Ritz analysis of vibrating rectangular and skew multilayered plates based on advanced variable-kinematic models Lorenzo Dozio a,⇑ , Erasmo Carrera b a Department of Aerospace Engineering, Politecnico di Milano, via La Masa, 34, 20156 Milano, Italy b Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy article info Article history: Available online 14 February 2012 Keywords: Free vibration Multilayered plates Variable-kinematic Ritz method Layerwise plate theories abstract A variable-kinematic Ritz formulation based on two-dimensional higher-order layerwise and equivalent single-layer theories is described in this paper to accurately predict free vibration of thick and thin, rect- angular and skew multilayered plates with clamped, free and simply-supported boundary conditions. The main result is the derivation at a layer level of so-called Ritz fundamental nuclei for the stiffness and mass matrices which are invariant with respect to both the assumed kinematic model and the type of Ritz functions. In this work, products of Chebyshev polynomials and boundary-compliant functions are cho- sen as admissible trial set. After studying the convergence of the method, its accuracy is evaluated, in terms of frequency parameters and through-the-thickness distribution of modal displacements, by com- parison with some reference results available in the literature. Results for sandwich plates with soft core are given for the first time, which may serve as benchmark values for future research. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction With growing use of laminated composite and sandwich plates as primary structural components in many engineering applications, accurate assessment of their response is becoming more and more crucial. Contrary to single-layer metallic structures made of isotropic materials, multilayered constructions are typically characterized by high shear deformation. Displacements in the thickness direction may exhibit discontinuous derivatives in correspondence to each layer interface (the so-called zig-zag behavior). In addition, for equi- librium reasons, transverse shear and normal stresses should satisfy appropriate interlaminar continuity conditions. Such complicating effects put great difficulties in achieving reli- able prediction of the laminate mechanical behavior with tradi- tional two-dimensional (2-D) models such as the classical plate theory (CPT) [1] or first-order shear deformation theory (FSDT) [2]. CPT and FSDT were originally proposed as axiomatic models for isotropic structures and then adapted to layered plates as equiv- alent single-layer (ESL) models with appropriate laminate stiffness properties [3]. Both theories rely on overly simplified assumptions concerning the three-dimensional (3-D) kinematics of deformation of the plate in accordance to the need of working with reasonable yet economical models that could be handled by hand or by the calculation capabilities available at the time they were derived. Nowadays, computing capabilities allow overcoming the limita- tions of CPT and FSDT by using more refined theories encompassing an enriched set of kinematic variables, while preserving the 2-D nature of the models. In this way, more complicated problems, including static and dynamic response of laminated and sandwich plates with moderate thickness-to-length ratios or high degree of orthotropy, can be solved with better accuracy, without resorting to fully 3-D cumbersome analysis. In the last three decades, much research has been made on ad- vanced 2-D models of multilayered plates. The most common at- tempts involve displacement-based higher-order ESL theories, where the conventional single-layer displacement form of FSDT is enriched with various high-order terms as power series expan- sion of the thickness coordinate, and layerwise (LW) or discrete- layer formulations, in which two-dimensional approximations of the kinematic field are introduced at a layer level [4]. Investiga- tions have been and are still currently focused on identifying which aspects of the 3-D plate behavior should be accounted for and properly modeled, with the aim of obtaining reliable formulations without unnecessary complexity. It is beyond the scope of the present study to review the extensive literature on laminated plate theories. Interested readers may refer, for example, to early survey papers [5,6] and to more recent review articles [7–10]. In contrast to classical plate models, the use of higher-order or layerwise theories typically lead to complex formulas and equa- tions describing the structural problem. Derivation and computer implementation of advanced formulations would be less cumber- some with the availability of appropriate techniques capable of handling in an efficient and unified way arbitrary refinements of 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2012.02.008 ⇑ Corresponding author. Tel.: +39 02 2399 8329; fax: +39 02 2399 8334. E-mail addresses: lorenzo.dozio@polimi.it (L. Dozio), erasmo.carrera@polito.it (E. Carrera). Composite Structures 94 (2012) 2118–2128 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct