Differential Quadrature Solution for Parabolic Structural Shell Elements

This work deals with the dynamical behaviour of complete parabolic shells of revolution and parabolic shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. At the moment it can only be pointed out that by using the GDQ technique the numerical statement of the problem does not pass through any variational formulation, but deals directly with the governing equations of motion. Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes of mid-surface displacements and rotations, in this paper the system of second-order linear partial differential equations is solved, without resorting to the one-dimensional formulation of the dynamic equilibrium of the shell. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. Several examples of parabolic shell elements are presented to illustrate the validity and the accuracy of GDQ method. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions. The effect of the distribution choice of sampling points on the accuracy of GDQ solution is investigated. GDQ results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Ansys, Femap/Nastran, Abaqus, Straus, Pro/Engineer.