a consistent nonlinear approach for analysing steel, cold-formed steel, stainless steel and...

Thin-Walled Structures 68 (2013) 1–17

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Thin-Walled Structures

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Review

A consistent nonlinear approach for analysing steel, cold-formed steel,stainless steel and composite columns at ambient and fire conditions

Ehab Ellobody n

Department of Structural Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt

a r t i c l e i n f o

Article history:

Received 28 January 2013

Received in revised form

27 February 2013

Accepted 27 February 2013Available online 25 March 2013

Keywords:

A consistent nonlinear approach

Cold-formed steel

Composite columns

Fire conditions

Stainless steel

Structural steel design

31/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.tws.2013.02.016

fax: þ2 40 3315860.

ail address: [email protected]

a b s t r a c t

This paper presents a consistent nonlinear 3-D finite element approach, adopted by the author over the

last ten years, for analysing steel, cold-formed steel, stainless steel and composite columns at ambient

and fire conditions. The main parameters affecting the finite element approach, which has accounted

for the nonlinear material properties of the column cross-sections at ambient and elevated tempera-

tures, initial local and overall geometric imperfections and residual stresses, are highlighted in this

paper. The finite element approach could be easily extended to study columns constructed from other

materials or built-up using different sections. This paper also presents up-to-date review for previously

published experimental and numerical investigations highlighting the stability of the aforementioned

columns at ambient and elevated temperatures. In addition, the paper highlights the design rules

specified in current codes of practice for the columns. Furthermore, this paper presents, as examples,

comparisons of finite element analysis results, previously reported by the author, with design values

calculated using of current codes of practice. In overall, the paper aims to stress the fact that consistent,

robust and efficient nonlinear 3-D finite element models could improve and assess the accuracy of

design rules specified in current codes of practice at ambient and elevated temperatures. Also, better

understanding of the structural performance of the columns in the cold condition is essential to analyse

the column behaviour under severe fire conditions.

& 2013 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1. Stability of steel columns at ambient and fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2. Stability of cold-formed steel columns at ambient and fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3. Stability of stainless steel columns at ambient and fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4. Stability of composite columns at ambient and fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2. Finite element modelling approach for columns at ambient temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2. Choice of finite elements and mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3. Modelling of initial local and overall geometric imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4. Modelling of residual stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5. Modelling of nonlinear material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6. Modelling of interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7. Load application and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3. Finite element modelling approach for columns under fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2. Thermal heat transfer analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3. Thermal–structural analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4. Verification of the finite element models at ambient and fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5. Current codes of practice for steel, cold-formed steel and composite columns at ambient and fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . 13

6. Comparisons of finite element results with design code results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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E. Ellobody / Thin-Walled Structures 68 (2013) 1–172

6.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.2. An example of stainless steel stiffened columns at ambient temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.3. An example of concrete-encased steel composite columns at fire conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7. Main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1. Introduction

1.1. Stability of steel columns at ambient and fire conditions

Extensive experimental and numerical investigations werereported in the literature at ambient temperature on normalstrength mild carbon steel columns having different cross-sec-tions, geometries and slenderness. Since the main objective of thispaper is to present recent developments and current researcheson steel columns, these well highlighted investigations will not berepeated in this review. Alternatively, recent researches on highstrength steel columns were presented in this section. Highstrength thin-walled steel columns have been widely used insteel structures as reported in [1–3] to reduce the self-weight ofstructures, as an example long-span steel bridges. High strengthsteels with high yield strength usually have little strain hardeningand low ductility. The stability of the high strength thin-walledsteel members is affected by the lack of strain hardening. Earlierstudies by Usami and Fukumoto [4] have presented valuableexperimental results on local and overall buckling behaviour ofwelded box columns fabricated using high strength steel with anominal yield strength of 690 MPa. Rasmussen and Hancock [5,6]studied plate slenderness limits for high strength steel sectionsand the buckling of high strength columns fabricated using highstrength steel with a nominal yield strength of 690 MPa. Follow-ing to the investigations [5,6], the section capacity and theinfluences caused by imperfections, residual stresses and yieldslenderness limits of very high strength circular steel tubes havebeen studied [7,8].

Gao et al. [9] have investigated experimentally and numeri-

cally the load-carrying capacity of thin-walled box-section stub

columns fabricated using high strength steel. The high strength

steel columns tested were axially loaded in compression having

different geometries. The measured load-carrying capacities of

the stub columns were considerably higher than normal-strength

mild steel, which suggests that the existing effective width

method should be revised for high strength steel stub columns.

Chen and Young [10] have extended investigating the buckling

behaviour of high strength steel columns to study the behaviour

at elevated temperatures. The authors [10] have studied the

behaviour and design of high strength steel columns at elevated

temperatures using finite element analysis. In the study, equa-

tions predicting the yield strength and elastic modulus of high

strength steel and mild steel at elevated temperatures were

proposed. Yang and Hsu [11] conducted a series of tests to

examine the behaviour of high strength steel columns subjected

to axial load at fire conditions. This experimental work investi-

gated the effect of the width-to-thickness ratio of flanges, the

slenderness ratio of steel columns and residual stresses on the

ultimate strength of the high strength steel columns at a specified

temperature. It was found [11] that the column strength

decreases with the increase of width-to-thickness ratio and

slenderness ratio. It was concluded that the column behaviour

is sensitive to width-to-thickness ratios and the slenderness ratio

at temperatures below 550 1C.Wang and Li [12] carried-out two fire tests on steel columns

with partial loss of fire protection. The steel columns were

connected by flush end-plates at two ends and the axial loadwas kept constant with a load ratio of 0.55 and was subjected toelevated temperatures. The specimens were protected with20 mm thickness of fire protection. The damaged length of fireprotection was 7% of the complete length of the column for onespecimen and 14% for the second specimen at the two ends of thesteel columns. It was concluded [12] that the damage length ofthe fire protection has a great effect on the fire resistance of steelcolumns. The failure of the specimens mainly resulted from thebuckling or yielding at the portion where the fire protection wasdamaged. Li et al. [13] studied experimentally the behaviour ofrestrained steel columns exposed to fire. Two fire tests on axiallyand rotationally restrained steel columns having different axialrestraint stiffness were conducted. The axial and rotationalrestraints were applied by a restraint beam. It was found thatthe axial restraint reduced the buckling temperature of arestrained column. Further to the study [13], Wang et al. [14]investigated behaviours of restrained steel columns in fire analy-tically. The results of parametric studies have shown that theaxial restraint resulted in a reduction in the failure temperature ofthe restrained column. Recently, Scullion et al. [15] conducted anexperimental investigation on the performance of elliptical sec-tion steel columns under hydrocarbon fire. The authors found thatalthough elliptical structural steel hollow sections represent arecent addition to the range of steel sections available to struc-tural engineers, a complete absence of fire resistance designguidance of the columns, which restrained its applications. Sixcolumns were tested under different loading levels and subjectedto a hydrocarbon fire curve. The paper provided the recorded dataof axial displacements and temperature profiles of the steelcolumns, while highlighting the unique local and overall bucklingfailure modes of the elliptical hollow section. Also, Dwaikat et al.[16] conducted fire tests to investigate the mechanics andcapacity of steel beam-columns that develop a thermal gradientthrough their depth when exposed to fire. The specimens inves-tigated were tested with several combinations of load level, firescenario, and direction of the thermal gradient.

