Modeling Aspects of the Twin Towers' Collapse
One might think that the unprecedented way in which the Twin Towers exploded into dust might warrant the creation of some computer models to better understand this behavior. FEMA's investigation appeared to be timed to coincide with the site cleanup. With the rapid pace of operations there apparently wasn't time for any computer models.
NIST's simulations model phenomena such as the impact fireballs and smoke plumes that rose from the Towers, and the pattern of damage caused by the impact of jetliners. Conspicuously missing from their study are simulations, models, or even calculations that attempt to describe how and why the Towers came down. With tens of millions of dollars at its disposal, NIST couldn't spend a few thousand dollars to study progressive collapse, the new-found phenomenon that accounted for the total destruction of all three skyscrapers, WTC 1, 2, and 7.
Modeling Total Collapse Time
One aspect of the destruction of the Twin Towers that suggests controlled demolition as a cause is the speed of their descent. In a passage noting the events' rapidity, an article in Scientific American quotes Eduardo Kausel as stating, "The towers' resistive systems played no role. Otherwise the elapsed time of the fall would have been extended." However, even if the crushing of the Towers' intact structures below the crash zones did not slow the rubble at all, the acceleration of the stationary mass would have. Thus, if one accepts the central assumption of the pile-driver theory -- that the tower's falling mass remained aligned over the tower's intact portion as the destruction progressed downward -- the total collapse time would have been extended considerably by the time required to accelerate each floor encountered by the falling mass.
This program, created by Jim Hoffman in 2003, computes total collapse times based on parameters describing the floor on which the collapse started. The program makes the following assumptions, all of which favor short collapse times.
- Each floor's support vanishes when touched by the falling block.
- Momentum is conserved.
- None of the kinetic energy of the falling mass is diverted to other sinks (concrete pulverization, steel bending, etc.)
- Each floor is an infinitely thin slab, and all the mass of a story is concentrated in the slab.
- The overhanging portion (eg: 14 floors in the North Tower) falls as a block, with its bottom floor accumulating pancaked slabs of the once-intact floors as it encounters them.
- The accumulation of floors is inelastic.
- Once the bottom of the block reaches the ground, the floors in it start to pancake from bottom to top, the roof of the tower falling at freefall at that point.
- Mass is uniformly distributed among the stories.
- The falling block remains perfectly centered over the intact portion.
The following table summarizes the results of running the program with parameters specifying that the collapse starts at the 80th and 95th floors:
|elapsed time in seconds|
|start floor||crash zone to ground
|roof to ground
In 2006, Hoffman created a generalized version of the program that allows the removal of the last two assumptions. In particular, it allows the specification of:
- A linear increase in the mass of stories from the roof to the ground.
- The movement of a fixed fraction of the mass falling within the Tower's profile to move outside of the profile with the collapse of each story. Once mass moves outside of the Tower's profile, it does not participate in the acceleration of mass downward.
The Twin Towers, as all large steel-framed skyscrapers, had columns that became less massive at increasing elevation. This means that the Towers' upper stories were considerably lighter than the lower ones. The assumption that the lowest stories were about 1.5 times as massive as the top stories seems like a reasonable assumption. Implementing this adjustment to the model means longer collapse times, of course, since more of the mass would initially be lower in the Tower where it would have less mass underneath it to accelerate downward. However, the simulation shows that even making the mass ratio of the bottom to the top story 2.0 has relatively little effect on total collapse times.
In contrast, the second adjustment of the new model has a pronounced effect on total collapse times. Assuming that just 6 percent of the mass above the impact zone is ejected outside of the Tower's footprint for each story crushed results in a total collapse time of nearly 20 seconds, assuming the collapse started at the 95th floor.
|input parameters||elapsed time in seconds|
|start floor||mass of bottom story relative to top||mass dispersal per story||mass dispersed by end||crash zone to ground||roof to ground|