Analytical solutions for vibrations of laminated and sandwich plates using mixed theory M.K. Rao, Y.M. Desai * Department of Civil Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India Abstract A semi-analytical method has been presented in this paper to evaluate the natural frequencies as well as displacement and stress eigenvectors for simply supported, cross-ply laminated and sandwich plates by using higher order mixed theory. Models based on equivalent single layer as well as layerwise (LW) theories have been formulated. By assuming a non-linear variation of axial dis- placements through the plate thickness, the warping of the transverse cross-section has been considered. HamiltonÕs principle has been employed to derive the equilibrium equations. The proposed LW model fulfills a priori the continuity of displacements as well as the transverse and the normal stress components at each interface between two adjacent layers. Results obtained by present higher order mixed theory have been found in good agreement with those obtained by three-dimensional elasticity solutions. After establishing the accuracy of present results for orthotropic plates, new results for thin and thick sandwich plates have been presented which can serve as benchmark solutions for future investigations. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Mixed theory; Free vibrations; Cross-ply laminates; Analytical solutions 1. Introduction Applications of composite materials have grown rapidly over the last three decades because of their high strength and light weight. Although composite materials offer many desirable properties, they present challenging problems in the understanding of their structural be- havior. Free vibration analysis is one amongst them for laminated composites. The simple laminate theories are most often incapable of determining the 3-D stress field at the ply level. Thus, the analysis of composite lami- nates may require the use of layerwise (LW) laminate theory or a 3-D elasticity theory. Exact three-dimen- sional solutions [1–4] have shown the fundamental role played by the continuity conditions for the displace- ments and the transverse stress components at the in- terfaces between two adjacent layers for an accurate analysis of multilayered composite thick plates. Further, these elasticity solutions demonstrated that the trans- verse normal stress r z plays a predominant role in these analyses. However, accurate solutions based on the three-dimensional elasticity theory are often intractable. On the other hand, displacement based equivalent single layer (ESL) theories [5–10] do not account for continuity of the transverse stress components. More- over, evaluation of the transverse stresses requires postprocessing procedures in ESL theories. Due to the mechanical coupling between the transverse and the in- plane normal stresses, all the proposed ESL studies cannot accurately describe the variation of the trans- verse stresses through the thickness. As a consequence, these analyses have shown severe limitations in the study of the vibration of thick plates with arbitrary layouts. A much better description can be obtained by the use of LW models in which two-dimensional approxima- tions are introduced at a layer level. The first attempt to consider each layer in a laminate as a separate beam or plate was by Kao and Ross [11] for multicore sandwich beams and also by Swift and Heller [12] for the bending of a laminated beam. Numerous displacement based, LW laminate theories have appeared in the literature (for e.g. [13–15]). All these LW models, in particular, those by Nosier et al. [13] and Cho et al. [15], showed good agreement with respect to the exact three-dimen- sional analysis to predict free vibration response. However, continuity of the transverse stresses at lamina interface cannot be enforced in the LW models based on displacement based formulations. * Corresponding author. Tel.: +91-22-2576-7333; fax: +91-22-2576- 7302. E-mail address: desai@civil.iib.ac.in (Y.M. Desai). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00185-5 Composite Structures 63 (2004) 361–373 www.elsevier.com/locate/compstruct