Lateral buckling of thin-walled beam-column elements under combined axial and bending loads

Based on a non-linear stability model, analytical solutions are derived for simply supported beam-column elements with bi-symmetric I sections under combined bending and axial forces. An unique compact closed-form is used for some representative load cases needed in design. It includes first-order bending distribution, load height level, pre-buckling deflection effects and presence of axial loads. The proposed solutions are validated by recourse to non-linear FEM software where shell elements are used in mesh process. The agreement of the proposed solutions with bifurcations observed on non-linear equilibrium paths is good. It is proved that classical linear stability solutions underestimate the real resistance of such element in lateral buckling stability especially for I section with large flanges. Numerical study of incidence of axial forces on lateral buckling resistance of redundant beams is carried out. When axial displacements of a beam are prevented important tension axial forces are generated in the beam. This results in important reduction of displacements and for some sections, the beam behaviour becomes non-linear without any bifurcation.     

Bending and buckling of inflatable beams: Some new theoretical results

The non-linear and linearized equations are derived for the in-plane stretching and bending of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure. The Timoshenko beam model combined with the finite rotation kinematics enables one to correctly account for the shear effect and all the non-linear terms in the governing equations. The linearization is carried out around a pre-stressed reference configuration which has to be defined as opposed to the so-called natural state. Two examples are then investigated: the bending and the buckling of a cantilever beam. Their analytical solutions show that the inflation has the effect of increasing the material properties in the beam solution. This solution is compared with the three-dimensional finite element analysis, as well as the so-called wrinkling pressure for the bent beam and the crushing force for the buckled beam. New theoretical and numerical results on the buckling of inflatable beams are displayed.     

A beam model for large displacement analysis of flexibly connected thin-walled beam-type structures

This paper presents a beam formulation for large displacement analysis of beam-type structures with flexible connections. Within the framework of updated Lagrangian incremental formulation and the nonlinear displacement field of thin-walled cross-sections, which accounts for restrained warping and the second-order displacement terms due to large rotations, the equilibrium equations of a straight beam element are firstly developed. Due to the nonlinear displacement field, the geometric potential of semitangential moment is obtained for both the internal torsion and bending moments, respectively. Material nonlinearity is introduced for an elastic-perfectly plastic material through the plastic hinge formation at finite element ends. To account for the flexible connection behaviour, a special transformation procedure is developed. The numerical algorithm is implemented in a computer programme and its reliability is validated trough several test examples.     

交叉加劲钢板深梁弹性屈曲分析

以交叉加劲钢板深梁为研究对象,利用有限元软件ANSYS分析其弹性屈曲性能,讨论了抗弯刚度比、跨高比、钢板深梁厚度对其弹性屈曲性能的影响;考虑钢板深梁在钢框架的弯剪受力特性,根据板的经典理论建立了交叉加劲钢板深梁屈曲荷载计算公式,提出了等效屈曲系数。结果表明:交叉加劲钢板深梁的临界屈曲荷载随抗弯刚度比增大而增大,但达到门槛刚度比后,增大幅度急剧减小,得到门槛刚度比约为10;临界屈曲荷载随跨高比和板厚的减小而减小,等效屈曲系数随板厚减小而增大;等效屈曲系数与跨高比关系曲线由二次抛物线形向波浪形渐变,交叉加劲钢板深梁受力特性由剪切主导向弯曲主导过渡。     

Lower bound estimation of load-carrying capacity of thin-walled structures with intermediate stiffeners

