Thermal Bowing
22
FUNDAMENTAL PRINCIPLES OF STRUCTURAL BEHAVIOUR UNDER THERMAL EFFECTS A.S.Usmani and J.M. Rotter School of Civil and Environmental Engineering, University of Edinburgh Keywords: Composite structures, thermal expansion, thermal bowing, restraint to thermal actions, non-linear geometrical responses. Abstract Behaviour of composite structures in fire has long been understood to be dominated by the effects of strength loss caused by thermal degradation and that large deflections and run- away resulting from the action of imposed loading on a ‘weakened’ structure. Thus ‘strength’ and ‘loads’ are quite generally believed to be the key factors determining structural response (fundamentally no different from ambient behaviour). The exceptional longevity of this view derives in no small measure from observations in the standard fire tests. These observations have little relevance to realistic structural configurations present in large multi-storey compos- ite frame structures. The investigations as part of the DETR, PIT project on composite structure behaviour in fire has clearly shown that this understanding is gravely in error. A ‘new under- standing’ has been produced from the PIT project, sponsored by DETR (UK) and executed by a consortium led by Edinburgh University and including British Steel (now CORUS) and Im- perial College. The key message from this new understanding is that, composite framed struc- tures of the type tested at Cardington possess enormous reserves of strength through adopting large displacement configurations, and that thermally induced forces and displacements, not material degradation, govern the response in fire. Degradation (mainly steel yielding and buck- ling) can even be helpful in developing the large displacement load carrying modes safely. This paper attempts to lay down some of the most important and fundamental principles that govern the behaviour of composite frame structures in fire in a simple and comprehensible manner. This is based upon the analysis of the response of single structural elements under a combina- tion of thermal actions and end restraints representing the surrounding structure. 1 Introduction The assessment of the adequacy of composite steel frame structures in fire continues to be based upon the performance of isolated elements in standard furnace tests. This is despite the widespread acceptance amongst structural engineers that such an approach is over-conservative and even more importantly, unscientific. Current codes such as BS 5950 Part 8 and EC3 (draft) allow designers to take advantage of the most recent developments in the field by treating fire related loading as another limit state. The advances in understanding of structural behaviour in fire achieved in the last few years have been considerable. In theory, these advances make it possible for designers to treat the design for fire in an integrated manner with the design of a structure for all other types of loading by using the numerical modelling tools that have been instrumental in developing this understanding. However the use of such tools, which are indispensable for research, is not practical in the design office. Exploitation of the new knowledge can only become feasible if this understanding is further developed into simpler analytical expressions, enabling consulting engineers and designers to undertake performance-based design of steel frame structures without having to resort to large scale computation. This paper builds upon earlier work presented at the INTERFLAM [1] and SiF [2] conferences. The most fundamental relationship that governs the behaviour of structures when subjected to
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Transcript of Thermal Bowing
FUNDAMENTAL PRINCIPLES OF STRUCTURAL BEHAVIOURUNDER THERMAL EFFECTS
A.S.Usmaniand J.M. RotterSchoolof Civil and Envir onmentalEngineering,University of Edinburgh
Keywords: Compositestructures,thermalexpansion,thermalbowing, restraintto thermalactions,non-lineargeometricalresponses.
Abstract
Behaviour of compositestructuresin fire haslong beenunderstoodto be dominatedbytheeffectsof strengthlosscausedby thermaldegradationandthat large deflectionsandrun-awayresultingfrom theactionof imposedloadingona ‘weakened’structure.Thus‘strength’and‘loads’ arequitegenerallybelieved to bethekey factorsdeterminingstructuralresponse(fundamentallyno differentfrom ambientbehaviour). Theexceptionallongevity of this viewderivesin no smallmeasurefrom observationsin the standardfire tests.Theseobservationshave little relevanceto realisticstructuralconfigurationspresentin largemulti-storey compos-ite framestructures.Theinvestigationsaspartof theDETR,PITprojectoncompositestructurebehaviour in fire hasclearlyshown that this understandingis gravely in error. A ‘new under-standing’hasbeenproducedfrom thePIT project,sponsoredby DETR(UK) andexecutedbya consortiumled by Edinburgh UniversityandincludingBritish Steel(now CORUS) andIm-perialCollege.Thekey messagefrom thisnew understandingis that,compositeframedstruc-turesof thetypetestedat Cardingtonpossessenormousreservesof strengththroughadoptinglarge displacementconfigurations,andthat thermallyinducedforcesanddisplacements,notmaterialdegradation,governtheresponsein fire. Degradation(mainlysteelyieldingandbuck-ling) canevenbehelpfulin developingthelargedisplacementloadcarryingmodessafely. Thispaperattemptsto lay down someof themostimportantandfundamentalprinciplesthatgovernthe behaviour of compositeframestructuresin fire in a simpleandcomprehensiblemanner.This is basedupontheanalysisof theresponseof singlestructuralelementsunderacombina-tion of thermalactionsandendrestraintsrepresentingthesurroundingstructure.
1 Intr oduction
Theassessmentof theadequacy of compositesteelframestructuresin fire continuesto bebasedupontheperformanceof isolatedelementsin standardfurnacetests.This is despitethewidespreadacceptanceamongststructuralengineersthatsuchanapproachis over-conservativeandevenmoreimportantly, unscientific. CurrentcodessuchasBS 5950Part 8 andEC3(draft) allow designersto take advantageof the mostrecentdevelopmentsin the field by treatingfire relatedloadingasanotherlimit state.Theadvancesin understandingof structuralbehaviour in fire achieved in thelast few yearshave beenconsiderable.In theory, theseadvancesmake it possiblefor designersto treat the designfor fire in an integratedmannerwith the designof a structurefor all othertypesof loadingby usingthenumericalmodellingtoolsthathavebeeninstrumentalin developingthis understanding.However the useof suchtools, which areindispensablefor research,is notpracticalin the designoffice. Exploitationof the new knowledgecan only becomefeasibleifthis understandingis further developedinto simpleranalyticalexpressions,enablingconsultingengineersanddesignersto undertake performance-baseddesignof steelframestructureswithouthaving to resortto largescalecomputation.
This paperbuilds uponearlierwork presentedat the INTERFLAM [1] andSiF [2] conferences.The most fundamentalrelationshipthat governsthe behaviour of structureswhen subjectedto
thermal�
effectsis:εtotal � εthermal
� εmechanical
with εmechanical� σ and εtotal � δ (1)
Thetotal strainsgovernthedeformedshapeof thestructureδ, throughkinematicor compatibilityconsiderations.By contrast,thestressstatein thestructureσ (elasticor plastic)dependsonly onthemechanicalstrains.
