Greiner Lindner EC3 PM Interaction

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

Interaction formulae for members subjected to bending and axial compression in EUROCODE 3—the Method 2 approach R. Greiner a,∗ , J. Lindner b a TU Graz, Institute for Steel Structures and Shell Structures, Lessingstr. 25, 8010 Graz, Austria b 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 buildings. 2005] for Steel Structures provides two alternatives for the buckling check of members subjected to axial compression and bending by interaction formulae, which are called there Method 1 and Method 2. This paper presents the characteristics, the background and the use of Method 2. The analogous presentation of Method 1 has already been given in [Boissonnade N, Jaspart J-P, Muzeau J-P, Villette M. New Interaction formulae for beam-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 ENV-rules; however with advanced numerical background and consistent classification of the buckling modes. In this respect new improved interaction factors were developed from a wide scope of numerical simulations and the concept of the formulae was focussed distinctly on describing the modes of in-plane and out-of-plane buckling for members susceptible to fail either in flexural buckling or in lateral–torsional buckling. As result two sets of formulae are provided, which each cover a clear scope of physical member behaviour. Hereby, the specific effects of intermediate lateral restraints—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 standard cases. c 2006 Elsevier Ltd. All rights reserved. Keywords: Interaction formulae; Buckling of beam–columns; EUROCODE 3

1. Introduction In the process of the conversion of the ENV-version of the EUROCODE 3 [2] to the final EN-version [1] basic developments were carried out by Technical Committee 8 (TC8) of ECCS, which resulted in new buckling rules for members subjected to axial compression and bending. This was caused by some physical inconsistencies and overconservatism found in the ENV-rules [4], which could lead to considerable underestimation of the capacity of beam–columns in certain cases. The use of modern computer techniques and FEprograms opened the possibility of simulating the geometrically and materially nonlinear buckling behaviour of imperfect members in much wider parametric scope than before [4,5]. On ∗ Corresponding author. Tel.: +43 316 873 6200; fax: +43 316 873 6707.

E-mail address: [email protected] (R. Greiner). c 2006 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2005.11.018

this basis the interaction factors could be directly recalculated from the numerical limit load results and the effect of different parameters be consistently identified. The final definition of the new rules was then based on calibration with the existing test results and the numerical simulations. The elaboration of the Method 2-rules therefore presents the consequent development of the former work on interaction formulae by use of the opportunities of the new computer techniques. At present they cover double-symmetric crosssections in EC3-1-1 [1], however an extension to monosymmetric sections has already been made in the meantime [6]. In this spirit of further development the traditional interactionformulae were maintained in format and notation as far as possible in order to facilitate the understanding of the changes and the practical application for the user. However, in a number of points new formulations were necessary, in particular to provide possible further developments.

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

3. Numerical calculations and statistical evaluation

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

The objective of this paper is to present the background of the buckling formulae of Method 2 in EC3-1-1, which are in the tradition of the simplified design rules of many existing codes. A short but not detailed enough explanation to this background was already given in [16]. EC3-1-1 will provide further methods for the stability check of members (Method 1 and General Method), which are more complex and are not dealt with in this paper. 2. Characteristics of the buckling rules of Method 2 of EC3-1-1 The buckling rules of Method 2 are directed to cover the following three main characteristics of the structural behaviour of steel members (Fig. 1): • Cross-section shapes: Depending on the shape of the cross-section members subjected to axial compression and bending may behave susceptible to torsional deformations or not. Therefore, specific sets of buckling formulae are provided for torsionally stiff members (hollow sections or torsionally restrained I-sections) and torsionally flexible members (I and H-sections). • Buckling modes: In general spatial buckling deformation of members subjected to axial compression and bending is traditionally split up into the two buckling modes about the y-axis and about the z-axis. In each set of formulae the first one describes the strong-axis mode and the second one the weak-axis mode. In this way the specific physical behaviour is transparently connected with the design process (lateral restraining etc.). • Intermediate lateral restraints: The buckling formulae were primarily derived for the free beam–column with end-forkconditions. However, at the same time the application to the practically frequent cases of members with intermediate lateral restraints has also been provided. This requires us to differentiate between the buckling lengths and moment diagrams along the span and along a segmental part between the lateral restraints.

