The model does not necessarily capture the actual mechanics of collapse. Part of the purpose is to see whether a basic stepwise model can fit the observables with a minimum of goal-directed tweaking. Of course, the model needs a fair shake which means reasonable accuracy on the target constraints (e.g., if it really were only 22 m/s or 6 seconds to reach speed there would be more relaxed solutions found).
No adjustment in stretch is made for mass loss, but it's not clear where the thickness is shaved from the debris zone or whether density changes, etc. This is a weak point in the process because the solutions probably don't have much accretion over many of the later floors, thus stretch is no longer a measure of compaction and will be unnaturally small. Moreover, the choice of mass loss as opposed to force increase is somewhat arbitrary, and one can substitute for the other within reason. The driving mass over time given by the solutions does not need to be strictly interpreted as such but, if it is, the stretch will lose some physical meaning and might also be an impediment to a good fit.
- 4.5 seconds to velocity sample time
- velocity greater than 25 m/s at sample time
- time to reach ground between 14 and 16 s
First pass is wide open (i.e., stretch down to zero, M0 up to 25) to get an idea of what's possible. The solutions, ordered by deviation from a target velocity of 28 m/s:
(M0) initial driving mass
(c) coefficient of mass reduction
The first conclusion is there are no solutions at an initial driving mass M0 less than 12 stories. That means, for the other mutual constraints, most of the mass above floor 96 needs to be the (rigid) body impacting initially.
The second conclusion is that the stretch parameter is really hostile to solutions, with little possible above 0.13. That's quite a lot of compaction if taken literally, especially early on, but this is not possible because of the implied mass loss.
The coefficient is never especially interesting, but it's small and that's good.
Restricting the stretch to be above 0.05 and M0 less than 17 gives these results:
(M0) initial driving mass
With the additional constraints on stretch, there are no solutions for M0 of 15 or less stories. Conversely, the constraint on maximum initial mass has diminished the upper end of stretch to less than 0.075, which as noted does not represent a compaction greater than 13x (except early in the collapse which is not realistic).
Taking two of the solutions - the top ranked (highest v at sample time) and a low rank at around 15 stories extra mass, these graphs compare upper block mass and velocity versus elevation in stories:
4.5 seconds to get to 25 m/s is tight, given the mechanics. By making mass variable, beyond that of the fixed accretion scheme, it is possible to squeak in under dubious and very narrow parametric ranges of 15 <= M0 <= 17 and 0.05 <= s <= 0.075.
Since the current results indicate a single or two floor collapse is far from a solution, these are the possibilities:
- the observation is not really a crush front but some other phenomena
- the mechanics of the crush front don't correspond to this model
- a large, intact chunk of mostly rigid upper block is the driver, not slabs
- the constraints need to be relaxed
- stretch must be made non-constant, perhaps dependent on mass loss
The last two options are the first to explore, naturally, because they're less radical and within reach.