Beam theories are exploited worldwide to analyze civil, mechanical, automotive, and aerospace structures. Many beam approaches have been proposed during the last centuries by eminent scientists such as Euler, Bernoulli, Navier, Timoshenko, Vlasov, etc. Most of these models are problem dependent: they provide reliable results for a given problem, for instance a given section and cannot be applied to a different one. Beam Structures: Classical and Advanced Theories proposes a new original unified approach to beam theory that includes practically all classical and advanced models for beams and which has become established and recognised globally as the most important contribution to the field in the last quarter of a century. The Carrera Unified Formulation (CUF) has hierarchical properties, that is, the error can be reduced by increasing the number of the unknown variables. This formulation is extremely suitable for computer implementations and can deal with most typical engineering challenges. It overcomes the problem of classical formulae that require different formulas for tension, bending, shear and torsion; it can be applied to any beam geometries and loading conditions, reaching a high level of accuracy with low computational cost, and can tackle problems that in most cases are solved by employing plate/shell and 3D formulations.

Beam Structures: Classical and Advanced Theories

On Timoshenko-like modeling of initially curved and twisted composite beams

The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the variational asymptotic method to split a three-dimensional nonlinear elasticity problem into a twodimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with transverse shear and Vlasov refinements, for initially twisted and curved beams with arbitrary geometry and material properties. Several validation cases are presented. First, an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method, and the solution is shown to be identical to that obtained from the theory of elasticity. The shear center locations calculated by VABS for various cross sections agree well with those obtained from common engineering analyses. Comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model. For numerical validation, values of the one-dimensional variables are extracted from an ABAQUS model and compared with results from a one-dimensional beam analysis using cross-sectional constants from VABS. Furthermore, point-wise three-dimensional stress and strain fields are recovered using VABS, and the correlation with the three-dimensional results from ABAQUS is excellent. Finally, classical theory is shown to be insufficient for general-purpose beam modeling. Appropriate refined theories are recommended for some classes of problems.

Validation of the Variational Asymptotic Beam Sectional Analysis

Refined beam elements with arbitrary cross-section geometries

Variational asymptotic beam sectional analysis – An updated version

On asymptotically correct Timoshenko-like anisotropic beam theory

There is no lack of composite beam theories. Quite to the contrary, there might be too many of them. Different approaches, notation, etc., are used by the authors of those theories, so it is not always straightforward to compare the assumptions made and to assess the quantitative consequences of those assumptions. Moreover, there is a serious lack of experimental results and benchmark problems. As a result, one finds that most theories perform about equally well on the few extant benchmark problems. This can obscure differences among theories and simultaneously create the false expectation that a specific theory will perform as well in all cases. The goal of this paper is to attempt to objectively assess theories within a common framework. The validity and relative importance of various assumptions that are present in the literature are discussed. It is hoped that this will be a first step toward the clearly desirable situation in which an engineer can safely and easily choose a composite beam theory based on the type of application and specific needs for fidelity.

Assessment of Beam Modeling Methods for Rotor Blade Applications

Early analyses for numerical cross-sectional analysis of anisotropic beams were based on linear elasticity theory. For more general treatment of non-linear phenomena, asymptotic formulations can serve as the basis for the numerical method, say a finite element method. When based on geometrically non-linear elasticity theory, an intermediate result of such analyses is frequently the splitting of the non-linear 3-D problem into a linear 2-D analysis of the cross section and a non-linear 1-D analysis along the beam. Thus, the published work to date cannot treat the so-called “trapeze effect”, because it stems from non-linearity of the cross-sectional analysis. Herein, a non-linear numerical cross-sectional analysis is presented, based on the variational-asymptotic method and capable of treating cross sections of arbitrary geometry and generally anisotropic material. This type of analysis is particularly important in rotating structures, such as helicopter rotor blades. Results from this model are compared with those available in the literature, both theoretical and experimental.

Bogdan Popescu

Papers

On asymptotically correct Timoshenko-like anisotropic beam theory

Assessment of Beam Modeling Methods for Rotor Blade Applications

Asymptotic treatment of the trapeze effect in finite element cross-sectional analysis of composite beams

Obliqueness effects in asymptotic cross-sectional analysis of composite beams

Composite beam cross-sectional modeling including obliqueness and trapeze effects