On this 15th anniversary of 9/11, I came across something that I’ve been repeatedly reading or hearing for the last 15 years.
“How do buildings with 250,000 tons of structural high-grade steel, four inches thick, collapse at the speed of gravity and accelerating in its speed of collapse as it came down. There is only one way that can happen and that is a controlled demolition. I lived it, I was there and I heard the explosions, but it seems every time that you share this, it gets edited out.”
What’s the answer to this question? I sometimes think of adapting the orbital simulation model I was using yesterday to look at the dynamics of the collapse of the WTC buildings.
The problem is one of working out what kind of stresses build up in the columns supporting a building when entire floors above them collapse.
Looked at abstractly, it’s a question of what happens when a ball bearing of mass m (representing the mass of the floors above) is dropped from a height h onto a spring (representing the columns in the floors beneath).
Normally, the mass is resting on top of the spring, exerting a downward force m.g on it such that where k is the spring constant, and x is the length by which the spring is compressed, according to Hooke’s law:
m.g = -k.x
The engineers who designed the WTC would have wanted to make x quite small, say 1 cm or 0.01 metres, by setting the value of k accordingly.
k = m.g/x = m.9.81/0.01 = 981.m
The amount of potential energy, Es, that is stored in a spring is ½.k.x². And since we have set k = 981.m it follows that the mount of energy stored in this spring is
Es = ½. 981.m.x² = 490.m.x² ………..(1)
Now let’s look at the ball bearing that falls onto the spring. In the WTC the height between floors was 12 feet or 3.66 metres. The speed of something accelerating at g, or 9.81 m/s², after travelling 3.66 m is given by
v = √(2. 9.81. 3.66) = 8.47 m/s
And the kinetic energy, Em, of a mass m moving at 8.47 m/s is
Em = ½.m.8.47² = 35.9.m …………..(2)
And when the ball bearing lands on the spring, its kinetic energy is transferred into potential energy stored in the spring. So we can equate equation (1) with equation (2)
Es = Em, or
490.m.x² = 35.9.m
So in order for the spring to absorb the entire kinetic energy of the falling ball bearing, the spring must be compressed in length by
x = √(35.9/490) = 0.27 metres
So a 3.66 metre high WTC column would have to be compressed 0.27 metres – 27 cm – to absorb the kinetic energy of a mass m that has fallen 3.66 metres onto it.
In principle that’s perfectly possible. But in practice it’s out of the question. And this is because civil engineers build in safety factors into their structures so that they can resist forces only one and a half times or twice what they are expected to experience. So the WTC columns would have only been designed to withstand being compressed in length by up to 0.02 m. After the compression had exceed 0.02 m, the column would have failed, by buckling or exploding. And once this had happened, it would have no compressive strength at all. There was no way whatsoever that a column could survive a 0.27 m compression, unless the engineers had built them with safety factors of 30 rather than 2.
So when the upper floors of the WTC fell 3.66 m onto the floors below, there was probably a fraction of a second (a few milliseconds) during which the columns below compressed like springs until they had been compressed by over 0.02 m, and they failed. And if they failed explosively (with a loud bang), there would have been a series of loud bangs as the columns on each floor failed, starting about one second apart, and getting more frequent as the collapse progressed more rapidly.
The WTC collapse only took a second or two longer than free fall. And that’s probably because the columns on each of its 110 floors lasted about 10 milliseconds before their maximum design load was exceeded, and they failed.
There was no need for any controlled demolition. The building was going to collapse that way anyway.
The real problem is that events like the WTC collapse are beyond most people’s experience, and they extrapolate from their personal experience (of watching controlled demolitions, for example).
But the next time a 100 storey building needs to be demolished, it might be an idea to simulate 9/11 by simultaneously blowing out all the columns on the 60th floor, and watching what happens next. I bet it would be just like 9/11.