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。     

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.     

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.     

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。     

Buckling mode identification of perforated thin-walled members by using GBT and shell FEA

The paper presents an original method based on the Generalised Beam Theory (GBT) whereby the general buckling modes, provided by the shell Finite Element Analysis (SFEA) of perforated thin-walled members, are expressed in terms of the fundamental (pure) buckling types (global, distortional and local). The contribution of each pure buckling mode to a coupled instability can be quantified, allowing a better understanding of the member buckling behaviour and post-buckling strength reserve. The main advantage of this method lies in using only the GBT cross-sectional pure deformation modes instead of member pure modal shapes. There are no restrictions regarding the element cross-sectional shape, loading and boundary conditions.     

基于广义梁理论的非线性材料薄壁受压构件屈曲荷载计算方法

提出适用于非线性材料的广义梁理论屈曲荷载计算方法,并对不锈钢薄壁受压构件屈曲荷载进行计算验证。通过定义材料非线性应力应变关系和瞬时弹性模量,对传统线弹性广义梁理论进行修正,建立非线性材料薄壁构件受压屈曲荷载计算方法,推导不锈钢薄板受压局部屈曲、冷弯薄壁不锈钢卷边槽形柱畸变屈曲及箱形不锈钢长柱弯曲屈曲荷载计算公式,并与既有试验数据对比。经验证,线弹性分析方法不适用于不锈钢材料;提出的修正GBT法具有较高精度,且本构关系采用变形法则结果偏于安全,可用于不锈钢等非线性金属材料薄壁构件受压屈曲荷载的确定,为研究和设计提供理论指导。     

Modal decomposition of thin-walled member collapse mechanisms

Following recent investigations on the decomposition of elastic buckling modes into combinations of structurally meaningful deformation modes, this work presents a novel extension of the above procedure to elastic–plastic collapse mechanisms and highlights the relevant role that this concept may play in the mechanical knowledge/interpretation of thin-walled member failures. In order to achieve the sought decomposition, a code based on a Generalised Beam Theory (GBT) formulation developed to perform first-order elastic–plastic analyses of thin-walled members is employed. Five illustrative examples are presented and discussed, and the results displayed, namely load-deflection curves, deformed configurations and stress contours, are validated through the comparison with values provided by shell finite element analyses. The most relevant modal results addressed consist of (i) load-deflection curves determined on the basis of pre-selected deformation mode sets, (ii) modal participation diagrams and (iii) modal amplitude functions. These results make it easy to characterise and interpret the mechanics associated with the thin-walled member elastic–plastic failures (as well as with the various loading stages), which may be of great importance in the improvement/development of existing/new design methods (e.g, yield-line theory, direct strength method).     

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。     

GBT-based buckling mode decomposition from finite element analysis of thin-walled members

This paper presents an original method based on the Generalised Beam Theory (GBT) capable to decompose the elastic buckling modes from a shell finite element analysis (FEA) of an isotropic thin-walled member, into pure buckling modes of global, distorsional or local nature. The main novelty lies in using only the GBT cross-sectional deformation modes instead of member base mode shapes. The contribution of each pure buckling mode can be calculated, allowing a better understanding of the member post-buckling behaviour and strength reserve. Following the GBT classical assumptions, the membrane shear strains and transverse extensions are neglected. The modal participations obtained from FEA are in good agreement with the values calculated via classical GBT approach.     

Influence of shear deformation on restrained torsional warping of pultruded FRP bars of open cross-section

Pultruded FRP bars of open cross-section possess relatively low transverse shear moduli in relation to their axial and flexural moduli. This can result in shear deformation constituting a significant proportion of the total deformation induced by non-uniform bending, and a reduction in the buckling loads of members subjected to axial compression and bending. Herein an approximate theory for quantifying the influence of shear deformation on the restrained torsional warping of pultruded FRP bars of open cross-section is presented. Contrary to expectations the theory indicates that the influence of shear deformation on the restrained warping torsional stiffness of such members is not significant. The theory is validated by a series of bending and torsion tests on three pultruded FRP I-beams.     

GBT-based elastic–plastic post-buckling analysis of stainless steel thin-walled members

When compared with carbon steel, stainless steel exhibits a more pronounced non-linearity and no well-defined yield plateau, as well as appealing features such as aesthetics, higher corrosion resistance and lower life cycle cost. Due to its considerably high ductility/strength and cost, stainless steel structural solutions tend to be adopted mostly for slender/light structures, thus rendering the assessment of their structural behaviour rather complex, chiefly because of the high susceptibility to instability phenomena. The first objective of this paper is to present the main concepts and procedures involved in the development of a geometrically and materially non-linear Generalised Beam Theory (GBT) formulation and numerical implementation (code), intended to analyse the behaviour and collapse of thin-walled members made of materials with a highly non-linear stress–strain curve (e.g., stainless steel or aluminium). The second objective is to validate and illustrate the application of the proposed GBT formulation, by comparing its results (equilibrium paths, ultimate loads, deformed configurations, displacement profiles and stress distributions) with those provided by shell finite element analyses of two lean duplex square hollow section (SHS) columns previously investigated, both experimentally and numerically, by Theofanous and Gardner (Eng Struct 2009; 31(12): 3047–3058.). The stainless steel material behaviour is modelled as non-linear isotropic and the GBT analysis includes initial geometrical imperfections, but neglects corner strength enhancements and membrane residual stresses. It is shown that the GBT unique modal nature makes it possible to acquire in-depth knowledge concerning the mechanics of the column behaviour, by providing “structural x-rays” of the (elastic or elastic–plastic) equilibrium configurations: modal participation diagrams showing the quantitative contributions of the global, local, warping shear and transverse extension deformation modes - moreover, this feature makes it possible to exclude, from future similar GBT analyses, those deformation modes found to play a negligible role in the mechanics of the behaviour under scrutiny, thus further reducing the number of degrees of freedom involved in a GBT analysis, i.e., increasing its computational efficiency.     

