The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free vibration analysis Francesco Tornabene ⇑ , Nicholas Fantuzzi, Michele Bacciocchi DICAM – Department, School of Engineering and Architecture, University of Bologna, Italy article info Article history: Available online 20 May 2014 Keywords: Doubly-curved shells and panels Laminated composites Higher-order shear deformation theory Thickness functions Local Generalized Differential Quadrature method abstract This paper presents a general two-dimensional approach for solving doubly-curved laminated composite shells using different kinematic expansions along the three orthogonal directions of the curvilinear shell model. The Carrera Unified Formulation (CUF) with different thickness functions along the three orthog- onal curvilinear directions is applied to completely doubly-curved shells and panels, different from spherical and cylindrical shells and plates. Furthermore, the fundamental nuclei for doubly-curved struc- tures are presented in their explicit form for the first time by the authors. These fundamental nuclei also allow to consider doubly-curved structures with variable thickness. In addition, the theoretical model includes the Murakami’s function (also known as zig-zag effect). For some problems it is useful to have an in-plane kinematic expansion which is different from the normal one. The 2D free vibration problem is numerically solved through the Local Generalized Differential Quadrature (LGDQ) method, which is an advanced version of the well-known Generalized Differential Quadrature (GDQ) method. The main advantage of the LGDQ method compared to the GDQ method is that the former can consider a large number of grid points without losing accuracy and keeping the very good stability features of GDQ method as already demonstrated in literature by the authors. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction It is well-known that curved structures have a higher strength than flat ones, due to the fact that membrane forces are coupled with bending ones. This aspect can be enforced by employing lam- inated composite structures. The first coupling effect states at the kinematic level, whereas the second one is at the constitutive one. It becomes clear why several researchers in the last 70 years [1–55] have focused their efforts in the study of these models. In particular, the most relevant milestones on the study of plates are the books by Timoshenko and Woinowsky-Krieger [7], Lekhnit- skii et al. [14], Leissa [15], Szilard [19] and Reddy [34,39]. On the other hand other important books about shells must be cited as well as Sanders [6], Flügge [8], Gol’denveizer [9], Novozilov [10], Ambartusumyan [11], Kraus [13], Leissa [17], Markuš [25], Ventsel and Krauthammer [38], Reddy [40], Qatu [42], Carrera et al. [50], Tornabene [54] and Tornabene and Fantuzzi [55]. The recent growing need of curved components in civil, mechanical, aerospace and naval engineering has pushed researchers even further. The first theoretical models were based on thin shells and shallow shells. Nowadays, these models can be considered as very well-known because they were deeply investigated. On the contrary the same cannot be said for thick shells, due to the fact that the mechanical behavior of these structures is more complex, especially when lam- inated composite shells are taken into account. The simplification of a two-dimensional model is clearly justified by a thin shell theory. The validity of thick shell theories is still an open topic, due to the increasing use of higher-order kinematic models [56–66]. A good starting point for studying higher-order theories is given by the works of Carrera [56–58], where the basics of the Carrera Unified Formulation (CUF) are given in detail. This formulation allows to study and compare several higher-order displacement fields when applied to beams, plates and shells. Furthermore, various shell geometries and mechanical shell complexities were conducted in the review article by Qatu et al. [59], where the dynamic behavior of laminated composite shells for different geometries has been examined in the first decade of the new millennium and a wide and complete bibliography on the subject is presented. http://dx.doi.org/10.1016/j.compstruct.2014.05.008 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +39 0512093500; fax: +39 0512093496. E-mail address: francesco.tornabene@unibo.it (F. Tornabene). URL: http://software.dicam.unibo.it/diqumaspab-project (F. Tornabene). Composite Structures 116 (2014) 637–660 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct