On the mechanics of distortion in thin-walled open sections

This paper addresses the distortional kinematics and mechanics of thin-walled sections and provides clear definitions of cross-section properties that characterise the distortional deformation, as it is usually done for conventional modes (axial, bending and torsion). In particular, a procedure to build the distortional displacement field of a given thin-walled section is described. The first part of the paper describes the essentials of distortion in comparison with the conventional modes of classical beam theories. It is shown that primary warping is the key factor that controls the distortion of thin-walled sections. Then, an analytical procedure to determine the distortional warping displacement distribution of a given cross-section is described, on the basis of orthogonality conditions existing between the distortional and conventional modes. Next, an overview of the kinematical assumptions underlying the distortional deformation is presented and a simple procedure to build distortional displacement fields of thin-walled sections is provided. This procedure is then applied to obtain the distortional displacement field of C-sections and general expressions of distortional cross-section properties are given. Finally, a simple example is presented to illustrate how the distortional displacement field of a C-section is built, on the basis of simple kinematics principles. The distortional critical stress and half wavelength are determined and good agreement with exact numerical estimates is found.     

Distortional theory of thin-walled beams

The classic thin-walled beam theory for open and closed cross-sections is generalized to include one distortional mode of deformation. Distortional cross-section parameters are introduced and the new orthogonality conditions for uncoupling of the axial displacement modes are given. A normalization technique for the distortional modes leads to unique distortional cross-section properties. The theoretical formulations for torsion and distortion are nearly similar and result in nearly identical equilibrium equations. However, for closed single- or multi-cell cross-sections the torsional and distortional shear flows may couple. A study of the order of magnitude of the governing torsional and distortional parameters shows the difference between open and closed cross-sections and the related solution types. The difference in the order of magnitude of the governing cross-section parameters also leads to approximate solution techniques. In the examples, section three cross-sections are used to illustrate variations of the theoretical parameters.     

Direct Strength Method for calculating distortional buckling capacity of cold-formed thin-walled steel columns with uniform and non-uniform elevated temperatures

This paper assesses the applicability of the Direct Strength Method (DSM) to calculating the distortional buckling strength of cold-formed thin-walled (CF-TW) steel members with uniform and non-uniform elevated temperature distributions in the cross-section. The assessment was carried out by checking the DSM calculation results with numerical simulation results using the general finite element software ABAQUS which was further validated against ambient and uniform elevated temperature tests on short lipped channel sections, in addition to the author's previous validation studies for thin-walled steel columns with non-uniform temperature distributions. The validated numerical model has been used to generate an extensive database (453) of numerical results of load carrying capacity of CF-TW members with different uniform and non-uniform temperature distributions in the cross-sections, under different boundary and loading conditions and with different dimensions and lengths. It is concluded that the existing DSM distortional buckling curve for ambient temperature application is also applicable for columns with uniform temperature distributions in the cross-section, but is un-conservative for columns with non-uniform temperature distributions in the cross-section. This paper proposes a modification to the distortional buckling curve to enable DSM to deal with distortional buckling in columns with non-uniform temperature distributions.     

Shear strain effects in flexure and torsion of thin-walled beams with open or closed cross-section

In this study a unified approach is presented for the analysis of the shear strain effects in thin-walled beams subjected to both non-uniform bending and torsion. Middle surface shear strains are taken into account for open as well as closed cross-sections. A suitable axial displacement field is introduced by making the basic choice that the solution to the St. Venant problems is to be reproduced for v = 0.By making use of a variational formulation, a system of differential equations is derived which rules the behaviour of a thin-walled beam with any cross-section. Hence the influence of the shear strains on the stress state as well as on the global deformation of the beam is shown through some significant examples.     

Nonlinear theory of non-uniform torsion of thin-walled open beams

A nonlinear theory of non-uniform torsion based on finite displacements is developed. Expressions for the finite nonlinear strains in Lagrangian coordinates and the Kirchhoff stresses for thin-walled open beams are presented. Using the principle of stationary total potential, the dual forms of the beam equilibrium equations are derived. For conservatively loaded thin-walled open beams a static stability criterion, based on the positive definiteness of the second variation of the total potential, is presented. The criterion developed takes into account the effects of changes in beam geometry such as initial bending curvature, prior to instability.     

