Mechanics of Collapse of WTC Towers Clari¯ed by RecentColumn Buckling Tests of Korol and Sivakumaran

Jia-Liang Le

Department of Civil, Environmental, andGeo- Engineering, University of Minnesota

500 Pillsbury Dr. S.E. MinneapolisMN 55455, USAjle@umn.edu

Zden!ek P. Ba!zant*

Department of Civil and EnvironmentalEngineering, Northwestern University

2145 Sheridan Rd. EvanstonIL 60201, USA

z-bazant@northwestern.edu

Received 2 October 2016Accepted 11 November 2016Published 22 December 2016

The previously formulated model of the gravity-driven collapse of the twin towers of the WorldTrade Center (WTC) on September 11, 2011 was shown to match all the existing observations,including the video record of the crush-down motion of the top part of tower during the ¯rst fewseconds, the seismically recorded duration of collapse, the size distribution of particles caused byimpact comminution of concrete °oor slabs, the loud booms due to near-sonic lateral ejectionvelocity of air and dust, and precedence of the crush-down collapse mode before the crush-up.Nevertheless, di®erent degrees of ductility, fracturing and end support °exibility of WTCcolumns could lead to an equally good match of these observations and remained uncertain, dueto lack of test data. Recently, Korol and Sivakumaran reported valuable experiments that allowclarifying this uncertainty. They revealed that, under the simplifying assumptions of rigid endsupports and unlimited ductility (or no fracturing) of unheated columns, the energy dissipationof the WTC columns would have been at maximum 3.5-times as large as that calculated by theplastic hinge mechanism normally considered for small-de°ection buckling. This increase wouldstill allow close match of all the aforementioned observations except for the ¯rst two seconds ofthe video. The proper conclusion from Korol and Sivakumaran's tests, based on close matchingof the video record, is that the fracturing of unheated columns and the °exibility of their endrestraints must have signi¯cantly reduced the energy dissipation in columns calculated underthe assumptions of no fracture and no end restraint °exibility.

Keywords: Column buckling; ductility; fracture; WTC collapse; 9/11.

*Corresponding author.

International Journal of Structural Stability and DynamicsVol. 17, No. 9 (2017) 1771011 (7 pages)#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0219455417710110

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1. Introduction

Previous studies1–4 led to a rigorous mathematical model which showed that agravity driven collapse of the twin towers of the World Trade Center (WTC) in NewYork on September 11, 2011 was inevitable. The model showed that, at the begin-ning of collapse, the energy dissipation by plastic buckling of columns was the mainmechanism of resistance. It also explained why the crush-down must have precededthe crush-up, and why air and debris particles were ejected laterally at almost thesound velocity, inevitably making loud booms and big dust clouds. It gave the correctsize distribution of the particles from impact comminution of the concrete °oor slabs,and agreed closely with the observed duration of collapse. Further, it explained whythe towers did not topple sideway like a tree,1 and why the motion observed in theinitial video was virtually smooth, without any velocity °uctuation detectable byeye.4 However, for lack of test data, it left two questions unanswered: (1) Did notextensive fracturing of the columns limit signi¯cantly the ductility of steel? (2) Didnot °exibility and plasticity of the spandrel plates and °oor trusses reduce therotations, and thus the energy dissipation, in the plastic hinges at column ends? Andif so, by how much?

Korol and Sivakumaran5 recently presented valuable experimental results thatallow answering these questions. They tested reduced-scale extruded H-shapedaluminum columns without end restraints which exhibited virtually unlimited duc-tility, i.e. no fracture. They found that the dissipation by a 180! rotation of theplastic hinge at mid height of the column was about 3.5-times as large as thatcalculated in Ba!zant and Zhou1 by extrapolating from small rotations the work ofplastic bending moment acting on a planar cross-section. This extrapolation ignoredthe local buckling and folding of column °anges, and large tensile °ange extension, asrevealed by these tests (Fig. 1).

(a) (b) (c)

Fig. 1. Experiments by Korol and Sivakumaran5: (a) and (b) specimen during the test and (c) ¯naldeformation shape of the specimen.

Source: Figures 3 and 7(a) in Korol Sivakumaran.

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2. Simpli¯cations of Original Analysis Due to Lack of Data

Because of uncertainties due to lack of experimental data, the original the demon-stration of necessity of gravity-driven collapse1,2 introduced four conservative sim-plifying assumptions:

(a) perfect ductility of steel with no fracture,(b) constancy of the bending moment in the plastic hinge up to 180! rotation, and(c) rigid support of column ends,(d) no multi-story buckling of columns of the °oor on ¯re as well as the underlying

cold °oors.

With these assumptions and using the plastic bending moment based on thestandard engineering theory of bending6 (with the cross-section remaining planar), itwas found that the energy, Wd, dissipated by buckling of the columns of the ¯rstcollapsing °oor represented about 1/8 of the kinetic energy Mv 2

0=2 of the impactingmass M of the upper part of tower,1 v0 being the impact velocity (0.19 m/s).According to these original assumptions, the buckling of the columns of the ¯rstimpacted °oor reduces the kinetic energy toMv 2

1=2 ¼ ð1$ 18ÞMv 2

0=2. So, the velocityafter impact drops to v1 ¼ v0

ffiffiffiffiffiffiffiffi7=8

p¼ 0:935 v0.

