Join Date: Nov 2006
Gumboot, it is possible to answer your question in a simple way, using only F=ma, if you make lots of simplifying assumptions. Let's start with your already simplified case of simply dropping the upper 1/5 of the tower through 1 storey of open air onto the lower tower.
In order for the lower portion of the tower to arrest the collapse, the upper portion of the tower, already in motion, now has to stop moving. (Even if it were possible for it to "topple to the side" at that point, its vertical motion still has to stop or else it will keep smashing downward through the lower tower instead.)
To go from falling to at-rest, the moving mass must accelerate upward. The structure it falls onto must provide sufficient force for a sufficient amount of time to accelerate the moving mass to zero downward velocity. A lot of force over a short time will do, or a lesser force over a longer time.
How much time does the lower structure have to decelerate the upper mass? Actually that's not quite the right question yet. Let's look at the moment the masses come in contact. The lower mass starts resisting, and the upper mass starts decelerating. But it can't decelerate instantly, that would take infinite force. So the upper mass is still moving after it comes in contact with the lower. The greater the resistive force, the faster the moving mass will accelerate (slow down), and the less distance it will therefore move before it comes to rest. If the resistive force is less, the moving mass will accelerate to zero over a greater distance. (If the resistive force is less than or equal to the gravitational force acting on the falling mass (that is to say, its weight), the moving mass won't slow down at all, it will keep going and possibly accelerating downward. But we expect the resistive force to be greater than the falling mass's weight, because the lower tower was designed to support that weight.)
But as long as the upper mass is still moving, the lower structure that's providing the force to slow it down is also being deformed by that movement. That deformation is going to weaken the lower structure, reducing the force with which it acts against the moving upper mass. So we can transform the question of how much time does it take for the lower structure to bring the upper portion to rest, into over what distance (amount of deformation) can the resistive force continue to act upon the upper mass? This is where it gets complicated, because different kinds of deformation (buckling, fracturing, etc.) will affect the resistance in different ways over different amounts of time and distances. But we can look at ultra-simplified models of this.
Let's say, for instance, that when the upper mass smashes into the next floor down, that next floor will fail completely, cease to offer any resistance, once it is deformed (pushed downward) by half a meter. And let's also say that until then, it resists with its full designed strength. So, the floor must resist with sufficient force to bring the upper mass to a stop in half a meter or less, or else the collapse will continue to the next floor.
In your floor-removed scenario, the upper mass accelerates freely at g for 3 meters. It must decelerate at 6g to come to rest again in .5 meters. (For any given change in velocity, the acceleration needed to cause that change in velocity over a given distance is inversely proportional to that distance.) Therefore the floor it lands on must provide a force of 6 times the (static) weight of the upper mass -- actually 7 times, because it also must resist the gravitational force that's still acting on the upper mass as it falls.
The structures, of course, were not built with 7x redundancy in the amount of upward force they could exert (that is, weight they could bear). The best estimates of their excess capacity seem to be about 1.6x. So, the floor cannot bring the upper mass to rest within .5 meters of deformation. Not even close.
In fact, over the .5 meter distance, in this scenario, the falling mass only is accelerated upward (that is, slowed down) by .6g over a distance of half a meter. Of the velocity it's attained by falling 3 meters at g, it loses only 1/10th of that velocity as a result of the floor's resistance -- after which it has another 3 meters to accelerate at g before hitting the next floor.
I find this a useful mental model to look at the collapses, because it's easy to examine what happens when you change some of the parameters. For instance, you can ask questions like "how far would each floor of the lower structure have to be able to deform, without losing any strength, in order for the falling mass to get progressively slower and eventually stop -- assuming that none of the mass in the lower tower gets added to the falling mass along the way? For a floor-floor distance of 3.5 meters and a "deformation at full strength" distance d, the increase in velocity is proportional to the square root of (g *(3.5 - d)) and the decrease in velocity is proportional to the square root of (.6 * g * d). The decrease becomes greater when the deformation distance is greater than ~2.2 meters.
Of the two biggest assumptions in this model, one works unrealistically in favor of collapse and the other against it. The one that's biased in favor of collapse is the assumption that the upper mass is able to accelerate at g for nearly a whole storey before meeting any resistance. The one that's biased against collapse is that the lower tower is able to resist the force of the falling mass with its full designed load strength, when the upper mass is actually displaced horizontally and tilting, concentrating forces instead of evenly distributing them.
While far short of adequate for any real engineering analysis, this kind of calculation does help me understand why structural engineers are so sure that "collapse is inevitable" once entire upper storeys of a tower start moving. If the structure were strong enough to arrest (literally, bring to rest) the collapsing floors, it would have been strong enough for the floors to not start moving in the first place, even as damaged as they were. (Which probably would have been way too over-buit for any practical economic use.)
That, of course, is why NIST focused on collapse initiation as the key engineering issue here.