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STRUCTURAL PERFORMANCE OF REDUNDANT STRUCTURES UNDER LOCALFIRES
Publishedin the proceedingsof Interflam99-EdinburghJ.M.Rotter, A.M.Sanad,A.S.Usmaniand M.Gillie
ABSTRACT
Mostrulesandinvestigationsof thestrengthof structuralmembersunderfireassumethatthememberactsaloneasanisolatedstructure.Thismatchesthetestingof individualmembersin a standardfurnacetest.Theconceptmayseemappropriatewherefire in acompartmenteffectively attacksonly theindivid-ual structuralmembersnearby. However, no accountis takenof theinteractionswhich inevitably occurwith thesurroundingstructure.Wherethecompletestructureis largeandredundant,theseinteractionscancompletelychangethestructuralresponseandeffectively invalidatethedesignassumptions.
This paperdiscussestheresponseof a structuralelementunderfire within a highly redundantstructure,suchasa large building. The behaviour of the elementunderfire is stronglyaffectedby the restraintprovided by thesurroundingpartswhich arenot subjectedto heating.A numberof responsesin quitesimplestructuresareshown, to illustratethe rolesof expansion,lossof materialstrength,the relativestiffnessof adjacentpartsof the structure,developmentof large deflections,buckling andtemperaturegradients. Theseaspectsare illustratedwith simpleexamples,and it is shown that thereare severalcounter-intuitive phenomenain structuresof this kind. Thesignificanceof thesefindingsfor thedesignof largebuildingsis exploredbriefly.
INTRODUCTION
Thefire resistanceassessmentof structuresis currentlybasedonfire testsonsingleelements,evalu-atedin termsof thetimeto failure.Theconditionsin thesefire testsarenotwell relatedto theconditionsprevailing in realfires. Not only is thefire testscenarioartificial, but thestructuralidealisationis alsounrealistic.
A determinatestructureis onein which thepatternof internalforcesandstressescanbedeterminedus-ing equilibriumconsiderationsalone.Most testson isolatedmembersmatchthiscondition.A redundantstructureis normallydefinedasonein whichthepatternof forcesandstresseswithin thestructurecannotbedeterminedby equilibriumalone,but dependinsteadon therelative stiffnessesof partsof thestruc-ture. Undercollapseconditions,determinateandredundantstructuresaremoresharplydifferentiated.Thedeterminatestructurecollapseswhenthemosthighly stressedregion reachesthelocal strength,ap-propriatelyreducedaccordingto its currentthermalstate.By contrast,providedit hasadequateductilityanddoesnot suffer from instability, the redundantstructuremayfind differentload pathsby which tosupportadditionalloadwhenits local strengthis reachedat a singlelocation.Wherea structureis veryredundantandtherearemany alternative loadpaths,largedeformationscandevelopwithouta lossof itscapacityto carrytheimposedloads,andfailuremustbedefinedin adifferentway. Thisproblemis facedin fieldsotherthanfire: researchersin pressurevesselsandrectangularstoragestructuresarealsotryingto find new failuredefinitionswhichcanaccommodatelargedisplacementconcepts.
Complex structuralinteractionstake placeduring fires in highly redundantframedstructures,whichtypically possessahighdegreeof staticindeterminacy or redundancy (andhenceoffer severalalternativeloadpaths)leadingto anextensive redistribution of loads. This phenomenoncreatessufficient reservecapacityto allow mostsuchstructuresto survive fireswith little structuraldamage.As this behaviour
resultsfrom interactionsbetweenmost of the structuralelementsas an ensemble,it is necessarytoconsidersuchstructuresasan integratedwholewhentheir fire resistanceis beingevaluated.Althoughthis facthasbeenrecognisedfor sometime (asdemonstratedby thefires in BroadgatePhase8 in June1990[1], MercantileCreditBankBasingstoke andMinsterCourt),it is only recentlythatfull-scalefiretestson realisticstructuralconfigurationshave beencarriedout. The Cardingtontestsundertaken byBritish Steel(SwindenTechnologyCentre)andBRErepresenta majorlandmarkin thefield.
Theresponseof structuresto fire dependsuponthematerialsusedin construction,theresponseof indi-vidual structuralelementsandthe behaviour of thestructuralsystemusedto connectthe elements.Inadditionto theabovetheresponseof structuresnaturallydependsuponthefire itself: its sequence,sever-ity, spreadandrateof development.Thecommonassumptionin structuralfire resistancedesignis thatfireswill belocalisedthrougheffectivecompartmentation.In theabsenceof thisassumption,it becomesverydifficult to predictthebehaviour of thestructure,mainlybecausethefire behaviour in suchcasesisquiteuncertain,sotheheatingregimeandsequencein thestructureis unknown. A primeobjectiveof thetestsconductedat Cardingtonon theBRE 8-storey framewasto provide datato aid thedevelopmentofunderstandingof the interactionsbetweendifferentstructuralmechanismswhich determinetheoverallbehaviour of compositesteelframesin fire.
Thedatafrom theCardingtontestsis currentlybeingusedto validatenumericalmodelswhichareunderdevelopmentin aDETRfundedproject(throughthePIT scheme)ledby theUniversityof Edinburghandin collaborationwith British Steel,ImperialCollege,SCI andBRE.Theoverall objective of theprojectis to develop analyticaltools that adequatelymodelthestructuralbehaviour of compositesteelframesin fire. Onekey observation from the testsis that compositesteel-concreteframeshave significantlymorefire resistancethanindividual structuralsteelmembers,dueto their ability to redistribute loadstorelatively stiffer partsof thestructure.Observationsof realfiresandfire experimentshave long shownthatsuchredistributionsclearlyexist. Theseredistributionsarenow beingobservedin largedisplacementnumericalmodelsof theCardingtontests.
If numericalmodelsareto reveal thechangingpatternsof load transferin thestructureduring thefire,they mustincludeadequaterepresentationsof all thephenomena.A demonstrationthatamodelpredictsanobserved deflectionpatternagainstthe known temperaturehistory doesnot necessarilyvalidatethemodel. It mustalsoidentify thekey eventsduringthefire, reveal thestructuralphenomenaresponsiblefor themandjustify theconclusionsusingthefundamentalprinciplesof structuralmechanics.
This papergivesa brief accountof someof the underlyingphenomenaoccurringin highly redundantframedstructuresexposedto fire which influencethe completeresponse.The individual phenomenadescribedhereall influencethebehaviour of thestructure,leadingto acomplex totalpatternwhichis notsimpleto interpret.
IMPORTANT PHENOMENA IN REDUNDANT COMPOSITE STRUCTURES
In thissection,somekey featuresof theresponseof a typicalelementof a redundantstructureunderfire areidentified.Thediscussionis restrictedto thosewhich are,perhaps,lessobvious. Thestructuralelementsin abuilding whichexperiencetheeffectsof fire mostdirectlyarethebeamsandfloor slabsofthefloor above thefire compartment.For this reason,thefollowing discussionis very muchfocusedonthebehaviour of beamsandslabs.
Thermal expansion
Beamsaredesignedto carrytheir loadsby bendingandshear. In fire, significantaxial forcesdevelopifthebeamis fully or partiallyrestrainedagainstaxialexpansion(or contractionduringcooling).Depend-ing on thesurroundingstructure,theseforcescanbeeitherbeneficialor deleteriousto theperformanceof thestructure.Floorsarealsodesignedto carry loadsin bendingandshear. Whenthey expand,floor
slabscanexert enormousforceson thesurroundingstructure.
Wherehigh levels of restraintexist, the thermalexpansionforcescanbecomevery large. For a fullyrestrainedsteelelementcompressive yield underthermalexpansionoccursat a temperatureof only
∆Ty� σy
Eα(1)
in which ∆Ty is the temperaturerise to causeyield, α is the thermalexpansioncoefficient and E isthe elasticmodulusof steel. This relationshipshows that a temperaturechangeof 102 degreesC for250 gradesteeland142 degreesfor 350 gradesteel(ignoring any materialdegradation)is neededtoachieve yield. Thesetemperaturesareso low that it is clearthat thereis plentyof scopefor high stressdevelopmentin realfiresevenwhentherestraintis only partial.
A heatedstructurecanrespondto thermalexpansionwith a variety of responses.Becausestructuralengineersaretrainedfromtheoutsetto relatestressestodeflectionsanddeflectionsto materialproperties,the morecomplex responseswith thermalexpansionssometimesgive surprises.Indeed,becausetheliteratureon fire responsesis mostlyconcernedwith determinatestructuresin which theseconnectionscontinueto exist, theimportanceof thermalexpansionstrainsareoftenlost.
Thekey relationshipswhichareneededto understandredundantstructuralbehaviour underfire are
εtotal� εthermal
� εmechanical
with εmechanical� σ and εtotal � δ (2)
Thetotalstrainsgovernthedeformedshapeof thestructureδ, throughkinematicor compatibilityconsid-erations.By contrast,thestressstatein thestructureσ (elasticor plastic)dependsonly onthemechanicalstrains.
Wherethethermalstrainsarefreeto developin anunrestrictedmannerandthereareno externalloads,axialexpansionor thermalbowing resultsfrom
εtotal� εthermal and εtotal � δ (3)
By contrast,wherethe thermalstrainsarefully restrainedwithout externalloads,thermalstressesandplastificationresultfrom
0 � εthermal� εmechanical with εmechanical� σ (4)
Most situationsin real structuresunderfire have a complex mix of mechanicalstrainsdueto appliedloadingandmechanicalstrainsdueto restrainedthermalexpansion.Theseleadto combinedmechanicalstrainswhichoftenfarexceedtheyield values,resultingin extensiveplastification.Thedeflectionsof thestructure,by contrast,dependonly on thetotal strains,sothesemaybequitesmallwherehigh restraintexists,but they areassociatedwith extensive plasticstraining.Alternatively, wherelessrestraintexists,largerdeflectionsmaydevelop,but with a lesserdemandfor plasticstrainingandsolessdestructionofthestiffnesspropertiesof thematerials.
Theserelationships,which indicatethat largerdeflectionsmayreducematerialdamageandcorrespondto higherstiffnesses,or thatrestraintmayleadto smallerdeflectionswith lower stiffnesses,canproduceexampleproblemsandstructuralsituationswhichappearto bequitecounter-intuitive for moststructuralengineers.
Thermal buckling
Whenanelasticbeamwith rigid axial restraintat its endsis heated,compressivestressesdevelopaccord-ing to Equation2. Thesecauseit to reachabifurcationstatewhenthethermalthrustattainstheclassicalbuckling load
EAα∆T � π2EIl2
� π2EA� r
l � 2(5)
wherel is theeffective lengthof thebeamanddependson therestraintconditions.Thecritical bucklingtemperaturerise∆Tcr in anelasticstructurewith unchangingelasticmodulusE is thus,
∆Tcr � π2
α
� rl � 2
(6)
For structuralelementsof theslendernesscommonlyfoundin slabsandbeams,this critical temperaturecaneasilybeaslow as100or 200degreesC. Thephenomenonis thusalsolikely to occurin mostfires.
Wheretheelasticmodulusis temperaturedependent,therelationshipcannotbesosimplydefined,sincethe thrust is a nonlinearintegral of the thermalexpansionandelasticmodulus,whilst the stability isgovernedby a tangentmoduluscondition�
ET � T � σ � α � T � dT � π2� r
l � 2ET � T � σ � (7)
in whichET � T � σ � is thetangentmoduluswhichvarieswith thetemperatureandstressstate.
Rigid axial restraintis generallyimpossibleto achieve, andthusrepresentsonly a limit: realstructuresoffer only finite restraint.Assumingthat therestraintto axial expansioncanberepresentedby a lineartranslationalspringof stiffnesskt , (Figure1(b))againfor anelasticbeamwith unchangingmodulus,thecompressive axialstressdevelopedby thermalexpansionbecomes
σ � Eα∆T�1
� EAkt L � (8)
Thecritical buckling temperatureincrementis modifiedfrom Equation6 to become
∆Tcr � π2
α
� rl � 2
1� EA
ktL (9)
Fromthis relationshipit canbeseenthatbuckling andpost-buckling phenomenashouldbeobservableatmoderatefire temperatures(say300degreesC) in structureswith translationalrestraintstiffnesses(kt )which arequite comparablewith the axial stiffnessof the member(EA
L ). This axial stiffnessitself isreducedby heatingthroughthereductionin E, so thesepost-buckling phenomenaarevery likely to beobservedin beamsin typical fires.
The effects of heatinga beambetweenrigid axial end restraints(Figure1(a)) whilst it is carryingaconstantuniformly distributedload,areshown in Figures2 & 3. A constantmoduluselasticbehaviouris adoptedfor clarity. Thegrowth of themidspandeflectionwith temperatureis shown in Figure2(a).Thedistributedloadsmoothesthebifurcationphenomenonslightly, but thecritical temperaturecanbeclearly identified,andthepost-buckling responseinvolving rapidgrowth of deflectionsinto a largede-formationstatecanbeseen.Thisgrowth of post-buckling deflectionsis ratherdifferentfrom traditionalpost-buckling understaticloads.Underambienttemperatureconditions,axial forcesappliedto abeam-columnarelargelyunaffectedby themember’s response.Here,theaxial forcesdeveloponly becausetheaxial displacementsarerestrained,andthe increasingdeflectionsin thepost-buckling stateallow axialshorteningthroughmembercurvature,andthusdo not correspondto anunstablecondition. The axialforcedevelopingin thebeamunderincreasingtemperatureis shown in Figure3(a). This force is closeto constantin thepost-buckling zoneandadditionalthermalexpansionis all absorbedin additionalde-flection,insteadof causingincreasedstresses(seeEquations2-4again).For localfiresin realstructures,
(a)
� �� �� �� �� �� �� �� �
���small transverse load
P
Pcr
P
Pcr
prebuckling state: � expansion develops axial compression
postbuckling state:� expansion produces deflections
length L, effective length l, properties E, A, I
ends restrained against axial translation
(b)
� �� �� �
� �� �� �P
Pcr
kt � �� �� �P
� �� �� �Pcr
prebuckling state: � expansion develops axial compression
postbuckling state:� expansion produces deflections
end restrained with stiffnesskt against axial translation
Figure1: Axially restrainedbeamssubjectedto increasingtemperature:(a) Rigid restraint,(b) Finiterestraint
Deflection at Mid-span
-1.80E+02
-1.60E+02
-1.40E+02
-1.20E+02
-1.00E+02
-8.00E+01
-6.00E+01
-4.00E+01
-2.00E+01
0.00E+00
0 50 100 150 200 250 300 350
Temperature (C)
Def
lect
ion
(m
m)
Deflection at mid-span
� � � � � � � � � � � � � � � � � � � Deflection at Mid-span
-3.00E+02
-2.50E+02
-2.00E+02
-1.50E+02
-1.00E+02
-5.00E+01
0.00E+00
0 50 100 150 200 250 300 350
Temperature (C)
Def
lect
ion
(m
m) D/L = 1.78%
D/L = 2.17%
D/L = 2.67%
! " # $ % & % ' & ( % ) * + , - % ' & % . ' % , , % / / % 0 1 2
(a) (b)
Figure2: Deflectionof axially restrainedelasticbeamssubjectedto heating:(a)Singlebeam,(b) Threebeamsof varyingslenderness
this is ahelpfuleffectasit limits theadditionalforcesgeneratedby therestrainedthermalexpansionandthusdoesnotdamageadjacentpartsof thestructuresomuch.Thus,it canbeconcludedthatbuckling isgoodfor thestructure! It shouldberememberedthatthematerialis assumedto beelasticandthestableaxial force is associatedwith this. The momentsdevelopingin the beamasthe temperaturerisesare
Normal Force at Mid-span
-7.00E+06
-6.00E+06
-5.00E+06
-4.00E+06
-3.00E+06
-2.00E+06
-1.00E+06
0.00E+00
1.00E+06
0 50 100 150 200 250 300 350
Temperature (C)
Fo
rce
(N)
Normal Force at support
Normal Force at mid-span
� � � � � � � � � � � � � � � � � � � Moment at Mid-span
-2.00E+08
0.00E+00
2.00E+08
4.00E+08
6.00E+08
8.00E+08
1.00E+09
1.20E+09
0 50 100 150 200 250 300 350
Temperature (C)
Mo
men
t (N
.mm
)
Moment due to Distributed Load
Moment due to Non-Linear geometricaleffect (P-Delta)
Moment Carried by the beam
3 4 5 6 7 5 8 9 : ; 5 8 5 < 8 = 5 6 > ?
(a) (b)
Figure3: Forcesin anaxially restrainedelasticbeamsubjectedto heating:(a)Axial Forces,(b) Moments
shown in Figure3(b). Thetransverseloadingmomentsareoverwhelmedby P @ ∆ momentsastheaxialforceanddeflectionrise: thestability of thepost-buckling axial forcemeansthatthis figurefollows thedeflectionpatternThegrowth of deflectionsin threebeamswith differentslendernessratiosis shown inFigure2(b). Thesameresponseis seenin all three,with buckling temperaturesat relatively low values.Spaceheredoesnot permit an expositionof the relevant theory, but the transversedeflectionsduetothermaleffectsδy in this largedisplacementpost-buckingregimecanbeverycloselymodelledby
δy� 2L A α∆T
π2 @ � rl � 2
(10)
Thermal gradients
As notedabove, mostrealfiresheatthefloor andbeamsfrom below, leadingto a regimein which tem-
peraturedifferentialsdevelopbetweentheupperandlowersurfaces.Thesedifferentialsleadto thermallyinducedbendingor thermalbowing,whichcanincreasedeflections.A hightemperaturegradientthroughthedepthof afloor slab(typically concrete)will induceeitherbendingmomentsor additionaldeflectionsor bothin theslab(seeEquations2-4again).
Thefirst examplegivenhereis of a beamwhich is axially androtationallyunrestrainedat its endsandsubjectto a linearthermalgradientwhich is constantalongthe length. It producesa uniform curvaturegiven by φ � αTB y, whereTB y is the gradientof temperaturethroughthe beamdepth. No stressesareinducedanda hot lower surfaceleadsto downward bowing. If instead,the beamis rigidly restrainedagainstendrotations(but axially freeto translate),no deflectionsdevelopat all in thebeam!It remainsperfectlystraight. Instead,a constantbendingmomentis inducedthroughoutthebeam(seeEquations2-4 again),given by EIαTB y. The stressesassociatedwith a hot lower surfacearethe bottomfibre incompression,andin concretefirst crackingoccurson thetopunheatedsurface(this is acounter-intuitiveresultfor moststructuralengineers).
