Lateral-distortional buckling of I-beams and the extrapolation techniques

Various extrapolation techniques are utilized for predicting the critical buckling load of structural members undergoing lateral buckling. For the application of these methods it is only necessary to have data relating load to a deformation characteristic. In order to obtain an experimental comparison of the Southwell, Modified, and Massey Plot methods, these have been applied on lateral deflections and web transverse strains of four full-scale simply supported I-beams subjected to a central concentrated load with an effective lateral brace at the midspan of the top flange, undergoing inelastic lateral-distortional buckling. Although there is some scatter due to the application of various deformation variables and extrapolation techniques, the agreement between the extrapolated and the maximum test loads is generally good. The smallest discrepancy is found in the case of the Modified Plot. Moreover, in addition to demonstrating direct coupling between the lateral deflections and the web transverse strains representing web distortion, it is also shown that good predictions are provided by applying the Meck Plot on these deformation variables.     

Elastic distortional buckling of doubly symmetric I-shaped flexural members with slender webs

Intermediate length I-section members, particularly those with slender webs and stocky flanges, have been shown to buckle in a distortional mode, in which the web distorts and the flanges displace sideways with less twist than for the lateral–torsional buckling mode. This, in turn, reduces the torsional rigidity of the member, and hence results in a lower buckling strength. The web distortional flexibility is not addressed explicitly in standards for flexural design of steel I–section members. However, in the 2005 AISC specification it has been tried to account for the influence of web distortional flexibility on the lateral–torsional buckling resistance for slender-web I-section members through implicit use of J=0 in the lateral–torsional buckling equations. This paper evaluates the amount of effectiveness of this consideration within the elastic limit. By comparing the AISC code predictions with the accurate finite strip analysis (FSA) distortional buckling solutions as well as the theoretical predictions of two elastic distortional buckling formulae developed by other researchers, it is found that the assumption of J=0 for slender-web members results in overconservative estimate of the buckling strength in some cases which may cause economic losses.     

Distortional buckling in monosymmetric I-beams: Reverse-curvature bending

This paper deals with distortional buckling in monosymmetric propped cantilever I-beams under two types of load: a central point load and a uniformly distributed load. Top-flange and bottom-flange load positions are considered. ABAQUS is used for the investigation. Moment modification factors are obtained and are compared with those based on Structural Stability Research Council Guidelines, which take into account lateral–torsion buckling only. It is seen that provisions in SSRC Guidelines seriously overestimate the critical load, especially for short beams.     

Distortional buckling in monosymmetric I-beams

This paper deals with distortional buckling of simply supported monosymmetric I-beams under three types of load: a central point load, a uniformly distributed load and a uniform sagging moment. ABAQUS is used for the investigation. Top-flange and bottom-flange load positions are considered for the first two load cases. It is found that for comparatively short beams, buckling may be governed by distortion of the web. Moment modification factors are calculated based on the present analysis, which accounts for the distortion of web and these are compared with those based on SSRC Guidelines, which are based on lateral-torsional buckling analysis only. It is seen that for short beams, provisions in SSRC Guide-1998 seriously overestimate the critical load.     

Distortional buckling in braced-cantilever I-beams

This paper deals with distortional buckling of braced-cantilever monosymmetric I-beams under three types of load: a tip point load, a uniformly distributed load and a moment at the end. Top-flange and bottom-flange load positions are considered for the first two load cases. ABAQUS is used for the investigation. The effect of different types of bracing on buckling load is investigated. Results are compared with results from previous experimental investigations. It is also found that top lateral bracings are very effective for beam sections having larger bottom-flanges when a point load or a uniformly distributed load acts at the top-flange, and for the uniform moment case, except for the T-section or the inverted T-section cantilever beams. On the other hand, bottom lateral bracings are very effective for beam sections having larger top-flanges. When loads are placed at the bottom-flange, position of any kind of lateral bracing has practically no effect on the buckling capacity of a monosymmetric cantilever beam, except for the inverted T-section cantilever beams.     

Thermal buckling of rotationally restrained steel columns

The in-plane elastic buckling of a steel column under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or an elastic foundation. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain-displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the buckling of columns in three different thermal loading cases is studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The thermal gradient plays a positive role in improving the stability of columns. Furthermore, the effect of thermal gradients decreases while increasing the rotational restraint stiffness and decreasing the slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the rotational stiffness of thermal restraints will increase the critical temperature.     

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.     

