Computations and evaluations of higher-order theories for free vibration analysis of beams

This paper deals with higher-order theories for the analysis of free vibration of beam structures. Refined theories are implemented by the application of the Unified Formulation by the first author which allows one to introduce any-order expansions of the displacement unknowns over the beam sections. The selection of the most appropriate theory is made by using a so-called axiomatic–asymptotic approach which permits one to retain only those terms of the displacement expansion which have been established to be significant with respect to an assigned control parameter. The finite element method is used to provide numerical solutions. Various beam sections as well as boundary conditions are considered. Depending on the vibration modes (bending, torsion, etc.), quite different theories are selected. In general, the number of the effective terms of the resultant theories is much lower than the full expansion case amount. The nature of these terms can differ very much as different beam geometries and boundary conditions are considered. It has been concluded that the method proposed appears to be suitable and convenient to establish the most appropriate beam theory for a given problem; it leads, in fact, to the cheapest computational model for a given accuracy.