It is difficult to achieve rigid restraintagainstrotation,exceptin thecaseof perfectsymmetryandsmallrotations. In fires, large rotationsin the slaboccurandcrackingoccursearly in the concreteover thesupports,reducingtherotationalrestraint.If theadjacentstructureis representedby a rotationalspringstiffnesskr at oneend(whilst theotherendis assumedto fixed),a momentis inducedin therotationalspringof,
Mk� EIαTB y�
1� 4EI
kr L � (11)
whereL is the lengthof the beam. Thus for substantialbendingmomentsto be inducedby thermalgradients,therotationalrestraintstiffnessmustbecomparablewith therotationalstiffnessof thebeamitselfwhenits farendis fixed(aperfectanalogywith theaxialcaseabove). If theunheatedadjacentspansare identicaland have effectively fixed endsand the systemis elastic,this restraintcanbe achieved.Wherethe hot surfaceis on the bottom, it shouldbe notedthat the thermalbowing deflectionis stilldownward,but that tensilestressesdevelopon the top surface,andin concretefirst crackingoccursonthetop,producinganimagewhichsurprisesmostengineers.
Whenthebeamis rotationallyunrestrained,thethermalcurvaturedueto auniformgradient(with nonettemperaturerise),givenby αTB y, causesadeflectionδy in anaxially freebeamof
δy� 1
αTB y 1 @ cos
αTB yL2 (12)
and,in a largedisplacementevaluation,thiscausesthedistancebetweenthesupportsto reduceby
δx� L @ 2
αTB y sin
αTB yL2 (13)
If thebeamendsarenow axially restrained,the lossof lengthin arcshorteningδx mustbereplacedbya stress-relatedextension,which requiresa uniform axial tension.Thus,for axially restrainedbut rota-tionally freebeams(closeto realconditions),thermalgradientswill produceaxial tension.By contrast,auniformtemperatureriseproducesaxialcompression.
Thus,thedeformedshapegiveslittle indicationof whetherpartof thestructureis in tensionor compres-sion,andamixtureof thermalgradientsandcentroidaltemperatureincreasescanleadto axial tensionoraxialcompressionwith quitesimilardeformations.Someof theseforcescanparticipatein load-carryingmechanisms(underlargedisplacementregimes),whilst othersarepurelyself-stressingin character.
Degradationof strengthand stiffness
Thestrengthof abeamcanbequantifiedfrom thegeometryof its cross-sectionandits materialstrength.High temperatureswill resultin lossof strength(bothyield andultimatestrengths)andstiffness(mod-uli of elasticity). Under inelasticconditions,the valueof E in the above equationsdependson otherparameters.In thecaseof fire E dependson thetemperature:mostmaterialslosestiffnesswith heating.
The samebeamas that of Figures2 and3, spanningbetweenrigid endrestraints,wasgiven elastic-plasticproperties.The changedbehaviour is shown in Figures4. The key differenceis that the axialforce rapidly dropsshortly after buckling, as the midspanandend cross-sectionsreachthe plasticityyield surfaceandareconstrainedto follow it. Increasingdeflectionsthenmeanthattheaxial forcemustdropeven if the momentwereto remainconstant,but theyield surfacepermitsan increasingmomentwith falling axial force(Figure5). Thecorrespondingeffectsin abeamwhichremainselasticbut whose
Deflection at Mid-span
-2.00E+02
-1.80E+02
-1.60E+02
-1.40E+02
-1.20E+02
-1.00E+02
-8.00E+01
-6.00E+01
-4.00E+01
-2.00E+01
0.00E+00
0 50 100 150 200 250 300 350
Temperature (C)
Def
lect
ion
(m
m)
Deflection at mid-span
C D E F G E H I J K E H E L H M G E E N O E F P Q R N F M G S T U N F M G J V O E W F X J S Y Z [ Normal Force at Mid-span
-4.50E+06
-4.00E+06
-3.50E+06
-3.00E+06
-2.50E+06
-2.00E+06
-1.50E+06
-1.00E+06
-5.00E+05
0.00E+00
5.00E+05
0 50 100 150 200 250 300 350
Temperature (C)
Fo
rce
(N)
Normal Force at support
Normal Force at mid-span
\ ] ^ _ ` ^ a b c d ^ a ^ e a f ` ^ ^ g h ^ _ i j k g _ f ` l m n g _ f ` c o h ^ p _ q c l r s t
(a) (b)
Figure4: Deflections(a),& Axial forces(b), in anaxially restrainedelastic-plasticbeam
Moment at Mid-span
-5.00E+07
0.00E+00
5.00E+07
1.00E+08
1.50E+08
2.00E+08
2.50E+08
3.00E+08
3.50E+08
0 50 100 150 200 250 300 350
Temperature (C)
Mo
men
t (N
.mm
)
Moment due to Distributed LoadMoment due to Non-Linear geometrical effect (P-Delta)Moment Carried by the beam
u v w x y w z { | } w z w ~ z � y w w � � w x � � � � x � y � � � � x � y | � � w � x � | � � � �
Figure5: Momentsin anaxially restrainedelastic-plasticbeam
propertiesdegradewith temperature(asfor steel)areshown in Figures6. Thesameeffectscanbeseen,but theconsequencesarelessdramatic.In realstructures,a mixtureof thermaldegradationof stiffnessandyieldingcausedby thermalexpansionoccurs,andtheseeffectsaremoredifficult to separateout.