1.2. Stability of cold-formed steel columns at ambient and fire

conditions

Cold-formed steel structures have many advantages in termsof their superior strength to self-weight ratio, ease of constructionand economic design. In recent years, developed manufacturetechniques and increased strength of materials gave the edge ofcold-formed steel over traditional hot rolled steel in the con-struction of wide range of structures. A summary of the majorrecent research developments in cold-formed steel structures isgiven in Yu [17] and Hancock [18]. Earlier researches by Mandu-gula et al. [19] reported the results of 16 tests, at ambienttemperature, on single equal-leg angle columns with hinged endconditions. The nominal slenderness ratios of the test specimensvaried from 90 to 250 and comparison of test results with designstrengths predicted from different codes of practice was investi-gated. The research was extended by Popovic et al. [20] whoreported the results of 12 fixed-ended and 18 pin-ended com-pression tests performed on cold-formed plain angle columns.

Nomenclature

D overall depth of cross-section (larger dimension)Fy yield stress is taken as 0.2% proof stress (s0.2)f equivalent uniaxial stressfc unconfined compressive cylinder strength of concretefcc confined compressive strength of concretefcu unconfined compressive cube strength of concretefl lateral confining pressurefy yield stressfys yield stress of structural steelfu ultimate stressGf fracture energy of concreteh crack band width

k3 coefficient for confined concreteL length of columnt thicknessr reduction factor for confined concretee equivalent uniaxial strainec unconfined concrete strainecc confined concrete strainet tensile straine elongation (tensile strain) after fracture based on

gauge length of 50 mmepl

true plastic true strains stressstrue true stress

E. Ellobody / Thin-Walled Structures 68 (2013) 1–17 3

The b/t ratio (flat flange width-to-thickness ratio) of 20.6, 12.2and 9.6, respectively. The residual stresses and material proper-ties of flat and corner portions of the angle sections weremeasured. It was concluded that the section capacities obtainedfrom the stub column tests are between 15 and 40% higher thanthose calculated using current codes of practice. Further to theresearch presented in [20], Young [21] conducted a series of testson cold-formed steel plain angle columns compressed betweenfixed ends. The angle sections were brake-pressed from highstrength structural steel sheets and had b/t ratio ranged from 35.8to 57.9. The test results were compared with design strengthsobtained from current codes of practice for cold-formed steelstructures. It was concluded [21] that the design strengths aregenerally very conservative for all column lengths. Hence, designequations for cold-formed steel plain angle columns wereproposed.

The behaviour of cold-formed steel columns is affected byinitial geometric imperfections and residual stresses. Weng andPekoz [22] carried out detailed experimental study of residualstresses effect on the strength of cold-formed steel members. It isfound that the residual stress distribution in cold-formed steelsections is different from that in hot-rolled steel sections. Schaferand Pekoz [23] explained that characterization of initial geo-metric imperfections and residual stresses is possible. The authorsprovided a simple set of guidelines for the representation ofgeometric imperfections and residual stresses of cold-formedsteel sections in the computational modeling process. Jiao andZhao [8] investigated the effect of initial geometric imperfections,residual stresses and yield slenderness limit on the behaviour ofvery high strength circular steel tubes. It is found that themeasured residual stresses of the tubes is approximately 4% ofthe yield stress with the outside surface in compression and theinside surface in tension for both longitudinal and transversedirections. On the other hand, the residual stresses of non-heat-treated tubes are approximately 50% and 20% of the yield stressalong longitudinal and transverse directions, respectively. Theresidual stresses of the non-heat-treated tubes are always incompression on the inside surface and tensile on the outsidesurface.

Accurate finite element model is needed to predict the complexbehaviour of thin-walled structures. A detailed finite element studywas carried out by Young and Yan [24] for the analysis and design offixed-ended plain channel columns. The four-node doubly curvedshell element with reduced integration and hourglass control (S4R5)was used in the model. The element has five degrees of freedom pernode. Bakker and Pekoz [25] investigated the basic principles for themodelling of thin-walled members. The authors focused on possiblesources of error in linear and nonlinear analysis. Suggestions on how

to check and avoid these errors were given. Sarawit et al. [26]studied the applications of finite element modelling on thin-walledmembers. Gardner [27] investigated the structural stainless steelhollow sections. The finite element models used the measured andpredicted values of initial plate imperfections and material proper-ties. Assumed values of residual stresses were also used in the finiteelement models. Enhanced material properties were employed inthe corner regions. The 9-noded reduced integration shell elementwith five degrees of freedom per node (S9R5) was used in theanalysis. It was found that the residual stresses had negligibleinfluence on the overall behaviour and ultimate load carryingcapacity.

Following the continuing developments in manufacturingtechniques and cold-formed steel materials, Kwon et al. [28]conducted a series of compression tests on cold-formed simplelipped channels and lipped channels with intermediate stiffenersin the flanges and web fabricated from high strength steel plate ofdifferent thicknesses. The high strength cold-formed steel channelsections of intermediate lengths generally displayed a significantinteraction between local and distortional buckling. Vieira Jr. et al.[29] investigated the stability and strength of cold-formed steellipped channel section columns. The authors reported a total of 26tests covering short, intermediate and long specimens, variedsheathing configurations, and varied end boundary conditions arecompleted. It was concluded that composite action between thestud and sheathing, and isolating direct loading of the sheathing,were shown to be significant in determining the strength andcontrolling limit state of the stud. Recent investigations on cold-formed steel columns in the cold condition have introducednew innovations in the field. Nguyen et al. [30] presentedcompression tests on cold-formed plain and dimpled steel col-umns. The authors carried out a series of compression and tensiletests on plain and dimpled steel of different geometries. Theauthors have found that enhancements in buckling and ultimatestrengths were observed in the dimpled steel columns caused bythe cold-work of the material during the dimpling process. It wasconcluded that the buckling and ultimate strengths of dimpledsteel columns were up to 33% and 26% greater than companionsof plain steel columns. Tong et al. [31] presented an experimentalstudy on cold-formed thick-walled square hollow sections withthickness greater than 6 mm. The investigation comprised squarehollow sections formed using two different forming processes.Two test methods were used to measure the magnitudes anddistributions of longitudinal residual stresses. It was shown thatthe longitudinal residual stresses are in tension at outer surfaceand in compression at inner surface, and present nonlineardistributions. Zhu and Young [32] presented numerical simula-tion and design of cold-formed steel oval hollow section columns.


The experimental column strengths and numerical results pre-dicted by the parametric study were compared with the designstrengths calculated using current codes of practice for cold-formed steel structures.

Although, the use of thin and high strength cold-formed steelsections in light weight floor and wall systems has increasedrapidly around the world due to the development of advancedmaterial and manufacturing technologies, its higher slendernessand the high thermal conductivity of steel lead to rapid steeltemperature rise during fires and hence result in lower fireresistance. Therefore, currently, the structural fire behaviour oflight gauge steel structures has become an important area ofresearch in order to improve their fire safety. In recent times,considerable progress has been made in this field by Feng et al.[33–36], Ranby [37], Kaitila [38], Ala-Outinen and Myllymaki [39],Outinen [40] and Chen and Young [41]. Their research rangedfrom local and flexural buckling of cold-formed steel columns atelevated temperatures to the effects of initial imperfections andnon-uniform temperature distributions, and mechanical proper-ties at elevated temperatures. Their research results provided astrong base for the fire safety research and design of light gaugecold-formed steel structures. It should be noted that, in practicalapplications, cold-formed steel studs are likely to be protected bygypsum board or similar boards, resulting in studs being subjectto a non-uniform temperature condition. However, if the max-imum temperature in the studs can be estimated for a fire event,the stud compression strength under fire conditions can beestimated using a uniform elevated temperature design method.Hence, past research has used this simpler uniform elevatedtemperature approach in their research [33,34]. However, Fenget al. [35] extended their work to include the important effects ofnon-uniform elevated temperatures on the axial strength of cold-formed steel channels using numerical studies. Ranawaka andMahendran [42] have detailed experimental studies on the dis-tortional buckling behaviour of light gauge cold-formed steelcompression members under simulated fire conditions. Theauthors have carried-out more than 150 axial compression testsat ambient and elevated temperatures. Two types of crosssections were selected with different nominal thicknesses. Theultimate loads of compression members subject to distortionalbuckling were then used to review the adequacy of the currentdesign rules at ambient and elevated temperatures. Recently,extensive research was presented by Wei and Jihong [43] toevaluate the material properties of 1-mm-thick G550 cold-formedsteel at elevated temperatures. The authors have conductedsteady and transient state tests on tensile coupons of G550cold-formed steel. It was concluded that the steady state testresults for G550 may lead to an overestimate of the fire resistanceof cold-formed steel structures.

1.3. Stability of stainless steel columns at ambient and fire

conditions

Stainless steel structural members have been significantlyused due to their superior characteristics including high strength,ductility and durability, high corrosion resistance, ease of con-struction and maintenance, good fire resistance as well asaesthetic appearance. Most of the published experimental inves-tigations have focused on normal strength stainless steel (stain-less steel types 304 and 316) and on unstiffened hollow sectioncolumns. These investigations were detailed by Rasmussen andHancock [44], Talja and Salmi [45], Macdonald et al. [46], Youngand Hartono [47], Gardner [27], Young and Liu [48], Gardner andNethercot [49,50] and Ashraf et al. [51]. An extensive review ofrecent experimental research on stainless steel tubular structureshas been performed by Rasmussen [52]. The review [52] focused

on cold-formed tubes from annealed austenitic stainless steel coilstrip. The tubes were square, rectangular and circular hollowsection columns and beams, as well as welded X- and K-joints insquare and circular hollow sections. A summary of the designrules proposed on the basis of the tests was presented. Highstrength stainless steel material has higher yield stress, higherultimate tensile strength and lower ductility than the normalstrength stainless steel. Tests on cold-formed high strengthstainless steel unstiffened square and rectangular hollow sectioncolumns were conducted by Young and Lui [53]. Gardner andNethercot [54] described numerical modelling of normal strengthstainless steel unstiffened hollow section columns. Ellobody andYoung [55] developed a numerical model for analyzing fixed-ended cold-formed high strength stainless steel unstiffened squareand rectangular hollow section columns. The behaviour of highstrength stainless steel stiffened slender square and rectangularhollow section columns has been covered by Ellobody [56]. Recently,Saliba and Gardner [57] presented an interesting study on a recentlydeveloped grade, known as lean duplex stainless steel (EN 1.4162),which has a lower nickel content and lower cost. The authors haveshown that the stainless steel has higher strength compared withtraditional austenitic stainless steels as well as good corrosionresistance and high temperature properties and adequate weldabil-ity and fracture toughness. The study [57] comprised experimentaland analytical investigations on lean duplex stainless steel. Theexperimental investigation comprised material tests, stub columntests and 3-point and 4-point bending tests. Based on the study [57],design recommendations for incorporation into European Code wereproposed.

Compared to traditional carbon steel, stainless steel has high fireresistance [58]. An extensive investigation of the material propertiesof stainless steel alloys at elevated temperatures was presented byGardner et al. [58]. The study [58] covered different ferritic,austenitic and austenitic–ferritic stainless steel types specified inEuropean Code. The characteristic superior performance of stainlesssteel at elevated temperatures compared with carbon steel isattributed to the difference in crystal structure of the two steels.The atoms of stainless steel microstructure are more closely packedand have a high level of alloying elements compared with carbonsteel. Alloying elements in stainless steel lower the diffusion rates ofatoms within the crystal lattice at a given temperature. Lowering thediffusion rates slows down softening, recrystallisation and creepdeformation mechanisms that control strength and plasticity atelevated temperatures. Furthermore, carbon steel undergoes trans-formation from ferrite to leanly alloyed austenite on heatingwhile, stainless steels maintains its structure at elevated tempera-tures. Hence, stainless steels exhibit better strength retention thancarbon steels above about 550 1C and better stiffness retention at alltemperatures, as presented by Ellobody [59] in a recent study.The study [59] showed a comparison between five stainless steelgrades investigated and carbon steel in terms of the strength andstiffness reductions at a given temperature with respect to thestrength and stiffness at ambient temperature as specified inEuropean Code. It was shown that stainless steel exhibits superiorstiffness retention at higher temperatures (approximately between500 and 900 1C) compared with carbon steel. As an example, at700 1C, the stiffness retention of stainless steel is more than fivetimes that of carbon steel. In addition, stainless steel also exhibitsgenerally better thermal material properties at elevated tempera-tures compared with carbon steel, as presented in [59].

1.4. Stability of composite columns at ambient and fire conditions

Concrete–steel composite columns have been increasinglyused in many modern structures. Their usage provides highstrength, high ductility, high stiffness, full usage of construction


materials and considerably increased fire resistance. In additionto these advantages, the fact that the steel sections surroundconcrete makes it possible to eliminate permanent formwork andto provide confinement to concrete. Composite columns may beconcrete-encased steel columns, partially concrete-encased steelcolumns or concrete-filled steel columns.

Earlier experimental researches have been carried out toinvestigate the strength and behaviour of concrete-filled steeltube columns in the cold condition. Schneider [60] studied thebehaviour of short axially loaded concrete-filled steel tube col-umns. Fourteen specimens were tested to investigate the effect ofthe tube shape and steel tube plate thickness on the compositecolumn strength. It was concluded that circular steel tubes offermuch more post-yield axial ductility than square and rectangulartube sections. Similar to Schneider [60], Huang et al. [61] tested17 concrete-filled steel tube column specimens but with highercolumn diameter-to-steel tube plate thickness ratio. The sameconclusion was achieved even for higher column diameter-to-steel tube plate thickness of 150. Sakino et al. [62] tested 114specimens of centrally loaded concrete-filled steel tube shortcolumns. In addition, Sakino et al. [62] studied the effect of steeltube tensile strength and concrete strength on the behaviour ofthe composite columns. Giakoumelis and Lam [63] carried out 15tests on circular concrete-filled tube columns. The effects of steeltube plate thickness, bond between steel tube and concrete aswell as concrete confinement on the behaviour of these columnswere studied. The test results were compared with columnstrengths calculated from current codes of practice. Little successhas been achieved so far in developing an accurate model due tothe complexity in modeling the concrete confinement. Schneider[60] developed a 3-D nonlinear finite element model for concrete-filled steel tube circular columns. Strain-hardening was notconsidered for the steel tube. Hu et al. [64] developed a nonlinearfinite element model to simulate the behaviour of concrete-filledsteel tube columns. The concrete confinement was achieved bymatching the numerical results by trial and error via parametricstudy.

Also, earlier experimental investigations on concrete-encased steelcomposite columns have been conducted by Anslijn and Janss [65],Matsui et al. [66], SSRC Task Group 20 [67], Mirza and Skrabek [68],Mirza et al. [69], Chen and Yeh [70], Tsai et al. [71], Chen et al. [72], El-Tawil and Deierlein [73] and Dundar et al. [74], with extensive reviewof most of these researches is given by Shanmugam and Lakshmi[75].These tests were carried out on concrete-encased steel compositecolumns having different slenderness ratios, different steel sectionsand different concrete and steel strengths. On the other hand,analytical studies on concrete encased steel composite columns havebeen performed by Furlong [76], Virdi and Dowling [77], Roik andBergmann [78], Kato [79], Munoz and Hsu [80,81], and Chen and Lin[82]. However, detailed nonlinear 3-D finite element model wererarely found in the literature to highlight the behaviour of concrete-encased steel composite columns. This is attributed to the complexityof the concrete confinement, steel–concrete interface, longitudinalreinforcement bar-transverse reinforcement bar interface, and rein-forcement bar-concrete interface as well as the nonlinear constitutivestress–strain curves of the composite column components.

