Greiner Lindner EC3 PM Interaction

Journal of Constructional Steel Research 62 (2006) 757–770www.elsevier.com/locate/jcsr

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Interaction formulae for members subjected to bending and axialcompression in EUROCODE 3—the Method 2 approach

R. Greinera,∗, J. Lindnerb

aTU Graz, Institute for Steel Structures and Shell Structures, Lessingstr. 25, 8010 Graz, Austriab Engineering office, Bismarckallee 4, 14193 Berlin, Germany

Received 9 August 2005; accepted 30 November 2005

Abstract

The final version of EN1993-1-1, EUROCODE 3 [EN1993-1-1. Eurocode 3. Design of steel structures, general rules and rules for b2005] for Steel Structures provides two alternatives for the buckling check of members subjected to axial compression and bending byformulae, which are called there Method 1 and Method 2. This paper presents the characteristics, the background and the use of Meanalogous presentation of Method 1 hasalready been given in [Boissonnade N, Jaspart J-P, Muzeau J-P, Villette M. New Interaction formulae fobeam-columns in Eurocode 3. The French-Belgian approach. Journal of Constructional Steel Research 2004;60;421–31].

The Method 2 formulae have been derived on the basis of the general format of the interaction concept of existing codes, e.g. the Ehowever with advanced numerical background and consistent classification of the buckling modes. In this respect new improved interactwere developed from a wide scope of numerical simulations and the concept of the formulae was focussed distinctly on describing thof in-plane and out-of-plane buckling for members susceptible to fail either in flexural buckling or in lateral–torsional buckling. As result tsets of formulae are provided, which each cover a clear scope of physical member behaviour. Hereby, the specific effects of intermedrestraints—as often found in steel structures—have also been included.

The Method 2 formulae aim at providing buckling rules with compact simplified interaction factors and transparent application for scases.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Interaction formulae; Buckling of beam–columns; EUROCODE 3

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1. Introduction

In the process of the conversion of the ENV-versionthe EUROCODE 3 [2] to the final EN-version [1] basicdevelopments were carried out by Technical Committee(TC8) of ECCS, which resulted in new buckling rules formembers subjected to axial compression and bending. Thiscaused by some physical inconsistencies and overconservfound in the ENV-rules [4], which could lead to considerablunderestimation of the capacity of beam–columns in cercases. The use of modern computer techniques andprograms opened the possibility of simulating the geometricaand materially nonlinear buckling behaviour of imperfectmembers in much wider parametric scope than before [4,5]. On

∗ Corresponding author. Tel.: +43 316 873 6200; fax: +43 316 873 6707E-mail address: [email protected](R. Greiner).

0143-974X/$ - see front matterc© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2005.11.018

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this basis the interaction factors could be directly recalculatefrom the numericallimit load results and the effect of differenparameters be consistently identified. The final definition of tnew rules was then based on calibration with the existing tesresults and the numerical simulations.

The elaboration of the Method 2-rules therefore presthe consequent development of the former work on interacformulae by use of the opportunities of the new computechniques. At present they cover double-symmetric crsections in EC3-1-1 [1], however an extension to monosymmetric sections has already been made in the meantim6].In this spirit of further development the traditional interactioformulae were maintained in format and notation as farpossible in order to facilitate the understanding of the chanand the practical application for the user. However, in a numof points new formulations were necessary, in particulaprovide possible further developments.

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http://dx.doi.org/10.1016/j.jcsr.2005.11.018

758 R. Greiner, J. Lindner / Journal of Constructional Steel Research 62 (2006) 757–770

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Fig. 1. Characteristics of Method 2 of EC3-1-1.

The objective of this paper is to present the backgrounthe buckling formulae of Method 2 in EC3-1-1, which are in thtradition of the simplified design rules of many existing codA short but not detailed enough explanation to this backgrowas already given in [16].

EC3-1-1 will provide further methods for the stability cheof members (Method 1 and General Method), which are mcomplex and are not dealt with in this paper.

2. Characteristics of the buckling rules of Method 2 ofEC3-1-1

The buckling rules of Method 2 are directed to coverfollowing three main characteristics of the structural behaviof steel members (Fig. 1):

• Cross-section shapes: Depending on the shape of thcross-section members subjected to axial compressionbending may behave susceptible to torsional deformator not. Therefore, specific sets of buckling formulaeprovided for torsionally stiff members (hollow sectionstorsionally restrained I-sections) and torsionally flexibmembers (I and H-sections).

• Buckling modes:In general spatial buckling deformatioof members subjected to axial compression and bendintraditionally split up into the two buckling modes about ty-axis and about thez-axis. In each set of formulae the firone describes the strong-axis mode and the second onweak-axis mode. In this way the specific physical behavis transparently connected with the design process (larestraining etc.).

• Intermediate lateral restraints: The buckling formulae weprimarily derived for the free beam–column with end-foconditions. However, at the same time the applicationthe practically frequent cases of members with intermedlateral restraints has also been provided. This requireto differentiate between the buckling lengths and momdiagrams along the span and along a segmental part betthe lateral restraints.

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3. Numerical calculations and statistical evaluation

The basis of the new developments of the buling interaction formulae was the numerical simulatiof the elastic–plastic buckling behaviour of single-spbeam–columns, which accounts for geometric bow impertions, residual stresses and a linear-elastic–plastic materiawithout hardening effects. The calculations, called geomecally, materially nonlinear analyses of the imperfect structur[GMNIA], were performed by a computer module for autmatic parameter variation and data evaluation, which allowus to cover a wide scope of member and load parameters (a25000 cases) [7]. The analyses were carried out by the FProgramme ABAQUS [8] using beam elements. Verificationwere made by comparison of the numerical simulation wbuckling tests and other computer programs [9].

