Laminated Beam Analysis by Polynomial, rigonometric, Exponential and Zig-Zag Theories

A number of refined beam theories are discussed in this paper. These theories were obtained by expanding the unknown displacement variables over the beam section axes by adopting Taylor's polynomials, trigonometric series, exponential, hyperbolic and zig-zag functions. The Finite Element method is used to derive governing equations in weak form. By using the Unified Formulation introduced by the first author, these equations are written in terms of a small number of fundamental nuclei, whose forms do not depend on the expansions used. The results from the different models considered are compared in terms of displacements, stress and degrees of freedom (DOFs). Mechanical tests for thick laminated beams are presented in order to evaluate the capability of the finite elements. They show that the use of various different functions can improve the performance of the higher-order theories by yielding satisfactory results with a low computational cost

A variable kinematic one-dimensional model for aeroelasticity and dynamic analysis of multi-layered rotors

Flutter is one of the most known instability phenomena. This condition occurs when a given structure exhibits sustained, harmonic oscillations, sometimes leading to catastrophic events. The prediction of flutter represents a crucial point for a correct and safe design. When fluid structure interactions produce dynamic instability, flutter analyses require accurate descriptions of body deformations and aerodynamic loads. To this end, aerodynamic theories have been coupled with structural models to develop aeroelastic analysis tools, whose reliability is the results of a trade-off between the accuracy and the computational efficiency. From a computational point of view, the most efficient formulation is based on the 1D assumption, where the problem is reduced to a set of variables that only depends on the beam-axis coordinate. Besides the well-known classical beam theories, several refined kinematic models have been proposed, to study the stability of rotating blades and shafts. However, when these structures are highly deformable or the material distribution involves non-classical structural couplings, 2D and 3D solutions are still required. Within this work, we propose an advanced 1D formulation to analyse the stability of rotating structures. The higher-order beam theories are obtained using the Carrera Unified Formulation (CUF), which allows to derive, at least theoretically, an infinite number of kinematic models. The Equations of Motion (EoM) for shafts and blades include the Coriolis term and the centrifugal effects (spin softening and geometrical stiffening). For the subsonic flow regime, aerodynamic loads are defined following the unsteady strip theories proposed by Theodorsen and Loewy. For the supersonic regime, the linear Piston theory is extended to structures rotating in compressed air flow. The Finite Element Method (FEM) is used to solve the weak form of the EoM. Firstly, to evaluate the accuracy of 1D CUF elements, static and free-vibration analysis are carried out on compact and thin-walled structures of isotropic, orthotropic and functionally graded materials. Then, higher-order elements are used to study the dynamics of laminated shafts, thin cylinders, discs and blades, which rotate about the longitudinal and transverse axis. Results show the improved performance of the 1D CUF theories compared to 2D and 3D solutions. In order to evaluate the proposed aeroelastic formulation, we test different wing configurations, including thin-walled box beams. The effects of the sweep angle and the lamination scheme on flutter conditions are evaluated, and results are compared with plate solutions, experimental tests and aeroelastic analysis carried out with the Doublet Lattice Method (DLM). Moreover, comparisons between Theodorsen and Loewy aerodynamic theories are presented for a realistic rotary-wing model. In the last numerical examples, the linear Piston Theory is used to describe the dynamics of thin plates with different aspect-ratio surrounded by compressed air. For this cases, results are compared with an existing solution based on a non-linear plate theory

Variable-kinematic finite beam elements for geometrically nonlinear dynamic analyses

This article investigates the dynamic nonlinear response of three-dimensional structures using variable-kinematics finite beam elements obtained with the Carrera Unified Formulation. The formalism enables one to consider the three-dimensional form of displacement–strain relations and constitutive law. The deformation mechanisms and the associated couplings are described consistently with the selected kinematic model. The Hilbert–Hughes–Taylor method and the iterative Newton–Raphson scheme are adopted to solve the motion equations derived in a total Lagrangian scenario. Various models have been obtained by using Taylor- and Lagrange-like expansions. The capabilities of the beam elements are assessed considering isotropic, homogeneous structures with compact and thin-walled sections

