GBT-based buckling analysis of thin-walled members with non-standard support conditions

This paper reports on the use of a recently developed Generalised Beam Theory (GBT) formulation, and corresponding finite element implementation, to analyse the local and global buckling behaviour of thin-walled members with arbitrary loading and support conditions — this formulation takes into account longitudinal normal stress gradients and the ensuing pre-buckling shear stresses. After presenting an overview of the main concepts and procedures involved in the performance of a GBT-based (beam finite element) member buckling analysis, one addresses in detail the incorporation of non-standard support conditions, such as (i) full or partial localised displacement or rotation restraints, (ii) rigid or elastic intermediate supports or (iii) end supports corresponding to angle connections. In order to illustrate the application and capabilities of the proposed GBT-based approach, one presents and discusses numerical results concerning cold-formed steel (i) lipped channel beams and (ii) lipped I-section beams and columns with various “non-standard” support conditions — while the beams are acted by uniformly distributed or mid-span point loads, applied at the shear centre axis, the columns are subjected to uniform compression. In particular, it is possible to assess the influence of the different support conditions on the beam and column buckling behaviour (critical buckling loads and mode shapes). For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the code Ansys.     

Global buckling analysis of plane and space thin-walled frames in the context of GBT

This paper reports research work concerning the use of Generalised Beam Theory (GBT) to analyse the global buckling behaviour of plane and space thin-walled frames. Following a brief overview of the main concepts and procedures involved in the performance of a GBT buckling analysis, one presents in detail the formulation and numerical implementation of a GBT-based beam finite element that includes only the first four (rigid-body) deformation modes — namely, one describes (i) the kinematical models developed to simulate the warping transmission at frame joints connecting two or more non-aligned U- and I-section members, (ii) the procedures adopted to handle the effects stemming from the non-coincidence of the member centroidal and shear centre axes (cross-sections without double symmetry), and (iii) the definition of joint elements, which involves providing a relation between the connected member GBT degrees of freedom and the joint generalised displacements. Finally, one presents and discusses numerical results that make it possible to illustrate the application and show the capabilities of the above GBT-based finite-element formulation and implementation. For validation purposes, the GBT-based results (critical buckling loads and mode shapes) are also compared with values yielded by shell (mostly) and beam finite element analyses carried out in the code ANSYS.     

Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes

A formulation of generalised beam theory (GBT) developed to analyse the elastic buckling behaviour of circular hollow section (CHS) members (cylinders and tubes) is presented in this paper. The main concepts involved in the available GBT are adapted to account for the specific aspects related to cross-section geometry. Taking into consideration the kinematic relations used in the theory of thin shells, the variation of the strain energy is evaluated and the terms are physically interpreted, i.e., they are associated with the geometric properties of the CHS. Besides the set of shell-type deformation modes, the formulation also includes axisymmetric and torsion deformation modes. In order to illustrate the application and capabilities of the formulated GBT, the local and global buckling behaviour of CHS members subjected to (i) compression (columns), (ii) bending (beams), (iii) compression and bending (beam-columns) and (iv) torsion (shafts), is analysed. Moreover, the GBT results are compared with estimates obtained by means of shell finite element analyses and are thoroughly discussed.     

对广义梁理论中的平面和空间薄壁构件的整体屈曲分析

采用广义梁理论（GBT）对平面和空间薄壁构件进行整体屈曲分析。简要概述主要概念和广义梁理论屈曲分析中所采用的程序。基于广义梁理论的梁有限元分析考虑了4种刚体变形模式,即：i）运动学模型,用于模拟连接2个或更多U型和I型构件节点的翘曲传输作用;ii）采用程序处理构件重心和剪力中心轴（横截面非双向对称）不重合的效应;iii）节点构件的定义,这些节点可提供连接构件的自由度和节点广义位移之间的联系。最后,介绍并讨论数值结果,其可能有助于基于广义梁理论的有限元公式的制定和实施。并将基于GBT分析的结果（临界屈曲载荷和模态）与ANSYS程序中壳单元和梁单元的建模分析结果进行了对比验证。     

Ultimate strength of submarine pressure hulls with failure governed by inelastic buckling

The analysis of submarine pressure hull assumes great importance among structural engineers due to the complexity involved in the collapse mechanism of stiffened shell structures. In most of the cases, the failure of stiffened shell structures occurs due to elastic buckling. But for some combinations of shell-stiffener geometry and material characteristics, the structure can fail by inelastic buckling, for which the methods of analysis are meagre. In this paper, the analysis of submarine pressure hull structure in which the failure gets governed by inelastic buckling is demonstrated. Three different approaches have been employed to investigate the ultimate strength of the ring stiffened submarine pressure hull structure with inelastic buckling modes of failure. The methods used are ‘Johnson–Ostenfeld inelastic correction’, ‘imperfection method’ and ‘finite element approach’. A typical submarine shell structure has been analysed for the inelastic buckling failure using these three approaches and the results are discussed.     