The design of thin structures must take into account the overall instability and the instability of component plates in the form of local buckling. This investigation is concerned with the interactive buckling of thin-walled structures with central intermediate stiffeners under axial compression and/or a constant bending moment. The structures are assumed to be simply supported at the ends. The lower bound estimation of load-carrying capacity on the basis of the post-buckling behaviour of thin-walled structures with imperfections is studied when the distortional deformations are taken into account. The asymptotic expansion established by Byskov and Hutchinson (AIAA J. 15 (1977) 941) is employed in the numerical calculations performed using the transition matrix method. The present paper is a continuation of previous work by the authors (Int. J. Solids Struct. 32 (1995) 1501; 33 (1996) 315; 37 (2000) 3323), where the interactive buckling of thin-walled beam-columns with central intermediate stiffeners in the first- and the second-order approximation were considered. In the solution obtained, the transformation of buckling modes with an increase in the load up to the ultimate load, the effect of cross-sectional distortions and the shear lag phenomenon are included. The results obtained are compared with data reported by other authors.     

Finite element formulation for inflatable beams

The discretized nonlinear equations for bending and buckling of inflatable beams are written by use of the virtual work principle with Timoshenko's kinematics, finite rotations and small strains. The linearized equations around a pre-stressed reference configuration are then deduced, giving rise to a new inflatable beam finite element. The stiffness matrix contains the shear coefficient and the internal pressure. Use is made of the particular 3-node beam element to investigate the bending and the buckling of a cantilever beam, the deflection of a pinched torus and the buckling of a torus submitted to a radial compressive force. The numerical results obtained with the beam element are shown to be close to analytical and three-dimensional (3D) membrane finite element results. The validity of the numerical results is discussed, in connection with the concepts of the crushing force or the wrinkling pressure of the inflated beam.     

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.     

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).     

Mechanics of shear deformable thin-walled beams made of composite materials

In this paper, a new theoretical model is developed for the generalized linear analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form the shear deformability by means of two features. The first one may be addressed as a mechanical aspect where the effect of shear deformability due to both bending and non-uniform warping is considered. The second feature is connected with the constitutive aspects, and it contemplates the use of different hypotheses adopted in the formulation. These topics are treated in a straightforward way by means of the Linearized Principle of Virtual Works. The model is developed by employing a non-linear displacement field, whose rotations are formulated by means of the rule of semitangential transformation. This model allows studying many problems of static's, free vibrations with or without arbitrary initial stresses and linear stability of composite thin-walled beams with general cross-sections. A discussion about the constitutive equations is performed, in order to explain distinctive aspects of the effects included in the theory. This paper presents the theoretical formulation together with finite element procedures that are developed with the aim to obtain solutions to the general equations of thin-walled shear deformable composite beams. A non-locking fourteen-degree-of-freedom finite element is introduced. Numerical examples are carried out in several topics of static's, dynamics and buckling problems, focusing attention in the validation of the theory with respect to experimental data and with 2D and 3D computational approaches. Also, new parametrical studies are performed in order to show the influence of shear flexibility in the mechanics of the thin-walled composite beams as well as to illustrate the usefulness of the model.     

Design of composite asymmetric cellular beams and beams with large web openings

The design of composite asymmetric cellular beams is not fully covered by existing guidance but is an area of important practical application. Asymmetry in the shape of the cross-section of cellular beams causes development of additional bending moments in the web-posts between closely placed openings. Furthermore, the development of local composite action influences the distribution of forces in the web-flange Tees. The design method presented in this paper takes account of high degrees of asymmetry in the cross-section and also the influence of elongated or rectangular openings.Web-post moments also influence buckling of the web-post between openings, which is accentuated by adjacent long openings. Simplified equations are presented for web-post buckling based on a compression field or ‘strut’ model, which is calibrated against the results of Finite Element Analyses (FEA). The FEA are also extended to cover the case of highly asymmetric sections and ring-stiffened openings. Closed solutions are presented that enable the designer to calculate the maximum shear force acting on the beam when its load resistance is limited by web-post bending or buckling.For long openings, high pull-out forces may exist in the shear connectors at the edge of the opening. When combined with possible second-order effects due to shear deflection across the opening, it is necessary to limit the magnitude of local composite action due to Vierendeel bending that can be considered in design.     

