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Journal of Constructional Steel Research 64 (2008) 725–731www.elsevier.com/locate/jcsr

Torsional and exural torsional buckling — A study on laterallyrestrained I-sections

Andreas Taras , Richard Greiner Institute for Steel Structures and Shell Structures, Graz University of Technology, Graz, Austria

Received 26 February 2007; accepted 18 January 2008

Abstract

Torsional and torsional–exural buckling of columns under axial compression – as dened in the Eurocode 3-1-1(2005) – may practicallyoccur in axially compressed I-sections, which are laterally restrained so that weak axis exural buckling is prevented and buckling is initiatedby torsional deformation about the axis of lateral restraint. The present study is focussed on the development and representation of specicbuckling curves for this type of buckling mode. This is based on numerical simulations taking into account material nonlinearities as wellas geometric imperfections and residual stresses. The study not only illustrates the typical buckling curves for torsional buckling, but alsothe transition from torsional buckling to main axis exural buckling when the column length increases. It also shows that the use of weak-axis buckling coefcients for torsional buckling – as given in the rules of Eurocode 3-1-1 – may lead to rather conservative results inmany cases.c 2008 Elsevier Ltd. All rights reserved.

Keywords: Torsional buckling; Buckling curves; Member stability; Column design; Eurocode 3; Laterally restrained beams

1. Introduction

Columns in the shape of I- or H-proles are frequentlyrestrained in the lateral direction in order to prevent bucklingabout the weak axis. The buckling mode is then describedby a rotation of the cross-section about the axis of lateralrestraint. Depending on the location of the restraints the columnmay react by a mere torsional deformation, or by a combinedtorsional and exural deformation.

The problem of the ultimate strength and buckling behaviourof laterally supported beam–columns is well known, andremarkable fundamental research work on the topic hasalready been published by Horne and his co-workers quitesome years ago [ 1,2]. These early studies mainly focused onthe determination of maximum slendernesses (or minimumsupport spacing and stiffness) for a fully plastic design of beam–columns under combined axial compression and in-planebending moment using the plastic hinge theory. They constitutethe theoretical background for the so-called “stable length”method found in Annex BB.3 of EC 3-1-1 [ 3], see also [4].

Corresponding author. Tel.: +43 316 873 6203; fax: +43 316 873 6707. E-mail address: [email protected] (A. Taras).

Nevertheless, the basic case of torsional and torsional–exural buckling, seen as a member instability phenomenoncomparable to exural or lateral torsional buckling and express-ible in the general form of a buckling curve χ = N /( A · f y)= f (λ) , seems to be treated with a certain degree of neglect inmany international design codes. In the Eurocode EC 3-1-1 [3],torsional and torsional–exural buckling is considered to be-have somehow similarly to out-of-plane exural buckling, withthe λ -dependent reduction factor χ being equal to that for out-of-plane exural buckling.

The present study is therefore aimed at developing andrepresenting specic buckling curves of members with double-

symmetric I- and H-sections with lateral restraints underpure axial compression, which tend to fail by torsional ortorsional–exural buckling. Thereby, single span members areconsidered and the lateral restraints are assumed to be appliedcontinuously along the span ( Fig. 1). Numerical simulationshave been carried out on the basis of geometrically andmaterially nonlinear analyses with imperfections (GMNIA).

The results of the GMNIA calculations are illustrated in theform of buckling curves for different cross-sectional shapes,and different positions of the lateral restraints in relation tothe centroid of the sections. A considerable inuence of the

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726 A. Taras, R. Greiner / Journal of Constructional Steel Research 64 (2008) 725–731

Fig. 1. Single span members under axial compression with lateral restraints. Geometric imperfections and residual stresses.

torsional rigidity of the cross-sections on the load carryingbehaviour could be seen.

The numerically obtained buckling curves are also comparedwith the rules given in Eurocode 3-1-1, leading to the insightthat, depending on the section shape and the position of the intermediate lateral support, the given rules may contain

signicant conservatism.

