Interaction of Buckling Modes in Castellated Steel Beams

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pFinite element modellingNormal and high strengthSteel structuresStructural design

the effects of the change in cross-section geometries, beam length and steel strength on the strengthand buckling behaviour of castellated steel beams. The parametric study has shown that the presenceof web distortional buckling causes a considerable decrease in the failure load of slender castellated steelbeams. It is also shown that the use of high strength steel offers a considerable increase in the failure loadsof less slender castellated steel beams. The failure loads predicted from the finite element model werecompared with that predicted from Australian Standards for steel beams under lateral torsional buckling.It is shown that the Specification predictions are generally conservative for normal strength castellatedsteel beams failing by lateral torsional buckling, unconservative for castellated steel beams failing by webdistortional buckling and quite conservative for high strength castellated steel beams failing by lateraltorsional buckling.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Castellated steel beams fabricated from standard hot-rolledI-sections have many advantages including greater bendingrigidity, larger sectionmodulus, optimum self-weightdepth ratio,economic construction, ease of services through the web openingsand aesthetic architectural appearance. However, the castellationof the beams results in distinctive failure modes depending ongeometry of the beams, size of web openings, web slenderness,type of loading, quality of welding and lateral restraint conditions.The failure modes comprise shear [1,2], flexural [2], lateraltorsional buckling [3], rupture of welded joints [4] and web post-buckling failure modes [5,6]. Investigation of these failure modeswas previously detailed by Kerdal and Nethercot [7]. Also, adetailed review of the experimental and theoretical investigationson the failure modes of castellated beams was presented byDemirdjian [8].

Extensive experimental and numerical investigations werefound in the literature highlighting the distortional bucklingbehaviour of doubly symmetric steel I-sectionsmainly by Bradford[9,10], Vrcelj and Bradford [11] and Zirakian [12]. However, veryfew tests were found in the literature on the distortional bucklingbehaviour of castellated beams. These tests were carried out by

Tel.: +20 40 3315860; fax: +20 40 3315860.E-mail addresses: [email protected], [email protected].

Zirakian and Showkati [13] and provided useful information in theform of failure loads, failure modes, loadlateral deflection curvesand loadstrain curves that could be used in developing finiteelement models. Finite element modelling could provide betterunderstanding for interaction of lateral torsional and distortionalbuckling behaviour of castellated beams and compensate the lackin the tests on this form of construction. However, accurate finiteelement modelling of the buckling behaviour of castellated steelbeams is quite complicated due to the presence of the initialgeometric imperfections, web openings, lateral buckling restraintsand loading conditions. Hence, to date, there is no detailed finiteelement model in the literature highlighting the interaction ofbuckling modes in castellated beams, which is addressed in thisstudy.

Current design rules specified in the American Institute forSteel Construction AISC [14], Australian Standards [15], BritishStandards 5950 [16] and Eurocode 3 (BS EN 1993-1-1) [17] areapplicable to normal strength steel Grades S235 to S460. Although,Eurocode 3 (BS EN 1993-1-12) [18] has proposed additional rulesfor the extension of Eurocode 3 up to steel grades S700, this wasnot investigated on the behaviour of steel beams under combinedbuckling modes. Furthermore, the behaviour of castellated beamsunder distortional buckling was not specified in the specifications,with only limited design rules were given in the AustralianStandards AS4100 [15] and American Standards AISC [14] fordoubly symmetric I-sections. Hence, the behaviour of normal andJournal of Constructional Stee

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Journal of Construct

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Interaction of buckling modes in castellaEhab Ellobody Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egy

a r t i c l e i n f o

Article history:Received 8 October 2010Accepted 13 December 2010

Keywords:Castellated beamsBuckling modes

a b s t r a c t

This paper investigates the belateral torsional and distortiodeveloped for the analysis owere carefully considered inon castellated beams havingof buckling modes as well asin this study. An extensive0143-974X/$ see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2010.12.012Research 67 (2011) 814825

le at ScienceDirect

onal Steel Research

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ed steel beams

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haviour of normal andhigh strength castellated steel beamsunder combinednal bucklingmodes. An efficient nonlinear 3D finite elementmodel has beenthe beams. The initial geometric imperfection and material nonlinearitiesthe analysis. The nonlinear finite element model was verified against testsdifferent lengths and different cross-sections. Failure loads and interactionloadlateral deflection curves of castellated steel beams were investigatedarametric study was carried out using the finite element model to study

E. Ellobody / Journal of Constructiona

Nomenclature

B Overall flange width of castellated beamb1 Dimension of castellated beamb2 Dimension of castellated beamb3 Dimension of castellated beamb4 Dimension of castellated beamCOV Coefficient of variationD Overall depth of cross-section (larger dimension)E Youngs modulus of steelfy Yield stress of steelfu Ultimate stress of steelG Shear modulus of steelH Overall height of castellated beamh Height of castellated beam in finite element modelh1 Height of openinghw Height of webIy Minor axis section moment of areaIw Warping section constantJ Uniform torsion section Warping section constantL Length of castellated beamLTB Lateral torsional buckling failure modelu Lateral unsupported length of castellated beamMb Nominal buckling moment strengthMP Proposed buckling moment strengthMpx Major axis full plastic momentMyz Elastic buckling momentPAS4100 Unfactored design load calculated using the Aus-

tralian StandardsPFE Failure load from finite element analysisPTest Failure load from testsPTheory Failure load from theoretical analysisSx Plastic section modulusSY Steel yielding failure modes Web thickness of speciment Flange thickness of specimenWD Web distortion failure modem Moment modification factors Slenderness reduction factory Yield stress factor Nondimensional slenderness.

high strength castellated steel beams under combined bucklingmodes including distortional buckling is also addressed in thisstudy.

The main objective of this paper is to develop an efficient non-linear 3D finite elementmodel highlighting the buckling behaviourof castellated beams. The finite element programABAQUS [19]wasused in the analysis, which considered the inelastic material prop-erties of flange and web portions of beams and initial geomet-ric imperfections. The failure loads, failure modes and loadlateraldeflection curves were predicted using the finite element modeland compared against published experimental results. Paramet-ric study was performed to investigate the effect of cross-sectiongeometries, steel beam length, steel strength and nondimensionalslenderness on the failure loads and buckling behaviour of castel-lated beams. The failure loads predicted from the parametric studywere compared with that predicted from Australian Standards forsteel beams under lateral buckling.

2. Summary of experimental investigation

The tests on simply supported castellated steel beams under

distortional buckling were conducted by Zirakian and Showkati[13]. The castellated beamswere loaded with central concentratedl Steel Research 67 (2011) 814825 815

load applied through 100 100 100 mm steel cubes. Lateraldeflections were prevented at mid-span and at a distance of165 mm from the supports using lateral bracing. The testingprogram included six full-scale tests having nominal depths of180 and 210 mm and lengths of 3600, 4400 and 5200 mm. Thecastellated beams were fabricated from hot-rolled standard IPE120 and IPE 140, with the castellation dimensions and notationsshown in Fig. 1 and summarized in Table 1. The test specimenswere labeled such that the nominal height and length couldbe identified from the label. For example, the label C180-3600identifies that the castellated beam denoted by the letter C hasa nominal height of 180 mm and a length of 3600 mm.

The castellated beam tests [13] were designed so that the topcompression flange of the beam (Fig. 1) was restrained against lat-eral buckling at mid-span and near the support. Hence, the cross-section at quarter-span was subjected to unrestrained distortionalbuckling while the cross-section at mid-span was subjected torestrained distortional buckling. More details regarding the twobuckling modes were previously given in [9,13]. According to thistest setup, measurements of lateral deflections and strains weretaken at the mid-span and quarter-span locations. The load wasapplied step-by-step until failure occurred. Failure was identifiedwhen the lateral deflections were large at quarter-span locationsand unloading took place. The material properties of flange andweb portions were determined from tensile coupon tests takenfrom the two hot-rolled standard IPE 120 and IPE 140 sections.The yield stresses (fy) of flange and web portions of IPE 120 were279 and 234 MPa, respectively, while that of IPE 140 were 280and332MPa, respectively. Further details regarding the castellatedbeam tests are given in [13].

3. Finite element modelling

3.1. General

In this study, the finite element program ABAQUS [19] wasused in the analysis of castellated steel beams tested by Zirakianand Showkati [13]. The model has accounted for the measuredgeometry, initial geometric imperfections and measured materialproperties of flange and web portions. Finite element analysisfor bucking requires two types of analyses. The first is knownas Eigenvalue analysis that estimates the buckling modes andloads. Such analysis is linear elastic analysis performed with theload applied within the step. The buckling analysis provides thefactor by which the load must be multiplied to reach the bucklingload. For practical purposes, only the lowest buckling modepredicted from the Eigenvalue analysis is used. The second is calledloaddisplacement nonlinear analysis and follows the Eigenvalueprediction. It is necessary to consider whether the post-bucklingresponse is stable or unstable. The nonlinear material propertiesand loading conditions are incorporated in the loaddisplacementnonlinear analysis. It should be noted that previous studies by theauthor [2023] have shown that the residual stresses had a smalleffect on the buckling behaviour of different structural members,hence it is not included in the present study.

3.2. Finite element type and mesh

A combination of 4-node and 3-node doubly curved shellelements with reduced integration S4R and S3R, respectively,were used to model the flanges and web of the castellatedsteel beams, as shown in Fig. 1. The elements are suitable forcomplex buckling behaviour. The S4R and S3R elements havesix degrees of freedom per node and provide accurate solutions

to most applications. The elements allow for transverse sheardeformation which is important in simulating thick shell elements

load was applied in increments as concentrated static load overthe spreader block area, which is also identical to the experimentalinvestigation. The nonlinear geometry was included to deal withthe large displacement analysis.

3.4. Material modelling of castellated steel beams

The stressstrain curve for the structural steel given in theEC3 [17] was adopted in this study with measured values of theyield stress (fys) and ultimate stress (fus) used in the tests [13]. Thematerial behaviour provided by ABAQUS [19] (using the PLASTICoption) allows a nonlinear stressstrain curve to be used. The firstpart of the nonlinear curve represents the elastic part up to theproportional limit stress with Youngs modulus of (E) 200 GPa andPoissons ratio of 0.3 were used in the finite element model. Sincethe buckling analysis involves large inelastic strains, the nominal

and 3 show examples of unrestrained between ends and restrainedbuckling modes along the compression flange of castellated steelbeams, respectively. Only the first bucklingmode (Eigenmode 1) isused in the Eigenvalue analysis. Since buckling modes predictedby ABAQUS Eigenvalue analysis [19] are generalized to 1.0, thebuckling modes are factored by a magnitude of Lu/1000, whereLu is the length between points of effective bracing. The mag-nitude of Lu/1000 is a generally accepted average value for testmeasurements as recommended in Refs. [2024]. The factoredbuckling mode is inserted into the loaddisplacement nonlinearanalysis of the castellated beams following the Eigenvalue predic-tion. It should be noted that the investigation of castellated beamswith different slenderness ratios could result in lateral torsionalbuckling mode with or without web distortional buckling mode.Hence, to ensure that the correct buckling mode is incorporatedin the nonlinear displacement analysis, the Eigenvalue bucklingadequate accuracy in modelling the web while a finer mesh ofapproximately 8 15 mmwas used in the flange.

3.3. Boundary conditions and load application

Only half of the castellated beam tests was modelled due tosymmetry as shown in Fig. 1. Since the lateral bracing systemused in [13] was quite rigid, the top compression flange wasprevented from lateral displacements atmid-span and at a distanceof 165 mm from the support, which is identical to the tests. The

members as a result of the fabrication process. Previous investi-gations shared by the author have successfully modelled the initialgeometric imperfections in different structural sections [2023].Buckling of castellated beams depends on the lateral restraint con-ditions to compression flange and geometry of the beams. Mainlytwo buckling modes detailed in [9,13] could be identified asunrestrained and restrained lateral distortional buckling modes.Following the same approach, [2023], the lateral distortionalbucklingmodes could be obtained by performing Eigenvalue buck-ling analysis [19] for castellated beams with actual geometry andactual lateral restraint conditions to the compression flange. Figs. 2816 E. Ellobody / Journal of Constructiona

Fig. 1. Definition of symbols and finit

Table 1Dimensions and material properties of castellated beams.

Test Dimensions (mm)H B t s L h

C180-3600 176.3 64 6.3 4.4 3600 170.0C180-4400 176.3 64 6.3 4.4 4400 170.0C180-5200 176.7 64 6.3 4.4 5200 170.4C210-3600 206.5 73 6.9 4.7 3600 199.6C210-4400 210.3 73 6.9 4.7 4400 203.4C210-5200 211.7 73 6.9 4.7 5200 204.8

(thickness is more than about 1/15 the characteristic length ofthe shell). The elements allow for the freedom in dealing withfurther parametric studies on slender and compact sections. Theelement also account for finite strain and suitable for large strainanalysis as recommended by ABAQUS [19] and Refs. [2023].Since lateral buckling of castellated beams is very sensitive tolarge strains, the S4R and S3R element were used in this studyto ensure the accuracy of the results. In order to choose thefinite element mesh that provides accurate results with minimumcomputational time, convergence studies were conducted. It isfound that approximately 15 15 mm (length by width ofS4R element and depth by width of S3R element) ratio provides(engineering) static stressstrain curves were converted to truestress and logarithmic plastic true strain curves. The true stressl Steel Research 67 (2011) 814825

e element mesh for castellated beam.

