how did the wtc towers collapse: a new theory …

HOW DID THE WTC TOWERS COLLAPSE: A NEW THEORY

A.S.Usmani,Y.C.ChungandJ.L.ToreroSchoolof EngineeringandElectronics,Universityof Edinburgh,

Edinburgh,EH9 3JN,UK

ABSTRACT

This paperusesa finite elementmodelto investigatethe stability of the Twin-Towersof the WorldTradeCenter, New York for anumberof differentfire scenarios.This investigationdoesnot take intoaccountthestructuraldamagecausedby theterroristattack.However thefire scenariosincludedarebaseduponthe likely fires thatcouldhave occurredasa resultof theattack. A numberof differentexplanationsof how andwhy the Towerscollapsedhave appearedsincethe event. Noneof thesehowever have adequatelyfocusedon themostimportantissue,namely‘what structuralmechanismsled to thestatewhich triggeredthecollapse’.Also, quitepredictably, therearesignificantandfunda-mentaldifferencesin theexplanationsof theWTC collapsesonoffer sofar. A completeconsensusonany detailedexplanationof thedefinitive causesandmechanismsof thecollapseof thesestructuresis well nigh impossiblegiventheenormousuncertaintiesin key data(natureof thefires,damagetofire protection,heattransferto structuralmembersandnatureandextent of structuraldamageforinstance).Thereis however a consensusof sortsthat thefires thatburnedin thestructuresafter theattackhadabig partto play in thiscollapse.Thequestionis how big?Takingthis to theextreme,thispaperposesthehypotheticalquestion,“had therebeennostructuraldamagewould thestructurehavesurvivedfiresof asimilar magnitude”?

A robustbut simplecomputationalandtheoreticalanalysishasbeencarriedout to answerthis ques-tion. Robustbecauseno grossassumptionshavebeenmadeandvaryingimportantparametersoverawiderangeshowsconsistentbehaviour supportingtheoverallconclusions.Simplebecauseall resultspresentedcanbe checked by any structuralengineereither theoreticallyor usingwidely availablestructuralanalysissoftwaretools. The resultsare illuminating andshow that the structuralsystemadoptedfor theTwin-Towersmayhavebeenunusuallyvulnerableto amajorfire. Theanalysisresultsshow a simplebut unmistakablecollapsemechanismthatowesasmuch(or more)to thegeometricthermalexpansioneffectsasit doesto thematerialeffectsof lossof strengthandstiffness.Thecol-lapsemechanismdiscoveredis asimplestability failuredirectly relatedto theeffect of heating(fire).Additionally, themechanismis notdependentuponfailureof structuralconnections.

KEYW ORDS

Structuresin fire, WTC Towers,collapsemechanisms,stabilityof tall buildings

INTRODUCTION

Fires in buildings inducegeometric(thermalexpansion)andmaterialeffects(reductionin strengthandstiffness)in structuralelements.Computationalmodellingandanalyticalstudies[1, 2] at Ed-inburgh University have shown that of thesetwo competingeffects in large frame structures,theformerdominatesthebehaviour of thestructurein theearlystageswhile the latterbecomesimpor-tantnearfailure. In this paperstructuralbehaviour is interpretedby analysingthe typical structuralresponsesto loading,namely, strainsandstressesandmoreimportantly their resultants(integrals),i.e. displacementsandmemberforces. Researchso far on compositesteelframestructuresin firehasshown thatsteelframecompositestructureshavea considerablymorerobustperformancein firethanaccountedfor in design.Thisconclusionhasbeensteadilygainingcurrency in thestructuralfireengineeringprofessionover the last few decades.Themain reasonfor this hasbeentheabsenceofstructuralcollapsesevenasaresultof majorfiresin largecompositeframestructures.In UK themost

importanteventwastheBroadgatePhase8 fire [3]. This eventuallyled BRE andBritish Steel(nowCorus)to carryout 6 full scalefire testson an 8-storey steelframecompositestructureat the BRELargeBuilding TestFacility at Cardington(Bedforshire,UK) [4]. Thesetestsconsistentlyproducedtheexpectedrobustbehaviour for a rangeof compartments(smallandmoderatelylarge)subjectedtoseverefires. To understandthemechanicsgoverningthis behaviour, a largeconcertedcomputationalmodellingexercisewascarriedoutby Universityof Edinburgh,British SteelandImperialCollege[1].This projectrevealedin greatdetail the wholestructureresponseto fire thatproducedtheobservedrobustbehaviour. Thiswork hasbeenreportedextensively in many papers[5–7].

Lessonsfr om CardingtonThedetailsof modellingof theCardingtonfire testsandsubsequentinterpretationsof behaviour aretoo voluminousto presenthere,but considerableinformationcanbedownloadedfrom thewebpagegivenin [1]. Very briefly, this work revealedthe following lessonsfor wholestructurebehaviour infire:

� Restraintto thermalstraindominatesthe structuralbehaviour (in termsof internalforcesanddisplacements)

� Relativeto thermalstrains,thecontributionof conventional(gravity) loadingis low (if nofailureoccurs)

� The resultsshow low sensitivity to variationsin strengthandstiffnesspropertiesof steel,asthermalexpansionswampstheseeffects

� At large deflections,tensilemembraneactionin the spansandcompressive membraneactionneartheperimetersupportsof floor slabsareobserved

� Thermalstrainsproducea beneficialload-carryingshapefor tensilemembraneactionin slabs,without largeanddamagingmechanicalstrains

� Theloadcapacityis furtherenhancedby thermallyinducedpre-stressingin laterallyrestrainedslabsandcompositefloor systems

� Local buckling (this is a misnomerasthis is in fact ‘local yielding’ [8]) of the lower flangesof steelbeams(actingcompositelywith the slab)occursat relatively low steeltemperatures(between100

�C to 200

�C) but this wasnot foundto beadetrimentalmechanism

A numberof generallyapplicablefundamentalprinciplesof structuralbehaviour in firewereidentifiedto be importantaspart of the modellingandanalysisexercise[2] andproved invaluablein makingsenseof thecomplex computationalresults.Theseare:

1. Unrestrainedthermalexpansioncausedby a rise in meantemperatureof a structuralmembercausesits endsto moveapart.Thethermalstrainproducingthis expansionis

εT� α∆T

whereα is thecoefficientof thermalexpansionand∆T is theaveragetemperatureincrement.

2. Thermalexpansion(or elongation)of a structuralelementin the presenceof end restraintsto lateral translationfrom the surroundingstructureproducescompressionforcesleadingtoyielding or buckling (dependinguponthe ‘slenderness’of thestructuralelement).For this tooccur, neithertherestraintstiffnessnor themeantemperaturerisehasto belarge.

3. A thermalgradientover the depthof a simply supportedstructuralelementleadsto ‘thermalbowing’ or curvaturein theelement,

φ � αT� zwhereT� z is theaveragetemperaturegradientthroughthedepth.Thethermallyinducedcurva-tureresultsin thepulling in of theendsin a simply supportedbeam.Thereductionin distancebetweentheendscanbewrittenasa “contraction”strain,

εφ� 1 � sin lφ

2lφ2

where,l is thelengthof thebeam.

