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For thin-walled sections, lateral and lateral-torsional buckling are often affected by distortion of the section and this can severely reduce the critical stress. A calculation method for the critical stress which takes account of distortional effects, based on Generalised Beam Theory (GBT), is presented in this paper. The limiting slenderness at which distortional effects begin to take effect is evaluated for channel- and hat-sections on the basis of parametric studies. Approximate formulae for the limiting slenderness are then given.

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... The origin of GBT can be traced back to the 1960s, when Schardt (1966) published his first work on this approach to solve structural problems involving prismatic thin-walled members (in German), which was inspired by the so-called "general variational method" proposed by Vlasov (1958). Subsequently, Schardt and his coworkers carried out an intense research activity that led to the development of several GBT formulations and applications (Schardt, 1989(Schardt, , 1994a(Schardt, , 1994b À see also the bibliography available at http://vtb.info/). However, it is fair to say that this theory had very little impact on the international technical/scientific community until the beginning of the 1990s, due to the fact that the overwhelming majority of the research on GBT was published in German and its dissemination (almost exclusively through technical reports of the Technical University of Darmstadt) was quite limited. ...

... Although this chapter is limited to "isotropic linear elastic buckling" problems (those most pertinent for mild steel structures), it should be noted that GBT formulations are available to perform bifurcation analyses of stainless steel or aluminum (e.g., Gonçalves, Le Grognec, & Camotim, 2010b), steel-concrete composite (e.g., Henriques, Gonçalves, & Camotim, 2016), and laminated FRP composite (e.g., Silvestre & Camotim, 2002bor Silva, Silvestre, & Camotim, 2010. Moreover, the "starting point" of this chapter are the problems handled by Schardt (1989Schardt ( , 1994aSchardt ( , 1994b and disseminated by Davies (1998), which involve essentially simply supported isolated steel members acted by uniform stress resultant diagrams. To keep the presentation systematic, the GBT developments concerning the (i) loading conditions, (ii) support conditions, (iii) extension to structural systems, and (iv) design applications are addressed separately and sequentially À the advances related to the crosssection analysis were already addressed in Section 5.3. ...

... Most of the treatments of the lateral buckling of tapered members have made recourse to numerical formulations, as closed-form solutions do not in general exist, even if some sweeping simplifying assumptions are made. Some research findings can be found in the work of Lee, Morrell, and Ketter (1972), Nethercot (1973aNethercot ( , 1973b, Morrell and Lee (1974), Taylor, Dwight, and Nethercot (1974), Horne and Morris (1977), Horne, Shakir-Khalil, and Akhtar (1979), Brown (1981), Nakane (1984), Wekezer (1985, Bradford and Cuk (1988), Bradford (1988aBradford ( , 1988bBradford ( , 1989, Chan (1990), Trahair (1993), Ronagh and Bradford (1994a, 1994b, and Ronagh, Bradford, andAttard (2000a, 2000b). Some of the more recent and rigorous studies have identified some anomalies between solutions for web-tapered I-section members developed by earlier researchers, and the most profound anomaly was found to be as a result of the omission of an important term in the formulation of the buckling analysis of web-tapered members. ...

This chapter presents a new design method for considering the bracing effect of sheathing boards that are attached to CFS (cold-formed steel) structural members. It also presents a detailed summary of history, development, and flaws of the available sheathing braced design guidelines of the American Iron and Steel Institute. Previous investigations have shown that the current American Iron and Steel Institute (AISI) design specification for sheathing bracing design of CFS wall panels is unconservative (design predictions>experimental strength) due to exaggerated sheathing stiffnesses calculated from ideal loading conditions rather than worst-case loading conditions. Therefore a new design method is suggested based on the performance (strength and stiffness) of the individual sheathing-fastener connections. The current test setup of the AISI is improved to simulate realistic failure modes of the sheathing-fastener connections. The test results revealed: (1) the prominent influence of tensile modulus of the sheathing board and the geometric dimensions of the CFS stud (lever arm), (2) the sudden and catastrophic failure modes and they should be considered in the form of coefficients in the design expressions to prevent an unsafe design, and (3) the sheathing thickness did not influence the performance of the sheathing board (strength, stiffness, and failure mode). Based on test results, new expressions are formulated to predict the stiffness and strength of the individual sheathing-fastener connections. Finally, the effectiveness of the proposed simplified design method is illustrated by a design example, which quantifies the benefit of adopting this method in AISI design guidelines.

... Later in 1994, Schardt publishes his publications in English language, [137,138], in which the focus are the non-linear analysis and the buckling from the coupling among different deformations modes. ...

... Non-linear analysis in GBT have been proving to be an alternative and powerful tool in describing and modeling complex problems in thin-walled beams. The first non-linear analysis in GBT was developed to solve coupled stability problems by Zhang [143,144,180] and Richard Schardt, [137,138], and later by Davies and his co-workers [47,48]. These publications introduce the concept of third-order coupling tensor i jk X and its physical meaning. ...

... 7.59 is the linear part of the virtual work of the initial stress, which leads to the initial stress stiffness matrix due to the membrane part. The studies of Richard Schardt, [137,138] consider only this part in non-linear GBT. The development of this integral is carried out by the introduction of GBT's separation of variable assumptions, given in equations 2.12, 2.13 and 2.14, which splits the domain of integration in longitudinal and transversal directions, x and s, respectively. ...

