Bifurcation of singly symmetric columns

The paper derives the governing equations for the fundamental and bifurcated states of members with singly symmetric cross-sections that locally buckle in the fundamental state. The members are subject to pure compression and assumed to be geometrically perfect in the overall sense. It is shown using the fundamental state equations that fixed-ended columns exhibit overall bifurcation behaviour while pin-ended columns do not. The bifurcation equation is applied to plain channel sections and the results are compared with tests of fixed-ended columns. The variation of the bifurcation loads with the length is shown to be in good agreement with the tests. The results are shown to be sensitive to the magnitudes of local and overall geometric imperfections.     

Inelastic bifurcation of cold-formed singly symmetric columns

The paper describes a technique for determining the overall flexural and flexural–torsional bifurcation loads of locally buckled cold-formed channel columns. The method of analysis uses an inelastic geometric non-linear finite strip local buckling analysis to determine the flexural and torsional tangent rigidities of a locally buckled section. These tangent rigidities are substituted into the flexural and flexural–torsional bifurcation equations to calculate the inelastic overall buckling loads. The members are assumed to be geometrically perfect in the overall sense but can include geometric imperfections and yielding in the local mode. The bifurcation analysis is applied to cold-formed plain channel columns. The bifurcation loads and failure modes are compared with tests of fixed-ended columns and shown to be in good agreement with the tests. The effect of yielding is highlighted in the paper.     

Flexural and flexural-torsional bifurcation of locally buckled beam-columns

The paper describes the overall bifurcation behaviour of locally buckled I-sections in combined compression and major axis bending. The overall bifurcation loads are calculated using the tangent rigidities of the locally buckled cross-section for minor axis flexure and warping torsion. The expression for the overall bifurcation load is analytic while the tangent rigidities are calculated numerically using an elastic nonlinear finite strip local buckling analysis. It is shown that while the interaction curves for local buckling and overall flexural- torsional buckling of the undistorted cross-section are convex, the interaction curves for the flexural torsional bifurcation of locally buckled beam-columns may be convex, linear or concave depending on the cross-section geometry and the extent of local buckling. It is demonstrated that the overall bifurcation load depends primarily on the local buckling of the flanges. The overall bifurcation load of sections that are locally buckled in the web only is virtually unaffected by local buckling for any combination of bending and compression.     

Bifurcation of locally buckled point symmetric columns—Experimental investigations

The bifurcation equations for locally buckled point symmetric sections are derived in a companion paper [Rasmussen KJR. Bifurcation of locally buckled point symmetric columns—analytical developments. Thin-walled Struct, submitted for publication]. In the present paper, two series of experiments are reported, one on narrow flange Z-sections and one on wide flange Z-sections. The main objective of the tests was to validate the bifurcation load predictions derived in [Rasmussen KJR. Bifurcation of locally buckled point symmetric columns—analytical developments. Thin-walled Struct, submitted for publication] against experimental values. A further objective of the tests on narrow flange sections was to investigate the change in the direction of overall buckling, as predicted by the theory, from principal axis directions to non-principal directions. A further objective of the tests on wide flange sections was to investigate the possible change of the critical overall buckling mode from a flexural mode to a torsional mode as a result of local buckling. Agreement is generally found between analytical and experimental results.     

Bifurcation of locally buckled point symmetric columns—Analytical developments

The paper derives the differential equations for the overall bifurcation of locally buckled point-symmetric columns. It is shown that flexural buckling about the minor and major principal axes are coupled in a locally buckled point-symmetric column whereas the buckling modes are uncoupled in a non-locally buckled column. The governing equations are solved for simply supported and fixed-ended columns and applied to Z-sections. Overall bifurcation curves are obtained for five Z-sections with increasingly slender flanges. It is shown that local buckling reduces the elastic torsional buckling load more so than the flexural buckling load for Z-sections, and that local buckling can cause a mode switch from the flexural to the torsional mode in Z-sections with very slender flanges.     

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.     

Elastic biaxial bending and torsion of thin-walled members

This paper presents a detailed treatment of the non-linear elastic biaxial bending and torsion of thin-walled open section members. The treatment is valid for uniform members of linear elastic material, and is limited to small strains and rotations, and moderate deflections. Shear straining of the mid-surface of the member wall is neglected, and it is assumed that the member does not distort or buckle locally. The effects of initial deformations, loads, stresses, and strains are incorporated.

The treatment is based on non-linear strain-displacement relationships, and these are used to derive the non-linear equilibrium and tangent stiffness equations in forms which are suitable for computer solution by the finite element method.

Approximate linear and non-linear differential equilibrium equations are derived, as are the differential equilibrium equations and the energy equation for neutral equilibrium at bifurcation buckling, and these are then related to the classical equations developed by Timoshenko, Vlasov, and others.     

