Closure to “Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions” by Zdenĕk P. Bažant and Mathieu Verdure

Newton’s Third Law: The discusser is not correct in repeatedly claiming that Newton’s third law is violated in the paper and particularly in concluding that the “two-phase collapse scenario is scientifically implausible because it ignores Newton’s third law and the equal but opposite upward force dictated by it.” As explained at the outset in every course on mechanics of materials, this law is automatically satisfied, since all the calculations are based on the concept of stress or internal force, which consists of a pair of opposite forces of equal magnitude acting on the opposite surfaces of any imagined cut through the material or structure. This concept is so central to the discipline of structural mechanics and self-evident to structural engineers that Newton’s third law is never even mentioned in publications.

Are the Internal Forces in Upper and Lower Parts of Tower Equal? Contrary to the discusser’s claim which is based on his understanding of Newton’s third law, these forces are not equal, as made clear by Fig. 2(g and h) of the original paper. Their difference is equal to the weight of the intermediate compacted layer B plus the inertia force attributable to the acceleration of layer B (for additional accuracy, one may also add the energy per unit height needed for the comminution of concrete and the expelling of air, which are secondary phenomena not taken into consideration in the original paper). When the compacted layer attains a sufficient mass, which occurs after the collapse of only a few stories, this difference becomes very large.

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 $v=\sqrt{E_t∕\rho }$ where $E_t=\text{tangential}$ modulus of steel in the loaded columns and $\rho =\text{mass}$ density, tends to zero as soon as the plastic or fracturing response is triggered, because in that case, $E_t\to 0$ . 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.

Can Crush-Up Proceed Simultaneously with Crush Down? It 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 downward. Denote $v_1,v_2=\text{current}$ velocities of the downward and upward crushing fronts (positive if downward); $x\left(t\right),z\left(t\right)=\text{coordinates}$ of the mass points at these fronts before the collapse began (Lagrangian coordinates); and $q\left(t\right)=\text{current}$ coordinate of the tower top. All the coordinates are measured from the initial tower top downward. After the collapse 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 algebraic equationswhere $v_3=\dot{q}=(1-\lambda )(v_1+v_{\mathit{cu}}),v_{\mathit{cu}}=[(1-\lambda )v_1-v_2]∕\lambda$ , $v_{\mathit{cu}}=\text{initial}$ crush-up velocity (positive if upward); $m_s=\text{mass}$ of one floor slab; $m_0,m_1=\text{masses}$ of the upper Part C and of the story that was the first to collapse (not including the floor slab masses), $m_2=\text{mass}$ of a single story; $\lambda =\mu \left(z\right)(1-{\kappa }_{\mathrm{out}})∕{\mu }_c=\text{mass}$ compaction ratio where ${\mu }_c=\text{specific}$ mass of compacted layer (per unit height), which is constant, $\mu \left(z\right)=\text{specific}$ mass at $z$ in the initial intact state ( ${\kappa }_{\mathrm{out}}=\text{mass}$ shedding ratio, as defined in the paper); and $\Delta E_c=\text{energy}$ loss attributable to comminution of materials, predominantly concrete, into small fragments during impact. This energy has been calculated as $0.35m_s(v_1{}^2+v_{\mathit{cu}}{}^2)$ by using the theory of comminution (Bažant et al. 2007). Eqs. (1) - (2) assume that the momentum density varies linearly throughout the compacted layer B, and that, when the crushing front starts to propagate upward, the falling Part C moves downward as a rigid body, except that its lowest story has momentum density varying linearly (i.e., homogenized) throughout the story.

During impact, $\lambda =0.2$ for the North Tower and 0.205 for the South Tower. For the North or South Tower: $m_0=54.18\cdot {10}^6$ or $112.80\cdot {10}^6\mathrm{kg}$ , $m_1=2.60\cdot {10}^6$ or $2.68\cdot {10}^6\mathrm{kg}$ , $m_2=3.87\cdot {10}^6$ or $3.98\cdot {10}^6\mathrm{kg}$ , and $m_s=0.627\cdot {10}^6\mathrm{kg}$ for both. For a fall through the height of the critical story, by solving Eq. (2) of Bažant et al. 2007, one obtains the crush-front velocity $v_0=8.5\mathrm{m}∕\mathrm{s}$ for the North Tower and $8.97\mathrm{m}∕\mathrm{s}$ for the South Tower.

The solution of Eqs. (1) - (2) yields the following velocities after impact: $v_1=6.43$ or $6.80\mathrm{m}∕\mathrm{s}$ , $v_2=4.70$ or $4.94\mathrm{m}∕\mathrm{s}$ , and $v_{\mathit{cu}}=2.23$ or $2.25\mathrm{m}∕\mathrm{s}$ 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 $x\left(t\right)$ and $z\left(t\right)$ of the crush-down and crush-up fronts are used, these equations can easily be shown to have the following forms:where the superior dots denote derivatives with respect to time $t$ ; $l=\int _x{}^z\lambda \left(s\right)ds=\text{current}$ height of the compacted layer of rubble; $m\left(x\right)=\int _0{}^x\mu \left(s\right)ds=\text{all}$ the mass above level $x$ ; $g=\text{gravity}$ acceleration; and $F_c$ and $F_c^{\prime }$ are the normal forces in the crush-down and crush-up fronts (note: these are internal forces, the use of which ensures that Newton’s third law will automatically be satisfied). The cold-steel strength is used for the story below the critical one, and a 15% reduction in steel strength due to heating is assumed for the story above the critical one.

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, $\dot{x}<0$ , 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 equivalent Maxwell line that corresponds to a uniform resisting force $F_c^{\prime }$ cannot be sufficiently accurate to study the beginning of two-way crush. Therefore, a solution more accurate than that in the paper has been obtained on the basis of Eqs. (3) - (4). In that solution, the variation of the crushing force $F_c^{\prime }$ within the story was taken into account, as shown by the actual calculated resistance force labeled $F\left(u\right)$ in Fig. 3 of the paper, by the force labeled $F\left(z\right)$ 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 $F\left(u\right)$ was calculated from Eq. 8 of Bažant 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., $\mid \dot{x}\mid$ 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 velocity 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 $37\mathrm{mm}$ for the North Tower and $26\mathrm{mm}$ 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.

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 iswhere $F_c=\text{normal}$ force at the crush-down front; $m_c=\text{mass}$ of the compacted zone B; $v_B=[(1-\lambda \left(z\right))\dot{z}+\dot{z}]∕2=\text{average}$ velocity of zone B; and ${\dot{v}}_B=\text{its}$ acceleration. The acceleration ${\dot{v}}_B$ rapidly decreases because of mass accretion of zone B and becomes much smaller than $g$ , converging to $g∕3$ near the end of crush down (Bažant et al. 2007). This is one reason that $F_c$ is much larger than $F_c^{\prime }$ . After the collapse of a few stories, mass $m_c$ becomes enormous. This is a further reason that the normal force $F_c^{\prime }$ in the supposed crush-up front becomes much smaller than $F_c$ in the crush-down front. When the compacted zone B hits the ground, $v_B$ suddenly drops to zero, the force difference $\Delta F$ suddenly disappears, and then the crush-up phase can begin.

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 (deformation 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 $F_c^{\prime }$ acting on Part C upward can easily be calculated from the dynamic equilibrium of Part C (see Fig. 2g), and it is found that $F_c^{\prime }$ 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.

Variation or Mass and Column Size along Tower Height: This variation was accurately taken into account by Bažant 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.

Were the Columns in the Stories above Aircraft Impact Hot Enough to Fail? At one point, the discusser argues that the “steel temperatures . . . may not have exceeded $250°\mathrm{C}$ ,” but at another point he argues for the opposite, namely that “the heating of the upper floors would mean that the steel components were, if anything, weaker and more likely to fail (crush up) than the relatively cooler components that made up the intact lower structure of each building.” If heating weakened these components, the steel temperature would have had to exceed $250°\mathrm{C}$ . The discusser cannot have it both ways.