In the last decades, Finite Element Method has become the main method in statics and dynamics analysis in engineering practice. For current problems, this method provides a faster, more flexible solution than the analytic approach. Prognoses of complex engineer problems that used to be almost impossible to solve are now feasible. Although the finite element method is a robust tool, it leads to new questions about engineering solutions. Among these new problems, it is possible to divide into two major groups: the first group is regarding computer performance; the second one is related to understanding the digital solution. Simultaneously with the development of the finite element method for numerical solutions, a theory between beam theory and shell theory was developed: Generalized Beam Theory, GBT. This theory has not only a systematic and analytical clear presentation of complicated structural problems, but also a compact and elegant calculation approach that can improve computer performance. Regrettably, GBT was not internationally known since the most publications of this theory were written in German, especially in the first years. Only in recent years, GBT has gradually become a fertile research topic, with developments from linear to non-linear analysis. Another reason for the misuse of GBT is the isolated application of the theory. Although recently researches apply finite element method to solve the GBT's problems numerically, the coupling between finite elements of GBT and other theories (shell, solid, etc) is not the subject of previous research. Thus, the main goal of this dissertation is the coupling between GBT and shell/membrane elements. Consequently, one achieves the benefits of both sides: the versatility of shell elements with the high performance of GBT elements. Based on the assumptions of GBT, this dissertation presents how the separation of variables leads to two calculation's domains of a beam structure: a cross-section modal analysis and the longitudinal amplification axis. Therefore, there is the possibility of applying the finite element method not only in the cross-section analysis, but also the development for an exact GBT's finite element in the longitudinal direction. For the cross-section analysis, this dissertation presents the solution of the quadratic eigenvalue problem with an original separation between plate and membrane mechanism. Subsequently, one obtains a clearer representation of the deformation mode, as well as a reduced quadratic eigenvalue problem. Concerning the longitudinal direction, this dissertation develops the novel exact elements, based on hyperbolic and trigonometric shape functions. Although these functions do not have trivial expressions, they provide a recursive procedure that allows periodic derivatives to systematise the development of stiffness matrices. Also, these shape functions enable a single-element discretisation of the beam structure and ensure a smooth stress field. From these developments, this dissertation achieves the formulation of its primary objective: the connection of GBT and shell elements in a mixed model. Based on the displacement field, it is possible to define the coupling equations applied in the master-slave method. Therefore, one can model the structural connections and joints with finite shell elements and the structural beams and columns with GBT finite element. As a side effect, the coupling equations limit the displacement field of the shell elements under the assumptions of GBT, in particular in the neighbourhood of the coupling cross-section. Although these side effects are almost unnoticeable in linear analysis, they lead to cumulative errors in non-linear analysis. Therefore, this thesis finishes with the evaluation of the mixed GBT-shell models in non-linear analysis.

... The origin of GBT can be traced back to the 1960s, when Schardt [1] published his seminal work (in German) on this approach to solving structural problems involving prismatic thin-walled members. Subsequently, Schardt and his co-workers carried out intense research that led to the development of several GBT formulations and applications [2]- [4]. However, it is fair to say that this theory had very little impact on the international scientifi c community before the beginning of the 1990s due to the fact that the overwhelming majority of the research on GBT was published in German and had a rather limited dissemination (mostly through technical reports of the University of Darmstadt, although [3], [4] are the exceptions to this rule). ...

... Subsequently, Schardt and his co-workers carried out intense research that led to the development of several GBT formulations and applications [2]- [4]. However, it is fair to say that this theory had very little impact on the international scientifi c community before the beginning of the 1990s due to the fact that the overwhelming majority of the research on GBT was published in German and had a rather limited dissemination (mostly through technical reports of the University of Darmstadt, although [3], [4] are the exceptions to this rule). This situation was altered by Davies and his collaborators (e.g. ...

... The solution to the diff erential equilibrium equation (Eq. (2)) may be obtained through various methods or techniques, namely (i) fi nite diff erences, originally employed by Schardt [2] (and later also used by Davies et al. [3], [4]), (ii) the methods of Galerkin or Rayleigh-Ritz (in elastic buckling problems both methods lead to exactly the same results if the same approximation functions are adopted), which provide exact solutions for locally and globally simply sup-activity concerning GBT formulations and applications was carried out at the Technical University of Lisbon (TU Lisbon) and has led to a large number of publications. The aim of this work is to provide a state-of-the-art report on the GBT formulations and applications developed at TU Lisbon to assess the buckling behaviour of members and structural systems built from thin-walled steel profi les, namely cold-formed ones. ...

This paper addresses the use of generalized beam theory (GBT) to analyse the local and global buckling behaviour of thin-walled steel members and structural systems. After a brief historical perspective of GBT developments, the main concepts and procedures involved in performing buckling analyses are summarized in a systematic fashion. That is followed by a state-of-the-art report concerning the most recent GBT formulations and applications that have been developed to assess the buckling behaviour of members, frames and trusses with various loading and support conditions. In order to illustrate the unique modal features and show the potential of the GBT approach to buckling analysis, numerical results and a few practical applications are presented and discussed; for validation purposes, most of these results are compared with values yielded by shell finite element analyses, performed in the ABAQUS and ANSYS programs. The paper closes with a few words on the future perspectives of GBT-based buckling analysis.

... Refs. [9][10][11]. ...

... Analysis of Steel-Concrete Composite Beams Fig.16. Orthogonal GBT deformation modes(6)(7)(8)(9)(10)(11)(12)(13).17 ...

This paper reports the most recent developments concerning Generalized Beam Theory (GBT) formulations, and corresponding finite element implementations, for steel-concrete composite beams. These formulations are able to perform the following types of analysis: (i) materially nonlinear analysis, to calculate the beam load-displacement response, up to collapse, including steel plasticity, concrete cracking/crushing and shear lag effects, (ii) bifurcation (linear stability) analysis, to obtain local/distortional bifurcation loads and buckling mode shapes of beams subjected to negative (hogging) bending, accounting for shear lag and concrete cracking effects and (iii) long-term service analysis including creep, cracking and arbitrary cross-section deformation (which includes shear lag) effects. The potential (computational efficiency and accuracy) of the proposed GBT-based finite elements is illustrated through several numerical examples. For comparison purposes, results obtained with standard finite strip and shell/brick finite element models are provided.

... Development of nonlinear Generalized Beam Theory (GBT) analysis was first attempted by Schardt [1,2] and his co-workers [3] to solve coupled stability problems. Here, the linear uncoupled combination of deformation modes in linear GBT analysis [4] was extended to a nonlinear analysis by introducing the concept of third order deformation modes coupling tensor ijk X which indicates if there is a coupling among the internal forces generated in mode j due to a virtual displacement in mode k and an initial stress in mode i. ...

... ABSTRACT Development of nonlinear Generalized Beam Theory (GBT) analysis was first attempted by Schardt [1,2] and his co-workers [3] to solve coupled stability problems. Here, the linear uncoupled combination of deformation modes in linear GBT analysis [4] was extended to a nonlinear analysis by introducing the concept of third order deformation modes coupling tensor ijk X which indicates if there is a coupling among the internal forces generated in mode j due to a virtual displacement in mode k and an initial stress in mode i. ...