Wherethethermalstrainsarefreeto developin anunrestrictedmannerandthereareno externalloads,axialexpansionor thermalbowing resultsfrom
εtotal � εthermal and εtotal � δ (2)
By contrast,wherethethermalstrainsarefully restrainedwithoutexternalloads,thermalstressesandplastificationresultfrom
0 � εthermal� εmechanical with εmechanical� σ (3)
Thesinglemostimportantfactorthatdeterminesarealstructuresresponseto heatingis themannerin which it respondsto the unavoidablethermalstrainsinducedin its membersthroughheating.Thesestrainstake the form of thermalexpansionto an increasedlength(underan averagecen-troidal temperaturerise) and curvature(inducedby a temperaturegradientthroughthe sectiondepth). If the structurehasinsufficient endtranslationalrestraintto thermalexpansion,the con-siderablestrainsare taken up in expansive displacements,producinga displacement-dominatedresponse.Thermalgradientsinducecurvatureleadingto bowing of amemberwhoseendsarefreeto rotate,againproducinglargedisplacements(deflections).
Memberswhoseendsare restrainedagainsttranslationproduceopposingmechanicalstrainstothermalexpansionstrainsandthereforelarge compressive stresses(seeEquation1). Curvaturestrainsinducedby the thermalgradientin memberswhoseendsare rotationally restrainedcanleadto large hogging(negative) bendingmomentsthroughoutthe lengthof the memberwithoutdeflection.Theeffect of inducedcurvaturein memberswhoseendsarerotationallyunrestrained,but translationallyrestrained,is to producetension.
Thereforefor the samedeflectionin a structuralmembera large variety of stressstatescanex-ist; largecompressionswhererestrainedthermalexpansionis dominant;very low stresseswheretheexpansionandbowing effectsbalanceeachother;in caseswherethermalbowing dominates,tensionoccursin laterallyrestrainedandrotationallyunrestrainedmembers,while largehoggingmomentsoccurin rotationallyrestrainedmembers.This varietyof responsescanindeedexist inrealstructuresif oneimaginesthemany differenttypesof fire a structuremaybesubjectedto. Afastburningfire thatreachesflashoverandhightemperaturesquickly andthendiesoff canproducehigh thermalgradients(hot steelandrelatively cold concrete)but lower meantemperatures.Bycontrast,aslow fire thatreachesonly modesttemperaturesbut burnsfor a longtimecouldproduceconsiderablyhighermeantemperatureandlower thermalgradients.
Most situationsin realstructuresunderfire have a complex mix of mechanicalstrainsdueto ap-plied loadingandmechanicalstrainsdueto restrainedthermalexpansion.Theseleadto combinedmechanicalstrainswhich often far exceedthe yield values,resultingin extensive plastification.Thedeflectionsof thestructure,by contrast,dependonly onthetotalstrains,sothesemaybequitesmall wherehigh restraintexists,but they areassociatedwith extensive plasticstraining. Alter-natively, wherelessrestraintexists, largerdeflectionsmaydevelop,but with a lesserdemandforplasticstrainingandso lessdestructionof the stiffnesspropertiesof the materials. Theserela-tionships,which indicatethat larger deflectionsmay reducematerialdamageandcorrespondto
higher�
stiffnesses,or thatrestraintmayleadto smallerdeflectionswith lowerstiffnesses,canpro-ducestructuralsituationswhichappearto bequitecounter-intuitive if viewedfrom aconventional(ambient)structuralengineeringperspective.
Theideaspresentedabovewill bemoreformally exploredin thefollowing sectionsin thecontextof simplestructuralconfigurationsandanalyticalexpressionswill bedevelopedfor many casesoffundamentalimportance.
2 Standard fir e testand runaway failur es
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Runaway
laterally fixedlaterally free
Unrestrained Laterally restrained
Figure1: Runaway in unrestrainedandrestrainedbeams
Figure1 shows a simplecomparisonof two geometricallynon-linearanalyses.The first caseisheated,simplysupported(laterallyunrestrained)steelbeamwith auniformly distributedloadandthe secondis a laterally restrained(but rotationallyunrestrained)beamwith the sameuniformlydistributedload. Initial deflectionsare lower in the first beambecausethe supportsareable totranslateoutwardsuponexpansion. However, ‘runaway’ occursat around450� C (even thoughconsiderablesteelstrengthremains)mainlybecauseof pulling in of thesupportswhentheflexuralstiffnessof the beamreducesto a point whereit cannotsustainthe imposedload and thereisnothing to restrainthe growing deflections. In the secondbeamlarger initial deflectionsoccurbecausethebeam‘buckles’dueto therestrainingforcesveryearlyon(70� C ) andfurtherincreasesin lengthdueto thermalexpansioncanonly beaccommodatedin deflection.But runawaydoesnotoccuruntil muchlater(800� C) whenthesteelpropertiesarecompletelylost. This illustratesthatthepresenceof restraintsto endtranslationdelays‘runaway’ to muchhighertemperaturesbecauseof developmentof catenaryactionto replacethehighly depletedflexuralstiffness.
Thesecondbeamis a muchmoreappropriatemodelfor beamsin large redundantstructures.Inrealstructuresnot only is this restraintavailablebut thesteelbeamis in compositeactionwith the
concreteslabwhichproducesamuchstrongerandmorerobuststructure.Thisstrengthandrobust-nessis enhancedby theredistributionmechanismspresentin redundantstructures(for instancetheloadmaybecarriedby thetransverseslabsupportedin tensilemembraneactionwhich retainsitsstrengthfor muchlongerthanthesteelbeams).Largedeflectionsseenin realstructuresareoftenmisinterpretedasimpendingrunaway failure. Figure1 clearlyshows that for temperaturesbelow300� C, thedeflectionsfor therestrainedbeamaremuchlargerthanthat for thesimply supportedbeam,however they havenothingto dowith runaway. Thesedeflectionsarecausedentirelyby theincreasedlengthof thebeamthroughthermalexpansionandarenot a signof lossof ‘strength’or‘stif fness’in thebeamuntil muchlater. In factapproximately90%of thedeflectionat 500� C and75%at600� C is explainedby thermalexpansionalone.Mostof therestis explainedby increasedstrainsdueto reducedmodulusof elasticity. However the behaviour remainsstableuntil about700� C whenthefirst signsof runawaybegin to appear.