GBT formulation to analyse the buckling behaviour of isotropic conical shells

This paper presents a Generalised Beam Theory (GBT) formulation developed to analyse the elastic buckling behaviour of isotropic conical shells with constant thickness under axial compression. The GBT approach provides a general solution for 1st and 2nd order analysis using bar elements capable of describing the global and local deformations. The kinematic relations of the theory of thin shells are greatly simplified by the assumptions of null membrane shear strain and transverse extension before they are used for the evaluation of the strain energy variation. Because of the cross-section variation specific to conical shells, the mechanical and geometric properties are no longer constant along the bar axis as it is the case of cylinders and prismatic thin-walled structures. This formulation is validated by comparison between GBT results and values obtained by means of shell finite element analyses.     

Distortional buckling formulae for cold-formed steel C and Z-section members: Part I—derivation

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.     

GBT buckling analysis of thin-walled steel frames: A state-of-the-art report

This paper presents a state-of-the-art report on the use of Generalised Beam Theory (GBT) to assess the buckling behaviour of plane and space thin-walled steel frames. After a very brief overview of the main concepts and procedures involved in performing a GBT buckling analysis, one addresses the development and numerical implementation of a GBT-based beam finite element formulation that is able (i) to unveil local, distortional and global buckling modes, (ii) to handle arbitrary loadings (namely those causing non-uniform member internal force and moment diagrams) and (iii) to incorporate the presence of several frame joint configurations and arbitrary end and/or intermediate support conditions (including those associated with the modelling of bracing systems). In particular, one describes the procedures employed to establish the frame linear and geometric stiffness matrices – special attention is paid to the constraint conditions adopted to ensure the local displacement compatibility at the frame joints. The paper closes with the presentation and discussion of a number of numerical results that make it possible to illustrate the application and show the potential of the GBT-based approach to perform frame buckling analyses – they concern both plane and space frames. In order to validate and assess the numerical efficiency and accuracy of the GBT analyses and results (critical buckling loads and mode shapes), the frames are also rigorously analysed in the commercial code Ansys – both the members and joints are discretised by means of fine shell finite element meshes.     

Interactive buckling and load carrying capacity of thin-walled beam–columns with intermediate stiffeners

The design of thin-walled beam–columns must take into account the overall instability and the instability of component plates in the form of local buckling. This investigation is concerned with interactive buckling of thin-walled beam–columns with central intermediate stiffeners under axial compression and a constant bending moment. The columns are assumed to be simply supported at their ends. The asymptotic expansion established by Byskov and Hutchinson (AIAA J. 15 (1977) 941) is employed in the numerical calculations performed by means of the transition matrix method and Godunov’s orthogonalisation. Instead of the finite strip method, the exact transition matrix method is used in this case. The most important advantage of this method is that it enables us to describe a complete range of behaviour of thin-walled structures from all global (flexural, flexural-torsional, lateral, distortional and their combinations) to local stability. In the presented method for lower bound estimation of the load carrying capacity of structures, it is postulated that the reduced local critical load should be determined taking into account the global pre-critical bending within the first order non-linear approximation to the theory of the interactive buckling of the structure. The paper’s aim is to expand the study of the equilibrium path in the post-buckling behaviour of imperfect structures with regard to the second order non-linear approximation. In the solution obtained, the transformation of buckling modes with an increase of the load up to the ultimate load, the effect of cross-sectional distortions and the shear lag phenomenon are included. The calculations are carried out for a few beam–columns. The results are compared to those obtained from the design code and to the data reported by other authors.The results discussed in the present study represent the most important results obtained by the authors in earlier investigations devoted to central intermediate stiffeners (Int. J. Solid Struct. 32 (1995) 1501; Eng. Trans. 43 (1995) 383; Int. J. Solid Struct. 37 (2000) 3323; Int. J. Solid Struct. 33 (1996) 315; Thin Wall. Struct. 39 (2001) 649; Arch. Mech. Eng. XLVIII (2001) 29).     