4. Restraintto end translation(while the rotationsare free) in the presenceof a through-depththermalgradient(without any rise in meantemperature,therefore∆T � 0) producestensionsin thebeamwhichgrow with growth in thethermalgradients.

5. Rigid restraintto endrotation,when∆T � 0 andTz� 0, producesa hoggingmomentof EIφ

over thewholelengthof thebeamwith no curvature.Finite rotationalrestraintsproducecom-binationsof hoggingmomentsandcurvature.

6. For a structuralelementrestrainedagainstend translation,but relatively free to rotate(as isoften the casefor many elementsin real structuresunderreal fires), combinationsof thermalelongationandthermalbowing producea wholerangeof internalforces,from largecompres-sionsto tensions.Theseinternalforceshoweverarealwaysaccompaniedwith largedeflectionstowardtheheatedsideof themember. If for theaforementionedstructuralelementin a givenfire thetwo effectsof elongationandbowing arenearlyequal,theinternalforces(andthereforethedamagingmechanicalstrains)in theelementwill beverysmall(thisagainis acommonsit-uationwith many steel-concretecompositemembers).This is a directresultof thewell knownequation:

εtotal� εthermal

� εmechanical

7. Compatibilityof displacementsin compartmentswith orthotropicstiffnessandtemperaturedis-tribution (for instance,steelonly in onedirectionor compartmentsof high aspectratio) influ-encesthe forcesanddisplacementsin orthogonaldirections(suchastensionsalongthe shortspanandcompressionsalongthelongspan)

Theseprinciples,whensetagainstthedetailedresultsfrom thecomputationalmodellingof theCard-ington Tests,provided excellent insights[8]. Furtherwork hasbeenunderway towardsdevelopingnew analysisanddesignmethods[9–11], to fully exploit thenew understandingdevelopedin futureanalysisanddesignof structuresfor fire.

Impact of 9/11While this remarkableprogressin understandingthebehaviour of structuresin fire wasachievedoverthepreviousfive yearsandhopesof greatnew stridesin the futuredesignof structuresin fire wererising, theatrocityof 9/11occurred.Initial reactionswerequitepredictablyemotionalandextreme,with somesuggestingthatskyscrapersshouldnotbebuilt usingsteelanymore.Theseinitial reactionshavethankfullygivenwayto coldanalysisandcalculationandadesireto understandtheexactnatureof themechanicsthatcausedsomany of thestructuresin theWTC complex to collapse.

Analyses of the collapseOneof the earliestsignificantattemptsat explaining the Twin Tower collapseswasby Bazantand

Zhou, laterpublishedin [12]. This work restrictsitself to explaining theprogressive collapseof thewholestructurecausedby dynamicoverloadsfrom thefalling superstructureasarigid body, oncetheinstability thatmadethesuperstructuremobilehadalreadyoccurred.Thisanalysisis interestingasitshows clearly thatoncethis point is reached,thedynamicoverloadswill leadto completecollapse.Howeverthebiggerandmuchmoreimportantquestionis ”what ledto theinstability thatinitiatedthecollapsein thefirst place”.Thebriefly statedreasonin theintroductionsectionof thispaperattributesthis to creepbucklingafterprolongedheatingof steelcolumnsto temperaturesof 800

�C (andtheloss

of fire protectionfrom theinitial impactandexplosion).

On the faceof it this soundscredible,however the FEMA report[13] suggeststhat therecouldnothave beenan explosion(only small over-pressures)andthat the temperaturescould not have beenthathigh over thewholefire affectedareaof thebuilding at thesametime. Thereportindicatesthattemperaturesrangedfrom a high of 1100

�C to a low of 400

�C, it alsosaysthat from photographic

evidencetherewereanumberof largelocal firesvaryingwith bothtimeandlocation.Thereforeit isquitepossiblethatwhenpartsof thestructureheatedotherpartsmayhave cooled.Thecontributionof the aircraft fuel to the fire is moderateand limited to the initial minutesafter impact leaving afire limited by ventilationandfueledmainly by andaverageoffice building fuel loadredistributedina mannerdifficult to assess[14]. Empirical dataon ventilationlimited fires supportthe conclusionthataveragetemperatureswithin thecompartmentswould have remainedwithin therangeprovidedby theFEMA report[15] andcouldhave led to energy releaseratesof theorderof a gigawatt [14].NumericalmodellingusingFDS validatedthis orderof magnitudeby comparingplumetrajectorycalculationswith photographicimages[14].

Thehottesttemperaturesareexpectedto exist neartheregionof theopeningswherethereis sufficientventilation.It is very unlikely thatthetemperaturescouldhavebeenthathigh in theinterior of com-partmentsof sucha largeaspectratio (heightto width) andtheconsequentresistanceto ventilationitwould create(becauseof thecomplex multi-cell flow patternsin suchgeometries).Internaldamageof thecorestructurecouldhaveresultedin asignificantincreasein theoxidizersupplybut numericalcomputationshaveshown thatthis would havealsoresultedin significantexterior flames.Thevideofootageof theeventsshows darksmoke spewing from theopeningscausedby theinitial impactsforthewholedurationwhile thebuildingsremainederect.Thus,theauthorsfavour thehypothesisthatsuchenhancedventilationdid not exist [14].

Theexternalcolumnshadthreeof their facesopento atmospherethatin theabsenceof externalflam-ing is inconsistentwith high temperatures.Thereforeonemustconcludethatevenif therewereareasof high temperatureinsidethe building, it is quite likely that thecolumnsdid not heatsignificantly(evenif they hadlostfire protection).

The FEMA reportalso indicatesthat the despitethe structuraldamage,the remainingcolumnsre-tainedsignificantloadcarryingcapacity(asmostof theexternalstructurewasdesignedto resistveryhigh wind loadingand in the absenceof wind, would have provided considerableredundancy forgravity loads).Theutilisationfactorof for gravity loadingis reportedto bea very low 20%for exte-rior columnsanda moderate60%for thecorecolumns.Consideringthesearguments(which clearlywerenot availableto theauthorsat thetime of thepublicationof their paper),thescenarioof a largenumberof columnssuffering creepbuckling almostsimultaneouslyis not credible. Structurallyaswell, a low temperaturefire scenariomakesmoresense.This is becauseif the temperatureswereto rise to 800

�C over the whole compartmentsimultaneouslywith columnsthat had lost their fire

protection,thebuildingswould not have survivedfor aslong asthey eventuallydid. This paperwillshow that thereis a betterexplanationfor initiation of thecollapsewhich doesnot requirevery hightemperatures.

