Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories Erasmo Viola ⇑ , Francesco Tornabene, Nicholas Fantuzzi Department of Civil, Environmental and Materials – DICAM, University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy article info Article history: Available online 23 January 2013 Keywords: Static analysis Doubly-curved laminated shells Differential geometry Higher-order shear deformation theory Generalized differential quadrature method abstract This paper investigates the static analysis of doubly-curved laminated composite shells and panels. A the- oretical formulation of 2D Higher-order Shear Deformation Theory (HSDT) is developed. The middle sur- face of shells and panels is described by means of the differential geometry tool. The adopted HSDT is based on a generalized nine-parameter kinematic hypothesis suitable to represent, in a unified form, most of the displacement fields already presented in literature. A three-dimensional stress recovery pro- cedure based on the equilibrium equations will be shown. Strains and stresses are corrected after the recovery to satisfy the top and bottom boundary conditions of the laminated composite shell or panel. The numerical problems connected with the static analysis of doubly-curved shells and panels are solved using the Generalized Differential Quadrature (GDQ) technique. All displacements, strains and stresses are worked out and plotted through the thickness of the following six types of laminated shell structures: rectangular and annular plates, cylindrical and spherical panels as well as a catenoidal shell and an ellip- tic paraboloid. Several lamination schemes, loadings and boundary conditions are considered. The GDQ results are compared with those obtained in literature with semi-analytical methods and the ones com- puted by using the finite element method. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Composite doubly-curved shells are used in many engineering applications, because they have a great capacity in carrying exter- nal loads due to the curvature effect operating on shells. The wide use of laminated composite shells can be ascribed to their higher strength and stiffness to weight ratios, when compared to most metallic materials. As far as the behavior of laminated composite shells is concerned, by acting on material type, fiber orientation and thickness, a designer can tailor different properties of a lami- nate to suit a particular application. Nowadays, composite shells constitute the largest percentage of aerospace, marine and auto- motive structures. Two dimensional linear theories of thin shells and plates have been developed during the last sixty years by many contributors, such as Timoshenko and Woinowsky-Krieger [1], Flügge [2]. Gol’denveizer [3], Novozhilov [4], Vlasov [5], Ambartusumyan [6], Kraus [7], Leissa [8,9], Markuš [10], Ventsel and Krauthammer [11] and Soedel [12]. All these contributions are based on the Kir- chhoff–Love theory, referred to as Classical Shell Theory (CST), which assumes that normals to the middle-surface remain straight and normal to it during deformations and unstretched in length. Generally, classical shell theories give inaccurate results when applied to moderately thick or laminated anisotropic shells and plates. In order to overcome this limitation, the shear deformation theory has been introduced by Reissner [13], named First-order Shear Deformation Theory (FSDT). By relaxing the assumption on the preservation of the normals to the middle surface after the deformation, a wide investigation for elastic isotropic shells and plates was made by Kraus [7] and Gould [14,15]. Later on, Higher-order Shear Deformation Theories (HSDTs) were introduced. According to the first two theories (CST and FSDT) a shear correction factor is required. The HSDTs overcome this limitation since a non-linear shear stress distribution through the thickness of the shell is assumed. A deep analysis of laminated composite plates and shells modeled within HSDTs has been done by Reddy [16]. However, it should be noted that Reddy’s hypothe- sis does not satisfy the top and bottom boundary conditions of the shell, so a posteriori strain and stress recovery procedure is needed. As regards classical theories, there are appropriate tools for solving structural engineering problems, as pointed out by Qatu in his book [17] about vibration of laminated shells and plates. Nevertheless, the increasing use of laminated shell structures in engineering applications requires adequate instruments in order to achieve refined solutions to the shell problems under investiga- tion. Recently, a unified approach to the theoretical and applied as- pects of shell analysis is reported in the book by Carrera et al. [18]. Over the last decades, a significant number of Higher-order Shear Deformation Theories for composite plates and shells has been 0263-8223/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.01.002 ⇑ Corresponding author. Tel.: +39 0512093510. E-mail address: erasmo.viola@unibo.it (E. Viola). Composite Structures 101 (2013) 59–93 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct