THE WTC COLLAPSE ARGUMENTS OF ZDENEK BAZANT


The vague nature of the NIST reports when describing the collapse progression mechanisms of the Twin Towers has led to a long series of misrepresentations of the WTC collapses in ASCE publications and elsewhere that continues to this day. An excellent example of what lies at the heart of these misrepresentations is a series of papers appearing in the Journal of engineering mechanics by Zdenek Bazant.

Dr Bazant wrote 5 papers on the WTC collapses which appear in ASCE journals. The first paper, published in 2002, is based on a simple argument that can be stated in a single sentence. In his own words: "The analysis shows that if prolonged heating caused the majority of columns of a single floor to lose their load carrying capacity, the whole tower was doomed."

He wrote 3 more papers in succession in 2007-8 in which he develops a general one dimensional mechanical model of the motion of the progressive collapse of a building and applies the model to the Twin Towers.

It is quite fascinating to observe how this model, based on the mechanics of interacting blocks and gross assumptions, had gone from being a simple 1-D formulation to something that has been taken quite literally. The block mechanics first derived in these papers is now often portrayed as the most realistic description of the collapse progressions of the Twin Towers to date. This has occurred even though it is easy to show that block mechanics grossly contradicts what is observable and verifiable within the visual record of the events themselves. Block mechanics is taken to represent the history of the towers so completely that overwhelming visual evidence that the real collapse modes were quite different goes virtually unmentioned in pretty much all professional and government literature. Even stranger, many groups and individuals that argue contrary to Bazant have adopted the same block mechanics formulation to express their own understanding of the collapse progression modes of the Twin Towers.


For those readers who find this section to be too technically complex, I have prepared a simpler way to spot the mistakes within these papers using direct quotes from the papers followed by simple, straightforward questions linked here.



CRUSH DOWN, THEN CRUSH UP BLOCK MECHANICS

Bazant's original crush down, then crush up conception of collapse progression was published in 2 parts in 2007. The first part, called "The Mechanics of Progressive Collapse", describes the general model. In his own words, in the paper:

" a dynamic one-dimensional continuum model of
progressive collapse is developed. The collapse, in which two phases'"crush-down followed by crush-up'"must be distinguished, is described in each phase by a nonlinear second-order differential equation for the propagation of the crushing front of a compacted block of accreting mass."

"First, let us review the basic argument (Bazant 2001; Bazant and Zhou 2002). After a drop through at least the height h of one story heated by fire, the mass of the upper part of each tower has lost enormous gravitational energy, equal to mgh. Because the energy dissipation by buckling of the hot columns must have been negligible by comparison, most of this energy must have been converted into kinetic energy K = m0v2/2 of the upper part of tower, moving at velocity v. Calculation of energy Wc dissipated by the crushing of all columns of the underlying (cold and intact) story showed that, approximately, the kinetic energy of impact K > 8.4 Wc (Eq. 3 of Bazant and Zhou 2002)."

"Introduction and Failure Scenario

The 110-story towers of the World Trade Center were designed to withstand as a whole the forces caused by a horizontal impact of a large commercial aircraft (Appendix I). So why did a total collapse occur? The cause was the dynamic consequence of the prolonged heating of the steel columns to very high temperature. The heating lowered the yield strength and caused viscoplastic (creep) buckling of the columns of the framed tube along the perimeter of the tower and of the columns in the building core.

The likely scenario of failure is approximately as follows. In stage 1 (Fig. 1), the conflagration, caused by the aircraft fuel spilled into the structure, causes the steel of the columns to be exposed to sustained temperatures apparently exceeding 800°C. The heating is probably accelerated by a loss of the protective thermal insulation of steel during the initial blast. At such temperatures, structural steel suffers a decrease of yield strength and exhibits significant viscoplastic deformation (i.e., creep'"an increase of deformation under sustained load). This leads to creep buckling of columns (Bazant and Cedolin 1991, Sec. 9), which consequently lose their load carrying capacity (stage 2). Once more than half of the columns in the critical floor that is heated most suffer buckling (stage 3), the weight of the upper part of the structure above this floor can no longer be supported, and so the upper part starts falling down onto the lower part below the critical floor, gathering speed until it impacts the lower part. At that moment, the upper part has acquired an enormous kinetic energy and a significant downward velocity. The vertical impact of the mass of the upper part onto the lower part (stage 4) applies enormous vertical dynamic load on the underlying structure, far exceeding its load capacity, even though it is not heated. This causes failure of an underlying multifloor segment of the tower (stage 4), in which the failure of the connections of the floor-carrying trusses to the columns is either accompanied or quickly followed by buckling of the core columns and overall buckling of the framed tube, with the buckles probably spanning the height of many floors (stage 5, at right), and the upper part possibly getting wedged inside an emptied lower part of the framed tube (stage 5, at left). The buckling is initially plastic but quickly leads to fracture in the plastic hinges. The part of building lying beneath is then impacted again by an even larger mass falling with a greater velocity, and the series of impacts and failures then proceeds all the way down (stage 5)."

stage 1) Airplane damage, fire and fuel
stage 2) Creep buckling
stage 3) Majority of columns lose strength, building starts to move downwards
stage 4) First significant collision
stage 5) Collapse propagation

The aircraft impacts and resulting fires
Early deformation
Collapse initiation sequence
Collapse progression
The debris layout and condition

"a dynamic one-dimensional continuum model of progressive collapse is developed. The collapse, in which two phases -crush-down followed by crush-up - must be distinguished, is described in each phase by a nonlinear second-order differential equation for the propagation of the crushing front of a compacted block of accreting mass."

"The mass of columns is assumed to be lumped, half and half, into the mass of the upper and lower floors. Let u denote the vertical displacement of the top floor relative to the floor below (Figs. 3 and 4), and F(u) the corresponding vertical load that all the columns of the floor transmit. To analyze progressive collapse, the complete load-displacement diagram F(u) must be known (Figs. 3 and 4 top left). It begins by elastic shortening and, after the peak load F0, curve F(u) steeply declines with u due to
plastic buckling, combined with fracturing (for columns heated above approximately 450°C, the buckling is viscoplastic). For single column buckling, the inelastic deformation localizes into
three plastic (or softening) hinges (Sec. 8.6 in Bažant and Cedolin 2003; see Figs. 2b,c and 5b in Bažant and Zhou 2002a). For multistory buckling, the load-deflection diagram has a similar shape but the ordinates can be reduced by an order of magnitude; in that case, the framed tube wall is likely to buckle as a plate, which requires four hinges to form on some columns lines and three on others (see Fig. 2c of Bažant and Zhou). Such a buckling mode is suggested by photographs of flying large fragments of the
framed-tube wall, which show rows of what looks like broken-off plastic hinges."

"Thus it appears reasonable to make four simplifying hypotheses:

(1) The only displacements are vertical and only the mean of vertical displacement over the whole floor needs to be considered.

(2) Energy is dissipated only at the crushing front (this implies that the blocks in Fig. 2 may be treated as rigid, i.e., the deformations of the blocks away from the crushing front may be neglected).

(3) The relation of resisting normal force F (transmitted by all the columns of each floor) to
the relative displacement u between two adjacent floors obeys a known load-displacement diagram (Fig.4), terminating with a specified compaction ratio ( which must be adjusted to take into
account lateral shedding of a certain known fraction of rubble outside the tower perimeter).

(4) The stories are so numerous, and the collapse front traverses so many stories, that a continuum
smearing (i.e., homogenization) gives a sufficiently accurate overall picture."

"First it needs to be decided whether crushed Zone B will propagate down or up through the tower. The equation of motion of Zone B requires that



where F1 and F2 are the normal forces (positive for compression) acting on the top and bottom of the compacted Zone B (Fig. 2(c)). This expression is positive if Zone B is falling slower than a free fall, which is reasonable to expect and is confirmed by the solution to be given. Therefore F2F1 always. So, neither upward, nor two-sided simultaneous, propagation of crushing front is
possible."

"This is true, however, only for a deterministic theory. A front propagating intermittently up and down would nevertheless
be found possible if Fc(z) were considered to be a random (autocorrelated ) field. In that case, short intervals (t may exist in which the difference Fc1-Fc2 of random Fc values at the bottom and top of crushed Block B would exceed the right-hand side of Eq. (10). During those short intervals, crush-up would occur instead of crush-down, more frequently for a larger coefficient of variation. The greater the value of s0, the larger the right-hand side of Eq. (10), and thus the smaller the chance of
crush-up. So, random crush-up intervals could be significant only at the beginning of collapse, when s0 is still small enough. Stochastic
analysis, however, would make little difference overall and is beyond the scope of this paper."

"The phase of downward propagation of the front will be called
the crush-down phase, or Phase I (Fig. 4(b)). After the lower crushing front hits the ground, the upper crushing front of the compacted zone can begin propagating into the falling upper part
of the tower (Fig. 4(d)). This will be called the crush-up phase, or Phase II"

"4. For the typical WTC characteristics, the collapse takes about
10.8 s (Fig. 6 top left), which is not much longer (precisely only 17% longer) than the duration of free fall in vacuum from the tower top to the ground, which is 9.21 s (the duration
of 10.8 s is within the range of Bazant and Zhou's (2002a) crude estimate). For all of the wide range of parameter values considered in Fig. 6, the collapse takes less than about double the free fall duration."


"Eqs. (12) and (17) show that Fc(z) can be evaluated from precise monitoring of motion history z(t) and y(t), provided that 
z and 
z are known. A millisecond accuracy for z(t) or y(t) would be required. Such information can, in theory, be extracted from a high-speed camera record of the collapse. Approximate information could be extracted from a
regular video of collapse, but only for the first few seconds of collapse because later all of the moving part of the WTC towers became shrouded in a cloud of dust and smoke (the visible
lower edge of the cloud of dust and debris expelled from the tower was surely not the collapse front but was moving ahead of it, by some unknown distance)."

"3. Distinction must be made between crush-down and crush-up phases, for which the crushing front of a moving block with
accreting mass propagates into the stationary stories below or into the moving stories above, respectively. This leads to a second-order nonlinear differential equation for propagation
of the crushing front, which is different for the crush-down phase and the subsequent crush-up phase.

4. The mode and duration of collapse of WTC towers are consistent with the present model, but not much could be learned because, after the first few seconds, the motion became obstructed
from view by a shroud of dust and smoke."

• Inability to verify predictions
• Limited understanding of WTC collapse features
• Inability to identify a specific WTC collapse mode
• Misrepresentation of the actual crushing process along the collapse front contact interface

The paper presents a very nteresting concept of an accidental demolition, whereby heavy damage sustained by an intermediate
story of a building leads to the upper part of the structure crushing the lower one in a equence of story collapse steps. The focus of
the paper is on the treatment of equations of motion and very few numbers are quoted; that is, numbers that relate to the physical properties of the structure discussed, namely the World Trade
Center (WTC) towers. The following comments are intended to fill that gap as well as to ascertain the likelihood of the applicability of this concept.

This discussion describes flaws in the modeling and analysis of the World Trade Center collapses by Ba�ant and Verdure in their paper entitled "Mechanics of Progressive Collapse: Learning
from World Trade Center and Building emolitions." First, the paper's two-phased approach to the collapse analysis will be considered. The writers will demonstrate that a two-phase collapse analysis is not representative of reality, because it disregards well-accepted laws of physics and therefore is not instructive. Second, the original paper's summary of the findings of the NIST report will be analyzed.

This discussion describes flaws in the modeling and analysis of the World Trade Center collapses by Ba�ant and Verdure in their paper entitled "Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions."

Discussion by James R. Gourley

The interdisciplinary interests of Gourley, a chemical engineer with a doctorate in jurisprudence, are appreciated. Although none
of the discusser's criticisms is scientifically correct, his discussion provides a welcome opportunity to dispel doubts recently voiced
by some in the community outside structural mechanics and engineering. It also provides an opportunity to rebut a previous similar discussion widely circulated on the Internet, co-authored by S. E. Jones, Associate Professor of Physics at Brigham Young University and a cold fusion specialist. For the sake of clarity, this
closure is organized into the points listed subsequently and rebutted one by one.

1. Newton's Third Law

2. Are the Internal Forces in Upper and Lower Parts of
Tower Equal?

3. Localization of Energy Dissipation into Crushing Front

4. Can Crush-Up Proceed Simultaneously with Crush Down?

5. Why Can Crush-Up Not Begin Later?

6. Variation or Mass and Column Size along Tower Height

7. Were the Columns in the Stories above Aircraft Impact
Hot Enough to Fail?

8. Steel Temperature and NIST Report

Discussion by G. Szuladzinski

The interest of Szuladzinski, a specialist in homeland security, is appreciated. After close scrutiny, however, his calculations are found to be incorrect, for reasons explained in the following.

1. Load-Displacement Curve of Columns and Energy Absorption Capacity

2. Does Excess of over Gravity Load Imply Arrest of Collapse?

3. Is the Equation of Motion for Calculating the Duration of Fall Correct?

4. Could Stress Waves Ahead of Crushing Front Destroy the Tower?

"Discussion by James R. Gourley
3. Localization of Energy Dissipation into Crushing Front:
In the discusser's opinion: the hypothesis that "the energy is dissipated at the crushing front implies that the blocks in Fig. 2 may be treated as rigid, i.e., the deformations of the blocks away from the crushing front may be neglected." This is a fundamental misunderstanding. Of course, blocks C and A are not rigid and elastic waves do propagate into them."

But the wave velocity, given by v = ?Et / ? where Et = tangential e modulus of steel in the loaded columns and ? = mass density, tends to zero as soon as the plastic or racturing response is triggered, because in that case, Et ? 0. Therefore, as explained in courses on stress waves, no wave attaining the material strength can penetrate beyond the crushing (or plastic) front. Only harmless elastic waves can. Propagation of the crushing front is not a wave-propagation phenomenon. Destruction of many stories at the rate corresponding to the elastic wave speed, which would appear as simultaneous, is impossible. This is why the collapse is called progressive. Blocks C and A can, of course, deform. Yet, contrary to the discusser's claim, they may be treated in calculations as rigid because their elastic deformations are about 1,000 times smaller than the deformations at the crushing front."

"4. Can Crush-Up Proceed Simultaneously with Crush Down?I

t can, but only briefly at the beginning of collapse, as mentioned in the paper."

"Statements such as "the columns supporting the lower floors . . . were thicker, sturdier, and more massive,"
although true, do not support the conclusion
that "the upper floors (i.e., the floors comprising Part C) would be more likely than the lower floors to deform and yield during collapse"� (deform they could, of course, but
only a little, i.e., elastically)."

"More-detailed calculations than those included in their paper were made by Bažant and Verdure to address this question. On the basis of a simple estimate of energy corresponding to the area between the load-deflection curve of columns and the gravity force for crush down or crush up, it was concluded at the onset that the latter area is much larger, making crush-up impossible."

We have now carried out accurate calculations, which rigorously justify this conclusion and may be summarized as follows.