Exact and approximate solutions for the flexural buckling of columns with oblique rotational end restraints

This paper is concerned with the elastic flexural buckling of doubly symmetric columns with oblique restraints under concentric loading. Oblique restraints cause coupling between the principal axis deflections and rotations, and the flexural buckling mode involves simultaneous bending about both principal axes.The paper discusses the nature of oblique restraints, and presents exact and approximate solutions for the buckling loads of columns with rigid or elastic restraints against rotations at the ends. The exact solutions are obtained by solving the governing differential equations and boundary conditions, while the approximate solutions are based on energy solutions with assumed buckling displacements. The approximate solutions are sufficiently simple that they can be used in design, and are shown to be within 1% of the exact solutions.     

Nonlinear elastic response of locally buckled thin-walled beam-columns

A theory is described which combines the finite strip method of nonlinear elastic analysis of locally buckled thin-walled sections with the influence coefficient method of analysis of beam-columns. The theory combines the advantages of both methods of analysis to produce a computationally efficient procedure for the estimation of the overall nonlinear elastic response of locally buckled thin-walled beam-columns.The theory is used to predict the previously reported measured behaviour of I-section test specimens of Bijlaard and Fisher. The theory is also compared with theoretical responses of eccentrically loaded thin-walled beam-columns analysed by an asymptotic theory. An example of a locally buckled thin-walled beam-column is studied in detail and some observations are made concerning the nonlinear response of members of this type.     

Thin-walled cold-formed sections subjected to compressive loading

The weakening effects of local buckling on the compressive load carrying capability of thin-walled cold-formed sections is examined and discussed. Theoretical results are presented for the pin-ended support condition and a study is made of the local buckling and overall bending interaction behaviour of compression members with different cross-sectional shapes. It is shown that singly symmetric plain channel sections exhibit considerably different interactive equilibrium characteristics to those of I-sections for which the shear centre and section centroid are coincident.

The theoretical results given have been obtained through the solution of a differential equation which governs, approximately, the overall bending behaviour of a locally buckled compression member. Both local and overall imperfections are considered in the theoretical approach and it is shown that the effect of these is to reduce the ultimate compressive carrying capacity of cold-formed sections.

An experimental investigation of the behaviour of concentrically loaded pin-ended I-section struts and columns is reported and the findings from this are compared with theoretical predictions. The theoretical predictions of load-deflection equilibrium behaviour and stress variations with load are shown to be in good agreement with those obtained from test.

The provisions made in the UK Code of Practice and the American Specification pertaining to the design of cold-formed compression members are briefly outlined and compared. It is shown that with regard to I-section struts and columns the American design procedure gives, in general, more conservative estimates of collapse.     

Elastic flexural-torsional buckling of thin-walled cantilevers

Previous studies by the authors revealed that the two representative theories with slight differences between, widely used in investigating the flexural-torsional buckling of thin-walled beams, have led to two different solutions in well-known literature for assessing critical loads of simply supported beams of monosymmetric cross section. With these two solutions, significant differences in critical loads may be found for these monosymmetric beams. Based on the classical variational principle for buckling analyses, a new theory on the flexural-torsional buckling of thin-walled members was proposed by the authors. In this paper, this new theory as well as the other two typical theories is employed to investigate the flexural-torsional buckling of cantilevers.This paper first gives a brief review and a careful comparative study on the flexural-torsional buckling of thin-walled cantilevers employing three different buckling theories. Differences between these theories are demonstrated with investigations on buckling of cantilevers under pure bending and two typical transverse loads. Explicit solutions, capable of considering variations of beam length and loading position along the vertical axis of cross section, are presented for predicting the critical loads of doubly symmetric cantilevers under two typical transverse loads. Advantages of presented solutions, such as good accuracy and ease of use, are exploited through the comparisons of critical results with those from existing solutions and finite element analyses.     

Flexural–torsional buckling of thin-walled composite box beams

Buckling of an axially loaded thin-walled laminated composite is studied. A general analytical model applicable to the flexural, torsional and flexural–torsional buckling of a thin-walled composite box beam subjected to axial load is developed. This model is based on the classical lamination theory, and accounts for the coupling of flexural and torsional modes for arbitrary laminate stacking sequence configuration, i.e. unsymmetric as well as symmetric, and various boundary conditions. A displacement-based one-dimensional finite element model is developed to predict critical loads and corresponding buckling modes for a thin-walled composite bar. Governing buckling equations are derived from the principle of the stationary value of total potential energy. Numerical results are obtained for axially loaded thin-walled composites addressing the effects of fiber angle, anisotropy and boundary conditions on the critical buckling loads and mode shapes of the composites.     