In this study the formulation of fully geometrically nonlinear GBT analysis is developed for thin-walled circular pipes. This formulation uses the Sanders definition of kinematic relations for circular cylindrical shells which satisfies the Love-Kirchhoff assumption. Here, the nonlinearity is only considered for the membrane strains since for thin-walled sections plate contributions are insignificant. Furthermore displacements are generally kept small since the formulation is based on a Total Lagrangian description. The initial second-order and third-order stress and displacement tangent stiffness matrices were developed using third-order and fourth-order GBT mode coupling tensors.

... [1]) or in-plane (distortional) (e.g. [2]) deformations take place. The reason is the assumption of rigid cross-section that makes them unable to take into account warping and distortion. ...

... GBT (Generalized Beam Theory), formulated by Schardt [2], [25], played an important role in generalizing Vlasov's theory by integrating flexural and torsional distortional effects. In what followed, Davies and co-researchers [27]- [30] disseminated GBT through a series of publications studying linear static or bucking problems of thin-walled beams. ...

Comparing Euler-Bernoulli or Tismoshenko beam theory to higher order beam theories, an essential difference can be depicted: the additional degrees of freedom accounting for out-of plane (warping) and in-plane (distortional) phenomena leading to the appearance of respective higher order geometric constants. In this paper, after briefly overviewing literature of the major beam theories taking account warping and distortional deformation, the influence of distortion in the response of beams evaluated by higher order beam theories is examined via a numerical example of buckling drawn from the literature.

... However, in many generalized beam models, the CS-modes are evaluated from linear or linearized problems and so it is not guaranteed that the reduced kinematics are capable to reproduce the true one in the nonlinear range. In fact, many proposals regard the linear ( Schardt, 1989;Camotim et al., 20 04;20 06;Vieira et al., 2014 ) and buckling ( Schardt, 1994;Camotim and Basaglia, 2013;Gonçalves and Camotim, 2013;Li and Schafer, 2010 ) analysis of TWMs while few studies analyze the complete post-critical behavior of such structures Gonçalves and Camotim, 2012 ). In our opinion, the reason is that, in this context, it is necessary to consider a significant number of CS-modes to obtain accurate results, much more than in linear or buckling analyses. ...

... Many proposals of generalized beam models exist in the context of linear ( Schardt, 1989;Camotim et al., 20 04;20 06;Vieira et al., 2014 ) and buckling ( Schardt, 1994;Camotim and Basaglia, 2013;Gonçalves and Camotim, 2013 ) analysis of TWMs. On the contrary, few studies analyze the complete post-critical behavior of such structures Gonçalves and Camotim, 2012 ) because, in this case, this approach seems to be less effective. ...

In this paper a semi-analytic solution for the post-critical behavior of compressed thin walled members with generic cross sections is presented. It is based on the Koiter approach and the method of separation of variables. The buckling solution is exactly evaluated using a single sinusoidal function and the initial post-critical behavior is obtained as a decoupled sinusoidal series solution along the beam axis. A specialized integration scheme allows obtaining the solution with only a few terms and a very low cost, with respect to standard finite element analyses. This tool is then used to highlight the reason of the poor behavior of beam models enriched with cross section deformation modes in reconstructing the post-critical solution. Successful strategies are proposed in order to overcome these limitations. A series of numerical tests are reported.

... In the 1990s, more studies using GBT started to be published in English concerning the stability analysis of thin-walled members by Schardt [132,133], Davies [39,40] and Leach [87] which have played a major role in introducing GBT to the international research community. ...

The detailed structural analysis of thin-walled circular pipe members often requires the use of a shell or solid-based finite element method. Although these methods provide a very good approximation of the deformations, they require a higher degree of discretization which causes high computational costs. On the other hand, the analysis of thin-walled circular pipe members based on classical beam theories is easy to implement and needs much less computation time, however, they are limited in their ability to approximate the deformations as they cannot consider the deformation of the cross-section. This dissertation focuses on the study of the Generalized Beam Theory (GBT) which is both accurate and efficient in analyzing thin-walled members. This theory is based on the separation of variables in which the displacement field is expressed as a combination of predetermined deformation modes related to the cross-section, and unknown amplitude functions defined on the beam's longitudinal axis. Although the GBT was initially developed for long straight members, through the consideration of complementary deformation modes, which amend the null transverse and shear membrane strain assumptions of the classical GBT, problems involving short members, pipe bends, and geometrical nonlinearity can also be analyzed using GBT. In this dissertation, the GBT formulation for the analysis of these problems is developed and the application and capabilities of the method are illustrated using several numerical examples. Furthermore, the displacement and stress field results of these examples are verified using an equivalent refined shell-based finite element model. The developed static and dynamic GBT formulations for curved thin-walled circular pipes are based on the linear kinematic description of the curved shell theory. In these formulations, the complex problem in pipe bends due to the strong coupling effect of the longitudinal bending, warping and the cross-sectional ovalization is handled precisely through the derivation of the coupling tensors between the considered GBT deformation modes. Similarly, the geometrically nonlinear GBT analysis is formulated for thin-walled circular pipes based on the nonlinear membrane kinematic equations. Here, the initial linear and quadratic stress and displacement tangent stiffness matrices are built using the third and fourth-order GBT deformation mode coupling tensors. Longitudinally, the formulation of the coupled GBT element stiffness and mass matrices are presented using a beam-based finite element formulation. Furthermore, the formulated GBT elements are tested for shear and membrane locking problems and the limitations of the formulations regarding the membrane locking problem are discussed.

... Moreover, Schardt proved that high order cross-section's warping shapes are combined to transversal bending cross-section distortion. Furthermore, GBT can be not only extended to geometrically non-linear analysis (Schardt 1994a, Schardt 1994b, Davies 1994, but also to physically non-linear analysis (Abambres 2013, Abambres 2014a, Abambres 2014b. ...

This paper presents alternative complementary shear and transversal elongation modes of Generalized Beam Theory (GBT) for thin-walled hollow circular cross-sections and compares them to the recent developments concerning membrane's shear and transversal elongation. The main features of the alternative complementary modes are: i) despite of Poisson's effect, each complementary mode is related to a clear membrane's behavior: transversal elongation and shear deformation; ii) The coupling between these complementary modes is minimized, as well as the coupling between theses modes and the respective original GBT's mode; iii) the remained coupling effect is limited to plate's behavior. To illustrate the present alternative, complementary modes and its limitations, the detailed examples applied in a short and deep pipe are carried out and their final results are compared with a full shell element model.

