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.     

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

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

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.     

Coupled instabilities with distortional buckling in cold-formed steel lipped channel columns

The paper addresses the elastic post-buckling behaviours of cold-formed steel lipped channel simply supported columns affected by mode interaction phenomena involving distortional buckling, namely local/distortional, distortional/global (flexural-torsional) and local/distortional/global mode interaction. The results presented were obtained by means of Abaqus shell finite element analyses adopting column discretisations into fine 4-node element meshes. In order to enable a thorough assessment of all possible mode interaction effects, the column lengths and cross-section dimensions were carefully selected to ensure similar local, distortional and/or global buckling loads. One analyses otherwise identical (elastic) columns having initial geometrical imperfections (i) with various configurations (combinations of the competing critical buckling mode shapes) and (ii) sharing the same overall amplitude.     

不同斜卷边檩条的局部屈曲和畸变屈曲

利用Hancock的简化模型分析了卷边槽形、卷边Z形檩条的畸变屈曲,用有限元方法计算了卷边槽形、卷边Z形檩条的畸变屈曲荷载以及翼缘局部屈曲系数,研究了卷边角度对畸变屈曲荷载的影响,提出斜卷边加劲板的屈曲系数取值以及屈曲系数关于卷边角度变化的公式。经对比验证,在卷边角度40°～90°的范围内,误差在2%以内。可供实际设计参考。     

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.     

碳纤维增强卷边槽形钢柱的非线性性能和承载力

对碳纤维增强卷边槽形钢柱的试验结果、非线性性能的数值分析以及承载力进行研究。基于GBT的基础研究,涉及柱弹性抗弯性能的研究集中在下列方面:i)布的位置(腹板、翼缘、卷边);ii)碳纤维的方向(纵向、横向或倾斜)以便对加固效果(性能价格比)进行优化。对19个端固定的短、长卷边槽形钢柱的试验过程进行描述。采用碳纤维布对钢柱的不同部位(腹板、翼缘、卷边)进行加固,纤维的方向包括纵向和横向:因为研究的目标是评估碳纤维加固对柱的结构响应的影响,对未加固钢试件也进行了试验。基于壳有限单元模型并采用Abaqus规定,由包含非线性平衡路径(荷载与轴向压缩的比值)和极限强度(大部分与局部/屈曲失稳破坏机理相关)组成的试验结果被用于校准和验证几何及物理非线性数值分析。最后,根据试验和数值结果,对于受到局部/屈曲失稳影响的卷边槽形钢柱,得出碳纤维布加固的有效位置和纤维方向的相关结论。     

Local/distortional mode interaction in cold-formed steel lipped channel beams

This paper reports a numerical investigation concerning the post-buckling behaviour of cold-formed steel lipped channel beams, subjected to uniform major axis bending, affected by local/distortional mode interaction – the results presented and discussed were obtained through shell finite element analyses performed using the code Abaqus. One analyses simply supported beams (locally/globally pinned and free to warp end sections) (i) with cross-section dimensions ensuring coincident local and distortional critical buckling moments and (ii) having critical-mode initial geometrical imperfections with different shapes and the same overall amplitude – local buckling is triggered either by the compressed flange (most common case) or by the web. The results reported consist of (i) elastic and elastic–plastic post-buckling equilibrium paths, (ii) curves and figures showing how the beam deformed configuration evolves along those paths and, for the elastic–plastic beams, (iii) figures making it possible to visualise (iii₁) the location and growth of plastic strains and (iii₂) the nature of the failure mechanisms detected.     

Post-buckling behaviour and direct strength design of lipped channel columns experiencing local/distortional interaction

This paper reports an investigation aimed at developing a Direct Strength Method (DSM) approach to estimate the ultimate strength of lipped channel columns affected by local/distortional buckling mode interaction. Following a brief presentation of a few relevant aspects concerning the shell finite element analysis of the geometrically and materially non-linear behaviour of thin-walled members, one illustrates the methodology adopted to obtain a lipped channel column ultimate load “data bank” intended to be used in the development and assessment of a DSM design approach. Next, the current DSM expressions to predict the load-carrying capacity of columns failing in local and distortional modes are briefly reviewed, devoting special attention to an approach that takes into account the above mode interaction. Then, the results of a parametric study, carried out by means of the code Abaqus, are presented and discussed — this study involves the evaluation of the “exact” ultimate loads of 276 lipped channel columns with various geometries and two boundary conditions (pinned and fixed end supports), all exhibiting local/distortional interaction. Finally, these ultimate strength data are compared with the estimates provided by the available DSM expressions and, on the basis of this comparison, one identifies several features that a DSM approach successfully accounting for local/distortional interaction must incorporate.     