The basis of the new developments of the buckling interaction formulae was the numerical simulation of the elastic–plastic buckling behaviour of single-span beam–columns, which accounts for geometric bow imperfections, residual stresses and a linear-elastic–plastic material law without hardening effects. The calculations, called geometrically, materially nonlinear analyses of the imperfect structure [GMNIA], were performed by a computer module for automatic parameter variation and data evaluation, which allowed us to cover a wide scope of member and load parameters (about 25000 cases) [7]. The analyses were carried out by the FEProgramme ABAQUS [8] using beam elements. Verifications were made by comparison of the numerical simulation with buckling tests and other computer programs [9]. The resulting limit load data were used to recalculate the interaction factors on the basis of the proposed interaction equations and afterwards to develop simplified formulae for them. These formulae were calibrated with the low number of available buckling tests and with the high number of numerical results using statistical evaluation [10–12]. 4. Concept of interaction formulae for N + My As already explained in chapter 2 two sets of buckling formulae are provided, which each describe the buckling behaviour of either torsionally stiff or torsionally flexible steel sections (Fig. 2). In the first case they concern flexural buckling, in the second one lateral–torsional buckling. The first formula of each set is related to buckling about the y-axis and the second one to buckling about the z-axis. This clear structure may assist the designer to connect the results of the buckling check with structural provisions. In this respect it has already been said, that the buckling formulae are appropriate for both free, single-span members as well as those supported laterally by intermediate restraints. In the present state such restraints are expected to be full restraints, which means that in the case of flexural buckling a pure lateral restraint would be sufficient, while the case of LT-buckling would require lateral plus torsional restraint. Different kinds of restraints are under investigation. The four interaction formulae are given in Fig. 3 for class 1 and 2 sections (analogous formulae for class 3 and 4 sections also exist in EC3-1-1). In order to accentuate the systematic structure of the two sets of design formulae for practical use the four formulae were numbered consecutively by (1) to (4) and marked by boxes. Eqs. (1) and (2) describe flexural buckling of torsionally stiff members and Eqs. (3) and (4) LT-buckling for torsionally flexible members. In the case of flexural buckling about the weak axis a simplified approach (2a) is allowed, which is based just on the axial compression term alone This traditional rule— used in many existing codes—has been maintained in EC3 for uniaxial bending and compression. The new interaction formulae of Method 2 use the following factors:


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

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

• Interaction factors k y , k z , kLT • Equivalent uniform moment factors Cmy , Cmz , CmLT . In addition the buckling reduction factors χ y , χz , χLT for the basic buckling modes under pure compression or pure bending are necessary. They cover the cases of separate flexural buckling caused by pure compression or LT-buckling by pure bending. In the following the derivation of these terms is presented. The interaction factors k y and k z apply to the uniform distribution of the bending moments M y and Mz , while the kLT factors include the non-uniformity of the moment diagrams of M y . This non-uniformity is accounted for by Cm -factors, which are based on the widely used concept of Austin [13], i.e. to take the constant moment (not the sinusoidal moment distribution) as reference, so that Cm = 1.0 holds for uniform moments and any other moment diagram leads to Cm -values lower than 1.0. In this respect this definition of Cm is different from the β M factors of the ENV-version. It may also be noted, that the coefficients χ y and χz (European column buckling curves) are the same as in the ENV-version, but that the coefficients χLT have been further

developed for the EN-version. The latter concerns mainly the classification of the LT-buckling curves connected with the torsional capacity of the cross-section (h/b ≥ or < 2) and the inclusion of the beneficial effect of non-uniform moment diagrams by the modified coefficient χLT,mod . It has to be pointed out, that the stability check by the interaction formulae (1) to (4) has to be supplemented by the cross-section check at the member ends, if the bending moments are non-uniform along the span and the moment maximum appears at the member end. This means, that the cross-section capacity at the end section may govern, if the stability 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 buckling behaviour leads to flexural buckling. This is illustrated by the example of an RHS 200/100/10 of a beam–column of 4 m

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Fig. 4. Flexural buckling of torsionally stiff members.

length under axial compression and uniform bending moment M y in the form of an interaction diagram (Fig. 4). In the case of sufficient intermediate lateral restraint in-plane buckling is the governing buckling mode, which is described by the interaction curve y–y and is approximated by Eq. (1). For an unrestrained member, which is free to deflect spatially along its span, outof-plane buckling will occur if N/N pl is larger than 0.4. This is described by the branch z–z of the interaction curve and is approximated by Eq. (2). The approximation by Eq. (2a) is also presented. The key-parameter of the interaction behaviour is the interaction factor k, whose development is explained in the following for the different buckling modes. As already said before, the interaction factors are the result of recalculations from a large scope of limit load-data resulting from GMNIA-calculations. The parameters used for this recalculation were taken from the general derivation of the buckling interaction formula on basis of the second order theory (see e.g. [4]), which shows that the three parameters λ¯ y , χ y and NNpl determine the interaction behaviour. The interaction factors were finally expressed by λ¯ y and n y , the latter of which connects NNpl and χ y as in the term for buckling under pure axial compression. 5.2. In-plane buckling under N + M y The design buckling formulae are as follows (Fig. 5): Cmy · M y,Ed N Ed + ky ≤1 χ y · N pl,Rd M pl,y,Rd k y = 1 + λ¯ y − 0.2 · n y ≤ 1 + 0.8 · n y N Ed ny = χ y N pl,Rd Cmy = 0.6 + 0.4ψ ≥ 0.4 (see Fig. 5 and also Table 1).

(1) (5) (6) (7)

The recalculations of the k y -factor from GMNIA-results were made on basis of (Eq. (1) for different cross section shapes. Examples are shown in Fig. 6 for an IPE and an RHS. They indicate a certain influence of the cross-section shape, however it turned out to be of minor effect on the end result, so that it could be ignored as a design parameter further on.

Fig. 5. Design diagrams for in-plane buckling under M y .