Analysis of Composite Space Structures Subjected to Loading Factor

Reinforced structures are mandatory in the space structures on which the lightweight is the main project parameter. The coupling between simple thin-walled plate and different systems of ribs or beams along one or more directions make it possible to meet the requirements of lightness and strength. During the project phase a structure is usually analysed via Finite Element Method (FEM), where different approaches can be used but the pointed out one common essential characteristic, a mesh discretization of a continuous domain into a set of discrete subdomains, usually called elements. Three main finite elements (FEs) are widely used in the commercial code, but only the Solid (3D) FE represents more faithfully the behaviour of a real structure. The solid FE models require a large number of degrees of freedoms (DOFs) and therefore the analyses are computational expensive [1]. For these reason that usually the reduced models are used as substitute of solid models. The reduced models are made using shell (2D) and beam (1D) FEs, and they are suitable to build a reinforced structure, in fact the shell are used for the skin and the beam for the stringers. The present work uses a refined 1D model based on the Carrera Unified Formulation (CUF) [2] to analyse space structures made coupling skin and stringers. Thanks to its refined cinematic the present model can be used to represent both skin and stringers. The whole structure is obtained connecting simple one-dimensional structures using a new approach called Component-Wise (CW) [3]. This is possible because the unknowns are only displacements. Free-vibration analysis of isotropic and composite space structures with non-structural masses and loading factor are considered. A space vehicle is inspired to Arian 5 with a central body, on which the cryogenic fuel and the payload are accommodated, and two lateral boosters, on which solid fuel is stored. The results show the quasi-3D capabilities of the present 1D CUF model and the coupling with the CW approach provide accurate results nearest to solid FE results than the classical refined FEs models. In conclusion the present 1D refined model appears suitable for the analysis of reinforced thin-walled structures, it provides accurate results with the benefit to reduce the computational costs with respect to the classical refined FE approaches. References [1] E. Carrera, E. Zappino and T. Cavallo. Accurate free vibration analysis of launcher structures using refined 1D models. International Journal of Aeronautical and Space Sciences,vol. 16(2) 206-222, 2015. [2] E. Carrera, G. Giunta and M. Petrolo. Beam Structures: Classical and Advanced Theories. Jhon Wiley & Sons Ltd, 2011. [3] E. Carrera, A. Pagani and M. Petrolo. Component-wise Method Applied to Vibration of Wing Structures. J Appl Mech, vol. 80(4), 041012-1-041012-15, 2013

Advanced models for free vibration analysis of laminated beams with compact and thin-walled open/closed sections

In this paper, refined one-dimensional beam theories are implemented for the free vibration analysis of laminated beams with compact and thin-walled cross-sections. The proposed models are based on the Carrera Unified Formulation, which was formerly introduced for the analysis of plates and shells and recently extended to beam structures by the first author and his co-workers. Carrera Unified Formulation is a hierarchical modelling technique leading to very accurate and computationally efficient finite element theories. According to the latest developments in the framework of Carrera Unified Formulation, refined beam models are implemented using either Taylor-like or Lagrange-like polynomials in order to expand the unknown kinematic variables on the cross-section of the beam. Equivalent single layer models result from the former approach. On the other hand, if Lagrange polynomials are used, layer-wise models are produced. In this work, a classical one-dimensional finite element formulation along the beam length is used to develop numerical applications. A number of laminated beam structures are analyzed, and particular attention is given to laminated box beams with open and closed cross-sections. The frequencies and the mode shapes obtained with the present refined beam elements are compared with solid/shell finite element solutions from the commercial code MSC/Nastran and, when possible, with those found in the literature. The modal assurance criterion is used for model-to-model comparisons so as to demonstrate the enhanced capabilities of the proposed formulation in investigating the free vibration characteristics of both compact and thin-walled box laminated beam