Variationally-based theories for buckling of partial composite beam-columns including shear and axial effects

This paper is focused on elastic stability problems of partial composite columns: the conditions for the axial load not to introduce any pre-bending effects in composite columns; the equivalence, similarities and differences between different sandwich and partial composite beam theories with and without the effect of shear, with and without the effect of axial extensibility, and also the effect of eccentric axial load application. The basic modelling of the composite beam-column uses the Euler-Bernoulli beam theory and a linear constitutive law for the slip. In the analysis of this reference model, a variational formulation is used in order to derive relevant boundary conditions. The specific loading associated with no pre-bending effects before buckling is geometrically characterized, leading to analytical buckling loads of the partial composite column. The equivalence between the Hoff theory for sandwich beam-columns, the composite action theory for beam-columns with interlayer slip and the corresponding Bickford-Reddy theory, is shown from the stability point of view. Special loading configurations including eccentric axial load applications and axial loading only on one of the sub-elements of the composite beam-column are investigated and the similarity of the behaviour to that of imperfect ordinary beam-columns is demonstrated. The effect of axial extensibility on kinematical relationships (according to the Reissner theory), is analytically quantified and compared to the classical solution of the problem. Finally, the effect of incorporating shear in the analysis of composite members using the Timoshenko theory is evaluated. By using a variational formulation, the buckling behaviour of partial composite columns is analysed with respect to both the Engesser and the Haringx theory. A simplified uniform shear theory (assuming equal shear deformations in each sub-element) for the partial composite beam-column is first presented, and then a refined differential shear theory (assuming individual shear deformations in each sub-element) is evaluated. The paper concludes with a discussion on this shear effect, the differences between the shear theories presented and when the shear effect can be neglected.     

Buckling of shear deformable thin-walled members—I. Variational principle and analyticalsolutions

A variational formulation for the buckling analysis of thin-walled members is developed based on the principle of stationary potential energy. The formulation is based on non-orthogonal coordinates and captures shear deformation effects due to bending and warping. It is applicable to members of doubly symmetric cross-sections subject to general axial and transverse forces and naturally incorporates the effect of load position relative to the shear centre. Applying the conditions of neutral stability to the variational expression, the governing differential equations of neutral stability and associated boundary conditions are formulated. The resulting field equations are exactly solved for benchmark cases involving column flexural buckling, column torsional buckling, and lateral-torsional buckling for beams, and the results are compared to closed form solutions based on classical and other modern theories.     

Experimental characterization and finite element analysis of inflated fabric beams

An airbeam is a high-strength fabric sleeve with a highly flexible internal bladder that can be used as a load-bearing beam or arch when inflated. Due to their fabric construction, airbeams are inherently thin-walled structures that are prone to local buckling. In this study, airbeams were tested in bending at different inflation pressures to quantify their load–deformation response and the effect of inflation pressure on response. Tension–torsion tests of the airbeam fabric were conducted to estimate the fabric shear modulus, and the bend test results were used in conjunction with Timoshenko beam theory to estimate the fabric elastic modulus. Three-dimensional membrane finite element (FE) models were then used to predict the beam load–deformation response given these moduli. The FE models successfully predicted localized fabric buckling and softening of load–deflection response. Comparison of FE model-predicted load–deflection response with beam theory shows that conventional beam theory is accurate prior to local buckling of the airbeam fabric. The FE model and test results indicate that the consideration of work done by pressure under deformation-induced volume changes may increase beam capacity beyond previously derived theoretical limiting values.     

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.     