利用广义梁理论分析钢筒及钢管的屈曲性能

采用广义梁理论(GBT)公式分析圆柱及管状等圆形截面(CHS)构件的弹性屈曲性能,其中主要应用横截面几何学的概念。考虑到从薄壳理论的运动学关系可以推断出应变能变化,也就是说它们与圆形截面(CHS)的几何特性联系紧密。除了壳变形,公式同时适用于轴对称及扭转变形模式。文中对CHS构件在(i)压力(柱),(ii)弯曲(梁),(iii)压弯(梁-柱),(iv)扭转情况下的局部和整体屈曲性能进行了分析,以论证GBT方法的可行性。此外,还将GBT计算的结果与壳体有限元分析的结果进行了对比分析。     

Some comments on the numerical analysis of plates and thin-walled structures

The paper briefly reviews the theoretical analysis of plates structures that might exhibit multiple ‘loading paths’ and highlights the need for engineers using non-linear numerical modelling to be aware of the multi-mode phenomenon and to ensure that the modelling is set up in such a manner that the various ‘loading paths’ and possible changes of path would be incorporated in the modelling response. The paper presents a simple example of numerical analysis of thin-plate buckling that involves ‘coupled buckling modes’ and provides comments on suitable methods for defining in a simple and straightforward way the numerical modelling that could ensure that results from computer analysis describe the physically correct relationship between applied loadings and deformations of thin-walled structural components.     

Design of thin-walled purlins for distortional buckling

Roof purlins and sheeting rails are generally loaded through the cladding members that they support and this provides both rotational and translational restraint. This restraint reduces the tendency to lateral torsional buckling and thus increases the importance of distortional buckling in the design procedure. This paper shows that the fundamental behaviour of restrained purlins under both downward and uplift load can be best understood with the aid of ‘Generalized Beam Theory’ (GBT). GBT also provides a yardstick by which approximate design methods can be assessed. The existing approximate methods are evaluated and an improvement is proposed. The proposed design approach is then validated by comparison with test results.     

Buckling of wide struts/plates resting on isotropic foundations

Starting from an Airy stress function in plane strain a new solution is presented for the buckling of plates on elastic foundations with in-plane loading. Saint Venant's Principle is shown to play an important part in plate buckling behaviour when ‘deep’ foundations are present. Results are compared with those from Winkler's and Pasternak's models and they demostrate that, provided certain restrictions are satisfied, the latter can give reasonably accurate results.     

On the incorporation of equivalent member imperfections in the in-plane design of steel frames

This work deals with the incorporation of equivalent member imperfections in the global analysis of steel frames and, in particular, is intended to clarify the Eurocode 3 (EC3) provisions involved in such procedure. In fact, these provisions stem from the well-known “European column buckling curves”, which means that they are based on the behaviour of simply supported isolated members under uniform compression (columns). First, one addresses the geometrically non-linear behaviour of isolated columns displaying arbitrary support conditions and different initial geometrical configurations. Then, the results obtained are used to propose a systematic and rational method to evaluate the appropriate “equivalent initial imperfections” that need to be incorporated in the second-order global elastic analysis of a frame or isolated compressed member. This method (i) is fully consistent with the EC3 column buckling curves and (ii) adopts critical buckling mode shape initial imperfections with amplitudes determined by means of closed-form analytical expressions. In order to enable a better grasp of the concepts involved and also to illustrate the capabilities of the proposed methodology, several numerical examples are presented and discussed throughout the paper.     

Recent research advances on ECBL approach.: Part I: Plastic–elastic interactive buckling of cold-formed steel sections

The objective of this two parts paper is to present some recent developments and applications of erosion of critical bifurcation load (ECBL) approach for the interactive buckling. Two different types of problems are analysed: (1) plastic–elastic interactive buckling which implements into the Ayrton–Perry interaction formula the plastic strength of the stub columns evaluated by means of local plastic mechanism analysis, and (2) elastic–elastic interactive buckling for members with perforations.The first part of the paper analyses the occurrence of local plastic mechanisms in cold-formed steel sections in compression, and how they can be implemented in the ultimate limit state analysis of the members. Actually, the failure of thin-walled cold-formed members in compression always occurs with a local plastic mechanism. Starting from this observation, the authors suggest to use in the interactive local-overall buckling analysis the sectional plastic mechanism strength instead of traditional ‘effective section’. The ECBL approach is used to implement the proposed interactive buckling model. Results are compared with those of other two recent methods, namely the direct strength method and plastic effective width approach. Relevant tests are used to evaluate the three methods. Comparisons with European and American design codes are also presented in the paper.     