The form of the k y -curves illustrates the physical behaviour of members with plastic cross-section: At λ¯ y = 0 they express the plastic cross-section interaction of N + M y . With increasing λ¯ y the second-order effect causes the growth of the bending moment and by that the rise of the k y -factor. Approximately at λ¯ y ∼ = 1.0 the point is reached, where a further increase of k y is not needed, since further stability effects are covered by the χ y in the term of axial compression. For λ¯ y > 1.0 the k y -values remain at approximately constant level, which may be explained by the fact, that the member in this slenderness range behaves increasingly elastically, and therefore, the effect of M pl,y used in Eq. (1) needs to be counter-compensated by k y (this effect is obviously more pronounced in RHS, than in IPE-section, as indicated by the diagrams in Fig. 6). The k y -curves have been transformed into a formula, which is given by Eq. (5). The bi-linear form of this formula, with a kink at λ¯ y = 1.0 (see Fig. 5) has been chosen with respect to user-friendliness. This means that related to λ¯ y ≥ or < 1.0 just one part of Eq. (5) needs to be determined. It may also be noted that the differences of the k y -curves in the range of λ¯ y > 1.0 are of just minor effect on the design, because the differences mainly occur at high values of n y , where the moment terms are accordingly small, so that k y has very little influence. The k y -factors of the ENV-version are also given in Fig. 6. Compared with the new k y -factors of Fig. 5 the economical improvement is obvious. Results for examples shown as interaction diagrams for different forms of moment distributions are illustrated in Fig. 7. The results of the new formulae are compared with the GMNIA-results and those of the ENV-version. The improved approximation in the cases (a) and (b) is obvious, the differences in case (c) are due to the conservatism of the Austinformula for bilinear moment diagrams (see chapter 5.6).


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Fig. 6. Interaction factor k y .

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

5.3. Out-of-plane buckling under N + M y The design buckling formula is as follows: Cmy · M y,Ed N Ed + 0.6 k y ≤1 χz · N pl,Rd M pl,y,Rd N Ed ≤ 1. χz · N pl,Rd

(2) (2a)

As discussed in par. Section 5.1 (Fig. 4) the spatial buckling tendency of a laterally free member may be described by two branches of the interaction curve, which are related to the in-plane or to the out-of-plane buckling mode. Formula (1) describes the first one and formula (2) the second one. The interaction factor of Eq. (2) has been defined as 0.6 · k y , which means that 60% of the in-plane bending term affects the out-of-plane buckling. As Fig. 4 illustrates, this definition is a conservative linear approximation of the branch z–z, which has been chosen firstly with respect to simple calculation and secondly because the factor 0.6 will be used later for describing the biaxial bending behaviour under M y and Mz . Fig. 8 presents a comparison of Eq. (2) with the GMNIAresults and those of the ENV-version. It shows the overconservative results of the ENV-version due to the use of χmin (instead of differentiating between χ y and χz ). It further shows the result of Eq. (2a), which is the widely used out-of-plane buckling check of many existing codes. In contrast to Eq. (2) it may locally be a bit unconservative, however TC8 supported keeping this traditional check for use with double-symmetric

sections under uniaxial bending, since the range of unconservatism (about 7%–9% maximum) has been considered tolerable with respect to the safety concept. It should be noted, that just torsionally stiff sections are concerned here, since torsionally flexible I-sections are treated by LT-buckling. 5.4. In-plane buckling under N + Mz The buckling behaviour of members in weak-axis bending is very similar to that of par. 5.2. The k z -factors recalculated from GMNIA-results of IPE- and HEB-sections on the basis of the design Eq. (8) are given in Fig. 10: N Ed Cmz · Mz,Ed + kz ≤1 χz · N pl,Rd M pl,z,Rd k z = 1 + 2λ¯ z − 0.6 · n z ≤ 1 + 1.4 · n z . . . I-profiles (see Fig. 9) k z = 1 + λ¯ z − 0.2 · n z ≤ 1 + 0.8 · n z . . . RHS-profiles (analogous to Fig. 5) N Ed nz = χz · N pl,Rd Cmz = 0.6 + 0.4ψ ≥ 0.4 (see Fig. 9).

(8)

(9) (10) (11) (12)

The k z -curves (see Fig. 10) show similar shapes as the k y curves of Fig. 6 in principle, however—understandably—the effect of the cross-section interaction at small λ¯ z is much larger and the level of the curves at λ¯ z > 1.0 is considerably higher.

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

The interaction formulae Eqs. (13) and (14) for biaxial bending have been developed by extending the given formulae (1) and (2) by the terms of Mz . For the buckling mode about z–z the full k z -factor is applied and for the buckling mode y–y the reduced value 0.6 · k z . By this, the two equations show the following properties: • their format is ‘symmetrical’ with respect to the axes y and z; this is significant for describing the transition from rectangular to square hollow sections consistently, • they allow by use of the factors 0.6 a qualitatively good approximation of the plastic cross-section interaction for biaxial bending M y + Mz . For N → 0 two linear equations appear, which approximate the convex interaction curve, see Fig. 12, compared to the linear interaction used before.

Fig. 9. Design diagrams for in-plane buckling under Mz .

