Friday, March 02, 2007

Energy sums for the twin towers

Are roof-to-ground collapses plausible?

WTC1
Height: 420 meters

Mean distance per floor: 3.82m

Collapse began at: floor 94

Mass of top block:
0.145M or less, where M is the mass of the entire building (we have neglected the mass of the airliner)

Energy required to keep top block in place:
mgy = 0.145M(9.8)(359.08)meters = (510.25M)Joules

Energy inherent in fail-safe design to keep top block in place:
2mgy or more = (1020.5M)J or more

Energy converted to entanglement or damage energy after the top block falls one story onto the bottom block:
Less than 1/2mv2 = 0.5(0.145M)(8.65)2 = (5.43M)J

This represents less than 1 percent of the normal force energy of (510.25M)J.

Remaining potential energy in top block at time of collision:
mgy - 1/2mv2 = (504.82M)J.

Considering that such a small amount of energy was available to inflict structural damage, it seems problematic that the damage energy was not dissipated rapidly near the top of the underlying block, implying a collapse of no more than a few floors.

Yes, there remained (504.82M)J that could be converted into the kinetic energy of a fall to ground level, but that was counterposed by the normal force energy. Only if the damage energy resulted in large-scale and swift dissipation of the normal force could the observed collapse have occurred. But the amount of damage energy seems inconsistent with that result.

WTC2
Height: 417m

Mean distance per floor: 3.79m

Collapse began at: floor 82

Mass of top block: 0.25M or less (again we neglect the airliner mass)

Energy required to keep top block in place:
mgy = 0.25M(9.8)(310.78) = (761.4M)J

Energy inherent in fail-safe design:
at least 2mgy = (1522.8M)J

Energy converted into entanglement or damage energy after the top block falls one story:
Less than 1/2mv2 = 0.5(0.25M)(8.62)2 = (9.29M)J

This represents 1.2 percent of the underlying structure's minimal normal force energy.

Remaining potential energy in top block at time of collision:
mgy - 1/2mv2 = (752.11M)J

Again, it seems problematic that the damage energy wasn't rapidly dissipated near the top of the underlying block, implying collapse of no more than a few floors. In other words, the potential energy and the normal force energy are counterposed prior to impact. So, for the observed collapse to occur, the damage energy must have inflicted swift and largescale dissipation of the normal force, a result that seems inconsistent with the low amount of damage energy available.

The NIST doesn't really start the collapses as described but says "shortened" core columns dragged the floors inward, precipitating full collapse. However, once collapse is under way, a top-down scenario must ensue. The question is then, at what floor does gravitational collapse begin?

But, this is a non-issue since the inequality 1/2mv2 greater than or equal to mgy requires, in our case, y less than or equal to about 3.8 meters.
That is, collapse would have had to have begun near the foundation (which would imply explosives). On the other hand, if y greatly exceeds 3.8m, then the quantity of normal force energy overwhelms the quantity of damage energy.

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