The resulting limit load data were used to recalculateinteraction factors on the basis of the proposed interacequations and afterwards to develop simplified formulae forthem. These formulae were calibrated with the low numbeavailable buckling tests and with the high number of numeriresults using statistical evaluation [10–12].

4. Concept of interaction formulae for N + My

As already explained in chapter2 two sets of bucklingformulae are provided, which each describe the buckbehaviour of either torsionally stiff or torsionally flexible stesections (Fig. 2).

In the first case they concern flexural buckling, in the secone lateral–torsional buckling. The first formula of eachis related to buckling about they-axis and the second onebuckling about thez-axis.

This clear structure may assist the designer to connecresults of the buckling check with structural provisions. In trespect it has already been said, that the buckling formare appropriate for both free, single-span members asas those supported laterally by intermediate restraints. Inpresent state such restraints are expected tobe full restraints,which means that in the case of flexural buckling a pure latrestraint would be sufficient, while the case of LT-bucklingwould require lateral plus torsional restraint. Different kindsrestraints are under investigation.

The four interaction formulae are given inFig. 3 for class 1and 2 sections (analogous formulae for class 3 and 4 secalso exist in EC3-1-1). In order to accentuate the systemstructure of the two sets of design formulae for practical usefour formulae were numbered consecutively by(1) to (4) andmarked by boxes.

Eqs. (1) and (2) describe flexural buckling of torsionallstiff members and Eqs.(3) and(4) LT-buckling for torsionallyflexible members. In the case of flexural buckling aboutweak axis a simplified approach(2a)is allowed, which is basedjust on the axial compression term alone This traditional rule—used in many existing codes—has been maintained in EC3uniaxial bending and compression.

The new interaction formulae of Method 2 use the followifactors:

R. Greiner, J. Lindner / Journal of Constructional Steel Research 62 (2006) 757–770 759

Fig. 2. Concept of interaction formulae of Method 2.

Fig. 3. Interaction formulae of Method 2, EC3-1-1.

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• Interaction factorsky , kz, kLT

• Equivalent uniform moment factorsCmy , Cmz , CmLT.

In addition the buckling reduction factorsχy , χz , χLT for thebasic buckling modes under pure compression or pure bendinare necessary. They cover the cases of separate flexural bucaused by pure compression or LT-buckling by pure bendin

In the following the derivation of these terms is presentThe interaction factorsky and kz apply to the uniformdistribution of the bending momentsMy andMz , while thekLT-factors include the non-uniformity of the moment diagramsMy . This non-uniformity is accounted for byCm-factors, whichare based on the widely used concept of Austin [13], i.e. to takethe constant moment (not the sinusoidal moment distributioas reference, so thatCm = 1.0 holds for uniform moments anany other moment diagram leads toCm -values lower than 1.0In this respect this definition ofCm is different from theβM -factors of the ENV-version.

It may also be noted, that the coefficientsχy and χz

(European column buckling curves) are the same as inENV-version, but that the coefficientsχLT have been furthe

ling

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developed for the EN-version. The latter concerns mainlyclassification of the LT-buckling curves connected with thtorsional capacity of the cross-section(h/b ≥ or < 2) andthe inclusion of the beneficial effect of non-uniform momediagrams by the modified coefficientχLT,mod.

It has to be pointed out, that the stability check by thinteraction formulae(1) to (4) has to be supplemented bthe cross-section check at the member ends, if the bendingmoments are non-uniform along the span and the mommaximum appears at the member end. This means, thacross-section capacity at the end section may govern, if thstability effect in the span is low.

5. Members not susceptible to torsional deformations—Class 1 and 2-sections

5.1. General buckling behaviour

If torsionally stiff members are present the bucklinbehaviour leads to flexural buckling. This is illustrated by thexample of an RHS 200/100/10 of a beam–column of 4


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Fig. 4. Flexural buckling of torsionally stiff members.

length under axial compression and uniform bending momMy in the form of an interaction diagram (Fig. 4). In the case ofsufficient intermediate lateral restraint in-plane buckling is thegoverning buckling mode, which is described by the interactcurvey–y and is approximated by Eq.(1). For anunrestrainedmember, which is free to deflect spatially along its span, oof-plane buckling will occur if N/Npl is larger than 0.4. Thisis described by the branchz–z of the interaction curve and isapproximated by Eq.(2). The approximation by Eq.(2a)is alsopresented. The key-parameterof the interaction behaviour isthe interaction factork, whose development is explained in thfollowing for the different buckling modes.

As already said before, the interaction factors are theresult of recalculations from a large scope of limit load-daresulting from GMNIA-calculations. The parameters usedthis recalculation were taken from the general derivation of thebuckling interaction formula on basis of the second order theory(see e.g. [4]), which showsthat the three parametersλy , χy

and NNpl

determine the interaction behaviour. The interactio

factors were finally expressed byλy andny , the latter of whichconnects N

Nplandχy as in the term for buckling under pure axi

compression.

5.2. In-plane buckling under N + My

The design buckling formulae are as follows (Fig. 5):

NEd

χy · Npl,Rd+ ky

Cmy · My,Ed

Mpl,y,Rd≤ 1 (1)

ky = 1 + (λy − 0.2

) · ny ≤ 1 + 0.8 · ny (5)

ny = NEd

χy Npl,Rd(6)

Cmy = 0.6 + 0.4ψ ≥ 0.4 (seeFig. 5and alsoTable 1). (7)

The recalculations of theky-factor from GMNIA-resultswere made on basis of(Eq. (1) for different cross sectionshapes. Examples are shown inFig. 6 for an IPE and an RHS.They indicate a certain influenceof the cross-section shapehowever it turned out to be of minor effect on the end ressothat it could be ignored as a design parameter further on.