Both may be explained by the much higher plastic reserve of I-sections in bending about the weak axis than about the strong axis. The k z -factor for I- and H-sections needs, therefore, to be defined separately (Fig. 9). In contrast to the I-sections the k z -factor for RHS may be kept as for buckling about the strong axis, which is understandable from the similarity of their behaviour about both axes. The Cmz -factor is now related to the moment diagram of Mz , which—in the case of intermediate lateral restraints—should be taken as the segmental diagram between the lateral restraints; see Figs. 9 and 11. 5.5. Buckling caused by biaxial bending and axial compression N + M y + Mz The design formulae are as follows (see also Fig. 11): Cmy · M y,Ed Cmz · Mz,Ed N Ed + ky + 0.6 · k z ≤ 1 (13) χ y · N pl,Rd M pl,y,Rd M pl,z,Rd Cmy · M y,Ed N Ed Cmz · Mz,Ed + 0.6 · k y + kz ≤ 1. χz · N pl,Rd M pl,y,Rd M pl,z,Rd (14)

The interaction formulae (13), (14) have been compared graphically and statistically with the GMNIA-results, which now lead to a three-dimensional graph. Fig. 13 illustrates the results of the new formulae compared with the GMNIAresults and those of the ENV-version. Although still an approximation, the simplified formulae may lead to a considerable improvement. In particular, the effect of the factor 0.6 is an amendment for biaxial bending. 5.6. Equivalent uniform moment factor Cm The Cm -factor has the purpose to take account of bending moment diagrams, which are non-uniform along the span or along a segment between lateral restraints, if relevant. It is assumed that the fictitious uniform bending moment Cm · M has the same effect on the buckling behaviour as the actual moment diagram. Non-uniform bending moment diagrams have generally more favourable effects than uniform ones—therefore, the Cm -factors are always smaller than or equal to 1.0. As already mentioned in chapter 4 the “equivalent uniform moment factor” Cm differs from the analogous β M -factor in the previous ENV-version and it differs also from the Cm -factor of Method 1, which takes the sinusoidal shape of the moment diagram as its basic form. In Method 2 the approach of the widely used “Austin formula” [13] has been applied for reasons of simplicity in practical use.


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Fig. 10. Interaction factor k z .

– Secondly the Cm -factor is related to the shape of the moment diagram of the relevant span between lateral restraints. While Cmy is always related to the overall span of the member, CmLT and Cmz may be determined by the segmental moment diagram between intermediate restraints. 6. Members susceptible to torsional deformations—Class 1 and Class 2 sections Fig. 11. Definition of Cm -factors.

The Austin-formula (Eqs. (7) and (12)) is valid for linear moment diagrams (Fig. 14(a)). Although very simple in format it is able to describe the buckling effects very closely. Recent investigations [14] showed that this concerns not only elastic behaviour, but also plastic behaviour with or without imperfections. It further showed, that the cut-off by Cm = 0.4 at ψ ≤ −0.5 is conservative since Cm -values down to about 0.2 may be reached by numerical simulation. This margin, however, has not been exploited in the design rules of Method 2, since firstly the cross-section check at the end of the member becomes frequently governing and secondly an additional reserve was considered appropriate for the cases of the bi-linear moment diagram. In more general cases of non-uniform moment diagrams, e.g. under end-moments and transverse loading no formula existed, so that the Cm -factor had to be recalculated from GMNIA-results of different moment distributions. Fig. 14(b) shows such recalculated values for the case of uniform transverse loading. Herein, the horizontal parts of the curves indicate that the cross-section resistance is governing. The solid line may be regarded as upper bound of the different curves, which are—analogously to the Austin formula—cut off by the conservative limit of 0.4. Formulae for the Cm -factors, which were derived from the calculated curves, are presented in Table 1. For the use of the Cm -factors the following two points should be noted (see Table 1): – Firstly, the given Cm -factors may only be used if the two ends of the member can be regarded as fixed. In cases of buckling in a sway mode the Cm -factor should be taken due to simplification as 0.9. This applies in particular to columns with a deformable upper end.

6.1. General buckling behaviour The buckling behaviour of torsionally flexible members is given by lateral–torsional buckling. It is illustrated by the example of an HEB 300 under axial compression and uniform bending moment M y (Fig. 15). The beam–column is free to deflect about the y-axis along its overall span, however it is laterally restrained at midspan against buckling about the z-axis. At N = 0 pure LT-buckling between the lateral restraints occurs. As the axial force grows, buckling about the y-axis—superposed by the local LT-buckling effect—becomes dominating. This behaviour is described by the interaction curve y–y and is approximated by Eq. (3). The difference from pure flexural buckling acc. to Eq. (1) is obvious. The LT-buckling behaviour about the z-axis (along the segment between the lateral restraints) is described by the interaction curve z–z and is approximated by Eq. (4); it is not relevant in the given case. An illustration of the different typical buckling modes is given in Fig. 16. While for members, which are laterally free along the span, LT-buckling about the z-axis always governs, members with intermediate lateral restraints may also fail in a buckling mode, where the deformation about the y-axis dominates and the LT-buckling effect between the restraints is just a superposed effect. The first case is described by the interaction Eq. (4), the second one by Eq. (3). If the lateral restraints are arranged at small distances, the buckling mode y–y tends to the case of in-plane flexural buckling. So it may be resumed, that in the case of free members just Eq. (4) for buckling about z–z has to be considered, while for members with intermediate lateral restraints both Eqs. (3) and (4) have to be checked, since depending on the specific in- and out-of-plane slenderness one of the two buckling modes may be decisive.