Consider that there are two crushing fronts, one propagating upward into the falling block, and the other down-ward. Denote v1 , v2 = current velocities of the downward and upward crushing fronts (positive if downward); x(t) , z(t)
= coordinates of the mass points at these fronts before the collapse began (Lagrangian oordinates); and q(t) = current coordinate of the tower top. All the coordinates are measured from the initial tower top downward. After the ollapse of the first critical story, the falling upper Part C with the compacted Part B impacts the stationary lower Part A. During that impact, the total momentum and the total energy must both be conserved. These conditions yield two lgebraic equations

During impact, ? = 0.2 for the North Tower and 0.205 for the South Tower. For the North or South Tower: m0 = 54.18· 106 or 112.80· 106 kg, m1 = 2.60· 106 or 2.68· 106 kg,
m2 = 3.87· 106 or 3.98· 106 kg, and ms = .627· 106 kg for both. For a fall through the height of the critical story, by solving Eq. (2) of Bazant et al. 2007, one obtains the rush-front velocity v0 = 8.5 m / s for the North Tower and 8.97 m / s for the South Tower. he solution of Eqs. (1) and (2) yields the following velocities after impact: v1 = 6.43 or 6.80 m / s, v2 = 4.70 or 4.94 m / s, and vcu = 2.23 or 2.25 m / s for the North or South Tower. These data represent the initial values for the differential equations of motion of the upper Part C and of the compacted layer B. If Lagrangian coordinates x(t) and z(t) of the crush-down and crush-up fronts are used, these equations can easily be shown to have the following forms:

"These two simultaneous differential equations have been converted to four first-order differential equations and solved
numerically by the Runge-Kutta method. The solution has been found to be almost identical to the solution presented in the paper, which was obtained under the simplifying assumption that the crush-up does not start until after the crush down is finished."

"The reason for the difference being negligible is that the condition of simultaneous crush-up, x ? 0, is violated
very early, at a moment at which the height of the first overlying story is reduced by about 1%."

"This finding further means that the replacement of the load-deflection curve in Fig. 3 of the paper by the energetically quivalent Maxwell line that corresponds to a uniform resisting force F? cannot be sufficiently accurate to study the beginning of two-way crush.)"

Therefore, a solution more ac-
curate than that in the paper has been obtained on the basis of Eqs. (3) and (4). In that solution, the variation of the crushing force F? within the story was taken into account, as shown by the actual calculated resistance force labeled F(u) in Fig. 3 of the paper, by the force labeled F(z) on top of Fig. 4 of the paper, and by the resistance curves for the crushing of subsequent stories shown in Fig. 5 of the paper. The precise curve F(u) was calculated from Eq. 8 of Bazant and Zhou (2002). Very small time steps, necessary to resolve the changes of velocity and acceleration during the collapse of one story, have been used in this calculation. Fig. 1 shows the calculated evolution of displacement and velocity during the collapse of the first overlying story in two-way crush. The result is that the crush-up stops (i.e., (x) )_ drops to zero? when the first overlying story is squashed by the distance of only about 1.0% of its original height for the North Tower, and only by about 0.7% for the South Tower (these values are about 11 or 8 times greater than the elastic limit of column deformation)."

"Why is the distance smaller
for the South Tower even though the falling upper part is much more massive? That is because the initial crush-up velocity is similar for both towers, whereas the columns are much stronger (in proportion to the weight carried)."

"The load-displacement diagram of the overlying story is qualitatively similar to the curve with unloading rebound sketched in Fig. (4c) of the paper and accurately plotted without rebound in Fig. 3 of the paper. The results of accurate computations are shown by the displacement and velocty evolutions in Fig. 1."

"So it must be concluded that the simplifying hypothesis of one-way crushing (i.e., of absence of simultaneous crush-up),
made in the original paper, was perfectly justified and caused only an imperceptible difference in the results."

"The crush-up simultaneous with the crush down is found to have advanced into the overlying story by only 37 mm for the North Tower
and 26 mm for the South Tower."

"This means that the initial crush-up phase terminates when the axial displacement of columns is only about 10 times larger than their maximum elastic deformation. Hence, simplifying the analysis by neglecting the initial two-way crushing phase was correct and
accurate."

"5. Why Can Crush-Up Not Begin Later? The discusser further states that "it is difficult to imagine, again from a basic physical standpoint, how the possibility of the occurrence of crush-up would diminish as the collapse progressed."�

"Yet the discusser could have imagined it easily, even without calculations, if he considered the free-body equilibrium diagram
of compacted layer B, as in Fig. 2(f) of the paper."

"After including the inertia force, it immediately follows from this diagram that the normal force in the supposed crush up front acting upward onto Part C is

"The discussers' statement that "the yield and deformation strength of . . . Part C would be very similar to the yield and
deformation strength of . . . the lower structure"� shows a misunderstanding of the mechanics of failure. Aside from the fact that "deformation strength"� is a meaningless term (de-
formation depends on the load but has nothing to do with strength), this statement is irrelevant to what the discussers try to assert. It is the normal force in the upper Part C that is much smaller, not necessarily the strength (or load capacity) of Part C per se."

"Force F? acting on Part C upward can easily be calculated from the dynamic equilibrium of Part C (see Fig. 2g), and it is found that F? never exceeds the column crushing force of the overlying story. This confirms again
that the crush-up cannot restart until the compacted layer hits the ground."

"6. Variation or Mass and Column Size along Tower Height:

This variation was accurately taken into account by Bazant et al. (2007). Those who do not attempt to calculate might be surprised that the effects of this variation on the history of motion and on the collapse duration are rather small. Intuitively, the main reason is that, as good design requires, the cross-section areas of columns increase (in multistory steps, of course) roughly in proportion to the mass of the overlying
structure. For this reason, the effect of column size approximately compensates for the effect of the columns’ mass."

Closing Comments Although everyone is certainly entitled to express his or her opinion on any issue of concern, interested critics should realize that,
to help discern the truth about an engineering problem such as the WTC collapse, it is necessary to become acquainted with the relevant material from an appropriate textbook on structural mechanics. Otherwise critics run the risk of misleading and wrongly influencing the public with incorrect information."

"3. Is the Equation of Motion for Calculating the Duration of Fall Correct? It is not. Under the heading “Duration of Fall,” the discusser writes the equation of motion (Newton's law) as d(mv) / dt = m0g (in the discusser's notation, m is M, and m0 is M 0). He states that “M 0 is the mass of the upper part of the building,” and argues that "the net effect of gravity applies now only to M 0.” This statement is incorrect. The accreted mass, which he denotes as ?z, does not disappear and thus is also subjected to gravity. Therefore, the discusser's equation of motion for the falling mass must be revised as d?m?t?v? / dt = m?t?g, and the solution is totally different from the last equation of the discusser. This is, of course, only the most simplified form of the equation of motion, originally applied to WTC collapse by E. Kausel of MIT (Kausel 2001). A realistic form of the equation of motion must take into account the energy dissipation Fc per unit height, the debris compaction ratio, and the mass shedding ratio, as shown in Eq. (12) of the paper."

"For the resistance to motion near the end of collapse, it is also necessary to include the energy per unit height required
for the comminution of concrete floor slabs and walls and for expelling air at high speed, which is found to be close to the speed of sound (Bažant et al. 2007). The discrepancy between the observed collapse duration and the collapse duration of 23.8 s calculated by the discusser does not support his conclusion that "the postulated failure mode is not a proper explanation of the WTC Towers collapse."�

"Rather, what this discrepancy means is that the discusser's calculations are erroneous. The collapse duration alculated in the paper for the most realistic choice of input
values is in agreement with the observations."

"Moreover, a more accurate analysis by Bazant et al. (2007) is found to be
in nearly perfect agreement with the video ecords of motion, available for the first few seconds of collapse, as well as with the available seismic records for both towers."

"Could Stress Waves Ahead of Crushing Front Destroy the Tower? They could not. The discusser is, of course, right in pointing out that the “stress wave . . . will partially reflect from all the discontinuities" ?though not only 'reflect'� but also 'diffract'). But while alluding to shock fronts, he is not right in stating that a "shock loading . . . will greatly magnify the effect of all discontinuities."
Since the stress-strain diagram of the steel used, as reported by FEMA (Figs. B-2 and B-3 in McAlister 2002), exhibits a long yield plateau, rather than hardening of gradu lly decreasing slope, the shock front coincides with the crushing front, which is not a wave phenomenon. The only waves than can penetrate ahead of this front are elastic. When these waves hit discontinuities such as joints, local energy-absorbing plastic strains and fractures will be created, and what will be reflected and iffracted will be weakened elastic waves.
Thus it is not true that "during such eflections, enhancements take place."� Rather, the energy of these waves ahead of the crushing front will quickly dissipate during repeated reflections and diffractions, and only noncatastrophic localized damage will happen to the structure until the crushing front arrives. To sum up, the existence of stress waves ahead
of the crushing front does not cast any doubt on the analysis in the paper."

"Conclusion Although closing comments similar to those in the preceding discussion could be repeated, let it suffice to say that the discusser's conclusion that "the motion will be arrested during the damaged story collapse and the building will stand"� is incorrect. Thus, the recent allegations of controlled demolition are baseless."

"One and the same mathematical model, with one and the same set of parameters, is shown capable of matching all of the observations, including: (1) the video records of the first few seconds of motion of both towers, (2) the seismic records for both towers, (3) the mass and size distributions of the comminuted particles of concrete, (4) the energy requirement for the comminution that occurred, (5) the wide spread of the fine dust around the tower, (6) the loud booms heard during collapse, (7) the fast expansion of dust clouds during collapse, and (8) the dust content of cloud implied by its size."