The research on concrete-filled steel tube columns continuedby Han et al. [83] who studied the behaviour of concrete-filledsteel tubular stub columns subjected to axially local compressionexperimentally. The authors conducted a total of thirty-twospecimens. The main parameters varied in the tests were sec-tional types (circular and square), local compression area ratio(concrete cross-sectional area to local compression area) andthickness of the endplate. Also, Portoles et al. [84] reported 37tests conducted on slender circular tubular columns filled withnormal and high strength concrete subjected to eccentric axial

load. The test parameters were the nominal strength of concrete,the diameter to thickness ratio D/t, the eccentricity ratio e/D andthe column slenderness (L/D). The results showed that for thelimited cases analyzed, the use of high strength concrete forslender composite columns is interesting since this achievesductile behavior despite the increase in load-carrying capacity isnot greatly enhanced. Recently, Ellobody and Ghazy [85,86] havecarried out an experimental investigation on pin-ended fibrereinforced concrete-filled stainless steel circular tubular columns.The investigation augmented available tests published in theliterature on concrete-filled stainless steel composite columns[87,88]. The columns tested in [85,86] had different lengths equalto 3D of short columns, 6D of relatively long columns and 12D oflong columns, where D is the external diameter of the stainlesssteel circular tubes. The circular tubes were cold-rolled from flatstrips of austenitic stainless steel. The tubes had diameter-to-plate thickness (D/t) ratio of 50.

Extensive investigations were presented in the literature[89–99] on concrete-filled steel tubes under fire conditions tounderstand the behaviour of the composite columns at elevatedtemperatures and to evaluate the fire resistance of the compositecolumns. These investigations were extended by Han [100] whoreported the research results on the fire resistance of concrete-filled steel tube columns with both circular and square cross-sections, subjected to axial compression or eccentric compressionloads. Further to the investigation reported in [100], Yin et al.[101] studied the behaviour of axially loaded square and circularconcrete-filled steel tube columns when exposed to elevatedtemperatures. The fire resistance of the composite columns werecalculated. Comparisons of the square and circular columns in thefire resistance have shown that for the columns with the samesteel and concrete cross-section areas, the circular column hasslightly better fire resistance than the square column.

Limited fire tests were found in the literature highlighting thebehaviour of concrete-encased steel composite columns at ele-vated temperatures. These were mainly the earlier fire testsconducted in Europe by Malhotra and Stevens [102], where 14fire tests were reported on concrete-encased steel stanchions andthe fire resistances of the stanchions were predicted. The fire testsreported in [102] were followed by fire tests on concrete-encasedI-section steel composite columns conducted in Singapore asdetailed in Huang et al. [103] and Eugene [104]. The tests[103,104] comprised seven fire tests having different cross-sectional sizes, different steel sections and different load ratios.Although, the tests were heated following a different fire curvethan the standard specified fire curves, it provided valuable datain the form of time–temperature curves of the reinforcement,concrete and encased steel section as well as time-axial displace-ment behaviour and test periods that could be used in theverification of finite element models.

Recently, Correia and Rodrigues [105] studied the behaviour ofcomposite columns constructed from partially encased steelsections subjected to fire experimentally. The authors presentedthe results of a series of fire resistance tests on the partiallyconcrete-encased composite columns with restrained thermalelongation. The parameters studied were the load level, the axialand rotational restraint ratios and the slenderness of the column.It was concluded that for low load levels, the stiffness of thesurrounding structure has a major influence on the behaviour ofthe column subjected to fire. Mao and Kodur [106] presented theresults from seven fire resistance experiments on concrete-encased steel composite columns under standard fire exposureconditions. The investigated parameters included column size,3- and 4-side fire exposure, load intensity and load eccentricity.It was shown that the load ratio and load eccentricity havea noticeable influence on the fire resistance of the composite


columns. In addition, spalling of the concrete decreases the fireresistance of the composite columns. It was concluded that thatthe current codes of practice may not be conservative in somesituations regarding the investigated concrete-encased steel com-posite columns.

Fig. 1. Buckling modes for a stainless steel stiffened rectangular hollow section

column having a length of 1500 mm developed by Ellobody [115] using shell finite

elements. (a) Local buckling of a stiffened column and (b) Overall buckling of a

stiffened column.

2. Finite element modelling approach for columns at ambienttemperature

2.1. General

As mentioned previously that this paper presents a nonlinear3-D finite element modelling approach used by the author overthe last ten years for analyzing different steel, cold-formed steel,stainless steel and composite columns. The finite element pro-gram ABAQUS [107] was used in the buckling and nonlinear load–displacement analyses of the aforementioned columns. The mod-elling approach was based on using the measured geometry,initial local and overall geometric imperfections, residual stressesand material properties from tests conducted by the author orpublished in the literature by other authors. To investigate thebehaviour of a column under compressive loads using finiteelement analysis, two types of analyses are required. The first isknown as Eigenvalue buckling analysis that estimates the buck-ling modes and loads. Such analysis is linear elastic analysisperformed using the (nBUCKLE) procedure available in ABAQUS[107] library with the live load applied within the step. Thebuckling analysis provides the factor by which the live load mustbe multiplied to reach the buckling load. For practical purposes,only the lowest buckling mode predicted from the Eigenvalueanalysis is used. The second is called load–displacement nonlinearanalysis and follows the Eigenvalue prediction. It is necessary toconsider whether the post buckling response is stable or unstable.The initial local and overall geometric imperfections, residualstresses and nonlinear material properties of the components ofthe columns were inserted in the nonlinear load–displacementanalysis.

2.2. Choice of finite elements and mesh

Buckling and nonlinear load–displacement analyses of thin-walled steel, cold-formed steel and stainless steel columnsinvolve complex deformations. Previous investigations by theauthor on steel members [108,109], cold-formed steel members[110–112] and stainless steel members [113–116], whichundergo complex buckling behaviour, require carefully chosenfinite elements. In Refs. [108–116], only the 4-noded doublycurved shell elements with reduced integration S4R was usedfor investigating the complex buckling behaviour of the columnsconstructed from different materials. The S4R element has sixdegrees of freedom per node and provides accurate solutions tomost applications. The element allows for transverse sheardeformation which is important in simulating thick shell ele-ments (thickness is more than about 1/15 the characteristiclength of the shell). Though it is recognized that thin-walledmembers have higher width-to-thickness ratios, the choice of S4Rfinite elements to represent their behaviour allows for the free-dom in dealing with different parametric studies. The elementalso account for finite strain and is suitable for large strainanalysis. On the other hand, concrete-encased or concrete-filledsteel composite members have two main components formingthe column, which are concrete elements and steel section.Adding concrete limits the column deformation especially in caseof compact steel sections. Therefore, concrete–steel compactcolumns may be modelled using combinations of solid 3-D

elements that have three degrees of freedom per node, as anexample the investigation reported in [117]. While, concrete–steel slender columns should be modelled using two differentfinite elements, which are shell elements for steel sections andsolid elements for concrete elements, with examples the investi-gations reported in [118–121].

