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Stability of columns supported laterally by side-rails
Abstract
The buckling conditions are derived for an I-section column supported laterally by uniformly spaced side-rails which provide rigid lateral support and elastic torsional restraint, the column being subjected to any combinations of axial load and uniform or non-uniform moment about the major axis. Criteria are developed to determine, for any loading condition and given spacing of lateral supports, the minimum torsional restraint which causes buckling to occur between supports rather than in an overall mode.
... For this reason, they must be taken into account in design. Torsional bracing of elastic columns is studied by Dooley [20] and Horne and Ajmani [21] where semi-analytical solutions for discrete intermediate torsional braces were carried out for simply supported columns. The torsional buckling behaviour of columns with eccentric lateral bracing was investigated in [22][23][24]. ...
... Horne and Ajmani [21] approximated discrete intermediate torsional braces from a continuous model of a column under elastic torsion restraints. The study considers the buckling behaviour of a column under eccentric torsion bracing (Fig. 4). ...
... According to Eq. (37), the torsional bracing stiffness threshold requirement can be obtained when the buckling load ( , ) reaches the higher buckling load ( ,(n+1) ). After rearranging, the torsional threshold brace stiffness was obtained in [21] and given by: ...
The present study investigates the buckling strength of braced thin-walled columns under bending and torsional modes. Unlike the well-known bending modes, the torsion buckling modes of braced columns are not adequately assessed in design and are often overlooked. In order to control the buckling behaviour, the effects of elastic discrete springs in bending and torsion are taken into account. Analytical solutions are derived for bending and torsion buckling modes in the case of simply supported struts with doubly symmetric cross sections. Further, the threshold bracing stiffness to achieve the higher buckling modes is found in both bending and torsion modes. For more general cases, the finite element approach of the model is implemented and it is able to carry out the buckling modes of columns under compression in the presence of 3D elastic springs and for any arbitrary cross sections. The analytical and the numerical results of the present models are compared to some available solutions in the literature and to finite element simulations of the commercial code Abaqus. The efficiency of the closed-form solutions and the numerical approach is successfully verified. The post-buckling of braced columns is considered at the end in the presence of elastic bending and torsion springs. The effect of braces to enhance the buckling capacity concludes the study.
... u = u n sin(nπx/L). Horne & Ajmani (1969) analysed the provision of multiple rigid discrete bracings to columns and used energy methods to solve for the buckling loads. Since a fixed axis of rotation was imposed, u and φ were directly related and as such the problem was reduced to a single DOF problem. ...
... In reviewing these two analyses,Wadee et al. (1997) mention that the effect of the nonlinearity in the foundation outweighs that of including the effects of large deflections and so, in their subsequent comparison of results from a perturbation scheme whereby the governing ODEs of the system are solved and from a Rayleigh-Ritz analysis, assume small deflections and include a nonlinear foundation. Another investigation of note is that of Horne & Ajmani (1971b) who expanded upon the case examined byHorne & Ajmani (1969) andHorne & Ajmani (1971a) of a column laterally restrained at its top flange to examine the postbuckling behaviour of the column by including a plastic hinge at mid-height. From this analysis, limiting slendernesses were calculated below which full plastic resistance can be still achieved despite the beam already having undergone flexural buckling. ...
... 0 − κ s and η = m/(n b + 1) as mode numbers are designated by m rather than n in the current section.An analytical solution for µ to Equation (4.41) is difficult to obtain, a problem that previous authors -such asHorne & Ajmani (1969),Nethercot & Rockey (1971),Nethercot & Rockey (1972),Nethercot (1973),, Trahair (1993) -have avoided by using finite element methods to provide data around which approximate design formulae were fitted. As mentioned in §4.1 these formulae do not cover both variable restraint height and multiple elastic braces.The implicit relationship between the critical moment and restraint stiffness provided by Equations (4.41) and (A.41) is exploited in Chapter 6 to provide values around which design formulae are fitted. ...
... Structures such as portal frames [3], [4] , multi-storey buildings with continuous composite beams or arched bridges may also rely on U-frame action to provide restraint against buckling. ...
