Numerical study of a steel sub-frame in fire

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Computers and Structures 86 (2008) 1619–1632

Numerical study of a steel sub-frame in fire

Aldina Santiago a,*, Luıs Simoes da Silva a, Paulo Vila Real b, Milan Veljkovic c

a Department of Civil Engineering, University of Coimbra, Polo II, Pinhal de Marrocos, 3030-290 Coimbra, Portugalb Department of Civil Engineering, University of Aveiro, Portugal

c Department of Civil Engineering, Technical University of Lulea, Sweden

Received 28 May 2007; accepted 16 January 2008Available online 10 March 2008

Abstract

Steel-framed buildings are generally designed with ‘‘simple” shear-resisting connections, and lateral forces are resisted by verticalbracing and shear walls. When a beam is considered then the effects of the longitudinal restraints by the adjacent structure and the rota-tional restraint by the connections has to be taken into account. Because of structural interaction, the beam behaviour at elevated tem-perature is rather complex.

This paper presents a numerical parametric study of a structural system consisting of an exposed steel beam restrained between a pairof fire protected steel columns. The structural sub-frame is modelled using 3D shell elements, thereby taking into account the effect of thelocal failure modes, and the realistic behaviour of the sub-frame exposed to natural fire. The numerical model accounts for the initialgeometrical imperfections, nonlinear temperature gradient over the cross-section, geometrical and material nonlinearity and temperaturedependent material properties.

Results obtained using a general Finite Element software – LUSAS and a fire dedicated software – SAFIR, are compared. The influ-ence of following variables: beam span/depth ratio, lateral restraint, gradient temperature within the cross-section and mechanical loadlevel is presented in the paper. The failure modes, the development of the internal forces and displacements throughout the analysis areconsidered to exemplify the effects of the variables considered.� 2008 Elsevier Ltd. All rights reserved.

Keywords: Fire behaviour; Natural fire; Numerical modelling; Steel structures; Connections; Local and distortional buckling

1. Introduction

The behaviour of steel members during fire is complexand very much dependent on the restraint at the memberends. In order to capture the beam behaviour during a fireit is therefore required to take a global view and to examinethe behaviour of a representative sub-structure [1] that canreproduce the redistribution of forces that takes place asthe temperature varies with time. In steel structures, therestraint at member ends depends on the following twoaspects: (i) the behaviour of the joints connecting, a beam

0045-7949/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2008.01.006

* Corresponding author. Tel.: +351 239 797257; fax: +351 239 797217.E-mail addresses: [email protected] (A. Santiago), [email protected]

(L.S. da Silva), [email protected] (P.V. Real), [email protected] (M.Veljkovic).

to a column that is usually semi-rigid and/or partialstrength and; (ii) the stiffness provided by the adjacentmembers (bending, axial or torsional stiffness).

Obtaining detailed experimental evidence of the behav-iour of steel members subjected to realistic fire conditionsis quite difficult and expensive. Natural fire tests, such asthe Cardington tests [1] are of course ideal as they reproducereality very closely but it is quite difficult to obtain detailedmeasurements of the mechanical response of individualmembers and to quantify the various parameters that con-trol their behaviour. Available sub-frame tests are few andare usually unable to reproduce the transient temperatureconditions that occur along the length of the members. Itis therefore useful to develop numerical models to assessthe behaviour of steel sub-frames subjected to fire.

Ideally, prescribed equations would be of greater inter-est for design purposes. However, the existing analytical

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Table 1Overview of the parametric study

Parameter Study cases

Gradienttemperaturewithin the beamcross-section

(i) Cardington beam temperature; (ii) largetemperature gradient within the cross-section; (iii)uniform temperature within the cross-section

Gradienttemperaturealong the beamspan

(i) Cardington beam temperature along the beamspan; (ii) lower temperature on 100 mm from thejoint

Lateral–torsionalrestraint

(i) top beam flange restrained; (ii) beamunrestrained

Beam length (i) span/depth ratio = 19.3; (ii) span/depthratio = 9.7; (iii) span/depth ratio = 14.5; (iii) span/depth ratio = 29

Mechanical loadfactor

(i) Pfi,d = 12% PRd; (ii) Pfi,d = 20% PRd; (iii)Pfi,d = 40% PRd; (iv) Pfi,d = 60% PRd; (v)Pfi,d = 80% PRd. PRd is the uniform distributedloading corresponding to the design momentresistance of the simple supported beam at roomtemperature

1620 A. Santiago et al. / Computers and Structures 86 (2008) 1619–1632

approaches to include temperature effects (that change themechanical properties of steel with temperature) such asBradford et al. [2] are still of no practical interest becausethey are not able to incorporate key effects and are onlyvalid in the elastic range (that in a fire event is only validfor the initial stage of the fire).

Steel sub-frames were studied in Huang et al. [3], Santi-ago et al. [4], Hao et al. [5]. The beam and columns are usu-ally modelled by beam elements with the appropriatenumber of degrees of freedom, allowing in plane and out-of-plane bending and shear deformation, while the jointbehaviour is simulated by an assumed moment-rotationrelationship. Large displacements are also required toreproduce catenary action. However, under fire conditions,large variable axial forces due to the restraint to thermalexpansion during the heating phase and to thermal con-traction during the cooling phase can be observed. Thisaxial force, in combination with bending moment andshear force in the connection area induces local bucklingand distortional buckling. Beam elements cannot realisti-cally model local and distortional buckling; therefore, shellor solid elements should be used. Liu [6] and Lui andDavies [7] used a three-dimensional model created by iso-parametric shell finite elements and beam-spring elementsto model an internal bolted joint consisting of two identicalcantilevers attached to a column. Comparisons againstavailable tests results have shown good agreement. Yinand Wang [8] emphasized the effect of the axial and rota-tional restraints at the beam ends in the survival tempera-ture in fire. They calibrated a numerical model againstexperimental results of fire tests on a restrained steel beam[7,9] and carried out a parametric study on the parametersthat affect the development of catenary action in a steelbeam. Beam and end-plate were simulated by shell ele-ments while the end restraints were modelled by elasticsprings: axial restraint at the centroid of the beam androtational restraints by two springs near the top and bot-tom of the beam. These studies were focused on the analy-sis of the influence of the joint and end restraints on thebeam behaviour, specially force redistribution and dis-placements at high temperatures. Furthermore, with theexception of Santiago et al. [4], all used a ‘‘nominal” fireloading, constantly increasing temperature (ISO curve),so that the effect of the cooling phase was not considered.However, during the cooling phase the plastically deformedbeam contracts significantly and some connection compo-nents experience tensile forces [10].

The main objective of this paper is to propose a three-dimensional numerical model for a welded steel sub-frameusing shell elements. The following phenomena are consid-ered: the local failure modes; the influence of the heatingand cooling phases and the response of the sub-structure;and the influence of several parameters relevant for thebeam design. The main parameters are: temperature varia-tion within the beam cross-section and along the beam, lat-eral torsional restraints on the top beam flange, beam span/depth ratio and mechanical load factor. The study cases are

listed in Table 1. The temperature results measured in the7th Cardington fire test, Wald et al. [11], are considered.

2. Numerical model

2.1. Introduction

In order to explore the behaviour of a sub-frame steelsystem under fire loading, the structure of Fig. 1 was cho-sen. It consists of two HEA300 cross-section columnswelded to an IPE300 cross-section beam with 5.7 m freespan, supporting a 150 mm thick steel–concrete compositeslab. The beam was assumed unprotected and the topflange unexposed to the fire load, while the column is ther-mally protected and unexposed to the fire load. This struc-tural model corresponds to an experimental fire testprogramme carried out at the University of Coimbra inPortugal [12]. It also corresponds to a representative sub-structure of the 7th Cardington fire compartment test[11,13]. The specification of the thermal loading (time–tem-perature–space variation) on the sub-structure was themeasured transient temperatures.

As stated in Section 1, the degree of restraint at thebeam ends is crucial for the redistribution of forces thattakes place throughout the fire. It is thus useful to establisha non-dimensional measure of the restraint provided by thecolumns to the beam. The axial and rotational stiffnesses ofthe column at the beam level are given by:

Ka ¼3Ec Ic

L3sup

þ 3EcIc

L3inf

¼ 100715:6 kN=m ð1Þ

Kr ¼3EcI c

Lsup

þ 3EcIc

Linf

¼ 174530:5 kN=m ð2Þ

IPE 30012

10

HE

A 3

00

1129

5700

300

HE

A 3

00

YZ

X

Fig. 1. Sub-structure model (mm).

A. Santiago et al. / Computers and Structures 86 (2008) 1619–1632 1621

where Ec is the Young modulus of the column, Ic the sec-ond moment of area of the column, Lsup the upper lengthof the column and Linf the lower length of the column.The corresponding non-dimensional axial and rotationalrestraint ratios are given by:

ba ¼Ka

EbAb

Lb

¼ 0:267 ð3Þ

br ¼Kr

4EbIb

Lb

¼ 7:460 ð4Þ

where Eb is the Young modulus of the beam, Ib the secondmoment of area of the beam, Ab the area of the beam andLb the length of the beam.

In the following section, the implemented numericalmodel will be described in detail. The numerical modelwas modelled and calculated using the finite element codeLUSAS [15].

2.2. Material properties

The assumed material of steel beam and columns atroom temperature corresponds to the nominal law ofS355. In order to perform a nonlinear geometrical analysis,this curve was modified to the true stress–logarithmic strainlaw as explained in Fig. 2. The steel is nonlinearly temper-ature dependent and therefore material properties, as givenin EN 1993-1-2 [14], were used in the FE model. The effect

0

90

180

270

360

450

0.00 0.02 0.04 0.06 0.08 0.10strain

stre

ss (

MPa

)

Nominal law

True stress-logarithmic strain law

Fig. 2. Stress–strain relationship of steel at room temperature.

of creep on the deformation of steel has been consideredindirectly through the use of the conventional stress–strain–temperature relationship suggested by EN 1993-1-2.

2.3. Finite elements and mesh discretization

The numerical analysis was performed with the FE codeLUSAS [15]. In order to reduce the model size and subse-quent processing time, symmetry conditions were imposedat the beam mid-span. Three-dimensional eight-nodedthick shell elements with five degrees of freedom per node(u,v,w,ha,hb) referred to the axes shown in Fig. 3 wereemployed. Its formulation accounts for shear, bendingand membrane internal forces. The FE mesh depicted inFig. 3 complies with the requirements for a reliable simula-tion and satisfies the mesh convergence study performedwithin the framework of this research work. This modelhas 2918 elements and 9068 nodes.

The concrete slab was not modelled as it only providedthermal boundary conditions for the temperature and lat-eral restraint to the beam top flange (non-compositebehaviour).

2.4. Imperfections

In the beam, lateral geometrical imperfections were con-sidered while the residual stresses were neglected. A lateralgeometrical imperfection given by expression (5) was con-sidered in accordance to prEN 1090-2:2007 [16] and Grein-er [17]. An initial rotation around the x-axis with amaximum value of L/1000 at mid-span and a curvatureon the beam web with a maximum value of 4 mm at themid-web were also introduced (Fig. 4).

yðxÞ ¼ L1000

sinpxL

� �ð5Þ

2.5. Loading and boundary conditions

Symmetry conditions were enforced at mid-span of thebeam, the axial displacement in the X-direction wasrestrained as well as the rotations around the Y and Z

Fig. 3. Finite element model.

dy4 mm

Fig. 4. Local geometric imperfection on the beam cross-section.

0

200

400

600

800

1000

0 25 50 75 100 125 150t (min)

T (

ºC)

Beam webBeam lower flangeBeam upper flangemaximum applied temperature

Fig. 5. Thermal loading.


directions. Displacements in the X, Y and Z directions wererestrained at the bottom of the column, while the top of thecolumns was only restrained in the X and Y directions. Inthe reference case it was considered that the beam wasrestrained by a concrete slab: displacement in the Y-direc-tion was thus prevented at the top flange of the beam.

The mechanical loading applied at room temperaturecorresponds to the self-weight and a uniform loading of10 kN/m distributed along the top flange of the beam,which corresponds to about 20% of the beam resistanceat room temperature. This loading was applied at roomtemperature and remained constant throughout the analy-sis. The assumed thermal loading on the beam is shown inFig. 5, and corresponds to a fire curve with heating andcooling phases that reproduces the beam mid-span temper-atures observed during the 7th Cardington fire test [11]. Toavoid numerical convergence problems related to very lowvalues of stiffness and yield stress of steel, slightly lowermaximum temperatures than those observed experimen-tally were adopted (dotted line). Because the columns werefire protected they are assumed to remain at roomtemperature.

2.6. Nonlinear solution strategies

The integration algorithm is of implicit type (implicitbackward Euler), which ensures a quadratic convergenceto the iteration procedure associated with the Newton–Raphson method. The kinematic description of shell ele-ments in a nonlinear geometrical analysis is based on theTotal Lagrangian formulation, which accounts for largedisplacements and small strains; in this formulation stressesand strains are output in terms of the ‘‘second Piola–Kirc-hoff stresses” and ‘‘Green Lagrange strains” with referenceto the undeformed configuration. This requires the use of atrue stress–logarithmic strain measure (rn � en) for the def-inition of the uniaxial material response, instead of theclassic engineering constitutive law (r � e) (Fig. 2). Bothmeasures are related by the following relationship, Malvern[18]:

rn ¼ rð1þ eÞ and en ¼ lnð1þ eÞ ð6Þ

To account for the material nonlinearity, the von Misesyield criterion with nonlinear isotropic hardening was used.


In the FE code LUSAS, the influence of the temperature onthe elastoplastic response of the materials is represented inan approximate manner by calculating the elastic stressesusing updated temperature dependent properties. The elas-tic predictor or trial stress r at time t + Dt is evaluated inthe following way:

(a) Elastically unloading the structure at time t, using thematerial modulus interpolated at temperature h, i.e.

r� ¼ tr� tD � tee where tee ¼ te� tep ð7Þ(b) Elastically re-loading the structure at time t + Dt,

using the material modulus interpolated at tempera-ture ht+Dt, i.e.

tþDtr ¼ r� þ tþDtD tee þ Deð Þ where De ¼ tþDte� te ð8Þwhere tr is the stress at time t, tD and t+DtD are the stiffnessmatrices at time t and t + Dt, respectively, and te and t+Dteare the strains at time t and t+Dt, respectively.