2. Critical torsional and torsional–exural buckling loads

Before the ultimate strength of members, subjected totorsional or torsional–exural buckling, can be representedin the familiar form of buckling curves, it is necessary todetermine the critical Euler buckling loads N cr for these casesof member instability. The dimensionless slenderness λ T orλ TF for torsional or torsional–exural buckling can then becalculated as the square root of the ratio N pl / N cr = A · f y/ N cras customary.

The critical Euler buckling loads N cr, T

or Ncr,TF

fortorsional and torsional–exural buckling of members withdouble-symmetric sections and enforced axis of rotation arewell known (see e.g. [5] or [6]) and can be expressed by thefollowing set of equations:

N cr,TF = N cr, z ·c2 + d 2

i 2 p + d 2

(1)

where

N cr, z =π 2 E I z

L2 (2)

i p = I p/ A = ( I

y + I z)/ A (3)

c = I w I z+

G I T

N cr, z(4)

d is the distance of the lateral restraint from the centre of gravity, see Fig. 1.

For practical use in design, the relationship between thebuckling loads N cr, T and N cr, z can be expressed equivalentlyto the procedure used in BS 5950 [ 7] by referring to theslenderness coefcients λ TF and λ z and combining themthrough a factor k T :

λ TF = k T λ z (5)

where k T is a factor taking account of the torsional rigidity of the section.

In [8], k T was evaluated for different cross section shapesand represented in form of diagrams. The diagram in Fig. 2,taken from [8], was developed under the geometric assumptionthat the thickness of the web is half the thickness of the

anges. This assumption was necessary in order to eliminatethe otherwise free parameter of web thickness from thechosen form of representation. By comparing the results of thecalculations carried out under these assumptions with resultsof calculations for actual rolled beam sections (IPE500 andHEB300), it could be shown that the parameter which bestreproduces the benecial effect of torsional rigidity on thecritical buckling behaviour of double symmetric I- and H-sections is the ratio h / t f l of section depth to ange thickness.The assumption concerning the web thickness was shownto yield slightly higher values of slenderness than the onescalculated for actual rolled beam sections and is thereforesomewhat on the safe side. Hence, the application of theprocedure illustrated in Fig. 2 allows for a simple and safeevaluation of critical torsional buckling loads over the moreaccessible buckling loads for out-of-plane exural instability.

3. Numerical simulations of the imperfect structure

The numerical simulations were carried out as geometricallyand materially non-linear analyses using the softwareABAQUS [9], taking into account imperfections in the formof residual stresses and initial geometric out-of-straightness.The boundary conditions of the member as well as the chosenshapes of imperfection are shown in Fig. 1. Only single-spanmembers with end fork boundary conditions and intermediatelateral supports were considered, i.e. members with in-plane(vertical), out-of-plane (lateral) and torsional restraints at theends and solely lateral restraints along the free span.

The residual stresses were assumed to vary linearly overthe single cross-section components. The maximum value of residual stress depends on the type of rolled cross-sectionand is expressed as a fraction of the yield strength f y.The stress–strain relationship was considered to follow anelastic–plastic path without strain hardening. All calculationswere conducted for mild steel, assuming a yield strength of f y = 235 N/ mm2. These assumptions closely follow theprocedure chosen by Beer and Schulz [ 10] for the development

of the European buckling curves for exural buckling.



A. Taras, R. Greiner / Journal of Constructional Steel Research 64 (2008) 725–731 727

Fig. 2. Critical loads for torsional and torsional exural buckling expressed as functions of λ z. (d = distance between the centroid axis and the position of the lateralrestraints).

Fig. 3. Shell model of the double-symmetric beam with end-fork boundary conditions and results of a GMNIA calculation.

The initial geometric imperfections were assumed to bedistributed in a sinusoidal shape along the length of the member,having their maximum value at mid-span. The possibility thatthe imposition of the restraints would inuence the initial shapeof the member was ignored for the purposes of this study.Both initial rotations and deections were considered. Theamplitude of these imperfections, as dened in Fig. 1, wasassumed to be equal to L/ 1000. Only global imperfectionswere considered, i.e. no imperfections involving cross-sectiondistortion were taken into consideration. Considering theseimperfections, as was skilfully performed in [ 11] for compactsections – however under a bending moment – with continuousrestraints along the tension ange, would additionally requiretaking into account the effects of web plate distortion. Forthe sake of a clearer representation of the effects inherent tothe beam theory alone, these effects were omitted from theanalyses.

Linear two-node beam elements with 7 degrees of freedom

per node were chosen for carrying out the parametric studies

presented in this paper. Since these elements do not takeinto account the contribution of shear stresses in plasticity,additional calculations were conducted using four-node linearshell elements with six degrees of freedom per node and nitestrain formulation to model the beam cross-section, see Fig. 3.These calculations were conducted in order to obtain evidenceof the small, or negligible, effect of torsional shear in plasticityfor the cases at hand.

In all calculations, the llet was omitted from the cross-section denition. This simplication can in some casesinuence the torsional response of a member. Nevertheless,it was shown in [12] that the llets may be ignored forthe determination of buckling curves, as the effect of thellets enters both the slenderness λ and the buckling capacityexpressed in terms of χ . The omission of the llets is therefore justiable in the light of the objective of this study, which isthe general representation of the ultimate strength of I-sectionproles subjected to torsional buckling in the familiar terms of

a buckling curve.




Fig. 4. Critical and ultimate (GMNIA) buckling loads, IPE 500 section, plotted over L .

4. Buckling behaviour for lateral restraints along thecentroid axis of the section

The rst case considered is the torsional buckling behaviourof members which are laterally restrained along the centroidaxis, which can be found in practice with columns supportedby centrally located bracing. Flexural buckling about the strongaxis of the section was either prevented or allowed for in theGMNIA calculations. This case can be found in practice whencolumns are braced laterally to shorten the exural bucklinglength for weak-axis buckling.

Fig. 4a shows the ratio between the critical Euler bucklingload N cr and the plastic resistance to axial compression N plfor an IPE 500 section of varying length. For this type of prole and boundary condition, torsional buckling yields lowercritical buckling loads than strong-axis exural buckling up toa length of approximately 26 m. This means that, within therealm of practicality, torsional buckling will be the instabilityphenomenon yielding the lowest critical buckling load for thegiven boundary condition.

This type of representation also illustrates a typical featureof the torsional buckling behaviour; with increasing length of the member, the critical torsional buckling load tends towardsa limit value. This is due to the fact that the warping rigidity

vanishes with increasing length, while the term in Eq. (1)related to the torsional stiffness GI T is independent of length.For the case of lateral restraint in the centroid axis, Eq. (1) canbe rewritten as follows, Eq. (6)

N cr, T = N cr, z · I w I z + G I T

N cr, z

I p/ A. (6)

For a member of innite length, the critical buckling load forweak axis exural buckling assumes a value of zero. The criticaltorsional buckling load in this case is given by Eq. (7)

N cr

,T

, L

=∞ =G I T

I p/ A. (7)

Fig. 4 shows the results of GMNIA calculations for an IPE500 section. Again, the results are plotted over the physicallength of the member. The curve representing the strong-axisexural buckling behaviour intersects the curve for torsionalbuckling at a length of approximately 23 m. Again, it can bestated that, within the realm of practicality, torsional bucklingwill be dominant for the given boundary condition. The ultimatebuckling resistance for the case of pure torsional buckling alsoreaches a limit value, which lies only slightly underneath thecritical value. Fig. 4b also features a third plot, which representsa buckling curve as it is calculated for the combined presence of a rotational imperfection and initial bow deection, both with

an amplitude of L / 1000. It could be shown that this combinedimperfection yields buckling reduction factors that lie wellbelow both the pure torsional and pure exural buckling curves.