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w0.5w0.2w
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Laterally restrained beam of lengthL and maximum flexural capacity Mp
� � � w = 8Mp
L2
0
w
2w
3w
1 2 3 4
800 C
Defl.(m)
Load
700 C
Figure2: Effectof increasingloadsonrunaway in a restrainedbeam
Figure2 showsanotherinterestinganalysis,wherethesamelaterallyrestrainedbeam(asin Figure1) hasbeenloadedwith differentmagnitudesof uniformly distributedload (w) in proportiontotheudl thatwill causea plastichingeto occurat midspan.TheFigureclearlyshows that tensilemembrane(or catenary)actionallows loadsmuchlargerthat theultimateto besupportedto veryhightemperatures.It alsoshowsthatdeflections(plottedatmidspan)remaininsensitiveto theloaduntil justbeforerunaway. This is trueevenfor thecaseof high loads(2w and3w) wheretheinitialdeflectionsbeforeheatingarealreadyvery large indicatingthat tensilemembraneactionis beingexploited. Thedifferencebetweenthemidspandeflectionfor the lowestload(0 02w) againstthehighestload (3w) stayspracticallyconstantupto approximately600� C, indicatingthat upto thistemperaturethe imposedloadshave little effect on theresponseof thestructure,which is almostcompletelygovernedby thethermalactions.Justbeforerunawayfailure,theloadsbegin to exertamuchgreaterinfluenceon theweakenedstructure.Thetemperatureat which runaway eventuallyoccurs,is sensitive to the magnitudeof load. The insetgraphshows the changein load againstdeflectionfor two temperatures,wherereducingslopeindicatesrunaway.
3 Thermal expansion
Heatinginducesthermalexpansionstrains(sayεT) in moststructuralmaterials.Thesearegivenby,
εT � α∆T (4)
If auniformtemperaturerise,∆T, is appliedto asimplysupportedbeamwithoutaxialrestraint,theresultwill simply beanexpansionor increasein lengthof lα∆T asshown in Figure3. Thereforethe total strain(sayεt) is equalto the thermalstrainandthereis no mechanicalstrain(sayεm)whichmeansthatnostressesdevelopin thebeam.
� � � � � � � l εT l
Uniform temperature rise ∆T
Figure3: Uniform heatingof asimplysupportedbeam
3.1 Thermal expansionagainstrigid lateral restraints
Clearly beamsin a real structuresdo not have the freedomto elongatein the mannerdescribedabove. Thereforea morerealisticcaseis to consideran axially restrainedbeamsubjectedto auniform temperaturerise,∆T (asshown in Figure4). It is clearto seethat in this casethe totalstrainεt is zero (no displacements).This is becausethe thermalexpansionis cancelledout byequalandoppositecontractioncausedby the restrainingforceP (i.e. εt � εT
� εm � 0 thereforeεT ��� εm). Thereexistsnow a uniform axial stressσ in thebeamequalto Eεm. Themagnitudeof therestrainingforceP is,
P � EAεm ��� EAεT ��� EAα∆T
� � �� � � � � �� � �P PUniform temperature rise ∆T
Figure4: Axially restrainedbeamsubjectedto uniformheating
If thetemperatureis allowedto riseindefinitely, therearetwo basicresponses,dependingupontheslendernessof thebeam:
1 If thebeamis sufficiently stocky theaxialstresswill sooneror laterreachtheyield stressσy
of thematerialandif thematerialhasanelastic-plasticstress-strainrelationship,thebeamwill continuetoyieldwithoutany furtherincreasein stress,but it will alsostoreanincreasingmagnitudeof plasticstrains. Theyield temperature increment∆Ty is,
∆Ty � σy
Eα
2 If the beamis slenderthenit will buckle beforethe materialreachesits yield stress.TheEulerbuckling loadPcr for abeam/columnasin Figure4 is,
Pcr � π2EIl2
equatingthis to therestrainingforceP, wehave,
EAα∆T � π2EIl2
which leadsto acritical buckling temperatureof,
∆Tcr � π2
α
� rl � 2
(5)
or
∆Tcr � π2
αλ2 (6)
wherer is theradiusof gyrationandλ is theslendernessratio ( lr ). This expressionis valid
for otherend-restraintconditionsif l is interpretedastheeffectivelength.
In this case,if thetemperatureis allowedto risefurther, thetotal restrainingforcewill stayconstant(assumingelasticmaterialandnothermaldegradationof properties)andthethermalexpansionstrainswill continueto beaccommodatedby theoutwarddeflectionof thebeamδ asshown in Figure5.� � � � � �� � �
Pcr
Uniform temperature rise ∆T
δ Pcr
Figure5: Bucklingof anaxially restrainedbeamsubjectedto uniformheating
The above casesrepresentthe two fundamentalresponsesin beamssubjectedto restrainedther-mal expansion. Either of the two (yielding or buckling) may occuron its own (basedupontheslendernessof thebeam)or amorecomplex responseconsistingof acombinationof yieldingandbucklingmayoccur.
Thepatternof developmentof deflections,axial compressionforcesandmomentswith increasein temperaturein slenderrestrainedelasticbeamsis asshown in Figures6and7. Thedeflectionandaxial force figuresclearly show a pre-buckling andpost-buckling type response.The sharpbifurcationpatternis absentasa uniformly distributedloadis imposedon thebeam,impartinganinitial displacementto it. Themidspanmomentcontinuesto riseevenafterbucklingasit consistsmainly of the P � δ momentgeneratedby the axial restraintforce timesthe midspandeflection(whichcontinuesto risebeyondbuckling).
Idealelasticpropertieswereassumedwhendiscussingthecaseof bucklingabove. If thepropertiesareidealelasto-plasticthedeflectionsandaxialcompressionvariationswill haveapatternasshownin Figure8. If the propertiesremainelasticalbeit with a uniform degradationwith temperature,the patternof deflectionandaxial compressionin the beamchangesto the oneshown in Figure9. Clearly the responseof real compositebeamssubjectto restrainedthermalexpansionwillconsistof a combinationof the responsesshown here. That this is indeedthe case,canbe seenin reportAM1, wherethe resultsof modellingthe British SteelrestrainedbeamTestareshown(which comesclosestto the ideal caseof rigid lateral restraint). Thereareother factorsin thatTestthat govern the responseof the heatedcompositebeam,particularlythe effect of deflectioncompatibility in the two directions,however the similarity of the developmentof axial forcesinthesteeljoist andthecompositebeamto thepatternsshown hereis clearto see.