-0.35

-0.27

-0.19

-0.11

-0.03 0 20 40 60 80 100 120 140 160 180t (min)

dz (

m)

Shell elements - FE code LUSASShell elements - FE code SAFIR

Beam elements - FE code SAFIRL/20 - convencional beam failure

Fig. 6. Development of the mid-span deflection during the fire.

2.7. Failure criteria

In order to assess the occurrence of failure throughoutthe fire event, a failure criteria is required. The failure con-ditions can be established by assuming that cracking occurswhen the ultimate strain eu is attained. The failure shouldbe assessed by comparing the maximum equivalent straineE,max with eu. Literature indicates that at high tempera-tures, the ultimate strain for high strength bolts is about1–6%, Li et al. [19], and for constructional steel, ultimatestrains of 20% may be expected EN 1993-1-2 [14].

3. Reference case: behaviour and validation of the numerical

model

3.1. Introduction

To provide the basis for the various parametric studies,a reference case was established. It consists of the structuralmodel of Fig. 1. With respect to Table 1, it corresponds to:

(i) gradient temperature within the cross-section thatcorresponds to the Cardington temperature at thebeam mid-span. Gradient temperature along thebeam span was not considered;

(ii) top flange of the beam continuously restrained;(iii) a span/depth ratio of 19.3, corresponding to a free

span of L0 = 5700 mm;(iii) a mechanical load factor Pfi,d = 20% PRd.

Because of the complexity of performing a transienttemperature nonlinear numerical analysis of a steel struc-ture with temperature varying material properties, this ref-erence case was analyzed using two different FE programs.LUSAS [15], a general purpose commercial nonlinear finiteelement code was used throughout this study, while SAFIR

[20], a specialized finite element code specifically developedfor the analysis of structures subjected to fire was only usedfor the reference case, to provide additional validation ofthe numerical results.

In the case of LUSAS, all calculations were carried outusing shell elements, as it was already described in Section2.3, while for SAFIR two alternative analyses were per-formed using beam and shell elements.

3.2. Global behaviour

Fig. 6 illustrates the variation of the mid-span deflec-tions of the beam throughout the fire, for the three FEmodels. For all cases, this variation is markedly differentduring the analysis. It starts gradually but, as the lowerbeam flange temperature rises above 550 �C (t = 25 min),the rate of deformation increases significantly. At the max-imum temperature, the mid-span deflection was about0.34 m, corresponding to L/8, L/20 being the usual maxi-mum deflection at the fire limit state, BS 476 [21]. Duringthe cooling phase, since the material has been stressedbeyond the elastic limit, the corresponding strain doesnot follow the same stress–strain path as during the heatingphase and a plastic deflection is observed: 0.27 m.

It is noted that the analysis using beam elements isunable to follow the cooling phase, the model becomingnumerically unstable before reaching the maximum tem-perature. Both analyses using shell elements show goodagreement.

Fig. 7 illustrates the development of the axial force andbending moment at mid-span of the beam (cross-section A)and at a cross-section 0.03 m away from the column flange– cross-section B. For the shell models, the axial force wascalculated by equilibrium and by the integration of the nor-mal stresses at the cross-section B while the bendingmoment is always calculated by taking moments withrespect to the beam bottom flange axis.

The analysis of the development of the internal forcesshould be divided into different stages. In Phase 1(t < 9 min) the thermal expansion was converted into ther-mal stresses that increased the level of axial compressionand sagging moment without large displacements(Fig. 6). Phase 2 that starts approximately after 9 minof the fire exposure, corresponds to the increase of the

- 700

- 500

- 300

- 100

100

300

500

700

900

0 20 60 80 100 120 140 160 180t (min)

N (k

N)

Shell elements (mid-span beam) - LUSASShell elements (mid-span beam) - SAFIRBeam elements (mid-span beam) - SAFIRShell elements (line 1) - LUSAS

-270

-220

-170

-120

-70

-20

30

80

130

0 20 40 60 80 100 120 140 160 180t (min)

M (k

Nm

)

Shell elem. (line 1) (def. position) - LUSASBeam elem. (line 1) - SAFIRShell elem. (line 1) (indef. position) - LUSASShell elem. (mid-span beam) - LUSAS

40

Fig. 7. Development of the beam axial force and bending moment during the fire.

Fig. 8. Deformed mesh (LUSAS – left side; SAFIR – right side). Scale factor is 1.0.



internal forces. In Phase 3 (26 min < t < 45 min) a decreaseof the internal forces is noted because of the materialstrength degradation and large deflections. In Phase 4(45 min < t < 55 min), the beam behaviour changes frombending to catenary action and tensile forces are devel-oped. Bending moments reduce and become similar tothose observed at room temperature. Phase 5 (t < 55 min)corresponds to the cooling phase – increasing of the tensileforces and hogging moment because of the recovering ofstrength and stiffness. At about t = 125 min, because thecooling rate reduces and the beam reaches temperatureslower than 400 �C, and the bottom flange temperature islower than 200 �C, the rate of development of the mid-span

Fig. 9. Development of the maximum equivale

deflection and internal forces also decrease. It is finallynoted that during cooling the axial force is not constantalong the beam.