If we directly apply Korol and Sivakumaran's results5 to WTC towers, the energydissipation of the columns would have to be 3.5-times the energy calculated by theplastic hinge mechanism6 (according to the foregoing assumptions (a)–(c)). There-fore, we would have Mv 2

1=2 ¼ ð1$ 716ÞMv 2

0=2, which gives v1 ¼ 0:750 v0. Obviously,this updated estimate again indicates a continuing collapse. In no way the energydissipation in the columns of one °oor could be large enough to exceed Mv2

0=2, whichwould be necessary to arrest the gravity-driven collapse.

After the crushing front advances by about 10 °oors down, the accumulatedcollapsing mass grows signi¯cantly and the kinetic energy of the falling mass dwarfsthe energy dissipated by the columns. It then ceases to matter whether or not thedissipation by plastic buckling is tripled. Therefore, the calculations of the overallduration of collapse, of the velocity of expelling air and debris shedding, and of theimpact comminution of concrete slabs into particles, would not change beyond therange of error in the observations made.

3. Non-Rigid Restraints at Column Ends

The Korol and Sivakumaran's columns developed no plastic hinges at the ends. Theirend supports had a °at free contact with the loading platens rather than perfectrestraint. Beginning with a certain small de°ection without end rotations, the col-umn ends pivoted freely about the end of one °ange (Fig. 1). This complicatescomparisons with the WTC columns.

The perimeter columns of the WTC towers were fabricated in units of three-story height. They consisted of three column sections and three spandrel plates

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(Fig. 2(a)). For each story, the rotation of the two ends of each column was re-strained by the spandrel plates. The spandrel plates must have deformed elasti-cally and plastically, rotating together with the column ends (Fig. 2(c)).Therefore, the plastic hinges at the ends of WTC columns must have dissipatedmuch less energy than the mid height hinge. This would make the estimate ofenergy dissipation per column much smaller than that calculated for a column with¯xed ends (Fig. 2(b)).

Meanwhile, Korol and Sivakumaran's experiments indicated that, at the plastichinge location, the columns experienced large plastic deformation on the tensile°ange and local buckling on the compressive °ange (Fig. 1). These local mechanismsmake signi¯cant contributions to the total energy dissipation. However, for columnsin the WTC towers, the two ends are not fully restrained and therefore the energydissipation due to plastic deformation and local °ange buckling at these two endswould be smaller than that at the mid-span. Therefore, we can conclude that, forcolumns in the WTC towers, the increase of the energy dissipation relative to theprediction by the plastic hinge model would be much smaller than that observed inKorol and Sivakumaran's experiments.

4. Limited Ductility and Fracture of Steel

To get a conservative estimate of the maximum possible dissipation, the ductilitylimitation and fracture of steel were neglected in previous studies.1,2 In reality, nu-merous column fractures were likely to occur, especially because a high rate of de-formation promotes the fracture of steel. For example, the tensile rupture strain oftypical A36 steel under static loading is only about 0.35.7 The fractures during WTCcollapse, which greatly reduced energy dissipation, have been documented by pho-tographs and videos showing many °ying fragments of columns.

Floor

Floor

Floor

Columns

Spandrelplate

(a) (b) (c)

Fig. 2. Deformation of the perimeter columns of WTC towers: (a) geometry of the prefabricated unit ofperimeter columns, (b) deformation of columns with fully restrained ends and (c) deformation of columnswith elastically restrained ends.

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The fracturing of columns must have been particularly intense in the columns oflower stories. They consisted of high strength steel (with the yield strength of 690MPa),which is more brittle and much more prone to fracture, especially at high rate.

5. Multi-Story Buckling of Columns During the Collapse Initiation Stage

Beside the °exibility of end restraints of columns and the limited ductility of steelmaterials, multi-story buckling of columns should also be considered for the esti-mation of energy dissipation (assumption (d)). Previous studies have investigatedthe behavior of the building upon the aircraft impact.3,8 It transpired that visco-plastic sagging and disconnect of the heated °oor trusses caused a multi-story bowingof some perimeter columns. During this process, the °oor trusses were not able toprovide lateral restraints to the columns. Therefore, the e®ective length of thesecolumns in the weak zone was much larger than the story height, and multi-storybuckling must have occurred. The multi-story buckling mechanism signi¯cantlyreduces the average energy dissipation of columns per story. This is especially im-portant for the analysis of the initial stage of the collapse.

It should also be mentioned that as the collapse motion proceeds, debris impact onthe °oor truss could also cause columns to lose the lateral restraints and inducemulti-story buckling. Various photographs have documented column fragments ofmulti-story height, which con¯rms that multi-story buckling must have occurredduring the collapse process, and thus diminished the energy dissipation.

