Mechanics Of Collapse Of WTC Towers Clari Ed By Recent Column Buckling .

4Journal of StructuresZone of highenergy dissipation6Journal of StructuresFigure 4: Compression flange folds of SR 35 in a state of largedisplacement.𝜃L La)(a)(b)Figure 3: Photographs of SR 42 specimen loaded well into theFigure 7: Experimental and idealized plastic-hinged column at maximum displacement.postmaximum range.b)Figure 4: Compression flange folds of SR 35 in a state of largedisplacement.c)[9], a referenceloadwhich simply validates the work ofspecimen SR 42 in the testing machine elementsin postmaximumafter test. The plastic hinge rotation associated with suchmany outstanding researchers over recent decades. But evena displacement can be estimated from the formula 𝜃 Continuedloading led to inlocalbucklingof theFigure 1: Experiments by conditions.Korol andSivakumaran:specimenduringc) final(𝐿 /𝐿), where𝜃 is the angle through which thethea)-b)inelasticresponserange, the importanceof thethebuck- test,𝜋 2sinandflanges on the compression side of the lingplasticFigure4 to boundary conditions for two segments are rotated, which represents the localizedplate hinge.coefficientwith respectdeformation shape of the ).(see Figure 7).rotation at mid-height associated with Δclassification ofsidesectionsinto design categories is dependentshows the folds that form on the compressionof the 1 maxThe plastic hinge rotations for all seven specimens wereboth geometrical and material properties [10].plastic hinge associated with specimenonSR35. All seven testcalculated and are tabulated in Column 8 of Table 1. Note thatThe establishment of the amount of energy associatedspecimensexhibited this compressionwithflangelocalbucklingFigure 3: Photographs of SR 42 specimen loadedwell into thethe rotations are expressed in radians and the correspondingconverting a straight column into one that has a kink atpostmaximum range.angles are given within brackets. As shown in Column relafollowed by folding. Such contact of the insides of the flangesSimplifications of OriginalAnalysisDuetoLackofDataof Table 1, the experimental hinge rotations, that is, thetionship.This wasachievedvia instrumentation describedwas noted for all specimens and is showninFigure5whichscissors angle just at the point of flange clashing, are in theearlier linking the loads to averaged LVDT readings ofwas assembled after completion of the experimentalprogram.range of 150–160 imen SR 42 in the testing machine in postmaximum ata,theoriginalanalysisofWTCcollapse [4, 3]Table 1 showthe peakloads and thepeak onceitwasconditions. Continued loading led to local buckling of therespectively,observedeach test. The energy absorbedevidentthat4 flanges above and belowthe hingewerein inflanges on the compressionside of the plastic during an entire displacement range is, of course, the area3. Discussion and Conclusionscontactwithtestandbeyondshows the folds that form on the compressionside ofthe one another. Had we continuedunder thethecurveis summarized in Column 5 of Table 1.Figure 5: Specimenstest completion.The energyafterdissipatedin a plastic hinge undergoing a plasticplasticassociatedductilitywith specimen ofSR 35.All seventesta)hingeperfectsteel,withno fracture,Note thatwouldthe energydissipated by the axially loaded columnscontactofthe flanges,the residual resistancecontinue𝜃 is given as 𝑀𝑝 𝜃, where 𝑀𝑝 is plastic momentspecimens exhibited this compression flange evenlocal bucklingrangedfromahighenergywith such severe folding, since another fold above orof 8.19 kN m to a low energy of rotationresistance based on the assumption that the cross-section6.78 kN m.The otheruppertinentin Figure 6 andb) constancyof thebendingmoment in the plastichingetoinformation180 notedrotation,followedby folding. Such contactof the insidesof the flangesbelowthe original can occur, as notedcrushloadtestson exhibited by the columns will indeed reach the plastic moment prior to flange localis theinaverageresistanceswas noted for all specimens and is shown in Figure 5 whichbuckling.Since bendingtheselengthof511mm,reacheda maximumloadin of124experimentskN, whichis about c)rigidsupportofcolumnends.was assembled after completion of the experimental program.minoraxis, we needofto 15.1computecorrespondingDataaugmentedfrom the sevenenergyplots, and as summarizedthe s plasticevenIn each test the experiment was terminatedonceshownit wasa rise of load resistance