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Fig. 5. Design diagrams for in-plane buckling underMy .

The form of theky-curves illustrates the physical behavioof members with plastic cross-section:

At λy = 0 they express the plastic cross-section interactof N + My .

With increasingλy the second-order effect causes the growof the bending moment and by that the rise of theky-factor.

Approximately atλy ∼= 1.0 the point is reached, whera further increase ofky is not needed, since further stabilieffects are covered by theχy in the term of axial compression

Forλy > 1.0 theky-values remain at approximately constalevel, which may beexplained by the fact, that the membin this slenderness range behaves increasingly elastically, atherefore, the effect ofMpl,y used in Eq.(1) needs to becounter-compensated byky (this effect is obviously morepronounced in RHS, than in IPE-section, as indicated bydiagrams inFig. 6).

Theky-curves have been transformed into a formula, whis given by Eq. (5). The bi-linear form of this formula, witha kink at λy = 1.0 (seeFig. 5) has been chosen with respeto user-friendliness. This means that related toλy ≥ or < 1.0just one part of Eq.(5) needs to be determined. It may albe noted that the differences of theky-curves in the rangeof λy > 1.0 are of just minor effect on the design, becauthe differences mainly occur at high values ofny , where themoment terms are accordingly small, so thatky has very littleinfluence.

The ky-factors of the ENV-version are also given inFig. 6.Compared with the newky-factors of Fig. 5 the economicalimprovement is obvious.

Results for examples shown as interaction diagramsdifferent forms of moment distributions are illustrated inFig. 7.The results of the new formulae are compared withGMNIA-results and those of theENV-version. The improvedapproximation in the cases (a) and (b) is obvious,differences in case (c) are due to the conservatism of the Auformula for bilinear moment diagrams (see chapter5.6).


Fig. 6. Interaction factorky .

Fig. 7. Comparison of in-plane flexural buckling.

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5.3. Out-of-plane buckling under N + My

The design buckling formula is as follows:

NEd

χz · Npl,Rd+ 0.6ky

Cmy · My,Ed

Mpl,y,Rd≤ 1 (2)

NEd

χz · Npl,Rd≤ 1. (2a)

As discussed in par.Section 5.1(Fig. 4) the spatial bucklingtendency of a laterally free member may be described btwo branches of the interaction curve, which are relatedthe in-plane or to the out-of-plane buckling mode. Form(1) describes the first one and formula(2) the second one.The interaction factor of Eq.(2) has been defined as 0.6 · ky ,which means that 60% of the in-plane bending term affecthe out-of-plane buckling. AsFig. 4 illustrates, this definitionis a conservative linear approximation of the branchz–z, whichhas been chosen firstly with respect to simple calculation ansecondly because the factor 0.6 willbe used later for describinthe biaxial bending behaviour underMy andMz .

Fig. 8 presents a comparison of Eq.(2) with the GMNIA-results and those of the ENV-version. It shows the overconvative results of the ENV-version due to the use ofχmin (in-stead of differentiating betweenχy andχz). It further showsthe result of Eq. (2a), which is the widely used out-of-planbuckling check of many existing codes. In contrast to Eq.(2)it may locally be a bit unconservative, however TC8 suppokeeping this traditional check for use with double-symme

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sections under uniaxial bending, since the range of unconvatism (about 7%–9% maximum) has been considered tolerwith respect to the safety concept. It should be noted, thattorsionally stiff sections are concerned here, since torsionaflexible I-sections are treated by LT-buckling.

5.4. In-plane buckling under N + Mz

The buckling behaviour of members in weak-axis bendinvery similar to that of par.5.2. Thekz-factors recalculated fromGMNIA-results of IPE- and HEB-sections on the basis ofdesign Eq.(8) are given inFig. 10:

NEd

χz · Npl,Rd+ kz

Cmz · Mz,Ed

Mpl,z,Rd≤ 1 (8)

kz = 1 + (2λz − 0.6

) · nz ≤ 1 + 1.4 · nz . . . I-profiles

(seeFig. 9) (9)

kz = 1 + (λz − 0.2

) · nz ≤ 1 + 0.8 · nz . . . RHS-profiles

(analogous toFig. 5) (10)

nz = NEd

χz · Npl,Rd(11)

Cmz = 0.6 + 0.4ψ ≥ 0.4 (seeFig. 9). (12)

The kz-curves (seeFig. 10) show similar shapes as theky-curves ofFig. 6 in principle, however—understandably—teffect of the cross-section interaction at smallλz is much largerand the level of the curves atλz > 1.0 is considerably higher


Fig. 8. In- and out-of-plane flexural buckling.

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Fig. 9. Design diagrams for in-plane buckling underMz .

Both may be explained by the much higher plastic reserveI-sections in bending about the weak axis than about the staxis. Thekz-factor for I- and H-sections needs, therefore, todefined separately (Fig. 9).

In contrast to the I-sections thekz-factor for RHS maybe kept as for buckling about the strong axis, whichunderstandable from the similarity of their behaviour abboth axes.

TheCmz-factor is now related to the moment diagram ofMz ,which—in the case of intermediate lateral restraints—shoultaken as the segmental diagram between the lateral restrseeFigs. 9and11.