It is not difficult to understand why, in the stories above the aircraft impact zone, the steel could not have attained a temperatures greater than $>350°\mathrm{C}$ , which are necessary to cause creep under stresses in the service stress range. although, according to NIST (2005), most of the thermal insulation of steel in the aircraft impact zone was stripped by flying fragments propelled by impact and fuel explosion, nothing comparable could have occurred in the higher floors. Therefore, it must be assumed that most of the steel in the stories above the aircraft impact zone did not lose its thermal insulation. Consequently, the steel temperature in those stories could not have become dangerously high in less than the duration of the standard ASTM fire, which is $4\text{hours}$ . Also, since the aircraft impact caused no serious damage to the columns in the higher stories, the stresses attributable to gravity load on these columns must have been in the service stress range, i.e., less then 30% of the yield strength of steel.

Steel Temperature and NIST Report: The discusser’s statement that the “steel temperatures . . . may not have exceeded $250°\mathrm{C}$ ” is not a fact but a conjecture. It is neither supported nor contradicted by observations. The NIST (2005) report (Part NCSTAR-1, Chapter 6, p. 90) states that only 1% of the columns from the fire floors were examined for paint cracking attributable to thermal expansion. Examination of 170 areas (spots of unspecified size) on 16 perimeter columns did show evidence of temperatures greater than $250°\mathrm{C}$ , but only on three perimeter columns, and it is not clear whether this temperature occurred before or after collapse. Only two core columns had sufficient paint to conduct such an examination, and on these no temperature greater than $250°\mathrm{C}$ was documented. But NIST cautions that “the examined locations represent less than about one percent of the core columns located in the fire-exposed region.” So it is a misrepresentation of evidence to assert that, among the remaining many hundreds of unexamined columns in the aircraft impact zone, none suffered higher temperatures.

Writing about the collapse process, the discusser misinterprets the NIST (2005) report in stating that “NIST documents that steel temperatures were below $600°\mathrm{C}$ .” Steel exposures to lower as well higher temperatures were documented, and NIST (2005) (Part NCSTAR 1-3, Sec. 9.4.5, p. 132) cautions: “It is difficult or impossible to determine if high-temperature exposure occurred prior to or after the collapse.” So nothing has been documented with certainty by direct observations, as far as steel temperatures prior to collapse are concerned.

Nevertheless, a potent logical argument that steel in the critical story was exposed to high temperatures before collapse is that the collapse calculations based on the idea of thermally influenced delayed failure of columns and on the knowledge of thermal properties of structural steel are in excellent agreement with the videos of initial motion history of the top part of both towers, with the durations of collapse deduced from seismic records, with the observed comminution (or pulverization) of concrete, and with the high velocity of ejected air implied by videos of rapidly expanding dust clouds (Bažant et al. 2007).

Were Very High Temperatures Necessary to Trigger Gravity-Driven Collapse? Not necessarily. It suits critics to claim that Bažant et al.’s conclusions are contingent on the hypothesis of very high steel temperatures and to attack this hypothesis as if it were the Achilles heel of these conclusions. However, the discussers overlook two crucial facts: (1) After the aircraft impact, the stresses in some columns must have increased much above the range of service stresses attributable to gravity, which are generally less than 30% of the yield strength (the stresses attributable to wind loading were zero); and (2) the yield strength of steel is not independent of temperature. The tests reported by NIST (2005, part NCSTAR 1-3D, p. 135, Fig. 6–6) show that, at temperatures $150°\mathrm{C}$ , $250°\mathrm{C}$ , and $350°\mathrm{C}$ , the yield strength of the steel used was reduced by 12%, 19%, and 25%, respectively. Hence, any column loaded to 88%, 81%, and 75% of its cold strength, respectively, must have lost its capability to resist load soon after it was heated to the respective temperature.

