Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method

In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of laminated composite doubly-curved shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Examples of hyperbolic, catenary, cycloid, parabolic, elliptic and circular shell and panel structures are presented to illustrate the validity and the accuracy of the GDQ method. Furthermore, GDQ results are compared with those presented in literature and the ones obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Thin and thick shells as structural elements occupy a leadership position in many branches of engineering technologies and, in particular, in civil, mechanical, architectural, aeronautical, and marine engineering. Examples of shell structures in civil and architectural engineering are large-span roofs, cooling towers, liquid-retaining structures and water tanks, containment shells of nuclear power plants and concrete arch domes. In mechanical engineering, shell shapes are used in piping systems, turbine disks and pressure vessels technology. Aircrafts, missiles, rockets, ships and submarines are examples of the use of shells in aeronautical and marine engineering. Shells have been widespread in many fields of engineering as they give rise to optimum conditions for dynamic behaviour, strength and stability. These structures support applied external forces efficiently thanks to their geometrical shape. In other words, shells are much stronger and stiffer than other structural shapes. The vibration effects on these structures caused by different phenomena can have serious consequences for their strength and safety. Therefore, an accurate frequency and mode shape determination is of considerable importance for the technical design of these structural elements. The aim of this paper is to study the dynamic behaviour of doubly-curved shell structures derived from shells of revolution, which are very common structural elements. It is well known that a shell may be considered as a threedimensional body and the methods of the linear theory of elasticity may be applied. However, a calculation based on these methods will generally be difficult and computationally expensive. In the theory of shells, an alternative simplified method is therefore used. Adapting some hypotheses, the 3D problem of shell equilibrium may be reduced to the analysis of its middle surface only and the given shell