#4664 
by Kat Dorman » Aug 22, 2011 7:47 pm
When you think you've caught a mistake of mine, the posts with the LOLs come in fast and furious. When it's shown that the mistakes are yours, there's silence. Do you want me to argue your side for you? I'll do a much better job.
Dorman: Therefore we see that, far from being mere name-dropping ("you expect people to be impressed by all of your complexity and talk about MAXWELL"), the Maxell construction is a very simple way of expressing the average force and likewise acceleration over the crushing distance of a single story. If the Maxwell line is a constant 38% of the static load, the average acceleration is 62% of g. It couldn't get any simpler than that. The whole point of the Maxwell line is to SIMPLIFY the problem.
Anti-Dorman: As a simplification, is it not also an approximation? How close is it to the non-simplified dynamics?
D: To within story granularity, it is very close.
A-D: But, as psikeyhackr pointed out, it does not account for the peak forces which are sub-story granularity, and this is a place where arrest can occur.
D: True. However, I can (and have) run both calculations and time-stepped simulations which account for the peaks, and so have already determined those configurations which can arrest. When examining a case which cannot arrest due to peak resistive forces, the use of the Maxwell construction to simplify the problem is perfectly valid and gives quite accurate results. Moreover, by smoothing out the peaks and valleys in resistive forces, one actually obtains a more accurate descent curve when comparing to a messy, non-ideal, non-axial 3D collapse. In real collapses, measurement locations (e.g. the roofline) do not exhibit the jolts predicted by a discrete undamped model because real structures deform and are damped.
A-D: Fine. But either method of computation relies on the accuracy of the assumed load displacement curve. An average force of 13% of the peak static capacity seems awfully low. If one accepts the applicability of the Maxwell construction, then everything hinges on having a valley much lower and/or longer than the peak so the average force ends up being less than the static load. How can you justify using such a small value?
D: I rely on Bazant for the form of the load-displacement curve for steel columns in axial compression. This is based on well-accepted properties determined by experiment and then backed by constitutive analysis. I admit, if he's wrong, then my analyses are off accordingly. But that's not the same thing as being wrong, because I never claimed my example above (or any others) represented an actual tower collapse. I presented it as what would happen in a system with a Maxwell line at 0.38mg.
Besides, the chances of Bazant being wrong are about nil! Pick up a textbook if you don't believe me. The load displacement relation is standard fare.
A-D: Weasel words. We're talking about the WTC collapses. If it's not applicable, it's a diversion at best and useless at worst.
D: Well, the initial accelerations of the towers were within a relatively close margin to those values, so it would appear it IS applicable despite the caveats.
A-D: Only the initial acceleration?
D: Yes, the acceleration dropped to zero in WTC1 after a time.
A-D: What does your Maxwellized approximation say about that?
D: It says nothing about it. Those models don't have velocity-dependent sinks like concrete comminution, etc, or the coefficients for those terms have been set to zero for the trials in question. It's an attempt to keep it simple; one thing at a time.
A-D: I still don't accept that 38% of static load is an accurate approximation. There are axial compression modes which produce load displacement curves which do not have a long trough like what Bazant depicts. Hinge buckling is, at its most generic, expressed as n-hinge buckling where n is odd. By choosing n to be 3, Bazant has deliberately opted for the lowest energy dissipation possible for a column fixed at both ends. Other modes, like an accordion configuration, present much greater resistance long before full compaction.
D: Also true. Cherapanov exploited this when he did calculations which suggested a first impact in the Bazant scenario would result in 5 or more hinges, with the result that Bazant greatly underestimated the energy dissipation. Cherapanov's flaw in reasoning is that there was no drop through empty space to produce the higher velocity and the impacts were not, and could not have been, between perfectly aligned column ends. Bazant's model was a limiting case which was unrealistically biased towards survival. Take away the perfectly aligned strike, and you don't even get 3-hinge buckling. As was clear from the columns observed in the debris pile, hinge buckling of any order was a rarity. All other applicable failure modes involve less KE loss than 3-hinge buckling.
A-D: Now you say there was no drop through empty space, indicating the upper portion impinged with a lower velocity than it would have had it been a free drop. That means there was less KE available to do mechanical work on the lower portion, and again your models fail to capture the true dynamics.
D: Now you're the one with weasel words. Of course these trials fail to capture the true dynamics, it's never been touted as anything but instructional simplifications. But I have done slow drops and low drops and even NO drops. That's right, NO drop. If one applies the Maxwell construction to all stories, including the failed one at the split point, it still collapses. Capacity of 38% of load means there will be inadequate support, and the model will spontaneously collapse at an acceleration of 0.62g. That is indeed 38% less than freefall. Doesn't matter much.
Bazant calculates that a free fall drop of only 0.5m is sufficient to overload the underlying story. My computations have confirmed the correctness of that claim.
A-D: How does the top begin to move in the first place? You conveniently ignore the first cause.
D: Loss of capacity occurred. I can see big gaping holes and substantial fires. This unequivocally means loss of capacity. Ongoing fires means an ongoing loss of capacity. This is only common sense. Beyond that, I've done no work to examine whether this is sufficient to initiate collapse, so I don't discuss it. It's not absurd, in any case, even if it is in question for some people.
A-D: I've run out of objections.
D: Well, at least you gave it a college try.
