Journal of Constructional Steel Research 105 (2015) 119–128

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Journal of Constructional Steel Research

Lateral–torsional buckling resistance of cellular beams

Delphine Sonck ⁎, Jan BelisDepartment of Structural Engineering-LMO, Ghent University, Technologiepark-Zwijnaarde 904, 9052 Ghent, Belgium

⁎ Corresponding author. Tel.: +32 9 2645469.E-mail addresses: Delphine.Sonck@UGent.be (D. Sonck

http://dx.doi.org/10.1016/j.jcsr.2014.11.0030143-974X/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 3 April 2014Accepted 3 November 2014Available online xxxx

Keywords:Cellular beamsLateral–torsional bucklingResidual stressesNumerical simulationsExperimentsAbaqus

The evenly spaced circular web openings in I-section cellular beams have an advantageous effect on thematerialuse if these beams are loaded in strong-axis bending. However, not all aspects of the behaviour of such beamshave been studied adequately, such as the lateral–torsional buckling failure. For the latter failure mode, theexisting design approaches conflict.Furthermore, the detrimental effect of the modification of the residual stresses by the production process, asrecently demonstrated by the authors in previous work, was never taken into account.In this paper, the lateral–torsional buckling behaviour of cellular beams is investigated using a numerical modelthat was validated based on experimental results. In this model, the effect of the modified residual stress patternwas aptly taken into account. Using the results of the parametric study, a preliminary design approach wasproposed. This approach is based on the currently existing European guidelines for the calculation of thelateral–torsional buckling resistance of I-section beams, but with a modified calculation of the cross-sectionalproperties and a modified buckling curve selection.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Cellular beams are typically made starting from a hot-rolled parentsection with an I-section. Usually, the parent section is cut accordingto a certain pattern, after which both obtained halves are shifted andrewelded (Fig. 1). In this manner, the cellular beam height H will be40 to 60% higher than its parent section height h, resulting in a moreeconomicmaterial usewhen the beam is loaded in strong-axis bending.This fabrication method was first used for castellated beams. Thesebeams have hexagonal web openings instead of the circular openingswhich are present in the webs of cellular beams. Different web openingshapes, asymmetric or even tapered members can be obtained byvarying the cutting pattern and the properties of the parent sectionsfrom which both halves are made. However, only symmetric cellularbeams will be considered in this paper.

Due to their economic material use, cellular beams are mostly usedfor applications in which they are loaded in bending about the strongaxis, both in steel and composite steel-concrete structures. The use ofcellular members in these applications is made even more advanta-geous by guiding service ducts through the web openings instead ofunder the beams, thus diminishing the necessary total floor and build-ing construction height. Additionally, their lighter appearance alsoentails certain aesthetic advantages.

The presence of the web openings complicates the failure behaviourof cellular beams. A full overview of the failure behaviour of castellatedbeams, which will be similar to that of cellular beams, is given in [1].

), Jan.Belis@UGent.be (J. Belis).

New, local failure modes around the openings come into existence,such as web post buckling or failure by formation of a Vierendeel mech-anism around an opening. Furthermore, the already existing failuremodes for I-section members without web openings will be modified,such as lateral–torsional buckling (LTB) (Fig. 2). As will be shown inthis paper, the currently existing design methods to calculate the LTBresistance of cellular members are conflicting and the correspondingresearch is incomplete.

During the production of cellular beams, a thermal cuttingprocedure, such as oxycutting or plasmacutting is used for the cuttingof the web. Later in the fabrication procedure, heat is introduced asecond time during the welding of the halves. As shown in [2,3], thiswill affect the already present residual stresses in the parent sectionsin amanner which is expected to be detrimental for the lateral–torsion-al buckling resistance. However, the effect of the modified residualstresses in cellular members has not been investigated yet, and wasnot considered during the different studies corresponding with theexisting design rule proposals for the lateral–torsional bucklingresistance (cf. Section 2).

In this paper, the lateral–torsional buckling behaviour of cellularmembers subjected to a constant bending moment will be examined,considering the effect of the modified residual stress pattern causedby the production procedure. A numerical model, validated bycomparing it with experimental results, will be used to study the later-al–torsional buckling behaviour of a large variety of cellular beamgeometries. Both the elastic critical buckling moment Mcr and the LTBresistance MRd will be determined, and a preliminary design rule willbe proposed for both moment values. This design rule proposal fits inthe framework of the current European standard for the calculation of

h H

Fig. 1. Standard fabrication method of cellular members starting from a plain-webbedparent section.

M

M

Fig. 2. Lateral–torsional buckling failure of a cellular member loaded by a bendingmoment.

120 D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

steel structures [4], which will be later referred to as EC3. Thisresearch was part of a wider doctoral research about the flexural andlateral–torsional buckling behaviour of castellated and cellular mem-bers, including the effect of the production process on the residual stresspattern [3].

2. Existing research for LTB of cellular members

In this section, the currently existing research about LTB of cellularmembers will be discussed. Based on this research, a plausible designapproach will be proposed at the end of the section.

2.1. Existing research

Due to their increased height, cellularmembers aremore susceptibleto lateral–torsional buckling than plain-webbed members. This failuremode will occur for beams of which the compressed flange is insuffi-ciently supported laterally. Currently, two design approaches exist forthe LTB resistance calculation of cellular members. However, bothapproaches yield conflicting results: the first is very conservative,while the second approach could possibly be unsafe.

The first approach for the LTB resistance calculation can be found inthe design guidelines of CTICM (Centre Technique Industriel de laConstruction Métallique) [5]. This approach reduces the LTB check of acellular member to a lateral flexural buckling check of the compressedtee of the beam (1T approach in Fig. 3). The flexural buckling load ofthe compressed tee can be calculated using the EC3 buckling curve c[4] and the cross-sectional properties of this tee. Since the stabilizingeffect of the part of the beam in tension is completely neglected, as ifthe torsional stiffness of the beam were non-existent, this approach isvery conservative [6–8]. Due to its conservative nature, this approachwill not be discussed further.