5.5. Buckling caused by biaxial bending and axial compressionN + My + Mz

The design formulae are as follows (see alsoFig. 11):

NEd

χy · Npl,Rd+ ky

Cmy · My,Ed

Mpl,y,Rd+ 0.6 · kz

Cmz · Mz,Ed

Mpl,z,Rd≤ 1(13)

NEd

χz · Npl,Rd+ 0.6 · ky

Cmy · My,Ed

Mpl,y,Rd+ kz

Cmz · Mz,Ed

Mpl,z,Rd≤ 1.

(14)

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ets;

The interaction formulae Eqs.(13) and (14) for biaxialbending have been developed by extending the given form(1) and(2) by the terms ofMz . For the buckling mode abouz–z the full kz-factor is applied and for the buckling modey–ythe reduced value 0.6 · kz .

By this, the two equations showthe followingproperties:

• their format is ‘symmetrical’ with respect to the axesyand z; this is significant for describing the transition fromrectangular to square hollow sections consistently,

• they allow by use of the factors 0.6 a qualitatively goapproximation of the plastic cross-section interactionbiaxial bendingMy + Mz . For N → 0 two linear equationsappear, which approximate the convex interaction curveFig. 12, compared to the linear interaction used before.

The interaction formulae(13), (14) have been comparegraphically and statistically with the GMNIA-results, whichnow lead to a three-dimensional graph.Fig. 13 illustrates theresults of the new formulaecompared with the GMNIA-results and those of the ENV-version. Although stillapproximation, the simplified formulae may lead toconsiderable improvement. In particular, the effect of the fa0.6 is an amendment for biaxial bending.

5.6. Equivalent uniform moment factor Cm

The Cm -factor has the purpose to take account of bendmoment diagrams, which are non-uniform along the spaalong a segment between lateral restraints, if relevant.

It is assumed that the fictitious uniform bending momCm · M has the same effect on the buckling behaviourthe actual moment diagram. Non-uniform bending momendiagrams have generally more favourable effects than unifones—therefore, theCm-factors are always smaller thanequal to 1.0.

As already mentioned in chapter4 the “equivalent uniformmoment factor”Cm differs from the analogousβM -factorin theprevious ENV-version and it differs also from theCm -factorof Method 1, which takes the sinusoidal shape of the momdiagram as its basic form. In Method 2 the approach ofwidely used “Austin formula” [13] has been applied for reasonof simplicity in practical use.


Fig. 10. Interaction factorkz .

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Fig. 11. Definition ofCm -factors.

The Austin-formula (Eqs.(7) and (12)) is valid for linearmoment diagrams (Fig. 14(a)).

Although very simple in format it is able to describe tbuckling effects very closely. Recent investigations [14] showedthat this concerns not only elastic behaviour, but also plasbehaviour with or without imperfections. It further showed, ththe cut-off by Cm = 0.4 atψ ≤ −0.5 is conservative sinceCm -values down to about 0.2 may be reached by numesimulation. This margin, however, has not been exploited in thdesign rules of Method 2, since firstly the cross-section chat the end of the member becomes frequently governingsecondly an additional reserve was considered appropriatthe cases of the bi-linear moment diagram.

In more general cases of non-uniform moment diagrae.g. under end-moments and transverse loading no forexisted, so that theCm -factor had to be recalculated froGMNIA-results of different moment distributions.Fig. 14(b)shows such recalculated values for the case of unifotransverse loading. Herein, the horizontal parts of the curveindicate that the cross-section resistance is governing. Theline may be regarded as upper bound of the different curwhich are—analogously to the Austin formula—cut off bthe conservative limit of0.4. Formulae for theCm -factors,which were derived from the calculated curves, are presein Table 1.

For theuse of theCm-factors the following two points shoulbe noted (seeTable 1):

– Firstly, the given Cm -factors may only be used if the twends of the member can be regarded as fixed. In casbuckling in a sway mode theCm -factor should be taken duto simplification as 0.9. This applies in particular to columwith a deformable upper end.

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– Secondly theCm -factor is related to the shape of the momdiagram of the relevant span between lateral restraWhile Cmy is always related to the overall span of themember,CmLT andCmz may be determined by the segmenmoment diagram between intermediate restraints.

6. Members susceptible to torsional deformations—Class 1and Class 2 sections

6.1. General buckling behaviour

The buckling behaviour of torsionally flexible membersgiven by lateral–torsional buckling. It is illustrated by thexample of an HEB 300 under axial compression and unifobending momentMy (Fig. 15). The beam–column is free tdeflect about they-axis along its overall span, howeveris laterally restrained at midspan against buckling aboutz-axis. At N = 0 pure LT-buckling between the laterrestraints occurs. As the axial force grows, buckling abouty-axis—superposed by the local LT-buckling effect—becomdominating. This behaviour is described by the interaccurve y–y and is approximated by Eq.(3). The differencefrom pure flexural buckling acc. to Eq.(1) is obvious. TheLT-buckling behaviour about thez-axis (along the segmenbetween the lateral restraints) is described by the interactioncurvez–z and is approximated by Eq.(4); it is not relevant inthe given case.

An illustration of the different typical buckling modesgiven in Fig. 16. While for members, which are laterally frealong the span, LT-buckling about thez-axis always governs,members withintermediate lateral restraints may also faila buckling mode, where the deformation about they-axisdominates and the LT-buckling effect between the restrainis just a superposed effect. The first case is described binteraction Eq.(4), the second one by Eq.(3). If the lateralrestraints are arranged at small distances, the buckling my–y tends to the case of in-plane flexural buckling. So it mabe resumed, that inthe case of free members just Eq.(4) forbuckling aboutz–z has to be considered, while for membewith intermediate lateral restraints both Eqs.(3) and(4) have tobe checked, since depending on the specific in- and out-of-pslenderness one of the two buckling modes may be decisiv


Fig. 12. Cross-section interaction atNEd → 0.

