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.
Highlights
Beam structures are widely used in many engineering applications
Two, three- and four-node finite elements have been derived according to Carrera Unified Formulation (CUF)
In a compact form, named fundamental nucleus, that does not depend on the theory approximation order
Summary
Beam structures are widely used in many engineering applications. Well-known examples are aircraft wings and helicopter rotor blades in aerospace engineering, and concrete made beams in civil constructions. Classical 1-D models for beams made of isotropic materials are based on the Euler-Bernoulli and Timoshenko theories The former does not account for transverse shear effects on cross-sections deformations. Kant et al [6] have formulated an analytical solution for natural frequencies of composite and sandwich beams based on a higher order refined theory Their model accounted for cubic axial, transverse shear and quadratic transverse normal strain. Shi and Lam [7] have conducted free vibrations analysis of composite beam using a finite element formulation based on a third-order shear deformation beam theory Their studies on flexural frequencies showed that the influence of higher-order terms is negligible on the fundamental frequencies, while it is significant on the frequencies of high flexural modes. The results are compared with benchmarks retrieved from the classical theories and with shell finite element and analytical models
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