3.3. Local behaviour at the joints

Fig. 8 shows the evolution of deformation with timearound the joint area. Local buckling of the beam bottomflange and shear buckling at the web in the vicinity of thejoint were the main failure mechanisms. This local bucklingoccurs during the heating phase after about 30 min of fire,due to the restraint to thermal elongation provided by theadjacent cooler column. The heated bottom flange of the

nt strains in the beam near the connection.

Table 2Beam span-to-depth ration: study cases

L/L0 L Span/depth Pfi,d (%) P (kN/m) ba br

0.5 2850 9.7 20 40.0 0.125 3.550.75 4000 14.5 20 17.8 0.180 4.981.0 5700 19.3 20 10.0 0.265 7.461.5 8550 29 20 4.44 0.380 10.63


beam is insufficient to transmit to the column the high nor-mal forces generated by the thermal stresses; subsequently,these forces are partially transferred to the heated web, thatis also unable to transfer them to the column and bucklingis developed. The beam could be assumed to behave as‘simply supported’, allowing larger mid-span deflectionsto develop (Fig. 6).

Fig. 9 shows the development of the equivalent strains,eE [22] in the vicinity of the joint. Fig. 10 depicts the corre-sponding axial forces, as well as the beginning of the yield-ing at each beam zone trE P tfy; where trE and tfy are theequivalent stress and the yield stress at time t. Also forthe joint behaviour distinct phases during the fire can beidentified: before the start of the fire, uniform tensile andcompressive bands along the alignment of the top and bot-tom flanges of the beam are present, due to the mechanicalloading. On the web panel a state of almost pure shear isobserved. These stresses and strains on the top and bottomflanges of the beam are mechanical strains. As the temper-ature starts to increase (phase 1), thermal strains are alsointroduced (positive strains) that increase the total strainin the top flange and decrease it in the bottom flange(t = 9 min.). Some of these thermal strains are convertedinto stresses which increase the axial compressive force.Beyond a temperature of 100 �C (phase 2), the elastic mod-ulus reduces with temperature and the mechanical strains

also increase. Beyond 400 �C (phases 3 and 4), total strains

continue to increase. In the joint these strains are convertedin stresses; however, due to the reduction of the yield stresswith temperature, stresses on the beam decrease. Due tothe high stiffness of the column, that remains at room tem-perature, and the local massivity of the joint comparedwith the beam, the high strains observed on the beam bot-tom flange move way from the joint – 100–110 mm(26 min). The maximum value of the equivalent strain iseE = 27% and it is observed in the interface bottom flange– web at t = 50 min (hbottom flange = 890 �C). Theses highstrains induce the local buckling observed in Fig. 8. Atthese high temperatures, the bottom flange has low com-pressive strength, the band of stresses and strains spreadsup to the web and shear buckling in the web is alsoobserved. Once the cooling phase starts (phase 5) the ther-

-500

-300

-100

100

300

500

700

0 20 40 60 80 100 120 140 160 180t (min)

N (

kN)

Beam bottom flangeBeam webBeam top flangeyielding

Fig. 10. Axial forces developed in the beam at 0.03 m from the columnflange (cross-section B).

mal strains begin to reduce. This induces a reversal of thedeflection (Fig. 6) generating tensile axial forces in theweb and bottom flange and compressive axial forces inthe top flange (166 min).

4. Influence of the beam span-to-depth ratio

In order to assess the influence of the beam span-to-depth ratio, four different spans were considered for thesub-structure model of Fig. 1: 2850 mm, 4275 mm,5700 mm and 8550 mm, corresponding to 50%, 75%,100% and 150% of the span of the reference case (L0).Given that the cross-section of the beam is kept constant(IPE300), the corresponding span-to-depth ratios varybetween 9.7 and 29, representing a range that is fairly typ-ical of practical designs. For consistency, the mechanicalload level on the beams was adjusted to give a constantload ratio of Pfi,d = 20% PRd. Table 2 lists the resulting uni-formly distributed loads, together with the correspondingaxial and rotational restraint ratios.

For the longer beam spans, the failure mode is similar tothe observed for the reference case. For the short beamspans, low bending and axial deformations are observedand large compressive deformations in the web and bottomflange are developed.

Fig. 11 shows that when the beam behaviour is con-trolled by bending at low temperatures, a longer beam spangives larger beam deflections. At high temperatures, whenthe beam behaviour is controlled by catenary action, allbeam deflections converge (45 min), except for L = 1.5L0,due to the previous large deflections that already occurredat low temperatures. During the cooling phase, the rate ofrecovery is dependent on the beam span. Long beam spansshow the highest rate of deflection reduction.