Strength of cold-formed lipped channel beams under interaction of local,distortional and lateral torsional buckling

Cold-formed thin-walled lipped channel steel beams may undergo buckling modes such as short half-wavelength local buckling, intermediate half-wavelength distortional buckling and long half-wavelength lateral-torsional buckling or a combination of these before failure. ABAQUS software based on finite element analysis is used to analyse the interaction behaviour of these buckling modes in this study. The finite element model, after calibration with experimental results available in the literature, is used to perform parametric studies, to evaluate the behaviour and strength of such beams under different types of interactions due to variation of material and member properties. The large volume of synthetic data thus generated over a range of failure modes along with the available test results are used to evaluate different equations for calculating the strength of such cold-formed lipped channel beams. Based on the comparison, a method for the design of lipped channel beams failing under the interaction of local, distortional and overall lateral torsional buckling is recommended.     

On the Direct Strength Method (DSM) design of cold-formed steel columns against distortional failure

This work reports the results of a numerical investigation on the influence of the cross-section geometry and end support conditions on the post-buckling behaviour and Direct Strength Method (DSM) design of cold-formed steel columns buckling and failing in distortional modes. The columns analysed exhibit (i) four end support conditions, (ii) different cross-section shapes, dimensions and lengths, and (iii) several yield stresses. These features were carefully selected in order to ensure, as much as possible, that all columns (i) buckle and fail in “pure” distortional modes and (ii) cover a wide (distortional) slenderness range. The post-buckling equilibrium paths and ultimate loads presented and discussed were obtained through Ansys elastic and elastic-plastic shell finite element analyses. Moreover, the ultimate strength data acquired are used to show that, regardless of the column geometry, the current DSM distortional design curve is not able to predict adequately (safely and accurately) the ultimate loads of columns with other than fixed end supports. The paper also includes a preliminary assessment/proposal of the modifications that must be included into this DSM design curve in order to overcome the above limitation.     

Lateral buckling strengths of cold-formed Z-section beams

This paper is concerned with the inelastic lateral buckling strengths of cold-formed Z-section (CFZ) beams. The point symmetry of the cross-section of a CFZ beam introduces characteristics that are not encountered in a doubly symmetric I-beam. Firstly, the effective section rotates after yielding, so that a CFZ beam under in-plane bending about the geometrical major principal axis is subjected to bending moments about the effective minor axis and bimoments. Secondly, the minor axis bending and warping strain distributions and therefore the lateral inelastic buckling behaviour and strengths of CFZ beams are related to the twist rotation and minor axis displacement directions. The stress–strain curves, residual stresses, initial imperfections, and lipped flanges of CFZ beams are all different to those of hot-rolled I-beams. This paper develops a realistic finite element model for the analysis of CFZ beams and uses it to investigate the elastic lateral-distortional buckling, inelastic behaviour, and strengths of CFZ beams with residual stresses and initial imperfections. The results of the study are used to develop improved design rules which are suitable for CFZ beams. The effects of moment distribution and load height on the lateral buckling strengths are also studied.     

局部/畸变屈曲相互作用下卷边槽钢柱的后屈曲性能及直接强度设计

对直接强度法(DSM)进行改进,以计算局部/畸变屈曲相互作用下卷边槽钢柱的极限强度。首先,采用壳体有限元分析方法分析薄壁构件的几何特性和材料的非线性特性,并简单描述该分析过程的主要内容。利用该方法建立卷边槽钢柱的极限荷载"数据库",以用于改进和评估DSM设计方法。其次,简要回顾了当前计算局部和畸变作用下柱破坏时承重能力的DSM表达式,特别是考虑局部和畸变交互作用的方法。此外,研究了两种边界条件下(铰接和固接)局部/畸变屈曲相互作用下几何尺寸不同的276根卷边槽钢柱的极限荷载,给出ABAQUS软件的参数研究结果,并对其进行了讨论。最后,将这些极限强度数据与DSM计算结果进行比较,结果表明,采用DSM计算局部/畸变屈曲交互作用时需考虑某些特性。     

Performance of cellular composite floor beams at elevated temperatures

This paper describes an experimental and numerical study at both ambient and elevated temperatures on the behaviour of full-scale composite floor cellular steel beams. A total of four specimens, comprising two different steel geometries and loading conditions were tested under monotonic loading and at elevated temperatures. All beams were designed for a full shear connection between the steel beam and the concrete flange using headed shear studs. The beams were designed to fail by web-post buckling, which was observed in all the tests. Failure temperature observed in the fire tests indicated that failure by web-post buckling of cellular beams in fire cannot simply be estimated by applying temperature dependent reduction factors on stiffness, as given in codes. A finite element model is then established with both material and geometrical non-linearity using shell elements to compare against the experimental results. The comparison between the finite element prediction and actual test results are quite good in terms of failure modes, load deflection behaviour and ultimate loads.     

