by OneWhiteEye on Fri Nov 21, 2008 6:13 pm
Dr. G:
Thank you for that awesome post. You have to know that going back to ET was high on my list. It's nice to have it quickly laid out like that. Had to take some time with that, but I hope what I have to say adds in some small measure.
The minimum 8 m/s^2 figure accounts for the slower areas of the roofline? I just want to be sure, I haven't worked through it and I don't recall your previous values. The NW corner is very fast, of course, but I become less convinced with time that it's a fair representation. I assume your figure covers the global characteristics.
A number of odd ideas have crossed my mind lately as a result of obtaining numerical solutions in two radically different ways on a variety of configurations, and assimilating the more theoretical end as a consequence. Organizing these thoughts now is a bit of a challenge, all I can do is throw them out for consideration.
The geometry, uniform gravitational field, and the generic nature of crush in this case dictate there will always be more PE liberated than can be accounted for by increase in KE. I've seen how you can get the whole bulding moving a little at the bottom, and it's (usually) all over from there, unless the building is insanely overbuilt. But I also see the energy distribution over time dictates that all energy sinks which result in an effective resisting force correspondingly deduct instantaneously from the KE, despite the available excess of PE, and so it goes.
...unless you're willing to admit complex solutions, one of the odd things that occured to me. Or at least have an E1 that is a function of many hidden variables, something you've tackled very well in other endeavors. Maybe my interpretation is wrong, but this formulation leads to arrest when the energy consumed at a level equals the current KE. But the real situation is that arrest is only achieved when the residual capacity exceeds the discrete impulse or continuous effective load. Energy spent blowing air/slurry out holes slows things down but doesn't do anything to keep the building up statically.
Take only structural resistance, ignore non-structural derived resistance like violent expulsion of fluid and dead load impact crushing. If an intact WTC7 had all the columns simultaneously removed on the bottom floor, it would almost certainly obey the relations you've set forth. At first, anyway. Pretty close.
The devil is definitely in the details. Regardless of mechanism, we see from direct observation that a substantial amount of PE was liberated in advance of the global collapse. There is little that can be reliably inferred about the resulting internal state of the building, the bottom-up vertical progression that appears to have occurred leaves us with an unknown energy balance.
Obviously, an internal crush-up does not apply. Intuitively I'd think the horizontal decoupling intrinsic to a localized, internal bottom-up failure would cap the energy absorption by the rest of the structure, except at the bottom. All that mass falls in succession from a slightly lower height than it was originally after breaking free... how on earth does that happen? Why is a building designed such that taking a chunk out towards the bottom results in chunks dropping out all the way to the top? OK, let's just accept that's what happened, it was after hours of fire and possible chemical degradation.
It leaves us with even more PE to KE because those chunks just drop, the work has been done. Air has places to go. Only collisions with the 'tunnel' wall will transfer energy to the structure, and these would provide localized impulses but not necessarily sustained loading on the standing portion. But what happens when they hit bottom, wherever that is? Think of the power generated and to what degree the rest of the structure could absorb it. (Edit: assuming any amount can be transmitted)
When I do sims with rigid but largely inelastic bodies, crap flies off the bottom unless I use slabs. But the slabs start oscillating vertically because I just can't get them to shut up! It's a high capacity, short term mechanical battery! Too much energy is left over after collision to throw away without artificially forcing the rubble pile to be fixed. My rubble pile can jump up and clobber three intact floors at the bottom of the upper block, or come up smooth and gentle and provide a nice cushion to spread out the impulse and allow arrest.
An internal collapse, with high acquired KE, could do significant work in the horizontal direction upon impact.
E1 of a damaged structure versus intact is then a primary consideration. How else can this excess PE be spent so as to reduce the structural capacity component of E1? Thought experiment: If we dropped an intact WTC7 from just high enough to fail the lower floor, then gently set the remaining 46 stories down on flat ground, would it stand? Or would the bottom three or four stories be so misshapen and compromised that it would drop like a stone? In the Japanese study you cited (something I've been wanting to comment on), we see strain energy being distributed all the way up the structure. It looks like a model crush-up, but it takes so loooooong. Elastic structure.
I guess you can see I'm working the angle of a highly variable E1, among other things. More in a bit.