The results of two calculations using shell elements to modelbeams of 10 and 20 m of length respectively are also plottedin Fig. 4b. The correspondence between the results of beamand shell element calculations is very clear in these two cases,with shell element calculations yielding slightly higher ultimatebuckling resistances. As has already been stated in Section 3,these calculations were conducted in order to obtain evidenceof the small, or negligible, effect of torsional shear in plasticity.

An interesting representation of the torsional bucklingbehaviour can be achieved by plotting the results of both thecritical and the GMNIA buckling loads over the dimensionlessslenderness λ , dened according to Eq. (8) as the square root of the quotient between the plastic resistance N pl = A · f y and thecritical buckling load N cr .

λ = N pl

N cr= A · f y

N cr. (8)

In Fig. 5a, the buckling loads are plotted over λ T , theslenderness for torsional buckling. The previously explainedcharacteristic of torsional buckling, i.e. the tendency of thebuckling loads to approach a certain non-zero value with

increasing length, causes the buckling curve to stop at a certain




Fig. 5. Critical and ultimate (GMNIA) buckling loads, IPE 500 section, plotted over λ T and λ y.

value of λ T . This is quite untypical for buckling curves, whichgenerally tend towards innity. Thus the numerically calculatedbuckling curve shown in Fig. 5a is interrupted at a certain valueof λ T for the simple reason that there is no value of λ T beyondthis point.

By combining Eqs. (8) and (7), we can calculate this limitslenderness as follows, Eq. (9):

λ T , L=∞ = N pl

N cr= f y · I p

G I T . (9)

The gure also shows the relevant buckling curve for a designcheck against torsional buckling according to EC3-1-1 [3].In the code’s provisions, torsional buckling is equated in itsbehaviour to weak-axis exural buckling and the bucklingcurve used for weak-axis buckling checks is also recommendedfor checks against torsional buckling. Thus, in the given caseof an IPE 500 and steel grade S235, the buckling curve baccording to EC3-1-1 applies. The discrepancy between thebuckling code according to the code and the numericallydetermined curve is clearly visible.

In Fig. 5b, results of GMNIA calculations for the sameIPE 500 beam are plotted over λ y, the slenderness for exuralbuckling about the strong axis. In these calculations, strongaxis buckling was not prevented by supports, thus the resultingnumerical buckling curve shows the transition from torsionalto strong axis exural buckling. Since λ y is a quantity thatincreases linearly with the length L of the member, thisrepresentation is quite similar to the one shown in Fig. 4b.The values of the relevant buckling curve for torsional andstrong-axis exural buckling according to EC3 are also plottedover λ y. While the numerical and the code buckling curveoverlap in the exural buckling range, the buckling curvesdiverge considerably for lengths dominated by the torsionalbuckling mode, with the numerical GMNIA buckling curveyielding signicantly higher buckling resistances than the code

provision.

The benecial inuence of high torsional rigidity on thebuckling behaviour of compressed members is illustratedin Fig. 6. In this gure, the previously described form of representation is used to show the torsional buckling behaviourof different types of I- and H- sections, ranging from aquite stocky HEB 200 to a slender (i.e. slender for torsionalbehaviour) welded I-section of 1000 mm height and smallanges of 100 mm width and 25 mm thickness.

Fig. 6a illustrates that members with HEB 200 sections,which are supported so to prevent strong- and weak-axisexural buckling, can be loaded up to their plastic resistancewithout incurring any torsional instability phenomenon.Furthermore, in the case of IPE 200 proles and lateralrestraints in the centroid axis, the reduction of the ultimate loaddue to torsional instability is all but negligible for membersof realistic physical length. On the other hand, the slenderwelded section, of which the geometric shape was obviouslychosen for the purpose of illustration, is quite susceptible totorsional buckling, producing similar reduction factors as weak-axis exural buckling. All buckling curves have in commonthat they stop at a certain level when plotted over the relevantslenderness λ T .