... Distortional buckling behavior has also been investigated using the generalized beam theory (GBT). A calculation method based on the GBT was developed by Schardt (1994), including approximate formulas for the limiting slenderness of hat and C-sections. Further study by Silvestre and Camotim (2004a) provided GBT development for ordinary C-and Z-sections, with particular attention given to distortional buckling. ...

The strength of cold-formed steel beams with stiffened flanges may be controlled by distortional buckling. Buckling stress prediction methods have been developed for flanges under uniform compression. However, channel sections are commonly used where bending occurs about the minor axis with flanges under a stress gradient, such that the edges that are in compression and the flanges may experience distortional buckling. Current design specifications do not explicitly address this failure mode, which could lead to unsafe designs. This paper presents and verifies an analytical approach for distortional buckling stress prediction for flanges under a stress gradient. The approach is consistent with the design method used for flanges under uniform compression in the American Iron and Steel Institute (AISI) specification for the design of cold-formed steel members. This consistency facilitates a straightforward incorporation into the design specification.

... Distortional buckling behavior has also been investigated using the generalized beam theory (GBT). A calculation method based on the GBT was developed by Schardt (1994), including approximate formulas for the limiting slenderness of hat and C-sections. Further study by Silvestre and Camotim (2004a) provided GBT development for ordinary C-and Z-sections, with particular attention given to distortional buckling. ...

The strength of cold-formed steel beams with stiffened flanges may be controlled by distortional buckling. Buckling stress prediction methods have been developed for flanges under uniform compression. However, channel sections are commonly used where bending occurs about the minor axis with flanges under a stress gradient, such that the edges that are in compression and the flanges may experience distortional buckling. Current design specifications do not explicitly address this failure mode, which could lead to unsafe designs. This paper presents and verifies an analytical approach for distortional buckling stress prediction for flanges under a stress gradient. The approach is consistent with the design method used for flanges under uniform compression in the American Iron and Steel Institute (AISI) specification for the design of cold-formed steel members. This consistency facilitates a straightforward incorporation into the design specification.

... The shear flow associated with non-uniform warping leads also to in plane deformation of the cross section, relaxing the assumption that the cross section shape does not change after deformation. For this purpose the so-called higher order beam theories have been developed taking into account shear lag [1] and distortional (in-plane deformation) effects [2]. In this paper, a higher order beam theory is employed for linear static analysis of beams of arbitrarily shaped, homogeneous cross-section, taking into account warping and distortional phenomena due to axial [3], shear, flexural, and torsional behavior [4]. ...

... The shear flow associated with non-uniform warping leads also to in plane deformation of the cross-section, relaxing the assumption that the cross-section shape does not change after deformation. For this purpose, the so-called higher order beam theories have been developed, taking into account shear lag (out-of-plane deformation) [1][2] and distortional (in-plane deformation) effects [3][4]. Higher order beam theories are of increased interest due to their important advantages over approaches such as 3-D solid or shell solutions as they [5][6] : a) require less modelling time, b) permit isolation of structural phenomena and results' interpretation (rotations, warping parameters, stress resultants etc. are also evaluated in addition to displacements and stress components), c) facilitate modelling of supports and application of external loading, d) require significantly less number of degrees of freedom (dofs) reducing computational time, e) facilitate parametric analyses without the construction of multiple models. ...

Both Euler-Bernoulli and Timoshenko beam theories maintain the assumptions that neither out-of-plane (warping) nor in-plane (distortion) deformation contribute to beams response. To account for shear lag effects, the inclusion of non-uniform warping is necessary, relaxing the assumption of plane cross section. The shear flow associated with non-uniform warping leads also to in-plane deformation of the cross-section, relaxing the no-distortion assumption. For this purpose, the so-called higher order beam theories have been developed taking into account shear lag and distortional effects. In this paper, the higher order beam theory developed in a previous work of the authors is employed for linear analysis of beams of arbitrarily shaped, homogeneous cross-section, including warping and distortional phenomena due to axial, shear, flexural, and torsional behavior. The beam is subjected to general load and boundary conditions. The analysis consists of two stages. The first stage is a cross-sectional analysis, establishing the possible distortional and warping deformation patterns (axial, flexural and torsional modes by means of the sequential equilibrium scheme and the Boundary Element Method). The second stage is a longitudinal analysis where the four rigid body displacements along with the extracted deformation patterns multiplied by respective independent parameters expressing their contribution to the beam deformation are included in the beam analysis (Finite Element Method). Axial warping and distortional modes are examined. In addition, numerical examples with practical interest are presented in order to highlight the importance of axial modes in linear analysis of beams-cross-sections of class 4 (EN1993-1-1), i.e., vulnerable to axial load are analyzed.

... Its creator, Richard Schardt [1], initially developed it as a generalization of Vlasov beam theory [2] in order to describe the linear behavior of open thin-walled beams. Schardt demonstrated not only the extension of GBT to non-linear analysis [3][4][5], he also showed that the incorporation of hollow circular cross-sections [1,6] is straightforward. ...

... Among them, generalized beam theory (GBT) is one of the most recent contributions. GBT originates from the work of Schardt [18,19], and has been extended into almost every field of structural analysis of thin-walled beams by Davies et al. [20], Silvestre et al. [21], and Camotim et al. [22]. By applying a piece-wise description of cross-sections and performing cross-section analyses, GBT is able to handle arbitrary prismatic cross-sections [23] and provide a set of deformation modes hierarchically organized into several families. ...

In this paper, a new approach is proposed to identify sectional deformation modes of the doubly symmetric thin-walled cross-section, which are to be employed in formulating a one-dimensional model of thin-walled structures. The approach considers the three-dimensional displacement field of the structure as the linear superposition of a set of sectional deformation modes. To retrieve these modes, the modal analysis of a thin-walled structure is carried out based on shell/plate theory, with the shell-like deformation shapes extracted. The components of classical modes are removed from these shapes based on a novel criterion, with residual deformation shapes left. By introducing benchmark points, these shapes are further classified into several deformation patterns, and within each pattern, higher-order deformation modes are derived by removing the components of identified ones. Considering the doubly symmetric cross-section, these modes are approximated with shape functions applying the interpolation method. The identified modes are finally used to deduce the governing equations of the thin-walled structure, applying Hamilton’s principle. Numerical examples are also presented to validate the accuracy and efficiency of the new model in reproducing three-dimensional behaviors of thin-walled structures.