-0.49

-0.42

-0.35

-0.28

-0.21

-0.14

-0.07

0.00

0 20 40 60 80 100 120 140 160 180t (min)

dz (

m)

L = L0L = 0,5 L0L= 0,75 L0 L = 1,5 L0


-700

-500

-300

-100

100

300

500

700

0 20 40 60 80 100 120 140 160 180t (min)

N (

kN)

L = 0,5 L0

L = L0

L = 0,75 L0

L = 1,5 L0Tensile capacity

Fig. 12. Development of the beam axial force during the fire.


At low temperatures, the compression force varies withthe beam span, due to different rate of thermal expansion(Fig. 12). As the temperature increases and the axial stiff-ness decreases, this different rate is more noticeable. Longbeams have higher compression forces. The compressionforce starts to drop at a slightly lower temperature forthe longer beam spans, but the start of the decrease of com-pression displacement is independent of the beam span. Athigh temperatures, because the beams have the same cross-sectional area, they behave similarly and have almost thesame catenary force, indicating a minor influence of thebeam bending resistance. The tensile axial force is indepen-dent of the beam span during the cooling phase. In allcases, the axial forces are within the bounds correspondingto the tensile capacity [14], but in the case L = 0.5L0, the

Fig. 13. Development of the maximu

beam fails when the maximum catenary force reaches thetensile capacity.

Fig. 13 shows that for the short beams, the maximumequivalent strains near the joint are located at the top ofthe cross-section. However, as the beam span increases,the maximum equivalent strains decrease and shiftstowards the bottom beam flange.

5. Influence of temperature gradient within the beam cross-

section

Gradient temperature within the beam cross-section isusually the result of the heat sink effect of the concrete slabon the top flange. Special reference on the study of the tem-perature gradient effect within the beam cross-section onthe structural behaviour is made to Yin and Wang [8] thatdemonstrate that the temperature gradient has a smallinfluence on the development of the catenary action at hightemperatures. In the following, the temperature gradienteffect on the local joint behaviour is discussed. The resultsfor the reference case (case a) are compared with two otherdifferent cases (Fig. 14):

(i) Case b: uniform temperature within the beam cross-section that corresponds to the bottom beam flangetemperature of the reference case; and

(ii) Case c: large gradient temperature, the bottom flangeand web having the same temperature as the referencecase but the top flange temperature is reduced.

m equivalent strains in the joint.

0

200

400

600

800

1000

0 25 50 75 100 125 150

t (min)

T (

ºC)

Beam webBeam bottom flangeBeam top flangemaximum applied temperature

0

200

400

600

800

1000

0 25 50 75 100 125 150t (min)

T (

ºC)

uniform beam temperature

maximum applied temperature

0

200

400

600

800

1000

0 25 50 75 100 125 150

t (min)

T (

ºC)

Beam web

Beam bottom flange

beam top flange

maximum applied temperature

Fig. 14. Thermal loading: (a) reference case; (b) uniform temperature and (c) large gradient temperature.


When the cross-section is subjected to a non-uniformtemperature profile, differential expansion occurs in eachfibre and curvature is observed. Fig. 15 compares the beammid-span deflection for the three cases. All curves have asimilar pattern, but some points are noticeably different:at high temperatures, case c shows the lowest maximumdeflection because the lower temperature in the top flange.Case b deflected faster, reaching the maximum deflection att = 45 min that corresponds to the maximum temperaturein whole cross-section; for case c, the bottom flange andthe web reached the maximum temperature at t = 45 min,but the maximum top flange temperature is only reachedat t = 54 min. During the cooling phase, for case c the bot-tom flange temperature decreased faster than the web andthe top flange, leading to a faster reduction of deflection.

-0.35

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-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 20 40 60 80 100 120 140 160 180t (min)

dz (

m)

case acase bcase c


Although case b showed the highest internal forces inthe beam and is considered the critical thermal load forthe design of steel members, the equivalent strain contourswere softened around the joint area, while in the non-uni-form temperature cases, large equivalent strains are con-centrated in the lower zone of the beam, where themaximum temperatures are introduced.

6. Influence of temperature gradient along the beam span

Variations of temperature are usually only studiedacross the cross-section and it is assumed that the gradienttemperature along the beam span is constant, EN 1993-1-2[14]. Experimental observations show that due to the con-centration of mass near the joints they exhibit differenttemperatures throughout the fire, Wald et al. [13]. In thisstudy, the reference temperature (case a) was comparedto a temperature gradient along the beam span that repro-duces the measured joint temperatures (case b), as shown inFig. 16.

