Rodrigo Gonçalves

GBT-based Structural Analysis of Thin-walled members: Overview, Recent Progress and Future Developments

This paper provides an overview of the Generalised Beam Theory (GBT) fundamentals and reports on the novel formulations and applications recently developed at the TU Lisbon: the use of conventional GBT to derive analytical distortional buckling formulae and extensions to cover (i) the buckling behaviour of members with (i1) branched, closed and closed/branched cross-sections and (i2) made of orthotropic and elastic-plastic materials, and (ii) the vibration and post-buckling behaviours of elastic isotropic/orthotropic members. In order to illustrate the usefulness and potential of the new GBT formulations, a few numerical results are presented and briefly discussed. Finally, some (near) future developments are briefly mentioned.

On the application of beam-column interaction formulae to steel members with arbitrary loading and support conditions

Abstract An investigation on dealing with the application of the so-called Level 1 and Level 2 beam-column interaction formulae to isolated steel members with arbitrary loading and end support conditions is reported. The issues addressed concern mostly the importance, both in terms of safety and accuracy, of adopting appropriate “equivalent moment factor” ( C m ) values. Only the in-plane behavior of members subjected to axial compression and major or minor axis bending is investigated, but several load combinations and support conditions are dealt with. The strength estimates provided by the Level 1 and Level 2 interaction formulae are presented, discussed and compared with “exact” values, obtained from second-order plastic zone analyses performed by means of the FEM code, ABAQUS. On the basis of such comparative studies, several conclusions are drawn concerning the safety and accuracy of the strength estimates yielded by the two interaction formulae and, in particular, it is shown that significant improvements can be achieved by adopting “exact” C m values. Moreover, some attention is paid to issues related to the proper choice or determination of these “exact” C m values.

Global–Local-Distortional Vibration of Thin-Walled Rectangular Multi-Cell Beams

This paper presents the results of an investigation concerning the free vibration behavior (undamped natural frequencies and vibration mode shapes) of thin-walled beams with rectangular multi-cell cross-section (assemblies of parallel rectangular cells in a single direction). Besides local (plate-type) and global (flexural, torsional and extensional) vibration modes, attention is paid to the relatively less-known distortional vibration modes, which involve cross-section out-of-plane (warping) and in-plane deformation, including displacements of the wall intersections. A computationally efficient semi-analytical Generalized Beam Theory (GBT) approach is employed to obtain insight into the mechanics of the problem. In particular, the intrinsic modal decomposition features of GBT — the fact that the beam is described using a hierarchical set of relevant cross-section deformation modes — are exploited to identify and categorize the most relevant vibration modes and deformation mode couplings.

GBT-Based Buckling Analysis Using the Exact Element Method

Generalized Beam Theory (GBT), intended to analyze the structural behavior of prismatic thin-walled members and structural systems, expresses the member deformed configuration as a combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude functions. The determination of the latter, usually the most computer-intensive step of the analysis, is almost always performed by means of GBT-based conventional 1D (beam) finite elements. This paper presents the formulation, implementation and application of the so-called “exact element method” in the framework of GBT-based linear buckling analyses. This method, originally proposed by Eisenberger (1990), uses the power series method to solve the governing differential equation and obtains the buckling eigenvalue problem from the boundary terms. A few illustrative numerical examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determination of buck...

Plastic Bifurcation of Thin-Walled Members: Thin Shell Elements vs. GBT-Based Beam Elements

