I'll try a verbal explanation of the "factor 2", if y'all don't mind. Numbered list of steps, so you can refer to a number if you have questions:
- The thing that's getting loaded is a "floor below". It's carrying itself already, plus any design live load (furniture, people, carpets, cubicle walls...), and probably built such that it's level throughout under that load. Let's define this situation as the null load.
- If, hypothetically, you now put another, identical floor on that "floor below", carefully, and wait a bit until everything has settled, you will find that the "floor below" now is sagging a bit - the floor surface is 1 mm or so lower than at null load. That is because the load (the stress) is causing an elastic response, a downward deflection in this case, called "strain". The higher the stress, the higher the strain.
- When a steel-supported structure is still well inside its load capacity, you roughly expect stress and strain to be proportional: You double the stress, you double the strain
- Ok, so we had loaded the "floor below" with 1 more floor, and observed a deflection of 1 mm. We also observe - because the load is static, so this is practically by definition - that the floors are not moving. Velocity 0. It's static, after all!
- Now let's do the dynamic loading instead! Preparation: We place the additional floor juuuuuuuuuuust above the "floor below" - with essentially 0 (unmeasurable) vertical distance, a nanometer or whatever, or juuuuust touching, but we are still holding it up with the exact amount of upward force that gravity pulls the additional floor downward. So the "floor below" experiences 0 stress and reacts with 0 strain.
- Then we let go! We suddenly cut all supports of the additional floor, so it starts loading the "floor below". The immediate stress is the weight of the entire floor - but as there is, at t=0, no strain yet, the floor below exerts 0 force - the additional floor falls at gravitational acceleration g
- But immediately, because the floors have moved down by an infinitesimally little bit, the floor below goes into tension, it starts to develop strain (deflects downward elastically), and thus starts to excert a force upward. That force at first is mininal and not enough to stop the floors from accelerating, however acceleration decreases a bit, it is no longer at g, it's just below g.
- Okay, so the floors keep dropping, strain keeps increasing, upward force from elastic resistance keeps increasing, downwarf acceleration keeps decreasing
- Now here comes the clou: at 1 mm deflection, so the value at which the statically loaded system was in equilibrium and didn't move, didn't accelerate or decelerate in any direction, the strain results in an upward force that is equal to the weight of the additional floor - and so the weight is exactly counterbalanced, net force becomes zero, acceleration becomes zero - and velocity reaches its maximum value. Notice: Stress and strain are now the same as in the static scenario - only difference is that the floors are still moving down at some v(max).
- And so, the deflection overshoots the 1 mm mark, strain keeps increasing and with it the elastic force response - upward force surpasses static weight, floors start to decelerate, and deceleration increases and increases...
- ...until velocity reaches zero, and the floors have reached a maximum downward deflection
- Here should go some rigid argument why, but I have none, so you just have to believe me that the plots of motion during the deceleration phase are symmetrical to acceleration phase, i.e. maximum deflection was 2 mm, and the time to get there was twice the time to reach 1 mm. The stress on the floor below, which started at 0, was 1 m*g at 1 mm (in both the static and dynamic case), increased by another 1 m*g at max. deflection - hence a load of 2 m*g - twice the static load.
I hope that makes sense!
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As for the "2/3 g" or whatever acceleration:
As econ41 said, much confusion reigns. Let me add to that confusion!
A value of "2/3 g" was the result of David Chandler, when he did a motion analysis for the first few stories that a tower collapsed - until his reference point (something on the roofline? The North Tower antenna? I don't even remember) disappeared behind dust. That "2/3 g" was essentially an average acceleration for that phase of collapse. Chandler's plot looked a bit like that's a constant value, but there was noise in the data, and the deduction of velocity and then acceleration is very sensitive to imprecision in grainy data.
Several people, myself included, have modelled the tower undergoing collapse progression as a stack of floors, each 0 mm thick, without any columns whatsoever, suspended up in the air by magic, separated by 12 feet of vacuum, and the collapse starting at floor 95 or 82 or whatever with 15 or 28 floors already lumped together. These lumped floors now start falling freely through the first 12 ft of vacuum, then hit a single floor below inelastically, so that floor below is accrued to the falling lump, and momentum is transfered between the lump and the floor below. At the exact same time, the floor below is released from its magical suspension, so it can fall in unison with the lump without any additional structural resistance.
In such a model, acceleration is always = g between any two floors, and negative infinity at the precise impact time between lump and floor below. If you take each floor - the combination of 12 feet of vacuum freefall and instant velocity reduction by momentum transfer, that takes time t and changes velocity by a value v, such that the average acceleration for that floor is v/t.
It turns out that this acceleration is something like 94% of g for the first floor, 80-something % for the second, and it decreases such that after something like 10 floors it is around 2/3 of g, and or after 15 or 20 floors averaged about 2/3 of g. The acceleration continues to decrease - but at long last it asymptotically approaches a value that appears to be 1/3 of g.
A Czech engineering professor, Ivan Nemec, did an analytical model, which he smoothed by, conceptually having infinitel many floors of 0 thickness spaced 0 apart from one another, and with that model, he rigorously showed that indeed the limit value is 1/3 of g - and because his model is smooth and already has infinitely many drop-collision cycles through the first 0.00000001 s, his model would predict 1/3 of g from the start.
And because that is only half of Chandler's observed 2/3 of g, they argue that the collapse was way faster than possible if pancaking was the driver.
But, of course, the towers did not have infinitely many floors, and thus the discrete model with 12 ft spacings gives the better idea of what would happen.
Now, why is this model at all of interest?
Because, as econ41 never tires to point out, the Bazant-model, where column-crush is the dominant mode of resistance, is not what happened in reality. The ends of columns falling from above did not realistically hit the ends of standing columns below - mostly, columns bypassed one another. And thus, pancaking was a dominating mechanism of collapse progression at least in the early stages. We can observe on some videos how floor after floor collapsed, making dust shoot out many windows on the same floors, while the walls were still standing many stories higher. In severalinstances, it is rather obvious that big, tall pieces of wall simple toppled over/outward as a result of all the floors connected to them having first pancaked down, deleting lateral support.
The model with the magically floating pancakes results in accelerations and total collapse times reasonably in line with what was observed from collapse videos: 2/3 g average early on, but then slowing behind dust clouds to explain collapse times of 12 to 15 seconds or thereabouts.