Material properties fy (MPa) Ref.h1 b1 b2 b3 b4 Flange Web

60 30 60 60 180 279 23460 30 60 60 180 279 23460 30 60 60 180 279 234 [13]70 35 70 70 210 280 33270 35 70 70 210 280 33270 35 70 70 210 280 332

(true) and plastic true strain (pltrue) were calculated using Eqs. (1)

and (2) as given by ABAQUS (2004):

true = (1+ ) (1)pltrue = ln(1+ ) true/Eo (2)where Eo is the initial Youngs modulus, and are the measurednominal (engineering) stress and strain values, respectively.

3.5. Modelling of initial geometric imperfections

Initial geometric imperfections are found in structural steelanalysis must be performed for each castellated beam with actualgeometry.

o(PTest/Calculated) as reported in [13] and finite element analysesperformed in this study (PFE). It can be seen that good agreementbetween the test/calculated and finite element results. The meanvalue of PTest/Calculated/PFE ratio is 1.01 with the coefficient ofvariation (COV) of 0.020, as shown in Table 2. Three failure modeswere observed experimentally and verified numerically using thefinite element model as summarized in Table 2. All the tested

The loadlateral deflection curves predicted experimentallyand numerically were also compared as shown in Fig. 4. The curveswere plotted as an example at quarter-span of test specimen C180-3600 at three locations. The locations are the top, middle andbottom points of the web of castellated beam. It can be shown thatgenerally good agreement was achieved between experimentaland numerical relationships. The maximum measured lateral

Table 2Comparison of test and finite element results.

Test/Calculated [Ref.] Test/Theoretical Finite element analysis PTest/CalculatedPFEPTest/Calculated (kN) Failure mode PFE (kN) Failure mode

C180-3600 [13] 21.58 LTB+ SY+WD 21.6 LTB+ SY+WD 1.00C180-4400 [13] 15.63 LTB+ SY+WD 16.0 LTB+ SY+WD 1.02C180-5200 [13] 14.48a LTB+WD 14.8 LTB+WD 1.02C210-3600 [13] 37.22 LTB+ SY+WD 37.9 LTB+ SY+WD 1.02C210-4400 [13] 28.91a LTB+ SY+WD 28.0 LTB+ SY+WD 0.97C210-5200 [13] 24.90 LTB+WD 25.4 LTB+WD 1.02Mean 1.01Fig. 3. Restrained elastic lateral distortional buckling mode (Eigenmode 1) for a castellated beam laterally restrained along the compression flange.

4. Verification of finite element model

The developed finite element model for castellated beamsunder distortional buckling was verified against the test resultsdetailed in [13]. The failure loads, failure modes and loadlateraldeflection curves obtained experimentally and numerically usingthe finite element model were compared. Table 2 shows acomparison between the failure loads obtained from the tests aswell as calculated using the design equation proposed in Ref. [25]

castellated beams [13] underwent lateral torsional buckling (LTB)and web distortion (WD), while steel yielding (SY) was observedin castellated beams with lengths of 3600 and 4400 mm. TheSY failure mode was predicted from the finite element modelby comparing the Von Mises stresses in the castellated beams atfailure against the measured yield stresses. On the other hand,the SY was judged in the tests by comparing the test failure loadsagainst the plastic collapse loads (Ppx) calculated according toAS4100 [15].Fig. 2. Unrestrained elastic lateral distCOV a Denotes predicted failure loads using proposed equation given in Ref. [25].rtional buckling mode (Eigenmode 1).E. Ellobody / Journal of Constructional Steel Research 67 (2011) 814825 817 0.020

xmode observed experimentally and confirmed numerically wasa combination of lateral torsional buckling (LTB), web distortion(WD) and steel yielding (SY). The data obtained from ABAQUS [19]has shown that the Von Mises stresses at the maximum stressed

(a) Numerical. (b) EFig. 5. Comparison of experimental and numerical bwith different steel yield and ultimate stresses. Groups G4G6 hadthe same steel stresses as G1G3, respectively. The web openingdimensions for the castellated beams in G4G6 were also identicalto C210-3600 [13].

perimental [13].818 E. Ellobody / Journal of Constructional Steel Research 67 (2011) 814825

Fig. 4. Comparison of loadlateral deflection curves at quarter-span of testspecimen C180-3600.

deflections from the tests at failure were 5.6, 2.8 and 0.8 mm attop, middle and bottom points of the web, respectively, comparedwith 6.4, 3.8 and 0.6 mm, respectively, from the finite elementanalysis as shown in Fig. 4. The positive sign represents the lateraldeflection in front of the web and the negative sign represents thelateral deflection in back of the web.

Furthermore, the deformed shapes of castellated beams atfailure observed experimentally and numerically were compared.Fig. 5 shows as an example of the buckled shape observed in thetest specimen C210-4400 in comparison with that predicted fromthe finite element analysis. It can be seen that the experimentaland numerical deformed shapes are in good agreement. The failure

fibers at the top and bottom flanges at mid-span exceeded themeasured yield stresses.

5. Parametric study

The verified finite element model was used to study theeffects of the change in cross-section geometries, beam length,steel strength and nondimensional slenderness on the strengthand buckling behaviour of castellated steel beams. Ninety-sixcastellated steel beams were analysed using the finite elementmodel. Tables 3 and 4 summarize the dimensions and materialproperties of the castellated steel beams. The beams were dividedinto 24 groups denoted G1G24. The first twelve groups G1G12had a length of 3600 mm, while groups G13G24 had a length of5200 mm. Group G1 had four castellated beams S1S4 having anoverall height (H) of 176.3 mm, a width (B) of 64 mm and a webthickness (s) of 4.4 mm, which is identical to the tested castellatedsteel beam C180-3600 [13], but with different flange thickness (t)of 2, 4, 6 and 8 mm, respectively. This has resulted in B/t ratiosof 32, 16, 10.7 and 8 for specimens S1S4, respectively. GroupG1 had a steel yield stress (fy) of 275 and an ultimate stress (fu)of 430. Groups G2 and G3 were identical to G1 except with fy of460, and 690 MPa and fu of 530 and 760 MPa, respectively. Theyield and ultimate stresses conform to EC3 [18]. The web openingdimensions in G1G3were also identical to C180-3600 [13]. GroupG4 had four specimens S13S16 having H of 210.9 mm, B of73 mm and s of 4.7 mm but with different t of 2, 4, 6 and 8 mm,respectively, which is identical to the tested castellated steel beamC210-3600 [13]. The B/t ratios of S13S16 were 36.5, 18.3, 12.2and 9.1, respectively. Groups G5 and G6 were identical to G4 butuckled shapes at failure for specimen (C210-4400).