Table 1Equivalent uniform moment factorCm

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n

6.2. In-plane buckling under N + My

The design formula is given by Eq.(3):

NEd

χy · Npl,Rd+ ky

Cmy · My,Ed

χLT · Mpl,y,Rd= 1. (3)

The interaction formula is equivalent to Eq.(1) apart fromreplacing the bending resistanceMpl,y,Rd by the bucklingresistance χLT · Mpl,y,Rd . The buckling reduction factorχLT accounts for the LT-buckling effect between the laterestraints. If the slendernessλLT decreases,χLT will approach1.0 and Eq.(3) becomes identical with Eq.(1), which meansthat the basically torsionally susceptible section will failthe flexural, in-plane buckling mode. Therefore, no separflexural buckling check is needed for I- or H-sections, becausit is always included in Eq.(3).

l

An example of a very slender column (e.g. to be understas a column with the equivalent buckling length 2· l) with IPE-section illustrates again this behaviour (Fig. 17). This formula(3) is a new one compared to the ENV-version and to mexisting codes, where it was meant to be physically represeby the response of Eqs.(1) and(4). It has been incorporated inthe EN-version mainly for reasons of consistency of the desigconcept.

6.3. Out-of-plane buckling under N + My

The design formulae are as follows:

NEd

χz · Npl,Rd+ kLT

My,Ed

χLT · Mpl,y,Rd= 1 (4)

kLT = 1 − 0.1 λz nz

CmLT − 0.25≥ 1 − 0.1 nz

CmLT − 0.25(15)


Fig. 13. Comparisons for flexural buckling under bi-axial bending.

Fig. 14. Equivalent uniform moment factorCm .

dhe

boutantentthe

s

λz < 0.4 : kLT = 0.6 + λz (16)

nz = NEd

χz . Npl,Rd(17)

CmLT = 0.6 + 0.4ψ ≥ 0.4 (see alsoTable 1). (18)

The interaction formula Eq.(4) is the traditional LT-bucklingformula. The interaction factorkLT has been recalculatefrom GMNIA-results in analogous form as explained for tky-factor above.Fig. 18 illustrates the recalculatedkLT-factorfor an IPE-section and three different moment diagrams.

The systematic behaviour is obvious. All thekLT-valuesare smaller than 1.0. Atλz = 0 they start with the cross-section interaction, then they increase moderately up to aλz ∼= 1 before they decline again to an approximately constlevel for large slenderness. The non-uniformity of the momdiagram has a beneficial effect, which, however, reduceskLT-factor significantly just for high nz-values, where—a

Fig. 15. LT-buckling of torsionally flexible members.


(a) LT—buckling modey–y. (b) LT—buckling modez–z.

Fig. 16. General buckling modes of torsionally flexible members.

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are

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already noted above—the influence is limited because of threduced magnitude of the bending moment. For most praccases thekLT-values will be closely below 1.0.

The formula for the interaction factorkLT has beenapproximated in bilinear form—analogously to those ofky andkz. Thekink was chosen again atλz = 1.0 and for higherλz

thekLT-values remain constant (Fig. 19). A difference exists bythe inclusion of the effect of non-uniform moments implicitby CmLT in thekLT-factor. The reason for this is, that using thCm-factor of the Austin-formula again, an explicit applicationof CmLT in the bending termis not appropriate.

As Fig. 19 shows, the simplified bi-linear form of thformula leads to a linear approach of thekLT-factor towards 1,0with λz → 0. Since this does not represent the moderate drothe actualkLT-curves to the cross-section resistance atλz = 0,a cut-off fo rmula Eq.(16) has been used, which allows usexploit the full resistance in this range of transition for smaslenderness belowλz = 0.4. This will be reached by applyingEq.(3), which accounts for the cross-section interaction in tslenderness range consistently. It may further be noted, thacut-off formula by approachingkLT = 0.6 for small slendernessallows us to exploit approximately the convex plastic crosection interaction for biaxial bendingMy + Mz (as alreadyshown in chapter5).

For practical use it should be indicated, that theλLT factorand theCmLT-factor should be related to the specific momediagram between the relevant lateral restraints. This is the fulspan for a laterally free member or the segmental part ofmember in the case of intermediate lateral restraints. It nnot be mentioned that this relation to lateral restraints shoalso be used for the determination ofλz andnz respectively.

6.4. LT-buckling under biaxial bending and axial compressionN + My + Mz

The design formulae are as follows:

NEd

χy Npl,Rd+ ky

Cmy · My,Ed

χLT · Mpl,y,Rd+ 0.6kz

Cmz · Mz,Ed

Mpl,z,Rd≤ 1 (19)

NEd

χz Npl,Rd+ kLT

My,Ed

χLT · Mpl,y,Ed+ kz

Cmz · Mz,Ed

Mpl,z,Rd≤ 1. (20)

The interaction formulae Eqs.(19) and (20) have beendeveloped by extending the given formulae(3) and(4) by theterms ofMz . This is in full analogy to the development of thprocedure in chapter5.4for flexural buckling.

al

of

she

-

t

ed

ld

Fig. 17. LT-buckling of member with intermediate lateral restraints.

The formulae—in the case of vanishing bendingMy—provide a consistent transition to flexural buckling about thez-axis.

Fig. 20 illustrates the results of the design formulaecomparison with those of the ENV-version and with GMNIresults.