Effects of anchoring tensile stresses in axially loaded plates and sections

Thin-walled compression members are commonly designed on the assumption that the loaded edges remain straight. Under this assumption, tensile stresses develop in the most flexible parts of the component plates at advanced local buckling deformation, and thus are assumed to be ‘anchored’ at the ends. However, current design rules for plate elements, such as the Winter formulae, are partly based on tests in which the load was applied by use of rigid platens that did not permit tensile stresses to develop. There exists an apparent inconsistency between the assumption of straight loaded edges and the use of a design curve calibrated from tests in which the loaded edges of component plates may not have remained straight.This paper addresses this apparent inconsistency by comparing finite element solutions for the conditions of straight loaded edges and loading by use of a contact surface between the plate edge and a non-deformable rigid body end platen, where there is no constraint for the plate edge to remain in contact with the rigid body. Solutions are provided for a single half-wavelength of unstiffened and stiffened plate elements simply supported along three and four edges, respectively. The effect of multiple half-wavelengths is also investigated, as is the effect of interaction between elements in practical sections comprising stiffened and unstiffened elements.     

A unified energy formulation for the stability analysis of open and closed thin-walled members in the framework of the generalized beam theory

Generalized beam theory—GBT—is among the most adequate tools for the analysis of thin-walled prismatic elements. It enables the analysis of the distortion of the element cross-section and local buckling of individual walls in a unified manner that incorporates the results from classical bending theory. The basis of this theory was developed in the 1960s by Schardt for first and second order elastic behaviour of thin-walled members.Open and closed thin-walled members present the distinctive difference of the unknown shear flow that characterizes the latter. More specifically, shear strains must follow an elasticity law, as opposed to the simplifying assumptions for open cross-sections.It is the purpose of the present paper to present a unified energy formulation for the non-linear analysis of both open and closed sections in the framework of GBT, able to deal with all modal interaction phenomena between local plate behaviour, distortional behaviour and the more classical global (flexural, torsional and flexural–torsional) response. Finally, an application to the stability analysis of a compressed thin-walled column is presented and discussed.     

GBT-based first-order analysis of elastic-plastic thin-walled steel members exhibiting strain-hardening

This paper addresses the development and illustrates the application of a generalised beam theory (GBT) formulation intended to perform first-order elastic-plastic analyses of thin-walled members made of isotropic non-linear materials exhibiting strain-hardening. After presenting an overview of the main concepts and procedures involved in the above GBT formulation, its application is illustrated through the analysis of (i) simply supported Z-section beams and (ii) fixed-ended lipped channel beams. In both cases, a bilinear elastic-plastic material model is adopted, which exhibits three strain-hardening levels, namely E_sh = 0 (perfectly plastic model), E_sh = E/100 and E_sh = E/50. The results presented and discussed consist of equilibrium paths, modal participation diagrams, displacement profiles, beam deformed configurations and stress diagrams and contours. For validation purposes, most of the GBT results are compared with values obtained from shell finite element analyses ? with a few relatively minor exceptions, a very good correlation is always found. Finally, the paper closes with some remarks concerning the influence of the strain-hardening slope on the structural behaviour of thin-walled steel beams.     

基于广义梁理论的薄壁构件屈曲模态有限元分析

提出基于广义梁理论(GBT)的新方法,将各向同性薄壁构件通过壳体有限元分析方法(FEA)获得的弹性屈曲模态分解成整体、畸变和局部屈曲模态。其创新之处在于仅使用GBT截面变形模态,而非构件变形模态。该方法能够单独计算各屈曲模态,更好地了解各构件的后屈曲特性和强度曲线。根据GBT的经典假设,忽略剪切应变和横向张力。通过有限元方法得到的各模态与经典GBT计算结果一致。     

A discrete method of limit analysis and its application to plastic stability problems of structural members

A method of discrete limit analysis which was proposed by Kawai in 1976 is extended to study structural behaviour of a set of solid blocks at the ultimate state of loading where the effect of finite rotation of these blocks should be taken into account. It is shown that Shanley's theory of plastic buckling can be simply demonstrated by using the present discrete models.