6. Calibration Based on Korol and Sivakumaran's Tests and VideoRecord of Collapse

The uncertainties in the estimation of the energy dissipation by column failures, byair and mass ejection and by comminution of concrete slabs, were recognized in theprevious analysis of WTC collapse.3 A sensitivity analysis was performed in whichplausible ranges of these dissipation terms were considered. For columns, a range of&20% of the mean energy dissipation capacity was used (although, in view of Koroland Sivakumaran's tests, it should have been broader). For air and mass ejection, arange of &50% of the mean energy dissipation capacity was considered.

The calculations showed that these variations make little di®erence in the pre-dictions of the total collapse duration, as well as the crush front propagation andconcrete slab comminution. A larger variation of the energy dissipation capacity ofcolumns (i.e. more than 50% increase) was recognized to cause noticeable deviationsfrom the video record of collapse during approximately the ¯rst two seconds (seeFig. 6 of Ref. 2). Yet, the match of the seismically recorded duration of collapsewould barely be a®ected.

Based on the aforementioned discussion, the increase of energy dissipation incolumns indicated by Korol and Sivakumaran's test data does not make an appre-ciable di®erence in the analysis of progressive failure of WTC columns. It makes an

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appreciable di®erence only for matching the video record of the ¯rst two seconds ofcollapse. Therefore, the proper way of using these data together with this videorecord is to exploit them for calibrating the energy dissipation per column, restricted,of course, to the realistic range of uncertainties in the material and structureproperties.

A calibration of this kind has already been done in the previous study,3 whichconcluded that, for the upper stories, the energy dissipation capacity of columns wasabout 2/3 of the value predicted by the simple plastic three-hinge model with per-fectly rigid end constraints. The 1/3 reduction is not unreasonable if we consider thedecrease in energy dissipation due to the °exibility of end restraints, likelihood ofmulti-story buckling of heated columns, and steel fracture.3 This decrease can greatlyo®set the increase of the energy dissipation due to local plastic deformation and localbuckling at the hinges. Anyway, note that, by using the calibrated energy dissipationcapacity of columns, the model was able to predict correctly all the other observa-tions such as the seismically documented collapse duration; the particle size distri-bution of fragmented concrete slabs; the wide spread of the ¯ne dust around thetower; the loud booms heard during the collapse; and the fast expansion of dustclouds during collapse. This multitude of data matching serves as a strong validationof the overall model.

7. Conclusion

The experiments of Korol and Sivakumaran help in clarifying the mechanics ofenergy dissipation in the columns of WTC and in reducing the previously statedrange of uncertainties of analysis. They indicate that if the column ends were rigidlysupported and if the ductility of steel were unlimited, then the simple plastic three-hinge mechanism with constant bending moments (Sec. 8.2 of Ref. 6), of the typeused for small-de°ection buckling, would have dissipated about 3.5-times more en-ergy than considered in previous studies.

But calibration by matching of the video record of initial collapse implies that thisenergy must have been reduced to about 2/3 of the energy predicted by the three-hinge model. This estimated reduction must have been caused by the fracturing ofsteel, by the °exibility of spandrel beams which reduced the rotations of the plastichinges at column ends, and by the multi-story buckling of columns during the initialdrop. With this update of input data, all the observed features of the WTC collapseremain to be closely matched by the gravity-driven mechanics of progressive collapse.

References

1. Z. P. Ba!zant and Y. Zhou, Why did the World Trade Center collapse? — Simple analysis,ASCE J. Eng. Mech. 128(1) (2002) 2–6, with Addendum, March (No. 3), 369–370.

2. Z. P. Ba!zant and M. Verdure, Mechanics of progressive collapse: Learning from WorldTrade Center and building demolitions, ASCE J. Eng. Mech. 133 (2007) 308–319.

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3. Z. P. Ba!zant, J.-L. Le, F. R. Greening and D. B. Benson, What did and did not causecollapse of WTC twin towers in New York, ASCE J. Eng. Mech. 134(10) (2008) 892–906.

4. J.-L. Le and Z. P. Ba!zant, Why the observed motion history of World Trade Center towersIs smooth? ASCE J. Eng. Mech. 137(1) (2011) 82–84.

5. R. M. Korol and K. S. Sivakumaran, Reassessing the plastic hinge model for energy dis-sipation of axially loaded columns, J. Struct. Article ID: 795257.

6. Z. P. Ba!zant and L. Cedolin, Stability of Structures: Elastic, Inelastic, Fracture andDamage Theories (Oxford University Press, New York); 2nd. edn. (Dover Publications,New York 2003), 1011 pp. +xxiv pp.; (3rd edn. World Scienti¯c Publishing, Singapore,New Jersey, London 2010).

7. C. G. Salmon, J. E. Johnson and F. A. Malhas, Steel Structures Design and Behavior,5th Edn. (Pearson Prentice Hall, United States, 2009).

8. NIST, Final Report on The Collapse of The World Trade Center Towers. S. ShyamSunder, Lead Investigator, Rep., Gaithersburg, MD.

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