Performance of CUF approach to analyze the structural behavior of slender bodies

This paper deals with the accurate evaluation of complete three-dimensional (3D) stress fields in beam structures with compact and bridge-like sections. A refined beam finite-element (FE) formulation is employed, which permits any-order expansions for the three displacement components over the section domain by means of the Carrera Unified Formulation (CUF). Classical (Euler-Bernoulli and Timoshenko) beam theories are considered as particular cases. Comparisons with 3D solid FE analyses are provided. End effects caused by the boundary conditions are investigated. Bending and torsional loadings are considered. The proposed formulation has shown its capability of leading to quasi-3D stress fields over the beam domain. Higher-order beam theories are necessary for the case of bridge-like sections. Various theories are also compared in terms of shear correction factors on the basis of definitions found in the open literature. It has been confirmed that different theories could lead to very different values of shear correction factors, the accuracy of which is subordinate to a great extent to the section geometries and loading conditions. However, an accurate evaluation of shear correction factors is obtained by means of the present higher-order theories

A variational model for anisotropic and naturally twisted ribbons

We consider thin plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental form of the stress-free configuration, if it exists, or a target curvature tensor. In the latter case, residual stress arises from the geometrical frustration involved in the attempt to achieve the target curvature: as a result, the plate is naturally twisted, even in the absence of external forces or prescribed boundary conditions. Here, starting from this kind of plate energies, we derive a new variational one-dimensional model for naturally twisted ribbons by means of Gamma-convergence. Our result generalizes, and corrects, the classical Sadowsky energy to geometrically frustrated anisotropic ribbons with a narrow, possibly curved, reference configuration

FORMAL ASYMPTOTIC ANALYSIS OF ELASTIC BEAMS AND THIN-WALLED BEAMS: A DERIVATION OF THE VLASSOV EQUATIONS AND THEIR GENERALIZATION TO THE ANISOTROPIC HETEROGENEOUS CASE

International audienceThe modelling of ordinary beams and thin-walled beams is rigorously obtained from a formal asymptotic analysis of three-dimensional linear elasticity. In the case of isotropic homogeneous elasticity, ordinary beams yield the Navier-Bernoulli beam model, thin-walled beams with open profile yield the Vlassov beam model and thin-walled beams with closed profile the Navier-Bernoulli beam model. The formal asymptotic analysis is also extensively performed in the case of the most general anisotropic transversely heterogeneous material (meaning the heterogeneity is the same in every cross-section), delivering the same qualitative results. We prove, in particular, the non-intuitive fact that the warping function appearing in the Vlassov model for general anisotropic transversely heterogeneous material, is the same as the one appearing in the isotropic homogeneous case. In the general case of anisotropic transversely heterogeneous material, the analysis provides a rigorous and systematic constructive procedure for calculating the reduced elastic moduli, both in Navier-Bernoulli and Vlassov theories

Composite Structural Materials

The development and application of filamentary composite materials, is considered. Such interest is based on the possibility of using relatively brittle materials with high modulus, high strength, but low density in composites with good durability and high tolerance to damage. Fiber reinforced composite materials of this kind offer substantially improved performance and potentially lower costs for aerospace hardware. Much progress has been made since the initial developments in the mid 1960's. There were only limited applied to the primary structure of operational vehicles, mainly as aircrafts

Prandtl’s formulation for the Saint–Venant’s torsion of homogeneous piezoelectric beams

AbstractThe Saint–Venant torsional problem for homogeneous, monoclinic piezoelectric beams is formulated in terms of Prandtl’s stress function and electric displacement potential function. The analytical approach presented in this paper generalizes the known formulation of Prandtl’s solution which refers to homogeneous elastic beams. The Prandtl’s stress function and electric displacement potential function satisfy the so called coupled Dirichlet problem (CDP) in the cross-sectional domain. A direct and a variational formulation are developed. Exact analytical solutions for solid elliptical cross-section and hollow circular cross-section and an approximate solution based on a variational formulation for thin-walled closed cross-section are presented