Anothernotablecauseof thecollapseis providedby Quintiereet. al. [16] which is almostidentical

to theonetentatively put forwardin theFEMA report[13]. Thispaperlooksatacompletelydifferentexplanationof thecollapse.They have concentratedon thebehaviour of thecompositefloor systemand the supportingtrusses.They correctly identify that the trusswould undergo large deflectionsdueto the effect of restrainedthermalexpansion. Beyond this point they conjecturethat the largedeflectionswould leadto tensilemembraneactionin theslabandtheresultanttensionwould leadtoconnectionfailuresandfloor collapse,thussettingoff a chainof progressivecollapse.This theoryisalsoimprobableasit reliesupona largenumberof connectionfailuresin averyshortspaceof timetosetoff thefloor collapseswith sufficient kinetic energy. Furthermore,Cardingtonexperimentshaveshown thatconnectionsremainundercompressionwhile thefiresburnandthetemperaturesincreaseasa resultof restrainedthermalexpansion(initially from steelandlater from concreteexpansion).They do snapin tension,but this occurson cooling,whenthefireshaveburnt out,which clearlywasnot thecasefor thesebuildings.Thediscussionof theattemptsto analysetheWTC collapsesindicatethat althoughtherehasbeenconsiderablerecentprogress,therearetoo many counter-intuitive andsubtlephenomenain the thermo-mechanicalresponseof large frame structuresto fires which arenot well understood,even in the profession.Therefore,quite predictably, therearesignificantandfundamentaldifferencesin theexplanationsof theWTC collapsesonoffer sofar.

The aim of this workAlthoughtheTwin Towersthemselveshadsustainedconsiderablestructuraldamage,WTC 7 whichhadnot, alsocollapsed(beingthefirst recordedcaseof thecollapseof sucha structureentirelybe-causeof fire). This poor responsefrom compositesteelframestructuresis not in accordancewiththe findingsof Cardington[1]. All of the conclusionsreachedin analysingthe Cardingtonexperi-mentswerereinforcedby considerabletheoretical/computationalanalysis,thusthis work choosestomake useof thesameprinciplesandanalysisthatprovidedsucha comprehensive explanationof theCardingtonteststo explain thecollapsesin theWTC complex.

Given the magnitudeof this task, this paperfocuseson WTC 1 & 2 and seeksa more plausibleexplanationof the collapseof thesestructures. The emphasiswill not be on comparative detailsbetweenthe computationsandactualevents,assuchan emphasiswill significantlycomplicatethetaskandperhapsalsoobfuscatethemainissue.For thispurpose,wehave limited our investigationtoa simple2D modelof thestructureandto assessa rangeof fire scenarios(from severeto mild) thatwill cover therangeof theplausiblefiresfor WTC 1 & 2. Any effectof damagewill beignored.Theaim of theinvestigationis not to predicttheexactsequenceof failureof thetwo Towersor to predictthefailuretimes(asin [16]. Themainpurposeof thiswork is to find out if thestructuralsystemof thetwo Towershadanunusualvulnerability to large fires. Thereforea hypotheticalfailuremechanismis proposed(basedon our understandingof the structureand its likely behaviour) and the modeltestedto seeif the mechanismwould actuallyoccur for a rangeof fires. It will hopefully be seenlater that the resultsproducea very crediblescenarioof collapse,which doesnot dependuponanygrossassumptionsaboutthefire or failureof connectionsor evenstructuraldamage.A cleanstabilityfailuremechanismis clearlyindicatedby asimplecomputationalanalysis.Not only this,theanalysisis entirelyconsistentwith thefundamentalprinciplesdevelopedduringtheCardingtonwork [2]. Thefollowing sectionsreportin reasonabledetailthevariousstepsof theanalysisundertaken

HYPOTHESIS

The work beganwith the hypothesisgraphicallyillustratedin Figures1 and2. Figure1 shows aframewith externalandinternalcolumnsconnectedby a long-spancompositetrussfloor system(theinternalcolumnsarepartof thebuilding coreandtheexternalcolumnsrepresentthecloselyspacedtubularcolumnsof theWTC Towers).Figure1 alsoshowstheclassicalbucklingcollapsemechanismof theexternalcolumnsunderoverloadsin absenceof any thermaleffects(assumingthat the truss-columnconnectionsarepins). For this mechanismto occurin themannershown, it is necessarythat

thecompositefloor systemhassufficient axial stiffnessto beableto provide a minimumrestrainttothecolumn[17]. This restraintstiffnessallowsusto representthewholesystemasacolumnlaterallysupportedby translationalsprings(alsoshown in Figure1). Giventhatsufficientstiffnessis availablein thesprings,theclassicalEulerbuckling mechanismwill beasshown in Figure1.

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InternalCORE

Typical composite floor

Typical Column in External TUBE

Figure1: Classicalbuckling of columnsin amulti-storey framewith sufficient lateralrestraint

Now considerthat for somereasonthe spring stiffnessesare reduced. One way this could occurwould beif therewasfire on oneor moreof thefloorsconnectingtheexternalandinternalcolumns.A patentlyobviousreasonfor this is thatif thefire burnt for longenough, it wouldeventuallysoftenthesteeltrussessupportingthefloors.Oncethesteelstiffness(modulus)reducessufficiently, theaxialstiffnessdemandfor lateralsupportto thecolumnswill nolongerbeavailable.Thismaterialsofteningeffect would have occurredto varyingdegreesdependinguponthefire andfire protectionappliedtothesteeltruss(assumingit hadnotbeendislodgedby theeventthatcausedthefire). Thereis howeveranothernotsoobviousreasonfor thelossof axial stiffnessthatdoesnotdependsomuchon thelevelof steelfire protection.Thiseffect is muchmoredamagingto thestabilityof theoverallstructurethanthe previous one. It comesfrom the very first two principlesof behaviour outlinedin the previoussectionandis essentiallya geometriceffect. Note that thefloor trusssystemis very long andquiteslenderandnot really designedfor in planeforces(membraneor axial compression).Initially thefloor systemmayexpand(for multiplefloor firesthermalexpansionwill dominateoverbowing asthe

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centreline

Columns pushed out as thermal expansion dominates

External column External column

ModeBuckling

New

centreline

Initial thermal expansion (and thermal bowing) Loss of stiffness in floors (by material softening andbuckling induced by restrained thermal expansion)

Figure2: Classicalbucklingof columnsin amulti-storey framewith reducedlateralrestraint

floor is heatedfrom topandbottom)resultingin thepushingoutof theexternalcolumns.Eventuallythe increasingmembranecompressionsfrom restrainedthermalexpansionandthehigh slendernessof thefloor will leadto bucklingof thefloor andfurtherincreasein deflections.Thesignificantchangein thegeometricshapeof thefloor systemwill leadto furtherreductionin axial capacity, leadingto arapidlossof lateralrestraintto thecolumn.If thefire hasaffectedseveralfloors,this lossof restraintwill allow amuchsmallerloadto buckletheexternalcolumnsin adifferentbucklingmodeasshownin Figure2.