Deflection at Mid-span
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m) D/L = 1.78%
D/L = 2.17%
D/L = 2.67%
% & ' ( ) * ) + * , ) - . / 0 1 ) + * ) 2 + ) 0 0 ) 3 3 ) 4 5 6
(a) (b)
Figure6: Deflectionof axially restrainedelasticbeamssubjectedto heating:(a) Singlebeam,(b)Threebeamsof varyingslenderness
Normal Force at Mid-span
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� � � � � � � � � � � � ! � " � � # $ Moment at Mid-span
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t (N
.mm
)Moment due to Distributed Load
Moment due to Non-Linear geometricaleffect (P-Delta)
Moment Carried by the beam
7 8 9 : ; 9 < = > ? 9 < 9 @ < A 9 : B C
(a) (b)
Figure7: Forcesin anaxially restrainedelasticbeamsubjectedto heating:(a) Axial Forces,(b)Moments
Deflection at Mid-span
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D E F G H F I J K L F I F M I N H F F O P F G Q R S O G N H T U V O G N H K W P F X G Y K T Z [ \ Normal Force at Mid-span
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] ^ _ ` a _ b c d e _ b _ f b g a _ _ h i _ ` j k l h ` g a m n o h ` g a d p i _ q ` r d m s t u
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Figure8: Deflections(a),& Axial forces(b), in anaxially restrainedelastic-plasticbeam
Deflection at Mid-span
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v w x y z { | } ~ { � � x � � x � � � { � z | x { ~ � z � | { � { � x � y � Normal Force at Mid-span
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� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
(a) (b)
Figure9: Deflections(a),& Axial forces(b), in a restrainedbeamwith reducingelasticstiffness
3.2 Thermal expansionagainstfinite lateral restraints
In the previous discussionwe have assumedthe axial restraintsto be perfectlyrigid. This is anupperlimit andpracticallyimpossibleto achievein realstructureswhichofferonly finite restraints.Figure 10 shows sucha beamrestrainedaxially by a translationalspring of stiffnesskt . Thecompressiveaxial stressdevelopedby thermalexpansionis,
σ � Eα∆T�1� EA
ktL � (7)
andcritical buckling temperatureis now givenby,
∆Tcr � π2
αλ2 � 1� EA
ktL � (8)
� �� �� �
� �� �� �� �
P
Pcr
kt P
¡ ¡¡ ¡¡ ¡¡ ¡
Pcr
prebuckling state: ¢ expansion develops axial compression
postbuckling state:¢ expansion produces deflections
end restrained with stiffnesskt against axial translation
δ
Figure10: Heatingof beamwith finite axial restraint
FromEquation8 it canbeseenthatbucklingandpost-bucklingphenomenashouldbeobservableat moderatefire temperaturesin structureswith translationalrestraintstiffnesses(kt ) which arequite comparablewith the axial stiffnessof the member(EA
L ). Figure11 shows a plot derivedfrom Equation8, wherecritical buckling temperaturesare plottedagainstslendernessratio fordifferentrestraintstiffnesses.Theresultsclearlyshow thattheamountof restraintrequiredis notlarge for slendersectionsto reachbuckling temperature.Bearingin mind that theaxial stiffness
of themember(EAL ) itself is reducedby heatingthroughthereductionin E, sothesepost-buckling
phenomenaarevery lik ely to beobservedin beamsin typicalfires.
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£ ª¥£ £ ®¨£ ¯¥£ ¦§£¥£ ¦°ª«£ ¦ £ ¦§®¥£ ¦§¯¨£ ª«£¨£
T±³²
λ
kt � ∞kt � 5EA
l
kt � 2EAl
kt � EAl
Figure11: Buckling temperaturesfor thermalexpansionagainstfinite lateralrestraint
4 Thermal bowing
In theprevioussectionwe discussedtheeffectsof uniform temperatureriseon axially restrainedbeams.In realfiresthetemperaturedistributionsareanythingbut uniform. In asmallto moderatesizecompartmentof a regular shapeonemay assumethat the compartmenttemperaturewill beroughlyuniform at a giventime. The temperatureof thestructuralmembersin thecompartmentdependsuponthematerialthey aremadeof andotherdetailsof geometry, constructionanddesign(suchasinsulation).Concretebeamsandslabsontheceilingof thecompartmentcanbesubjectedto veryhightemperaturegradientsdueto theveryslow ratesof heattransferto concrete.Thereforethesurfacesexposedto fire will beatamuchhighertemperaturethanthesurfacesontheoutsideofthecompartment.This causestheinnersurfacesto expandmuchmorethantheoutersurfacesin-ducingbendingin themember. Thiseffect is calledthermalbowingandis oneof themainreasonsof thedeformationsof concreteslabsandmasonrywalls in fires. Anothervery importantsourceof thermalbowing in compositebeams/slabsis the large differencebetweenthe temperaturesofthesteeljoist andtheslab. This effect is muchmoreimportantin theearlystagesof thefire whensteelretainsmostof its strength.
Relationshipscanbederivedfor thermalbowing analogousto theonederivedearlierfor thermalexpansion.Figure12showsa beamsubjectedto a uniformtemperaturegradientthroughits depth(d) alongits wholelength(l ). Assumingthebeamis simplysupported(asshown in Figure12)wecanderive thefollowing relationships:
1 Thethermalgradient(T́y) over thedepthis,
T́y � T2 � T11d
2 A uniformcurvature(φ) is inducedalongthelengthasa resultof thethermalgradient,
φ � αT́y
3 Dueto thecurvatureof thebeamthehorizontaldistancebetweentheendsof thebeamwillreduce. If this reductionis interpretedasa contractionstrain(not literally) εφ (analogousto the thermalexpansionstrainεT earlier), the valueof this straincanbe calculatedfromanalysingFigure12as:
εφ � 1 � sin lφ2
lφ2
(9)
T1
T2
d
θ
R
NA
T2 T1>
µ µ µ µµ µ µ µ ¶ ¶ ¶¶ ¶ ¶Figure12: Simplysupportedbeamsubjectedto auniformthermalgradient
Now considerthelaterallyrestrainedbeamof Figure5. If auniformthermalgradientT́y is appliedto thisbeam(asshown in Figure13), theresult(in theabsenceof any averagerisein temperature,i.e.meantemperatureremainingconstant)is a thermallyinducedtensionin the beamandcorre-spondingreactionsat thesupport(oppositethethepurethermalexpansioncasediscussedearlier).This is clearlycausedby therestraintto endtranslationagainstthecontractionstrain(εφ) inducedby thethermalgradient.