"Abstract: Previous analysis of progressive collapse showed that gravity alone suffices to explain the overall collapse of the World Trade Center (WTC) towers. However, it remains to be checked whether the recent allegations of controlled demolition have any scientific merit. The present analysis proves that they do not. The video record available for the first few seconds of collapse is shown to agree with the motion history calculated from the differential equation of progressive collapse but, despite uncertain values of some parameters, it is totally out of range of the free fall hypothesis, on which these allegations rest. It is shown that the observed size range (0.01 mm�"0.1 mm) of the dust particles of pulverized concrete is consistent with the theory of comminution caused by impact, and that less than 10% of the total gravitational energy, converted to kinetic energy, sufficed to produce this dust (whereas more than 150 tons of TNT per tower would have to be installed, into many small holes drilled into concrete, to produce the same pulverization). The air ejected from the building by gravitational collapse must have attained, near the ground, the speed of almost 500 mph (or 223 m/s, or 803 km/h) on the average, and fluctuations must have reached the speed of sound. This explains the loud booms and wide spreading of pulverized concrete and other fragments, and shows that the lower margin of the dust cloud could not have coincided with the crushing front. The
resisting upward forces due to pulverization and to ejection of air, dust and solid fragments, neglected in previous studies, are found to be indeed negligible during the first few seconds of collapse but not insignificant near the end of crush-down. The calculated crush-down duration is found to match a logical interpretation of seismic record, while the free fall duration grossly disagrees with this record."

"Previous analysis of progressive collapse showed that gravity alone suffices to explain the overall collapse of the World Trade Center (WTC) towers. However, it remains to be checked whether the recent allegations of controlled demolition have any scientific merit. The present analysis proves that they do not."

"The video record available for the first few seconds of collapse is shown to
agree with the motion history calculated from the differential equation of progressive collapse but, despite uncertain values of some parameters, it is totally out of range of the free fall hypothesis, on which these allegations rest."

"It is shown that the observed size range (0.01 mm�"0.1 mm) of the
dust particles of pulverized concrete is consistent with the theory of comminution caused by impact, and that less than 10% of the total gravitational energy, converted to kinetic energy, sufficed to produce this dust (whereas more than 150 tons of TNT per tower would have to be installed, into
many small holes drilled into concrete, to produce the same pulverization)."

"The air ejected from the building by gravitational collapse must have attained, near the ground, the speed of almost 500 mph (or 223 m/s, or 803 km/h) on the average, and fluctuations must have reached the speed of sound. This explains the loud booms and wide spreading of pulverized concrete and other fragments, and shows that the lower margin of the dust cloud could not have coincided with the crushing front."

"The resisting upward forces due to pulverization and to ejection of air, dust and solid fragments, neglected in previous studies, are found to be indeed negligible during the first few seconds of collapse but not insignificant near the end of crush-down. The calculated crush-down duration is found to match a logical interpretation of seismic record, while the free fall duration grossly disagrees with this record."

"Generalization of Differential Equation of Progressive Collapse

The gravity-driven progressive collapse of a tower consists of two phases�"the crush-down, followed by crush-up (Fig. 2 bottom), each of which is governed by a different differential equation (Ba?zant and Verdure 2007, pp. 312-313). During the crush-down, the falling upper part of tower (C in Fig. 2 bottom), having a compacted layer of debris at its bottom (zone B), is crushing the lower part (zone A) with negligible damage to itself. During the crush-up, the moving upper part C of tower is being crushed at bottom by the compacted debris B resting on the ground. The fact that the crush-up of entire stories cannot occur simultaneously with the crush-down is demonstrated by the condition of dynamic equilibrium of compacted layer B, along with an estimate of the inertia force of this layer due to vertical deceleration or acceleration; see Eq. 10 and Fig. 2(f) of Bazant and Verdure (2007). This previous demonstration, however, was only approximate since it did not take into account the variation of crushing forces Fc and F0c during the collapse of a story. An accurate analysis of simultaneous (deterministic) crush-up and crush-down is reported in Ba?zant and Le (2008) and is reviewed in the Appendix, where the differential equations and the initial conditions for a two-way crush are formulated. It is found that, immediately after the first critical story collapses, crush fronts will propagate both downwards and upwards. However, the crush-up front will advance into the overlying story only by about 1% of its original height h and then stop. Consequently, the effect of the initial two-way crush is imperceptible and the hypothesis that the crush-down and crush-up cannot occur simultaneously is almost exact."

"Variation of Mass and Buckling Resistance along the Height

Based on the area under the buckling curve in Fig. 3 of Bazant and Verdure (2007), the energy dissipation due to column buckling at the impact zone of the North Tower (96th story) is Fb(1 - )h = 0.51 GJ (or approximately 0.5 GJ, as estimated by Bazant and Zhou 2002). For other stories, this quantity is scaled according to the approximate cross section area of columns."

"It is nevertheless interesting to check the amount of explosives that would be required to produce all of the pulverized concrete dust found on the ground. Explosives are notoriously inefficient as a comminution tool. At most 10% of their explosive energy gets converted into the fracture energy of comminution, and only if the explosive charges are installed in small holes drilled into the solid to be comminuted. Noting that 1 kg of TNT releases chemically about 4 MJ of energy, the total mass of TNT required to pulverize 14.6 �— 107 kg concrete material into dust of the sizes found on the ground would be 316 tons. So, in order to achieve solely by explosives the documented degree of concrete pulverization, about 1.36 tons of TNT per story would have to be installed into small holes drilled into the concrete slab of each story, and then wired to explode in a precise time sequence to simulate free fall."

"Note in Fig. 7 that the motion identified from the videos is generally seen to pass well inside the predicted band of uncertainty of the motion calculated from Eq. (2). This fact supports the present analysis. The main point to note is that the curve identified from the video record grossly disagrees with the free fall curve, for each tower. The belief that the towers collapsed at the rate of free fall has been a main argument of the critics claiming controlled demolition.
The video record alone suffices to prove this argument false."

