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.     

A more accurate evaluation of buckling loads of thin-walled members in torsion and torsion-bending

The objective of this paper is to present a more accurate evaluation of torsional and torsional-flexural buckling loads of thin-walled members. Numerical examples, which exhibit the accuracy of the formulae obtained in the buckling analysis of thin-walled members, are given.     

薄壁悬臂构件的弹性弯—扭屈曲作用

以往的研究表明,广泛用于分析薄壁梁的弯一扭屈曲作用的两个差异不大的具有代表性的理论,用于评估具有单轴对称横截面的简支梁的临界荷载时,会导致两种不同的解决方案。这两种解决方案可能会导致在这些单轴对称梁中出现有明显差异的临界荷载。基于屈曲分析中所采用的经典变分原理,作者提出一个新的理论,用于分析薄壁构件的弯一扭屈曲作用。文中采用了这三个理论来分析悬臂结构的弯一扭屈曲作用。首先进行简短回顾,并使用三种不同理论对薄壁悬臂结构的弯一扭屈曲作用进行详细的对比分析。在纯弯曲和两种典型的横向分布荷载作用下对悬臂结构的屈曲进行分析,表明三种理论的确存在差异。采用第三种理论,考虑梁长度的变化和沿梁横截面纵轴的加载位置,对两种典型的横向分布荷载下双向对称悬臂结构的临界荷载进行预测,与现有解决方案和有限元分析所得到的关键结果进行对比,可知将新方法具有：良好的准确性和易用性等优点。     

Lateral-torsional buckling capacity assessment of web opening steel girders by artificial neural networks — elastic investigation

Bridge girders exposed to aggressive environmental conditions are subject to time-variant changes in resistance. There is therefore a need for evaluation procedures that produce accurate predictions of the load-carrying capacity and reliability of bridge structures to allow rational decisions to be made about repair, rehabilitation and expected life-cycle costs. This study deals with the stability of damaged steel I-beams with web opening subjected to bending loads. A three-dimensional (3D) finite element (FE) model using ABAQUS for the elastic flexural torsional analysis of I-beams has been used to assess the effect of web opening on the lateral buckling moment capacity. Artificial neural network (ANN) approach has been also employed to derive empirical formulae for predicting the lateral-torsional buckling moment capacity of deteriorated steel I-beams with different sizes of rectangular web opening using obtained FE results. It is found out that the proposed formulae can accurately predict residual lateral buckling capacities of doubly-symmetric steel I-beams with rectangular web opening. Hence, the results of this study can be used for better prediction of buckling life of web opening of steel beams by practice engineers.     

Effect of higher order constitutive terms on the elastic buckling of thin-walled rods

In this paper we revisit an elastic constitutive equation proposed in two previous works and extend it in order to include all higher-order terms on the deformations. Our purpose is to assess the influence of these terms on the elastic buckling of thin-walled rods. The resulting material model was incorporated into a geometrically exact rod formulation and implemented into a nonlinear finite element code. By means of simple numerical examples we show that the higher order terms may play a significant role on the values of the buckling loads and on the post-buckling behavior of thin-walled beams and columns.     

冷弯薄壁梁的屈曲及优化设计综述

介绍了冷弯薄壁梁的整体-局部屈曲和优化,包括:简化分析与计算、数值分析及试验。详细阐述了薄壁梁翼缘和腹板的屈曲问题。采用质量检测方法,对不同横截面薄壁梁进行比较。     

Lateral buckling of web-tapered I-beams: A new theory

This paper presents a new theory for the lateral buckling of web-tapered I-beams. Linear analysis is first conducted by taking account into the tapering effects of web-tapered I-beams, where the deformation compatibilities of the two flanges and web are considered in terms of the basic assumptions of thin-walled members. Subsequently, the total potential for the lateral buckling analysis of web-tapered I-beams is developed, based on the classical variational principle for buckling analysis. The lateral buckling loads of web-tapered cantilevers and simply supported beams of I-sections from the proposed theory are compared with those from the finite element (FE) analyses using two shell element models and two widely used beam element models. The two beam element models respectively represent the equivalent method using prismatic beam elements and the typical tapered beam theory in existing literature. These comparisons show that the results based on the total potential proposed in this paper are more accurate in predicting the lateral buckling loads of web-tapered I-beams than those in existing theories, indicating that the theory proposed in this paper is superior to existing theories. It is also found that the equivalent method using prismatic beam elements may yield unreliable buckling loads of tapered beams.     