Fig. 17 compares the mid-span deflection for both cases.A lowest maximum temperature (10–15%) near the jointleads to a maximum displacement 15% lower than the ref-erence case. During cooling, the recovery rate of deflectiondoes not depend on the temperature gradient along thebeam span. In contrast to what is observed for the mid-span deflection, the influence of the gradient temperaturealong the beam is irrelevant on the internal forces along

0

200

400

600

800

1000

0 25 50 75 100 125 150t (min)

T (

ºC)

Beam webBeam bottom flangeBeam top flangemaximum applied temperature

0

200

400

600

800

1000

T (

ºC)

Beam webBeam bottom flangeBeam top flangemaximum applied temperatureJoint top flangeJoint webJoint bottom flange

0 25 50 75 100 125 150t (min)

Fig. 16. Thermal loading: (a) reference case; (b) temperature gradient along the beam span.

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beam length - gradient temp



the beam. Small differences are observed during the coolingphase, because the temperature at the bottom flange of thejoint decreased slower than the beam bottom flange, lead-ing to a small increase of the sagging moment due to ther-mal bowing; although, the maximum strains are movedaway from the joint.

7. Effect of lateral restraints on the beam top flange

Lateral torsional buckling is a relevant mode of failurefor a steel beam at ambient and elevated temperatures.Under fire conditions a number of relevant research studieson the lateral torsional buckling of steel beams have beencarried out. Bailey performed a numerical analysis forbeams under a uniform temperature distribution [23]. VilaReal and Piloto conducted a few experiments of slenderbeams at uniform elevated temperatures [24] and Vila Realand Franssen [25] carried out a numerical study and pro-posed a design method for lateral torsional buckling ofsteel beam under uniform temperature, which has beenadopted in the current version of EN 1993-1-2. This designmethod was improved by Vila Real et al. [26]. Recently,Yin and Wang presented a numerical study for beam later-ally unrestrained with uniform and non-uniform tempera-ture and different axial end restraints [8], in which someconsiderations about the axial reactions and mid-spandeflection are also made.

In the following, lateral torsional buckling is discussed.Initial imperfections were the same for restrained and unre-

strained beam (see Section 2.4). Fig. 18 compares thedeformed mesh of the sub-structure restrained at the topflange with the evolution of the deformed mesh of the samemodel without lateral restraint.

At low temperatures, the beam behaviour is governedby thermal expansion only, and low vertical and lateraldisplacements are observed (Fig. 19). Approximately after15 min the beam stiffness starts to decrease and anincrease of lateral displacement without vertical displace-ments is observed. The axial force reduces (Fig. 20) asthe beam deflects laterally and no local buckling nearthe joint is observed (Fig. 18). Due to the lateral move-ment of the unrestrained beam, the bending moment resis-tance is lower than for a beam with lateral restraints.Therefore, catenary action contributes more to the later-ally unrestrained beam resistance. As the temperatureincreases, to the influence of catenary action is biggerthan the bending and the lateral displacement of beamreduces so that the beam takes the original position whilethe large mid-span deflection is observed. Beyond thispoint, mid-span deflection is larger than the unrestrainedbeam (Fig. 19). The presence of lateral restraints delaysthe temperature at which the catenary action is achieved.At elevated temperatures, the lateral displacement is neg-ligible, and a torsional failure mode is evidenced until theend of the fire (Fig. 18). The vertical deflection of bothmodels converges during the cooling phase, reaching thesame value at the end of the fire.

Fig. 21 compares the growth of bending moment againstthe fire evolution in the beam. At low temperatures, thebending moment development is similar in both cases.Bending moment starts to reduce as the beam starts todeflect laterally. Simultaneously, the lateral bendingmoment and the lateral rotation increase and reach maxi-mum values at the instant corresponding to the maximumsagging moment of the restrained beam. The unrestrainedbeam reaches similar maximum lateral bending momentand maximum sagging bending moment. At elevated tem-peratures, the lateral bending moment reduces to negligiblevalues and the hogging moment gets close to values similarto those observed on unrestrained beam. During the cool-ing phase, the lateral bending moment increases again.

Fig. 18. Failure modes: (i) restrained beam; (ii) unrestrained beam (29 min.); (iii) unrestrained beam (36 min.); (iv) unrestrained beam (end of the fire –166 min). Scale factor 1.0.

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Fig. 20. Development of the beam axial force during the fire.

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Fig. 21. Development of the joint bending moment during the fire.


8. Effect of mechanical load factor

In order to examine the effect of catenary action on thecritical temperature of the beam, numerical simulations ofrestrained beams at different mechanical load levels wereconducted by Liu [7] and Yin and Wang [8]. The resultswere compared to those obtained from experimental testsshowing good agreement.

In this study, the reference case (Pfi,d = 20%Prd) wascompared to four other cases: Pfi,d = (12%; 40%; 60%;80%) Prd.


Due to the large torsional deformations at high load lev-els, a nominal deformation failure criteria was accepted.The numerical failure is assumed when the cross-sectionrotation is larger than 90�. This failure condition wasobserved in the case of a mechanical load higher than60% Prd and occurred during the heating phase. As forthe other cases, the failure modes are similar to thoseobserved for the reference case.