The representation of the buckling curves over theslenderness λ y in Fig. 6b further claries the behaviourof members with varying torsional rigidity which are onlyrestrained laterally. For a member with an IPE 200 section,torsional buckling is only relevant for a quite restricted lengthrange, and exural buckling about the strong axis is usually thepredominant stability criterion. In the case of a slender weldedsection, torsional buckling is the determining stability criterionfor a large length range. The ultimate strength of such a memberunder pure axial compression drops almost immediately to alevel that is considerably lower than the plastic resistance.

Fig. 6b gives further evidence that the provisions for thedesign check against torsional buckling in EC 3-1-1 areconservative for stocky proles of high torsional rigidity, whilethey are fairly accurate, albeit still on the safe site, in the case

of slender sections.




Fig. 6. Ultimate (GMNIA) buckling loads for various sections plotted over λ T and λ y.

Fig. 7. Ultimate (GMNIA) buckling loads for various sections plotted over λ TF and λ y.

5. Buckling behaviour for lateral restraints along the angeof the section

The second case considered is the buckling behaviour underaxial compression of members which are laterally restrained atthe height of one of the two anges. This case can be foundin practice when columns are braced laterally by secondarystructural members (such as rail beams used to support wallcladding) running on the outside of the prole. The shape of theinstability mode that occurs in this case, and features torsionaldeformations, is always coupled with a lateral deection of thecentroid axis of the compressed member. Hence, this type of buckling is referred to as torsional–exural buckling.

A number of studies have considered this problempreviously, mainly in relationship to the combined action of bending and axial force and the possibility of applying plastichinge design; see the basic work by Horne [ 1,2].

The Euler critical torsional–exural buckling load is

expressed by Eq. (1) and represented in Fig. 2. No signicant

qualitative difference exists between torsional–exural andtorsional buckling. Again, the critical load approaches a certainlimit value with increasing member length. Quantitatively, thebuckling loads for this type of instability are generally lowerthan the ones for pure torsional buckling. The same was shownto be true for the ultimate buckling loads calculated by GMNIAnumerical simulations.

Again, the benecial inuence of high torsional rigidityon the buckling behaviour of compressed members can beobserved. In Fig. 7a, the ultimate torsional buckling resistance(expressed by the ratio χ = N R,k / N pl) of different types of I-and H- sections is plotted over the relevant slenderness λ TF .

In the case of the stocky HEB 200 prole, no lateralinstability occurs, and the plastic load carrying capacity can beachieved if strong-axis exural buckling is prevented.

In the case of IPE proles, a strong decrease in loadcarrying capacity is recorded. For an IPE 500 prole, theGMNIA results lie close to the relevant buckling curve for

torsional and torsional exural buckling according to EC3-1-




1, and fall slightly underneath this curve in the last portion of the plot.

The physical length does not enter the parameter λ TF in alinear fashion. The actual length range of members is thereforebetter represented in a plot over λ y. Fig. 7b illustrates the resultsof GMNIA calculations for torsional–exural buckling when

they are plotted over λ y. This representation better illustratesthat, in the case of an IPE 500 prole, the buckling curveaccording to EC3-1-1 lies somewhat higher for a large rangeof physical member lengths. Flexural buckling about the strongaxis is only relevant for lengths corresponding to a value of λ yabove 2.5.

6. Summary and conclusion

In this paper, the torsional and torsional–exural bucklingbehaviour under pure axial compression of single spanmembers with double-symmetric cross-section was analysedby means of numerical simulations. These simulations takeinto account imperfections in form of residual stresses andgeometric out-of-straightness.

A typical feature of this type of instability phenomenon isthe fact that both the critical and the ultimate buckling loadsapproach a certain non-zero value with increasing memberlength. This is due to the independence of the torsional rigidityto the length of the member. A remarkable implication of thisphenomenon is the fact that buckling curves for torsional andtorsional–exural buckling stop at a certain limit slenderness.