Ricardo Pimentel of the SCI discusses the consideration of U‐Frame action to restrain members susceptible to flexural and lateral torsional buckling according to Eurocodes.
Buckling phenomena frequently govern the design of steel members under compression or for elements partially compressed. To achieve a good compromise between steel tonnage and performance, discrete restraints along the compressed member (or along the compressed part of the member) can be used. However, for certain cases, introducing restraints as part of an orthodox bracing system is not feasible and designers must use other options to achieve a capable structural solution. The use of U-frame action offers this opportunity.
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... The results of many studies of the effects of elastic restraints on the elastic flexural-torsional buckling of beams have been summarized in the Handbook of Structural Stability (1971), Trahair and Nethercot (1984), Guide to Stability Design Criteria for Metal Structures (1988), and in Trahair (1993). The effects of central torsional restraints summarized in Fig. 2 have been studied by Taylor and Ojalvo (1966), Horne and Ajmani (1969), Nethercot (1973), Mutton and Trahair (1973), Milner (1975), and Medland (1980), all of whom considered beams under a number of different loading conditions. When a simply supported beam with equal and opposite end moments M and a central torsional restraint of stiffness ot.R: buckles laterally as shown in Fig. I, the restraint exerts a restraining moment on the beam, in which <l>u2 = central twist rotation. ...
The vertical deflections perpendicular to the plane of a horizontal beam curved in plan are coupled with its twist rotations, and its axial deflections are coupled with its horizontal radial deflections. Because of the first of these couplings, a horizontally curved beam subjected to vertical loading has both primary bending and torsion actions. In the nonlinear range, second-order couplings between the vertical and horizontal deflections and the twist rotations are developed, and the nonlinear behavior of the curved beam becomes more complicated. This paper studies the linear, neutral, and nonlinear equilibrium of elastic horizontally curved I-beams under vertical loading and develops a curved finite-element model for their analysis. It is found that when the included angle of a curved beam is small, the primary coupling is also small and bending is the major action. In this case, the nonlinear behavior is similar to the elastic flexural-torsional buckling of a straight beam. However, if the included angle of the curved beam is not small, the primary coupling becomes significant and both torsion and bending are major actions. In this case, nonlinear behavior develops very early and is quite different from the flexural-torsional buckling behavior of a straight beam.
... The results of many studies of the effects of elastic restraints on the elastic flexural-torsional buckling of beams have been summarized in the Handbook of Structural Stability (1971), Trahair and Nethercot (1984), Guide to Stability Design Criteria for Metal Structures (1988), and in Trahair (1993). The effects of central torsional restraints summarized in Fig. 2 have been studied by Taylor and Ojalvo (1966), Horne and Ajmani (1969), Nethercot (1973), Mutton and Trahair (1973), Milner (1975), and Medland (1980), all of whom considered beams under a number of different loading conditions. When a simply supported beam with equal and opposite end moments M and a central torsional restraint of stiffness ot.R: buckles laterally as shown in Fig. I, the restraint exerts a restraining moment on the beam, in which <l>u2 = central twist rotation. ...
It is well known that a central elastic torsional restraint restricts the lateral buckling shape of an elastic I-beam and increases the elastic flexural-torsional buckling moment However, the effect of torsional restraints on inelastic buckling has not been studied, and it is not known whether the limiting stiffness for elastic buckling can be applied to beams that buckle inelastically. This paper develops a finite-element model for the inelastic nonlinear flexural-torsional analysis of steel I-beams and uses it to investigate the effects of central elastic torsional restraints on the inelastic flexural-torsional buckling of steel I-beams. It is found that a central elastic torsional restraint increases the inelastic strength of the beam, but that the increase in the inelastic strength decreases as the beam's modified slenderness decreases, The limiting value of the stiffness of a central elastic torsional restraint at which the inelastic strength of the beam is equal to that of the corresponding beam with a rigid restraint is related to the modified slenderness of the beam. For a beam with a low modified slenderness that buckles inelastically the limiting restraint stiffness is much smaller than that for a beam that buckles elastically. For slender beams the moment transferred by a central torsional restraint of the limiting stiffness exceeds the strength design requirement for the restraint implied by Standards Australia's AS4100 as a result of excessive central twist rotation.