双层有吊车厂房框架的弹性屈曲

将单跨有吊车厂房模拟成单跨两层框架,中间框架梁两端铰接,模拟吊车桥架的联系作用,对其弹性屈曲进行了分析。分析发现,此时层与层的相互作用导致的同一层每个柱子的临界荷载的增加或者减小具有相同的比例。借助此规律,利用合并解法对框架进行了屈曲分析,发现合并解法具有很高的精度,这说明同层各柱是作为一个整体参与层与层之间的相互作用的,同时还说明梁的左、右两端提供给左、右柱子的约束可能不同,但是总和是不变的。     

薄壁钢梁稳定性计算的争议及其解决

在规范修订过程中,对钢梁的整体稳定性计算,有学者对《钢结构设计规范》(GBJ17—88)所依据的临界弯矩计算公式提出了质疑,并提出了新的计算公式。本文对规范所依据的公式和新提出的公式所依据的理论进行了对比,对新公式所依据的理论提出了疑问。为避免薄壁构件理论引入的各种假设带来思辨上难以解决的争议,本文采用ANSYS通用有限元程序的三维板壳单元SHELL63进行了6组共36根梁的整体失稳分析。有限元分析结果表明,采用板稳定理论求得的梁临界弯矩与《钢结构设计规范》(GBJ17—88)所依据的公式符合较好,而与新提出的公式比较相差较大。本文根据薄壁梁弯扭稳定理论,推导了考虑非线性正应变和剪应变的梁失稳过程应变能的变化公式,从而解释了考虑非线性剪应变的理论与ANSYS分析结果不符的原因,并对传统的稳定理论得到的结果进行了肯定。     

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.     

Inelastic buckling analyses of beams, columns and plates using the spline finite strip method

A method of inelastic buckling analysis of thin-walled structural members and plates is described. The method is based on the spline finite strip method of structural analysis. The analysis takes into account the non-linear material stress-strain properties, strain hardening and residual stresses. The plastic theories used in the study are the flow theory of plasticity and the deformation theory of plasticity. The method of inelastic buckling analysis is applied to a variety of instability problems including plates, cold-formed columns, hot-rolled columns and welded tee section beams. The buckling modes and loads computed are compared with theoretical values and test results.     

Kranbahnträger aus warmgewalzten Mannesmann‐Stahlhohlprofilen (MSH)

Crane runway made of hot‐rolled Mannesmann steel hollow sections (MSH). A new crane runway over a couryard on the production site of Vallourec & Mannesmann Deutschland GmbH in Düsseldorf, Germany should be realised using quadratic hot‐rolled hollow sections. For this purpose a new design was to be developed and realised. Until now crane runways for large spans and high loads have been built using box girders or individual designed special solutions. The production of these girders as well as the erection on site is complex and expensively in terms of time and costs. On the basis of a close collaboration of all involved project partners and under the leadership of Dittmann & Pollmann, Hagen, Germany the technical challenge was mastered and a new type of crane runway was realised.     

Stabilität und Bemessung bisymmetrischer schlanker Stahlträger

The paper presents the influence of geometrical characteristics of the cross sections of beams upon their load capacity with the use of a simplified solution taking account of the influence of lateral torsional and local buckling. Bisymmetrical, simple supported steel beams loaded at ends by moments and by distributed or concentrated loads imposed in the centres of gravity of the cross sections have been analysed. The influence of local and global buckling of beams and interactive local and global buckling of flanges upon the load capacity of beams is assessed. For beams the local buckling capacities and reduced distortional buckling capacities are compared by approximate manners and according to Eurocode 3. The aim of the paper is to present the assessment of the influence of geometrical characteristics of beams, modes of load upon the ultimate load capacity of beams taking account of local and global buckling making use of approximate solutions and formulas given in Eurocode 3 and in the Polish Standard PN‐90/B‐03200.     

Recent advances on postbuckling analyses of thin-walled structures: Beams, frames and cylindrical shells

The lateral postbuckling response of thin-walled structures such as bars and frames with members having steel rolled shapes as well as circular cylindrical shells under axial compression is thoroughly reconsidered. More specifically via a simple and very efficient technique it is found that beams with rolled shapes (symmetric or non symmetric) under uniform bending and axial compression exhibit a stable lateral-torsional secondary path with limited margins of postbuckling strength. New findings for the static and dynamic stability of frames with crooked steel members-due to the presence of residual stresses-are also reported. It is comprehensively established that the coupling effect due to initial crookedness and loading eccentricity may have a beneficial effect on the load-carrying capacity of the frames. Moreover, simple mechanical models are proposed for simulating the buckling mechanism of axially compressed circular cylindrical shells. Very recently Bodner and Rubin proposed an 1-DOF mechanical model whose buckling parameters correlated to those of the shells by using an empirical formula based on experimentally observed shell buckling loads. In the present analysis a new 2-DOF model for the static and dynamic buckling of axially compressed circular cylindrical shells, which can include mode coupling, is presented.     