According to the second approach, which will be referred to as the2T approach, the LTB resistance of a cellular member can be calculatedsimilar as for plain-webbed beams, but using the cross-sectionalproperties calculated at the centre of the web opening. This approachis based on the experimental work described in [9,10], and is repeatedin annex N of the European pre-standard ENV3 [11]. However, in this

1T approach: flexural buckling of compressed tee

Fig. 3. 1T and 2T approach to ca

pre-standard, no specifications are given with regard to the bucklingcurve choice, which renders this specification rather unclear.

Almost all current design guideline proposals for LTB of cellular andcastellated members are based on the 2T approach [12–15]. However,the buckling curve choice varies between the different proposals. Anoverview of the different LTB curve proposals based on the EC3 LTBdesign approach, is given in Table 1 [6,16,17]. The boundary of H/b isbased on the currently used selection criteria for the lateral–torsionalbuckling curve choice for plain-webbed members in EC3 (Table 3).These proposals are based on experimental and numerical results, asdescribed in each work. For the proposal in [16], a value of γM1 = 1.1was used, while γM1 = 1.0 is now prescribed and was also used in thelater work in [17] and [6]. It can be seen that the buckling curveproposals vary. Additionally, the effect of the modified residual stressesin cellular members was not taken into account in any of the describednumerical studies, which makes these proposals possibly unsafe.

2.2. Plausible design approach

Based on the existing research and literature, it is expected that aplausible design approach will consist of using the 2T approach, i.e.using the EC3 design method for LTB of I-section members, in whichall cross-sectional properties should be calculated at the centre of theopening. Nevertheless it is still unclear what the effect of the modifiedresidual stress pattern and the modified geometry on the bucklingcurve choice will be. A short overview of this design approach will begiven in the next paragraphs.

According to EC3, the LTB resistance of a beamMRd can be calculatedusing Eq. (1), using a reduction factor χLT. In this expression, fy is thenominal yield stress, Wy the appropriate section modulus and γM1 thepartial factor for the resistance of members to instability assessed bymember checks (forwhichEC3 recommends a value of 1.0). The authorspropose to use the plastic section modulus Wpl,y of the cross-section atthe opening for all Wy calculations, using the full plastic moment ofthe cross-section at the opening (cf. Section 4.3).

MRd ¼ χLT Wy f yγM1

ð1Þ

Currently, two methods are present in EC3 for the calculation of χLT.The GeneralMethod, described in Section6.3.2.2 of EC3uses theflexuralbuckling curves, while the Specific Method (Section 6.3.2.3 of EC3) usesa modified expression of the buckling curves and takes into account anon-constant moment distribution. The design method in this paperuses the General Method, because the Specific Method has been provento be unsafe in some cases [18–21].

According to the General Method of EC3, the reduction factor χLT iscalculated according to Eqs. (2)–(3). The relevant buckling curvedetermines the value of the imperfection factorα (Table 2). These buck-ling curves reflect the effect of various imperfections such as thegeometric imperfections or the residual stresses on the buckling resis-tance. This effect is most pronounced for intermediate slendernessvalues, as can be seen in Fig. 4. For a specific buckling curve, the reduc-tion factor can be calculated from the non-dimensional slenderness λLT

(4), using the elastic critical LTB moment Mcr. For low slendernessvalues, the bending resistance will be determined by the plastic

2T approach: lateral-torsional buckling of 2T-section

lculate the LTB resistance.

Table 1Different buckling curve proposals according to literature, in function of the height-to-width ratio H/b of the completed cellular member.

H/b ≤ 2.0 H/b N 2.0

Maquoi et al. [16] a bLakusic et al. [17] b cNseir et al. [6] c c

0 1 2 3

LT

1

LT

abcd

pl

el

Fig. 4. EC3 buckling curves.

Table 3

121D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

moment of the cross-section, and χLT = χpl = 1. For high slendernessvalues, the reduction factor χLT ¼ χel ¼ 1=λ

2LT (Fig. 4), so that the LTB

resistanceMRd is equal to the elastic critical buckling momentMcr.

χLT ¼ 1

ϕþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϕ2−λ2

LT

q ≤1 ð2Þ

ϕ ¼ 0:5 1þ αLT λLT−0:2� �þ λ2

LT

h ið3Þ

λLT ¼ffiffiffiffiffiffiffiffiffiffiffiffiWy f yMcr

sð4Þ

In this work, only the basic case of simply-supported beams withfork-supports, loaded by a constant bending moment, will be consid-ered. In this case, the elastic critical moment Mcr can be determined byEq. (5), in which L is the length of themember,GIt the torsional stiffnessof the cross-section, EIz its weak-axis bending stiffness and EIw itswarping stiffness. According to the proposed 2T design approach,the cross-sectional properties It, Iz and Iw should be calculated for thecross-section at the centre of the opening, so that a value Mcr,2T of thecritical LTB moment is obtained.

Mcr ¼ πL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGItEIz þ

π2

L2EIzEIw

sð5Þ

The buckling curves that should be used for plain-webbed sectionsare given in Table 3, with h the total depth of the cross-section and bthe cross-section width. The depth-to-width ratio h/b should take intoaccount that slender sections (h/b N 2) are more susceptible to LTBthan stocky sections.

Using the results of the numerical parametric study, the effect of themodified geometry on the critical LTB moment Mcr will be investigatedin Section 5. Furthermore, the effect of the modified geometry andresidual stresses on the LTB resistance MRd will be determined inSection 6.

3. Description of LTB experiments

In this paper, the LTB behaviour is examined using a numericalparametric study. The utilized numerical model is validated using theresults of three LTB experiments executed at Ghent University (laterreferred to as UGent) and two LTB experiments executed at the Univer-sity of Applied Sciences and ArtsWestern Switzerland in Fribourg (laterreferred to as EIA-FR). The former experiments were executed in theframework of the doctoral study described in [3], while the latter exper-iments are described in [6]. In all considered experiments, the cellularbeams were loaded in a four-point bending test setup. By considering

Table 2Imperfection factors corresponding with different buckling curves, according to EC3.

