David B. Benson wrote:OneWhiteEye --- Write out your system of ODEs and I'll solve them against your data.
Since I don't think there's anything wrong with B&L's analysis, I'll point you to that text for the system of ODEs, the only difference being, instead of a 0.15 reduction in capacity in the upper floor, plug in 0.2. If it crushes up to the next floor, stop there. We'll cross that bridge if we come to it.
Ah, verinage. Well, about two stories are stripped out almost entirely with columns replaced by pistons or something, yes?
Columns displaced horizontally but that's the idea.
So in that case there is a definite inhomogeneity...
Actually, if you look at them, they're about as close to homogeneous, outside of the two stripped stories, as you're likely to find. What I've seen so far are apartment buildings, uniform cellular structures. The stripped floors are different from the unprepped, but appear homogeneous across their extent. What's more, the impact is about as symmetrical and even as possible (i.e, no tilt), like against like. NO asymmetrical damage.
...and a two story (nearly) free fall.
This is a difference. Of course, the demo company wants to ensure complete destruction. From the looks of it (and this is something you need to know), they haven't quite got the hang of that part yet. Even two stories of drop, with an upper block consisting of the entire upper half of the building, is not always enough to finish the job. Sure, the construction and materials are different, the upper block might only be 8 stories, whatever. These are parameters. Until they're plugged into the model and compared with experiment, it's just talk.
More important, it really is not freefall through two stories. Though I've not measured it, there's no need to, the mechanics forbid it. It's usually a concertina collapse, as Dr. G describes it. Here, the walls of the two floors go from vertical to hinged at the interface like two tipping pencils end on end. After a short distance, the effective resistance drops to null but isn't that the case for steel columns buckling under compression, too?
Yes, jumping off a two story building is likely to hurt rather badly when you land on your belly. Don't need fancy physics to know that. Nor videos.
Well, are you surprised then that some cases arrest?
I see your point. However, I will observe that the solution given in B&L, which remains the sole justification for application of d'Alembert's principle, looks very much like a knife-edge solution. It comes very close to exceeding the yield of the columns above. If it does, there won't be freefall, but it could descend independently, crushing up on Zone B, while B+C crush down A. The solution in B&L may be highly insensitive to the choice of relative capacities, but my gut tells me this is not the case.
The point is that the structure of WTC 1, damaged as it was, certainly did not result in any free fall, none at all.
I'll remind you Bazant himself has calculated only 0.5m of freefall is required to initiate crush down. It would seem, even for crush-up with an accelerated base below for impact, freefall is not something that's required. The mechanics will indicate whether the energy is sufficient to initiate and continue crush-up. B&L have done it for an assumed parameter of 0.15% reduction as a margin of error. Looks like it just about crushed up, if you look at their graph. To not carry through in an exploratory fashion, to the point where crush-up does initiate, should be considered an omission, given the subject matter.
So now we do need some physics in the attempt to understand the progressive collapse.
Before the equations come the reasoning. Interestingly, we already have the equations, we need to finish the reasoning.