The design checks according to EC3-1-1 were veriedthrough the results of the numerical calculations presentedin this study. In the case of pure torsional buckling, i.e. the

case of members supported laterally at the height of thecentroid axis, it was shown that the provisions of EC3-1-1are generally conservative. Specically, the study conrms thatstocky sections with h / b ratios around unity do not fail intorsional buckling at all.

When the lateral restraint is placed at the height of one of the two anges, the lateral deformation can occur more freely,larger rotation can take place and the critical, as well as theultimate, buckling loads fall compared to the pure torsionalbuckling case. For this torsional–exural instability mode, itwas shown that the provisions in EC3-1-1, which link thetorsional and torsional–exural buckling resistance to the weak-axis exural buckling resistance, are more accurate in somecases.

In Fig. 8, the effect on the buckling strength of compressedmembers of the transition from pure torsional buckling totorsional–exural buckling with eccentric lateral restraint isillustrated. With increasing eccentricity of the lateral restraint,the buckling resistance decreases until it approaches thebuckling curve of the EC3-1-1 provisions – valid for lateralexural buckling about the weak axis –, falling underneath itin some cases.

As a concluding remark, it can be pointed out that itwas not possible to nd a single buckling curve in theform of χ = f (λ) which appropriately describes the

torsional and torsional–exural buckling behaviour under pure

Fig. 8. Ultimate (GMNIA) buckling loads for IPE 500 sections braced laterallyat different heights.

axial compression of the double-symmetric I- and H-sectionsconsidered in this study. This is presumably caused by acertain insufciency, or inadequacy, of the parameters whichare commonly considered in design practice to describe thisbehaviour. Since the practical relevance of this buckling modefor certain ranges of member slenderness could clearly bedisplayed, further investigations should aim at enhancing theunderstanding of the inuence of such parameters as torsionalrigidity and plastic resistance for coupled lateral and torsionaldeections.

References

[1] Horne MR, Ajmani JL. Design of columns restrained by side-rails. TheStructural Engineer 1971;49(8):339–45.

[2] Horne MR, Ajmani JL. The post-buckling behaviour of laterally restrainedcolumns. The Structural Engineer 1971;49(8):346–52.

[3] EN 1993-1-1, 2005. Eurocode 3 — Design of steel structures. Part 1-1,General rules and rules for buildings. CEN Brussels; 2005.

[4] King C. Member stability at plastic hinges. In: Eurosteel 2005 conferenceproceedings. vol. B. p. 3 / 25–32.

[5] Timoshenko SP, Gere JM. Theory of elastic stability. In: Engineeringsocieties monographs. New York (Toronto, London): McGraw-Hill Book Company Inc; 1961.

[6] Bleich F. Buckling strength of metal structures. In: Engineering societiesmonographs. New York (Toronto, London): McGraw-Hill Book CompanyInc.; 1952.

[7] BS 5950-1:2000. Structural use of steelwork in building — Part 1: Codeof practice for design—Rolled and welded sections. BSi; 2000.

[8] Greiner R, Ofner R, Salzgeber G. LT-buckling of beam–columns.ECCS-Validation Group. Report 5. September 1999. Seehttp://www.shf.tugraz.at/report.html .

[9] Abaqus. Version 6.5. Abaqus Inc.[10] Beer H, Schulz G. Bases th eoriques des courbes europ´eennes de

ambement. Construction M etallique 1970;3:37–57.[11] Bradford MA. Strength of compact steel beams with partial restraints.

Journal of Constructional Steel Research 53, Elsevier 2000;183–200.[12] Kaim P. Spatial buckling behaviour of steel members under bend-

ing and compression. Dissertation. Institute for Steel Structuresand Shell Structures. Graz University of Technology; 2004. See

http://www.shf.tugraz.at/thesis.html .

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