Closure to “What Did and Did Not Cause Collapse of World Trade Center Twin Towers in New York?”
by Zdeněk P. Bažant, Jia-Liang Le, Frank R. Greening, and David B. Benson
The discusser’s interest is appreciated. However, he presents no meaningful mechanics argument against the gravity driven progressive collapse model of our paper. His claim that “the authors’ theory is wrong” is groundless. Briefly, the reasons are as follows:
Equations of Motion
The discusser claims that no differential equations are required to model the collapse. This is incorrect. The intuitive guesses emanating from his disconnected quantitative estimates prove nothing. Although the discusser uses some mechanics terms such as velocity and acceleration, nothing can be deduced without actually formulating and solving the equations of motion. If the discusser rejects the differential equation form of the equations of motion based on a smeared continuum approximation, he could be credible only if he formulated and solved discrete equations of motion.
Energy Dissipation Sources
The discusser claims that the progressive collapse model we developed in the paper does not consider the energy required to compress the rubble. This claim is absurd. He apparently overlooked that this energy is included in parameter γ of Eq. (11). On p. 898 of the discussed paper it is stated that, aside from the energy of comminution, parameter γ includes “the energy of plastic fracturing deformations of floor trusses with their connections and of horizontal steel beams connecting the perimeter columns, the energy dissipated by inelastic deformation and friction of colliding fragments, the energy of crushing the equipment, drywalls, perimeter walls, furniture, piping, etc.”
However, based on simple estimates of the surface areas of all the fractures, fracture energies on these surfaces, plastic strain magnitudes, magnitudes of frictional forces in collisions, and frictional slip distances, it transpires that the combined energy dissipated by the aforementioned processes is much smaller than the energies required for the comminution of concrete into particles, for the ejection of air, dust, and fragments at high speed (representing the work of F_a and F_e). The reason for the dominance of the energy of comminution of concrete is the extremely small size of the particles, ranging from 10 to 100 μm in size, which causes the combined surface energy of these particle to be enormous. All these energies, in turn, are small compared to the energy of plastic buckling of the massive stocky columns (work of F_b), and that energy is again smaller than the energy required to accelerate downward the accreted stationary mass at the crushing front [F_m in Eq. (5)].
Therefore, it is not important to know parameter γ accurately. It was mentioned in the paper that, within the range γ between 0.6 and 1, the calculation results for the motion history, the time to hit the ground, and the amount and size distribution of particles are virtually indistinguishable. So it makes no sense to argue about the precise energy dissipation by the aforementioned secondary processes.
The overall energy balance is ensured by deriving the differential equations of motion from an energy potential [see Eqs. (25)–(30) of Bažant and Verdure 2007]. The necessity of gravitydriven progressive collapse is demonstrated by the fact that the kinetic energy of impact on each floor far exceeds the energy absorption capability of the underlying columns [Eq. (3) in Bažant and Zhou 2002].
Crushing of Columns
The discusser further claims that, for the continuation of the crush-down phase, the columns in the part C (upper part) must be assumed to be in contact with the columns of part A (lower part). This claim is erroneous. During the crush-down collapse, part C and part B (compacted layer) are moving together as a whole while part A is being crushed by the compacted layer B [schematically, see Fig. 2(b) of the paper]. The energy condition for the crush-down phase to continue is given by Eq. (6) of Bažant and Verdure (2007) (and the gravity driven crush-down is actually guaranteed to occur by Eq. (3) in Bažant and Zhou 2002).
Video and Direction of Crushing
Observation of the upper margin of the cloud of dust and smoke in the videos somehow makes the discusser conclude that the tower top motion is caused by “part C becoming shorter while part A remains intact.” This is a delusion. Part A remaining intact would violate the principles of conservation of momentum and of energy. The writers’ analysis of the initial two-way collapse shows that the columns of part C get plastically squashed by only 1% of their original length and afterward the collapse proceeds in a one-way crush-down mode (Bažant and Le 2008).
The compacted layer cannot be expected to be seen in the video record. Similar to construction demolitions, it is not, and cannot be, located just under the upper margin of the cloud because the rapidly ejected air and dust spreads both downward and upward [Fig. 3(a) in the paper].
Rubble Pile
Based on the profile of the rubble pile shown in Fig. 3(b) of the paper, the discusser estimates the rubble density to have an unrealistic value (3.075 t/m³). Since this figure is only schematic, his point is meaningless. Besides, he ignores the fact that much of the rubble (characterized by mass shedding coefficient κ_out ≈ 0.2 in the paper) has been ejected during the crush-down and that the tall and narrow pile as sketched exists only for a split second just before the moment at which layer B hits the ground. At that moment, the pile immediately begins to spread rapidly outside the tower footprint. If one assumes the rubble pile density to remain constant during spreading, a simple calculation shows the rubble to spread about 60 m outside the tower footprint. This gives for the rubble pile a slope of about 20°, which agrees well with the typical slope of rubble piles seen in the demolitions of buildings.
References
- Bažant, Z. P., and Le, J.-L. (2008). “Closure to ‘Mechanics of progressive collapse: Learning from World Trade Center and building demolitions’ by Zdeněk P. Bažant and Mathieu Verdure.” J. Eng. Mech., 134(10), 917–923.
- Bažant, Z. P., and Verdure, M. (2007). “Mechanics of progressive collapse: Learning from World Trade Center and building demolitions.” J. Eng. Mech., 133, 308–319.
- Bažant, Z. P., and Zhou, Y. (2002). “Why did the World Trade Center collapse? Simple analysis.” J. Eng. Mech., 128(1), 2–6.