Although the stress values in various columns of the critical story have not been determined, it cannot be ruled out that the loads of many of the remaining columns were raised after aircraft impact above 90% of their yield strength. So, if the stress in a critical column was close enough to yield stress, it is not inconceivable that even a rise of steel temperature to mere $150°\mathrm{C}$ might have triggered progressive collapse of the whole tower.

The fact that some perimeter columns showed gradually increasing lateral deflections, reaching as much as $55\mathrm{in.}$ (or $1.40\mathrm{m}$ ) [NIST (2005), part NCSTAR-1, Chapter 2, p. 32 and Fig. 2–12], cannot be explained as anything other than creep buckling of heated columns. In this regard, it should further be noted that the multistory bowing implies a great decrease of the critical load for creep buckling, $P_{cr,t}\approx R_t{\pi }^2∕L_{\mathrm{eff}}^2$ , where $R_t=\text{effective}$ long-time bending stiffness of column, taking into account creep; and $L_{\mathrm{eff}}=\text{effective}$ buckling length. The visible bowing of columns appears to have spanned about three stories, which means that $L_{\mathrm{eff}}$ approximately tripled, indicating that $P_{cr,t}$ may have decreased by a factor of $1∕9$ and thus may have become much less than the plastic limit load of column.

What must have caused the loads of many of the remaining columns to be raised far above the service stress range and close to their load capacity is the load redistribution among the columns of the aircraft-impacted story. The asymmetry of damage within this story caused a shift of the stiffness centroid far away from the geometrical center of the tower, and thus the gravity load resultant $m_{0}g$ in that story developed a large eccentricity $e$ with respect to the stiffness centroid. The resulting bending moment $\left(m_{0}g\right)e$ reduced the column loads on the less damaged side of the critical story but greatly increased them on the heavily damaged side, where the load was carried by fewer remaining columns (the fact that the collapse came earlier for the South Tower, in which the eccentricity of aircraft impact was greater, corroborates this viewpoint). During the fire, the stresses in many columns on the more damaged side of the critical story were probably very close to the yield strength value of cold steel. Therefore, even a mild decrease of yield strength, by 5% to 20% after prolonged heating, sufficed to trigger progressive collapse.

The decrease of yield stress upon heating depends strongly on the rate of loading or on its duration, and is properly described as time-dependent flow, or viscoplastic deformation. For $1\text{hour}$ of loading, the decrease is much greater than it is for the typical duration of laboratory tests of strength, which is of the order of $1\text{minute}$ . In columns, the flow leads to time-dependent buckling, which is in mechanics called viscoplastic buckling or creep buckling. A temperature rise to $250°\mathrm{C}$ at high stress level can greatly shorten the critical time $t^{*}$ of creep buckling.

Some critics do not understand the enormous destabilization potential of creep buckling. The Dorn-Weertmann relation indicates that $\dot{ϵ}=A{\sigma }^n{e}^{-Q∕kT}$ (where $\dot{ϵ}=\text{strain}$ rate; $A=\text{constant}$ ; $n\approx 5$ ; $Q=\text{activation}$ energy of interatomic bonds; and $k=\text{Boltzmann}$ constant; Hayden et al. 1965, Eq. 6.8; Courtney 2000; Cottrell 1964; Rabotnov 1966). According to Choudhary et al. (1999), the typical value of $Q∕k$ for ferritic steel alloys is about $\mathrm{10,000}°\mathrm{K}$ (and about $\mathrm{20,000}°\mathrm{K}$ according to Frost and Ashby 1982). Using $\mathrm{10,000}°\mathrm{K}$ , one may estimate that, upon heating from $25°\mathrm{C}$ $(T_0=298°\mathrm{K})$ to $250°\mathrm{C}$ $(T=523°\mathrm{K})$ , the rate of deformations attributable to dislocation movements increases about ${10}^6$ times, and more than that when using $\mathrm{20,000}°\mathrm{K}$ . For heating to $150°\mathrm{C}$ , the rate increases about ${10}^4$ times. This rate is what controls the rate of flow and, indirectly, the yield strength upon heating.