In this paper, one compares plastic bifurcation results concerning thin-walled members made of non-linear elastic-plastic materials, which are obtained by means of two independent approaches, namely (i) a Total Lagrangian thin shell finite element formulation, developed by the second author, and (ii) a computationally efficient beam formulation based on Generalised Beam Theory (GBT), developed by the remaining two authors it is worth mentioning that the latter neglects the effect of pre-buckling deflections. Initially, one addresses the fundamental concepts, procedures and underlying assumptions involved in the application of the above two formulations, focusing on the similarities and differences existing between them. Then, one presents and thoroughly discusses a set of numerical results, determined through analyses based on the two alternative approaches and concerning (i) aluminium lipped channel and (ii) stainless steel rectangular hollow section (RHS) thin-walled columns (i.e., uniformly compressed members). In the first case, a very good correlation was found between the results (bifurcation loads/stresses and buckling mode shapes) yielded by the two formulations (e.g., see Fig. 1). In the second case, a non negligible discrepancy was observed, as the bifurcation loads provided by the shell formulation consistently lay below the GBT values and the differences, due to the combined influence of relevant pre-buckling deflection and a high imperfectionsensitivity, reached 19% however, the RHS buckling mode shapes exhibited again an excellent agreement. Open image in new window Figure 1 Plastic distortional buckling mode shapes of a lipped channel column of length L=55cm (sheel FEA and GBT).

GBT-Based Vibration Analysis Using the Exact Element Method

Generalized Beam Theory (GBT), intended to analyze the structural behavior of prismatic thin-walled members and structural systems, expresses the member deformed configuration as a combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude functions. The determination of the latter, often the most computer-intensive step of the analysis, is almost always performed by means of GBT-based “conventional” 1D (beam) finite elements. This paper presents the formulation, implementation and application of the so-called “exact element method” in the framework of GBT-based elastic free vibration analyses. This technique, originally proposed by Eisenberger (1990), uses the power series method to solve the governing differential equations and obtains the vibration eigenvalue problem from the boundary terms. A few illustrative numerical examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determina...

Local and Global Vibration Analysis of Thin-Walled Members Subjected to Internal Forces – Application of Generalised Beam Theory

This paper presents the results of a comparative study on the effects of various internal forces/moments on the vibration behaviour of thin-walled members. The analyses are based on Generalised Beam Theory (GBT), a thin-walled bar theory which accounts for crosssection in-plane deformations – its main distinctive feature is the representation of the member deformed configuration by means of a linear combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude functions. The study concerns simply supported cold-formed steel lipped-channel cold-formed steel members exhibiting a wide range of lengths and subjected to four uniform internal force/moment diagrams: (i) axial force, (ii) major-axis bending moment, (iii) minor-axis bending moment and (iv) bi-moment – their magnitudes are specified as percentages of the corresponding critical (buckling) values. The influence of the internal force/moment diagrams (applied loadings) on the member vibration behaviour is assessed in terms of the (i) the frequency drop and (ii) changes in the vibration mode shape. The GBT-based results, obtained with the code GBTUL 2.0 (developed by the authors and available online) are validated by means of available analytical formulae and values provided by analyses carried out by means of the ANSYS and CUFSM codes. Rui A.S. Bebiano, Dinar R.Z. Camotim, Rodrigo M. Gonçalves

A Large Displacement and Finite Rotation Thin-Walled Beam Finite Element Formulation

In this paper, one presents, implements and validates a total Lagrangian beam finite element formulation capable of handling large displacements and finite rotations, as well as cross-section in-plane (distortion and local bending) and out-of-plane (warping) deformations. This formulation can be viewed as a generalisation of the well-known Reissner-Simo beam theory that includes warping/transverse bending deformation modes and in which the member walls can undergo in-plane finite relative rotations. When compared with a shell finite element discretisation, the proposed beam formulation has the distinct advantage of drastically reducing the number of degrees-of-freedom required to achieve equally accurate results. Initially, the kinematical description of the member is addressed, devoting special attention to the assessment of the transverse plate bending effects associated with cross-section distortion. Next, the equilibrium equations and the corresponding symmetric tangent operator are obtained, by assuming an additive update of the rotational parameters. In order to illustrate the application, provide validation and show the capabilities of the proposed formulation, several numerical results are presented, discussed and compared with values yielded by large displacement shell finite element analyses (see Fig. 1) - one obtains a good correlation in all cases, which clearly demonstrates the vast potential of the proposed formulation. Open image in new window Figure 1 Beam deformed configurations yielded by the (a) proposed beam formulation and (b) shell element model