Rui Bebiano

GBT FORMULATION TO ANALYZE THE BUCKLING BEHAVIOR OF THIN-WALLED MEMBERS SUBJECTED TO NON-UNIFORM BENDING

In this paper, one investigates the local-plate, distortional and global buckling behavior of thin-walled steel beams subjected to non-uniform bending moment diagrams, i.e. under the presence of longitudinal stress gradients. One begins by deriving a novel formulation based on Generalized Beam Theory (GBT), which (i) can handle beams with arbitrary open cross-sections and (ii) incorporates all the effects stemming from the presence of longitudinally varying stress distributions. This formulation is numerically implemented by means of the finite element method: one (i) develops a GBT-based beam finite element, which accounts for the stiffness reduction associated to applied longitudinal stresses with linear, quadratic and cubic variation, as well as to the ensuing shear stresses, and (ii) addresses the derivation of the equilibrium equation system that needs to be solved in the context of a GBT buckling analysis. Then, in order to illustrate the application and capabilities of the proposed GBT-based formulation and finite element implementation, one presents and discusses numerical results concerning (i) rectangular plates under longitudinally varying stresses and pure shear, (ii) I-section cantilevers subjected to uniform major axis bending, tip point loads and uniformly distributed loads, and (iii) simply supported lipped channel beams subjected to uniform major axis bending, mid-span point loads and uniformly distributed loads — by taking full advantage of the GBT modal nature, one is able to acquire an in-depth understanding on the influence of the longitudinal stress gradients and shear stresses on the beam local and global buckling behavior. For validation purposes, the GBT results are compared with values either (i) yielded by shell finite element analyses, performed in the code ANSYS, or (ii) reported in the literature. Finally, the computational efficiency of the proposed GBT-based beam finite element is briefly assessed.

2 – GBT-based local and global vibration analysis of thin-walled members

Publisher Summary This chapter discusses the use of numerical techniques to perform vibration analysis, the concepts of finite element analysis (FEA) and finite strip analysis (FSA). The literature review in the chapter has been organized according to the particular methodology employed: there are separate sub-sections dealing with investigations carried out by means of (1) the finite element method (mostly shell element discretizations), (2) the finite strip method and (iii) the generalized beam theory (GBT) – because the aim of the chapter is to present the fundamentals and illustrate the application of a GBT formulation to analyze the vibration behavior of thin-walled members, the last sub-section also includes a brief outline of its content. Silvestre and Camotim (2003) formulated, implemented, and validated an efficient beam finite element intended to perform GBT-based buckling analyses in the context of arbitrarily orthotropic thin-walled members. The most relevant steps involved in the formulation of this finite element, specialized for the vibration analysis of isotropic thin-walled members, are described briefly in the chapter.

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...

GBT-Based Finite Element Formulation to Analyse the Buckling Behaviour of Thin-Walled Members Subjected to Non-Uniform Bending

In this paper, one investigates the local-plate, distortional and global buckling behaviour (critical bifurcation loads and buckling mode shapes) of thin-walled steel beams subjected to non-uniform bending moment diagrams, i.e., under the presence of longitudinal stress gradients. In order to achieve this goal, one begins by developing and numerically implementing a beam finite element formulation based on Generalised Beam Theory (GBT), which (i) can handle beams with arbitrary open cross-sections and (ii) incorporates all the effects stemming from the presence of longitudinally varying stress distributions. After presenting the main concepts, procedures and assumptions involved in the above formulation, one addresses the derivation of the equilibrium equation system that needs to be solved in the context of a GBT buckling analysis. Particular attention is devoted to the main steps involved in the determination of the elementary linear and geometric stiffness matrices, as they must incorporate the stiffness reduction stemming from the presence of the non-uniform bending moments (longitudinal stress gradients) and also of the pre-buckling shear stresses caused by them - the inclusion of this last effect constitutes an original contribution within the context of GBT buckling analyses. Then, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, one presents and discusses numerical results concerning thin-walled steel Ibeams acted by various (uniform and non-uniform) bending moment diagrams. In particular, one analyses (i) cantilevers subjected to uniform major axis bending (Fig. 1(a)), tip point loads (Fig. 1(b)) and uniformly distributed loads (Fig. 1(c)), as well as (ii) simply supported lipped beams subjected to uniform major axis bending, mid-span point loads and uniformly distributed loads by taking full advantage of the GBT modal features, one is able to acquire a much deeper understanding about the influence of the longitudinal stress gradients and shear stresses on the beam local and global buckling mode shapes. For validation purposes, some GBT-based critical loads/moments and buckling mode shapes are compared with values either (i) yielded by shell finite element analyses, performed in the code ANSYS, or (ii) reported in the literature. Finally, one assesses the computational efficiency of the buckling analyses carried out using the proposed GBT-based beam finite element, by comparing the number of degrees of freedom involved with those required to obtain equally accurate results with discretisations in shell finite elements (note that “uniform stress” GBT-based beam finite elements are no longer applicable). Open image in new window Figure 1 Local-plate buckling mode shapes of I-section cantilevers subjected to (a) uniform major axis bending, (b) a tip point load and (c) a uniformly distributed load.

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