A closed-form solution for elastic buckling of thin-walled unstiffened circular cylinders in pure flexure

To date, despite the significant development in the field of structural mechanics, there still remains a paradox in the solutions available for a classical shell buckling problem. The difference in strength between a cylindrical shell under uniform axial compression and that under pure bending is not quite well investigated. This lack of research is reflected in the wide variations in the elastic bending strength and the slenderness limits given in current international design standards. The discrepancies in the available classical solutions and hence the design rules have initiated the current research. The main aim of this paper is to present a closed-form solution for the elastic buckling strength of unstiffened circular cylinders under pure bending using a new simplified energy approach employing the well-known Ritz method. Two types of analyses are presented for cylinders with large (D/t>200) and medium (100<D/t<200) diameter-to-thickness ratios. A unique testing rig was used to experimentally verify the new theory using a Moiré fringe film. The theoretical results are compared against the available and present test results and the existing classical solutions. The current design rules for thin-cylinders in international steel specifications are also compared, and the newly derived design curve is proposed which was found in a good agreement with the available test results.     

Effect of aspect ratio on the elastic buckling of uniaxially loaded plates with eccentric holes

The Finite Element Method (FEM) has been employed to determine the elastic buckling load of uniaxially loaded rectangular perforated plates with length a and width b. Plates with simply supported edges in the out-of-plane direction and subjected to uniaxial end compression in their longitudinal direction are considered. Integer plate aspect ratios, a/b=1, 2, 3 and 4, have been chosen to assess the effect of aspect ratio on the plate buckling load. Two perforation shapes of different sizes are considered; circular, and rectangular with curved corners. The rectangular perforation is oriented such that either its long or its short side is parallel to the longitudinal direction of the plate. The center of perforation was chosen at different locations of the plate. The study shows that the buckling load of a rectangular perforated plate that could be divided into equal square panels is not the same as that of the square panel that contains the perforation when treated as a separate square plate. For rectangular plates, the study recommends not to have the center of a circular hole placed in a critical zone defined by the end half of the outer square panel, to try always to put the hole in an interior panel of the plate, and to have the distance between the edge of a circular hole and the nearest unloaded edge of the plate not less than 0.1b. The study concludes also that the use of a rectangular hole, with curved corners, with its short dimension positioned along the longitudinal direction of the plate is a better option than using a circular hole, from the plate stability point of view.     

On the buckling of axially restrained steel columns in fire

This paper describes the behaviour of restrained steel columns in fire. It follows the introduction of extra load into the column through the axial restraint of the surrounding cooler structure and the consequential buckling. Key to this understanding is the post-failure behaviour and re-stabilisation of the column, which is discussed with reference to a finite element model and an analytical model. Through bi-directional control of the temperature, the finite element model allows the snap-back behaviour to be modelled in detail and the effects of varying slenderness and load ratio are investigated. The analytical model employs structural mechanics to describe the behaviour of a heated strut, and is capable of explaining both elastic and fully plastic post-buckling behaviour.Through this detailed explanation of what happens when a heated column buckles, the consequences for steel-framed building design are discussed. In particular, the need to provide robustness is highlighted, in order to ensure that alternative load paths are available once a column has buckled and re-stabilised. Without this robustness, the dynamic shedding of load onto surrounding structures may well spread failure from a fire’s origin and lead to progressive collapse.     

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.     

On the buckling of cylindrical shells with through cracks under axial load

Presence of cracks or similar imperfections can considerably reduce the buckling load of a shell structure. In this paper, the buckling of cylindrical shells with through cracks has been studied. A general finite element model has been proposed, verified and applied to some novel cracked shell buckling problems for which documented results are not available. A special purpose program has been developed for generating finite elements models of cylindrical shells with cracks of varying length and orientation. The buckling behavior of cracked cylinders in tension and compression has been studied. The results of the analysis are presented in parametric form when it seems to be appropriate. Sensitivity of the buckling load to the crack length and orientation has also been investigated.     

Elastic distortional buckling of tee-section cantilevers

The paper presents the results of a finite element study of the elastic distortional buckling of tee-section cantilevers, which can be thought of as beams fully braced at one end and unbraced at the other. The finite element procedure is described briefly, and then three loading cases, viz., a tip moment, a tip load and a uniformly distributed load are considered. All of these loading cases place the unstiffened or free edge of the stem or web into compression. The effects of distortion are quantified for the three loading cases, as are the effects of fully restraining the top flange against lateral deflection and twist by a discrete brace positioned anywhere along the cantilever. It is shown that the effects of distortion during buckling cannot be ignored in a tee-section cantilever with even a moderately slender web.     