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.     

半离散法分析薄壁柱的畸变屈曲

采用广义梁理论(GBT)的耦合差分方程解决了半离散法分析薄壁柱的畸变屈曲问题。作者近期发表的两篇文章对类似GBT的新型半离散分析方法进行了阐述。对横截面进行离散分析,寻找沿梁变化的解析解。采用新方法,利用齐次和非齐次方程建立确定梁全部变形的一般GBT方程并求解,从而使GBT的(复杂)变形方程变形为可降阶的微分方程。提出的半离散方法在广义梁理论(GBT)基础上增加了用于柱的失稳分析和失稳形态识别的几何刚度因素。通过势能的变化并在梁理论中引入约束条件,对初始应力下建立的GBT齐次微分方程进行修正,以分析其变形特性。对简支梁梁端施加横向位移和轴力,建立GBT初始应力方程,通过该方程寻求失稳的解决方法。根据已知的边界条件,利用三角函数关系式和求解特征值的方法求解这些可降阶的微分方程,使得屈曲形态和相关特征值与分叉荷载因素相符。因此,无需通过模态分解,可由耦合的GBT方程直接求得屈曲形态的解析解。通过实例分析了柱的整体失稳、屈曲变形和局部纵弯失稳,以及如何将新方法用于描述特征曲线和弹性屈曲曲线。将该方法的分析结果与ABAQUS、GBTUL和CUFSM软件的分析结果进行对比,验证了该方法的正确性。     

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.     

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.     

Distortional buckling modes of semi-discretized thin-walled columns

This paper presents distorting buckling solutions for semi-discretized thin-walled columns using the coupled differential equations of a generalized beam theory (GBT). In two related papers recently published by the authors a novel semi-discretization approach to GBT has been presented. The cross section is discretized and analytical solutions are sought for the variation along the beam. With this new approach the general GBT equations for identification of a full set of deformation modes corresponding to both homogeneous and non-homogenous equations are formulated and solved. Thereby giving the (complex) deformation modes of GBT which decouple the state space equations corresponding to the reduced order differential equations.In this paper the developed semi-discretization approach to generalized beam theory (GBT) is extended to include the geometrical stiffness terms, which are needed for column buckling analysis and identification of buckling modes. The extension is based on an initial stress approach by addition of the related potential energy terms. The potential energy of a single deformation mode is formulated based on a discretization of the cross section. Through variations in the potential energy and the introduction of the constraints related to beam theory this leads to a modified set of coupled homogeneous differential equations of GBT with initial stress for identification of distortional displacement modes. In this paper we seek instability solutions using these GBT initial stress equations for simply supported columns with constrained transverse displacements at the end sections and a constant axial initial stress. Based on the known boundary conditions the reduced order differential equations are solved by using the trigonometric solution functions and solving the related eigenvalue problem. This gives the buckling mode shapes and the associated eigenvalues corresponding to the bifurcation load factors. Thus the buckling modes are found directly by the analytical solution of the coupled GBT-equations without modal decomposition. Illustrative examples showing global column buckling, distortional buckling and local buckling are given and it is shown how the novel approach may be used to develop signature curves and elastic buckling curves. In order to assess the accuracy of the method some of the results are compared to results found using the commercial FE program Abaqus as well as the conventional GBT and FSM methods using the software packages GBTUL and CUFSM.     

On the local and global buckling behaviour of angle,T-section and cruciform thin-walled members

This paper reports the results of an investigation aimed at providing fresh insight on the mechanics underlying the local and global buckling behaviour of angle, T-section and cruciform thin-walled steel members (columns, beams and beam-columns). Due to the lack of primary warping resistance, members displaying these cross-section shapes possess a minute torsional stiffness and, therefore, are highly susceptible to buckling phenomena involving torsion – moreover, it is often hard to distinguish between torsion and local deformations. Almost all the numerical results presented are obtained by means of Generalised Beam Theory (GBT) analyses and, taking advantage of its unique modal features, it is possible to shed some new light on how to characterise and/or distinguish the local and global buckling modes of the above thin-walled members. Finally, some comments are made concerning the development of a rational and efficient (safe and economic) approach for their design.     

通过有限元法计算受压构件的纯畸变弹性屈曲荷载

基于有限元法,提出一个分析方法,可以计算纯畸变弹性屈曲荷载。计算中采用的有限元模型为非耦合屈曲变形模式。具体方法分为2步:第1步,采用一般梁理论(GBT)分析构件横截面,在此可以给有限元模型施加约束条件;第2步,对受约束有限元模型进行线性屈曲分析,确定纯畸变荷载。将此法应用于开口薄壁构件和冷弯构件,得到的畸变荷载非常准确,与一般梁理论和约束有限条法(cFSM)计算的荷载值一致。     

Buckling modes identification from FEA of thin-walled members using only GBT cross-sectional deformation modes

This paper presents the latest developments of an original method based on Generalized Beam Theory (GBT) capable to identify the fundamental deformation modes of global, distorsional or local nature, in general buckling modes provided by the shell finite element analysis (FEA) of isotropic thin-walled members. This method has the advantage of using only the GBT cross-sectional deformation modes instead of the member base mode shapes. The participation of each fundamental buckling mode can be calculated, allowing an accurate and quantitative evaluation of the coupled instability. There are no restrictions regarding the element cross-sectional shape, loading and quite recently discovered, boundary conditions.     

Calculation of pure distortional elastic buckling loads of members subjected to compression via the finite element method

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).