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

This paper presents the formulation of a Generalised Beam Theory (GBT) developed to analyse the structural behaviour of composite thin-walled members made of laminated plates and displaying arbitrary orthotropy. The main concepts and procedures involved in the available isotropic first-order GBT are revisited and adapted/modified to account for the specific aspects related to the member orthotropy. In particular, the orthotropic GBT fundamental equilibrium equations and corresponding boundary conditions are derived and their terms are physically interpreted, i.e., associated with the member mechanical properties. Moreover, different laminated plate material behaviours are dealt with and their influence on the GBT equations is investigated. Finally, in order to clarify the concepts involved in the formulated GBT and illustrate its application and capabilities, a thin-walled orthotropic beam is analysed and the results obtained are thoroughly discussed.

Second-order generalised beam theory for arbitrary orthotropic materials

This paper presents a new and general approach for the calculation of cross-section deformation modes in thin-walled beams, to be used in the framework of generalized beam theory (GBT). The proposed approach subdivides and hierarchizes the cross-section deformation modes by employing several kinematic hypotheses. This makes it possible to discard a priori specific types of deformation modes and consequently reduce the number of cross-section degrees-of-freedom. The approach is applicable to arbitrary (with open and closed parts) polygonal cross-sections with external and internal constraints and allows for the a posteriori inclusion of particular deformation modes (e.g., shear deformation modes in a part of the cross-section). Although only GBT applications are dealt with, the deformation modes obtained may be straightforwardly incorporated in other thin-walled beam formulations that include cross-section deformation.

A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory

This paper presents the derivation, validates and illustrates the application of a Generalised Beam Theory (GBT) formulation developed to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Following a brief overview of the conventional GBT, one addresses in great detail the modifications that must be incorporated into its cross-section analysis procedure, in order to be able to handle the ‘branching’ points — they concern mostly issues related to (i) the choice of the appropriate ‘elementary warping functions’ and (ii) the determination of the ‘initial flexural shape functions’. The derived formulation is then employed to investigate the local-plate, distortional and global buckling behaviour of (i) simply supported and fixed asymmetric E-section columns and (ii) simply supported I-section beams with unequal stiffened flanges. For validation purposes, several GBT-based results are compared with ‘exact’ values, obtained by means of finite strip or shell finite element analyses.

GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections

When analysing the structural behaviour of a thin-walled member by means of Generalised Beam Theory (GBT) – a one-dimensional folded-plate theory expressing the member deformed configuration as a linear combination of cross-section deformation modes with amplitudes varying along its length – the performance of the so-called “cross-section analysis” is the key step. Indeed, it consists of determining the deformation modes and evaluating the corresponding modal mechanical properties. However, the available procedures to perform this task are strongly dependent on the cross-section geometry type, a feature precluding its general, efficient and systematic numerical implementation. In order to overcome this difficulty, this paper presents the development and illustrates the application of a novel procedure to perform “GBT cross-section analyses” that (i) is able to handle arbitrary (flat-walled) cross-section shapes, (ii) can be numerically implemented in a systematic and straightforward fashion, and (iii) provides a rational set of deformation modes, which are hierarchically organised into several families, each with well-defined structural/mechanical characteristics. Both the analytical derivations and the underlying mechanical reasoning are explained in detail, and the selected illustrative examples cover the various types of relevant cross-section geometries.

A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses

First-order generalized beam theory describes the behaviour of prismatic structures by ordinary uncoupled differential equations, using deformation functions for bending, torsion and distortion. In second-order theory, the differential equations are coupled by the effect of deviating forces. The basic equations for second-order generalized beam theory are outlined. Solutions for pin-ended supports are presented, demonstrating the coupling effect by modes and by loads. In the different ranges of length, the individual modes are sufficient approximations for the critical load. The application to a thin-walled bar with C-section under eccentric normal force demonstrates the quality of the single-mode compared to the exact solution.

R. Schardt

Papers

Generalized beam theory—an adequate method for coupled stability problems