Free Vibrations of Tapered AFG Timoshenko Beams

Abstract

Beams are probably the most widespread component of resistant structures. Consequently, enormous research effort has been developed about them. When they are in vibratory environments, it is important to know their dynamic parameters. In that case, the inertia effect magnifies the influence of the constituent materials. In recent years, it has become increasingly important to use advanced materials whose properties vary gradually in some of its dimensions (Functionally Graded Material -FGM). Initially, the research in the field established great progress in the domain of elasticity theory and the study of plates and shells built with FGM. Advance in its application to beams (Functionally Graded Beams - FGB) came later. However, most of early work on FGB has considered the gradation of material properties in the thickness direction. Far fewer researchers have considered the variation of material properties in the axial direction (Axially Functionally Graded - AFG). Probably because the problem becomes more complicated as variable coefficients appear in the governing differential equations, even for Bernoulli-Euler simplest theory. Therefore, except for a few specific situations, these problems are solved by applying approximate methods. In this development, the well-known Rayleigh-Ritz method, the Generalized Differential Quadrature method and an approach by the Finite Element method are used.Beams of materials whose properties vary functionally along the axis (AFG) and the cross section varies gradually (tapered) are studied. The flexural behavior of the beam is described by the Timoshenko theory which takes into account the rotary inertia and shear deformation effects. General boundary conditions are considered, by means of translational and rotatory springs at both ends which allow us to represent all possible combinations of classical boundary conditions. The objective of this study is twofold: from the point of view of structural mechanics to solve a difficult elastodynamic problem that has innumerable technological applications and from the point of view of computational mechanics the demonstration of the aptitude of classical numerical procedures to attack the problem as well as the possibility of comparing its accuracy.A variety of numerical examples are evaluated by combining boundary conditions, variations in cross section and material composition. Dynamic stiffening effect is achieved and discussed in some cases. The results are in good agreement with particular situations of the model, available in the scientific literature.