7. Member buckling for Class 3 and Class 4-sections

The fully analogous derivation of interaction formulaenumerical simulation for elastic cross-sections as shownplastic sections above causes a number of basic difficuThese are connected with the abstract definition of “elassections in EC3 on the one hand and with the limited knowledon the partial-plastic buckling behaviour of such sectioon the other. In this sense, the definition of fully elasbehaviour excludes the consideration of residual stressesalso does not allow us to recalculate buckling reduction facof such sections. Recent numerical simulations of Classections have, however, confirmed partial-plastic resistancsuch sections [17,18]. They indicate that plastic interactionbehaviour—although related to the resistanceMel,Rd —exists.On this basis the derivation of the design formulae for Classand Class 4 sections takes a moderate account of partial-peffects, in order to use some of this beneficial behaviourdesign.

The interaction formulae for sections of Class 3 and 4fully analogous with Eqs.(1) to (4), however the momentresistancesMpl have been replaced byMel or Me f f and theshif t of the neutral axis of Class 4-sections has been accoufor by�M = eN · N .


Fig. 18. Interaction factorkLT.

Fig. 19. Design values of interaction factorkLT.

icb

-

ct

%

ousivftetil

1eheeem

) ictdthodnin

s,e,

e of

eughreor’sthe

finge

m”nt

ical

ions

hg

e

sesss-nd

tion

The interaction factorsky andkz have been derived by elastsecond order theory and have been developed in simplifiedlinear format as above. The interaction factor for the out-ofplane buckling mode underMy has been defined by 0.8ky ,in which the factor 0.8 accounts for the partial-plastic effeSimilarly such an effect has been used for the factorkLT byapplying a reduced beneficial plastic effect of about 50Especially for the calculation of the reduction factorχLT forlateral torsional buckling an enhanced method is given in [17],which is also compared to test results.

These design formulae, presently take just moderate accof the plastic effects existing in these sections. A more extenuse of this load carrying behaviour may be expected afurther studies and test confirmation, which however are sunder way at present.

8. Differences between Method 1 and Method 2

As said in the introduction of this paper, EN1993-1-(EUROCODE 3) [1] also provides besides Method 2 thalternative of Method 1 for member buckling. However tcode does not give any indication on the differences betwthe two methods, which had been formally poured into the satype of basic interaction formulae Eqs. (6.61) and (6.62EN1993-1-1 by the authors of Eurocode 3. In this respeshould also be noted that in addition to Method 1 and Methothe code EN1993-1-1 provides two more methods, i.e.so-called General Method and the Stable-Length MethTherefore, in total there are four alternative methods opethe designer, which concern member buckling under bendand axial compression.

Therefore some comments might be useful for designerthere are differences in the application range of the formula

i-

.

.

nter

l

nenit2e.

tog

ifin

the accuracy or safety of the results and in the practical usthe two methods.

In the following explanations are given to the abovmentioned topics, concerning Method 1 and Method 2 althobeing awarethat full answering of all aspects would requia paper at least so long as the existing one. The authobjective for this paper has originally been focussed just onbackground of Method 2, so that together with [3] both methodsare presented similarly by their authors.

A first explanation refers to differences in the meaning otheCm -factors, which are used in the case of flexural bucklto account for non-uniform moment distribution. While thCm factors of Method 2 are based on equivalent “uniformoments, theCm factors of Method 1 are based on equivale“sinusoidal” moments. TheseCm factors of Method 1—beingderived from the elastic buckling theory—depend on the critflexural buckling load, while those of Method 2 follow theAustin-formula. Because of these different basic assumpttheCm -factors are generally different in magnitude.

It is important to mention that theCmLT-factors of bothmethods to account for non-uniform moment distribution inthe case of lateral torsional buckling are also quite different inprinciple. WhileCmLT in Method 1 is a factor connected witCmy , theCmLT in Method 2 is an independent factor coverinthe effect of the relevant moment diagrambetweenintermediatelateral restraints. Again theCmLT-factors of the two methods argenerally different.

The application range of the two methods is in both cabasically a single span member of doubly symmetrical crosection subjected to axial compression, end-moments atransverse loading. In the case of Method 1 the transifrom flexural buckling to lateraltorsional buckling has been


,assly

ofhnts,

frM

raint

eodis

teeranodth

wtednestorn

s or

ions

for

Fig. 20. Comparison for LT-buckling under biaxial bending.

approximated according to the torsional rigidity of the memberso that, therefore, a smooth transition between the two cis given. Method 2 distinguishes the two standard casetorsionally stiff and torsionally flexible members separatewhich seems to be in line with the traditional thinkingdesign engineers. It should be mentioned that Method 2been developed principally insuch a way that the applicatiorange also covers members with intermediate lateral restrainwhich may not be effective for inplane flexural buckling. Theexplicitly given guidelines for the use of specificCmLT-factorsmakes the practical application for these cases easy.

The accuracy of the two methods has been checkedeach of them by statistical evaluations of the small numbeavailable test resultsand the big number of theoretical FEcalculations, reported in several TC8-documents, e.g. [12]. Sofrom this point of view the methods are equivalent in geneHowever, this doesnot exclude differences of the resultsspecific examples. Since Method 1 has been based ontheoretical derivation of the spatial flexural buckling modand uses several specific factors it covers both flexural mmost closely, in particular out-of-plane buckling. Method 2based on the theoretical derivation of the in-plane bucklingformula with just one compact factor, which was calibraon the elastic–plastic member-capacity. Therefore, it covin-plane buckling most closely and approximates out-of-plbuckling. In the case of lateral–torsional buckling both methare extensions from flexural buckling calibrated mainly atresults of elastic–plastic numerical simulations.