In order to choose the finite element mesh that providesaccurate results with minimum computational time, convergencestudies should be performed. The convergence studies assess thefinite element results against test results for an investigatedcolumn until a reasonable mesh size is achieved. The reasonablemesh size is a (length by width) ratio for shell elements and a(length by width by depth) ratio for solid elements and shouldprovide adequate accuracy in modeling the cross-section compo-nents. It should be noted that the chosen mesh size is an averagemesh and has to be monitored in most of the regions of the cross-section, however it might be slightly exceeded or reduceddepending on the geometries of the cross-section. Figs. 1 and 2

Step (1) Modelling ofreinforcement

Step (2) Modelling ofunconfined concrete

Step (3) Modelling ofencased steel section

Step (4) Modelling ofhighly confined concrete

Step (5) Modelling ofpartially confined concrete

Step (6) Modelling ofloading plates

Longitudinal bar

Transversebar

Unconfined concreteEncased steel section

Highly confined concrete Partially confined concrete Loading plate

S

1 2

3

Fig. 2. Modelling steps of concrete-encased steel composite columns developed by Ellobody and Young [130] using combinations of solid finite elements.


show examples of different finite element meshes developedpreviously by the author in published papers.

2.3. Modelling of initial local and overall geometric imperfections

Most of steel, cold-formed steel, stainless steel and compositecolumns have initial local and overall geometric imperfections asa result of the manufacturing and transporting processes. Theseimperfections govern the buckling behaviour of columns andtheir simulation in the finite element modelling is of greatimportance. If the initial local and geometric imperfections werenot included in finite element modelling, the results will not beaccurate and good agreement with test results will not beachieved.

Measurements of initial local imperfections should be carriedout prior to tests. Fig. 3 shows the Coordinate Measuring Machine(CMM) used by the author in Refs. [110–112] and [114,116].The CMM machine uses the standard touch probe for inspectionand measurement of any objects. It can also employ a laserscanner to trace the profile of three-dimensional objects.A specific length cut of the specimen can be used to measurethe initial local imperfections at marked sections on the speci-men. An automatic feed was used to rotate the t7ouch probeindicator around the specimen at a specific cross-section. Themeasurements can be taken at the middle and quarter lengths ofthe specimen. Readings can be taken at regular intervals andmaximum magnitude of local plate imperfection can be deter-mined and identified as a percentage of the plate thickness of thespecimen. On the other hand, measurements of initial overall

Fig. 3. Measurements of Initial Local Imperfections using Mitutoyo coordinate measuring machine [114,116]. (a) Mitutoyo Coordinate Measuring Machine and

(b) A stainless steel circular specimen.

Lowest bucklingmode

x

y

z

Fig. 4. Buckling modes of a fibre reinforced concrete-filled stainless steel tubular

columns developed by Ellobody [121] using combinations of shell and solid finite

elements.

0

Positions of cutting

Positions of measurement

Tip

Corner

10

-250

-200

-150

-100

-50

0

50

100

150

200

0

Distance from tip (mm)

Stre

ss (

MPa

)

Bending stressMembrane stress

Corner10 20 30 40 50 60 70

20 30 40 50 60 70mm

Fig. 5. Measurement of residual stresses as an example in a plain angle specimen

by Ellobody and Young [110]. (a) Positions of cutting and measurements of a cold-

formed steel plain angle cross-section and (b) Distribution of membrane and

bending residual stresses along the plain angle cross-section.


geometric imperfections of the columns should be taken prior totesting. The geometric imperfections can be measured along twospecified marked gauge lines on the columns. Theodolites can beused to obtain readings in two directions at mid-length and nearboth ends of the specimens. The measured overall geometricimperfections at mid-length over the specimen length can be thendetermined and identified as a percentage of the columns length.The average maximum overall imperfections at mid-length can bealso known.

Slender steel, cold-formed steel or stainless steel columns thathave very high overall depth-to-plate thickness (D/t) ratio arelikely to fail by pure local buckling. On the other hand, columns thathave very low (D/t) ratio are likely to fail by overall buckling. Both

0

200

400

600

800

0Strain (%)

Stre

ss (

MPa

) Flat

Corner

2 4 6 8 10 12

Fig. 6. Stress–strain curves of flat and corner portions for a plain angle column

measured by Ellobody and Young [110].


initial local and overall geometric imperfections are found incolumns as a result of the fabrication and transportation processes.Hence, superposition of local buckling mode as well as overallbuckling mode with measured magnitudes is recommended foraccurate finite element analysis. These buckling modes can beobtained by carrying Eigenvalue analysis of the column with veryhigh (D/t) ratio and very low (D/t) ratio to ensure local and overallbuckling occurs, respectively. The shape of a local buckling mode aswell as overall buckling mode is found to be the lowest bucklingmode (Eigenmode 1) in the analysis. This technique is used in thisstudy to model the initial local and overall imperfections of thecolumns reported by the author in [110–112]. Stub columns havingvery short length can be modeled for local imperfection only. Sinceall buckling modes predicted by ABAQUS [107] Eigenvalue analysisare generalized to 1.0, the buckling modes are factored by themeasured magnitudes of the initial local and overall geometricimperfections. Figs. 1 and 4 show examples of local and overallbuckling modes of different columns obtained previously by theauthors in published papers.

2.4. Modelling of residual stresses

Residual stresses are found in most of cold-formed steel sectionsas a result of manufacturing process. To model the behaviour of thecolumns accurately, knowledge of the magnitude and distribution ofresidual stresses is required. The common method to determine theresidual stresses is the method of sectioning that requires cuttingthe cross-section of the column into strips to release the internalresidual stresses. By measuring the strains before and after cutting,consequently residual stresses can be determined. A specified lengthof the column specimen can be used to measure the residualstresses. Each cross-section has its characteristic distribution andmagnitudes of residual stresses. A gauge length of, as an example100 mm, has to be marked on the outside and inside mid-surfaces ofeach strip along the length. The residual strains can be measuredusing an accurate Extensometer, which has a low sensitivity over agauge length of 100 mm. The initial readings before cutting can berecorded for each strip together with the corresponding tempera-ture. The cutting should be carried out using a wire-cutting methodin the water to eliminate additional stresses resulting from thecutting process. The readings can be then taken after cutting and thecorresponding temperature can be recorded recorded. The readingshave to be corrected for temperature difference before and aftercutting. The residual strains can be measured for both inner andouter sides of each strip. The membrane residual strain can becalculated as the mean of the strains (Inner strainþOuter strain)/2.The bending strain can be calculated as the difference between theouter and inner strains divided by two (Outer strain–Inner strain)/2.A compressive membrane strain (negative value) normally occur atthe corners of the cross-section, while a tensile membrane strain(positive value) normally occur at the flat portions of the cross-section. Positive bending strain indicates compressive strain at innerfiber and tensile strain at outer fiber. Residual stresses are calculatedby multiplying residual strains by Young’s modulus of the testspecimen. The distribution of membrane and bending residualstresses along the cross-section of the test specimen can be thendetermined.

To ensure accurate modeling of the behaviour of the columns,the residual stresses should be included in the finite elementmodel. Measured residual stresses are implemented in the finiteelement model by using the ABAQUS [107] (nININTIAL CONDI-TIONS, TYPE¼STRESS) parameter. It should be noted that the flatand corner coupons material tests consider the bending residualstresses effects, hence, only the membrane residual stressesshould be included in the finite element model. The averagevalues of the measured membrane residual stresses can be

calculated for corner and flat portions. A preliminary load step toallow equilibrium of the residual stresses has to be defined before theapplication of loading. Fig. 5 shows, as an example, the distribution ofthe measured membrane and bending residual stresses along a plainangle cross-section by Ellobody and Young [110].