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Fig. 12. Cross-section interaction at N Ed → 0. Table 1 Equivalent uniform moment factor Cm

6.2. In-plane buckling under N + M y The design formula is given by Eq. (3): Cmy · M y,Ed N Ed + ky = 1. χ y · N pl,Rd χLT · M pl,y,Rd

(3)

The interaction formula is equivalent to Eq. (1) apart from replacing the bending resistance M pl,y,Rd by the buckling resistance χLT · M pl,y,Rd . The buckling reduction factor χLT accounts for the LT-buckling effect between the lateral restraints. If the slenderness λ¯ LT decreases, χLT will approach 1.0 and Eq. (3) becomes identical with Eq. (1), which means that the basically torsionally susceptible section will fail in the flexural, in-plane buckling mode. Therefore, no separate flexural buckling check is needed for I- or H-sections, because it is always included in Eq. (3).

An example of a very slender column (e.g. to be understood as a column with the equivalent buckling length 2 · l) with IPEsection illustrates again this behaviour (Fig. 17). This formula (3) is a new one compared to the ENV-version and to many existing codes, where it was meant to be physically represented by the response of Eqs. (1) and (4). It has been incorporated in the EN-version mainly for reasons of consistency of the design concept. 6.3. Out-of-plane buckling under N + M y The design formulae are as follows: M y,Ed N Ed + kLT =1 χz · N pl,Rd χLT · M pl,y,Rd 0.1 n z 0.1 λ¯ z n z ≥1− kLT = 1 − CmLT − 0.25 CmLT − 0.25

(4) (15)


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

Fig. 14. Equivalent uniform moment factor Cm .

λ¯ z < 0.4 : kLT = 0.6 + λ¯ z N Ed nz = χz . N pl,Rd CmLT = 0.6 + 0.4ψ ≥ 0.4 (see also Table 1).

(16) (17) (18)

The interaction formula Eq. (4) is the traditional LT-buckling formula. The interaction factor kLT has been recalculated from GMNIA-results in analogous form as explained for the k y -factor above. Fig. 18 illustrates the recalculated kLT -factor for an IPE-section and three different moment diagrams. The systematic behaviour is obvious. All the kLT -values are smaller than 1.0. At λ¯ z = 0 they start with the crosssection interaction, then they increase moderately up to about λ¯ z ∼ = 1 before they decline again to an approximately constant level for large slenderness. The non-uniformity of the moment diagram has a beneficial effect, which, however, reduces the kLT -factor significantly just for high n z -values, where—as

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

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(a) LT—buckling mode y–y.

(b) LT—buckling mode z–z.

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

already noted above—the influence is limited because of the reduced magnitude of the bending moment. For most practical cases the kLT -values will be closely below 1.0. The formula for the interaction factor kLT has been approximated in bilinear form—analogously to those of k y and k z . The kink was chosen again at λ¯ z = 1.0 and for higher λ¯ z the kLT -values remain constant (Fig. 19). A difference exists by the inclusion of the effect of non-uniform moments implicitly by CmLT in the kLT -factor. The reason for this is, that using the Cm -factor of the Austin-formula again, an explicit application of CmLT in the bending term is not appropriate. As Fig. 19 shows, the simplified bi-linear form of the formula leads to a linear approach of the kLT -factor towards 1,0 with λ¯ z → 0. Since this does not represent the moderate drop of the actual kLT -curves to the cross-section resistance at λ¯ z = 0, a cut-off formula Eq. (16) has been used, which allows us to exploit the full resistance in this range of transition for small slenderness below λ¯ z = 0.4. This will be reached by applying Eq. (3), which accounts for the cross-section interaction in this slenderness range consistently. It may further be noted, that the cut-off formula by approaching kLT = 0.6 for small slenderness allows us to exploit approximately the convex plastic crosssection interaction for biaxial bending M y + Mz (as already shown in chapter 5). For practical use it should be indicated, that the λ¯ LT factor and the CmLT -factor should be related to the specific moment diagram between the relevant lateral restraints. This is the full span for a laterally free member or the segmental part of the member in the case of intermediate lateral restraints. It need not be mentioned that this relation to lateral restraints should also be used for the determination of λ¯ z and n z respectively. 6.4. LT-buckling under biaxial bending and axial compression N + M y + Mz The design formulae are as follows: Cmy · M y,Ed N Ed Cmz · Mz,Ed + ky + 0.6 k z ≤ 1 (19) χ y N pl,Rd χLT · M pl,y,Rd M pl,z,Rd M y,Ed N Ed Cmz · Mz,Ed + kLT + kz ≤ 1. (20) χz N pl,Rd χLT · M pl,y,Ed M pl,z,Rd The interaction formulae Eqs. (19) and (20) have been developed by extending the given formulae (3) and (4) by the terms of Mz . This is in full analogy to the development of the procedure in chapter 5.4 for flexural buckling.