... Generalized Beam Theory, GBT, is a numerical approach, which was initially developed to describe open thin-walled beams by Richard Schardt in Darmstadt, Germany. This approach is applicable in linear analysis [1], but it has been further extended to geometric non-linear analysis [2][3][4]. As an introductory point, this theory can be understood as a generalization of Vlasov Beam Theory [5]. ...

This work presents a procedure to couple shell and Generalized Beam Theory, GBT, elements. The main focus of this procedure is the possibility to model mixed beam frame structures, which the traditional shell elements are applied at the joints and GBT elements are used to model the beams/columns. Such modeling technique can use the benefits of both elements. At the joints, shell elements can easily simulate different types of geometry conditions and details, such as stiffeners and holes; meanwhile, for the beams and columns, GBT can provide high performance, accuracy and an easy modeling approach with clear results.

... The shear flow associated with non-uniform warping leads also to in plane deformation of the cross section, relaxing the assumption that the cross section shape does not change after deformation. For this purpose the so-called higher order beam theories have been developed taking into account shear lag [1] and distortional (in-plane deformation) effects [2]. ...

... The model does not involve any springs and the critical stress can be calculated analytically. Apart from the analytical models mentioned above, finite strip methods [12][13][14][15][16][17][18], finite element methods [19][20][21], generalized beam theory (GBT) [22][23][24][25][26], neural network [27][28][29] and experimental methods [30][31][32] have also been used to analyze the distortional buckling of CFS columns and beams. ...

... where n is the number of half-waves and ℓ n / is the half-wavelength of the buckling mode. Plugging this solution in the bifurcation equation leads to the standard eigenvalue problem [31] ...

This paper compares two distinct approaches for obtaining the cross-section deformation modes of thin-walled members with deformable cross-section, namely the method of Generalized Eigenvectors (GE) and the Generalized Beam Theory (GBT). First, both approaches are reviewed, emphasizing their differences and similarities, as well as their resulting semi-analytical solutions. Then, the GE/GBT deformation modes for four selected cross-sections are calculated and examined in detail. Subsequently, attention is turned to the efficiency and accuracy of the GE/GBT mode sets in typical benchmark problems, namely the calculation of the global–local–distortional first-order and buckling (bifurcation) behaviors of bars with the previously analyzed cross-sections. It is concluded that GE and GBT, both based on the method of separation of variables, yield accurate results although they use different structural models and mode selection strategies. Therefore they offer complementary advantages, which are put forward in the paper.

... Distortional buckling involves deformations of the junctions between plate elements. Several studies have been conducted on cold formed steel buckling modes [6][7][8][9][10][11][12][13] by various authors. ...

In today's world, the construction industry both structural and non-structural elements are fabricated from thin gauges of steel sheets. These thin walled sections are being used as columns, beams, joists, studs, floor decking, built-up sections and other components for lightly loaded structures. Unlike hot rolled sections, the design of Cold-Formed Steel (CFS) section for beam is predominantly controlled by various buckling modes of failure, thereby drastically reducing their load carrying capacity. Hence there is an urgent need in the CFS industry to look beyond the conventional CFS beam sections and investigate newly proposed innovative CFS beam sections, which seem to prove structurally much more efficient. Prior to any experimental investigation of innovative beam sections, there is a need to carry out theoretical design using some of the most appropriate available methods applicable to the case under consideration. This paper focuses on such theoretical designs for various innovative sections using available analytical design tools together with appropriate codal guidelines.

... Indeed, together with his co-workers Leach and Jiang, Davies applied GBT extensively to investigate the buckling behaviour of cold-formed steel members and, in particular, showed that it is a valid and often advantageous alternative to fully numerical (finite element or finite strip) analyses [3,4] . Moreover, it seems fair to believe that Davies also had something to do with Schardt's decision to start publishing in English [5,6] . ...

This paper provides an overview of the Generalised Beam Theory (GBT) applications and formulations recently developed at the TU Lisbon. The conventional GBT is (i) applied to derive analytical distortional buckling formulae and (ii) extended to cover (ii 1) orthotropic and elastic-plastic materials, (ii 2) closed and branched cross-sections and (ii 3) vibration and post-buckling analyses. In order to give an idea about the potential of the new GBT formulations, a few numerical results are presented and briefly discussed. Key Words: Generalised beam theory (GBT), GBT stability analysis, GBT vibration analysis, GBT post-buckling analysis, distortional buckling formulae, thin-walled steel members, thin-walled aluminium members, thin-walled FRP members.

... Seah and Khong presented a semi-analytical and semi-numerical approach for predicting the critical lateral-torsional buckling moments of cold-formed steel channel section beams subjected to uniform bending [4]. Schardt investigated the lateraltorsional buckling behaviour of channel and hat sections also taking account of distortional effects [5] using the generalized beam theory. Based on the parametric studies, approximate formulae for the limiting slenderness were proposed. ...

This chapter addresses the use of Generalized Beam Theory (GBT) to analyze the local, distortional, and global linear stability (or bifurcation, or buckling) behavior of thin-walled steel members and structural systems. After presenting a brief historical perspective of the GBT developments, the main concepts and procedures involved in performing buckling analyses are summarized in a systematic fashion – particular attention is paid to a recently developed cross-section analysis procedure that provides automatically “hierarchically ordered” deformation mode families (the GBT “trademark”). Then, the chapter addresses the most recent GBT formulations and applications developed to assess the buckling behavior of thin-walled members, frames, and trusses exhibiting arbitrary flat-walled or circular cross sections and a wide range of loading and support conditions–a detailed list of the key references on the various topics involved is also included. To illustrate the application and show the potential, stemming from its unique modal features, of the GBT approach to perform buckling analysis, numerical results involving several problems of practical interest are presented and discussed. These numerical results concern isolated members with various cross-section shapes and different loading/support conditions, space building frames, and plane roof-supporting trusses. For validation purposes, most of these results are compared with values yielded by shell finite element analyses, performed in the code Ansys. Finally, the chapter closes with a few concluding remarks.

