David B. Benson wrote:Major_Tom offered two criticisms of the B&V crush-down equation. Neither hits the mark.
I think both are valid questions to raise. Further inspection may prove the objections to be untrue or irrelevant, but I think they need to be addressed.
...this crush-down equation is simply the continuous analogue of Frank Greening's floor-by-floor model;
In reality, the continuum limit with homogenous (or linearly varying) mass distribution is not much like the towers. Slabs of near the compaction size are closer. True or false?
Major_Tom objects to (2), implying, I think, two crushing fronts.
Yes, that's the idea.
Obviously Frank Greening does not think so...
Frank, what do you think? Is 'rigid body' a simplifying assumption, or is it dictated by the mechanics (exclude B&L for a moment, just your research)?
...and indeed Bazant & Le thoroughly remove that possiblity in the idealized, analytic case being considered just now.
They only prove their result for the case of homogeneous mass distribution and axial impact, which bears little resemblance to the actual situation. One can suppose it extends to the real situation, or that tilt somehow corrects it back to the valid domain of these assumptions, but that's all this is - supposition. How did the actual entanglement and subsequent punch-through dismember the upper block? What about any 1D analysis gives one confidence to proclaim the real eccentric impacts, possibly encompassing several floors, would act in any way similar to such analysis?
The whole point of both the exposition in B&V and Major_Tom's objections is rendered moot by the homogenization assumption;
The homogenization assumption is one of the things being scrutinized here. It is NOT representative of the tower's fine-grained structure. While I agree such detail should not be required for a global analysis, there is nothing I've seen that assures this is the case for early progression. In fact, what I have seen (barely above anectdotal at this point, but compelling all the same) indicates just the opposite. Initial conditions can have quite an influence, depending on how the system is modeled.
I don't expect you to buy into tinker-toy models, so let's talk about B&L. How much farther did the crush-up have to go before the plastic limit was exceeded (edit - removed text from here*)? We've got to be fair. Could the upper block drop a story on a (RIGID!) debris zone under one-third gravity and immediately arrest or even survive? You tell me. Looks like a pretty serious knife edge to me, regardless of the model. It seems that any significant asymmetry or uncertainty in parametric input has the potential to initiate a concurrent crush-up for at least a period. Tilt, non-uniform residual capacity, unknown damage... these things seem like major uncertainties.
How does the argument change when we're not talking about column-on-column impacts at all? Or an inhomogenous structure and mass distribution? These are valid questions.
I'm all for simplifying assumptions, genuinely; but I never lose sight that they are there for simplification and not improved physical description. If there is no difference, or the difference is acceptable for the purposes at hand, fine.
Edit - text removed:
" and by the same model's simplification, qualifies the resistance from the lowest columns in the upper block to be set to zero"