The effect of semi-rigid joints and an elastic bracing system on the buckling load of simple rectangular steel frames

A linear stability analysis for establishing the combined effect of joint flexibility and an elastic bracing system on the buckling load of steel plane frames is presented herein. Based on the beam-to-column model of Eurocode 3, the subsequent study shows that joint flexibility is a very important parameter that needs to be incorporated into the stability analysis of frames with semi-rigid connections. It was found that assuming flexible connections in such frames always leads to a reduction of their buckling load, which is proven to be significant in many characteristic cases. Numerical results for simple elastically braced or unbraced frames with semi-rigid connections, in tabular and graphical form, reveal the pronounced effect of joint flexibility and elastic bracing on their buckling load. The results presented herein can be readily used for the design of simple frames against buckling, while the flexible connection concept can be applied to all types of steel frames.     

Effects of imperfections of the buckling response of composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling response and failure of unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells are presented. The shells considered in the study have six different shell-wall laminates two different shell-radius-to-thickness ratios. The shell-wall laminates include four different orthotropic laminates and two different quasi-isotropic laminates. The shell-radius-to-thickness ratios includes shell-radius-to-thickness ratios equal to 100 and 200. The numerical results include the effects of traditional and nontraditional initial imperfections and selected shell parameter uncertainties. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The nontraditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, nonuniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. The cylinder parameter uncertainties considered include uncertainties in geometric imperfection measurements, lamina fiber volume fraction, fiber and matrix properties, boundary conditions, and applied end load distribution. Results that include the effects of these traditional and nontraditional imperfections and uncertainties on the nonlinear response characteristics, buckling loads and failure of the shells are presented. The analysis procedure includes a nonlinear static analysis that predicts the stable response characteristics of the shells, and a nonlinear transient analysis that predicts the unstable response characteristics. In addition, a common failure analysis is used to predict material failures in the shells.     

Shell element of relative degree of freedom and its application on buckling analysis of thin-walled structures

Shell element of relative degrees of freedom (SERDF) is a special transformation of solid element. It can be used in finite element (FE) analysis of both thin and thick shell structures and the formulation is simpler than normal shell elements. Introduction of additional internal degrees of freedom will improve the calculation precision.FE formulation of SERDF in the buckling analysis is derived in this article. Different measures of tackling additional internal degrees of freedom for different kinds of buckling problems and different stages of numerical analysis are presented. Some numerical examples are given to illustrate the validity of this element and the method.     

A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling

A computational model for analysis of local buckling and postbuckling of stiffened panels is derived. The model provides a tool that is more accurate than existing design codes, and more efficient than nonlinear finite element methods. Any combination of biaxial in-plane compression or tension, shear, and lateral pressure may be analysed. Deflections are assumed in the form of trigonometric function series. The deformations are coupled such that continuity of rotation between the plate and the stiffener web is ensured, as well as longitudinal continuity of displacement. The response history is traced using energy principles and perturbation theory. The procedure is semi-analytical in the sense that all energy formulations are derived analytically, while a numerical method is used for solving the resulting set of equations, and for incrementation of the solution. The stress in certain critical points are checked using the von Mises yield criterion, and the onset of yielding is taken as an estimate of ultimate strength for design purposes.     

An experimental method for measuring the buckling shapes of thin-walled structures

If plates are loaded alternately from both sides beyond their yield limit, permanent inelastic deformations can occur. Furthermore, pressure loadings rectangular to the deflections of inelastically deformed plates can cause buckling. In order to predict the deformations during these loading cycles, it is important to know the kind of snap-through. It can be symmetric or unsymmetric depending on the loading velocity. In the present study, the deformations and the buckling behaviour of metal plates, which are loaded by shock waves from both sides are investigated experimentally and numerically. The aim is to measure the buckling shapes and to predict the deflections in simulations. For comparison the snap-through of quasi-statically loaded plates are measured, too.     

Comparative study of the buckling of steel beams in Eurocode 3 and the Russian code

Lateral torsional buckling is the main limit state that must be checked for steel beams. Design codes provide different models for the analysis of this complex phenomenon. The present paper describes a comparative study of the interpretation of beam buckling of symmetrical sections in EN 1993-1-1 (EC3), its preliminary ENV-version and a Russian steel structure design code. In order to choose the nationally determined parameters in EC3, the impact of parameters on the safety level and overall design results has to be carefully studied. Modification factors, introduced in EC3, have moved the resistance for certain load distribution from ENV-based values closer to the Russian code.     

多种勘察手段在高速公路勘察中的综合应用

通过对王繁高速公路的勘察及各种勘察手段综合应用,准确查明了该项目工程地质特征,为设计提供了详实依据,以期为以后类似山区高速公路勘察提供指导。