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).     

任意开口薄壁截面圆弧曲梁弯扭精确分析

现有薄壁曲梁弯扭理论缺乏严密的理论推导。本文基于薄壁构件分析的两个基本假定，导出了任意开口薄壁截面圆弧曲梁的翘曲位移、正应力、剪应力及其各自合力以及平衡微分方程的精确表示式。为便于应用，本文还给出了一套具有良好精度的内力简化公式及相应的平衡微分方程。     

Analytical Method for thin-walled members in general bending and torsion

The classical theory of thin-walled members has been applied extensively in practice. Since the theory was based on the assumption of no shear deformation, it is unable to reflect some of the important phenomena such as shear lag in structures.In mixed variational principles, both stresses and displacements are taken as variables, and they create equal possibilities to yield good results both in stresses and in displacement. Based on a mixed variational principle and introducing the co-ordinate functions in the cross-section, a mixed variational method has been presented.¹ Following this method, the method of solution for thin-walled members of open cross-sections in general bending and torsion is derived in this paper. This method is more general than the classical one and can be applied to members with rows of openings. It can also be applied to problems involving tension, bending and torsion actions, and simple analytical solutions in closed form can be obtained. Both warping and shear lag phenomena can be dealt with.     

A theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion

Results of a theoretical analysis of the local buckling in thin-walled bars with open cross-section subjected to warping torsion are presented. The local critical bimoment, which generates local buckling of a thin-walled bar and constitutes the limit of the applicability of the classical Vlasov theory, is defined. A method of determining local critical bimoment on the basis of critical warping stress is developed. It is shown that there are two different local critical bimoments with regard to absolute value for bars with an unsymmetrical cross-section depending on the sense of torsion load (sign of bimoment). However, for bars with bisymmetrical and monosymmetrical sections, the determined absolute values of local critical bimoments are equal to each other, irrespective of the sense of torsional load. Critical warping stresses, local critical bimoments and local buckling modes for selected cases of thin-walled bars with open cross-section are determined.     

单箱双室箱梁对称弯曲时的局部扭转效应

为研究单箱双室箱梁在对称竖向荷载作用下的受力特征,基于横截面的荷载等效原理与荷载分解,分析了单箱双室箱梁对称弯曲时的局部扭转效应。基于截面的剪力流平衡和箱室受力分解,得到了局部扭转的等效荷载及应力计算式。通过与有机玻璃模型试验和有限元模拟结果的对比,验证了局部扭转效应计算式的正确性,并获得了单箱双室简支箱梁的局部扭转效应下的应力分布规律。研究表明,单箱双室箱梁在仅中腹板作用竖向荷载及两边腹板作用相等竖向荷载时,均存在纵向弯曲和局部扭转的组合变形模式。局部扭转由约束扭转、畸变和横向弯曲效应组成,在截面引起纵向应力和横向应力。局部扭转效应的理论计算结果与模型试验和板壳有限元分析结果吻合良好,表明基于截面剪力流等效的局部扭转荷载求解方法对双室箱梁是适用的;单箱双室箱梁的局部扭转效应在荷载作用点附近截面最为突出,截面上、下缘纵向应力和横向应力以中腹板为拐点呈折线分布,其应力分布和大小与荷载在横截面上作用的部位紧密关联;对算例箱梁,局部扭转效应产生的纵向应力可达初等梁弯曲应力的25%。     

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Derivation

This paper provides the first detailed presentation of the derivation for a newly proposed method which can be used for the decomposition of the stability buckling modes of a single-branched, open cross-section, thin-walled member into pure buckling modes. Thin-walled members are generally thought to have three pure buckling modes (or types): global, distortional, and local. However, in an analysis the member may have hundreds or even thousands of buckling modes, as general purpose models employing shell or plate elements in a finite element or finite strip model require large numbers of degrees of freedom, and result in large numbers of buckling modes. Decomposition of these numerous buckling modes into the three buckling types is typically done by visual inspection of the mode shapes, an arbitrary and inefficient process at best. Classification into the buckling types is important, not only for better understanding the behavior of thin-walled members, but also for design, as the different buckling types have different post-buckling and collapse responses. The recently developed generalized beam theory provides an alternative method from general purpose finite element and finite strip analyses that includes a means to focus on buckling modes which are consistent with the commonly understood buckling types. In this paper, the fundamental mechanical assumptions of the generalized beam theory are identified and then used to constrain a general purpose finite strip analysis to specific buckling types, in this case global and distortional buckling. The constrained finite strip model provides a means to perform both modal identification relevant to the buckling types, and model reduction as the number of degrees of freedom required in the problem can be reduced extensively. Application and examples of the derivation presented here are provided in a companion paper.     