2.5. Modelling of nonlinear material properties

The nonlinear material properties of different regions of thecross-sections of steel, cold-formed steel and stainless steelcolumns can be determined from standard coupon tests. It shouldbe noted that in cold-formed steel sections, tensile couponsshould be taken from corner and flat portions of the cross-sections since they are considerably different. The corner couponspecimen can be taken from the corner strip after the wire cuttingof the specimen used to measure the residual stresses. The cornercoupon specimen is curved before testing, and the effect ofbending residual stresses is included in the stress–strain curveof the corner tensile coupon test. The testing procedures of thecorner coupon should follow an available specification. As anexample, the tensile coupons tested by the author in [110–112]followed the Australian Standard AS 1391 [122]. The testingmachine can be with a MTS displacement controlled testingmachine with a reasonable capacity. A 50 mm gauge lengthextensometer can be used to measure the longitudinal strain.Two linear strain gauges can be attached on each face at thecentre of the coupon specimen to determine the Young’s mod-ulus. The load and strain readings can be recorded at regularintervals using a data acquisition system. The static load canobtained by pausing the applied straining for 1.5 min near the0.2% tensile proof stress and the ultimate tensile stress. Thisallows the stress relaxation associated with plastic straining totake place. The main important material properties measured areYoung’s modulus (E), the static 0.2% proof stress (s0.2), the statictensile strength (su) and the tensile strain after fracture (e). Fig. 6shows, as an example, the stress–strain curves of the flat andcorner portions for a cold-formed steel plain angle columndetailed in Young and Ellobody [110]. It should be noted thatthe ductility of the corner portion is much less than that of the flatportion. On the other hand, the static 0.2% proof stress of thecorner portion is around 15% more than that of the flat portion.

The measured stress–strain curves for flat/corner portions ofthe column specimen should be used in the finite elementanalysis. The material behaviour provided by ABAQUS [107]allows for a multi-linear stress–strain curve to be used. The firstpart of the multi-linear curve represents the elastic part up to theproportional limit stress with measured Young’s modulus and


Poisson’s ratio. Since the analysis of post-buckling involves largein-elastic strains, the nominal (engineering) static stress–straincurve was converted to true stress and logarithmic plastic straincurve. The true stress strue and plastic true strain epl

true cancalculated using Eqs. (1) and (2):

strue ¼ sð1þeÞ ð1Þ

epltrue ¼ lnð1þeÞ�strue=E ð2Þ

where E is Young’s modulus, s and e are the measured nominal(engineering) stress and strain values.

For concrete–steel composite columns, the nonlinear materialproperties of concrete and confined concrete have to be carefullydealt with. Previous investigations by the author have introduced aconsistent and robust models for confined and unconfined concrete,these were detailed in [117–121]. Fig. 7 shows an idealized uniaxialresponse for the compressive stress–strain curves of both unconfinedand confined concrete, where fc is the unconfined concrete cylindercompressive strength which is equal to 0.8(fcu), and fcu is theunconfined concrete cube compressive strength. The correspondingunconfined strain (ec) is taken as 0.003 for plain concrete asrecommended by the ACI Specification [123]. The confined concretecompressive strength (fcc) and the corresponding confined stain (ecc)can be determined from Eqs. (3) and (4), respectively, proposed byMander et al. [124].

f cc ¼ f cþk1f l ð3Þ

ecc ¼ ec 1þk2f l

f c

� �ð4Þ

Fig. 7. Response of concrete to uniaxial loading. (a) Compression and (b) Tension.

where fl is the lateral confining pressure imposed by the steel tube.The approximate value of (fl) can be obtained from empiricalequations given by Hu et al. [125]. The factors (k1) and (k2) are takenas 4.1 and 20.5, respectively, as given by Richart et al. [126]. Knowing(fl), (k1) and (k2), the values of equivalent uniaxial confined concretestrength (fcc) and the corresponding confined strain (ecc) can bedetermined using Eqs. (3) and (4).

To define the full equivalent uniaxial stress–strain curve forconfined concrete as shown in Fig. 7, three parts of the curve haveto be identified. The first part is the initially assumed elastic rangeto the proportional limit stress. The value of the proportional limitstress is taken as 0.5(fcc) as given by Hu et al. [125]. While theinitial Young’s modulus of confined concrete (Ecc) is reasonablycalculated using the empirical Eq. (5) given by ACI [123].The Poisson’s ratio (ucc) of confined concrete is taken as 0.2 [125].

Ecc ¼ 4700ffiffiffiffiffiffif cc

qMPa ð5Þ

The second part of the curve is the nonlinear portion startingfrom the proportional limit stress 0.5(fcc) to the confined concretestrength (fcc). This part of the curve can be determined fromEq. (4) which is a common equation proposed by Saenz [127].This equation is used to represent the multi-dimensional stressand strain values for the equivalent uniaxial stress and strainvalues. The unknowns of the equation are the uniaxial stress (f)and strain (e) values defining this part of the curve. The strainvalues (e) are taken between the proportional strain, which isequal to (0.5fcc/Ecc), and the confined strain (ecc) which is corre-sponding to the confined concrete strength. The stress values (f)can be determined from Eq. (6) by assuming the strain values (e).

f ¼Ecce

1þðRþRE�2Þð eeccÞ�ð2R�1Þð eecc

Þ2þRð eecc

Þ3

ð6Þ

where RE and R values are calculated from Eqs. (7) and (8),respectively:

RE ¼Eccecc

f cc

ð7Þ

R¼REðRs�1Þ

ðRe�1Þ2�

1

Reð8Þ

While the constants Rs and Re are taken equal to 4 asrecommended by Hu and Schnobrich [128].

The third part of the confined concrete stress–strain curve isthe descending part from the confined concrete strength (fcc) to avalue lower than or equal to rk3fcc with the corresponding strainof 11ecc. The reduction factor (k3) depends on the D/t ratio and thesteel tube yield stress (fy). The approximate value of k3 can becalculated from empirical equations given by Hu et al. [125].The reduction factor (r) was introduced by Ellobody et al. [117],based on the experimental investigation conducted by Giakou-melis and Lam [129], to account for the effect of different concretestrengths. The value of r is taken as 1.0 for concrete with cubestrength (fcu) equal to 30 MPa. While, the value of r is taken as0.5 for concrete with fcu greater than or equal to 100 MPa. Linearinterpolation is used to determine the value of r for concrete cubestrength between 30 and 100 MPa.

Following the investigations reported in [121,130,131], con-crete can be modeled using the damaged plasticity model imple-mented in the ABAQUS [107] standard and explicit materiallibrary. The model provides a general capability for modellingplain and reinforced concrete in all types of structures. Theconcrete damaged plasticity model uses the concept of isotropicdamaged elasticity, in combination with isotropic tensile andcompressive plasticity, to represent the inelastic behaviour of

0

200

400

600

800

1000

1200

1400

0Time (minutes)

Tem

pera

ture

(°C

)

Standard fire curve

60 120 180 240 300 360

Fig. 8. Temperature–time relationships for the standard fire curve specified in EC1

[133].


concrete. The model assumes that the uniaxial tensile andcompressive response of concrete is characterized by damagedplasticity. Under uniaxial compression the response is linear untilthe value of proportional limit stress is reached. Under uniaxialtension the stress–strain response follows a linear elastic relation-ship until the value of the failure stress (fct) is reached. Thesoftening stress–strain response, past the maximum tensilestress, was represented by a linear line defined by the fractureenergy and crack band width. The fracture energy Gf (energyrequired to open a unit area of crack) was taken as 0.12 N/mm forplain concrete as recommended by CEB [132]. The fracture energydivided by the crack band width (h) was used to define the areaunder the softening branch of the tension part of the stress–straincurve, as shown in Fig. 7b. The crack band width was assumed asthe cubic root of the volume between integration points for asolid element, as recommended by CEB [132]. Fig. 7 shows theresponse of concrete to uniaxial loading in compression andtension used in this study. The dilation angle can be taken forplain concrete as 20 degrees. The default viscosity parameter,which is zero, can be used.

