We've talked about this before, and I'm still waiting for you to propose a more optimistic scenario. The root of your statement above is in confusion over the definition of the word "scenario". Considering additional energy loss terms in the BZ scenario has no bearing on whether the scenario itself is more or less optimistic than any other, unless you can demonstrate that these additional energy terms are absent from other scenarios, and you have made no attempt to do so.
Dave
In the BZ scenario, all of the kinetic energy of the WTC top can potentially be used to buckle column segments in the topmost floor of the WTC bottom. There are three reasons why this is false.
1) Elastic energy loading would not mysteriously and magically stop at the bottom of the topmost story (until the collapse front descends below it), but rather continue propagating downward at 5100 m/s. Where does the elastic strain energy come from, below the impact floor? Ultimately, it comes from the Kinetic Energy of the top.
2) Elastic energy loading of the impacting top would not magically and mysteriously be zero, which necessarily follows from assuming a rigid top, as BZ do. Rather, it would also propagate at 5100 m/s
upwards (as Gordon Ross told you, long ago). Where does the elastic strain energy of the top come from? It comes from the Kinetic Energy of the top.
3) There is
no accounting in BZ for Kinetic Energy imparted to the WTC bottom, while in reality, in an axial strike, the column lengths bounded by the elastic wave fronts described in 1) and 2) would undergo a macroscopic acceleration, hence they would have a kinetic energy imparted to them. Where does this Kinetic Energy come from? It comes from the Kinetic Energy of the top. (I.e., the kinetic energy that the top had before impact.)
The energy sinks described in 1), 2), and 3) mean that much less energy is available to buckle columns, anywhere along the WTC bottom, including the area that BZ claimed would buckle first, which is along the topmost floor of the bottom. So, even without calculating their exact value, I still know that the Kinetic Energy source in BZ's scenario has to be adjusted downwards in the more physically realistic scenario. Hence, BZ is
not the "most optimistic" scenario.
Frankly, at least for a technical audience, it shouldn't be necessary to elaborate what a "more optimistic" scenario would be, given the above. If I were attempting to be quantitative, then I'd have to elaborate a great deal, including mathematically. However, since you ask, and since most people at JREF are not technical (or so I would assume):
A scenario that is far more optimistic than the Bazant Zhou scenario, given an axial strike, is the
more physically realistic one where elastic waves propagate along the lengths of the columns (until reflection from the ground and the top of the WTC top). Furthermore, it'd be both more optimistic and more physically realistic to account for the macroscopic acceleration of the columns. (Note: peak stress would probably occur after first reflection, but guesstimating from Goldsmith part(c), noted below, it is still nowhere near what BZ implies.)
In other words, this more optimistic, yet also more realistic scenario looks very similar to the BZ scenario, except that the columns are treated as elastic rods -
all along their lengths, for both the top and the bottom of the WTC.
Have you seen the graphs that I uploaded,
here, part (c), for the case of elastic rods colliding longitudinally, a free one against another fixed at it's opposite end? Notice that the elastic waves propagate in
both directions, away from the impact surface, until first reflection. Notice also that these elastic waves define a region of kinetic energy corresponding to velocity = 1/2(v1,0 + v2,0).
I will be posting graphs of Love's solution of a rigid mass hitting a fixed elastic rod within a week, or so, at the 911forum.freeforums.org, and here. I believe that I can also get my hands on the analytical solution graphed by Goldsmith in part (c), within a couple of weeks. If so, I can graph that, also, with values roughly corresponding to the WTC scenario. Based on calcs I have already done, I expect to find that BZ implies
almost 2 orders of magnitude higher
peak strain energy density than a more realistic treatment would predict. *
Ultimately, I want to solve a more realist axial strike scenario, while still modeling as a single column line. That means variable mass at every h, and variable spring constant every h, also, and gravity loading. However, even to do that, I need to learn more theory. I'm not sure, but as an interim step, I think that if I relearn my LaGrangian analysis and model each column length as a spring, this problem would be tractable analytically and numerically on my PC, and provide a reasonable approximation. Ultimately, though, I'd like to solve using coupled wave equations with non-column masses modeled as rigid, considered attached every h. Right now, I don't know how to do that, and at the rate I'm going, it'll probably take me at least a year. For a professional applied mathematician who works in PDE's, it would probably take 2 days, maximum.....
In case any mathematician or quant type person reads this, besides looking at clawpack, there is another PDE solver that might allow for an easy solution called diffpack.
* I haven't proved this, but I believe that static gravity loading of the bottom will have negligible efflect on the propagation of the elastic wave. I expect that it
will have a significant effect on the macroscopic kinetic energy transfer to the base, but in terms of peak stress, because the loading is in the same direction as initial elastic wave propagation, I expect that these effects will be most significant only after the first reflection from the ground level. That is more than enough time for the speed of the top to have been halved.