Buckling curve a b c d

Imperfection factor α 0.21 0.34 0.49 0.76

both groups of experiments for the validation, the numerical modelwill be validated over a wide slenderness range, as can be seen inTables 5 and 7, in which the numerically obtained slenderness λLT;abq

is given. The aforementioned experiments will be described in thissection.

3.1. LTB experiments at UGent

At Ghent University, the authors executed three LTB experiments oncellular beam specimens. Additionally, three castellated beamspecimens were tested as well [3]. Only the experiments on cellularbeams will be described below.

3.1.1. SpecimensThe geometry of the three cellular beam specimens, according to

Fig. 5, is given in Table 4, and the measured values of the yield stressare given in Table 5. Each specimen had the same cross-sectionalgeometry, but the length and loading points of the specimens differed.The slenderness λLT of each specimen, corresponding with the loadsetup, is given in Table 5.

In contrast with the most commonly used fabrication methoddepicted in Fig. 1, the cellular beamspecimensweremadeby oxycuttinga circular opening around each hexagonal opening of a completedcastellated member. The utilized castellated members were madestarting from an IPE160 parent section, by oxycutting the web, shiftingand rewelding the obtained beam halves. Due to these supplementarycutting operations, the residual stresses are more detrimental thanthose measured in cellular and castellated members made accordingto the standard fabrication procedure, in which the final geometry isobtained after only one cutting and welding sequence. Both residualstress patterns are depicted in Fig. 6: the residual stress patternproposed for cellular members made according to the standard fabrica-tionmethod is shown in Fig. 6a, while themeasured residual stresses inthe specimenwith the non-standard fabricationmethod are depicted inFig. 6b.

The proposed residual stress patterns are based on the measuredresidual stresses in the flanges of the castellated and cellular beamsused for the LTB experiments, as described in [2,3]. For membersmade according to the standard production method, the flange stresseswere observed to be constant along the member length, as depicted inFig. 6a. However, for the cellular members made according to the non-standard fabrication method, the compressive residual stresses in theflanges at the tee section were higher than those at the web post(Fig. 6b), due to the effect of the additional local heat introduction

Buckling curve selection for LTB according to EC3 (hot-rolled sections).

Buckling curve

h/b ≤ 2 ah/b N 2 b

a w

H

btftw

h

parent section cellular member

Fig. 5. Cellular member geometry.

Table 5Overviewof specimen lengths, number of openingsn,measured yield stress (topflange tfl,bottom flange bfl and web) and slenderness (UGent experiments).

Specimen L LF n fy,tfl fy,bfl fy,web Fcr, Fpl,abq λLT ;abq Fexp

122 D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

during the cutting of the circular openings. Since the compressiveresidual stresses in theflanges have the dominant detrimental influenceon the global buckling resistance [22–24], themeasured residual stress-es in the web are replaced by constant tensile stresses along the teesection web height, of which the magnitude can be determined byexpressing normal equilibrium. More details about the fabricationprocedure of the castellated and cellular beam specimens and theresidual stress measurements can be found in [2].

3.1.2. Experimental setup and instrumentationA sketch of the test-setup is depicted in Fig. 7. The distance between

the loads LF and the span of each specimen L are given in Table 5. TheLTB experiments were four-point bending tests executed on simplysupportedmemberswith fork-bearings at their ends. At the two loadingpoints, a steel sphere transferred the load from the load distributingbeam to the specimen. This load distributing beamwas loaded centrallyby a loading frame welded to it.

Through a tension cable connection, the loading frame was pulleddownwards by a manually operated hydraulic jack which could movelaterally by means of a roller mechanism. Consequently, the loadingframe could move freely in lateral direction and the loading wasunrestrained in lateral direction.

During the experiment, the load was measured and recorded by aload cell placed on the hydraulic jack. Secondly, the displacementswere measured at various locations using displacement transducers.Thirdly, electrical strain gages were used to measure the longitudinalstrains above the central opening, to observe more easily when thebeam started to buckle.

Each experiment started with a preload sequence of approximately10% of the expected failure load. This way, possible tolerances at thesupports were reduced. After the preload was removed, the beam wasloaded until failure, after which the beam was finally unloaded.

3.1.3. ResultsAll members failed in a lateral–torsional buckling mode. The

obtained maximum loads Fexp are given in Table 5. As expected, thefailure loads were the largest for the members with the lowest lengths(and corresponding slenderness values).

The load displacement curves of each specimen are drawn in Fig. 8,marked ‘exp’. In these curves, w is the mean vertical displacement ofthe two loading points, while φ is the mean torsional rotation of thetwo loading points. The curves clearly demonstrate the plasticbehaviour of the shorter specimen, for which a considerable plasticdeformation is visible after the complete unloading of the member.For the other two specimens, the plastic deformation is considerablysmaller, which was to be expected based on the slenderness of thespecimens.

Table 4Measured dimensions (in mm) of specimens (UGent experiments).

tf tw H b a w

7.3 5.5 220.0 83.1 142.8 67.2

For the longer CS2_L6 specimen, the load and displacement werestill slightly increasing just before the removal of the load. In this case,the manual operation of the hydraulic jack made it very difficult toincrease the load further, necessitating a premature load removal.Even if this resulted in a possible incorrect value of the failure load,the results of this experiment can still be comparedwith the numerical-ly obtained results before failure, by means of the load displacementscurves, depicted in Fig. 8.

3.2. LTB experiments at EIA-FR

At the EIA-FR in Fribourg (Switzerland), three LTB experimentswereexecuted on beams with large web openings. Since these experimentsalready have been extensively described in [6], the two experimentsexecuted on cellular beam specimens will be only concisely describedbelow.

3.2.1. SpecimensThe two cellularmember specimens each had a different parent sec-

tion. The first specimen wasmade from a HEA340 parent section, whilethe second cellular beam specimen was made starting from an IPE330parent section. The measured dimensions of both specimens are givenin Table 6, while the measured yield stress values fy and lengths of thespecimens are given in Table 7. The cellular beam specimens weremade according to the standard production procedure shown in Fig. 1.