Vibrations of an Inhomogeneous Rectangular Plate

Low-frequency vibrations of a thin multifibrous plate are analysed. Asymptotic homogenization and finite element methods are used to get the vibration frequencies. Approximate formulas for the lowest frequencies of thin inhomogeneous rectangular plate are found. The comparison of numerical and asymptotic results is performed

Much ado about shear correction factors in Timoshenko beam theory

AbstractMany shear correction factors have appeared since the inception of Timoshenko beam theory in 1921. While rational bases for them have been offered, there continues to be some reluctance to their full acceptance because the explanations are not totally convincing and their efficacies have not been comprehensively evaluated over a range of application. Herein, three-dimensional static and dynamic information and results for a beam of general (both symmetric and non-symmetric) cross-section are brought to bear on these issues. Only homogeneous, isotropic beams are considered. Semi-analytical finite element (SAFE) computer codes provide static and dynamic response data for our purposes. Greater clarification of issues relating to the bases for shear correction factors can be seen. Also, comparisons of numerical results with Timoshenko beam data will show the effectiveness of these factors beyond the range of application of elementary (Bernoulli–Euler) theory.An issue concerning principal shear axes arose in the definition of shear correction factors for non-symmetric cross-sections. In this method, expressions for the shear energies of two transverse forces applied on the cross-section by beam and three-dimensional elasticity theories are equated to determine the shear correction factors. This led to the necessity for principal shear axes. We will argue against this concept and show that when two forces are applied simultaneously to a cross-section, it leads to an inconsistency. Only one force should be used at a time, and two sets of calculations are needed to establish the shear correction factors for a non-symmetrical cross-section

Evaluation of the accuracy of classical beam FE models via locking-free hierarchically refined elements

It is well known that the classical 6-DOF (Degrees of Freedom) beam theories that are incorporated in commercial finite element (FE) tools are not able to foresee higher-order phenomena, such as elastic bending/shear coupling, restrained torsional warping and three-dimensional strain effects. In this work, the accuracy of one-dimensional (1D) finite elements based on the classical theories (Euler-Bernoulli and Timoshenko theories as well as a 6-DOF model including torsion) is evaluated for a number of problems of practical interest and modelling guidelines are given. The investigation is carried out by exploiting a novel hierarchical, locking-free, finite beam element based on the well-known Carrera Unified Formulation (CUF). Thanks to CUF, the FE arrays of the novel beam element are written in terms of fundamental nuclei, which are invariant with respect to the theory approximation order. Thus, results from classical as well as arbitrarily refined beam models can be formally obtained by the same CUF beam element. Linear Lagrange shape functions are used in this paper to interpolate the generalized unknowns and shear locking phenomena are avoided by adopting an MITC (Mixed Interpolation of Tensorial Components) scheme. Different sample problems are addressed, including rectangular and warping-free circular cross-sections as well as thin-walled beams. The results from classical theories and the 6-DOF model are compared to those from higher-order refined beam models, both in terms of displacement and stress fields for various loading conditions. The discussion focuses on the limitations of the commonly used 1D FEs and the need for refined kinematics beams for most of the problems of common interest. The research clearly depicts CUF as a valuable framework to assess FE formulations such as the 6-DOF model herein considered, which is one of the most known and used finite element for the analysis of structures

Elasticity-based vibrations of hollow anisotropic beams and an evaluation of the shape factor for hollow anisotropic sections

2012 Summer.Includes bibliographical references.This study considers the transverse vibrations and natural frequencies of hollow anisotropic beams free from end restraints using full three-dimensional elasticity solutions and common one-dimensional beam theory approximations. Calculations of the natural frequencies are made for a number of hollow beam dimensions using the one-dimensional Euler-Bernoulli, Rayleigh, and Timoshenko beam theories. Complete derivations of the elasticity solutions and beam theories are presented. The accuracy of the approximate methods is determined by comparison to elasticity solutions. Subsequent discussion on the limitations of each approximate beam theory in calculating natural frequencies is made. Mode shapes and cross-section deformations for the first five modes of vibration are presented. Additionally, the shape factor for the Timoshenko beam theory is analyzed for hollow-anisotropic sections