2.6. Modelling of interfaces

For concrete–steel composite columns, modelling interfacesbetween steel tubes and surrounding concrete as well as betweenreinforcement bars and concrete is of great importance. Theinterfaces can be modelled by interface elements (using thenCONTACT PAIR option) available within the ABAQUS [107]element library. The method requires defining two surfaces thatare the master and slave surfaces. The master surface within thismodel can be defined as the steel tube surface confining theconcrete infill that is the slave surfaces. The interface elementsare formed between the master and slave surfaces and monitorthe displacement of the slave surface in relation to the mastersurface. When the two surfaces remain in contact, the slavesurface can displace relative to the master surface based on thecoefficient of friction between the two surfaces, which is taken as0.25 [121,130,131]. When the two surfaces are in contact, theforces normal to the master surface can be transmitted betweenthe two surfaces. When the two surfaces separate, the relativedisplacement between the two surfaces can still be monitored butthe forces normal to the master surface cannot be transmitted.However, the two surfaces cannot penetrate each other.

2.7. Load application and boundary conditions

Accurate finite element models should simulate the appliedloads and boundary conditions identically. As an example, forfixed-ended steel, cold-formed steel, stainless steel and compositecolumns, the ends of the columns must be fixed against alldegrees of freedom except for transitional displacement at theloaded end in the direction of the applied load. The nodes otherthan the two ends must be free to translate and rotate in anydirections. On the other hand for pin-ended columns, rigid steelloading plates have to be modelled and attached to the ends ofthe columns. Two nodes on the outside surfaces of the loadingplates (support point and loading point) must be fixed against alldegrees of freedom except for transitional displacement at theloaded end in the direction of the applied load. The load can beapplied in increments using the modified RIKS method availablein the ABAQUS [107] library. The RIKS method is generally used topredict unstable and non-linear collapse of a structure such aspost-buckling analysis. It uses the load magnitude as an addi-tional unknown and solves simultaneously for loads and displa-cements. The load was applied as static uniform loads at eachnode of the loaded end which is identical to the experimental

investigation. The non-linear geometry parameter (NLGEOM) wasincluded to deal with the large displacement analysis. Forconcrete–steel composite columns, the rigid loading plates mustbe attached to the steel and concrete elements to allow foruniform load application for the composite column components.

3. Finite element modelling approach for columns underfire conditions

3.1. General

Modeling of the column behaviour at ambient temperatureprovides a good insight into the column strength, failure model,load–displacement relationships and load–strain relationships.The column strength at ambient temperature can be used todefine the static load applied to the column at elevated tempera-tures. In order to model the behaviour of the columns under fireconditions, two analyses have to be performed. The first is knownas thermal heat transfer analysis, which determines the tempera-ture distribution within the columns and the temperature mag-nitudes. In this analysis, the columns are heated following aspecific fire curve and the heated surfaces (exposed surfaces tofire) and unheated surfaces (unexposed surfaces to fire) have tobe defined. The thermal material properties of the columncomponents have to be defined in this analysis. The secondanalysis is known as the thermal–structural analysis, whichdetermines the failure modes, fire resistances, and load–displacement relationships. In this analysis, the columns aresubjected to a static initial load that is kept constant during thefire exposure and then subjected to heat and temperatures savedfrom the thermal heat transfer analysis. The mechanical materialproperties at elevated temperatures of the column componentshave to be defined in this analysis.

3.2. Thermal heat transfer analysis

A thermal 3-D finite element analysis has to be performed forthe columns, using the heat transfer option available in ABAQUS[107]. The temperature distribution in the columns can bepredicted and calibrated against the measured temperatures fromthe tests. The columns are heated using an experimental firecurve, which should follow a specific standard fire curve as that ofthe EC1 [133] shown in Fig. 8. A constant convective coefficient(ac) of has to be defined for the exposed and unexposed surfaces.Also, the radiative heat has to be calculated knowing the surface

Fig. 10. Strength reduction factors specified in EC3 [136,140].

Fig. 9. Examples of thermal material properties specified in EC3 [136,140].

(a) Thermal elongation, (b) Specific heat and (c) Specific heat.

Fig. 11. Stiffness reduction factors specified in EC3 [136,140].


emissivity (e). The specific heat, thermal conductivity and thermalexpansion of the column components have to be defined eitherexperimentally or using current codes of practice, with examplesgiven for steel material in comparison with stainless steelmaterials as shown in Fig. 9. Measuring moisture content ofconcrete is important for modelling concrete–steel compositecolumns since it is used in the calculation of the specific heat ofconcrete.

3.3. Thermal–structural analysis

Following the heat transfer analysis, a thermal–structural analy-sis has to be performed in two steps for the columns investigated. Inthe first step, the column has to be subjected to the static load atambient temperature. In the second step, the column has to beheated using the temperatures predicted from the heat transferanalysis, with the static load remained constant all the time. Thetemperatures were applied using the nTEMPERATURE option

available in ABAQUS [107]. The initial overall geometric imperfec-tion has to be included in the thermal–structural analysis withmeasured magnitudes from the tests. The stress–strain curves of thecolumn components have to be predicted at elevated temperaturesand inserted in the thermal–structural analysis. The strength andstiffness reductions at elevated temperatures can be predictedexperimentally or using current codes of practice, with examplesof strength and stiffness reductions as well as stress–strain curves ofconcrete at elevated temperatures as shown in Figs. 10–12. By theend of the thermal–structural analysis, the fire resistance of thecolumn can be predicted as well as the failure modes and time–displacement relationships and time–strain relationships.

4. Verification of the finite element models at ambient andfire conditions

After developing the finite element models at ambient and fireconditions, the results of the finite element models has to beassessed against the test results. Verification of the finite elementmodel at ambient temperature should ensure that the finite elementmodel predicts, with acceptable tolerances, the column strengths,failure modes, load–displacement and load–strain relationships. Onthe other hand, verifying the finite element models at elevatedtemperatures should ensure that the model predicts the fireresistance, the failure modes, time–displacement and time–strainrelationships. Once the finite element models are validated, it can beused to perform parametric studies investigating different column

Fig. 12. Stress–strain curves of concrete at elevated temperatures specified in EC4

[142]. (a) Compression and (b) Tension.


geometries, loading and boundary conditions, different fires, etc.,which provides a data base that can be used to improve design rulesspecified in current codes of practice.

5. Current codes of practice for steel, cold-formed steel andcomposite columns at ambient and fire conditions

This section mentions current codes of practice for steel, cold-formed steel, stainless steel and composite columns at ambientand elevated temperatures, which the author has used previouslyin Refs. [59,108–121,130,131] for comparisons with test and finiteelement results. The mentioned codes are that of the AmericanSpecification and European Code. The current design rules speci-fied in the American specification for steel columns at ambienttemperatures is covered by the American Institution for SteelConstruction (AISC) [134], while that of the European Code isEurocode 3 (EC3, BS EN 1993-1-1) [135]. The design rules for steelcolumns at elevated temperatures, once again used previously bythe author, is that of the European Code Eurocode 3 (EC3, BS EN1993-1-2) [136]. The current design rules specified in the Amer-ican specification for cold-formed steel columns at ambienttemperatures is covered by the North American Specification(NAS) [137], while that of the European Code is Eurocode 3(EC3, BS EN 1993-1-3) [138]. The design rules for cold-steelcolumns at elevated temperatures, is also covered by the Euro-code 3 (EC3, BS EN 1993-1-2) [136]. The current design rulesspecified in the American specification for stainless steel columnsat ambient temperatures is covered by ASCE [139], while that ofthe European Code is Eurocode 3 (EC3, BS EN 1993-1-4) [140]. Thedesign rules for cold-steel columns at elevated temperatures, isalso covered by the Eurocode 3 (EC3, BS EN 1993-1-2) [136].