The residual stresses were not measured for these specimens. It ishowever expected that the residual stress magnitudes will be higherfor the HEA340 specimen than for the IPE330 specimen, due to theexpected higher initial residual stresses for the more stocky HEA340parent sections [2].

3.2.2. Experimental setupAll executed experiments were four-point bending tests, for which

the points of load application and the supportswere laterally supported.At these four locations, the web was stiffened to avoid local buckling.The circular openings were only present in the central part of thebeams, since the openingswerefilled in the high shear regions betweenthe supports and the points of load application.

More details about the experimental setup, instrumentation andloading procedure can be found in [6].

3.2.3. ResultsAgain, bothmembers failed in a lateral–torsional bucklingmode. The

obtained maximum loads Fexp are given in Table 7. A curve displayingthe load in function of the vertical displacement is given in Fig. 9.Furthermore, the obtained torsional rotation φ, lateral displacement vand vertical displacement w at mid span, at the moment of failure, aregiven in Table 8. These results will be used for the validation of themodel.

4. Finite element modelling and validation

Using the validated numerical model, a parametric study wasexecuted to study the LTB behaviour of a simply supported cellular

abq

[m] [m] [MPa] [MPa] [MPa] [kN] [kN] [kN] [kN]

CS2_L3 3.15 0.21 15 342 341 329 27.8 74.6 1.64 22.34CS2_L4 3.99 1.89 19 348 351 339 25.9 104 2.00 23.62CS2_L6 6.09 1.89 29 342 341 329 8.5 52.8 2.49 6.41a

a Maximum load not reached during experiment.

H a

-100 MPa

+50 MPa

σres,web

-150 MPa

+100 MPa

σres,web

h/b>1.2 h/b≤1.2

--

--

--

--

-150 MPa

0 MPa

σres,web

- -

--

non-standard fabrication measured res cellular beams σ

standard fabrication (WP and TS)

-100 MPa

+50 MPa

σres,web

--

--(TS) (WP)

(a) (b)

Fig. 6. Residual stress patterns for cellular members derived from the results in [2]: standard fabrication method (a) and measured residual stresses in specimens made according to thenon-standard fabrication method (b). Values are given for the tee section (TS), at the opening, and for the web post (WP), between two openings.

123D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

beam. In this section, the numerical model used in this study will bedescribed first, after which the validation of the model using bothgroups of experiments will be illustrated. Lastly, the executed paramet-ric study to investigate the LTB behaviour of cellular beams will bedetailed. More details about the numerical study can be found in [3].

4.1. Numerical model used for parametric study

The LTB behaviour of the cellular beamswas investigated numerical-lywith thefinite element softwareAbaqus [25]. For each beam, the elas-tic critical moment Mcr,abq of a perfect beam was determined using aLinear Buckling Analysis (LBA). Additionally, the LTB resistance MRd,abq

of each beam was calculated through a modified Riks analysis. In thisgeometrically nonlinear analysis, a geometrically imperfect beam withresidual stresses and elastic–plastic material behaviour was loadeduntil the maximum load was reached, which will be considered thefailure load. This latter analysis type is referred usually to as a Geomet-rical and Material Nonlinear Analysis with Imperfections (GMNIA).

Fig. 7. Test setup for UGe

The geometry was simulated using quadratic shell elements withreduced integration (S8R) for both the flanges and the web, of whichthe element size was chosen after a mesh refinement study. The filletbetween the flanges and the web was not taken into account. Theboundary conditions and loads corresponded with the classical LTBsetup: a simply supported beamwith fork supports loaded by a constantbending moment. At the beam ends, the bending moments wereapplied by means of line loads acting on both the flanges and the web.

In the parametric study, only steel of grade S235 was considered,because themost detrimental effect of the residual stresses was expect-ed for this steel grade and a safe design rule would be obtained. Thematerial was modelled by a bilinear stress strain-curve without strainhardening, with a nominal yield stress fy of 235 MPa and a Young'smodulus of 210 GPa. For the elastic steel, the Poisson's ratio was 0.3.

For the nonlinear GMNIA calculations, the geometric imperfectionconsisted of a half-sine wave lateral imperfection with amplitude L/1000. For the residual stresses, the pattern derived from the measure-ments in earlier work [2,3], depicted in Fig. 6a, was used [3]. Thisresidual stress pattern, constant along the member length, depends on

nt LTB experiments.

0

10

20

-20-15-10-50

F [kN]

w [mm]

CS2_L3

abq TS

exp

0

10

20

-20-15-10-50

F [kN]

w [mm]

CS2_L4

abq TS

exp

0

10

20

-20-15-10-50

F [kN]

w [mm]

CS2_L6

abq

exp

WP

WP

WP

TS

TS

TS

-15-10-50

F [kN]

φφ [°]

CS2_L3

abq TS

exp

-15-10-50

F [kN]

φ [°]

CS2_L4

abq TS

exp

-15-10-50

F [kN]

φ [°]

CS2_L6

abq

exp

WP

WP

WP

TS

TS

TS

Fig. 8. Load displacement curves UGent: total load F vs. vertical displacement w and total load F vs. torsional rotation φ.

Table 6Measured dimensions (in mm) and yield stress fy (in MPa) of specimens (EIA-FRexperiments).

Specimen tf tw H b a w

HEA340 16 10.35 470 297.4 345 170IPE330 10.75 7.75 460 161.55 345 50

Table 7Overview of specimen lengths, average measured yield stress fy and slenderness (EIA-FRexperiments).

Specimen L LF n fy Fcr,abq Fpl,abq λLT ;abq Fexp

[m] [m] [−] [MPa] [kN] [kN] [kN] [kN]

HEA340 7.5 5.3 10 (+4 filled) 480 6397 2082 0.57 1977IPE330 11.0 7.11 17 (+10 filled) 373 233.8 361 1.24 176.9

124 D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

the height-to-width ratio h/b of the parent section. The residual stresseswere introduced using a user subroutine.