Figure14showsafixedendedbeam(by addingendrotationalrestraintsto theBeamof Figure13)subjectedto a uniform temperaturegradientthroughits depth.Recallingthata uniform curvatureφ � αT́y exists in a simply supportedbeamsubjectedto gradientT́y. If thatbeamis rotationallyrestrainedby supportmomentsM (uniformalonglength)anequalandoppositecurvatureinduced
· · · ·· · · · ¸ ¸ ¸ ¸¸ ¸ ¸ ¸Uniform temperature gradient T,y
δP P
Figure13: Laterallyrestrainedbeamsubjectedto auniformthermalgradient
by the supportmomentscancelsout the thermalcurvatureand thereforethe fixed endedbeamremains‘straight’ with a constantmomentM � EIφ alongits length.
¹ ¹¹ ¹¹ ¹¹ ¹º ºº ºº ºº º
Uniform temperature gradient T,y
M M
Figure14: Fixedendbeamsubjectedto auniformthermalgradient
Fromtheabove discussionit is clearthat theeffect of boundaryrestraintsis crucial in determin-ing theresponseof structuralmembersto thermalactions.Thekey conclusionto bedrawn fromthediscussionsofar is that, thermalstrainswill bemanifestedasdisplacementsif they are unre-strainedor asstressesif they are restrainedthroughcounteractingmechanicalstrainsgeneratedby restraining forces.
As discussedearlierfor lateralrestraint,perfectrotationalrestraintis alsonotveryeasilyachievedin realstructures(otherthanfor symmetricloadingonmembersovercontinuoussupports,withoutany hingesfrom strengthdegradation).Figure15showsabeamrestrainedrotationallyat theendsby rotationalspringsof stiffnesskr . In this casetherestrainingmomentin thespringsasa resultof auniformthermalgradientT́y canbefoundto be,
Mk » EIαT́y�1 ¼ 2EI
kr l � (10)
This equationimpliesthat if therotationalrestraintstiffnessis equalto therotationalstiffnessofthebeamitself (EI
l ) thenthemomentit will attractwill beaboutathird of afixedsupportmoment.
½ ½ ½½ ½ ½¾ ¾ ¾¾ ¾ ¾ ¿ ¿ ¿¿ ¿ ¿
kr krUniform temperature gradient T,y
Figure15: Beamwith finite rotationalrestraintwith auniformthermalgradient
5 Deflections
In theprevioussectionswe have lookedat theoverall behaviour of beamssubjectedto expansionandbowing for variousrestraintconditions.Oneinterestingaspectof structuralresponseto fireis the largedeflectionsthatarefound in structuralmemberssuchasbeamsandslabs.Largede-flectionsarenormallyassociatedwith lossof strengthin structuresunderambientconditions.In
caseof fire sucha simpleinterpretationcanbehighly misleading.Both the thermalmechanismsdiscussedearlier(thermalexpansionandthermalbowing) resultin largedeflections,however thestateof stressassociatedwith a membersubjectedto varying degreesof thesetwo mechanismsis not uniquefor a givendeflectionanda large rangeof stressstatesexist (largecompressionortensionor very low stresses)dependinguponthe temperaturedistribution in the memberanditsmaterialpropertiesandrestraintconditions.
Thechief reasonfor largedeflectionsis thatthestructuralmembertriesto accommodatetheaddi-tional lengthgeneratedby thermalexpansion,giventhatit is not possiblefor it to expandlongitu-dinally dueto endrestraints.Considera slenderbeam(very low buckling temperature)subjectedto uniformheatingagainstrigid lateralrestraints(asin Figure5). Bucklingwill occurveryearlyon(atverylow elasticstrains)afterwhichany furtherexpansionwill makethebeamdeflectoutwards.Theresultingmidspandeflectionδ canbeapproximatedquiteaccuratelyby,
δ » 2lπ
εT ¼ ε2T
2(11)
which is an approximationof the deflectionof a sin curve of length l À 1 ¼ εT Á , whereεT is thethermalexpansionstrain(α∆T).
If the samebeamis subjectedto a uniform thermalgradientproducingno net expansion,onlybowing asin Figure13, the responseis thendeterminedby the flexure-tensioninteraction. ThetensileP � δ momentsrestrainthecurvatureimposedby thethermalgradientsandlimit deflections.Thedeflectionsresultfrom thetensilestrainsproducedin thebeam,i.e.
εt » PEA
(12)
andthedeflectionscanthenbedeterminedby
δ » 2lπ
εt ¼ ε2t2
(13)
ThetensileforcePt canbedeterminedfrom substitutingEquation12 in Equation13andsolvingaquadraticequationfor Pt,
Pt »ÃÂÄ 12 � πδ
l � 2 ¼ 1 � 1ÅÆ EA (14)
To determinethedeflectiony À xÁ in thebeamof Figure13 for a givencurvatureφ (arisingfrom athermalgradient),adifferentialequationsolutioncanbewrittenasfollows:
For asimplysupportedbeamsubjectedto auniformcurvatureφ onecanwrite,
d2ydx2 » φ
If the beamis laterally restrainedasin Figure13, a tensileforce P will be generatedcausingamomentPyover thelengthof thebeam,therefore,
d2ydx2 » φ ¼ Py
EI(15)
ord2ydx2 � k2y » φ
where,
k »ÈÇ PEI
Thesolutionof thisequationis,
y À xÁ » � φk2 � coshkl � 1
sinhklsinhkx � coshkx ¼ 1� (16)
It maybeseenthatEquations14and16form asetof nonlinearequations.Theseequationscanbesolvedusinganappropriateiterative technique(bisection,Newton-Raphson)to obtainthetensileforcesanddeflectionsfor thermalgradientdominatedproblems.
6 Combinationsof thermal expansionand thermal bowing
In theprevioussectionstheresponseof beamsto eitherthermalexpansionor thermalbowing havebeenconsideredin isolation.To studythecombinedresponseletusfirst considerthecaseof afixedendedbeamasshown in Figure16which is bothrotationallyandtranslationallyrestrainedatbothends.If this beamis subjectedto a meantemperatureriseanda throughdepththermalgradient,it will experiencea uniform compressive stressbecauseof restrainedexpansionanda uniformmomentbecauseof thethermalgradient.Thestresseson any typical cross-sectionbecauseof thecombinedeffectof thetwo thermalactionsarealsoshownin Figure16. It is clearthatthebottomof
ÉÉÉÉ
Uniform temperature gradient T,y
Uniform temperature increase ∆TÊ ÊÊ ÊÊ ÊÊ ÊM M
P P
+ =Tens.