"These conclusions show the allegations of controlled demolition to be absurd and leave no doubt that the towers failed due to gravity-driven progressive collapse triggered by the effects of fire."

"These durations match reasonably well the durations of the crush-down phase calculated from Eq. (2), which are 12.81 s and 10.47 s for the North and South towers, under the assumption that the reduction factor  applied to Fb is 2/3. If the full uncertainty range,  2 [0.5, 0.8], is considered, the calculated mean durations are 12.82 s and 10.49 s, respectively. This uncertainty is shown by error bars in Fig. 8. Now note that these durations are, on the average, 65.5% and 47.3% longer than those of a free fall of the upper part of each tower, which are 7.74 s for the North Tower and 7.11 s for the South Tower. So, the seismic record, too, appears to contradict the hypothesis of progressive demolition by timed explosives."

"Since the initial crush-up phase terminates at very small axial deformation, it must be concluded that the simplifying hypothesis of one-way crushing is perfectly justified and causes only an imperceptible difference in results."

"If random fluctuation of column strength is taken into account, the crush-up resisting force F0c in the first overlying story may be lower or higher than indicated by the foregoing deterministic analysis. If it is lower, the crush-up will penetrate deeper. But even for the maximum imaginable standard deviation of the average column strength in a story, the crush-up will get arrested before it penetrates the full story height."

"In comparison with all these calculations, the claim that the observed fineness, extent and spread of pulverized dust could be explained only by planted
explosives has been found to be absurd. Only gravity driven impact could have produced the concrete dust as found on the ground."

"Some lay critics claim that out should be about 95%, in the (mistaken) belief that this would give a faster collapse and thus vindicate their allegation of free fall."

"Previously Refuted Hypotheses of Critics

Some other hypotheses have already been refuted in the discussions at the U.S. National Congress of Theoretical and Applied Mechanics in Boulder, June 2006. This includes the hypothesis that the structural frame was somehow brought to the brink of strain-softening damage and then destroyed by a peculiar phenomenon called the ”fracture wave”, causing the collapse to occur at the rate of free fall."

"There are three serious problems with this hypothesis: 1) It treats strain-softening as a local, rather than nonlocal, phenomenon (Bazant and Verdure 2007); 2) it considers the structural frame to have somehow been brought to a uniform state on the brink of strain softening, which is impossible because
such a state is unstable and localizes as soon as the strain softening threshold at any place (Bazant and Cedolin 2003, Sec 13.2); 3) the ‘fracture wave’ is supposed to cause comminution of concrete but the energy required for comminution cannot be delivered by this wave."

"Another previously refuted hypothesis of the lay critics is that, without explosives, the towers would have had to topple like a tree, pivoting about the base (Bazant and Zhou 2002) (Fig. 6b or c)."

"This hypothesis was allegedly supported by the observed tilt of the upper part
of tower at the beginning of collapse (Fig. 6a). However, rotation about a point at the base of the upper part (Fig. 6c) would cause a horizontal reaction approximately 10.3�— greater than the horizontal shear capacity of the story, and the shear capacity must have been exceeded already at the tilt of only 2.8 (Bazant and Zhou 2002). Thereafter, the top part must have been rotating essentially about its centroid, which must have been falling almost vertically.
The rotation rate must have decreased during the collapse as further stationary mass accreted to the moving block. So, it is no surprise at all that the towers collapsed essentially on their footprint. Gravity alone must have caused just that (Bazant and Zhou 2002)."

"In the structural engineering community, one early speculation was that, because of a supposedly insufficient strength of the connections between the floor trusses and the columns, the floors ‘pancaked’ first, leaving an empty framed tube, which lost stability only later. This hypothesis, however, was invalidated at NIST by careful examination of the photographic record, which shows some perimeter columns to be deflected by up to 1.4 m inward. This cannot be explained by a difference in thermal expansion of the opposite flanges of column. NIST explains this deflection by a horizontal pull from catenary action of sagging floor trusses, the cause of which has already been discussed. This pull would have been impossible if the floor trusses disconnected from the perimeter columns."

"Several of the parameters of the present mathematical model have a large range of uncertainty. However, the solution exhibits small sensitivity to some of them, and the values of others can be fixed on the basis of observations or physical analysis. One and the same mathematical model, with one and the same set of parameters, is shown capable of matching all of the observations, including: (1) the video records of the first few seconds of motion of both towers, (2) the seismic records for both towers, (3) the mass and size distributions of the comminuted particles of concrete, (4) the energy requirement for the comminution that occurred, (5) the wide spread of the fine dust around the tower, (6) the loud booms heard during collapse, (7) the fast expansion of dust clouds during collapse, and (8) the dust content of cloud implied by its size."

"At the same time, the alternative allegations of some kind of controlled demolition are shown to be totally out of range of the present mathematical model, even if the full range of parameter uncertainties is considered."

"These conclusions show the allegations of controlled demolition to be absurd and leave no doubt that the towers failed due to gravity-driven progressive collapse triggered by the effects of fire."