梯度材料薄壁钢梁的热弹性动力稳定性研究

研究剪力作用下梯度材料薄壁钢梁的动力稳定性,分析基于小变形和二阶非线性位移引起的中等旋转。钢梁承受外部轴向动力荷载。模型考虑了热弹性影响,推导了热量传导公式,用于计算钢梁横截面域的温度。采用Galerkin和Bolotin方法,详细地推导出控制方程和动力稳定的区域,以比例的形式评估并阐述了稳定区域,研究非稳定区的纵向振动影响。结果表明,荷载频率接近纵向自振频率的影响最大,并有更广的稳定区域。在分析中,也考虑了温度梯度、剪力弹性、轴向惯量、不同截面和不同梯度材料的影响。     

An effective four node degenerated shell element for geometrically nonlinear analysis

The development of a new four node degenerated shell element is presented for the analysis of the shell structures undergoing large deformations. In the formulation of the new element, the assumed covariant transverse shear strains are used to avoid the shear locking problem, and the assumed covariant membrane strains which are separated from the covariant inplane strains by mid-surface interpolation, are applied to eliminate the membrane locking problem and also to improve the membrane bending performance. This element is free of serious shear and membrane locking problems and undesired spurious kinematic deformation modes. An incremental total Lagrangian formulation is presented which allows the calculation of arbitrarily large displacements and rotations. The resulting nonlinear equilibrium equations are solved by the Newton-Raphson method combined with load or arc-length control. The versatility and accuracy of this new degenerated shell element is demonstrated by solving several numerical examples.     

Influence of longitudinal stiffeners on elastic stability of girder webs

In this paper, linear buckling analyses of plates with longitudinal stiffeners having various shapes and positions and subjected to axial force, in-plane bending and shear are developed. The aim is to give some new practical insights about the shape and optimum position of longitudinal stiffener in webs when axial force, bending moment and shear act. By means of a comprehensive numerical investigation, some practical issues for buckling phenomena in stiffened plates, taking into account (a) dimensions and shape (square and rectangular) of the plate, (b) dimensions and shape of the stiffener (with open and closed cross-sections), (c) location of the stiffener, and (d) load configuration (uniform compression, pure bending, combinations of axial compression and bending, and shear) are developed.     

Local buckling of thin-walled girders of triangular cross-section

A theoretical study of the local buckling of thin-walled girders under compound loading is presented. Girders of triangular cross-section with one axis of symmetry are considered. The following load cases are examined: compression, eccentric compression, pure bending and bending with shear. A parabolic distribution of shear stress in the webs is assumed when lateral forces are present.

The problem is solved using a semi-energy method, described in Jakubowski, S., Buckling of thin-walled girders under compound load, Thin-walled Structures,6 (1988) 129-50. Numerical results are presented in the form of diagrams. The analysis of buckling modes for the more complicated cases such as bending with shear has also been performed.     

Lateral–torsional buckling of cold-formed channel sections subject to combined compression and bending

This paper presents an analytical study of the flexural buckling and lateral–torsional buckling of cold-formed steel channel section beams subject to combined compression and bending about their major and minor axes. For channel section beams a bending about the minor axis creates a non-symmetric pre-buckling stress distribution, which has a significant influence on the lateral–torsional buckling of the beams. This kind of feature has not been discussed in the existing literature. The focus of this present study is the interaction between the compression load and the bending moments about the major and minor axes. It has been found that for a section subject to combined compression and the major-axis bending the bending moment will decrease the critical compression load, although the critical value of the largest compressive stress in the section actually increases with the applied bending moment. However, for a section subject to combined compression and the minor-axis bending the effect of the bending moment on the critical compression load depends on the direction of bending applied. For bending that creates a compressive stress in the lips the bending moment will reduce the critical compression load. However, for bending that creates a compressive stress in the web the bending moment has almost no influence on the critical compression load.     

钢筋混凝土框架结构设计有关问题的初步探讨

依据GB 50020-2002混凝土结构设计规范和GB 50011-2001建筑结构抗震设计规范,对轴压比限值、框架柱侧移二阶效应、荷载效应与地震作用效应组合、梁端弯矩调幅、风荷载作用下柱的剪力5个问题在结构设计中应注意的几个方面进行了初步探讨,并取得了较好的效果,可供设计人员参考。