S26 176.3 64 6.3 4.0 3600 1S27 176.3 64 6.3 6.0 3600 1S28 176.3 64 6.3 8.0 3600 1

G8 S29 176.3 64 6.3 2.0 3600 1S30 176.3 64 6.3 4.0 3600 1S31 176.3 64 6.3 6.0 3600 1S32 176.3 64 6.3 8.0 3600 1

G9 S33 176.3 64 6.3 2.0 3600 1S34 176.3 64 6.3 4.0 3600 1S35 176.3 64 6.3 6.0 3600 1S36 176.3 64 6.3 8.0 3600 1

G10 S37 210.9 73 6.9 2.0 3600 2S38 210.9 73 6.9 4.0 3600 2S39 210.9 73 6.9 6.0 3600 2S40 210.9 73 6.9 8.0 3600 2

G11 S41 210.9 73 6.9 2.0 3600 2S42 210.9 73 6.9 4.0 3600 2S43 210.9 73 6.9 6.0 3600 2S44 210.9 73 6.9 8.0 3600 2

G12 S45 210.9 73 6.9 2.0 3600 2S46 210.9 73 6.9 4.0 3600 2

S47 210.9 73 6.9 6.0 3600 2S48 210.9 73 6.9 8.0 3600 270.0 60 30 60 60 180 275 43070.0 60 30 60 60 180 275 43070.0 60 30 60 60 180 275 430

70.0 60 30 60 60 180 460 53070.0 60 30 60 60 180 460 53070.0 60 30 60 60 180 460 53070.0 60 30 60 60 180 460 530

70.0 60 30 60 60 180 690 76070.0 60 30 60 60 180 690 76070.0 60 30 60 60 180 690 76070.0 60 30 60 60 180 690 760

04.0 70 35 70 70 210 275 43004.0 70 35 70 70 210 275 43004.0 70 35 70 70 210 275 43004.0 70 35 70 70 210 275 430

04.0 70 35 70 70 210 460 53004.0 70 35 70 70 210 460 53004.0 70 35 70 70 210 460 53004.0 70 35 70 70 210 460 530

04.0 70 35 70 70 210 690 76004.0 70 35 70 70 210 690 760E. Ellobody / Journal of Constructional Steel Research 67 (2011) 814825 819

Group G7 had four castellated beams S25S28 having H of176.3 mm, B of 64 mm, t of 6.3 mm, which is identical to thetested castellated steel beam C180-3600 [13], but with different sof 2, 4, 6 and 8 mm, respectively. This has resulted in h/s ratiosof 85, 42.5, 28.3 and 21.3 for specimens S25S28, respectively.GroupsG8 andG9were identical toG7butwith different steel yieldand ultimate stresses. Groups G7G9 had the same steel stressesas G1G3, respectively. The web opening dimensions in G7G9were also identical to C180-3600 [13]. Finally, Group G10 had fourspecimens S37S40 having H of 210.9 mm, B of 73 mm and t of6.9mmbutwith different s of 2, 4, 6 and 8mm, respectively, whichis identical to the tested castellated steel beam C210-3600 [13].The h/s ratios of S37S40 were 102, 51, 34 and 25.5, respectively.Groups G11 and G12 were identical to G10 but with different steelyield and ultimate stresses. Groups G10G12 had the same steelstresses as G1G3, respectively. The web opening dimensions forthe castellated beams in G10G12 were also identical to C210-3600 [13]. The castellated steel beams S49S96 inGroupsG13G24

were identical to S1S48 in G1G12 but with different steel beamlength of 5200 mm instead of 3600 mm, respectively. The detaileddimensions of the castellated steel beams are shown in Tables 3and 4. The investigated castellated steel beams had differentnondimensional slenderness () calculated based on AS4100 [15]ranged from 1.03.1. The nondimensional slenderness () is equalto the square root of the major axis full plastic moment divided bythe elastic bucklingmoment, and is considered as a guide for beamslenderness in this study.

The failure loads (PFE) and failure modes of the castellated steelbeams predicted from the finite element analyses are summarizedin Tables 5 and 6. Looking at Tables 5 and 6, it can be seen that thefailure loads of the castellated beams showed logical and expectedresults, with less slender beams followed a more plastic collapsemode and are obviously driven by the steel strength. The moreslender the beam the more elastic buckling we will have andcollapse behaviour is dependent on the lateral torsional and webdistortional buckling behaviour of the beam. It can also be seen

Table 3Dimensions and material properties of castellated beams in the parametric study.

Group Specimen Dimensions (mm) Material propertiesH B t s L h h1 b1 b2 b3 b4 fy (MPa) fu (MPa)

G1 S1 176.3 64 2.0 4.4 3600 170.0 60 30 60 60 180 275 430S2 176.3 64 4.0 4.4 3600 170.0 60 30 60 60 180 275 430S3 176.3 64 6.0 4.4 3600 170.0 60 30 60 60 180 275 430S4 176.3 64 8.0 4.4 3600 170.0 60 30 60 60 180 275 430

G2 S5 176.3 64 2.0 4.4 3600 170.0 60 30 60 60 180 460 530S6 176.3 64 4.0 4.4 3600 170.0 60 30 60 60 180 460 530S7 176.3 64 6.0 4.4 3600 170.0 60 30 60 60 180 460 530S8 176.3 64 8.0 4.4 3600 170.0 60 30 60 60 180 460 530

G3 S9 176.3 64 2.0 4.4 3600 170.0 60 30 60 60 180 690 760S10 176.3 64 4.0 4.4 3600 170.0 60 30 60 60 180 690 760S11 176.3 64 6.0 4.4 3600 170.0 60 30 60 60 180 690 760S12 176.3 64 8.0 4.4 3600 170.0 60 30 60 60 180 690 760

G4 S13 210.9 73 2.0 4.7 3600 204.0 70 35 70 70 210 275 430S14 210.9 73 4.0 4.7 3600 204.0 70 35 70 70 210 275 430S15 210.9 73 6.0 4.7 3600 204.0 70 35 70 70 210 275 430S16 210.9 73 8.0 4.7 3600 204.0 70 35 70 70 210 275 430