"On another matter, we ordinarily start with the simplest hypothesis and stik with it until some evidence shows the hypothesis must be modified. In the case of the top portion, the simplest is that it stayed on top most of the way down; say with the roof at around floor 25. Until someone develops some actual evidence to the contrary, I'll stick with that rather than unending speculation and new simulations of the resulting hypothesis."

"Better to call the section cushed, rather than compressed, as it is inelastic. It did contain, for the most part, the core columns; only a few were bypassed."

"Albert Einstein once said something to the effect that a model should be as simple as possible, but no simplier. The B&V crush-down equation meets that criterion as long as one only considers measurements taken on the antenna mast. With your careful observations of perimeter wall sections breaking off at and above floor 98 and OneWhiteeEye's observation earlier on this thread to the effect that this led to a inhomogeneity in the structure, I then, as reported earlier on this thread, in effect moved zone C up to start at floor 102. That fits the antenna tower measurements and also (approximately) the additional observation that OneWhiteEye posted earlier on this thread, regarding the SW corner of WTC 1."

"So, the simplest possible model for WTC 1 collapse works very well even though I now conclude that some 4+ floors of early crush-up occurred due to the inhomogeneity introduced by missing perimeter wall sections. But not more early crushup than that. Once those were crushed, the homogeneity is re-introduced so that Bazant & Le then applies. I think. It's a point that needs checking."

"More complex equations simply are not required. Parsimony suggests the B&V crush-down equation with vertical avalanche resisting force together with starting the crushing front around floor 102, being good enough for the data in hand, is indeed good enough."

"Assuming homogeneity, Bazaant & Le show thaqt zone C is almost industrucible. That's mechincs for you. The sturcture obviiously was not homogeneous and you have, in other threads, shown some distruction along the west and north walls. In of itself that mass loss is not important, but it does mean the floor trusses in those areas have been weakened. So an average of about 4--6 stories above floor 98 do not come close to satisfying the homogeneity condition. Fine. consider then that zone C is from floor, say, 102 up. To keep the equation simple, assume crush-down begins from there. As I mentiioned in this thread yesterday, this works well enough to match the additional observations by OneWhiteEye.
"

"Zone C simply disappears into the obscuring dusts. Not sufficient reason to assume it is being crushed first. If sufficiently close to homogeneous, then from Bazant & Le it is not being crushed at all."

"OneWhiteEye --- I've been thorugh all this before. Homogenization is fine when the tilt is taken into account; crushing proceeded on 3+ floors simultaneaously which is surely better represented by homogenization that by stepwise floor-by-floor model. However, both give essentially ythe same results; shagster actually went to the effort of running his own version of Greening's ideas using minifloors to demonstrate this; although, after some study, this is analytically obvious.

"The issue of early crush-up never seems to die, does it? The problem is that it would have to proceed against the force of gravity, not with it. Instead what you seem to have noticed in frame 1007 is a lack of one dimensionality, with zone C west perimeter wall going outside the lower portion, yes? That actually does not trouble me, yet.
"

"No sign of zone C falling aprat as long as it can be seen. Unlike the case of WTC 2."

"Major_Tom ---
Do you doubt Newton's Laws?
Do you doubt http://en.wikipedia.org/wiki/D'Alembert's_principle?
Do you doubt the applicability of the four simplifying assumptions in B&V?
If not, the conclusion of little early crush-up of zone C follows.
Further, the timing studies in BLGB show that most of zone C mass must have stayed on top most of the way down."

"As for the core punching through the roof, I conjecture this occurred when the upper mechanical floors and up to the roof encountered the greater resistance offered around floors 75--79, about 30 stories (about 110 meters) down. No air escaping through such a puncture will be separately observable in any of the photos, IMO."

"The west and north walls peeled away sufficiently rapidly that deebris tended to move west and north near the spire. Similarly, but to a lesser extent, to south and east. There actually wasn't a gaping hole, just less density and in particular no structural steel to break connections."

"OneWhiteEye --- B&L show little inital crush-up, not none at all. Since it is so small, the argument is that the crush-down only in B&V is a valid approximation."

"Major_Tom --- B&V have four simplifying assumptions which lead to the crush-down ODE. These assumptions are reasonable for WTC 1 but not, by video timing, for WTC 2 after a few seconds. In the case of WTC 2 it is clear from the ABC video of the collpase proceeding down to the Mariott rooftop level that the collapse was proceeding much too slowly; the inference is that the top section broke apart and fell off rather early on.

But as BLGB indicates, this could not have happened to WTC 1 or the timing would be off."

"OneWhiteEye --- I'm not the one with any doubts about the matter: there can be no significant early crush-up."

"Read Bazant & Le to understand why zone C can be consired to be essentially rigid during crush-down.
I offered to start a thread about how to build a table-top demonstrator that will allow one to see that,
indeed, zone C remains intact during crush-down. I didn't bother when I realized that nobody here would bother to actually build it, test it, and in the process dicover that the application of Newton's laws and
d'Alembert's principle in Bazant & Verdure agrees with reality."

"See Bazant & Le for a further exposition of why early crush-up is very small. It is, I admit, a difficult
point. But it is similar to a house riding down a landslide for which many examples have occurred in southern California."