A computationalandanalyticalinvestigationhasbeencarriedoutto testthehypothesisoutlinedaboveanddeterminepossiblecollapsemechanismsfor theWorld TradeCenterTowerswhensubjectedtoa largefire. This investigationignoresthestructuraldamageto theTowersasa resultof theaircraftimpactsandanalysesasimplemodelof theWTC Towersfor theeffectof fire alone.Thefire scenariosconsideredconsistof a setof credibletemperaturedistributionsdeducedby theauthors(from otherpublishedwork andtheirown investigations)andincludeawiderangeof maximumfire temperatures.Structuraleffectsof Singleandmultiplefloor firesareanalysed.

COMPUTATION AL MODEL

Thecomputationalmodelchosenis a typical 2D sliceof theTwin-Tower structure(asshown in theplanof thebuilding in Figure3). Theheightconsideredincludes12floorsaroundtheimpactlevel intheNorthTower.

Structural modelIt is assumedthat the columnsin the coreregion of the building would be relatively cool andthatcollapsewouldinitiateattheexternalcolumns.Furthermoreasthefloor systemis likely to belaterallyrestrainedby the floor systemin the core, thereforeall translationsof the compositetrussesandconcreteslabat thecoreendarerestrained(rotationremainsfree). This is a reasonableassumptionandsimplifiesthemodelconsiderably. Thecompositefloor is pinnedto theexternalcolumnsat theotherend.Figure10 shows a schematicof themodel.Typical dimensionsof all membersthatmakeupthemodel(thetrusscomponents,concretedeckslabandtheexternalcolumn)areshown in Figure4 [13]. The concretedeck slab is modelledto act compositelywith the truss. The trusstop andbottomchordweremadeof two anglesbackto back. An equivalentrectangularsectionis usedinthemodelbasedon Figure2.9of theFEMA Report[13]. Figure5 shows thefinite elementmeshofthemodel. The columnsaremodelledusing2D linear Timoshenko beam-columnelements,andsowerethe concreteslabandthe top andbottomchordsof the truss. The concreteslabelementsandthetopchordelementswererigidly connectedto modelthecompositebehaviour usingmultiplepointconstraints.The diagonalelementsof the trussweremodelledusing2D axial elements.Only onelinearaxial elementwasusedto modelonetrussdiagonal,to enabletheanalysistimesfor theverylargenumberof analysescarriedoutto berelativelysmall.As thiswasalargedisplacementnon-linearanalysis,the lack of sufficient nodeswithin the trussdiagonalswould enablethemto remainmuchstiffer thanactuallywouldhavebeenpossibleandwouldhaveover-estimatedthestructurescapacity.The top and bottom chordshowever had sufficient nodesthat could displaceinto the appropriatebuckling modes. The useof stiffer diagonalsalsoallowed the analysisto be carriedconsiderablyfurtherinto thepost-failureregime.Temperaturedependentmaterialpropertiesfor steel(stress-straincurvesandcoefficient of thermalexpansion)weretakenfrom Eurocode3 [18]. For thepost-elasticbehaviour of theconcreteslabaDrucker-Pragermodelwasused.Theanalyseswerecarriedoutusingthecommercialfinite elementsoftwarepackageABAQUS.

Loadinghasbeenappliedbasedon theapproximatenumbersmentionedin theFEMA Report[13].A total loadof approximately1300to 1400tonnesis assumedto be uniformly distributedover thefloors.TheFloor loadingareafor thesliceof structuremodelledis assumedto be2mwide(consistentwith thespacingof thedoublefloor trusses).This givesa loadof 8kN/m on thehorizontalelements.

CORE

A typical 2D Slice of the frame

Figure3: Planview of a typical2D sliceof thebuilding frameselectedfor analysis

40%of this loadon eightstoreys (representingthestoreys above themodel)is appliedto the top ofthe columns. Furthermore,asthe exterior columnsarespacedat approximately1m centres,whilethe trussesframing into themarespacedat 2m centres,two extremecasesareanalysed.In thefirstcaseit is assumedthatthecolumnin themodelhasanareaof cross-sectionequalto a singlecolumn(350mmsquarehollow sectionwith a platethicknessof 6mm). This is highly conservative asloadwill beredistributedto theneighbouringcolumnsby theactionof thespandrelbeamsconnectingthecolumns.In thesecondcasethecross-sectionareais doubledassumingfull sharingof loadbetweenneighbouringcolumns.

equivalent of (1.5 " x 1.5 ") equivalent of (1.5 " x 1.5 ")

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Rod diameter 1.09 "

60 ’

29 "

12 ’

40 "

The external column(square hollow section 14 " x 14 ", 0.25 " thick plate)(Either one or two used in models) Concrete deck thickness 4 "

The top chord The bottom chord

(Two trusses used in all models composite with 2m wide strip of concrete deck )

Figure4: Typicalmodeldetailsanddimensions(shown in feetandinches)

Fir escenariosThe fire scenariosdescribedin the introductionindicatea rapid onsetof a generalizedventilationcontrolledfire that reachesmoderatepeaktemperatures.The bestrepresentationof sucha fire willbe throughthe direct modellingof the heatexchangebetweengasandsolid phases.Nevertheless,for this particularscenariotheseexchangesarehighly uncertain,thusa gasphasetemperaturecurvethat incorporatesthe two main parametersof the problem(the heatingrateandthe maximumtem-perature)wasconsideredadequate.A generalisedexponentialcurve is chosento representthe firetime-temperaturerelationship,andis givenby:

T * t + � T0� * Tmax � T0 +,* 1 � e - at + (1)

Where,Tmax is maximumcompartmenttemperature,T0 is the initial or ambienttemperature,andais anarbitrary‘rate of heating’parameter. A typical fire curve usingEquation1 is shown in Figure7. The time variableis thusanartificial “time” that is generatedto provide a sensitivity analysistothecurrentcomputations.These“temperaturevs. time” curvesareintendedto representcolder/hotterfiresor differentlevelsof insulation.Neverthelesstheseartificial timesshouldnot beusedfor com-parisonwith the actualtime scalesof the eventsof September11th,2001. The authorsacceptthatissuingtemperaturetime curvesis inappropriateandthey shouldbesubstitutedby heat-fluxvs. timecurvescorrespondingto the specificfire. Thesetemperaturesare an arbitrary gasphasetempera-tureassignedto thecompartmentfrom which a heat-fluxis extractedvia a convective-radiativeheattransferboundarycondition.Thedifferentconvectiveandradiativeheatexchangesbetweensurfaces,

Figure5: Finite elementmeshof the2D computationalmodel

fire andcombustionproductsarenot taken into account.The only justificationfor this approachisthatby doinga parametricstudya significantrangeof heatfluxesandheat-fluxincreaseratescanbeachieved.Thesewill beexpectedto coverall thefire scenariosof interest.

The columnswereassumedto be restrictedto a maximumtemperatureof 400�C, rampedlinearly

up from ambientfor all theanalysesregardlessof themaximumcompartmenttemperatureassumed.In all theanalyseswherecollapseoccurred,thecolumnsdid not reach400

�C andthereforeretained

practically their full materialstrength. This is in view of the reasonsdiscussedearliersuggestingthat the external columnscould remainrelatively cool in a major fire. This is also a deliberatelyconservative assumptionto show that thefailuremechanismdoesnot rely uponthereducedcolumncapacity. Theopenwebjoist (or truss)temperatureswereassumedto bethesameasthecompartmenttemperatureasdescribedabove. For thecompositedeckslab,a 1D heattransferanalysiswascarriedout to determinethetemperaturedistributionover thedepthof theslab.

