Calculation utilizing Calladine and English support Ross
Back of the envelope calc using Calladine and English, supporting the plausibility of the Ross paper
What follows is a rough application of their results to a contrived variation of the Gordon Ross scenario (which, in turn, is based on Bazant Zhou), to see if his results are plausible.
Calladine and English state that the effects of Kinetic Energy scale linearly with volume. I assume constant density, and thus calculate mass ratios.* CE's experiment furthermore shows non-catastrophic bending for a Type II device in the case of a drop from .72 h, with a weight that is 41x the weight of the impacted device.
NEU-FONZE has recently stated in a physorg thread that the weight of the columns constitute about 6% of the weight of a floor (near an impact region.)
In the Bazant Zhou scenario, m(14 storeys) impacts a floor. The weight ratio, compared to CE, are:
(14x / 6%) / 71x = 3.28
Thus, we have roughly 3.28x the mass ratio tested in the CE {.72h-41x} scenario.
Furthermore, in BZ's scenario, the free fall is through a height of h, not .72h. Since gravitational potential energy is linear with height, and since no impedance is assumed for the inital free fall through height h, we have 4.56x the relative kinetic energy available in CE's experiment.
Now, we don't know how much more of an impact the device in question in CE could have withstood before failing. But, for the sake of argument, let's assume that it absorbed the maximum amount possible. (In other words, with an infinitessimal amount more energy of impact, it would have failed completely).
(If this were true
and if the columns of the impacted floor were fixed at the bottom of the storey, then there would have been failure of the topmost impacted storey columns.
However, while a fixed lower end matches matches the CE scenario very well, it does not match reality very well. There is no "magic wall" between column splices that fixes them in vertically in space, until such time as a collapse descends to their level.)
From Gordon Ross' paper, we see that in the time it takes for the topmost impacted floor columns to undergo their 3% shortening phase ( = .013 seconds; call this t0 ), the impact force would be felt by 16 storeys.
Now, assume that energy dissipation effects for the first t0/2 seconds are entirely confined to the topmost impacted storey. Also, assume that energy dissipation effects ala CE are entirely applicable to the next 8 storeys for the next t0/2 seconds (when the topmost storey will still be carrying it's load effectively). Finally, assume that the CE energy dissipation effects will be equally "spread" over these 8 storeys.
With these assumptions, we see that energy dissipation per storey, for these 8 storeys, is roughly
4.27x / 8 = 53% of the energy necessary for a non-catastrophic buckling.
IOW, Ross' conclusion of an arrested collapse are sustained.
(In this calc, I've ignored the energy dissipation "lost" by the topmost, impacted storey in the .2% elastic compression phase, as well as the energy lost by the topmost impacted storey during the first half of it's 3% shortening phase.)
The Calladine and English paper is not exactly applicable to the BZ/Gordon Ross scenario, because the geometries of the CE Type II apparatus is not an I-Beam or box column. Some of the references I posted recently may allow us to make a similar computation for at least box columns.
* From CE:
For the sake of definiteness, suppose that we have (1) a prototype structure and (2) an accurate scale model of it, made from the same material and with every linear dimension equal to beta (beta < 1) times the mass of the prototype. Let the prototype be designed to withstand an impact from a moving rigid mass having kinetic energy Omega and velocity V0. What kinetic energy and velocity should be used in testing the scale model, in order that the final deformed configuration of the model should be an exact small-scale representation of the prototype?
The kinetic energy is easily dealt with. It should be equl to (beta^^3)(Omega), so that the energy input per unit volume of material is equal to both prototype and model...
I do not actually consider the dimensions involved, but note that for equal density, a volume ratio will exactly equal a mass ratio. Although, in the WTC collapses, the top was not solid metal, in terms of figuring out whether catastrophic buckling would occur, it seems to me that assuming it is cannot affect results much.