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

The formulae—in the case of vanishing bending M y — provide a consistent transition to flexural buckling about the z-axis. Fig. 20 illustrates the results of the design formulae in comparison with those of the ENV-version and with GMNIAresults. 7. Member buckling for Class 3 and Class 4-sections The fully analogous derivation of interaction formulae by numerical simulation for elastic cross-sections as shown for plastic sections above causes a number of basic difficulties. These are connected with the abstract definition of “elastic” sections in EC3 on the one hand and with the limited knowledge on the partial-plastic buckling behaviour of such sections on the other. In this sense, the definition of fully elastic behaviour excludes the consideration of residual stresses and also does not allow us to recalculate buckling reduction factors of such sections. Recent numerical simulations of Class 3sections have, however, confirmed partial-plastic resistance of such sections [17,18]. They indicate that plastic interaction behaviour—although related to the resistance Mel,Rd —exists. On this basis the derivation of the design formulae for Class 3 and Class 4 sections takes a moderate account of partial-plastic effects, in order to use some of this beneficial behaviour for design. The interaction formulae for sections of Class 3 and 4 are fully analogous with Eqs. (1) to (4), however the moment resistances M pl have been replaced by Mel or Me f f and the shift of the neutral axis of Class 4-sections has been accounted for by M = e N · N.


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Fig. 18. Interaction factor kLT .

Fig. 19. Design values of interaction factor kLT .

The interaction factors k y and k z have been derived by elastic second order theory and have been developed in simplified bilinear format as above. The interaction factor for the out-ofplane buckling mode under M y has been defined by 0.8 k y , in which the factor 0.8 accounts for the partial-plastic effect. Similarly such an effect has been used for the factor kLT by applying a reduced beneficial plastic effect of about 50%. Especially for the calculation of the reduction factor χLT for lateral torsional buckling an enhanced method is given in [17], which is also compared to test results. These design formulae, presently take just moderate account of the plastic effects existing in these sections. A more extensive use of this load carrying behaviour may be expected after further studies and test confirmation, which however are still under way at present. 8. Differences between Method 1 and Method 2 As said in the introduction of this paper, EN1993-1-1 (EUROCODE 3) [1] also provides besides Method 2 the alternative of Method 1 for member buckling. However the code does not give any indication on the differences between the two methods, which had been formally poured into the same type of basic interaction formulae Eqs. (6.61) and (6.62) in EN1993-1-1 by the authors of Eurocode 3. In this respect it should also be noted that in addition to Method 1 and Method 2 the code EN1993-1-1 provides two more methods, i.e. the so-called General Method and the Stable-Length Method. Therefore, in total there are four alternative methods open to the designer, which concern member buckling under bending and axial compression. Therefore some comments might be useful for designers, if there are differences in the application range of the formulae, in

the accuracy or safety of the results and in the practical use of the two methods. In the following explanations are given to the above mentioned topics, concerning Method 1 and Method 2 although being aware that full answering of all aspects would require a paper at least so long as the existing one. The author’s objective for this paper has originally been focussed just on the background of Method 2, so that together with [3] both methods are presented similarly by their authors. A first explanation refers to differences in the meaning of the Cm -factors, which are used in the case of flexural buckling to account for non-uniform moment distribution. While the Cm factors of Method 2 are based on equivalent “uniform” moments, the Cm factors of Method 1 are based on equivalent “sinusoidal” moments. These Cm factors of Method 1—being derived from the elastic buckling theory—depend on the critical flexural buckling load, while those of Method 2 follow the Austin-formula. Because of these different basic assumptions the Cm -factors are generally different in magnitude. It is important to mention that the CmLT -factors of both methods to account for non-uniform moment distribution in the case of lateral torsional buckling are also quite different in principle. While CmLT in Method 1 is a factor connected with Cmy , the CmLT in Method 2 is an independent factor covering the effect of the relevant moment diagram between intermediate lateral restraints. Again the CmLT -factors of the two methods are generally different. The application range of the two methods is in both cases basically a single span member of doubly symmetrical crosssection subjected to axial compression, end-moments and transverse loading. In the case of Method 1 the transition from flexural buckling to lateral torsional buckling has been

768


Fig. 21. System and internal forces.

user to choose, which format would be more beneficial in his or her specific design situation. 9. Worked examples 9.1. General

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

approximated according to the torsional rigidity of the member, so that, therefore, a smooth transition between the two cases is given. Method 2 distinguishes the two standard cases of torsionally stiff and torsionally flexible members separately, which seems to be in line with the traditional thinking of design engineers. It should be mentioned that Method 2 has been developed principally in such a way that the application range also covers members with intermediate lateral restraints, which may not be effective for in plane flexural buckling. The explicitly given guidelines for the use of specific CmLT -factors makes the practical application for these cases easy. The accuracy of the two methods has been checked for each of them by statistical evaluations of the small number of available test results and the big number of theoretical FEM calculations, reported in several TC8-documents, e.g. [12]. So from this point of view the methods are equivalent in general. However, this does not exclude differences of the results in specific examples. Since Method 1 has been based on the theoretical derivation of the spatial flexural buckling modes and uses several specific factors it covers both flexural modes most closely, in particular out-of-plane buckling. Method 2 is based on the theoretical derivation of the in-plane buckling formula with just one compact factor, which was calibrated on the elastic–plastic member-capacity. Therefore, it covers in-plane buckling most closely and approximates out-of-plane buckling. In the case of lateral–torsional buckling both methods are extensions from flexural buckling calibrated mainly at the results of elastic–plastic numerical simulations. With respect to the practical use of the formulae the two methods have different objectives. Method 2 was definitely aimed at the use by hand-calculation and, therefore, used the same simple structure of the formulae as traditional interaction formulae of many existing codes. Method 1 aimed at a structure of the formulae, which is as far as possible based on elastic, theoretical derivation described by a larger number of factors. Accordingly, the use of computer aids seems to be useful and necessary for practical application. In this view it is up to the