Thegasphasetemperatureis thenappliedfor a rangeof valuesof maximumtemperature(Tmax) andparametera. Theeffectof internalventilationdueto coredamageis addressedwith variousassumedspatialtemperaturedistributions(asshown in Figure6) alongthelengthof thecompositefloor. Thesetemperaturesdistributionsareappliedto fire scenariosencompassingone,two or threefloors. Thewholerangeof (over100)analysesareshown in Table1. In additiononefurtheranalysiswascarriedout for theparticularcaseof a tendingto infinity asa boundingcasecorrespondingto instantaneousattainmentof themaximumtemperature.Thiswasdoneusingfire scenario‘C’ for a 2-floorfire withamaximumtemperatureof 700

�C andthemodelwith strongercolumn.

Core

Scenario C Scenario B

Scenario A

T

Distance from external column1/2 span

Tmax

Tem

pera

ture

01/4 span

Figure6: Assumedtemperaturedistributionsalongthelengthof thefloor

ANALYSIS OF RESULTS

Figure8 shows anenvelopeof failureobtainedfrom all theanalysescarriedout for themodelwiththe assumptionof a single column. The lines themselves and the areato the right or above thelinesrepresentsthescenariosfor which thestructurecollapsed.For otherscenarios,it cameclosetofailure,howeveroverall collapsedid not occur. FromFigure8, it canbeseenthatevena singlefloorfire, if correspondingto scenariosA or B, collapsesthe structureat relatively modestcompartment

0

100

200

300

400

500

600

700

800

900

1000

0 600 1200 1800 2400 3000 3600

Tem

pera

ture

(C

)

.

Time (s)

Fire time-temperature curves

a=0.005a=0.004a=0.003a=0.001

Figure7: Temperature-timecurvesfor differentvaluesof a with Tmax � 1000�C, T0

� 20�C

Fir e Number of floors under fir e andScenario range of Maximum temperatures

Tempr. distr. a 1 2 3A 0.005 400

�C-1000

�C 400

�C-1000

�C 1000

�C

B 0.005 400�C-1000

�C 400

�C-1000

�C -

C 0.005 400�C-1000

�C 400

�C-1000

�C -

C 0.004 400�C-1000

�C 400

�C-1000

�C -

C 0.003 400�C-1000

�C 400

�C-1000

�C -

C 0.001 400�C-1000

�C 400

�C-1000

�C 400

�C-1000

�C

Table1: Thefull rangeof fire scenariosfor all analysescarriedout

temperaturesof 600�C and700

�C. At this point it is importantto statethattheword “collapse”does

meanactualfailureseenin thenumericalmodel,which is anacceleratingandirreversibledownwarddisplacementof thenodesconnectingthecolumnandthefire floors.Thecaseswherethefire floor(s)buckle but the structurestabilisesafter this event have not beencountedas a failure (suchas thesecondspecificanalysisof the modelwith doublecolumnand2 floor fire scenarioC with a=0.005andTmax � 700

�C, discussedbelow). Thebestindicatorof collapsehasbeentherateof changeof

displacement(pseudo-velocity)againsttimeplots(suchasFigure14).

It shouldbenotedthat thetemperaturesof thetrussareassumedto bethesameasthecompartmenttemperaturein the analysis.The columntemperaturesarerampedfrom ambientto 400

�C over the

whole periodof the fire. The maximumtemperatureof 400�C waschosenfor the columnson the

basisof the video imagesthat showed minimal external flaming (thereforein all of the analyseswherecollapseoccurredthe columntemperatureswerelower than400

�C). For the concreteslaba

total heatflux (convectionandradiation)boundarycondition is imposedand in-depthtemperaturedistributionscalculatedas a function of time using a one-dimensionalfinite elementheattransferanalysis(includingtheeffectof waterevaporation).

Fire scenarioC, which is perhapsthe mostrealisticone,doesnot show failure for any singlefloorscenario,however it shows collapseto occurat a very low temperatures(400

�C) whentwo or more

floorsareinvolved. At suchlow temperatures,thereis negligible lossof steelstrengthandthecauseof failureis entirelybecauseof instability createdby geometricchangesin thestructureasa resultofthethermalexpansion.As thesegeometricstructuralstability phenomenaareof thegreatestinteresthere,we will attemptto explain this in considerabledetail in discussinga few individualanalyses.It

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WTC TWIN-TOWERS COLLAPSE ANALYSIS (Summary)

Fire Scenario A (a=0.005)Fire Scenario B (a=0.005)Fire Scenario C (a=0.005)Fire Scenario C (a=0.001)

Figure8: Failureenvelopefrom all scenariosanalysedfor thesinglecolumnmodel

is beyondthescopeof this paperto discusseachof theover a hundredanalysescarriedout. This isnotnecessaryeitherasall theanalysesshow primarily thesamecollapsemechanism.Thereforethreeanalysesarechosenfor detaileddiscussionin thefollowing sub-sections.

Model with a singlecolumn, 2-floor fir e scenarioC, a � 0 0 001, Tmax � 600�C

This scenarioinvolvesa slowly developingfire (consistentwith someundamagedfire protectiononthetrusses)leadingto amaximumcompartmenttemperatureof justunder600

�C. Figure9 showsthe

time-temperaturecurvefor thefire usedfor thisanalysis.This temperatureis appliedontwo floorsasshown in Figure10. Thesefloorsareassumedto belocatedat thecentreof impactin WTC 1 (96thfloor, North Tower). Thestructuralloadappliedto thetop of thecolumnin themodel(node1156inFigure10) is consistentwith this location.

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WTC TWIN-TOWERS COLLAPSE ANALYSIS (C-F2-600-0.001)

Compartment temperature

Figure9: Compartmentfire evolution

Thisfire is appliedto two floors(asshown in Figure10). Figures11,12and13show thedisplacementvariationat variouslocationsin the model. Theseinclude the vertical deflectionsof the midspanof the main fire floor and the floors above andbelow it (node51018,61018and71018); columnhorizontalandverticaldisplacementsatthefire floor (node1072in Figure10);andcolumnhorizontaldisplacementstwo floors above andbelow the fire floor (nodes1060,1048,1072,1084and1096).Figure11 shows that the displacementof the main fire floor and the floor above it arepracticallyidentical until 1700 seconds,beyond which all the deflectionsincreasesignificantly and collapseoccurs.Thereis however a differencein the temperaturedistribution in thesetwo floors. Themainfire floor is heatedfrom bothtopandbottomandthereforeis subjectprimarily to ameantemperatureriseandnegligible thermalgradients.Thefloor above is only heatedfrom below andwill thereforebesubjectto both,a meantemperatureriseanda thermalgradientover thedepth.Thedeflectionofthefloor below themainfire floor is quite low until collapseinitiatesasit is heatedonly from above(thesteeltrussis notaffected).

