einsteen wrote:This is a nice thread. I think we've had this discussion in circles at physorg and we didn't have a solution. I don't buy the domino theory.
It's not an accurate dynamic model of the towers, for sure, but it is a system with which most people have some familiarity and presents an example of self-sustaining mechanical
reaction. Understanding such a system is helpful in the same way practicing arpeggios assists in playing a composition.
There aren't many examples of small energy input initiating a cascading mechanical reaction. Chemical and nuclear, yes, but mechanical? Avalanches, landslides, dominoes, and progressive collapse... am I missing any? There are contrived systems people make that are like domino chains: something rolls down a ramp and lands on a balance, which tips and strikes a match that burns through a string which releases a marble that rolls down a track, etc, etc.* I don't know what these things are called, but I take them as special cases of a more general system type which includes domino chains. Avalanches and landslides, as well, strike me as being special cases of a more general system. Toppling would seem to be in here somewhere. **
There are, then, very few general types of mechanical systems that display a cascading or self-sustaining effect, and I'm not sure they really are distinct categories. If you look at Seffen's paper, you'll see a graph depicting capacity over displacement - the initial spike of capacity until failure, where capacity drops off rapidly, then a third and final phase of rapid capacity increase tending towards infinite. Unbuckled column, buckling column, squashed column. Domino tipping, falling, coming to rest. How different are they?
Now, observing the parallels in terms of activation energy, sustaining energy, and arresting resistance, I have to wonder what else can be gleaned from study of such a system. Domino chains are relatively simple (yet quite an analytical nightmare if you look at it) so may yield some useful insights into the nature of propagation of instability. One thing that jumps out at me is the ability to 'snowball' without continual accrual of the mass of previously involved members. The peculiarities of a physical system dictate how energy can be distributed, how quickly transition from one form of energy to another can occur, and how much power (defined by energy loss
per unit time) can be generated by the system at each instant.
I'm particularly absorbed by the last item, power. This is technically a result of a proper dynamic solution, not a dictum or causative factor. Nevertheless, the power-limiting nature of a domino chain seems to be one of the most elegant dynamical values to consider, perhaps most directly reflective of the geometry of the structure. While we often consider losses at the elementary level, it is a different matter to consider gains at the global level: the domino chain's instantaneous power is only limited by the amount of 'reactants' currently engaged, which can grow at least geometrically.
A pancaking floor collapse is a vertical domino chain, precisely. I think dominoes are a better model than avalanche for pancaking. Both are discrete systems fundamentally based on unit transactions, the floors being initially separated and finally layered. We see immediately from domino chains that such systems can have energy in excess of that required to sustain the reaction, even without mass accumulation beyond a certain constant amount.
This means that pancaking is
a plausible scenario for a somewhat slower progressive collapse that has energy to expel debris, shatter concrete, shed the perimeters and laterally destroy most of the core. Initial energy deficit (if any is found in comparison with observation) in a pure lower block pancaking scenario can come from the upper block. If not much is needed, let the mass shedding be high and have the upper block fall off early - as it seems to have done anyway. Thus you have a collapse initiated by tilting and upper block descent but propagated in the later stages by internal debris flow, resulting in collapse front ahead of shed perimeters and standing spire, also in agreement with visual evidence.
Not saying it is the case, but surely an interesting thought.
* I saw a video of one of these things, constructed by university students in a warehouse, that perpetuated itself for about 20 minutes, as I recall. The camera followed the reaction as it progressed, only the active portions were lit. I wish I could remember the name of the university or what these Wile E. Coyote / Rube Goldberg machines are called, I'm sure it's on YouTube.
** I put toppling in the domino category. Toppling can be a single domino, tree, or antenna falling over. Or it can be like this jenga tower: http://www.youtube.com/watch?v=RdNeaSgqcFg
Wow, is all I can say.