The buckling check is based on the design values of actions and resistances as defined in [1]. The safety factors γ M0 and γ M1 have been taken as 1.0 as recommended in EC3 for buildings. 9.2. Flexural buckling of member under bending and axial compression Forces and buckling lengths, Fig. 21 N Ed = 300 kN M y,Ed = 40 kN m

(parabolic moment diagram)

L = L cr,y = L cr,z = 5.6 m cross-section: RHS 200/100/10, S235, hot-finished N Rd = 54.9 · 23.5/1.0 = 1290 kN M y,Rd = 341 · 23.5/1.0 = 8010 kN cm slenderness ratio: 560 = 0.857 λy = 6.96 · 93.9 560 = 1.50 λz = 3.98 · 93.9 buckling reduction factors χ y = 0.762

buckling curve a

χz = 0.372

buckling curve b.

Equivalent uniform moment factor and interaction factor: Cmy = 0.95 (see Table 1) ny =

300 = 0.305 0.762 · 1290

(6)

for λ y ≤ 1.0 follows k y = 1 + (0.857 − 0.2) · 0.305 = 1.20.

(5)

Verification: 0.305 + 1.20 ·

0.95 · 4000 = 0.874 ≤ 1 8010

300 = 0.625 ≤ 1. 0.372 · 1290

(1) (2b)


769

Fig. 23. System and internal forces. Fig. 22. System and internal forces.

9.3. Lateral–torsional buckling of a column in a two-storey frame The lower column of a two-storey frame with a canopy (see Fig. 22) should be checked on the basis of the equivalent column method. First order internal forces and buckling length have been taken from example 8.17 in [15]. Forces and buckling length: N Ed = 664 kN

non-uniform moment diagram to χLT,mod = χLT / f , where f has approximatively been based on kc = 0.752

for ψ = 0. f = 1 − 0.5(1 − 0.752) 1 − 2(0.784 − 0.8)2 = 0.876

χLT,mod =

0.774 = 0.884. 0.876

Equivalent uniform moment factor and interaction factors: Cmy = 0.9 for sway mode, see Table 1

L cr,y = 2.25 · 4.2 = 9.45 m

CmLT = 0.6 664 = 0.154 ny = 0.906 · 4760

L = L cr,z = 4.2 m

for λ y ≤ 1.0 follows:

cross-section: IPE 550, S355

k y = 1 + (0.555 − 0.2) · 0.154 = 1.05 664 = 0.305 nz = 0.457 · 4760

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

N Rd = 134 · 35.5/1.0 = 4760 kN M y,Rd = 2787 · 35.5/1.0 = 98900 kN cm. Critical buckling moment Mcr : Ncr,z =

π 2 · 21000 · 2670 4202

2

hereby, the factor C1 for the given moment diagram has been taken as 1.6 acc. to [15]. Slenderness ratio: 945 = 0.555 22.3 · 76.4 420 = 1.24 λz = 4.45 · 76.4 98900 = 0.784 λLT = 161000 λy =

buckling reduction factors χ y = 0.906

buckling curve a

χz = 0.457

buckling curve b

χLT = 0.774

LT-buckling curve c.

(5) (17)

0.1 · 0.305 = 0.913. 0.6 − 0.25

(15)

Verification:

420 · 124 1884000 + 2 2670 π · 2.6 · 2670 = 161000 kN cm

Mcr = 1.6 · 3140 ·

(6)

for λz ≥ 1.0 follows: kLT = 1 −

= 3140 kN

(18)

The reduction factor χLT has been determined according to EC3-1-1, 6.3.2.3, based on λ¯ LT,0 = 0.4 and β = 0.75. It has further been modified by use of the f -factor accounting for the

0.9 · 66800 = 0.876 ≤ 1 0.884 · 98900 66800 = 1.00 ≤ 1. 0.305 + 0.913 · 0.884 · 98900

0.154 + 1.05 ·

(3) (4)

9.4. Lateral–torsional buckling of a column under bi-axial bending and compression The column of a single-storey building should be checked on basis of the equivalent column method (see Fig. 23). Firstorder internal forces and buckling lengths have been taken from example 8.7 in [15]. Forces and buckling length: N Ed = 620 kN M y,Ed = 202 kN m (triangular moment diagram) Mz = Mz,Ed = 7.16 kN m (parabolic moment diagram) L cr,y = 2.3 · 4.6 = 10.6 m L = L cr,z = 4.6 m cross-section: IPE 450, S355 N Rd = 98.8 · 35.5/1.0 = 3510 kN