Comp.
Comp.
σ=Eα∆T σ=EαT,y y
y
Figure16: Combinedthermalexpansionandbowing in afixedendedbeam
thebeamwill experienceveryhighcompressivestresses,while thetopmaybeanywherebetweensignificantcompressionto significanttension.
The above scenariois a commonone in compositeframe structuressuchas Cardington. Thecompositeactionof a steeljoist, framing into an interior column,with a continuousslabover it,producesconditionsvery closeto a fully fixedsupport(asin Figure16). Thehigh compressionsresultingfrom thecombinedeffectof thermalactionsasdescribedabovealmostinvariablyproducelocalbuckling in thelowerflangeof thesteeljoist veryearlyon in a fire. Thiswhy localbucklingof the lower flangesis sucha commonoccurrencein fires(seenin all Cardingtontestsandotherfires).
Oncelocalbucklinghasoccurredthepatternof stressesattheendsof thecompositebeamchanges.The hoggingmomentis relieved by the hinge producedby local buckling andthe endrestraintconditionschangeto theoneshown in Figure13. As thishappensquiteearlyin realfires,theendconditionsdescribedby Figure13aretheonesthatgovernthebehaviour of acompositebeamformostof thedurationof thefire.
6.1Ë
Combinedthermal expansionand bowing in laterally restrainedbeam
Thefundamentalpatternof behaviour of a beamwhoseendsarelaterallyrestrained(but rotation-ally unrestrained,seeFigure13) subjectedto thermalexpansionandthermalbowing separatelywasestablishedin previous sections.Restrainedexpansionresultedin compressionandbowingresultedin tension. This helpedto illustratethat two oppositestressregimescanoccurdepend-ing upon the thermalregime applied,however the apparentresponseof the beamis the same(i.e.downwarddeflection).
To studythe effectsof the applyingcombinationsof thermalexpansionandthermalbowing wedefineandeffectivestrainasfollows,
εeff » εT � εφ (17)
Thevariationof εeff (for variousthermalregimes)canproducea largevarietyof responses.Pos-itive valuesof εeff imply compression(or the effect of meantemperaturerise is dominant)andnegativevaluesimply tension(or theeffectof thermalgradientsis dominant).Figure17showsthevariationof εeff for differentvaluesof thermalgradientwhenthetemperatureis increasedfrom 0to 400Ì C (εeff is plottedfor a linearincreasein gradientagainsttemperature).
T́y =5 Ì C/mmT́y =4 Ì C/mmT́y =3 Ì C/mmT́y =2 Ì C/mmT́y =1 Ì C/mm
T ( Ì C)
ε eff
400350300250200150100500
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
Figure17: Effectiveexpansionstrains
Case1: Zero stressin the beam(εeff » 0)
Figure18showsaninterestingtheoreticalcase.If theimpliedcombinationof εφ andεT areapplied:Í Thereareno stressesin the beam. All thermalstrainsareconvertedinto displacementasseenin thefigure.Í The deflectionof the beamis entirely dueto thermalbowing to accommodatethe excesslengthgeneratedby expansion.Í thedeflectionresponseof this beamcanbeanalyticallyexpressedastheincreasein lengthof theelasticcurve of thebeamversusits deflection.Figure19 shows a numberof lengthincreasevs midspandeflectionplotsbasedon assumedcurve shapes.Thefigureshows thattheshapeof thecurvechosendoesnotmattermuch,thereforetheformulagivenearlierbased
Step 2 : Impose a curvature αT,y to return support to the original position
Step 1 : Impose a temperature rise ∆T
Figure18: Case1: ZeroStress
uponthesin curve (Equation11) canbeusedto get a goodapproximationof themidspandeflectionym, i.e.
ym » 2lπ
εT ¼ ε2T
2
triangularparabolic
circulararchalf sin-wave
εT
y m l
0.0350.030.0250.020.0150.010.0050
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
Figure19: Strainv non-dimensionalmidspandeflection(perunit length)
Case2: Thermal expansiondominant (εeff Î 0)
If εT ÎÏÎ εφ, thermalexpansiondominatesandatwo stageresponseis producedconsistingof Pre-bucklingandPost-bucklingphases.Thethermalexpansionproducedispartlyusedupin generatingmechanicalstrainsandpartly in generatingdeflections.This is governedby themagnitudeof εeffwhich is the componentthat generatesstressesto progressthe beamtowardsbuckling. The εφcomponentannihilatespartof theexpansionandproducesdeflectionsby imposingcurvaturewiththeavailableexcesslength.Thepre-bucklingdeflections(yÐm) will for a smallpartresultfrom theelasticbendingof thebeamandalargerpartwill generallycomefrom thedeflectionresultingfrom
theÑ
imposedcurvature(ym À φ Á ).yÐm » y0
1 � ∆T∆Tcr
¼ ym À φ Á (18)
Herey0 canbe interpretedasthe initial elasticdeflectionbeforethe fire becauseof the imposedloadson thebeamandym À φ Á is theextradeflectiondueto thermalbowing givenby,
ym À φ Á » 2lπ
εφ ¼ ε2φ
2(19)
Thepresenceof thegradientclearlydelaysthebucklingeventandthecritical bucklingtemperature(∆Tcr) is increasedto,
∆Tcr » 1α � π2
λ2 ¼ εφ � (20)
Figure20 shows the typical variationin buckling temperaturewith the changein gradient(for abeamof slendernessratio l
r equalto 70).
Tcr
T́y ( Ì C/m)
T cr
(
Ò C)
5000450040003500300025002000150010005000
1000
900
800
700
600
500
400
300
200
Figure20: Critical Buckling temperaturesvsThermalgradient
Thepostbuckling deflectionswill carryonincreasingbecauseof all furtherexpansionstrainsε ÓT asin Equation11, i.e.
yÓm » 2lπ
ε ÓT ¼ ε ÓT 2
2
Thethermalbowing deflectionaddedto theelasticdeflections(dueto P � δ momentsandloading)will againact as ‘imperfections’ to ‘straightness’of the beamand producea smoothvariationof beammidspandeflectionwith temperatureuntil the large displacementpost-buckling modebegins (identifiedby thechangeof curvatureof the temperaturedeflectioncurve). This alsohastheeffect of reducingthedevelopmentof compressionforcesin thebeam(asthebeamdisplacesmorefor lowercompressionsbecauseof theadditionalbowing displacementsincreasingtheP � δmoments).