Lar gedeflections
Deflection at Mid-span
-1.8E+02
-1.6E+02
-1.4E+02
-1.2E+02
-1.0E+02
-8.0E+01
-6.0E+01
-4.0E+01
-2.0E+01
0.0E+00
0 50 100 150 200 250 300 350
Temperature (C)
Def
lect
ion
(m
m)
Deflection at mid-span
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Normal Force at Mid-span
-6.0E+06
-5.0E+06
-4.0E+06
-3.0E+06
-2.0E+06
-1.0E+06
0.0E+00
1.0E+06
0 50 100 150 200 250 300 350
Temperature (C)
Fo
rce
(N)
Normal Force at support
Normal Force at mid-span
¡ ¢ £ ¤ ¥ ¦ § ¨ © ¦ ª « £ ¬ £ « ® ¯ ¦ « ¥ § £ ¦ © ° ¥ ± § ¦ ° ¦ « £ ² ¤ ³
(a) (b)
Figure6: Deflections(a),& Axial forces(b), in a restrainedbeamwith reducingelasticstiffness
Under fire conditions,axially restrainedbeamsdevelop large deflectionsfor several reasons. Chiefamongtheseis thermalexpansion,which inducesenormousthrustsleadingto post-buckling states(de-pendinguponthe restraintavailable)which force thebeamdown. In compositesteel-concreteframes,this actionbeginsin therolled steeljoist, until somecross-section(s)reachestheultimatecapacity. Theprincipalsourceof restraintis theextensive andmassive concreteslab,whichhasahigh thermalinertia,andprovidesanextensive cold surroundto thefire event. Theactioncontinuesin theheatedpartof theconcreteslab,which heatsmoreslowly, with highergradients,but to a muchlower mean(middlesur-face)temperature.However, the forcestheslabcanexert arevery high becausea largecross-sectionalareais involved. Sincethe slabexpansionoccursin its own plane,andthis is the planein which thecoldsurroundingslabprovidesrestraint,largethrustscandevelopin two directions.Slabdeflectionscanincreasefurtherthroughcrackingleadingto significantreductionsin stiffnessandendrotationalrestraint.
Paradoxically, reductionsin the stiffnessesof the systemmay reducedeflections,becausedeflectionsaredrivenby thestiff restraintof strongthermallyexpandingstructuralelements.A reductionin axialstiffnessreducesthethruststhatcausedeflection.
Sourcesof restraint
The issueof how much restraintis available in a compositeframedstructureis important. A struc-tural memberwhich is far from the boundariesof the structurecanbe reasonablythoughtof asstifflyrestrained.Therestraintto interior elementscomesfrom thesurroundingcold compositeslabandsteelframe.If thefirecompartmentis sufficiently distantfromtheboundary(perhapsbyonebay),therestraintthisprovidescanbeconsideredasalmostrigid.
It is currentlydifficult to determinehow largeanedgezonemustbeif it is to provide a sufficiently stiffrestraintandadequateanchoragefor theslabforcesrequiredfor alternative loadcarryingmechanisms.In thesimplestglobalterms,theCardingtontestssuggestthatevenedgepanelscanbenefitenormouslyfrom transferof loadsfrom the weakeningsystemsof bendingandsheartowardsothermechanisms.Whenthe fire compartmentis at an edgeor a corner, restraintmay be provided by the in-planeshearstiffnessof a protectedcompositeedgebeam.This maybeassistedby a ”tensionring” forcepatternintheedgecompositebeams,leadingto acoupledflexure-tensionbehaviour. Thiseffectcanbeseenclearlyin Figure7, which shows a plot of majorprincipalstressesin theslabof a compositeframe(modelledhereusingelasticshellelements).Theinternalheatedzoneis surroundedby aring of membranetension.An expandingheatedsteelbeamwhichactscompositelywith aslabis restrainedby therelatively colderslabthroughouta fire andcanbeseverelyplastified.Theslabthenbecomesthemajorcauseof thrustin
12
3
RESTART FILE = eslab STEP 4 INCREMENT 8
TIME COMPLETED IN THIS STEP 400. TOTAL ACCUMULATED TIME 1.001E+03
ABAQUS VERSION: 5.8-1 DATE: 07-APR-1999 TIME: 10:11:34
12
3
Figure7: Tensionring restraintin theslaboveracornercompartmentunderfire
thesteelbeamfor mostof thedurationof thefire. Thisthrustleadsto largeplasticstrainsin thesteeljoist,causingsignificantshortening.On cooling,thereductionin lengthis not easilyrecovered,especiallyasthecoolingsteelgainsstiffnessandstrengthfasterthanthenecessarytensilestressesareinduced.Thus,plastificationof thesteelbeamcausedby slabrestraintleadsin duecourseto very high tensilestressesduringthecoolingphase.If rupturedamageis to beavoidedduringthis period,connectionsandweaklinkagesmustbe designedwith tensileductility in mind. It is normally far from a designer’s mind todesignbeamto columnor beamto beamconnectionsin structuralsteelwork to behighly ductilewhenlarge tensionsare appliedto the joist (this connectionis thoughtto be in compressionasa result ofnegative bending!).