G5 S17 210.9 73 2.0 4.7 3600 204.0 70 35 70 70 210 460 530S18 210.9 73 4.0 4.7 3600 204.0 70 35 70 70 210 460 530S19 210.9 73 6.0 4.7 3600 204.0 70 35 70 70 210 460 530S20 210.9 73 8.0 4.7 3600 204.0 70 35 70 70 210 460 530

G6 S21 210.9 73 2.0 4.7 3600 204.0 70 35 70 70 210 690 760S22 210.9 73 4.0 4.7 3600 204.0 70 35 70 70 210 690 760S23 210.9 73 6.0 4.7 3600 204.0 70 35 70 70 210 690 760S24 210.9 73 8.0 4.7 3600 204.0 70 35 70 70 210 690 760

G7 S25 176.3 64 6.3 2.0 3600 170.0 60 30 60 60 180 275 43004.0 70 35 70 70 210 690 76004.0 70 35 70 70 210 690 760

G16 S61 210.9 73 2.0 4.7 5200 204.0 70 35 70 70 210 275 430S62 210.9 73 4.0 4.7 5200 204.0 70 35 70 70 210 275 430S63 210.9 73 6.0 4.7 5200 204.0 70 35 70 70 210 275 430S64 210.9 73 8.0 4.7 5200 204.0 70 35 70 70 210 275 430

G17 S65 210.9 73 2.0 4.7 5200 204.0 70 35 70 70 210 460 530S66 210.9 73 4.0 4.7 5200 204.0 70 35 70 70 210 460 530S67 210.9 73 6.0 4.7 5200 204.0 70 35 70 70 210 460 530S68 210.9 73 8.0 4.7 5200 204.0 70 35 70 70 210 460 530

G18 S69 210.9 73 2.0 4.7 5200 204.0 70 35 70 70 210 690 760S70 210.9 73 4.0 4.7 5200 204.0 70 35 70 70 210 690 760S71 210.9 73 6.0 4.7 5200 204.0 70 35 70 70 210 690 760S72 210.9 73 8.0 4.7 5200 204.0 70 35 70 70 210 690 760

G19 S73 176.3 64 6.3 2.0 5200 170.0 60 30 60 60 180 275 430S74 176.3 64 6.3 4.0 5200 170.0 60 30 60 60 180 275 430S75 176.3 64 6.3 6.0 5200 170.0 60 30 60 60 180 275 430S76 176.3 64 6.3 8.0 5200 170.0 60 30 60 60 180 275 430

G20 S77 176.3 64 6.3 2.0 5200 170.0 60 30 60 60 180 460 530S78 176.3 64 6.3 4.0 5200 170.0 60 30 60 60 180 460 530S79 176.3 64 6.3 6.0 5200 170.0 60 30 60 60 180 460 530S80 176.3 64 6.3 8.0 5200 170.0 60 30 60 60 180 460 530

G21 S81 176.3 64 6.3 2.0 5200 170.0 60 30 60 60 180 690 760S82 176.3 64 6.3 4.0 5200 170.0 60 30 60 60 180 690 760S83 176.3 64 6.3 6.0 5200 170.0 60 30 60 60 180 690 760S84 176.3 64 6.3 8.0 5200 170.0 60 30 60 60 180 690 760

G22 S85 210.9 73 6.9 2.0 5200 204.0 70 35 70 70 210 275 430S86 210.9 73 6.9 4.0 5200 204.0 70 35 70 70 210 275 430S87 210.9 73 6.9 6.0 5200 204.0 70 35 70 70 210 275 430S88 210.9 73 6.9 8.0 5200 204.0 70 35 70 70 210 275 430

G23 S89 210.9 73 6.9 2.0 5200 204.0 70 35 70 70 210 460 530S90 210.9 73 6.9 4.0 5200 204.0 70 35 70 70 210 460 530S91 210.9 73 6.9 6.0 5200 204.0 70 35 70 70 210 460 530S92 210.9 73 6.9 8.0 5200 204.0 70 35 70 70 210 460 530

G24 S93 210.9 73 6.9 2.0 5200 204.0 70 35 70 70 210 690 760S94 210.9 73 6.9 4.0 5200 204.0 70 35 70 70 210 690 760S95 210.9 73 6.9 6.0 5200 204.0 70 35 70 70 210 690 760S96 210.9 73 6.9 8.0 5200 204.0 70 35 70 70 210 690 760

that the use of high strength steel offered a considerable increasein the failure loads of less slender castellated steel beams. It shouldbe noted that when presenting the failure loads as a percentage ofthe plastic collapse load of each castellated steel beam, the failureload ratios are decreased for high strength less slender steel beams.Lateral torsional buckling (LTB)was predicted for all the castellatedsteel beams, except for specimens S25, S29, S37, S41, S45, S85,S89 and S93 where web distortion dominated the failure mode.Steel yielding (SY)was predicted for castellated steel beams S4 andS16 having a length of 3600 mm. Finally, combination of LTB andWD was also predicted for some castellated beams as shown as inTables 5 and 6.

6. Comparison with design guides and discussions

To date, there is no design guides in current codes of

guides were found in the AS4100 [15] that considers lateraltorsional buckling of doubly symmetric I-sections as well asdesign guides in the AISC [14] that controls the errors associatedwith neglecting web distortion in doubly symmetric I-sections.In a recent study by Zirakian and Showkati [13], it is concludedthat the AISC [14] predictions are overconservative and in somecases may cause economic losses for doubly symmetric I-sectionsunder distortional buckling. In this study, the failure loads ofthe castellated beams investigated in the parametric study werecompared with the design guides given in the AS4100 [15].

Following the AS4100 design guides, the nominal bucklingmoment strength (Mb) of compact doubly symmetric I-sectionbeams is given by:

Mb = msMpx (3)where, Mpx = fySx is the major axis full plastic moment820 E. Ellobody / Journal of Constructiona

Table 4Dimensions and material properties of castellated beams in the parametric study.