Figure12showsthedisplacementsof thecolumnfor thefirst 1500secondsof analysis(pseudo-time,asthis is nota dynamicanalysis,andtime is simply linkedto thecompartmenttemperatureapplied).Until approximately1200secondsthecolumnexpandsoutwardsby about15mmat thejunctionwiththemainfire floor (wherethefloor is heatedfrom two firesaboveandbelow andthegreatestthermal

11036

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41036

51036

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101036

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41018

51018

61018

71018

81018

91018

101018

111018

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Middle nodes in concrete floor slab

Two floor fire

One floor fire

Three floor fire

Figure10: Locationof nodesatwhich outputsareplotted(hottestfloor nodesfor 2-floorfire in bold)

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Figure11: Floor deflections

expansionis generated).It mayalsobenoticedthatat thesametime this joint is alsomoving slowlyupwardbecauseof thermalexpansionof thecolumnitself. At 1200secondstheoutwardexpansionslowsdown andreverses.Therateof this reversedisplacementincreasesandshootspasttheoriginalpositionafter1300seconds.

Figure13showsall displacementsbegin to increasedramaticallyatapproximately1400seconds.Lat-eraldisplacementof over1m is achievedin thecolumnat themainfire floor level andapproximately700mmat thefloorsaboveandbelow this level. Thereis relatively little displacementin thenon-firefloors. Thereasonsfor this behaviour will becomeapparentin subsequentdiscussion.Theincreasedrateof downwarddisplacementof thecolumnandfloor midspanalsoin theFigure,indicatestheonsetof overallcollapse.

Figure 14 shows the rate of displacementwith time (velocity) of the column at the junction withthemain fire floor (at node1072,Figure10). Until roughly 1200seconds,the velocity is negative,indicatingoutwardmovementof thecolumn. Beyond this it becomespositive andconsiderableac-celerationis apparent.As anaside,it maybenotedthathadthis beena dynamicanalysisthis wouldcreateconsiderableinertial destabilisingforces.However, acounterargumentto this is thatthemassof thestructurecouldalsohavereducedthevelocity in thefirst place.Thereforeadynamicanalysisisdesirableto understandthis effect. Returningto theFigure14,a decelerationis noticedaround1500secondswhich lastsuntil roughly1700seconds,beyondwhich collapsetruly begins. The‘wiggles’in thecurve arean indicationof thedifficulty thenumericalalgorithmis experiencingin finding anequilibrium position. The lower curve shows the columnstartingto acceleratedownwardsbeyond1800seconds.Again thereasonfor this behaviour will becomeapparentwith furtheranalysis.

Clearly one of the purposesof the floors is to provide lateral supportto the columns. Figure 15shows the variation in the axial forcesin the main fire floor and two floors above andbelow this.

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Vertical displacement (at main fire floor)Lateral displacement (at main fire floor)

Figure12: Columndisplacementsat node1072

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Floor deflectionColumn vertical displacement

Column lateral displacement (main fire floor)Column lateral displacement (floor above)

Column lateral displacement (non-fire floor, above)Column lateral displacement (floor below)

Column lateral displacement (non-fire floor, below)

Figure13: Columnandfloor displacement

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Figure14: Columnandfloor pseudo-velocity

This Figureproducesconsiderablenew information that helpsus understandthe figuresdiscussedearlier. The forcesshown aresimply thehorizontalreactionprovidedby thesupportson right handsideof themodel(nodes41036,51036,61036,71036and81036in Figure10) asthis reactionmustequalthetotal axial forcein thefloors.Negativevaluesof thesereactionsindicatescompressionandpositive valuesindicatetensionin thefloor. We canseethat initially themainfire floor experiencesincreasingcompression(from restrainedthermalexpansion). However as thesefloor systemsareprimarily designedandoptimisedfor out of planeloads(for an18mspan!)they areveryslenderandcanonly resistverysmallamountof compressiveforcebeforebuckling. This is whatbeginsto occursatapproximately1000secondsandthelateralrestraintprovidedby thisfloor rapidlydisappears.It isinterestingto notethatthesteel-trusstemperature(sameascompartmenttemperature)at this stageisunder400

�C (seeFigure9) at which steelandconcreteretainover 90%of its strengthandstiffness.

Themid-spandeflectionof themainfloor at this time is approximately130mm(seeFigure13). Thisdeflectionleadsto considerablereductionin thegeometricstiffnessof thefloor to axial forcesto theextent thatno further lateralrestraintto thecolumncanbeprovidedat themainfire floor level. Thefloor thenmovesinto tensionatapproximately1300seconds.Correlatingthis to thepreviousFigures12 and14 of displacementandvelocity, it canbeseenthat thechangein signof themainfire flooraxial forcecoincideswith thechangein signof thecolumnlateraldisplacementat this level andtheaccelerationof thecolumn. Thefire floor above, heatedfrom below, alsoin compressionandunderlarge deflectionsuffers the samefate. The floor below, in contrastis heatedfrom above, thereforeonly theconcreteis heatedandtheaveragetemperaturesarevery low andsoarethedeflections(seeFigure11). This leadsto theaxial forcein this floor to betensioninitially (asit reactsto thecolumnmoving outward),which changesto compressionwhenthetwo floorsabove buckleinward. Thetopnon-firefloor doesthesame.Theresistanceofferedby thesefloorsaround1500secondscorrespondsto thedecelerationof thecolumnin Figure14. Howeverthefloor below themainfire floor eventuallybucklesaswell atapproximately1700secondsandmovesinto tension.Thetensileforcesin all three

fire floorsreducepracticallyto zeroby theendof theanalysis.

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Main fire floorFloor above

Non-fire floor, aboveFloor below

Non-fire floor, below

Figure15: Floor lateralreactionto column

Figure16 shows theaxial load-displacementpathsof thethreefire floorsthrougha rangeof columndisplacementfrom -15mmto 50 mm. The main fire floor shows a reasonablysteadycompressivemembranestiffnessuntil it buckles. Beyond this point the pathis not particularlymeaningfultherearetensilemembraneforcesin the floor which may contribute to the inward movementbut arenotentirelyresponsiblefor it. Therelativelycoolfloor below themainfire floor initially hastensileforces(asseenin Figure15),howeverassoonasthelateralrestraintstiffnessof thetwo floorsabove is lost,this floor resiststhecolumncollapseandbucklesat amuchhigherload(asthemidspandeflectionofthis floor is low thereforetheaxial restraintstiffnessis muchhigher).