Lateral–distortional buckling of monorails

This paper is concerned with the elastic lateral–distortional (LD) buckling of single span steel monorail I-beams and its influence on their design strengths. The distortion of a slender web reduces the elastic buckling resistance of an intermediate length beam below its flexural–torsional (FT) resistance. A finite element computer program was used to study the elastic LD buckling of single span beams. The LD to FT buckling moment ratios were generally higher for simply supported beams with bottom flange central concentrated loads than for uniform bending, and lower for shear centre concentrated loads. For beams with bottom flange loading and unrestrained bottom flanges, there were very significant reductions in these ratios, but they increased when rigid web stiffeners or top flange torsional restraints were provided at the supports. For beams with bottom flange loading and unrestrained bottom flanges, the reductions in the elastic buckling resistance were greater for beams with stocky flanges than for slender flanges. Approximations were found for estimating the reduced resistances which were generally of high accuracy or conservative, and for estimating the increased resistances caused by elastic and rigid top flange end torsional restraints. A method of designing steel beams against LD buckling was proposed and its use demonstrated by a worked example.     

Distortional buckling of thin-web beam-columns

An approximate finite element method of analysis is developed for the distortional buckling of beam-columns. The member consists of a flexible web, to which is connected two rigid flange assemblies of arbitrary shape which remain undistorted during buckling. The method permits an economic computer analysis to be made of elastic distortional buckling of members under various conditions of loading, end support and restraint.Following studies of the convergence and accuracy of the buckling solutions, the method is used to demonstrate the effects of web distortion on the elastic buckling of lipped channel beams under moment gradient.     

Behaviour,failure and DSM design of cold-formed steel beams: Influence of the load point of application

This work presents and discusses the results of an ongoing numerical investigation on the buckling, post-buckling, collapse and design of cold-formed steel beams subjected to non-uniform bending due to transverse loadings acting away from the shear centre (either at the top or bottom flange). These results consist of (i) elastic buckling loads and modes, obtained through analyses based on Generalised Beam Theory (GBT), and (ii) elastic–plastic equilibrium paths and collapse loads, yielded by Ansys geometrically and materially non-linear shell finite element analyses. The numerical ultimate strength values are compared with their estimates provided by the current Direct Strength Method (DSM) strength curves and, on the basis of this comparison, it is possible to assess the merits of the DSM approach to design beams subjected to transverse loadings acting away from the shear centre – moreover, novel features that may improve the performance of this approach are identified.     

考虑剪切变形影响的框架结构稳定性

巨型框架结构由于框架梁柱截面的剪切变形较大,在分析巨型结构整体稳定性时,应当同时考虑巨型梁柱的弯曲变形与剪切变形。本文研究了框架梁和框架柱截面的剪切变形对框架结构整体稳定性的影响。分别研究了简单对称框架柱顶作用有相同集中荷载时、简单对称框架柱顶作用有不等集中荷载时、以及不对称框架柱顶作用不等集中荷载时,框架结构的整体稳定性。分析了不对称框架柱剪切变形对框架稳定性的影响及框架发生整体有侧移失稳的内在规律。着重讨论了框架梁截面的剪切变形对框架整体稳定性的影响规律,根据这一规律提出了简化方法以考虑横梁剪切变形的影响。本文提出了计算框架临界荷载的近似计算方法,与有限元法的结果对比,具有很好的精度,近似算法均考虑了框架梁和框架柱截面的剪切变形及弯曲变形。     

卷边翼缘工形截面梁的稳定性能及设计方法

基于大挠度弹塑性板壳有限元分析方法,系统地研究了卷边工形梁在两端等弯矩作用下的失稳机理与稳定承载力,揭示了卷边工形截面构件的局部屈曲、畸变屈曲、整体侧向失稳以及它们之间的相互作用。系统地分析了截面宽高比、腹板高厚比、翼缘宽厚比、长细比等构件参数对侧向稳定承载力的影响,并基于大量的数值计算数据给出了卷边工形梁在两端等弯矩作用下的稳定承载力设计公式,可为规范修订以及工程设计提供依据。     

Non-linear model for stability of thin-walled composite beams with shear deformation

A geometrically non-linear theory for thin-walled composite beams is developed for both open and closed cross-sections and taking into account shear flexibility (bending and warping shear). This non-linear formulation is used for analyzing the static stability of beams made of composite materials subjected to concentrated end moments, concentrated forces, or uniformly distributed loads. Composite is assumed to be made of symmetric balanced laminates or especially orthotropic laminates. In order to solve the non-linear differential system, Ritz's method is first applied. Then, the resulting algebraic equilibrium equations are solved by means of an incremental Newton–Rapshon method. This paper investigates numerically the flexural–torsional and lateral buckling and post-buckling behavior of simply supported beams, pointing out the influence of shear–deformation for different laminate stacking sequence and the pre-buckling deflections effect on buckling loads. The numerical results show that the classical predictions of lateral buckling are conservative when the pre-buckling displacements are not negligible, and a non-linear buckling analysis may be required for reliable solutions.