Membrane action
Two separatestructuralstresspatternsin slabsaretermed”membraneaction”. Both involve forcesintheplaneof theslab(membraneforces)actingat themiddleplaneof theslabplane. Both requiretheboundariesof the slabto be restrainedto supportforcesin the planeof the slab(in the above discus-sion this was termedaxial restraint). At small displacements,compressive membraneactionoccurs.Whencrackingoccursin concrete,theneutralaxisor zerostrainaxisis displacedin thedirectionof thecompressionface. Themiddleplaneof theslabis thuseffectively subjectedto anexpansion.Suchaneffectcanoccuratbothmidspanin saggingbendingandatsupportsin hoggingbending,giving additiveexpansive displacements.Wheretheseexpansionsareresistedby a stiff boundary, additionalcompres-sive forcesdevelop,andwheretheslabis thick, theeccentricityof thecompressive force transmissionproducesanarchingactionwhichcancarryagreatlyincreasedload.Thismechanismis presentin steel-concretecompositebeamsin highly redundantstructuresevenunderambientconditions,dueto theverylargedisparitybetweentheirhoggingandsaggingneutralaxes.However, thisactionis morepowerfullydemonstratedin thethermallyexpandingslabof thecompositestructure,becausethethermalexpansionsarevery largeandcancausemajorchangesin theload-carryingmechanism.
At largedisplacements,tensilemembraneactionbegins. In tensilemembraneaction,thelargedeforma-
tionsof aslabwith axial tensions(throughthicknesstensileforces),leadto achangein load-carryingbychangeof geometry:effectively asmallcomponentof thetensioncarriestransverseloaddirectly. Underambientconditions,suchlarge displacementsmeanthat large mechanicalstrainshave developed,andthereis a dangerof rupturedueto lossof ductility. Underfire conditions,muchof the necessaryde-formation(Equation4) is derivedfrom thermalexpansion,placinga lessonerousburdenon mechanicalstrains,andconsequentplasticstrainingor ductility. Thepost-bucklingdeformationsdescribedearlierinthispaperassistin developinglargedisplacements,andthethreedimensionalcharacterof theslab,withthedisplacementfield requiredto becontinuous,permitstensilemembraneactionto developevencloseto areaswhich arein a post-buckling compressive state.Therestrictionswhich buckling placeson thedevelopmentof compressionforces,togetherwith theincreaseddeformations,make thedevelopmentoftensilemembraneactioneasierto achieve.
Thesemembranemechanismsmake thefloor slabsthestrongeststructuralelementsin thebuilding (inthesensethat,underextremeconditions,they possessconsiderablygreaterstrengththanis requiredtocarry thedesignloads). However, for thesemechanismsto bemobilised,thefloor mustbeadequatelyrestrainedalongits edgesby asurroundingstructurethatis relatively unaffectedby fire.
WHOLE STRUCTURE RESPONSEAND IMPLICA TIONS FOR FIRE DESIGN
Framedstructuresconsistingof a grid of beamsandcolumnssupportingfloor slabsrepresentthemostcommonstructuralsystem.If adequatecompartmentationis provided, this systemis very effec-tive in containingfiresandredistributing loadsaway from thefire affectedstructuralmembersto thoselessaffected. In steelandconcretestructuresthis is achieved principally throughthe slab,which pos-sessesextraordinaryreservesof strength,andbridgestheloadsoverweakenedbeamsandcolumnsto thestrongermembers.Thebeam-columnstructuralgrid is alsovery effective in redistributing loads.Theseredistribution mechanismsallow a framedstructureto continueto carry loadseven thoughsomeof itsmembersreachtheirultimatestrength:otherlessheavily loadedmemberstake additionalloads.