Figure17 shows anexplicit calculationof thestiffnessof themainfire floor andthefloor above, bydividing the axial force incrementin a time interval with the lateraldisplacementincrement. ThisFigureshows that theaxial restraintstiffnessof themain fire floor disappearsafter 1000seconds.Itreappearslater, but this is really tensilestiffnessto supportthefloor loadsin tensilemembraneaction,which also contributesto destabilisingthe column. The floor above initially reducesin stiffnessas most of the lateral supportis being provided by the main fire floor (due to the largestcolumndisplacementsat thatlevel). Whenthemainfire floor stiffnessdisappears,thereis a largepeakin thestiffnessof thefloor above asit takes-over thecolumnlateralsupportfrom themainfire floor. Thisleadsto buckling of the floor above aswell andits lateralrestraintstiffnessdisappearsaswell andthenreappearsin thetensilemembraneform.

Thediscussionaboveprovidesa completedescriptionof how thestructurecollapses.This is exactlyconsistentwith theoriginal hypothesis.Figure18 shows themodelcollapsingexactly in themannerpredictedby thefiguresin thehypothesis.

Model with doublecolumn, 2-floor fir e scenarioC, a � 0 0 005, Tmax � 700�C

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Figure16: Floor lateralreactionagainstcolumnlateraldisplacement

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Main fire floorFloor above

Figure17: Floor lateralstiffness

Figure18: Modelshowing collapse,fire scenarioC, Tmax � 600�C anda � 0 0 001for fire on2 floors

Thecasediscussedpreviously assumedthata singleoutercolumnwassupportingthe loadof a 2mwide floor strip. We now considerthe modelwherethis load is assumedto be supportedby twocolumns. As mentionedearlier, this is a betterrepresentationof the structureandthe analysesarethereforeof greaterrelevance.Thefundamentalcollapsemechanismdoesnot however change.Thefailure envelopefor the two columnmodel is presentedin Figure19. It may be noticednow thatno collapseoccursfor any of the 1-floor fire scenarios.However collapsestill occursfor 2-floorfire scenariosbut at relatively highertemperatures(700

�C andover). However for the3-floor fires,

collapsestill occursat temperaturesaslow as500�C, evenwith scenarioC.

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WTC TWIN-TOWERS COLLAPSE ANALYSIS (Summary)

Fire Scenario A (a=0.005)Fire Scenario B (a=0.005)Fire Scenario C (a=0.005)

Figure19: Failureenvelopefrom all scenariosanalysedfor thedoublecolumnmodel

Theresultsof this modelarepresentedhereasit shows a contrastto theunstablebehaviour seeninthepreviousanalysis.Figurereference20 shows thedeflectionsof thefire floorsandcolumnlateralandverticaldisplacementsat its connectionwith thehottestfire floor (mainfloor). Althoughthefloordeflectionscontinueto increase,the columnlateraldisplacementstabilises,after the expansionandreversalasnoticesin theanalysisdiscussedpreviously. Furthermore,thecolumncontinuesto displaceupward(positive)right to theendof theanalysis,suggestingthestructureis stable.Thesamegeneralconclusionis offeredwhenthefloor lateralreactionforcesareexamined(seeFigure21). Thecolumnlateralpseudo-velocity (Figure22) is seento becomenearlyzeroafter about1500seconds,againsuggestingthat thestructureremainsstable.Finally, Figure23 shows thefloor lateralreactionplotsagainstcolumn lateraldisplacement.This plot shows againthat the floors expandlaterally with aconstantstiffnessat first andthenbuckleandbegin to actin tensionasseenbefore.

Model with doublecolumn, 3-floor fir e scenarioC, a � 0 0 005, Tmax � 500�C

Thisfinal discussionfocusesona 3-floorfire casewith amoderatetemperature,whichclearlyshowstheonsetof collapse.Figuresreference24and25show thedeflectionsof thefire floorsandassociatedcolumnlateraldisplacements.All displacementsareseento beacceleratingto theendof theanalysis,suggestingimpendinginstability. Figure 26 clearly shows this as columnsbeginning to displace

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Upper floor deflectionMain floor deflection

Lower floor deflectionColumn vertical displacement

Column lateral displacement (main floor)

Figure20: Floor andColumndisplacements

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Non-fire floor, below


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Figure22: Pseudo-velocity

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downwardsafter about3000 seconds.The floor lateral reactionforces(seeFigure 27) have alsocomecloseto zeroat this time. The variationsin floor reactionshave a direct link to the pseudovelocity plots (Figures29 and28). This canbeseenby matchingall the largereversalsin velocitieswith variousfloorsmoving from compressionto tensionin Figure27. This providesan insight intotheprogressive natureof theeventualcollapse.Figure29 shows thatafterabout2800secondsbothcolumnsbegin to acceleratedownwards. Figure30 providesan insight into floor stiffnessvariationandtheinteractionstakingplacebetweenfloorsandcolumns.

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Bottom fire floor

Figure24: Floor deflections

DISCUSSION

If themodelsabove areconsidereda reasonablestructuralequivalentof theWTC Towers,it is clearthattheoverall structuralstability failureof thecolumnswouldoccurevenat low temperatures.Thisis primarily becauseof thedegradationof lateralsupportfrom thecompositetrussfloor systemwhichwastooslenderto continueto supplythecolumnlateralsupportdemand.Themaximumtemperatureof thecolumnsthemselveshasbeenlimited to 400

�C (oftennotreachedasfailureoccurredbeforethis

temperatureis attained).At suchtemperaturesthecolumnsareexpectedto retainover 90%of theiraxial stiffness,which would besufficient to continuesupportingthe loads(giventhe low utilisationfactorsof columnaxial capacitywhengravity loadresistanceis theonly demand).However thelossof lateralsupportfrom oneor moreof thecompositetrussfloorswill clearly reducethegravity loadcapacity(Eulerbuckling loadis reducedto 25%for thelossof justonefloor). Thispotentiallypointsto themostlikely failureinitiation mechanism(notethereis norequirementof connectionfailuresforthismodeof failureto occur- clearlytheconnectionswill fail oncethecollapseis initiated,but this is‘effect’ not ‘cause’). To elaboratefurther, themainmechanismsby which the lossof lateralsupportoccursarediscussedbelow:

1. Themostobviousmeansby whichthelateralsupportto thecolumnsmaydegradewouldbethe

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At top fire floorAt hottest upper floorAt hottest lower floor

At bottom fire floor

Figure25: Columnlateraldisplacements

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Figure26: Columnverticaldisplacements

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Column at hottest upper floorColumn at hottest lower floor

Figure28: Lateralpseudo-velocityof columnsat hottestfloors

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Hottest upper floor midspanHottest Lower floor midspanColumn at hottest upper floorColumn at hottest lower floor

Figure29: Verticalpseudo-velocityof columnsandhottestfloor midspans

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softeningof thesteeltrussmembersfrom heatingresultingin thelossof stiffnessandstrength.This softeningwould be particularlysevere if it is assumedthat the trussmembershad losta significantproportionof their fire protectionat impact (as all the trusscomponentswereratherlight andwould heatup very quickly without protection). This caseis coveredby theanalyseswhich assumerapidriseof compartmenttemperature(valuesof a=0.005).Evenwiththestrongertwo-columnmodels(seeFigure19)collapseoccursat trusstemperaturesof 500