with,forthesectionshown in Figure 1. Forsectionmodulus,𝑍suggestsmomentthat as the slendernessratioincreases,average𝑝Withthese assumptions andfor the plastic bendingbasedon t mid-heighta consequentincludesuch decreases,a clashevident that flanges above and below the absorption.hinge were inHowever, attempting to loadan idealizedH-shape that andneglectsthe corner radius effects,of resistancethat is, reducing from 20 kNcontactofwithbendingone another. Hadcontinued thetest beyondthe crossseveredrop-offin loadtheresistanceas energy,theuppercrosscomputedhead using theminor-axisplasticsectionmodulus,of (withflange-on-flangewas deemedto be(SRbeyondthescope33)to 12.7(SRplanar),42).ofOf tionremainingit wasfoundthattheFigure5: Specimensaftertest kNcompletion.22contact of the flanges, the residual resistance wouldcontinue 𝑡𝑤 (𝑑is 2𝑡is 13,162 mm3 .expression𝑍𝑝 1/4[𝑏thatbe dissipationmade in this ofregard, butit does suggestthatmovementprogressed.It is evidenta𝑓 )],pre𝑓 2𝑡𝑓 therereassessingthe plastic hinge model forcannotenergywith such severebyfolding,since anotheroffoldtheabove columnsorpostbucklingresistance woulddecreaseas a givenSincebytheaboutyieldstresson aof0.2%offset was determinedWd bout1/8ofthecipitousloss columnof strengthan basedordermagnitudeaxiallyloadedcolumns. of the firstsection increases in length, inferring the importance ofto be 58 MPa, we compute the fully plastic moment aboutbelow the original can occur, as noted in crush tests oncomparedwiththevaluesnotedin theslendernessratioin assessingtheenergydissipationpotentialthe minor strengthaxis0.763kN m.We thendetermine thelengthof 511 mm,a maximumloadof 124 kN,kineticenergyM[8].v02Continued/2 of themassM imumbeingtheto beimpactsquare hollowsectionstesting impactingmight havediagramsinFigureandTable eirenergyindissipationthe largeplastic 𝑏hingemultiplying by 𝜃occurred at an axial displacement of 15.1 mm. This was𝑓 /𝑡𝑓byratioseven shown a rise of load resistance with augmented energylengths.for each of -heightandaconsequentabsorption. However, attempting to include such a clashThe final lengths of the test specimens were derived basedresults are given in Column 9 of Table 1. Column 10 ofsevereduringdrop-offtestingin 2loadofresistanceas specimens.the1 upper crossheadof flange-on-flange was deemed to be beyondthe scopeaxiallycompressedunsupportedflangesbuckle elasticallyat with theputerdata energyvaluesimpactedfloorkineticenergyto progressed.MSR33,v1 /2which isgiven(1 ovementItareevidenttherev0is1./2.a prereassessing theplastic reduceshingeenergydissipationof the8originalmuchthando thosethathaveForexample,specimenhadin ergyvaluesbasedconnectingon the hinge model. It isp model forthecipitous loss of strength whichby aboutanoriginalorder lengthof magnitudeaxially loaded columns.had anof 549 mm was axially loadedevident that regardless of the slenderness ratio, the ratio ofdropsto v1 v0 7/8 0.935 v0 .compared with the maximumstrengthdisplacementvalues notedfor a maximumof in422themm, resulting in a finalexperimental amount of energy absorbed by an H-columnExperimental Results of H-Shapes Subject to Axial Loading.Figure 6 shows the load-displacement plots from the computer data generated during testing of all seven specimens.For example, the specimen SR33, which had an original/𝑡𝑓 Inratiosdiagrams in Figure 6 andin Table1. Fortolarge𝑏𝑓mm.lengthof specimenof 127thiscase the crushtest energyunder pure dissipationaxial compression is three to four times greaterBy directly applying Korol and Sivakumaran’sresults[5]WTCtowers,thewas conductedthe displacementthan what the plastic hinge model analysis predicts.this result is to be expectedsince ituntilis wellknown thatinto a oximately77%ofItmustbe acknowledged that the plastic hinge ticallyatcalculated by the plastic hinge mechanism [7, Sec.8.6]the(accordingtoFigurethe7 pecimen SR35calculations employed a yield stress value of 58 MPa, whichmuch lower stresses than dothoseheightconnecting7 that have22would have to increase by 3.5-times, i.e., M v1 /2 (1 16 )M v0 /2, which gives v1 0.750 v0 . Obviously, this updated estimate again indicates a continuing collapse. In no way the energy dissipation inthe columns of one floor could be large enough to exceed M v02 /2, which would be necessary to arrestthe gravity-driven collapse.After the crushing front advances by about ten floors, the collapsing mass grows significantly andthe kinetic energy of the falling mass dwarfs the energy dissipated by the columns. It then ceasesto matter whether or not the dissipation by plastic buckling is tripled. Therefore, the calculationsof the overall duration of collapse, of the velocity of expelling air and debris shedding, and of theimpact comminution of concrete slabs into particles, would not change beyond the range of error inthe observations made.Non-Rigid Restraints at Column EndsThe Korol and Sivakumaran’s columns developed no plastic hinges at the ends. Their end supportshad a flat free contact with the loading platens rather than perfect restraint. Beginning with a certain2