With respect to the practical use of the formulae the tmethods have different objectives. Method 2 was definiaimed at the use by hand-calculation and, therefore, usesame simple structure of the formulae as traditional interactioformulae of many existing codes.Method 1 aimed at a structurof the formulae, which is as far as possible based on elatheoretical derivation described by a larger number of factAccordingly, the useof computer aids seems to be useful anecessary for practical application. In this view it is up to the

esof,

as

orof

l.

heses

dsese

olythe

ic,s.d

Fig. 21. System and internal forces.

user to choose, which format would be more beneficial in hiher specific design situation.

9. Worked examples

9.1. General

The buckling check is based on the design values of actand resistances as defined in [1]. The safety factorsγM0 andγM1 have been taken as 1.0 as recommended in EC3buildings.

9.2. Flexural buckling of member under bending and axialcompression

Forces and buckling lengths,Fig. 21

NEd = 300 kN

My,Ed = 40 kN m (parabolic moment diagram)

L = Lcr,y = Lcr,z = 5.6 m

cross-section: RHS 200/100/10, S235, hot-finished

NRd = 54.9 · 23.5/1.0 = 1290 kN

My,Rd = 341· 23.5/1.0 = 8010 kN cm

slenderness ratio:

λy = 560

6.96 · 93.9= 0.857

λz = 560

3.98 · 93.9= 1.50

buckling reduction factors

χy = 0.762 buckling curvea

χz = 0.372 buckling curveb.

Equivalent uniform moment factor and interaction factor:

Cmy = 0.95 (seeTable 1)

ny = 300

0.762· 1290= 0.305 (6)

for λy ≤ 1.0 follows

ky = 1 + (0.857− 0.2) · 0.305= 1.20. (5)

Verification:

0.305+ 1.20 · 0.95 · 4000

8010= 0.874≤ 1 (1)

300

0.372· 1290= 0.625≤ 1. (2b)


pylengt

en

to

ed

rom


9.3. Lateral–torsional buckling of a column in a two-storeyframe

The lower column of a two-storey frame with a cano(seeFig. 22) should be checked on the basis of the equivacolumn method. First order internal forces and buckling lenhave been taken from example 8.17 in [15].

Forces and buckling length:

NEd = 664 kN

My,Ed = 668 kN m(triangular moment diagram with internal step)

Lcr,y = 2.25 · 4.2 = 9.45 m

L = Lcr,z = 4.2 m

cross-section: IPE 550, S355

NRd = 134· 35.5/1.0 = 4760 kN

My,Rd = 2787· 35.5/1.0 = 98900 kN cm.

Critical buckling momentMcr :

Ncr,z = π2 · 21000· 2670

4202= 3140 kN

Mcr = 1.6 · 3140·√

1884000

2670+ 4202 · 124

π2 · 2.6 · 2670= 161000 kN cm

hereby, the factorC1 for the given moment diagram has betaken as 1.6 acc. to [15].Slenderness ratio:

λy = 945

22.3 · 76.4= 0.555

λz = 420

4.45 · 76.4= 1.24

λLT =√

98900

161000= 0.784

buckling reduction factors


χz = 0.457 buckling curveb

χLT = 0.774 LT-buckling curvec.

The reduction factorχLT has been determined accordingEC3-1-1, 6.3.2.3, based onλLT,0 = 0.4 andβ = 0.75. It hasfurther been modified by use of thef -factor accounting for the

th


non-uniform moment diagram toχLT,mod = χLT/ f , where fhas approximatively been based on

kc = 0.752 forψ = 0.

f = 1 − 0.5(1− 0.752)[1 − 2(0.784− 0.8)2

]= 0.876

χLT,mod = 0.774

0.876= 0.884.

Equivalent uniform moment factor and interaction factors:

Cmy = 0.9 for swaymode, seeTable 1

CmLT = 0.6 (18)

ny = 664

0.906· 4760= 0.154 (6)

for λy ≤ 1.0 follows:

ky = 1 + (0.555− 0.2) · 0.154= 1.05 (5)

nz = 664

0.457· 4760= 0.305 (17)

for λz ≥ 1.0 follows:

kLT = 1 − 0.1

0.6 − 0.25· 0.305= 0.913. (15)

Verification:

0.154+ 1.05 · 0.9 · 66800

0.884· 98900= 0.876≤ 1 (3)

0.305+ 0.913· 66800

0.884· 98900= 1.00≤ 1. (4)

9.4. Lateral–torsional buckling of a column under bi-axialbending and compression

The column of a single-storey building should be checkon basis of the equivalent column method (seeFig. 23). First-order internal forces and buckling lengths have been taken fexample 8.7 in [15].Forces and buckling length:

NEd = 620 kN

My,Ed = 202 kN m (triangular moment diagram)

Mz = Mz,Ed = 7.16 kN m (parabolic moment diagram)

Lcr,y = 2.3 · 4.6 = 10.6 m

L = Lcr,z = 4.6 m

cross-section: IPE 450, S355

NRd = 98.8 · 35.5/1.0 = 3510 kN


en

to

eresonsughl

dng

d

and

tionh.

ion1.

3,

itte.

g.

-

rns.

dben

l 1ts—

l of

ghell

tural

s

s.