�C

for all three-floorfire scenariosandat 700�C for two-floorscenariosA andB

2. Thesecondandevenmoresignificanteffect of fire on structuralmembersis thatof restrainedthermalexpansion(thermalbowing is of lower importancefor multiplefloor fires).Thesignif-icanceof this effect is oftenunder-estimatedby mostfire professionals.In conventionalsteelframecompositestructures(suchasCardington),this effect is generallybeneficial.It reducesthe internal forcesthat a structuralmemberexperienceswhenheated(while restrainedat theends)asout-of-planedisplacementslimit the mechanicalstressesandconsequentdamage.Italsohelpsto createa favourabledeformedshapeof thecompositefloor systemthatallows thestructureto effectively deploy secondaryloadcarryingmechanismsof membranetension(andcompression)whenthebendingcapacityis reducedor exhaustedby theeffectsof heating.Thissamebehaviour howevercanhaveaseriouslydetrimentaleffectonthelateralsupportprovidingcapacityof thefloor system.If thefloor systemis very slender(aswasthecasefor theWTCTowers),only asmallamountof heatingwill make it buckleoutof plane(becauseof restrainedthermalexpansion).This is clearly seenin the analysispresentedabove. This leadsto a de-gradedaxial stiffness(and thus the capacityto provide lateralsupport)due to the deformedgeometry. For slenderfloor systemsthis occursmuchearlierthanthematerialsofteningeffect.It is alsoquitepossible,if this effect is largeenough,thatthelateralsupportcapacitycouldbelostwell beforethematerialpropertiesaresignificantlyreduced.For many of thecasesstudiedabove, collapseis initiatedbeforethesteeltrussreaches500

�C. This suggeststhatevenif the

fire protectionto the steeltrusseshadsurvived the impact, the failure temperaturesrequiredcouldhavebeenattained

3. Thethird andfurtherdestabilisingmechanismis thatof theeffect of thermalexpansionof thefloor systemagainstthecolumns.Theinternalboundaries(core)wouldhavebeenconsiderablystiffer becauseof the continuity of the structurebeyond the compositetrussfloor systemandbecauseof the considerablylower temperaturein the coreregion (asdiscussedearlier). Theexternalboundariesby contrastweremuchsofter(columnsin flexure/shearagainstthefloor inmembranecompression).Thereforeto begin with, the columnswould displacelaterally out-ward.Meanwhileasthecolumnsarealsoheatedtheoverallcompressionforcesin thecolumnsarealsoincreased.Thecombinationof theaddedeccentricityfrom theoutwarddisplacementandthe increasedcompressionsaddto thedestabilisationeffect on thecolumn. However thisis essentiallya stableandself limiting configuration,becausetheadditionaleccentricityis es-sentiallyacompatibilityphenomenonandthecolumnsdisplaceto accommodatetheexpandingfloor (which is alwaysin compressionin thissituationthereforetheconnectionsarenotat risk).In noneof theanalysescarriedout failureoccurredby outwardbuckling of thecolumns

4. It is clearthatthethermallyexpandingtrussfloor systeminitially pushesthecolumnout (whileit is sufficiently stiff in membranebehaviour). As the membranestiffnessreduces(througheithergeometricor materialeffects),theoutwardmovementof thecolumnis arrestedandthestoredstrainenergy in thecolumnmakesit recoil with an increasingrateof inwarddisplace-mentpushingthesoftenedfloor systembackin. Thisis seenin thepseudo-velocityplotsearlier.A point to notehereis that thereis potentiallya dynamicmagnificationeffect here(not con-sideredin this analysis)which couldbeanotherpossiblefactorcontributing to the instability.

Thecolumneventuallyover-shootstheoriginal positionsignificantlyandtheeccentricitythuscausedis quitedifferentfrom therelatively stableconfigurationwhenthefloorswerepushingthecolumnsout. Thefloor meanwhilehasbeenpushedbeyondits originalpositionanddeflec-tionsthereforeincreaseto theextentthatthemembranecompressionnow changesto membranetension,addingfurtherto thedestabilisingforceonthecolumn,potentiallytriggeringtheinsta-bility andprogressivecollapse(asseenin Figure28)

Thecumulativeeffectof theabove-mentionedphenomenaprovideshighly compellingreasonsfor thecollapseof theWTC Towers.Thestudydoesnot involve any uncertainassumptionswith regardstothe fire (asa large numberof scenarioshave beenincluded,someproducingcollapsesat relativelylow temperatures).Furthermore,no grossassumptionsof aboutstructuralbehaviour havebeenmade(suchasthepersistentclaimsaboutconnectionfailuresandresultant‘pancaking’of floors,withoutany supportingquantitative analysis).The analysisis very simpleandshouldbe reproduciblewithmostproprietaryfinite elementstructuralmechanicssoftwarecapableof modellingnon-linearlargedisplacementbehaviour.

CONCLUSIONS

Thechief conclusionsare:

1. Theanalysispresentedpointsto a compellingfire inducedcollapsemechanismratheruniqueto thetypeof structurethattheWTC Twin-Towersrepresented

2. This analysisalsoshows thatthecollapseis initiatedprincipally by a stability mechanismasaresultof geometrychangesin thestructurecausedby thermalexpansioneffects

3. Furthermoreit is quitepossiblethatthegeometricchangesrequiredto precipitatecollapsecouldresultfrom very low temperaturesnot high enoughto inducesignificantreductionin themate-rial properties

4. It canthereforebeprovisionallyconcludedthatthesebuildingscouldhavecollapsedasaresultof amajorfire event.This is of courseassumingthatany of theactivefire suppressionsystemswouldeitherfail orbeunableto controlthedevelopmentof thefire. Thisis anormalassumptionwhendesigningfire protectionfor buildings

To achieveafirmerconclusiona3D analysiswouldbenecessary, alsowith alargerangeof reasonablyrealisticfire scenariosascarriedoutfor thisanalysis.Thecurrentanalysisclearlyassumesthatthereisno supportforthcomingfrom thedirectionalorthogonalto theplaneof theanalysis.It is very likely,particularly for low temperaturesthat this additionalsupportwill delay failure and the the failingcolumn will unloadand the load will redistribute. This however could not carry on ad-infinitumandcollapsewould eventuallyoccur. Furthermoredynamiceffectshave not beenconsideredin thisanalysis,which could eitheraddto the destabilisingforcesor delaythemor both. A 3D dynamicanalysisof this problemis thenext logical stepto take this investigationfurther.

References

[1] Behaviour of steel framedstructuresunderfire conditions. TechnicalReportMain Report,DETR-PITProject,Schoolof Civil andEnvironmentalEngineering,Universityof Edinburgh,2000.availableat www.civ.ed.ac.uk/research/fire/technicalreports.html.

[2] A.S.Usmani,J.M.Rotter, S.Lamont,A.M.Sanad,andM.Gillie. Fundamentalprinciplesof struc-turalbehaviour underthermaleffects.Fire Safety Journal, 36:721–744,2001.

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