Furthermore, the equations in the aforementioned sources and those in Sec. 9.3 of Bažant and Cedolin (2003) make it possible to calculate that raising the column load from $0.3P_t$ to $0.9P_t$ (where $P_t=\text{failure}$ $\text{load}=\text{tangent}$ modulus load) at temperature $250°\mathrm{C}$ $(T=523°\mathrm{K})$ shortens the critical time $t^{*}$ of creep buckling from $\mathrm{2,400}\text{hours}$ to about $1\text{hour}$ (note the differences in terminology: material scientists distinguish between the microstructural mechanisms of creep, occurring at low stress, and of time-dependent flow, occurring near the strength limit, whereas in structural mechanics, the term creep buckling or viscoplastic buckling applies to any time-dependent buckling regardless of microstructural mechanism; thus the source of creep buckling of steel columns at high stress is actually not creep, as known in materials science, but time-dependent flow of heated steel at high stress).

Recently reported fire tests (Zeng et al. 2003) have demonstrated that structural steel columns under a sustained load of about 70% of their cold strength collapse when heated to $250°\mathrm{C}$ . However, creep of structural steel in the service stress range begins only after the steel temperature rises above $350°\mathrm{C}$ (Cottrell 1964, Frost and Ashby 1982, Huang et al. 2006).

The aforementioned crude estimates suffice to make it clear that the combination of asymmetric load redistribution among columns in the aircraft impacted stories with the heating of steel to about $250°\mathrm{C}$ (or even less) was likely to lead to a loss of stability attributable to creep buckling of the most overloaded columns within the observed time. Given the sustained elevated temperature caused by the stripping of insulation and the severe and asymmetric damage to many columns, as estimated in the NIST report, it would, in fact, be rather surprising if the towers did not collapse.

It would certainly be interesting to find out whether the steel temperatures were nearer $250°\mathrm{C}$ or $600°\mathrm{C}$ ; but for deciding whether the gravity-driven progressive collapse is a viable hypothesis, the temperature level alone is irrelevant. It is a waste of time to argue about it without knowing the stresses. If the stress in the column whose failure caused the critical floor to lose stability was greater than 90% of the cold yield strength, a mere $150°\mathrm{C}$ would have sufficed to trigger overall collapse; and if this stress was 75%, $350°\mathrm{C}$ would have been necessary. None of these situations can be excluded without precise calculations of the stress evolution in all the columns in the heated critical story. Feasible though such calculations are, they would necessitate a laborious extension of the study by NIST.

It was hypothesized that the lateral bowing of perimeter columns was caused mainly by a horizontal pull from steel trusses sagging because of differential thermal expansion. However, this hypothesis is not credible. As simple calculations show, the temperature difference between the lower and upper flanges of a floor truss would have to exceed $\mathrm{1,000}°\mathrm{C}$ to produce a curvature that would shorten the span of a sagging floor truss by $52\mathrm{in.}$ $\left(1.40\mathrm{m}\right)$ . Such a temperature difference is inconceivable. The differential thermal expansion must have been only a secondary triggering factor, which created a small initial imperfection in the overloaded columns, to be subsequently drastically magnified by creep buckling.

Load-Displacement Curve of Columns and Energy Absorption Capacity: The discusser’s curve of axial load $P$ versus axial displacement $u$ of a column (sketched in his figure 1(c) and redrawn to scale as the upper curve in Fig. 2 right is not correct and grossly overestimates the energy dissipation in the column (note that what the discusser denotes as $P$ is in the paper denoted as $F$ ). The correct curve (Fig. 2, left), based on the theory of plastic large-deflection buckling (Bažant and Cedolin 2003, Sec. 8.6), is given by Eq. 8 in Bažant and Zhou (2002) and readsfor $0\le u<\lambda L$ ( $\lambda =\text{compaction}$ ratio, assumed to be 0.2; $m_0=\text{mass}$ of falling top part of tower; $A=\text{cross}$ -sectional area of column; $L=\text{its}$ height; $M_0=\text{its}$ plastic yield moment; $u=\text{axial}$ shortening of column;). The expression ${\Pi }_1=4M_0\theta$ , which is used in the discussion to calculate the energy dissipation in the plastic hinges of each column, is correct. But the subsequent discussion consists of incorrect arguments that enormously exaggerate the estimate of the kinetic energy of the upper part of tower required to trigger progressive collapse.