A box beam finite element for the elastic analysis of thin-walled structures

The purpose of this paper is to present a formulation of a curved thin-walled box beam finite element having a variable cross-section with at least one vertical axis of symmetry.

To allow for the effects of warping and distortion, three degrees of freedom have been included in the formulation. These degrees of freedom have been designated as the rate of change of twisting angle, the distortional angle of the cross-section, and the rate of change of distortional angle. The effect of shear lag has also been included.

The element may be used for the elastic analysis of a variety of thin-walled structures and in particular for the preliminary analysis of box bridge decks where a three-dimensional analysis may be unnecessary. The accuracy of the element has been demonstrated by comparison of the results obtained with known results from other methods for some examples.     

受弯的冷弯卷边槽钢畸变屈曲强度和其相关屈曲承载力

受弯的冷弯薄壁卷边槽钢基本上有板件局部屈曲,截面畸变屈曲和构件弯扭屈曲三种屈曲模式,随后有它们之间的相关屈曲。由于畸变屈曲模式对缺陷的敏感度高,因此其屈曲后强度提高的幅度远低于局部屈曲模式。但是与局部屈曲模式相比,畸变屈曲模式抵抗破坏的能力却很强。可以用有限单元法计算受弯卷边槽钢截面的畸变屈曲强度。本文介绍了澳大利亚-新西兰标准AS/NZS4600-2005,用手算法计算受弯卷边槽钢截面的弹性畸变屈曲应力,並用直接强度法计算其相关的屈曲承载力。     

An experimental investigation of the behaviour of thin-walled box beams

The purpose of this paper is to present the results of an experimental investigation of the behaviour of four types of thin-walled box beam and to compare the results with those obtained from theoretical analyses.Three steel models consisting of a straight single cell cantilever, a curved single cell cantilever and a simply supported twin box have been constructed, in addition to a continous prestressed concrete two-span double cell beam.Details are given of the methods of construction, instrumentation and experimental procedure for all the models.The behaviour of the individual models has been studied, with particular attention being given to the torsion and distortion of the box sections, the cross-sectional distributions of the longitudinal and transverse bending stresses and the deflections. Appropriate experimental results are presented, therefore, and are compared with those obtained from the specially developed thin-walled box beam finite element which has been presented previously in this journal.     

550MPa高强冷弯薄壁卷边槽钢受弯构件畸变屈曲试验研究

为了研究屈服强度550MPa高强冷弯薄壁型钢受弯构件的畸变屈曲性能,分别对直卷边、斜卷边和复杂卷边3种卷边形式的12组高强冷弯薄壁槽钢受弯试件进行了静力试验研究,其中纯弯试验6组,非纯弯试验6组。试验结果表明,卷边形式是影响试件发生畸变屈曲或局部和畸变相关屈曲的重要因素。相同卷边形式下,非纯弯试件的承载力均高于纯弯试件的承载力,且提高幅度与试件屈曲破坏模式有关,只发生畸变屈曲的试件承载力提高幅度最大,而在发生局部和畸变相关屈曲的试件中,由畸变屈曲引起破坏的试件承载力提高幅度次之,由局部屈曲引起破坏的试件承载力提高幅度最小。     

Distortional eigenmodes and homogeneous solutions for semi-discretized thin-walled beams