FloorFloorFloorSpandrelplateColumnsa)b)Fig. 2c)Figure 2: Deformation of the perimeter columns of WTC towers: a) geometry of the prefabricatedunit of perimeter columns, b) deformation of columns with fully restrained ends, c) deformation ofcolumns with elastically restrained ends.small deflection without end rotations, the column ends pivoted freely about the end of one flange(Fig. 1). This complicates comparisons with the WTC columns.The perimeter columns of the WTC towers were fabricated in units of three-story high. Theyconsisted of three column sections and three spandrel plates (Fig. 2a). For each story, the rotationof the two ends of each column was restrained by the spandrel plates. The spandrel plates must havedeformed elastically and plastically, rotating together with the column ends (Fig. 2c). Thereforethe plastic hinges at the ends of WTC columns must have dissipated much less energy than the midheight hinge. This would make the estimate of energy dissipation per column much smaller than thatcalculated for a column with fixed ends (Fig. 2b).Meanwhile, Korol and Sivakumaran’s experiments indicated that, at the plastic hinge location,the columns experienced large plastic deformation on the tensile flange and local buckling on thecompressive flange (Fig. 1). These local mechanisms make significant contributions to the totalenergy dissipation. However, for columns in the WTC towers, the two ends are not fully restrainedand therefore the energy dissipation due to plastic deformation and local flange buckling at these twoends would be smaller than that at the mid-span. Therefore, we can conclude that, for columns inthe WTC towers, the increase of the energy dissipation relative to the prediction by the plastic hingemodel would be much smaller than that observed in Korol and Sivakumaran’s experiments.Limited Ductility and Fracture of SteelTo get a conservative estimate of the maximum possible dissipation, the ductility limitation andfracture of steel were neglected in previous studies [4, 3]. In reality, numerous column fractures werelikely to occur, especially because a high rate of deformation promotes the fracture of steel. Thefractures during WTC collapse, which greatly reduced energy dissipation, have been documented byphotos and videos showing many flying fragments of columns.3

The fracturing of columns must have been particularly intense in the columns of lower stories.They consisted of high strength steel (with the yield strength of 690 MPa), which is more brittle andmuch more prone to fracture, especially at high rate.Calibration Based on Korol and Sivakumaran’s Tests and Video Record of CollapseThe uncertainty in the estimation of the energy dissipation by column failures, by air and mass ejectionand by comminution of concrete slabs, was recognized in the previous analysis of WTC collapse [2]. Asensitivity analysis was performed, in which plausible ranges of these dissipation terms were considered.For columns, a range of 20% of the mean energy dissipation capacity was used (although, in viewof Korol and Sivakumaran’s tests, it should have been broader). For air and mass ejection, a range of 50% of the mean energy dissipation capacity was considered.The calculations showed that these variations make little difference in the predictions of the totalcollapse duration, as well as the crush front propagation and concrete slab comminution. A largervariation of the energy dissipation capacity of columns (i.e., more than 50% increase) was recognizedto cause noticeable deviations from the video record of collapse during approximately the first twoseconds (see Fig. 6 of [3]). Yet the match of the seismically recorded duration of collapse would barelybe affected.Based on the aforementioned discussion, the increase of energy dissipation in columns indicated