Reply #25: Of course...

[Carefulplease]

If some floor had resisted 1/4 of a second then the mass falling on it would have essentially come to rest. This means that the total kinetic energy of the falling mass would have been absorbed before that floor gave way. Is that a reasonable assumption? It seems merely to posit without argument that an average floor is capable of absorbing the energy of a significant portion of the tower (10 to 25 floors) falling on it from more than 12 feet. This is merely begging the question.

Consider that after collapse has been initiated, all the 287 core and perimeter columns have given way. So, during the fall, when the lowest intact floor of the upper block meets the highest intact floor of the lower section, the corresponding column segments from above and below are no longer supporting each other. The bulk of dynamic load of the upper block gets applied to one single floor below. This floor is attached to the columns of the lower section. The beams and trusses and their connections to the columns are designed to resist not much more than the dynamic load on *one* floor directly above falling on them. But now there are not only 10 office floors falling on them but also ten floors worth of core and perimeter columns plus other heavy structures such as the hat truss and the upper mechanical equipments. The resisting floor can at most remove some fraction F of the kinetic energy mgh = ((1/2)mv^2) of the falling mass (where F < 1). This means that the speed of the falling mass just after that floor gave way isn't smaller than v*sqrt(1-F). So, even if fully 25% of the kinetic energy of the falling upper block was consumed in disconnecting that floor, the speed of the of the upper block would only be reduced by 13.5%. (I am putting aside for the moment the effect of momentum transfer -- the energy needed to accelerate the floor below from rest to the speed of the falling stack)

Even if we disregard that the falling mass gets progressively heavier as the collapse proceeds we can see that when the upper block has fallen through n lower floors, it has gained the total kinetic energy K = n*mgh - n*F*mgh. Assuming constant loss of kinetic energy ((25%)*mgh) at each step, its final speed is thus also reduced by only 13.5%. This means that the effect of the constant resistance is to reduce the effective force of gravity by 13.5%. That is not much of an effect. The effects of momentum transfer are much more significant than this. They are discussed and calculated in a paper by Franck P. Greening referenced in another message of this thread. These effects also include the energy used up in crushing concrete and other stuff during the first stage of the collapse before the falling pile hits the ground.