770


M y,Rd = 1702 · 35.5/1.0 = 60400 kN cm Mz,Rd = 276 · 35.5/1.0 = 9800 kN cm

0.214 + 0.406 + 0.07 = 0.69 ≤ 1 0.495 + 0.859 ·

critical buckling moment Mcr : Ncr,z =

π 2 · 21000 · 1680 4602

Mcr = 1.77 · 1650 · = 82600 kN cm

= 1650 kN

4602 · 67.1 791000 + 2 1680 π · 2.6 · 1680

Acknowledgements

buckling reduction factors: χ y = 0.824 buckling curve a χz = 0.357 buckling curve b χLT = 0.729 LT-buckling curve c. The reduction factor χLT has been determined according to EC3-1-1, 6.3.2.3 based on λ¯ LT,0 = 0.4 and β = 0.75. It has further been modified by use of the f -factor accounting for the triangular moment diagram with ψ = 0. f = 1 − 0.5(1 − 0.752) 1 − 2(0.855 − 0.8)2 = 0.877 0.729 = 0.831. 0.877 Equivalent uniform moment factor and interaction factors: χLT,mod =

Cmy = 0.9 for sway mode, see Table 1 Cmz = 0.95 for parabolic moment CmLT = 0.6 for triangular moment 620 = 0.214 0.824 · 3510

(6)

for λ y ≤ 1.0 follows k y = 1 + (0.750 − 0.2) · 0.214 = 1.12 620 nz = = 0.495 0.357 · 3510

(5) (11)

for λz ≥ 1.0 follows k z = 1 + 1.4 · 0.495 = 1.69 0.1 · 0.495 = 0.859. kLT = 1 − 0.6 − 0.25

(9) (15)

Verification: 0.214 + 1.12 ·

(20)

0.495 + 0.346 + 0.117 = 0.96 ≤ 1.

hereby, the factor C1 for the given moment diagram has been taken as 1.77. slenderness ratio: 1060 = 0.750 λy = 18.5 · 76.4 460 = 1.46 λz = 4.12 · 76.4 60400 = 0.855 λLT = 82600

ny =

0.95 · 716 20200 + 1.69 · ≤1 0.831 · 60400 9800

0.95 · 716 0.9 · 20200 + 0.6 · 1.69 · ≤ 1(19) 0.831 · 60400 9800

The developments of the new interaction formulae were based on a wide scope of numerical investigations, comparisons and statistical evaluations and could only be carried out through the efforts of the collaborators of the two authors. Special thanks are given to Robert Ofner, Günther Salzgeber and Peter Kaim of TU Graz and to Andreas Rusch, Junping Wang/Kunming and Stefan Heyde of TU Berlin. References [1] EN1993-1-1. Eurocode 3. Design of steel structures, general rules and rules for buildings. 2005. [2] ENV1993-1-1. Eurocode 3. Design of steel structures, general rules and rules for buildings. 1993. [3] Boissonnade N, Jaspart J-P, Muzeau J-P, Villette M. New Interaction formulae for beam-columns in Eurocode 3. The French-Belgian approach. Journal of Constructional Steel Research 2004;60:421–31. [4] Greiner R. Background information on the beam–column interaction Formulae at Level 1. ECCS Report No. TC8-2001-021, 20; Sept. 2001. [5] Greiner R, Lindner J. Die neuen Regelungen in der europäischen Norm EN1993-1-1 für Stäbe unter Druck und Biegung. Stahlbau 2003, Heft 3, p. 157–72. [6] Greiner R, Kaim P. Erweiterung der Traglastuntersuchungen an Stäben unter Druck und Biegung auf einfach—symmetrische Querschnitte. Stahlbau 2003;72(Heft 3):173–80. [7] Ofner R. Traglast von Stäben aus Stahl bei Druck und Biegung. Dissertation, Institut für Stahlbau, Holzbau und Flächentragwerke der TU Graz, Heft 9; 1997. [8] ABAQUS Software. Hibbitt, Karlsson & Sorensen Inc., Version 5.7. [9] Greiner R, Ofner R, Salzgeber G. Verification of GMNIA-results. ECCSvalidation group. Report 2; July 1998. [10] Lindner J, Rusch A, Heyde S. Evaluation of different design concepts for flexural buckling with regard to test results and ultimate load calculations. Report 2131E, TU Berlin; November 1998. [11] Lindner J. Interaktionsgleichungen für das Biegeknicken bei Druck und zweiachsiger Biegung. Schlussbericht zum DIBt-Forschungsvorhaben IV 1-5-866/98, Bericht 2135 des Instituts für Baukonstruktionen und Festigkeit der TU Berlin, 10.6.19999. [12] Lindner J, Heyde S. Evaluation of interaction formulae at Level 1 approach with regard to ultimate load calculations and test results— flexural 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. Journal of Structural Division 1961;ASCE 87(ST 4):1–32. [14] Kaim P. Spatial buckling behaviour of steel members under bending and axial compression. Ph.D. Institute for Steel Structures and Shell Structures, TU Graz; Heft 12-2004. [15] Lindner J, Schmidt H, Scheer J. Stahlbauten, Erläuterungen zur DIN 18800 Teil 1 bis 4. Berlin: Ernst & Sohn, Beuth, 3. Auflage; 1998. [16] Lindner J. Design of beams and beam columns. Progress in Structural Engineerings and Materials. 2003;5:38–47. [17] Rusch A, Lindner J. :Application of Level1 interaction formulae to class 4 sections. Thin-walled Structures 2004;42:279–93. [18] Lechner A. Plastic cross-section capacity of semi-compact steel sections. Ph.D. Institute for Steel Structures and Shell Structures, TU Graz; 2005.

Greiner Lindner EC3 PM Interaction

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