4.2. Validation of numerical model

The utilized numerical model was validated by comparing its resultswith the results from the experiments described at UGent and EIA-FR,described below.

4.2.1. LTB experiments at UGentThe model used for the validation of the numerical model was

similar to the model described above used for the parametric study,except for the adaptations described below. As dimensions, themeasured dimensions from Table 4 were used. For thematerial proper-ties, the measured values of the yield stress fy (Table 5) and a meanvalue of the modulus of elasticity E, equal to 205 GPa, were used. Theboundary conditions were still the same fork-end supports, but thebeam was now loaded by two point loads.

It was chosen to take into account the variation in flange stressesalong the member length by considering two different patterns for theresidual stresses, both constant along the member's length instead ofone pattern varying between both, to avoid complicating the residualstress introduction in the numerical model. The first pattern corre-sponds with the measured residual stresses at the web post (WP)between two openings, and the second one corresponds with the resid-ual stresses at the tee section (TS) at the centre of the opening (cf.Fig. 6b).

The applied geometric imperfection corresponded with an LTBeigenmode imperfection with amplitude L/1000, of which the shapewas derived for a simply supported member of the same length anddimensions as the considered specimen, but loaded by a constant bend-ing moment.

In Fig. 8, the comparison between the measured and numericallyobtained load–displacement diagram is depicted, using the vertical

displacement w and the torsional rotation φ described in Section 3.1.3.A comparison between the obtained failure loads for the shorterCS2_L3 and CS2_L4 specimens is given in Table 9. For the longestCS2_L6 specimen, however, only the pre-failure elastic deformationscan be compared.

Overall, a good agreement is obtained between the numerical modeland the experiments. The experimental failure loads lie between thenumerical failure loads obtained using both residual stress patterns, asexpected. For the shortest CS2_L3 specimen, the residual stress patternmeasured at the TS seems to provide the better match, while the resid-ual stress pattern at the web post seems to be better suited for theintermediate length specimen CS2_L4.

4.2.2. LTB experiments at EIA-FRSimilar as for the validation study above, the numerical model used

for this validation study was similar to that used for the parametricstudy, with a few exceptions which will be described below. For the

ww

Fig. 9. Load displacement curves EIA-FR: total load F vs. vertical displacementw.

125D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

geometry, the measured dimensions (Table 6) were used and thevertical stiffeners and filled openings were included in the model.For the material properties, the measured yield stress values fywere used (Table 7), combined with a modulus of elasticity E of205 GPa. The sideways supports at the points of load applicationand the supports weremodelled by preventing the lateral movementof the outer stiffener and flange nodes, while the vertical supportswere modelled by preventing the vertical movement of the nodesof the bottom flange at the supports. Since no residual stress mea-surements were done, the residual stress patterns proposed formembers made according to the standard fabrication method(Fig. 6a) were used. Finally, as geometric imperfection an LTB eigen-mode shaped imperfection was applied in the middle part, of whichthe amplitude was LF/1000, with LF being the length of the middlepart between both loads. The shape of the eigenmode was calculatedanalytically based on the eigenmode shape of the classical case of asimply supported member loaded by a constant bending moment,of which the length was LF.

A comparison of the obtained failure loads is given in Table 10, whilethe load–displacement behaviour measured during the experimentscan be compared with the numerical results in Fig. 9. For the latter,the vertical displacement w of the centre of the beam was used. InTable 8, the deformations at failure are compared. Again, the numericalmodel agreed well with the experimental results.

4.3. Parametric study

For the parametric study, the numerical model described inSection 4.1 was used. Six different hot-rolled parent sections wereconsidered, of which the minimum and maximum heights H and theH/b ratio of the resulting cellular members are given in Table 11. Theopening diameter a = fa.h, the width between the openings w = fw.aand the minimum length Lmin = fL.H of the beam were varied indepen-dently (Table 12). The factors fL were determined for each parentsection so that the slenderness values λLT of the corresponding groupof geometries varied around 0.5, 1, 1.5, 2 or 2.5, with a minimumvalue of fL of 5.

Taking into account the geometric constraints given in [5,11], allfeasible 167 geometries from the six parent sections were considered.According to the EC3 and ENV3 standards [4,11], all these geometrieshave class 1 or 2 cross-sections, which means that the full plasticmoment of the cross-section can be reached before local buckling effectsarise. As a result, it is acceptable to use the plastic moment in the designrule (cf. Section 2.2).

Table 8Deformations at failure: rotation φ, lateral displacement v and vertical displacementw at mid

Specimen Experimental Numerical

φexp vexp wexp φabq v

[°] [mm] [mm] [°] [

HEA340 1.88 15.6 74.7 2.36 1IPE330 4.52 24.5 62.3 4.33 3

5. Results and discussion for elastic critical LTB moment Mcr

Qualitatively, the observed failure behaviour was similar to that ofplain-webbed beams: all considered beams failed in a pure or predom-inantly lateral–torsional buckling mode and no pure local bucklingeffects were observed.

First, the obtained values of Mcr,abq were compared with the analyt-ical values Mcr,2T, obtained by using the 2T-approach and a wire modelfor the cross-section geometry, which corresponds with the use ofshell elements in the numerical model. In Fig. 10, it can be seen thatthe analytical 2T approach overestimates the numerically obtainedvalues for the smaller slenderness values λLT , rendering the proposeddesign approach unsafe. A check of the cross-sectional deformationdemonstrates that this is due to web distortion (Fig. 11). This phenom-enon occurs due to the interaction of local buckling and global LTB [26,27], which is expected to bemore distinct for beamswithweb openingsthan for plain-webbed beams [8,12,28,29]. While this phenomenon canbe observed clearly during an LBA analysis of perfectly elastic beams, itis expected that the effects of plasticity will diminish the importance ofthis unsafety. After all, the failure of the concerned geometries will begoverned by plastic yielding instead of elastic buckling. Thiswas alreadyconcluded in earlier work [7], and it is confirmed further by the resultsof the GMNIA calculations in Section 6, where no local buckling effectscould be observed. As a result, web distortion will not be taken intoaccount in the proposed design rule.