Group Specimen Dimensions (mm)H B t s L h

G13 S49 176.3 64 2.0 4.4 5200 1S50 176.3 64 4.0 4.4 5200 1S51 176.3 64 6.0 4.4 5200 1S52 176.3 64 8.0 4.4 5200 1

G14 S53 176.3 64 2.0 4.4 5200 1S54 176.3 64 4.0 4.4 5200 1S55 176.3 64 6.0 4.4 5200 1S56 176.3 64 8.0 4.4 5200 1

G15 S57 176.3 64 2.0 4.4 5200 1S58 176.3 64 4.0 4.4 5200 1S59 176.3 64 6.0 4.4 5200 1S60 176.3 64 8.0 4.4 5200 1practice [1417] that account for the distortional buckling ofcastellated normal and high strength steel beams. Only designl Steel Research 67 (2011) 814825

Material propertiesh1 b1 b2 b3 b4 fy (MPa) fu (MPa)

70.0 60 30 60 60 180 275 43070.0 60 30 60 60 180 275 43070.0 60 30 60 60 180 275 43070.0 60 30 60 60 180 275 430

70.0 60 30 60 60 180 460 53070.0 60 30 60 60 180 460 53070.0 60 30 60 60 180 460 53070.0 60 30 60 60 180 460 530

70.0 60 30 60 60 180 690 76070.0 60 30 60 60 180 690 76070.0 60 30 60 60 180 690 76070.0 60 30 60 60 180 690 760corresponding to collapse load Ppx, fy is the yield stress, Sx is theplastic sectionmodulus, m is amomentmodification factor which

aS15 17.6 12.2 43.6 1.10 41.6 38.0 LTB 39.5 0.91 0.95 0.96S16 17.6 9.1 43.6 1.01 49.8 48.5 LTB+ SY+WD 49.8 0.97 1.00 0.97

G5 S17 17.6 36.5 43.6 1.84 41.3 23.4 LTB 18.0 0.57 0.44 1.30S18 17.6 18.3 43.6 1.57 55.6 38.8 LTB 31.9 0.70 0.57 1.22S19 17.6 12.2 43.6 1.42 69.6 53.4 LTB 46.8 0.77 0.67 1.14S20 17.6 9.1 43.6 1.31 83.4 74.6 LTB+WD 63.2 0.89 0.76 1.18


G7 S25 21.2 10.2 85.0 1.09 25.0 17.8 WD 24.0 0.71 0.96 0.74S26 21.2 10.2 42.5 1.14 29.1 28.2 LTB+WD 26.5 0.97 0.91 1.06S27 21.2 10.2 28.3 1.14 33.2 31.3 LTB 30.2 0.94 0.91 1.04S28 21.2 10.2 21.3 1.10 37.3 34.4 LTB 35.4 0.92 0.95 0.97






Mean 1.18COV 0.242

allows for non-uniform moment distributions (taken 1.75 forsimply supported beams under concentrated load at the middle)ands is a slenderness reduction factorwhich allows for the effectsof elastic buckling, initial geometric imperfections, initial twist andresidual stresses, and which is given by Trahair [26] as follows:

s = 0.6Mpx

Myz

2+ 3

MpxMyz

1.0 (4)where Myz is the elastic buckling moment of a simply supportedbeam in uniform bending given by:

Myz =2EIyL2u

GJ +

2EIwL2u

(5)

where, E and G are the Youngs modulus and shear modulus of

constant, respectively. The nondimensional slenderness () of thecastellated steel beam according to AS4100 is equal to

MpxMyz

, andis considered as a guide for beam slenderness in this study. Thedesign load of castellated steel beams, with simply supported endsunder a concentrated load at mid-span, based on AS4100 (PAS4100)is calculated from (Mb). The plastic collapse load (Ppx) and thedesign failure load (PAS4100) calculated according to AS4100 [15]are presented in Tables 5 and 6.

Looking at Tables 5 and 6 that summarize the design failureloads (PAS4100), it can be seen that the Specification predictionsare generally conservative for the castellated beams failing byLateral Torsional Buckling (LTB) and having steel yield stressof 275 MPa, except for S13, S14, S15, S28, S39 and S40 thatexperienced unconservative specification predictions ranged from1% to 9%. This is attributed to the fact that this study covered awiderange of castellated steel beams having different combinationsE. Ellobody / Journal of Constructiona

Table 5Failure loads obtained from finite element analysis and design rules for castellated be

Group CSB L/h B/t h/s Ppx (kN)

G1 S1 21.2 32.0 38.6 1.50 16.8S2 21.2 16.0 38.6 1.29 23.0S3 21.2 10.7 38.6 1.16 29.1S4 21.2 8.0 38.6 1.06 35.0

G2 S5 21.2 32.0 38.6 1.94 28.1S6 21.2 16.0 38.6 1.67 38.5S7 21.2 10.7 38.6 1.50 48.6S8 21.2 8.0 38.6 1.37 58.5

G3 S9 21.2 32.0 38.6 2.38 42.1S10 21.2 16.0 38.6 2.05 57.7S11 21.2 10.7 38.6 1.84 72.9S12 21.2 8.0 38.6 1.67 87.7

G4 S13 17.6 36.5 43.6 1.43 24.7S14 17.6 18.3 43.6 1.22 33.2elasticity, Iy, J and Iw are the minor axis section moment of area,the uniform torsion section constant and the warping sectionl Steel Research 67 (2011) 814825 821

ms in the parametric study.

FE analysis (kN) Design (kN) PFEPpxPAS4100Ppx

PFEPAS4100

PFE Failure mode PAS4100

11.8 LTB 10.4 0.70 0.62 1.1319.7 LTB 17.8 0.86 0.77 1.1127.7 LTB 25.8 0.95 0.89 1.0735.0 LTB+ SY+WD 34.7 1.00 0.99 1.0116.7 LTB 11.2 0.59 0.40 1.4928.1 LTB 19.9 0.73 0.52 1.4139.2 LTB 30.0 0.81 0.62 1.3150.6 LTB+WD 41.8 0.86 0.71 1.2119.2 LTB 11.5 0.46 0.27 1.6735.8 LTB 20.8 0.62 0.36 1.7252.0 LTB 32.0 0.71 0.44 1.6368.2 LTB+WD 45.4 0.78 0.52 1.5016.0 LTB 16.5 0.65 0.67 0.9727.5 LTB 27.9 0.83 0.84 0.99of L/h, B/t, h/s and ratios that may not be studied by thespecification. The Specification predictions were unconservative

aS63 25.5 12.2 43.6 1.47 28.8 25.4 LTB 18.4 0.88 0.64 1.38S64 25.5 9.1 43.6 1.33 34.5 32.8 LTB+WD 25.5 0.95 0.74 1.29



G19 S73 30.6 10.2 85.0 1.44 17.3 15.6 LTB 11.5 0.90 0.66 1.36S74 30.6 10.2 42.5 1.49 20.2 17.8 LTB 12.6 0.88 0.62 1.41S75 30.6 10.2 28.3 1.46 23.0 19.8 LTB 14.9 0.86 0.65 1.33S76 30.6 10.2 21.3 1.38 25.8 21.8 LTB 18.2 0.84 0.71 1.20