My,Rd = 1702· 35.5/1.0 = 60400 kN cm

Mz,Rd = 276· 35.5/1.0 = 9800 kN cm

critical buckling momentMcr :

Ncr,z = π2 · 21000· 1680

4602 = 1650 kN

Mcr = 1.77 · 1650·√

791000

1680+ 4602 · 67.1

π2 · 2.6 · 1680= 82600 kN cm

hereby, the factorC1 for the given moment diagram has betaken as 1.77.slenderness ratio:

λy = 1060

18.5 · 76.4= 0.750

λz = 460

4.12 · 76.4= 1.46

λLT =√

60400

82600= 0.855

buckling reduction factors:


χz = 0.357 buckling curveb

χLT = 0.729 LT-buckling curvec.

The reduction factorχLT has been determined accordingEC3-1-1, 6.3.2.3 based onλLT,0 = 0.4 andβ = 0.75. It hasfurther been modified by use of thef -factor accounting for thetriangular moment diagram withψ = 0.

f = 1 − 0.5(1 − 0.752)[1 − 2(0.855− 0.8)2

]= 0.877

χLT,mod = 0.729

0.877= 0.831.

Equivalent uniform moment factor and interaction factors:

Cmy = 0.9 for swaymode, seeTable 1

Cmz = 0.95 for parabolic moment

CmLT = 0.6 for triangular moment

ny = 620

0.824· 3510= 0.214 (6)

for λy ≤ 1.0 follows

ky = 1 + (0.750− 0.2) · 0.214= 1.12 (5)

nz = 620

0.357· 3510= 0.495 (11)

for λz ≥ 1.0 follows

kz = 1 + 1.4 · 0.495= 1.69 (9)

kLT = 1 − 0.1

0.6 − 0.25· 0.495= 0.859. (15)

Verification:

0.214+ 1.12 · 0.9 · 20200

0.831· 60400+ 0.6 · 1.69 · 0.95 · 716

9800≤ 1(19)

0.214+ 0.406+ 0.07 = 0.69 ≤ 1

0.495+ 0.859· 20200

0.831· 60400+ 1.69 · 0.95 · 716

9800≤ 1 (20)

0.495+ 0.346+ 0.117= 0.96≤ 1.

Acknowledgements

The developments of the new interaction formulae wbased on a wide scope of numerical investigations, compariand statistical evaluations and could only be carried out throthe efforts of the collaborators of the two authors. Speciathanks are given toRobert Ofner, Günther Salzgeber anPeter Kaim of TU Graz and to Andreas Rusch, JunpiWang/Kunming and Stefan Heyde of TU Berlin.

References

[1] EN1993-1-1. Eurocode 3. Design of steel structures, general rules anrules for buildings. 2005.

[2] ENV1993-1-1. Eurocode 3. Design of steel structures, general rulesrules for buildings. 1993.

[3] Boissonnade N, Jaspart J-P, Muzeau J-P, Villette M. New Interacformulae for beam-columns in Eurocode 3. The French-Belgian approacJournal of Constructional Steel Research 2004;60:421–31.

[4] Greiner R. Background information on the beam–column interactFormulae at Level 1. ECCS Report No. TC8-2001-021, 20; Sept. 200

[5] Greiner R, Lindner J. Die neuen Regelungen in der europäischen NormEN1993-1-1 fur Stabe unter Druck und Biegung. Stahlbau 2003, Heftp. 157–72.

[6] Greiner R, Kaim P. Erweiterungder Traglastuntersuchungen an Stäbenunter Druck und Biegung auf einfach—symmetrische QuerschnStahlbau 2003;72(Heft 3):173–80.

[7] Ofner R. Traglast von Stäben aus Stahl bei Druck und BiegunDissertation, Institut fur Stahlbau, Holzbau und Flächentragwerke der TUGraz, Heft 9; 1997.

[8] ABAQUS Software. Hibbitt, Karlsson & Sorensen Inc., Version 5.7.[9] Greiner R, Ofner R, Salzgeber G. Verification of GMNIA-results. ECCS

validation group. Report 2; July 1998.[10] Lindner J, Rusch A, Heyde S. Evaluation of different design concepts fo

flexural buckling with regard to test results and ultimate load calculatioReport 2131E, TU Berlin; November 1998.

[11] Lindner J. Interaktionsgleichungen für das Biegeknicken bei Druck unzweiachsiger Biegung. Schlussbericht zum DIBt-ForschungsvorhaIV 1-5-866/98, Bericht 2135 des Instituts für Baukonstruktionen undFestigkeit der TU Berlin, 10.6.19999.

[12] Lindner J, Heyde S. Evaluation of interaction formulae at Leveapproach with regard to ultimate load calculations and test resulflexural buckling and lateral torsional buckling. Report 2144E, TU Berlin.ECCS Report No. TC8-2001-017, 30.7.2001.

[13] Austin WJ. Strength and design of metal beam-columns. JournaStructural Division 1961;ASCE 87(ST 4):1–32.

[14] Kaim P. Spatial buckling behaviour of steel members under bendinand axial compression. Ph.D. Institute for Steel Structures and SStructures, TU Graz; Heft 12-2004.

[15] Lindner J, Schmidt H, Scheer J. Stahlbauten, Erläuterungen zur DIN18800 Teil 1 bis 4. Berlin: Ernst & Sohn, Beuth, 3. Auflage; 1998.

[16] Lindner J. Design of beams and beam columns. Progress in StrucEngineerings and Materials. 2003;5:38–47.

[17] Rusch A, Lindner J. :Application of Level1 interaction formulae to clas4 sections. Thin-walled Structures 2004;42:279–93.

[18] Lechner A. Plastic cross-sectioncapacity of semi-compact steel sectionPh.D. Institute for Steel Structures and Shell Structures, TU Graz; 2005.

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