But his assertion that Bazant's calculation is a limiting case is a naked assertion. There is no detailed analysis demonstrating how it must by necessity limit or box in the actual case.
Here is an excerpt from a paper by Mackey where he discusses the point:
On the other hand, there have
been several published results in support of the progressive collapse hypothesis. Perhaps
the best known is from Drs. Bazant and Zhou, who concluded the following, regarding
the situation after the first floor’s collapse:
To arrest the fall, the kinetic energy of the upper part, which is equal to the potential energy
release, would have to be absorbed by the plastic hinge rotations, i.e., Wp would have to be larger
than Wg. Rather,
Wg / Wp = 8.4 (3)
So, even under the most optimistic assumptions by far, the plastic deformation can dissipate only a
small part of the kinetic energy acquired by the upper part of the building.
And, regarding the second and successive floor collapses:
When the next buckle with its group of plastic hinges forms, the upper part has already traveled
many floors down and has acquired a much higher kinetic energy; the percentage of the kinetic
energy dissipated plastically is then of the order of 1%. The percentage continues to decrease
further as the upper part moves down. (64)
This is part of the reason why NIST did not consider the entire duration of the collapses.
Early results from engineers and scientists indicated that, once the upper stories began to
fall, the complete collapse of the structure was not in doubt, and there was no credible
result to the contrary. There still are none.
http://www.911myths.com/drg_nist_review_2_1.pdfBazant's statement that his calculation looks at the event "even under the most optimistic assumptions by far" is not demonstrated. It is merely asserted.
Bazant's alleged limiting case artificially creates an "all or nothing" straight-on, perfectly vertical collision between the upper and lower blocks. While it may be true that this maximizes the possibility of resistance in one way, it also minimizes the possibility of resistance in other ways. It maximizes by allowing the lower block to present its strongest possible face. But it minimizes by focusing the face-off into the shortest possible interval of time. And finally, it ignores areas of uncertainty that would need to be treated in their best case if a limit is to be established.
We all agree, I believe, that the actual interface between the upper and lower blocks was extremely different than Bazant's idealized version. Where we disagree is on the point of the relationship between the idealized version and the real version. Mackey claims that the former is limiting of the latter. But that would be demonstrated only if all aspects of the idealized version skew away from collapse and then collapse still occurs. But some aspects of the idealized version skew toward collapse and some away from it, and no demonstration has been provided for which of these unrealistic assumptions in opposite directions outweigh the others. Therefore the idealized version simply does not shed much light on the real version because there is no analytical way to establish toward which side (collapse or no collapse) the idealized version skews in the net.
On the timing aspect of the collision, Bazant's idealized straight-on, vertical collision artificially causes the two blocks to contact each other in one perfect instant of time as all the columns of the upper block contact all the columns of the lower block at precisely the same time. The artificially rigid upper block then applies all its momentum at that one instant and it is a sudden-death playoff that lasts a fraction of a second. In reality no such perfect collision would have occurred. The jagged and twisted structures would have engaged each other at a few points first and started their face-off. Then as the collision progressed some of the earliest contacting members would already fail or be failing as other members begin to make contact. The collision would obviously not occur at a single precisely timed instant as Bazant assumes but would rather extend over some larger interval of time during which a partial application of the upper block's total momentum would be applied by the imperfect, sometimes failing leading edge of the upper block.
This idealized, single-instant-of-time contact skews in favor of collapse in two ways: by focusing all the energy transfer into one fraction of a second and by erroneously assuming there is no structure failure in the upper block. The energy transfer would in fact be spread out over time, allowing the early-failing members to be sacrificial elements that potentially save later contacting elements. And the structure failures in the upper block would be an additional energy sink that is unaccounted for; another way in which Bazant's idealized version skews toward collapse.
Bazant's idealized case ignores at least one obvious aspect of uncertainty introduced by the totally unknown mechanics of how the upper and lower blocks actually interface with each other. The particulars of this interface could accidentally cause the lower block members to fail first, the upper block members to fail first, or some combination of the two. It is at least arguable that the visual evidence shows that the upper block tended to fail more than the lower and that the failure of the upper block structure lasted for at least several seconds during which the upper block descended through the height of multiple floors. If true, such a collision would provide a buffering effect such that the application of force by the upper block onto the lower block was spread out over several seconds. This spreading out of the force application may be as much as several orders of magnitude less focused than Bazant's artificially precise straight-on perfect collision.
A more likely candidate for a truly limiting case would be an assumption that the upper block behaved in a way that was the opposite of Bazant's assumption that it is rigid. The best case for lower block survival might be that the upper block is unlucky in each of its unpredictable encounters with the lower block and that the upper block members tended to fail quickly and without being able to get a good purchase on the lower and give it a shove. A series of glancing blows, if you will, in which the upper is always unlucky and snaps off relatively harmlessly. The net effect of this limiting case could then be approximated by treating the upper block as a totally disconnected swarm of individual elements that are going to rain down on the lower block. If Bazant or someone else can show that such a model would result in collapse then they may have succeeded in producing a limiting case. But Bazant, Zhou is not a limiting case; rather, it is a case that they knew all along was rigged to result in collapse.