Affiliations: Department of Aeronautics and Space Engineering,
Politecnico di Torino, Torino, Italy
Note: [] Corresponding author: E. Carrera, professor of Aerospace
Structures and Aeroelasticity, Department of Aeronautic and Space Engineering,
Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Tel.:
+39 11 090 6836; Fax: +39 11 090 6899; E-mail: erasmo.carrera@polito.it
Abstract: This paper presents hierarchical finite elements on the basis of the
Carrera Unified Formulation for free vibrations analysis of beam with arbitrary
section geometries. The displacement components are expanded in terms of the
section coordinates, (x, y), using a set of 1-D generalized displacement
variables. N-order Taylor type expansions are employed. N is a free parameter
of the formulation, it is supposed to be as high as 4. Linear (2 nodes),
quadratic (3 nodes) and cubic (4 nodes) approximations along the beam axis,
(z), are introduced to develop finite element matrices. These are obtained in
terms of a few fundamental nuclei whose form is independent of both N and the
number of element nodes. Natural frequencies and vibration modes are computed.
Convergence and assessment with available results is first made considering
different type of beam elements and expansion orders. Additional analyses
consider different beam sections (square, annular and airfoil shaped) as well
as boundary conditions (simply supported and cantilever beams). It has mainly
been concluded that the proposed model is capable of detecting 3-D effects on
the vibration modes as well as predicting shell-type vibration modes in case of
thin walled beam sections.