A new systematic approach is presented to derive the cross-section deformation modes of thin-walled beams with arbitrary sections within the framework of a higher-order beam theory (HoBT). New sets of higher-order modes, in this case warping and distortion, are derived hierarchically from the lowest mode set by considering the consistency between the strain field and the stress field generated by the modes in lower sets. Warping modes are derived by the shear stress of in-plane modes while distortion modes are induced by out-of-plane deformations via Poisson's effect. Higher-order modes are shown to be built as a linear combination of the integrated functions of lower-order modes, where the combination coefficients are determined by the orthogonality condition among the higher-order modes. Because the proposed method does not require any approximation when determining sectional mode shapes, no cross-section discretization, commonly used in existing studies, is needed. The effectiveness of the proposed mode derivation process is verified by comparing the static and modal analysis results of thin-walled beams with open, closed, and flanged cross-sections obtained by the proposed method, other HoBTs, and shell finite elements.

In this work, a finite element based optimization methodology is developed to obtain the optimal designs of thin-walled open cross-section columns for maximum buckling load. As a constraint for the optimization study, the total material volume of the column is kept constant. At first, an analytical formulation based on Bleich's (1952) approach, which considers the combined effect of both torsional and flexural buckling, is used to validate the finite element buckling load computation in ANSYS. Subsequently, these finite element buckling results are coupled with a Genetic Algorithm (GA) based optimization routine in MATLAB to obtain the optimal design of the cross-section of the columns. Optimal results are compared with a base model of the column having a cruciform cross-section. The optimization of the cross-sections results in remarkable enhancement, up to as high as 236%, in the maximum buckling load capacity compared to the base model.

Warping and distortion are relevant kinematic features of thin-walled beam structures, which have a non-trivial analysis. On this basis, this paper not only evaluates the possible kinematic transmissions involving high-order warping and distortion, but also presents a procedure to analyze structures using mixed models based on shell and Generalized Beam Theory (GBT) elements. In this mixed beam-shell structure, the traditional shell elements are applied at structural detailing points, such as joints, and GBT elements are used to model the beams/columns. Such a modeling technique uses the benefits of both elements. Shell elements can easily simulate different types of geometry conditions and details, such as stiffeners and holes; meanwhile, for the beams and columns, GBT can provide high performance, accuracy, and an easy modeling approach with clear results. The numerical formulation is based on multi-freedom constraint techniques. Special attention is given to the Master–Slave method, which is developed based on GBT kinematic assumptions. Furthermore, there is a discussion concerning the choice of the master degrees of freedom and its implications in numerical performance. An example of a thin-walled hollow circular cross section illustrates the proposed approach and is compared with fully shell element models.

In this paper, a higher order beam theory is employed for linear local buckling analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural, and torsional behavior. The beam is subjected to arbitrary concentrated or distributed loading, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross-sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the buckling analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters constitute the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. The buckling load is calculated and is compared with beam and 3d solid finite elements analysis results in order to validate the method and demonstrate its efficiency and accuracy.

This article presents an analytical investigation on the effects of shear stress and compressive stress gradient on the distortional buckling of cold-formed steel channel- and zed-section beams subjected to uniformly distributed transverse load. The study is performed by using the principle of minimum potential energy. It is shown that for beams subjected to a uniformly distributed transverse load the buckling wave coupling in distortional buckling modes caused due to compressive stress gradient is very important, particularly for long beams. The effect of shear stress on the critical stress of distortional buckling exists but only in short beams. For beams longer than 3 m the shear stress effect can generally be ignored.

This paper proposes an analytical model for analyzing the interaction between web distortion and lateral-torsional buckling of partially restrained I-section beams under transverse distribution loading. The analysis is performed by using Rayleigh-Ritz method, in which the web is modelled as a plate and the two flanges are treated as two independent beams. The total potential energy functional of the system is derived using three-dimensional strain-displacement relationships in solid mechanics. The critical buckling stress and critical buckling moment of the I-section beam are calculated by solving a 3x3 eigen-matrix equation. For the validation of the present model the finite element analysis using three-dimensional shell elements is also carried out. The comparison between the analytical and numerical results demonstrates the correctness and rigorous of the proposed analytical model despite its simplicity.

Usual design practice for distortional buckling considers a lower bound solution as the actual buckling load. In reality, this practice is inconsistent with actual case since the obtained buckling load is a constant value no matter how long the column is and whatever the end condition is. According to available literature, the research dealt with such a problem is found quite rare. In this scenario, this paper presents an analytical approach to establish a new distortional buckling formula, which takes both the effects of column length and end condition into consideration. The formula was derived based on an edge stiffened plate model. The model was assumed to be pin-ended and fix-ended so as to investigate their effects. The Galerkin method was employed to derive the distortional buckling formula. Further, simplifications to the rigorous formula were made to allow them to be easily used by the engineers. Subsequently, in order to verify the accuracy of the derived formula, the results obtained from the derived formula were compared with the numerical results obtained from the computer software GBTUL. In addition, the performance of the derived formula was further verified by comparing the corresponding ultimate strength based on Shafer's DSM expressions with numerical result from the literature. The comparison and validation result shows that the derived formula (i) can be used successfully in estimating the distortional buckling load for both pin-ended and fix-ended columns with practical length and (ii) can general more rational buckling strength estimation due to the consideration of column length and end condition effect.

This paper focuses on computational efficiency aspects related to Generalised Beam Theory (GBT) displacement-based finite elements. In particular, a cross-section node-based DOF approach is proposed, which makes it possible to (i) deal, straightforwardly, with discrete variations of the thickness of the walls (including holes) in the longitudinal direction and (ii) achieve significant computational savings in non-linear problems, with respect to the conventional GBT approach (based on cross-section deformation modes). The proposed approach leads to a beam-like finite element that is equivalent to an assembly of flat quadrilateral shell elements and, therefore, (i) much smaller matrices are handled and (ii) the resulting element stiffness matrix is significantly sparse. The deformation mode participations, which are the trademark of GBT, are recovered through post-processing. Several numerical examples are provided, involving both linear and non-linear (static) problems, to highlight the capabilities and efficiency of the proposed approach.

This paper presents a novel stiffened plate buckling model for describing the distortional buckling of cold-formed steel zed- and channel-section beams when they are bent about their major axis. In the model the compression flange and lip together with the web are treated as a plate with an angle stiffener. By using the classical principle of total potential energy, an analytical expression for the critical buckling stress of the stiffened plate is obtained. In order to validate the present model, finite strip analysis is also carried out for 59 CFS channel-section beams used in the UK market. The comparison of the critical stresses calculated from the present stiffened plate buckling model and those obtained from the finite strip analysis demonstrates that the present model provides an excellent prediction for the critical stress of distortional buckling of CFS section beams.