Case3: Thermal bowing dominant (εeff Ô 0)
Whenεφ ÎÕÎ εT , thedeflectionresponsewill bethesumof two components,
1 Deflectioncausedby bowing of the excesslengthgeneratedthroughexpansion,asbeforei.e. À ym Á 1 » 2l
πεT ¼ ε2
T
2
2 Thetensilestrainεt producedby the tension(Peff) causedby theexcesscontractionstrain(εeff) asin thecaseof purethermalgradients.
εt » PeffEA
whichwill producefurtherdeflectionsas,
À ym Á 2 » 2lπ
εt ¼ ε2t2
ThetensionPeff andthedeflectionscanthenbedeterminedaccordingto theiterativeproce-duresuggestedin theprevioussectionondeflections.
Finally Figure21shows themaintypesof deflectionresponsesthatmaybeobservedif a laterallyrestrainedbeamis exposedto combinationsof thermalactionsdiscussedabove.
Pre−buckling Bifurcation
Post−buckling
T
δ
εT (εφ =0)
εT = εφ (zero stress)
εT > εφ(or y
0 > 0)Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö
Tensile forces in the beam
Compressive forces in the beam
Tcr
Tcr (εφ)
εT < εφ
Figure21: Temperaturedeflectionresponsesfor combinationsof εT andεφ
7 Criterion for the various typesof responses
Fromthediscussionaboveasimplecriterionfor all thevarioustypesof responsesobservedcanbedeveloped.If εeff is closeto π2
λ2 thentherewill beno bucklingasnot enoughcompressionis gen-erated.A dimensionlessnumberζ maybedefinedasfollows to categorisethevariousresponses:
ζ » εT × εφπ2
λ2
(21)
To summarise,
1 ζ ÎÏÎ 1typically generatespreandpostbuckling type deflectionresponseswith thermalexpansionandcompressiondominant.Thecompressionforcepatternsareasdiscussedearlierin therestrainedthermalexpansionsection.
2 ζ Ø 1generatesresponseswheremostof the thermalexpansionis convertedinto deflectionbuttherearenegligible stressesin thebeam(closeto thezero stresscasediscussedearlier).
3 ζ ÔÏÔ 1generatesthermalbowing dominatedresponsewith deflectionpatternssimilar to the zerostresscaseandwith considerabletensileforcesin thebeamwhichgrow with theincreaseinthegradient.
8 Numerical analysisof the simplebeammodel
Theanalyticalapproachdevelopedaboveto fully understandthestructuralresponseto theheatingregimehasbeencheckednumericallyby modellingthesamesimplebeamexamplesin ABAQUS.Thedatafor beamanalysedwasasfollows:Í length(l )=9000mmÍ Modulusof Elasticity(E)=210000N/mm2Í Coefficientof thermalexpansion(α)=8 Ù 10Ð 6Í AreaA=5160mm2Í Secondmomentof areaI =8Ú 55 Ù 107 mm2
Thereforethe slendernessratio lr of the beamis approximatelyequalto 70. This calculationis
limited to investigatedthesimplebeammodelrestrainedlaterallybut free to rotateat its endsasin Figure13. The resultsconfirm the theoreticalsolutionsderived for the responseof the beamto thermalbowing andthermalexpansion. Figure22 shows the resultsof the numericalanaly-sis in termsof the deflectionsandaxial forcesproducedwhenthe beamis subjectedto a meantemperaturerise (uniform over the length) of 400Ì C and an effective thermalgradientthroughthe depthof the beam.The temperatureincrease,∆T andthermalgradient,T Û y wereappliedtothe simple numericalbeammodel at a constantrate from zero to their maximumvalues. Thedeflectionasa resultof purerestrainedthermalexpansionshows the doublecurvatureshapeofthe pre-buckling/post-buckling response(seeFigure22). Whena gradientis alsoappliedto themodelandthe responseof thebeamis governedby the interactionbetweenthermalbowing andrestrainedthermalexpansionthedeflectedshapebecomessmoother, indeedfor aspecificcombina-tion of meantemperatureriseandtemperaturegradienttheresponsewill beverycloseto linear. Atlargegradients(T Û y » 10Ì C/mm)whentheresponseis dominatedby thermalbowing thedeflectedshapeis verynon-linear.
The correspondingaxial forcesarealsoplottedin Figure22. Whena meantemperaturerise of400Ì C aloneis appliedto themodelandtheresponseof thebeamis governedpurelyby restrainedthermalexpansionandtheaxial force is in high compression.Whenthemodelis subjectedto acombinationof meantemperatureriseandtemperaturegradientthe axial forcebecomessmallerin compressionandat high gradientsmovesinto tension.Theaxial forceat thebeginningof theanalysisis alwaysin compressionbecausethe meantemperatureandthe gradientareappliedto
Axial Force in the model
-2.50E+06
-2.00E+06
-1.50E+06
-1.00E+06
-5.00E+05
0.00E+00
5.00E+05
1.00E+06
1.50E+06
2.00E+06
2.50E+06
3.00E+06
0 50 100 150 200 250 300 350 400
Temperature (°C)
Axi
al F
orce
(N
)
400°C400°C_1°C/mm400°C_3°C/mm400°C_5°C/mm400°C_10°C/mm
(a)
(b)
Deflection at mid-span due to thermal expansion and thermal bowing
-450.00
-400.00
-350.00
-300.00
-250.00
-200.00
-150.00
-100.00
-50.00
0.00
50.00
0 50 100 150 200 250 300 350 400
Temperature (°C)
Def
lect
ion
(mm
)
400°C400°C_1°C/mm400°C_3°C/mm400°C_5°C/mm400°C_10°C/mm
Figure22: Numericalmodel resultsfor combinedthermalexpansionand thermalbowing: (a)Deflections(b) Axial forces
theÑ
modellinearly from zeroto theirmaximumvaluesandaregovernedby thedevelopmentof theeffectivestrains,εeff , asshown in Figure17.
Theactualvaluesof deflectionsandforcesin thenumericalexerciseabovecanbeestimatedusingtheformulasgivenhere.For instancefor thecaseof a temperatureriseof 400Ì C thecompressionforceis simply theEulerbuckling load(π2EA
λ2 ) equalto 2170kN (approx.).Thedeflectionfor this
casecanbeobtainedfrom subtractingtheelasticcompressionstrain(π2
λ2 ) from the thermalstrain(εT) to obtainthestrainthatproducesthedeflections,therefore,
ym Ø 2lπ Ç εT × π2
λ2
whichproducesavalueof approximately200mm(within 10%)of thenumericalcalculationabove.The differenceis becausethe numericalcalculationis fully geometricallynon-linearwhile theabove formulasarebasedon1storderdefinitionsof strain.