薄壁复合箱型梁的弯曲-扭转屈曲分析

对一轴心受压薄壁复合构件的屈曲进行研究。提出一个广义的分析模型,可用于分析轴心受压薄壁复合箱型梁的弯曲、扭转以及弯扭屈曲作用。此模型基于经典层压理论,考虑了任意层压堆积规律,结构的弯曲和扭转模式的耦合问题,如非对称以及对称和各种边界条件。采用一个基于位移的一维有限元模型来预测薄壁复合钢筋的临界荷载和随后的屈曲模式。从总势能的平稳值原则中推导出屈曲控制方程。轴心受压薄壁复合件的数值计算结果可用于估测纤维角、各向异性和边界条件对临界屈曲荷载和复合件模态的影响。     

Distortion (and Torsion) of rectangular thin-walled beams

A study of rectangular cross-section thin-walled beams under torsional, distortional and bimomental loads is presented. The assumed displacement field describes both torsional and distorsional behaviour, the shearing strain in the walls being taken into account. By a variational principle the equilibrium and boundary equations are derived and a physical interpretation of the natural conditions is given.A closed form solution is given with reference to the matrix form of the governing differential system together with a discussion of a sixth-order ‘uncoupled’ differential equation generalizing the BEF analogy.The present formulation shows that, in general, torsion and distortion are pair of coupled problems and a study of distortion without considering torsion would therefore not be legitimate.The comparison of the results predicted by the theory is in good agreement with experimental evidence.     

Study on the influence of the initial deflection and load combination on the collapse behaviour of continuous stiffened panels

Using the finite element analysis, a series stiffened panels under combined normal loads and biaxial compressions are conducted to investigate the effect of several influential factors on the ultimate limit states. Two spans/bays FE model with periodical boundary condition is adopted to consider the interaction between adjacent structural members. The initial deflections assumed as Fourier components including symmetric and asymmetric modes are used to identify the half-wave number of collapse of the local plate, which is compared with half-wave number of buckling calculated by formula. Based on the numerical results, the influences of half-wave number assumed in the equivalent initial imperfection and loads combination on the collapse behaviours of stiffened panels are discussed. It is found that lateral pressure might increase the ultimate strength of stiffened panels for the stiffener-induced failure modes. The one half-wave region of local plate influences significantly the load carrying capacity of stiffened panels.     

双参数弹性地基上周边固支叠层厚板的解析解

基于三维弹性力学的基本方程，引入Dirac函数，导出了周边固支叠层板的状态方程。采用Cayley-Hamilton定理，得出了双参数地基上叠层厚板在任意荷载作用下的封闭解。此解满足弹性力学全部方程和层间连性条件，适用于任意厚跨比，可求出全部位移的应力分量，计算结果吻合较好。     

Buckling formulation for shear deformable thin-walled members—II. Finite element formulation

The variational principle developed in a companion paper is adopted to formulate a new finite element for the buckling analysis of members of doubly symmetric cross-sections. A series of examples demonstrate the convergence characteristics of the element and its applicability to a wide variety of problems. The examples include column flexural buckling, lateral–torsional buckling of beams subjected to moment gradients, buckling of members under combinations of axial force and bending moment, and eccentrically supported structural members. In all cases, the validity of the solutions is assessed through comparisons to well-established closed-form and/or other established numerical solutions.     

Nonlinear buckling of microfabricated thin annular plates

Annular plate structures are commonly used in MEMS devices, particularly in pumps and valves. In MEMS applications, large nonlinear deflections are routinely achieved. In this paper, the nonlinear buckling of a thin elastic annular plate under a compressive radial force acting in the plane of the plate is considered. Although the critical loads at which buckling starts can be determined by solving a linear eigenvalue problem, the large deflection behavior of a buckled plate beyond the critical loads can be described by the nonlinear theory proposed by von Kármán. The buckling loads of the annular plate for various boundary conditions are calculated and the post-buckling behaviors are examined. Interestingly, the buckling of an annular plate exhibits a supercritical pitchfork bifurcation. Thus, the post-buckling behavior is stable and will not collapse catastrophically. Several design implications for microfabricated structures, in particular microvalves, are also given.     

Local buckling and inelastic cyclic behavior of tubular members

Closed form expressions are obtained for the load-deflection behavior of the locally buckled circular tubular braces under the reversed loading conditions. These expressions are obtained by supplementing the deflected shape of the brace with closed form expressions for moment-curvature relationship of locally buckled circular tubular section. The developed closed form expressions are used to study the effects of the slenderness ratio and diameter-to-thickness ratio on the behavior of braces under reversed loading conditions.     

Asymmetric interactive buckling of thin-walled columns with initial imperfections

In this paper the effect of the interaction between two or more simultaneous buckling modes on the postbuckling behaviour of uniformly compressed thin-walled members (TWM) is analysed by means of the general theory of elastic stability. The analysis is restricted to third-order terms of the energy expansion and therefore can be fruitfully applied to the investigation of structures with asymmetric postbuckling behaviour only. Initial imperfection effect is taken into account. A simplified procedure is suggested for solving the nonlinear equations relative to the evaluation of the bifurcated paths. By using the finite strip method an extensive parametric analysis is performed. It is found that when the flexural-torsional (FT) buckling interacts with a local symmetric and antisymmetric mode, sensitivity to initial inperfections is remarkable and is comparable to the one arising from the interaction between the Euler (E) and any local buckling.