devised for vibration analysis of such members. Rayleigh–Ritz method is the most common method, which have been widely used [2,3,5–7]. Other conventional numerical methods such as finite elements, finite strip method, spline element method, and boundary element methods have also been used. High accurate solutions by these methods usually require large 0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.02.014 * Corresponding author. Tel.: +1 701 231 5859; fax: +1 701 231 8913. E-mail address: G.Karami@ndsu.nodak.edu (G. Karami). www.elsevier.com/locate/compstruct Composite Structures 74 (2006) 115–125