Analytical solutions using a higher order refined computational model with 12 degrees of freedom for the free vibration analysis of antisymmetric angle-ply plates K. Swaminathan * , S.S. Patil Department of Civil Engineering, National Institute of Technology Karnataka, Srinivasnagar, Karnataka 575 025, India Available online 9 January 2007 Abstract Analytical formulations and solutions to the natural frequency analysis of simply supported antisymmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher order refined computational model with 12 degrees of freedom 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 displace- ments 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, another higher order computational model with five degrees of freedom already available in the literature is also considered for comparison. The equations of motion are obtained using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s technique by solving the eigenvalue equation. Plates with varying slenderness ratios, number of layers, degrees of anisotropy, edge ratios and thickness of core to thickness of face sheet ratios are considered for analysis. Numerical results with real properties using above two computational models are presented and compared for the free vibration analysis of mul- tilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Free vibration; Higher order theory; Shear deformation; Angle-ply plates; Analytical solutions 1. Introduction Laminated composite and sandwich plates and shells are finding extensive usage in the aeronautical and aerospace industries as well as in other fields of modern technology. It has been observed that the strength and deformation characteristics of such structural elements depend upon the fibre orientation, stacking sequence and the fibre con- tent in addition to the strength and rigidities of the fibre and matrix material. Though symmetric laminates are simple to analyse and design, some specific application of composite and sandwich laminates requires the use of unsymmetric laminates to fulfil certain design requirements. Antisymmetric cross-ply and angle-ply laminates are the special form of unsymmetric laminates and the associated theory offers some simplification in the analysis. The Classi- cal Laminate Plate Theory [1] which ignores the effect of transverse shear deformation becomes inadequate for the analysis of multilayer composites. The First Order Shear Deformation Theories (FSDTs) based on Reissner [2] and Mindlin [3] assume linear in-plane stresses and displace- ments respectively through the laminate thickness. Since FSDTs account for layerwise constant states of transverse shear stress, shear correction coefficients are needed to rec- tify the unrealistic variation of the shear strain/stress through the thickness. In order to overcome the limitations of FSDTs, higher order shear deformation theories (HSDTs) that involve higher order terms in the Taylor’s expansions of the displacement in the thickness coordinate were developed. Hildebrand et al. [4] were the first to intro- duce this approach to derive improved theories of plates and shells. Using the higher order theory of Reddy [5] free vibration analysis of isotropic, orthotropic and laminated 0263-8223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2007.01.001 * Corresponding author. Tel.: +91 824 2474340; fax: +91 824 2474033. E-mail address: swami7192@yahoo.co.in (K. Swaminathan). www.elsevier.com/locate/compstruct Composite Structures 82 (2008) 209–216