The classical Vlasov theory for torsional analysis of thin-walled beams with open and closed cross-sections can be generalized by including distortional displacement fields. We show that the determination of adequate distortional displacement fields for generalized beam theory (GBT) can be found as part of a semi-discretization process. In this process the cross-section is discretized into finite cross-section elements and the axial variation of the displacement functions are solutions to the established coupled fourth order differential equations of GBT. We use a novel finite-element-based displacement approach in combination with a weak formulation of the shear constraints and constrained wall widths. The weak formulation of the shear constraints enables analysis of both open and closed cell cross-sections by allowing constant shear flow. We use variational analysis to establish and clearly identify the homogeneous differential equations, the eigenmodes, and the related homogeneous solutions. The distortional equations are solved by reduction of order and solution of the related eigenvalue problem of double size as in non-proportionally damped structural dynamic analysis. The full homogeneous solution is given as well as transformations between different degree of freedom spaces. This new approach is a considerable theoretical improvement, since the obtained GBT equations found by discretization of the cross-section are now solved analytically and the formulation is valid without special attention also for closed single or multi-cell cross-sections. Further more the found eigenvalues have clear mechanical meaning, since they represent the attenuation of the distortional eigenmodes and may be used in the automatic meshing of approximate distortional beam elements. The magnitude of the eigenvalues thus also gives the natural ordering of the modes.     

Analytical method for thin-walled members of closed sections

In a previous paper,¹ based on a mixed variational principle, an analytical method for thin-walled members of open cross-section in general bending and torsion has been derived. In this paper the solution of thin-walled members of closed sections will be presented.     

Buckling mode decomposition of single-branched open cross-section members via finite strip method: Application and examples

The objective of this paper is to provide implementation details of, and practical examples for, modal decomposition of the cross-section stability modes of thin-walled members by constraining a traditional finite strip method (FSM) solution. The theoretical development of the proposed method is provided in a companion to this paper [Ádány S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-walled Structures, submitted for publication, companion to this paper.] The constraint matrix, which is directly applied to the elastic and geometric stiffness matrices of a traditional FSM solution in order to constrain the deformations, is provided along with all formulae necessary in its construction. In addition, a completely worked out numerical example is provided to aid in implementing the constrained FSM solution. The authors implemented the constrained FSM in the open source program CUFSM. This modified version of CUFSM is then used to provide a series of numerical examples that illustrate (i) the advantages of performing modal decomposition, (ii) the importance of understanding and defining the deformation fields related to a desired mode, and (iii) the behavior of constrained FSM stability solutions compared with classical analytical solutions, GBT, and unconstrained FSM. Decomposition of the cross-section buckling classes related to global and distortional modes is demonstrated. Further, the impact of how to select the deformation fields and perform modal decomposition for cross-section stability modes within a class, e.g., for the traditional three global modes (weak-axis flexure, strong-axis flexure and flexural-torsional buckling), is explored and the impact of the deformation field definitions demonstrated. Comparisons of the constrained FSM solutions with other available solutions demonstrate the importance of properly determining when beam theory and plate theory should apply to the cross-section stability of thin-walled members.     

A full modal decomposition of thin-walled, single-branched open cross-section members via the constrained finite strip method

This paper derives a new method for fully decomposing the elastic stability solution, of a thin-walled single-branched open cross-section member, into mechanically consistent buckling classes associated with global, local, distortional, and shear and transverse extension buckling modes. The method requires a set of formal mechanical definitions for each of the buckling classes. For global and distortional buckling the definitions employed successfully by generalized beam theory are utilized herein, while for local and other (shear and transverse extension) buckling, new definitions are provided. The mechanical definitions for a given buckling class represent a series of constraint conditions on the general deformations that the thin-walled cross-section may undergo. These constraint conditions are derived as explicit constraint matrices within the context of the finite strip method, and provide the desired decomposition of the buckling deformations of the member. The decomposition is full in the sense that the union of the deformation spaces of the decomposed buckling classes is the same as the general deformation space in the original finite strip method. The resulting method is termed the constrained finite strip method (cFSM). The two primary applications for cFSM are modal decomposition and modal identification. Modal decomposition reduces the general finite strip solution to a desired set of buckling classes and performs a useful model reduction that allows the results to focus on a particular buckling class, e.g., distortional buckling. Modal identification provides a means to quantify the extent to which a given buckling class is contributing to a general buckling deformation. Application of cFSM, including graphical representation of the buckling classes, and the advantages of modal decomposition and modal identification, are provided in a series of numerical examples.