byKorol and Sivakumaran’s test data does not make an appreciable difference in the failure analysis ofprogressive failure of WTC columns. It makes an appreciable difference only for matching the videorecord of the first two seconds of collapse. Therefore, the proper way of using these data together withthis video record is to exploit them for calibrating the energy dissipation per column, restricted, ofcourse, to the realistic range of uncertainties in the material and structure properties.A calibration of this kind has already been done in the previous study [2], which showed that, forthe upper stories, the energy dissipation capacity of columns was about 2/3 of the value predictedby the simple plastic three-hinge model with perfectly rigid end constraints. The 2/3 reduction isnot unreasonable if we consider the decrease in energy dissipation due to the flexible end restraints,material fracture, and possible multi-story buckling [2]. This decrease can greatly offset the increaseof the energy dissipation due to local plastic deformation and local buckling at the hinges. Anyway,note that by using the calibrated energy dissipation capacity of columns, the model was able to predictcorrectly all the other observations such as the seismically documented collapse duration, the particlesize distribution of fragmented concrete slabs; the wide spread of the fine dust around the tower; theloud booms heard during the collapse; and the fast expansion of dust clouds during collapse. Thismultitude of data matching serves as a strong validation of the overall model.ConclusionThe experiments of Korol and Sivakumaran help in clarifying the mechanics of energy dissipation inthe columns of WTC and in reducing the previously stated range of uncertainties of analysis. Theyindicate that if the column ends were rigidly supported and if the ductility of steel was unlimited, thenthe simple plastic three-hinge mechanism with constant bending moments [7, Sec.8.2], of the type usedfor small-deflection buckling, would have dissipated about 3.5-times as much energy than consideredin previous studies.But calibration by matching of the video record of initial collapse implies that this energy musthave been reduced to about 2/3 of the energy predicted by the three-hinge model. This estimated 2/3reduction must have been caused by the fracturing of steel and by the flexibility of spandrel beamswhich reduced the rotations of the plastic hinges at column ends. With this update of input data,all the observed features of the WTC collapse remain to be closely matched by the gravity-drivenmechanics of progressive collapse.4

References[1] Bažant, Z. P., and Le, J.-L. (2008). “Closure to “Mechanics of Progressive Collapse: Learning from WorldTrade Center and Building Demolitions” by Zdeněk P. Bažant and Mathieu Verdure”, J. Eng. Mech.,ASCE, 134, No. 10, 917-923.[2] Bažant, Z. P., Le, J.-L., Greening, F. R., and Benson, D. B. (2008). “What did and did not cause collapseof WTC twin towers in New York”, Journal of Engineering Mechanics, ASCE, 134, No. 10, 892-906.[3] Bažant, Z. P., and Verdure, M. (2007). “Mechanics of progressive collapse: Learning from World TradeCenter and building demolitions.” J. Eng. Mech., ASCE 133, pp. 308–319.[4] Bažant, Z. P., and Zhou, Y. (2002). “Why did the World Trade Center collapse?—Simple analysis.” J.Eng. Mech., ASCE 128 (No. 1), 2–6; with Addendum, March (No. 3), 369–370 (submitted Sept. 13, 2001,revised Oct. 5, 2001).[5] Korol, R. M., and Sivakumaran, K. S. (2014). “Reassessing the plastic hinge model for energy dissipationof axially loaded columns.” J. Struct., Vol. 2014, Article ID 795257.[6] Le, J.-L., and Bažant, Z. P. (2011). “Why the observed motion history of World Trade Center towers Issmooth?”, J. Eng. Mech., ASCE, 137, No. 1, 82-84.[7] Bažant, Z.P., and Cedolin, L. (1991). Stability of Structures: Elastic, Inelastic, Fracture and DamageTheories, Oxford University Press, New York; 2nd. ed. Dover Publications, New York 2003 (1011 pp. xxiv pp.); 3rd ed. World Scientific Publishing, Singapore–New Jersey–London 2010.5

Mechanics of Collapse of WTC Towers Clari ed by Recent Column Buckling Tests of Korol and Sivakumaran Jia-Liang Le1, and Zden ek P. Ba zant2 Abstract: The previously formulated model of the gravity-driven collapse of the twin towers of the World Trade Center on 9/11/2001 was shown to match all the existing observations, including the video