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WHAT IS IT?
This is a model of the fall of a building such as a world trade center tower. A free fall is compared with a collapse, since the recorded fall times for the world trade center buildings correspond to a free fall, while the falls have been explained officially as a pancake type of collapse. In the simulation shown here, there are two mechanisms by which a collapse is retarded. First, the fall is retarded in a somewhat passive way by the presence alone of stationary mass in lower floors which opposes the momentum of the falling mass. Second, the fall is retarded in a somewhat active way, for example, by the fastening of each floor or by the bonds within materials which require energy for pulverization. The user may examine these effects by controlling how much mass is turned to dust and how strong the floors are. These effects may be controlled by selecting pulverization and floor strength which seem natural as well as by raising supplemental factors which enhance pulverization and which weaken floors. It remains for the user to decide whether the pancake collapse theory agrees with the recorded free fall times.
HOW IT WORKS
Pressing the setup button builds two towers and crashes an airplane into each. After the start button is pressed, the tower on the left falls by collapse and the tower on the right falls freely. The airplanes crash each building at the same height, and the floors above the crash point are assumed to fall downward as a single mass. In the collapsing building on the left, the combined mass of upper floors starts with zero initial velocity and falls freely until the next floor is reached. After the next collision, part of the mass is pulverized and ejected to the side of the building. If the momentum of the falling mass is large enough to overcome the floor below, the falling mass is increased by the unpulverized portion of the next floor, and the combined mass has a downward velocity determined by the conservation of momentum, necessarily less than the velocity of the falling mass before the collision. This collapse process continues until the crash point hits the ground, and then the remaining floors continue in free fall. In the free-falling building on the right, the fall proceeds as a stone falling from the top floor with zero initial velocity.
HOW TO USE IT
The number-of-floors of the buildings is by default 110, but it may be set up to 120. The crash-floor is by default 100, and it may also be set up to 120, but the code constrains the crash floor to be not greater than the number of floors. The height of a single floor, floor-height, is by default 3.79m, and it may be set between one and four meters. The mass-fraction-turned-to-dust is the fraction of the next floor which is pulverized by collision with the falling mass. On the one hand, this fraction can be expected to depend upon the kinetic energy of the falling mass; however, since so much of the world trade center towers was turned to dust, this parameter is assumed to be constant with each collision and can be set by the user. Maximum pulverization can be selected by the user by choosing either mass-fraction-turned-to-dust = 1 or supplemental-mass-pulverization = 100%. The floors-needed-to-break-next-floor is the minimum number of floors required to free fall from a minimum height of floor-height in order to overcome the next floor. The associated downward force is equal but opposite to the force exerted upward when the next floor is overcome by falling mass with sufficient momentum. The floor may be weakened maximally by choosing either floors-needed-to-break-next-floor = 0 or supplemental-floor-weakening = 100%. The fall time of the collapsing building is shown in the Collapse-Time monitor while the fall time of the free falling building is shown in the Freefall-Time monitor. The respective fall trajectories of the two buildings are shown in the plot as Collapse-Height and Freefall-Height.
THINGS TO NOTICE
The fall time of the collapsing building is of course never greater than that of the free falling building. The fall time and the trajectory of the free falling building are affected only by the number of floors and the floor-height. The fall time and the trajectory of the collapsing building are affected by all parameters. In particular, the number of dust clouds increases as pulverization is increased by whatever means. Notice the dependence of the fall time of the collapsing building on the parameters in the following graph:
Specifically, as the number of floors-needed-to-break-next-floor approaches 0, the Collapse-Time decreases as the mass-fraction-turned-to-dust increases. On the other hand, as the number of floors-needed-to-break-next-floor exceeds 1, the Collapse-Time increases strongly as the mass-fraction-turned-to-dust increases - note that the Collapse-Time scale runs up to 100 seconds.
THINGS TO TRY
Try to determine the parameters with which the collapsing building falls at the same speed as the free falling building. (These can also be read from the graph above.) Specifically, all resistance to the falling mass must be eliminated.
EXTENDING THE MODEL
As seen in the paper by Kenneth Kuttler and the references cited therein, there are many natural ways in which to refine the model. The assumption that the floors above the crash point fall downward as a single mass can be refined to allow these floors to collapse independently. The assumption that the mass-fraction-turned-to-dust is constant with each collision can be refined with a dependence of this fraction upon the kinetic energy of the falling mass. Also, the fastening strength of each floor could be estimated in terms of standard safety factors, and the kinetic energy required to pulverize concrete could be estimated in terms of a distribution of sizes in dust particles produced.
NETLOGO FEATURES
The present model is also essentially one-dimensional and thus does not take advantage of the multi-agent simulation capability of netlogo which should allow the independent computation of the motion of many building components.
RELATED MODELS
In addition to the paper by Kenneth Kuttler cited above, see also the paper by Judy Wood. Her models correspond to the case in the present model that mass-fraction-turned-to-dust = 1. For an extensive physical criticism of the official explanations of the events of September 11, 2001, see this paper by Steven Jones as well as other articles at the Journal of 911studies. See other discussions of scientific interest in the film Improbable Collapse with homepage here and viewable directly here.
CREDITS AND REFERENCES
This model has been developed in the course of modeling classes at the University of Graz as well as at the modeling week in Styria. Further mathematical details can be found here.
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