Semi-empirical design of complex metal beams by buckling analysis

Structural elements with complex geometries, boundary conditions and load patterns cannot be designed against buckling using empirical formulae because of uncertain elastic buckling moments or unknown buckling effective lengths, which are basic parameters for these equations. This article proposes a shell finite element procedure for buckling design of metal beams of complex configurations with codified initial imperfections assumed in the Perry–Robertson formula. The advantage of the proposed method lies in the use of elastic buckling moment with empirical design formulae for determination of design moment capacity of a beam; thereby eliminating the uncertainty of modelling initial imperfections. More importantly, the moment modification factor and assumption of effective length can be avoided because all second-order and yield effects have been considered in the computer model. Numerical examples demonstrate that the simplified method has a high level of accuracy, versatility and flexibility for the design of complex beams.     

The ultimate load sensitivity of lipped channel columns to column axis imperfection

An analytical investigation is made of the effect of column axis imperfection on the equilibrium behaviour and ultimate carrying capacity of thin-walled lipped channel columns. The Rayleigh-Ritz method is used to obtain local buckling loads and a semi-energy method is employed to describe post-buckling interaction behaviour. In the analysis a facility of allowing the locally deflected form to change after buckling is provided, resulting in asymptotic equilibrium curves tending towards continually changing reduced Euler loads as opposed to fixed reduced Euler loads based on an unchanging locally deflected form.The imperfection sensitivity of column designs which produce stable and unstable equilibrium behaviour after local buckling is investigated as is that of columns which exhibit neutral equilibrium behaviour before local buckling. Results are presented in the form of non-dimensional load-deflection equilibrium curves and ultimate load-slenderness plots covering these three equilibrium states. A comparison of the theory with independent experimental work is also included.     

Buckling of cylindrical shells under transverse shear

This work concerns with experimental studies on buckling of thin-walled circular cylindrical shells under transverse shear. The buckling loads are also obtained from finite element models, empirical formulae and codes and are compared. Experiments are conducted on 12 models made of stainless steel by rolling and longitudinal seam welding. In situ initial geometric imperfection surveys are carried out. The tests are conducted with and without axial constraint at the point diametrically opposite the loading. Theoretical analyses are carried out using ABAQUS finite element code. Two finite element models considered are: (i) geometry with real imperfection (FES-I) and (ii) critical mode imperfect geometry (FES-II). In the former, the imperfections are imposed at all nodes and in the latter, the imperfection is imposed by renormalizing the eigen mode, using the maximum measured imperfection. General nonlinear option is employed in both the cases for estimating the buckling load. Galletly and Blachut’s expressions, design guidelines of Japan for LMFBR main vessel expressions (empirical formulae), ASME and aerospace structural design codes are used for comparing with experimental loads.The comparisons of experimental, numerical and analytical buckling loads reveal the following. The numerical results are always higher than the experimental values; the percentage difference depends on the wall thickness. FES-II predicts somewhat a lower load than that of the FES-I. The Japanese guidelines predict the lowest load, which is conservative. Experimental loads are lower than that predicted by both ASME and aerospace structural design codes.     

Shear-flexible thin-walled element for composite I-beams

A shear-flexible finite element based on an orthogonal Cartesian coordinate system is developed for the flexural and buckling analyses of thin-walled composite I-beams with both doubly and mono-symmetrical cross-sections. Using the first-order shear deformable beam theory, the derived element includes both the transverse shear and the restrained warping induced shear deformations. Governing equations are derived from the principle of minimum total potential energy. Three different types of finite elements, namely, linear, quadratic and cubic elements are developed to solve the governing equations. The geometric stiffness for the buckling analysis of axially loaded, thin-walled composite beams is developed. The resulting linearized buckling problem is solved using a shifted inverse iteration algorithm. A parametric study of the effects of the aspect ratio and the fibre orientation on the tip displacement is presented. The convergence of the elements is also investigated. The elastic buckling loads for mono- and doubly-symmetric I-beam cross-sections are compared with other results available in the literature and with solutions using shell elements in a commercially available finite element program.     

Distortional mechanics of restrained steel sections

This paper presents an overview of the mechanics of distortion in members which have restrained thin-walled steel sections. First, the basic features that characterize the distortion of thin-walled sections under compression and/or bending are explained and the generally adopted kinematical assumptions are presented and discussed. On the basis of these assumptions, a simple procedure to build the distortional displacement fields is proposed. Then, two illustrative examples of restrained sections are given and their distortional displacement fields are built. The I-section illustrates the case of a symmetric section having a distortional displacement field with a single d.o.f. The Z-section exemplifies the case of a non symmetric section having a distortional field with two d.o.f. Based on an energy formulation, the equilibrium equation for buckling analysis of simply supported thin-walled members is derived. For two illustrative examples, the distortional displacement field is used to obtain distortional analytical formulae. Finally, some results (buckling loads and moments) were determined and validated by means of comparison with fully numerical obtained from finite strip analysis.