Thesinglemostimportantissuethatdeterminesa redundantstructure’s responseto heatingis theman-ner in which it dealswith the inevitable thermalstrainsinducedin its membersthroughheating.If thestructurelacks restraintto thermalexpansion,the considerablestrainsare taken up in expansive dis-placements,producinga displacement-dominatedresponse.This responseoccursin structureswhosebehaviour is primarily determinate,andcanhave disastrousconsequences,suchasmasonrywalls being‘pushedover’ by expandingsteelmembers.This kind of event canbe foreseenanddesignedagainstwithoutcompromisingstructuralintegrity.
In structuressuchasthecompositesteelframeat Cardington,theslabstronglyrestrainsthethermalex-pansionstrainsandconsequentlydevelopslargemembranecompressionandtensionforcesin thecom-positesteel-concretefloor system.Themembranecompressionscanbelimited by the largedownwarddeflectionswhichoccurthroughthermo-mechanicalpost-bucklingeffectsandthermalbowing (thesearenonlinearlyadditive). Theresultingbehaviour is thenacombinationof displacementandforceresponses.Theheatedsteelpartof thiscompositesystem,if unprotected,rapidlyreachesits axialcapacity(throughlocal buckling andstrengthdegradation),andproducesa beneficialeffect by limiting andthenreducingthe total membranecompression,so allowing increasedexpansionof the steelthroughsofteningandductility. This is clearlya desirablebehaviour here,asit reducestheforce imposedon thestructurebytheexpansionforcesandallows thedamageto belocalised.
It is clearthattheworstscenariofor afire in acompositeframebuilding structureis compartmentbreach.Structuralfire designshoulddefinecompartmentbreachasan‘ultimate limit state’andensurethat it isprevented. The only structuralmemberin a compositeframethat actsasa compartmentboundaryisthecompositefloor slab. As mostof theslabnearthefire is in membranecompressionthroughoutthefire, a compartmentbreachof the slab is unlikely. However, local areasin membranetensionlate inthefire couldcausetensionscrackswhich arelargeenoughto causecompartmentbreach.Appropriate
reinforcementshouldbeprovidedto prevent this,which will alsoaid loadredistribution to coolerpartsof thestructureandenhanceits intrinsic redundantbehaviour.
Finally, it shouldbe notedthat whereextensive plastic strainingoccurs(as in any steeljoists whichdo not deflectgreatly: seeEquations2-4 again),the plasticstrainsaredifficult to recover without thedevelopmentof veryhightensilestresses.Shearconnectionsatbeamendsshouldbedesignedto beveryductilewhensubjectedto a tensileforcein thebeam.
CONCLUSIONS
Thispaperhaspresentedseveralphenomenawhichoccuronly in highly redundantbuildingstructuresunderlocalisedfire conditions. They strongly influencethe behaviour of the structure,anda lack ofawarenessof themcanleadto seriousmisunderstandingsaboutthestructuralphenomenaoccurringinfire tests.
Whilstmaterialdegradationis thekey phenomenonin determinatestructuresunderfire, for highly redun-dantstructuresthesinglemostimportantfactoris theeffect of thermalexpansion.Thermalexpansioncoupleswith large displacementeffectsto producea numberof effectswhich appearcounter-intuitiveto theconventionallytrainedstructuralengineer. Thekey rolesof Equations2-4 in decouplingthedis-placementandstressfields shouldbe consideredcarefully wherever an understandingof behaviour issought.
It is clearthat largedisplacementsin thesestructuresarenot alwaysassociatedwith degradationof thestructure,but may indeedbe beneficial. Large displacementsarecommonlyassociatedwith bendingfailures,but herethey mayoccurwith membranethrusts,or with membranetensions,dependingon thethermalregimein thestructure.Thedevelopmentof largedeflectionsis helpful in thatit permitstheslabto transferloadto thealternative loadcarryingmechanismof tensilemembraneaction.This actioncanbe regardedasmorereliableunderthermally induceddeformationsbecausethe ductility requirementplacedonmechanicalstrainingis reduced.
Thesefindingsareof fundamentalsignificanceto thedevelopmentof understandingof compositeframesin fire. Theimplicationsfor thedevelopmentof designphilosophiesandproceduresareconsiderable.
Acknowledgements:The supportof DETR for funding this work throughthe PIT schemeis gratefullyacknowledged. Theauthorsarealsogratefulfor thesupportof British Steelwhoarecollaboratorsin thisproject(throughDrMark O’ConnorandDr Xiu Feng).
REFERENCES
1. S.C.I. Investigationof BroadgatePhase8 Fire. In Structural FireEngineering. SteelConstructionInstitute,Ascot,Berkshire,1991.
2. J.M.Rotter, A.M.Sanad,A.S.Usmani,andM.A.O’Connor. Structuralperformanceof redundantstructuresunderlocalfires. In Proceedingsof Interflam’99, Edinburgh,Scotland,June-July1999.