The deflection curves for the five levels of loading areshown in Fig. 22. At low temperatures, beam deflection isdependent on the mechanical loading. A further rise intemperature leads to a progressive run-away of beamdeflection as the loss of stiffness and strength accelerates;however, in the cases with low loading levels, the rate ofrun-way is reduced where the axial force is changing fromcompression to tension – catenary action (Fig. 23). In thecases with high loading levels, excessive deflection accom-panied by torsional deformation is observed and the beamfails. As for the other cases, whenever the cooling phase isreached, the heated beam begins to recover strength andstiffness, together with a reduction of thermal strains whichinduces a reversal of the deflection. The rate of deflectionreduction is not dependent on the mechanical load level.

Fig. 23 plots the growth of the axial force and bendingmoment against time for the five loading levels. Axial forcecurves have similar gradients up to a fairly similar maxi-mum of compressive force, confirming that they have sim-ilar rates of thermal expansion irrespective of the applied

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Psd = 0,2 PrdPsd = 0,12 PrdPsd = 0,4 PrdPsd = 0,8 PrdPsd = 0,6 Psd


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N (

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Psd = 0,2 PrdPsd = 0,12 PrdPsd = 0,4 PrdPsd = 0,8 PrdPsd = 0,6 PrdAxial capacity

Fig. 23. Development of the beam axial force

load level. When the stiffness starts to decrease, lower com-pression forces at higher loading levels are observed. Thecompression force starts to decrease at a slightly lower tem-perature in cases with higher loading level. This is followedby the onset of the vertical deflection run-away. At highloading levels, the compressive force quickly changes to atensile force and failure is attained due to excessive mid-span deflection. At lower loading levels, the beam is sup-ported by the catenary action which takes place earlierfor beams with lower applied loading and a more obviouschange of deflection gradient can be seen (Fig. 22). Ahigher tensile force at higher loading levels is reached. Dur-ing the cooling phase, the heated beam recovers strengthand stiffness and lower tensile forces at higher loading levelare observed.

As the temperature increases, the bending moment alsoincreases and the maximum bending moment is reachedwhen the beam resistance started to decrease after 25 minat temperature of 400 �C. The beams with high mechanicalloading do not reach a sagging moment. Beyond that, thetemperature continues to increase but the bending momentdecreases until the room bending moment. In the caseswhere the beam failure is not observed before cooling,the bending moment and the rotation remain constant athigh temperatures. The discontinuity of the bendingmoment curve due to the reversal thermal bowing duringcooling is more noticeable in the high mechanical loadcases. As the temperature continues to decrease to valuesbelow 500 �C the hogging moments have reversed. Resid-ual effects are clearly very important in these cases.

The effect of the mechanical loading on the distributionof strains is similar to the effect of the beam span: maxi-mum strains spread up in the cross-section as the mechan-ical loading increase. However, the high equivalent strainsobserved on the upper zone of the cross-section of shortbeams occurred because of the shear buckling of the beamweb due to high compression in the beam, while the highvalues on the higher load factor cases result from the hightensile forces due to the large bending deformations. As themechanical loading increases, the critical componentmoves from the bottom flange in compression to the topflange in tension.

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and joint bending moment during the fire.


9. Concluding remarks

This paper presented the results of a numerical studyusing LUSAS, FE software package, to analyse the behav-iour of a beam-to column steel system under fire. In orderto provide credibility to the FE modelling results obtainedby LUSAS they were compared to a fire dedicated program– SAFIR. A numerical parametric study was conducted toinvestigate possible behaviour of the sub-frame. In thisway, insight into the behaviour of the steel girder, whichis the most usual structural element of a floor in multi-sto-rey buildings, exposed to natural fire is gained. The maincharacteristics of every particular failure modes that corre-spond to certain geometry of the girder are discussed in thetext. The constant shape of the initial imperfections, con-sisting of the local and global imperfections, was used inde-pendently of the geometry of the girder. This approach isconsistent with proposed implementation of initial imper-fections according to EN 1993-1-5. In some cases, whenclear discrepancy between the realistically assumed imper-fections and the final failure mode exists, one should repeatthe computation assuming the initial imperfection in ashape of the failure mode obtained in order to obtain thelower boundary of the fire resistance. However, in the casesconsidered in this paper, no reasons were found eligible toperform the additional analysis. There were no doubtsabout the finding that no fire protection is necessary forthe composite girders exposed to the natural fire scenario,as measured in 7th Cardington test. The parametric studyalso highlighted the influence of the end-restraints of thebeam, thus indicating that behaviour is very much depen-dent of the joint characteristics and the degree of forceredistribution that takes place as the fire progresses.

Acknowledgement

Financial support from the Portuguese Ministry of Sci-ence and High Education (Ministerio da Ciencia e doEnsino Superior) under FCT research project POCI/ECM/55783/2004 is acknowledged.

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Numerical study of a steel sub-frame in fire

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