As a result of all these erroneous arguments, it must now be concluded that the energy dissipation, ${\Pi }_1$ , of one column is about 44% of that calculated from the discusser’s input values but the correct $P\left(u\right)$ curve. This becomes only about 15% if the excessive yield strength of upper columns assumed by the discusser is corrected and if the $P\left(u\right)$ curve is scaled down to approximately account for the average column weakening by fracture and local flange buckling (as done in Bažant et al. 2007).

Does Excess of ${\mathit{P}}_{\mathit{cr}}$ over Gravity Load ${\mathit{m}}_0\mathit{g}$ Imply Arrest of Collapse? Not at all, and this point is generally misunderstood by critics. In the discussion section entitled “Initial Phase of Collapse—Heavily Damaged Story,” the premise that “to cause initiation of failure, the buckling force $P_{\mathit{cr}}$ had to be reduced to the level of applied load,” would be correct in statics but not in dynamics, where the inertia forces must be taken into account, according to the d’Alembert principle.

The fact that $P_{\mathit{cr}}$ exceeds the applied load (i.e., $P_{\mathit{cr}}>m_{0}g$ ) does not mean that the motion of the falling mass would get instantly arrested (which would require an infinite upward acceleration $\mid \ddot{u}\mid$ and thus an infinite resisting force). Rather, it simply means that the downward motion will continue as decelerated (Fig. 3) until the sum of the resisting forces $P\left(u\right)$ of all columns of the story (which begins with the plastic critical load $P_{\mathit{cr}}$ ) drops below $m_{0}g$ . After that, the resistance $P\left(u\right)$ becomes less than $m_{0}g$ , which means that the downward motion will be accelerated. This is clear from the calculated diagrams shown in the second and third rows of Fig. 4 in the paper.

Misled by the omission of inertia force (Fig. 3), the discusser reduces the critical load $P_{\mathit{cr}}$ by the factor of 5.59 to make it equal to the gravity load (Fig. 2, right). This is impossible. The column strength is an objective property of the material and of the column geometry and not some fictitious property that can be adjusted according to the load to achieve static equilibrium.

Equally arbitrary and incorrect is the discusser’s scaling down the entire descending part of resisting curve $P\left(u\right)$ , in which he assumes that the minimum of $P$ and the entire rising part of $P\left(u\right)$ should be scaled down by the factor of 2 (the lower curve in Fig. 2, right). The resulting column resistance curves $P\left(u\right)$ are compared in Fig. 2. Note that, in spite of the scaling down, the area under this $P\left(u\right)$ -curve (Fig. 2, right), representing the energy absorption capability, is still much greater than the area under the correct $P\left(u\right)$ -curve (Fig. 10, left). The reason is that the parabolic shape is very different from the correct shape for large-deflection buckling, and that the rising part of the curve should not be present at all.

The present corrections to the calculation of energy absorption capability of a column are consistent with the value originally given in Eq. 3 of Bažant and Zhou (2002). The energy absorption capability of all the columns of the first cold story in crush down represents only about 12% of the kinetic energy of impact of the upper part.

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 $d\left(mv\right)∕dt=m_{0}g$ (in the discusser’s notation, $m$ is $M$ , and $m_0$ 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 $\mu z$ , 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 $d\left[m\left(t\right)v\right]∕dt=m\left(t\right)g$ , 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 $F_c$ 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 $23.8\mathrm{s}$ 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 calculated in the paper for the most realistic choice of input values is in agreement with the observations. Moreover, a more accurate analysis by Bažant et al. (2007) is found to be in nearly perfect agreement with the video records 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 gradually 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 diffracted will be weakened elastic waves.

Thus it is not true that “during such reflections, 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.