Vibration Analysis of Laminated Doubly-Curved Shells and Panels Using Higher-Order Equivalent-Single-Layer and Layer-Wise Theories

An accurate determination of the modal parameters is required for the technical design of structural shell elements. In this study, the free vibrations of laminated doubly-curved shells are investigated. There are three different ways to study anisotropic shell structures: the 3D Elasticity [1-3], Equivalent-Single-Layer (ESL) [1-3] and Layer-Wise (LW) [3,4] theories. The mechanical model proposed in this study, for the two ESL and LW approaches, is based on the Carrera’s Unified Formulation (CUF) [4]. The investigated kinematic models are presented in the following: The LD4 model of the expressions (1) refers to a fourth order Layer-Wise approach in terms of displacements (LD4), where the generalized displacements are defined at the -th layer level and the thickness functions are assumed as Legendre polynomials [4]. The other three model EDZ4, ED4 and FSDT can be classified as fourth order Equivalent Single Layer models in terms of displacements with (EDZ4) or without (ED4) the zig-zag effect (Z). In particular, the last one of the expressions (1) is the well-known First-order Shear Deformation Theory (FSDT). The curvature effect is included in the formulation of the constitutive equations. The fundamental operators, concerning a laminated composite doubly-curved shell in orthogonal curvilinear coordinate system, are obtained for the first time by the authors. Using the Differential Geometry [1,2] to completely describe the doubly-curved shell, the Differential Quadrature (DQ) [5] rule leads to numerically evaluate all the derivatives involved in the numerical calculations. The governing equations are expressed as functions of various kinematic parameters, when the constitutive and the kinematic relationships are used [6-11]. The system of second-order linear partial differential equations is solved considering the Generalized Differential Quadrature (GDQ) method [5]. In order to verify the formulation accuracy, the worked out results are compared with the ones obtained using analytical and numerical solutions. Tables 1, 2 and 3 show the first five natural frequencies for six different types of shell structures. Some mode shapes of the considered shell structures are illustrated in Figure 1. Various boundary conditions and general lamination schemes are investigated.