For the larger slenderness values λLT (and relatively longer lengths),an underestimation of the numerical critical moment is visible, causedby the underestimation of the torsion constant It by the 2T approach,of which the effect increases with increasing beam lengths (cf.Eq. (5)). It was decided to use a weighted average approach for thetorsion constant It to obtain a better fit for the larger lengths. In thisapproach, the torsion constant is calculated as the average torsionconstant for a beam with equivalent rectangular openings, as shownin Fig. 12. The best agreement was found for an opening length of0.9a, which results in an average torsion constant It,avg given byEq. (6). In this expression n is the number of openings, L the totalbeam length and It,full the torsion constant of the cellular beam cross-section without opening. The agreement with the analytical expressionusing theweighted average approach is also depicted in Fig. 10. It can beseen that the agreement is considerably improved for the longer beamlengths. As a result, this weighted average approach will be used inthe remainder of this paper for Mcr.

It;avg ¼ 0:9n:aL

It;2T þ 1−0:9n:aL

� �It;full ð6Þ

span.

Comparison experimental–numerical

abq wabq φexp/φabq vexp/vabq wexp/wabq

mm] [mm] [−] [−] [−]

8.7 70.7 0.80 1.06 0.835.4 62.6 1.04 1.00 0.69

Table 9Experimentally and numerically obtained failure load (UGent experiments).

Specimen Fexp Fcr,abq Fpl,abq Fabq,WP Fexp/Fabq,WP-1 Fabq,TS Fexp/Fabq,TS-1

[kN] [kN] [kN] [kN] [%] [kN] [%]

CS2_L3 22.34 27.8 74.6 23.84 −6.3 21.81 2.4CS2_L4 23.62 25.9 104 23.95 −1.3 22.06 7.1

Table 10Experimentally and numerically obtained failure load (EIA-FR experiments).

Specimen Fexp Fcr,abq Fpl,abq Fabq Fexp/Fabq-1

[kN] [kN] [kN] [kN] [%]

HEA340 1977 6397 2082 1775 11IPE330 176.9 233.8 361 179.2 −1

Table 11Overview of considered geometries in the parametric study:minimumandmaximumob-tained heights.

Parent section HE320A HE320M IPE300 HE650A HE650M IPE600

Hmin [m] 0.387 0.450 0.374 0.811 0.847 0.760Hmax [m] 0.487 0.530 0.471 1.014 1.059 0.950H/b [−] ≤2.0 ≤2.0 N2.0 N2.0 N2.0 N2.0

Table 12Overview of the independently varied parameters in the parametric study.

Parent sections HE320A HE320M IPE300 HE650A HE650M IPE600

fa 0.8 1.0 1.2fw 0.1 0.4 0.7

Fig. 11. Distortion of the web during LTB.

126 D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

6. Results and discussion for LTB resistance MRd

Again, all members failed by lateral–torsional buckling, without anyperceivable local buckling effects. Formost of the longer geometries, theload-deflection diagram kept increasing past the critical LTB moment,due to the stabilising effect of the considerable pre-buckling deforma-tions [30]. Since no value of MRd,abq could be determined for thesecases, the corresponding results were not taken into account.

The numerically obtained resistances MRd,abq can be compared withthe different possible analytical values for MRd,2T, obtained using aweighted average approach for It and a 2T approach for the othercross-sectional properties. For an easier comparison, the numericallyobtained resistanceswill be directly comparedwith the buckling curves.For each geometry, the numerically obtained slenderness λLT;abq can becalculated with Eq. (4) using the plastic section modulus of the 2TsectionWpl,2T. Furthermore, the reduction factor χabq for each geometryis calculated using Eq. (1), with MRd = MRd,abq.

comparison with Mcr,2T

0 10 20 30 40L [m]

6

4

2

0

2

4

6

Mcr

,2T

=M

cr,a

bq/M

cr,2

T-1

[%]

50

Fig. 10. Critical LTB moment: comparison of numerically obtained v

The obtained values of λLT;abq and χabq are compared with theexisting lateral–torsional buckling curves in Fig. 13. It can be seen thatvery high reduction factors χabq are obtained for the HE320M sections,while the reduction factors for the HE320A sections are much lower.This is due to the relatively high torsional stiffness of the HE320Msections compared with HE320A sections. Nevertheless, based on theH/b criterion used in the design rule proposals for LTB of cellularmembers and the original parent sections, one would expect that theresults for both parent section geometries would be similar. This devia-tion illustrates that the H/b criterion is ill-suited for the buckling curvechoice. This concern was already voiced in earlier work [31], wherereference is made to the work of Taras [32–34]. In the latter works, itis proposed to use a different LTB curve formulationwith amechanicallyconsistent background instead of just recycling the flexural–bucklingcurves whichwere derived for the flexural buckling case, as is currentlydone in EC3.

TheminimumandmaximumdeviationsΔMRd (according to Eq. (7))between the numerical results and the different buckling curves aregiven in Table 13. It can be seen that buckling curve c would be well-suited for the IPE300, HE650A, HE650M and IPE600 geometries. Forthe HE320A geometries, a buckling curve situated between curve cand d would give satisfactory results, while for the HE320M geometriesbuckling curve a could be used.

ΔMRd ¼ MRd;abq

MRd−1

� �:100% ð7Þ

If only one buckling curve should be proposed for all sections, thenbuckling curve c would overall give satisfactory results, disregardingthe small unsafe deviations for the HE320A geometries and the verylarge (safe) deviations for the HE320M geometries. However, before adefinitive proposal can be made, it should be checked whether theassumed residual stress pattern, based on [2] is also valid for heavierparent sections than those used for the residual stress measurements.Additionally, more bending moment patterns should be considered inthe numerical simulations. Using this additional information in similarstatistical evaluation as the one performed by Rebelo et al. [20], adefinitive buckling curve choice can be proposed.