Mean 1.63COV 0.164

for specimens S25, S29, S33, S37, S41, S45 and S85 failing mainlybyWebDistortional Buckling (WD), having higher h/s greater thanor equal to 85 and L/h less than or equal 25.5. The specificationswere also unconservative for specimens S16 and S38 failing bycombined (LTB + SY + WD) and (LTB + WD), respectively, andreaching or approaching full plasticization, respectively.While, theSpecification predictions were quite conservative for all remainingcastellated steel beams particularly beamswith high strength steelfy of 460, and 690 MPa. In overall, the mean value of PFE/PAS4100ratio for the castellated steel beams having a length of 3600 mm is1.18 with the coefficient of variation (COV) of 0.242, as shown inTable 5. On the other hand, the mean value of PFE/PAS4100 ratio forthe castellated steel beams having a length of 5200mm is 1.63withthe coefficient of variation (COV) of 0.164, as shown in Table 6.

Fig. 6 plotted the failure loads of castellated steel beamsin groups G1G3 predicted from the finite element analysis

nondimensional flange width-to-thickness ratio (B/t). Looking atthe castellated beams in G1, it can be seen that S4, having a B/tratio of 8 and a nondimensional slenderness ratio of 1.06, failedat the plastic collapse load. By increasing the B/t ratio from 8 to16 the PFE/Ppx and PAS4100/Ppx ratios are reduced in a nonlinearrelationship. On the other hand, the failure load ratios are reducedlinearly as B/t ratios are increased above 16. It can also be seenthat as the steel strength are increased (beams in G2 and G3)the failure loads are increased and the PFE/Ppx and PAS4100/Ppxratios are decreased significantly compared to beams in G1. Thecomparison of the numerical and design predictions has shownthat the AS4100 design guides are generally conservative for thecastellated steel beams with normal yield strength (beams in G1)while it is quite conservative for the beams with higher yieldstresses (beams in G2 and G3). Similar conclusions could be drawnfor the castellated steel beams inG4G6, G13G15 andG16G18 asshown in Figs. 79, respectively, except for castellated steel beams822 E. Ellobody / Journal of Constructiona

Table 6Failure loads obtained from finite element analysis and design rules for castellated be

Group CSB L/h B/t h/s Ppx (kN)

G13 S49 30.6 32.0 38.6 1.95 11.6S50 30.6 16.0 38.6 1.70 15.9S51 30.6 10.7 38.6 1.51 20.1S52 30.6 8.0 38.6 1.36 24.2

G14 S53 30.6 32.0 38.6 2.52 19.4S54 30.6 16.0 38.6 2.20 26.6S55 30.6 10.7 38.6 1.96 33.6S56 30.6 8.0 38.6 1.75 40.5

G15 S57 30.6 32.0 38.6 3.09 29.2S58 30.6 16.0 38.6 2.69 39.9S59 30.6 10.7 38.6 2.40 50.5S60 30.6 8.0 38.6 2.15 60.7

G16 S61 25.5 36.5 43.6 1.88 17.1S62 25.5 18.3 43.6 1.63 23.0(PFE) and design guides (PAS4100). The failure loads were plotted,as a percentage of the plastic collapse load (Ppx), against thel Steel Research 67 (2011) 814825

ms in the parametric study.

FE analysis (kN) Design (kN) PFEPpxPAS4100Ppx

PFEPAS4100

PFE Failure mode PAS4100

7.2 LTB 4.6 0.62 0.40 1.5712.2 LTB 8.0 0.77 0.5 1.5317.4 LTB 12.3 0.87 0.61 1.4122.8 LTB+WD 17.5 0.94 0.72 1.308.4 LTB 4.7 0.43 0.24 1.79

15.9 LTB 8.4 0.60 0.32 1.8923.8 LTB 13.2 0.71 0.39 1.8032.0 LTB+WD 19.3 0.79 0.48 1.668.7 LTB 4.8 0.30 0.16 1.81

17.3 LTB 8.5 0.43 0.21 2.0426.8 LTB 13.5 0.53 0.27 1.9937.2 LTB+WD 20.1 0.61 0.33 1.8511.0 LTB 7.2 0.64 0.42 1.5318.2 LTB 12.4 0.79 0.54 1.47S13, S14, S15 and S16 of G4 that experienced unconservativespecification predictions ranged from 1% to 4%.


Fig. 6. Comparison of finite element analysis and design predictions for castellatedbeams in groups G1G3.



The impact of the web distortional buckling on the failureloads of castellated steel beams can be shown in Figs. 1013.Fig. 10 plotted the PFE/Ppx and PAS4100/Ppx ratios against thenondimensional web height-to-web thickness ratio (h/s) for thecastellated steel beams in G7G9. Looking at the numerical failureloads of the castellated beams in G7, it can be seen that S25,having a h/s ratio of 85 and a nondimensional slenderness ratio of 1.09, failed prematurely owing to the predicted WD buckling.As the h/s ratios are decreased from 85 to 42.5 the PFE/Ppxratios are increased linearly. On the other hand, the failure load

ratios are decreased approximately nonlinearly as the h/s ratiosare decreased below 42.5. Once again, as the steel strength arel Steel Research 67 (2011) 814825 823


Fig. 10. Comparison of finite element analysis and design predictions forcastellated beams in groups G7G9.


increased (beams in G8 and G9) the failure loads are increased andthe PFE/Ppx ratios are decreased significantly compared to beamsin G7. The comparison of the numerical and design predictionshas shown that the AS4100 design guides are unconservative forthe castellated steel beams failing by distortional buckling (S25,S29 and S33). It can also be seen that the Specification generallyaccurately predicted the failure loads of castellated steel beamswith normal yield strength (beams S26S28 failingmainly by LTB).The specification predictions were quite conservative for beamswith higher yield stresses (beams in G8 and G9) failing mainly by

LTB. Similar conclusions could be drawn for the castellated steelbeams in G10G11 and G22G24 as shown in Figs. 11 and 13.

824 E. Ellobody / Journal of Constructiona



Fig. 14. Comparison of finite element analysis and design predictions forcastellated beams in groups G1G3 and G13G15.

Interestingly, unlike (S25, S29 and S33) in G7G9 failing byWD, thecastellated steel beams (S73, S77 and S81) in G19G21 did not failprematurely owing to the predicted LTB failure as shown in Fig. 12.It can be seen from Fig. 12 that the failure load ratios remainedapproximately constant for the beams having h/s ratios greaterthan 42.5. It can also be seen that the specification predictions arequite conservative for the castellated steel beams in G19G21.