A computational model was developed to study the nonlinear steady state static response and free vibration of thin-walled carbon nanotubes/fiber/polymer laminated multiscale composite beams and blades. A set of nonlinear intrinsic equations describing the response of rotating cantilever composite beams undergoing large deformations was established. The main assumptions were small local strains and local rotations, large deflections and global rotations. Halpin–Tsai equations and fiber micromechanics were used to predict the bulk material properties of the multiscale nanocomposite. The carbon nanotubes (CNTs) were assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. Discretized by the Galerkin approximation, eigenvalues and vectors and nonlinear steady state static response of the nanocomposite beams and blades were calculated. The volume fraction of fibers, weight percentage of single-walled and multi-walled carbon nanotubes (SWCNTs and MWCNTs) and their aspect ratio were investigated through a detailed parametric study for their effects on the nonlinear response of nanotubes-reinforced moving beams. It was found that natural frequencies are significantly influenced by a small percentage of CNTs. It was also found that the SWCNTs reinforcement produces more pronounced effect in comparison with MWCNTs on the nonlinear steady state static response and natural frequencies of the composite beams.

This paper addresses the determination of deformation modes for curved thin-walled cross-sections through the polygonal approximation of the cross-section mid-line, in the framework of Generalised Beam Theory (GBT). A new GBT cross-section analysis procedure is proposed, which discretises the geometry independently of the number of cross-section DOFs considered to obtain the deformation modes. This procedure is more efficient than the classic GBT one, since curved geometries may be accurately described without increasing the number of deformation modes. In particular, polygonal sections with rounded corners can be easily handled. For illustrative purposes, the procedure is applied to several cross-sections with curved walls and it is shown that it leads to accurate results with coarse cross-section DOF discretisations.

Analysis and design of axially compressed cold formed steel channel section presented in this paper was conducted through experimental study, design based on British standards and North American Specification for the design of cold-formed steel structural members. More than 18 laboratory experiments were undertaken first on these steel channel columns under axial compression. A series of parametric studies were also carried out by varying the thickness and column length. All of these columns failed by local and distortional buckling. The test results were compared with American (AISI-2007) and British Standards (BS5950-Part5) for the design of cold-formed steel structural members. The details of these investigation and the outcomes are presented in this paper.

A buckling loads formula based on Generalised Beam Theory (GBT) was proposed, which could be used in non-linear elastic metallic materials thin-walledcompressed members, such as stainless steel. By introducing non-linear stress-strain relations and instantaneous elastic modulus, the modifications were incorporated in the conventional GBT, and the expressions were formulated to calculate buckling loads of stainless steel members buckling in local, distortional and global modes.compared with the existed test results, it is shown that linear elastic method cannot deal with stainless steel, while the results of proposed method are much more reliable. Moreover, the modified GBT method with deformation plasticity theories produces safer results, which could be used in determining buckling loads of non-linear metallic materials thin-walled members incompression, as well as structural design and further researches.

This paper present a study on behavior of cold formed steel (CFS) built up hat section by varying depth under flexure. This study involves in examination of theoretical, numerical and experimental investigations of specimens in series. Overall four specimens were tested with same configuration length of 1200 mm and depth is varied by keeping all other parameters constant. The section properties of the specimens are obtained using CUFSM 4.0 software. The theoretical data are calculated using Indian Standard code IS 801-1975. All specimens are tested under two point loading with simply supported condition. The experimental results are verified finite element analysis using ANSYS V11 software. The basic buckling classes such as local buckling, distortional buckling and flexural torsional of the sections are compared by these three investigation and failure modes are also discussed.

The paper derives, validates and illustrates the application of GBT-based formulae to estimate distortional critical lengths and bifurcation stress resultants in cold-formed steel rack-section columns, beams and beam-columns with arbitrarily inclined mid-stiffeners and four support conditions. After a brief review of the Generalised Beam Theory (GBT) basics, the main concepts and procedures employed to obtain the formulae are addressed. Then, the GBT-based estimates are compared with exact results and, when possible, also with values yielded by formulae due to Lau and Hancock, Hancock and Teng et al. A few remarks on novel aspects of the rack-section beam-column distortional buckling behaviour, unveiled by the GBT-based approach, are also included.

This paper is a review of the most influential approaches to developing beam models that have been proposed over the last few decades. Essentially, primary attention has been paid to isotropic structures, while a few extensions to composites have been given for the sake of completeness. Classical models - Da Vinci, Euler-Bernoulli and Timoshenko - are described first. All those approaches that are aimed at the improvement of classical theories are then presented by considering the following main techniques: shear correction factors, warping functions, Saint-Venant based solutions and decomposition methods, variational asymptotic methods, the Generalized Beam Theory and the Carrera Unified Formulation (CUF). Special attention has been paid to the latter by carrying out a detailed review of the applications of 1D CUF and by giving numerical examples of static, dynamic and aeroelastic problems. Deep and thin-walled structures have been considered for aerospace, mechanical and civil engineering applications. Furthermore, a brief overview of two recently introduced methods, namely the mixed axiomatic/asymptotic approach and the component-wise approach, has been provided together with numerical assessments. The review presented in this paper shows that the development of advanced beam models is still extremely appealing, due to the computational efficiency of beams compared to 2D and 3D structural models. Although most of the techniques that have recently been developed are focused on a given number of applications, 1D CUF offers the breakthrough advantage of being able to deal with a vast variety of structural problems with no need for ad hoc formulations, including problems that can notoriously be dealt with exclusively by means of 2D or 3D models, such as complete aircraft wings, civil engineering constructions, as well as multiscale and wave propagation analyses. Moreover, 1D CUF leads to a complete 3D geometrical and material modeling with no need of artificial reference axes/surfaces, reduced constitutive equations or homogenization techniques.

This paper discusses aspects related to the mechanics underlying the distortion of thin-walled members with symmetric and periodic open cross-section, such as those commonly employed in cold-formed steel construction. The Generalised Beam Theory (GBT) framework is employed to determine the cross-section distortional deformation modes and obtain insight into the problem under consideration. Besides reviewing the well known case of reflectional symmetry, the implications of rotational symmetry and periodicity through translation or glide reflection are examined. For each case, computationally efficient procedures to obtain the distortional modes are provided. Several examples are presented throughout the paper, in order to enable a better grasp of the concepts and procedures addressed.