Furthermore, these results can be compared with the resultspresented in [31], depicted in Fig. 13. In the latter work, the LTBresistance of cellular members was also studied, using the same

0 10 20 30 40

L [m]

6

4

2

0

2

4

6

Mcr

,avg

=M

cr,a

bq/M

cr,a

vg-1

[%]

comparison with Mcr,avg

IPE300

HE320M

HE320A

IPE600

HE650M

HE650A

50

alues Mcr,abq with analytically obtained valuesMcr,2T and Mcr,avg.

a

0.9a

Fig. 12. Equivalent rectangular openings for calculation of weighted average It,avg andcorresponding Mrc,avg.

Table 13Agreement between different buckling curves and numerical results.

ΔMRd a b c d

Min Max Min Max Min Max Min Max

HE320A −20.6 −2.6 −11.6 4.9 −5.9 13.8 3.1 30.4HE320M −2.4 5.2 4.2 16.9 11.5 29.0 22.3 48.6IPE300 −15.4 5.8 −9.3 11.9 −2.7 18.8 8.1 31.7HE650A −16.8 2.8 −10.5 9.2 −3.8 17.1 6.8 31.6HE650M −14.4 8.3 −7.8 15.7 −0.6 24.2 11.1 40.2IPE600 −20.2 3.1 −12.0 8.2 −3.5 14.8 10.5 28.1

127D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

numerical model but with a different residual stress pattern. In contrastwith the modified residual stresses used in the current work, thesimulations from [31] were executed using the original residual stresspattern in the flanges of the parent sections, as proposed by the ECCSin [35]. Thus, the influence of the production process on the residualstresses and the corresponding buckling resistance can be estimated.From the comparison of both sets of results shown in Fig. 13, it can beseen that the influence of the production process decreases the bucklingresistance with approximately one buckling curve.

The currently obtained results can also be compared with thedifferent proposals made in literature (Table 1). First of all, it shouldbe repeated that the distinction based on H/b does not seem well-suited for the LTB buckling curves. The selection proposed by Maquoiet al. [16] seems highly unsafe for all sections expect the HEM320geometries with their exceptionally high reduction factors. This couldbe due to the assumed residual stress pattern in the numericalsimulations (which corresponds with the previously described patternused in [31]), combined with the higher value of γM1 = 1.1. The buck-ling curves proposed by Lakusic et al. [17] correspondwith the findingsin thiswork forH/b≥ 2.0, but are still unsafe for theHE320A sections forwhichH/b≤ 2.0. Finally, the buckling curves proposed by Nseir et al. [6]match the current findings.

7. Conclusions

In this work, the LTB behaviour of cellular beams was investigated.Using a validated numerical model, a parametric study was executedfor a large number of cellular beam geometries. In this study, allbeams were simply supported and loaded by a constant bendingmoment. The results were used to make a preliminary proposal for adesign approach for LTB, based on the existing EC3 approach for plain-webbed I-section beams [4].

0 1 2 3 4 LT [ -]

0.5

1.0

LT [ -]

el

a

b

c

d

Lmodified σres (Fig. 6)

Fig. 13. LTB resistance: comparison of numerically obtained values χabq with buckling curves. Tright figure was obtained during a similar parametric study in which the original residual stres

In this paper, it is numerically confirmed that the already existing 2Tdesign approach is suitable to calculate the critical bucklingmomentMcr

of cellular beams. In this approach, the approach for I-section beamswithout openings is used, butwith all necessary cross-sectional proper-ties calculated at the 2T section at the centre of the opening. Possibly, aneven better agreement could be obtained by using a weighted averageapproach for the calculation of the torsion constant It, using a weightedaverage torsion constant It,avg calculated in the length-direction of abeam of which the circular openings are replaced by equivalent rectan-gular openings.

This 2T approach could also be used for the calculation of the LTBresistanceMRd, although it was unclear which buckling curve to choose.Conflicting criteriawere presented in the already existing literature, andthe effect of the influence of the modification of the residual stresses bythe production process was never taken into account. In the numericalmodel used in this work, the results from earlier residual stressmeasurements were used to propose a residual stress pattern for cellu-lar members made according to a standard production method. Usingthis model, the LTB resistance MRd of cellular beams was studied. Asalready expected, it was found that the residual stress modificationhad a detrimental effect on the buckling resistance. As a preliminaryproposal, buckling curve c is chosen for all sections. However, furtherexperimental research will have to confirm whether the used residualstress pattern is also valid for heavier sections, and whether theproposed buckling curve is valid for a wider variety of bendingmomentdiagrams.

Acknowledgements

The author would like to acknowledge Huys-liggers (Venlo, TheNetherlands) for the production of the cellular and castellatedmembersused for the residual stress measurements and LTB experiments atGhent University.

Part of the computational resources (STEVIN SupercomputerInfrastructure) and services used in this work were kindly provided byGhent University, the Flemish Supercomputer Center (VSC), theHercules Foundation and the Flemish Government — department EWI.

0 1 2 3 4 LT [ -]

0.5

1.0T [ -]

el

a

b

c

d

IPE300

HE320M

HE320A

IPE600

HE650M

HE650A

original σres parent sections

he left figure was obtained during the parametric study described in this paper, while thes pattern was used in the flanges [31].

128 D. Sonck, J. Belis / Journal of Constructional Steel Research 105 (2015) 119–128

References

[1] Kerdal D, Nethercot DA. Failure modes for castellated beams. J Constr Steel Res 1984;4(4):295–315.

[2] Sonck D, Van Impe R, Belis J. Experimental investigation of residual stresses in steelcellular and castellated members. Construct Build Mater 2014;54:512–9. http://dx.doi.org/10.1016/j.conbuildmat.2013.12.045.

[3] Sonck D. Global buckling of castellated and cellular steel beams and columns. (Phddissertation) Ghent University; 2014 (http://hdl.handle.net/1854/LU-4256332).

[4] CEN. EN 1993-1-1 Eurocode 3: design of steel structures — part 1–1: general rulesand rules for buildings; 2005.