Finally, the latest design rules specified in the American specifica-tion for composite columns at ambient temperatures is coveredby the American Institution for Steel Construction (AISC) [134],while that of the European Code is Eurocode 4 (EC4, BS EN 1994-1-1) [141]. The design rules for concrete–steel composite columnsat elevated temperatures is that of the European Code Eurocode 4(EC4, BS EN 1994-1-2) [142]. It should be noted that detailing thedesign rules specified in the aforementioned current codes ofpractice is out of the scope of this paper. This is attributed to thatthey are already presented in Refs. [59,108–121,130,131].

6. Comparisons of finite element results with designcode results

6.1. General

This section provides examples of comparisons reported pre-viously by the author for different columns. The comparisonswere held between numerical results obtained using the finiteelement modelling approach presented in this study and designresults predicted using current codes of practice. The mainobjective is not to repeat previously reported research, but tohighlight the significance of developing nonlinear finite elementmodels in enhancing and assessing the design rules specified incurrent codes of practice at ambient and fire conditions. Twoexamples are presented in this section, The first example is forstiffened stainless steel columns at ambient temperature. While,the second example is for concrete-encased steel compositecolumns at fire conditions. In overall, it is interesting to showhow nonlinear finite element models could generate more dataoutside the limits specified in current codes of practice, whichwidens its use.

6.2. An example of stainless steel stiffened columns at ambient

temperature

The finite element approach presented in this study waspreviously used by the author in investigating the compressivebehaviour of stainless steel stiffened hollow section columns asdetailed by Ellobody [115]. The study [115] was a completelynumerical study that has incorporated stiffeners in a previouslydeveloped and verified finite element model detailed by Ellobodyand Young [113]. A total of 84 stainless steel stiffened andunstiffened columns were analysed in [115] to show how incor-porating stiffeners in stainless steel slender hollow sectioncolumns can considerably change the compressive behaviour ofthe columns. The study has shown that the stainless steelstiffened slender hollow section columns offer an average of128% increase in the column strength than that of the unstiffenedslender hollow section columns. Also, the study [115] has shownthat the stiffened column has a considerably higher stiffness thanthat of the unstiffened column which has resulted in sharperload-axial shortening behaviour as shown Fig. 13. The figureshows a comparison between the load-axial shortening curvesof the stiffened and unstiffened slender rectangular hollow sec-tion columns. In addition, the failure modes of the stiffened andunstiffened square and rectangular hollow section columns werealso compared in [115]. It was shown that although the failuremodes were unchanged, the out-of-plane local buckling modeswere considerably limited by the presence of the intermediatestiffeners in the stiffened column specimens. Furthermore, thestudy [15] has assessed the finite element strengths of the 84stainless steel stiffened and unstiffened hollow section columnsagainst the design rules specified in the American and Europeanspecifications. It has been shown [115] that the design strengths

Fig. 14. Comparison of time-axial displacement relationships for unrestrained and

axially restrained concrete encased steel composite columns as detailed by Young

and Ellobody [144].

Fig. 13. Comparison of load-axial shortening curves for stainless steel stiffened

and unstiffened rectangular hollow section columns as detailed by Ellobody [115].


calculated using the American and European specifications aregenerally conservative for the stainless steel unstiffened slendersquare and rectangular hollow section columns, but slightlyunconservative for the stiffened slender square and rectangularhollow section columns.

6.3. An example of concrete-encased steel composite columns at fire

conditions

The finite element approach presented in this study waspreviously used by the author in investigating the compressivebehaviour of unrestrained and axially restrained concrete-encased steel composite columns as detailed by Ellobody andYoung [143] and Young and Ellobody [144]. The studies [143,144]were, once again, completely numerical studies that haveextended a previously developed and verified finite elementmodel detailed by Ellobody and Young [130] at ambient tem-peratures. A total of 76 unrestrained and axially restrainedconcrete-encased steel composite columns were analysed atelevated temperatures using the finite element approach pre-sented in this paper. The investigated composite columns haddifferent load ratios during fire, different coarse aggregates,different slenderness ratios and different axial restraint ratios tothermal expansion, which are not presented in current codes ofpractice up-to-date. The studies [143,144] have shown thataxially restrained concrete encased steel composite columnsbehaved differently in fire compared to those unrestrained

columns as shown in Fig. 14. the figure compared an unrestrainedconcrete encased steel composite column specimen S37 and arestrained column specimen S38, as detailed in [144]. It wasshown that the unrestrained column experienced the typical‘‘runaway’’ failure, while the restrained column behaved differ-ently. The columns started to buckle at points A and B forrestrained and unrestrained columns, respectively. For therestrained column, the time from the start of heating to point Ais known as the expanding zone where the column is experien-cing axial expansion. On the other hand, the time from point A topoint C is known as the contracting zone where the column isexperiencing axial shortening. The axial shortening is the reflec-tion of the initiation of a large horizontal displacement in themid-height due to second order effects. However, bearing in mindthat the restraint to the composite column during the expansionstage must also remain during the contraction stage after thecolumn has buckled. Hence, in this study [144], the time from thestart of heating until point C, where axial displacement went backto initial state will be defined as the fire resistance of the concreteencased steel composite column. This is because at point C thecolumn will behave similar to unrestrained column and failuretakes place.

Also, the studies [143,44] have assessed the finite element fireresistances of the 76 unrestrained and axially restrained concrete-encased steel composite columns against the design fire resis-tances specified in the European Code (EC4) [142]. The study[143] has shown that the EC4 is conservative for all the investi-gated unrestrained concrete-encased steel composite columns,except for the columns having a load ratio of 0.5 as well as thecolumns having a slenderness ratio of 0.69 and a load ratio of 0.4.On the other hand, the study [144] has shown that the EC4 isgenerally conservative for all the axially restrained concrete-encased steel composite columns, except for some columns withhigher load and slenderness ratios.

7. Main conclusions

Review of previously published experimental and numericalinvestigations on the stability of steel, cold-formed steel, stainlesssteel and composite columns at ambient and fire conditions hasbeen presented in this paper. The review of the previousresearches of the columns has been presented up-to-date andprovided a collective useful materials for researches and aca-demics interested in this field. A consistent, robust and efficientnonlinear finite element modelling approach, adopted by theauthor over the last ten years, for analyzing the compressivebehaviour of the different columns at ambient and elevatedtemperatures has been presented. The finite element approachhas accounted for the initial local and overall geometric imperfec-tions, residual stresses and nonlinear material properties atambient and elevated temperatures of the different columns.Current codes of practice for predicting the column strengthshas been highlighted. In addition, comparisons of the finiteelement analysis results with the design predictions of currentcodes of practice have been presented. The paper has shown thatthe finite element approach presented provided better under-standing for the behaviour of the columns at ambient and fireconditions and assessed the accuracy of design rules specified incurrent codes of practice.

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a consistent nonlinear approach for analysing steel, cold-formed steel, stainless steel and...

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