The PFE/Ppx and PAS4100/Ppx ratios are plotted against nondi-mensional slenderness () as an example for the castellated steel

beams in G1G3 and G13G15 as shown in Fig. 14. Once again, itcan be seen that the failure loads of the castellated beams withl Steel Research 67 (2011) 814825

normal steel strength approached the plastic collapse load at of1.06. On the other hand, the more slender the beam the more elas-tic buckling we will have and collapse behaviour is dependent onthe lateral torsional andweb distortional buckling behaviour of thebeam.

7. Conclusions

The interaction of buckling modes in castellated normal andhigh strength steel beams has been investigated and reported inthis paper. A nonlinear finite element model for the analysis ofsimply supported castellated steel beams has been developed. Theinitial geometric imperfection and nonlinear material propertiesof steel have been incorporated in the model. The failure loadsof castellated steel beams, buckling behaviour, failure modesand loadlateral deflection relationships were predicted fromthe nonlinear finite element analysis and verified well againstpublished tests. Ninety-six castellated steel beams were analysedin an extensive parametric study highlighting the effects of thechange in cross-section geometries, beam length, steel strengthand nondimensional slenderness on the failure loads and bucklingbehaviour of the beams.

The parametric study has shown that the presence of web dis-tortional buckling causes a considerable decrease in the failure loadof slender castellated steel beams. It is also shown that the useof high strength steel offers a considerable increase in the failureloads of less slender castellated steel beams. The failure loads pre-dicted from the finite element analysis were compared with thatpredicted from Australian Standards for steel beams under lateralbuckling. It is shown that the Specification predictions are gener-ally conservative for normal strength castellated steel beams fail-ing by lateral torsional buckling, except for some castellated steelbeams that experienced unconservative Specification predictionsranged from 1% to 9%. On the other hand, the Specification pre-dictions were unconservative for castellated steel beams failing byweb distortional buckling and quite conservative for high strengthcastellated steel beams failing by lateral torsional buckling.

References

[1] Altifillisch MD, Cooke BR, Toprac AA. An investigation of open web expandedbeams. Welding research council bulletin, vol 47. 1957. p. 7788.

[2] Toprac AA, Cooke BR. An experimental investigation of open-web beams.Welding research council bulletin, vol 47. 1959. p. 110.

[3] Nethercot DA, Kerdal D. Lateral-torsional buckling of castellated beams. TheStructural Engineering 1892;60B(3):5361.

[4] Husain MU, Speirs WG. Experiments on castellated steel beams. Journalof American Welding Society, Welding Research Supplement 1973;52(8):32942.

[5] Zaarour WJ. Web buckling in thin webbed castellated beams. M. Eng. thesis.Montreal (Canada): Department of Civil Engineering and Applied Mechanics,McGill University; 1995.

[6] Zaarour W, Redwood R. Web buckling in thin webbed castellated beams.Journal of Structural Engineering, ASCE 1996;122(8):8606.

[7] Kerdal D, Nethercot DA. Failure modes for castellated beams. Journal ofConstructional Steel Research 1984;4:295315.

[8] Demirdjian S. Stability of castellated beam webs. M. Eng. thesis. Montreal(Canada): Department of Civil Engineering and Applied Mechanics, McGillUniversity; 1999.

[9] BradfordMA. Lateraldistortional buckling of steel I-sectionmembers. Journalof Constructional Steel Research 1992;23:97116.

[10] Bradford MA. Distortional buckling of elastically restrained cantilevers.Journal of Constructional Steel Research 1998;47:318.

[11] Vrcelj Z, Bradford MA. Elastic distortional buckling of continuously restrainedI-section beamcolumns. Journal of Constructional Steel Research 2006;62:22330.

[12] Zirakian T. Elastic distortional buckling of doubly symmetric I-shaped flexuralmembers with slender webs. Journal of Constructional Steel Research 2008;46:46675.

[13] Zirakian T, Showkati H. Distortional buckling of castellated beams. Journal ofConstructional Steel Research 2006;62:86371.

[14] AISC. Specification for structural steel buildings. American institute for steelconstruction. Reston (Chicago, Illinois, USA): ANSI/AISC 360-05; 2005.[15] Australian Standards AS4100. Steel structures. Sydney (Australia): StandardsAustralia, AS4100-1998; 1998.


[16] BS 5950. Structural use of steelwork in building part 1: code of practice fordesign rolled and welded sections. London: British Standards Institution;2000.

[17] EC3. Eurocode 3: design of steel structures part 11: general rules and rules forbuildings. London (UK): British Standards Institution, BS EN 1993-1-1; 2005.

[18] EC3. Eurocode 3: design of steel structures part 112: additional rules for theextension of EN 1993 up to steel grades S 700. London (UK): British StandardsInstitution, BS EN 1993-1-12; 2007.

[19] ABAQUS standard users manual. Hibbitt, Karlsson and Sorensen, Inc. vols. 1,2 and 3. Version 6.8-1. USA. 2008.

[20] Ellobody E, Young B. Behaviour of cold-formed steel plain angle column.Journal of Structural Engineering, ASCE 2005;131(3):45766. USA.

[21] Young B, Ellobody E. Buckling analysis of cold-formed steel lipped anglecolumns. Journal of Structural Engineering, ASCE 2005;131(10):15709. USA.l Steel Research 67 (2011) 814825 825

[22] Ellobody E, Young B. Structural performance of cold-formed high strengthstainless steel columns. Journal of Constructional Steel Research 2005;61(12):163149.

[23] Ellobody E. Buckling analysis of high strength stainless steel stiffened andunstiffened slender hollow section columns. Journal of Constructional SteelResearch 2007;63(2):14555.

[24] Chen S, Jia Y. Numerical investigation of inelastic buckling of steelconcretecomposite beams prestressed with external tendons. Thin-Walled Structures2010;48:23342.

[25] Nethercot DA, Trahair NS. Inelastic lateral buckling of determinate beams.Journal of the Structural Division, ASCE 1976;102(ST4):70117.

[26] Trahair NS. Flexuraltorsional buckling of structures. London: E and FN Spon;1993.

Interaction of buckling modes in castellated steel beamsIntroductionSummary of experimental investigationFinite element modellingGeneralFinite element type and meshBoundary conditions and load applicationMaterial modelling of castellated steel beamsModelling of initial geometric imperfections

Verification of finite element modelParametric studyComparison with design guides and discussionsConclusionsReferences

Documents

Interaction of Buckling Modes in Castellated Steel Beams