Torsional bracing of columns and beam-columns to prevent a lower torsional buckling was discussed. Adequately designed braces to prevent twist of the cross section were found to prevent the torsional mode of buckling. Finite element results were used to develop the stiffness and strength requirements for torsional bracing. Solutions for continuous torsional bracing were modified and simplified for application to discreet torsional bracing systems.
In the past decade structural research has concentrated on the design of tall buildings, and has resulted in more rational and economical design procedures for such structures. In contrast, research related specifically to the design of low buildings has lagged somewhat, even though these buildings form a large portion of the market for the steel industry. This paper is a state-of-the-art paper for light industrial buildings in structural steel. Topics covered are loads, economics of different layouts and framing schemes, design of flexural members, design of compression members, and resistance to lateral loads.
It is well known that a central elastic torsional restraint restricts the lateral buckling shape of an elastic I-beam and increases its elastic flexural-torsional buckling resistance. However, available information on the effects of torsional restraints on elastic buckling are either incomplete, or in some cases inaccurate. This paper investigates the effects of moment distribution and load height on the elastic flexural-torsional buckling of beams with central torsional restraints. The effects of off-center and continuous restraints are also studied, and design approximations and procedures are developed.
A rational model for predicting the elastic buckling load of thin-walled Isection columns, restrained fully against translation and elastically against twist at one flange, is presented. The energy method used results in a third-order eigenproblem. The numerical method is illustrated by considering the elastic buckling of doubly-symmetric and monosymmetric columns with various degrees of elastic twist restraint. It is shown that the buckling mode is of a lateral-distortional type. The accuracy of the U -frame buckling model used in through girders for predicting the lateral-distortional buckling load is assessed in the study.
Many column bracing details employed in steel construction do not prevent twist, and subsequently, torsional buckling may control the column capacity. This mode of buckling is not adequately considered in design codes and is often overlooked. This paper documents a finite-element investigation of the torsional buckling behavior of columns with lateral bracing located at different points on the cross section. The location of the lateral bracing on the cross section has a significant effect on torsional buckling. Equations are developed for the stiffness and strength requirements of bracing to control torsional buckling. Details for torsional bracing are discussed and presented. A connection detail must be provided between the column and the brace that controls cross-sectional distortion. A design example illustrates the use of the bracing recommendations.
General closed-form solutions of distributional differential equations of the elastic flexural-torsional buckling of monosymmetric, constant-section, thin-walled members fitted with discrete elastic bracings over their length and subjected to compression and bending in their symmetry plane have been worked out. The bracings confine the warping and displacements of the intermediate sections of the members from their symmetry plane. Besides the general case, members with a prescribed axis of rotation are considered. Particular solutions of spatial stability problems for members optionally supported at their ends have been worked out on the basis of the general solutions. The practical application of the presented solutions, incorporated into appropriate computer programs, is illustrated with numerical examples in which the effect of the particular bracings on the critical load of the flexural-torsional buckling of columns or that of the lateral buckling of steel structural beams is investigated. The theoretical solutions have been validated experimentally by the author's own tests on laterally and longitudinally braced steel columns and by Milner's tests on laterally braced beams [24].
The Wagner-Kappus theory of torsion is applied to an axially loaded column attached at finite intervals to sheeting rails and shear-stiff cladding. It is shown that the restraint may be treated as continuous except in the most exceptional conditions, thus bringing the phenomenon within the range of much previous analysis and facilitating its extension to the more complex problems of eccentricity about one or both axes and various forms of end-condition.
A solution is derived for the elastic buckling load of a symmetrical I-section member subjected to an axial thrust applied
through the centroid combined with unequal terminal bending moments acting in the plane of the web. An energy method is employed,
using an estimate of the deflected form obtained by a method of successive approximation. The solution may be obtained to
any required degree of accuracy. Approximate formulae which always give conservative results are also derived.