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.

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 thin-walled beam model that considers higher order effects is presented in this paper. The beam displacement field is approximated through a linear combination of products between a set of linear independent functions, which are defined over the beam cross-section, and the associated amplitudes that are only dependent on the beam axis. The beam model governing equations are then obtained through the integration over the cross-section of the corresponding elasticity equations weighted by the cross-section approximation functions. A set of uncoupled beam deformation modes is obtained from a non-linear eigenvalue problem that stems directly from the general solution of the differential homogeneous equilibrium equations. The classic deformation modes are obtained without prior assumptions, being associated with null eigenvalues, which requires an adequate computation of a Jordan chain, whereas the higher order modes correspond to the non-null eigenvalues, which allows to measure the mode decay along the beam axis. A numerical example is presented in order to verify the model capabilities to simulate the non-classic effects associated with thin-walled beam higher order deformation modes.

This thesis presents a series of analytical models, based on the Generalized Beam
Theory (GBT), to describe the buckling and post-buckling behaviour of thin-walled prismatic
cold-formed steel structural members under compression and/or bending. GBT has a unique
feature of enabling an theoretical significance to the structural analysis of these members,
which can not be achieved by any other known method.
Initially, a review of the current state of the art in GBT is carried out, together with a
review on the most recent bibliography of alternative methods for post-buckling analysis of
thin-walled structures, allowing to define the specific goal of the present work – the setting up
of a consistent GBT-based methodology for post-buckling analysis. Next, a consistent
formulation based on the concept of Total Potential Energy in the framework of the classical
GBT theory, for post-buckling analysis, was created, enabling the rigorous study of open nonbranched
and closed mono-cellular sections. Subsequently, a series of refinements in the GBT
theory and in the adopted numerical strategies, namely in the Rayleigh-Ritz method and in the
bifurcational calculus techniques, were made in order to analyze the perfect structural
member, without making resource to imperfections, made by plane plates rigidly connected
along the folding lines with a general cross section. Finally, the developments were illustrated
and validated by the resolution of several examples, which were compared to other methods
of analysis for the critical behaviour and for the post-buckling equilibrium paths, like the
Finite Strip and the Finite Elements Method.

A finite element procedure to carry out linear buckling analysis of thin-walled members is developed on the basis of the existing Generalised Beam Theory (GBT) and constrained Finite Strip Method (cFSM). It allows designers to uncouple the buckling modes of a finite element model and, consequently, to calculate pure elastic buckling loads. The procedure can easily be applied to members with general boundary conditions subjected to compression or bending. The results obtained are rather accurate when compared to the values calculated via GBT and cFSM. As a consequence, it is demonstrated that linear buckling analyses can be performed with the Finite Element Method in a similar way as can be done with the existing GBT and cFSM procedures.

Hat-sections are often used to experimentally investigate building sheeting subject to a concentrated load and bending. In car doors, hat-sections are used for side-impact protection. Their crushing behaviour can partly be explained by only observing their cross-sectional behaviour [1]. This crushing behaviour includes the initial and progressive location of moving yield lines and will be explained in this article by an analytical model. The model predicts the results of finite element simulations [1] well, and gives good insight in the complex cross-section crushing behaviour of hat-sections.

This paper presents the derivation of generalised beam theory (GBT)-based fully analytical formulae to provide distortional critical lengths and bifurcation stress resultant estimates in cold-formed steel C and Z-section members (i) subjected to uniform compression (columns), pure bending (beams) or a combination of both (beam–columns), (ii) with arbitrary sloping single-lip stiffeners and (iii) displaying four end support conditions. These formulae incorporate genuine folded-plate theory, a feature which is responsible for their generality and high accuracy. After a brief outline of the GBT fundamentals and linear stability analysis procedure, the main concepts and steps involved in the derivation of the distortional buckling formulae are described and discussed. Moreover, the paper also includes a few remarks concerning novel aspects related to the distortional buckling behaviour of Z-section beams and C-section beam–columns, which were unveiled by the GBT-based approach. Finally, note that, in a companion paper [Thin-Walled Struct., 2004 doi: 10.1016/j.tws.2004.05.002], the formulae derived here are validated and their application, accuracy and capabilities are illustrated. In particular, the GBT-based estimates are compared with exact results and, when possible, also with values yielded by the formulae developed by Lau and Hancock, Hancock, Schafer and Teng et al.

An analysis procedure is presented which allows to calculate pure distortional elastic buckling loads by means of the finite element method (FEM). The calculation is carried out using finite element models constrained according to uncoupled buckling deformation modes. The procedure consists of two steps: the first one is a generalised beam theory (GBT) analysis of the member cross-section, from which the constraints to apply to the finite element model are deduced; in the second step, a linear buckling analysis of the constrained FEM model is performed to determine the pure distortional loads. The proposed procedure is applied to thin-walled members with open cross-section, similar to those produced by cold-forming. The distortional loads obtained are rather accurate. They are in agreement with the loads given by GBT and the constrained finite strip method (cFSM).

This technical note presents a study on the calculation of the critical stress of distortional buckling of cold-formed sigma purlins using EN1993-1-3. The discussion is focussed on the determination of the spring stiffness of the stiffened element, a problem which has not yet been addressed in most design codes. Different support conditions at both the tension and compression ends of the web are employed and their influences on the critical stress of distortional buckling of sigma purlins are investigated. Comparison with finite strip analysis indicates that the model having a fixed support for the tension end and a roller support for the compression end of the web provides the best fit to the finite strip analysis.

This paper introduces the second-order terms associated with geometric nonlinearity into the basic equation of Generalised Beam Theory. This gives rise to simple explicit equations for the load to cause buckling in individual modes under either axial load or uniform bending moment. It is then shown how the explicit procedure can be extended to consider the interaction between local, distortional and global buckling modes. More general load cases require the use of numerical methods of analysis and the finite difference method offers a suitable procedure. The success of Generalised Beam Theory for a wide range of situations is demonstrated by comparing the results obtained using it with both test results and other analyses. It is shown that it offers particular advantages in the analysis of buckling problems in cold-formed sections.