Carrera Unified Formulation for Free-Vibration Analysis of Aircraft Structures

Advanced structural models, based on variable one-, two-, and three-dimensional kinematics, are proposed in this paper and applied to the analysis of the free vibration of reinforced aircraft shell structures. The used models go beyond classical structural theories, that is, Euler-Bernoulli (for one-dimensional beams) and Kirchhoff (for two-dimensional plates) type assumptions. The order of the expansion of the displacement fields over the cross section (one-dimensional case) and along the plate thickness (two-dimensional case) is, in fact, a free parameter of the problem. In this paper, Lagrange polynomials are used to build such expansions, and as a consequence, only displacements are used as the problem unknowns (no rotations or derivatives of displacements, which are typical of one-dimensional/two-dimensional classical theories, are introduced). The finite-element method is used to provide numerical solutions. The related arrays and the governing dynamical equations are written in terms of a few fundamental nuclei according to the Carrera unified formulation. Classical three-dimensional finite-element solid models are also considered. One-, two-, and three-dimensional finite elements are easily connected to each other to make the most appropriate computational model of the reinforced shell structures. The capability to use the same fundamental nucleus to derive finite-element matrices of one-, two-, and three-dimensional elements of the present model is unique because it is usually not available in other finite-element formulations, that is, no ad hoc techniques are required in the present case to couple finite elements with different kinematics. Three main benchmarks have been analyzed: a plate stiffened by means of bidirectional I-stiffeners, a simplified model of a complete aircraft, and a fuselage-wing connection. Comparison with commercial finite-element software (MSC Nastran) is provided for most of the quoted numerical investigations. The modal assurance criterion has been used to compare the free-vibration modes of the different models. The present mathematical models appear closer to reality and cheaper, from the computational point of view, than those of other existing formulations. Carrera unified-formulation-based finite elements do not require the definition of virtual lines (beam axes) or virtual surfaces (plate reference surfaces), and only physical lines/surfaces are therefore used.