If all thethermalstrainswereto producedeflection(by appropriatecombinationof εT andεφ) thenthe internalforceswould bevery low andthedeflectionwould beapproximately324mm (fromEquation11) which lies betweenthecasesof T Û y » 3Ì C/mmto T Û y » 5Ì C/mm. It maybenotedfrom Figure22 thattheaxial forcein thebeammovesfrom compressionto tensionbetweenthesevalues(suggestingthatfor thedeflectionsin theregion of 324mm) theforcesin thebeamwill beinsignificant.
Thetensileforcesanddeflectionsfor largegradientscanbecalculatedfrom iteratively solvingthesetof nonlinearEquations14and16.
Theaboveanalysisclearlyhighlightsthelargerangeof deflectedshapesandaxial forcespossibleasa resultof theinteractionbetweenthermalexpansionandthermalbowing.
9 Other important factors
Thediscussionabovehasfocussedupontheeffectsof thermalexpansionandthermalbowing andillustratedthe large variety of responsespossiblein real compositeframestructures.Analyticalexpressionshavebeenpresentedwhichallow agoodquantitativeestimateof forcesanddeflectionsto bemadefor simplestructures.
Effectsof thermaldegradationandimposedloadingwerefoundto berelatively lessimportantinthe modellingof CardingtonTests(seereportsSM1-2). The effect of strengthdegradationwasshown to changethe developmentof compressive forcesin a restrainedbeamin Figures8 and9. The loadingon a beamin a largedisplacementconfiguration(throughthermaleffects)will becarriedvery effectively in a catenary(or tensilemembrane)behaviour. It is clearfrom theabovediscussionthat for the most likely combinationsof thermalactions(εT andεφ) the mechanicalstrainsin a memberare likely to be very low (compressionor tension). If this is true andthatthermaldegradationhasbeencontainedin the surfacelayers,thenthe tensilestrainsinducedbymembranemechanismsshouldbe carriedquite reliably. This is however an areawhich needsfurtherextensiveinvestigationusingall standardresearchtechniques,experimental,computationalandtheoretical.
Restraintconditionscancertainlyhave a major effect on the distribution of the internal forcesandthe displacementsthat occurashasbeenillustratedby the simpletheoreticalandcomputa-tional analysesin this paper. The degreeof restraintavailablealso changesduring a fire , forinstancetherotationalrestraintavailableto thecompositebeamat thebeginningis lostquiteearlyon (around200C) [3] dueto the local buckling of thesteeljoist andtensilecapacityof theslab
beingÜ
reached.Rotationalrestraintsresultin increasinghoggingmomentsuntil a ‘plastic hinge’is achieved. Lateral translationrestraintsproducecompressionforcesif thermalexpansionwasdominantandtensionforcesif thermalbowing is dominant.Theamountof restraintrequiredisnot largeto producebuckling asfloor structuresusuallyvery slender. Thesourceof this restraintis obvious for interior compartments- the colderandstiffer surroundingstructure. For exteriorcompartmentsit is not soclearif sufficient restraintsarestill available. It is likely thatsufficientrestraintto lateralexpansionis availableatexteriorboundariesthroughtheactionsof tensionrings[1]. At largedeflectionslateralrestraintsprovideananchorto thetensilemembranemechanisms.Again, it is likely that sufficient lateral restraintis availableat exterior boundariesthroughtheactionof compressionrings [4]. This however is a matterof muchgreaterimportancethantherestraintto thermalexpansionasthesurvival of thefloor systemultimatelydependsuponthereli-ability of thetensilemembranemechanism.Thisagainis akey questionfor furtherinvestigation.
Anothervery importantfactorthathasnot beeninvestigatedhereis theeffectof thecompartmentgeometry. This canhave a large effect on the developmentof thermallyinducedforcesandde-flectionsin the heatedstructuralmembers.The principle that allows oneto make a quantitativeassessmentof theeffectof compartmentgeometry, is compatibility . For instancefor arectangularfire compartment,thethermalexpansionin theshorterdirectionwill besmallerthantheexpansionthe longer direction. This can lead to an increasein compressionin the longer direction (be-causecompatibilitydoesnot allow it to deflectasmuchasits thermalexpansiondemands).In theshorterthe reversehappens,compatibility forcesthedeflectionsin this directionto besomewhatlargerthanthermalexpansionwould allow resultingin lowercompressionsor eventensileforces.Thishasbeenidentifiedclearlyin themodellingof theBritish Steelrestrainedbeamtest(3m Ù 8m)wherethemidspanribsarein tension.Thisallowsredistributionof thethermallyinducedforces.
10 Conclusions
Thefundamentalprinciplespresentedin this paperprovide a meansof estimatingforcesanddis-placementsin realstructureswith appropriateidealisations.Suchestimatescanbeof considerableusein assessingtheresultsfrom morerigorousnumericalanalysesor they canbeusedin designcalculations.Examplesof suchusagewill bepresentedin a subsequentpaper. Therearehowevera considerablenumberof very importantissuesthatremainto beinvestigatedasmentionedin theprevious section. Considerableeffort is requiredto addresstheseissuesto satisfactionbeforeacompletesetof principlescanbedeveloped.
References
[1] J.M.Rotter, A.M.Sanad,A.S.Usmani,andM.Gillie. Structuralperformanceof redundantstruc-turesunderlocal fires. In Interflam’99,8th InternationalFire ScienceandEngineeringCon-ference, pages1069–1080,Edinburgh,Scotland,29June- 1 July1999.
[2] J.M.RotterandA.S.Usmani. Fundamentalprinciplesof structuralbehaviour underthermaleffects.In Proceedingsof theFirst InternationalWorkshoponStruturesin Fire, Copenhagen,Denmark,June2000.
[3] A.M.Sanad,J.M.Rotter, A.S.Usmani,andM.O’Connor. Compositebeamin buildingsunderfire. 2000.Resubmittedafterrevision.
[4] Y.C. Wang. TensileMembraneAction in Slabsand its Application to the CardingtonFireTests.Technicalreport,Building ResearchEstablishment,1996.Paperpresentedto thesecondCardingtonConference12-14March1996.