[5] CTICM. Arcelor cellular beams: detailed technical description; 2006.[6] Nseir J, Lo M, Sonck D, Somja H, Vassart O, Boissonnade N. Lateral torsional buckling

of cellular steel beams. Proceedings of the structural stability research councilannual stability conference (SSRC2012). Grapevine, Texas; 2012.

[7] Sonck D, Vanlaere W, Van Impe R. Influence of plasticity on lateral–torsional buck-ling behaviour of cellular beams. Mater Res Innov 2011;15(Suppl. 1):158–61.http://dx.doi.org/10.1179/143307511X12858956847958.

[8] Sonck D, Vanlaere W, Lagae G, Van Impe R. Influence of web openings on failure bylateral–torsional buckling of cellular beams. XV lightweight structures in civil engi-neering: contemporary problems (LSCE2009); 2009 170–3.

[9] Nethercot DA, Kerdal D. Lateral–torsional buckling of castellated beams. J Struct Eng1982;60B(3):53–61.

[10] Gietzelt R, Nethercot DA. Biegedrillknicklasten von Wabenträgern. Stahlbau 1983;11:346–9.

[11] CEN. ENV 1993-1-1. Eurocode 3: design of steel structures. Part 1–1: general rulesand rules for buildings. Annex N: openings in webs; 1998 (CEN (European Commit-tee for Standardization); Brussels, Belgium).

[12] Zirakian T, Showkati H. Distortional buckling of castellated beams. J Constr Steel Res2006;62(9):863–71. http://dx.doi.org/10.1016/j.jcsr.2006.01.004.

[13] Mohebkhah A. Themoment-gradient factor in lateral–torsional buckling on inelasticcastellated beams. J Constr Steel Res 2004;60(10):1481–94. http://dx.doi.org/10.1016/j.jcsr.2004.02.002.

[14] Sweedan AMI. Elastic lateral stability of I-shaped cellular steel beams. J Constr SteelRes 2011;67(2):151–63. http://dx.doi.org/10.1016/j.jcsr.2010.08.009.

[15] El-Sawy K, Sweedan A, Martini M. Moment gradient factor of cellular steel beamsunder inelastic flexure. J Constr Steel Res 2014;98:20–34. http://dx.doi.org/10.1016/j.jcsr.2014.02.007.

[16] Maquoi R, Heck C, de Ville de Goyet V, Braham M, Richard C, Müller C, et al. Lateraltorsional buckling in steel and composite beams — final report — books 1 and 2.European Commision; 2003.

[17] Lakusic VT, Dzeba I, Androic B. The buckling curve for lateral–torsional bucklingresistance of castellated beams. Proceedings of 5th European Conference on Steeland Composite Structures (Eurosteel 2008); 2008. p. 1587–92 (ECCS (EuropeanConvention for Constructional Steelwork); Graz, Austria).

[18] Snijder HH, Hoenderkamp JCD. Buckling curves for lateral torsional buckling of un-restrained beams. In: de Ville de Goyet V, Jaspart JP, Rondal J, editors. Proceedingsof the Hommages á René Maquoi birthday anniversary 2007:239–48 (Universitéde Liège; Liège).

[19] Snijder HH, Hoenderkamp JCD, Bakker MCM, Steenbergen HMGM, Bruins RHJ. LTBof steel beams with restraints between the supports. Proceedings of Eurosteel2008, vol. 1. 2008 (ECCS (European Convention for Constructional Steelwork);Graz, Austria).

[20] Rebelo C, Lopes N, Simões da Silva L, Nethercot DA, Vila Real PMM. Statisticalevaluation of the lateral–torsional buckling resistance of steel I-beams, Part 1:variability of the Eurocode 3 resistance model. J Constr Steel Res 2009;65(4):818–31. http://dx.doi.org/10.1016/j.jcsr.2008.07.016.

[21] Villette M. Propositions pour de nouvelles courbes de déversement. Constr Mét2002;2:17–39.

[22] Beedle LS, Huber AW. Residual stress and the compressive properties of steel — asummary report. Fritz laboratory reports, paper 46; 1957.

[23] Beedle LS, Tall L. Basic column strength. J Struct Div 1960;86(7):139–73.[24] Galambos TV. Inelastic lateral buckling of beams. J Struct Div ASCE 1963;89(236).[25] Dassault Systèmes. Abaqus v6.12; 2011.[26] Bradford MA. Lateral–distortional buckling of steel I—section members. J Constr

Steel Res 1992;23(1–3):97–116.[27] Schafer BW, Ádány S. Understanding and classifying local, distortional and global

buckling in open thin-walled members. Proceedings of the Structural StabilityResearch Council Annual Stability Conference (SSRC2005); 2005. p. 27–46[Montreal].

[28] Ellobody E. Interaction of buckling modes in castellated steel beams. J Constr SteelRes 2011;67(5):814–25. http://dx.doi.org/10.1016/j.jcsr.2010.12.012.

[29] Ellobody E. Nonlinear analysis of cellular steel beams under combined bucklingmodes. Thin-Walled Struct 2012;52:66–79. http://dx.doi.org/10.1016/j.tws.2011.12.009.

[30] Trahair NS. Flexural–torsional buckling behaviour of structures. In: Chapman & Hall,editor. E & FN Spon; 1993.

[31] Sonck D, Boissonnade N, Van Impe R. Instabilities of cellular members loaded inbending or compression. Proceedings of the structural stability research councilannual stability conference (SSRC2012). Grapevine, Texas; 2012.

[32] Taras A, Greiner R. New design curves for lateral–torsional buckling. Proposal basedon a consistent derivation. J Constr Steel Res 2010;66(5):648–63. http://dx.doi.org/10.1016/j.jcsr.2010.01.011.

[33] Taras A, Greiner R. Development of consistent buckling curves for torsional andlateral–torsional buckling. Steel Constr 2008;1(1):42–50. http://dx.doi.org/10.1002/stco.200890005.

[34] Taras A. Contribution to the development of consistent stability design rules for steelmembers. Graz University of Technology; 2010 (PhD dissertation).

[35] ECCS. Ultimate limit state calculations of sway frames with rigid joints; 1984 (no.33).