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Old 5th August 2006, 03:49 PM   #576
R.Mackey's Avatar
Join Date: Apr 2006
Location: The armpit of L.A.
Posts: 7,857
Why do Greening and Ross disagree?

For brumsen, there is a difference between Greening's paper ( and Ross's paper ( about whether or not the initial collapse of the WTC towers had enough energy to initiate a progressive collapse. Greening shows that the kinetic energy of a single floor collapse is more than sufficient to break all of the support columns on a single floor, therefore the collapse can proceed just like we saw on TV (twice). Ross concludes differently, that Greening has neglected linear strain in the columns below, sufficient to provide a "shock absorber" effect as the standing columns compress like springs, and that the recalculated energy is insufficient.

There's a couple of problems I see with Ross's paper. Page numbers start from 1, not the number printed at the bottom of each page.

Originally Posted by Gordon Ross, pp. 1-2
Bazant/Zhou [1] show in their analysis that elastic and plastic behaviour of a steel column under a dynamic buckling load can be shown to consist of three distinct phases. These can be
shown on a load against vertical deflection graph and consist of an initial elastic phase, a shortening phase and a rapid plastic deformation phase.
Ross is trying to bring out material properties of individual structural elements, more detailed than Greening's analysis. However, it's important to point out that Ross is trying to treat this problem one dimensionally. In reality, the columns will be subjected to a side force. The floor trusses are either failed or intact -- if failed, the top of the column is no longer constrained, and is free to deflect to the sides; if intact, the stronger core columns will experience less strain and the outer columns will be pulled inwards, balanced by an outward pull on the core. We also know from video evidence that both towers did not collapse, nor were hit, symmetrically.

I point this out because the yield strength of a column that is free on one end is considerably less than the yield strength where both ends are pinned.

Originally Posted by Gordon Ross, pg. 3
Because these columns suffer a vertical deflection, the attached floors move downwards and they will therefore have a velocity and momentum.

Energy Losses:

A simple conservation of momentum calculation, ignoring these movements, would have, 16 falling storeys moving at 8.5 m/sec before impact, changing to 17 storeys moving at (8.5 * (16/17)) = 8 m/sec after impact. This does not reflect the fact that a minimum of 24 further storeys will be caused to move downwards at varying speeds.
I don't understand why he is considering momentum. He claims to be doing an energy balance equation. The energy loss due to plastic deformation of the columns (assuming we accept his model) can be found simply by multiplying the applied force on the columns (== yield strength) times the distance of motion (== plastic deformation limit, assumed to be 3% x column height).

Ross has made an error claiming that the intial impact would lose energy because it takes some energy to accelerate the floors below impact, that are not in contact with the falling upper floors but are pulled along by the compressing columns. If Ross wants to bring this into the equation, he has to account for the downside, too -- this acceleration applied to the lower floors, by virtue of their inertia, will have the effect of further twisting the columns where the floors are attached. Far from being "dissipated" harmlessly (Gordon is claiming this accounts for a loss of 66% of the energy), this energy is being pumped into deforming the support pillars.

Finally, if that mass is accelerated, it must be stopped as well. That energy didn't disappear. It was briefly converted into kinetic energy, but then it must be converted back. I reject Ross's assumption that you can neglect all of this energy. I will grant that it spreads the energy over several floors, but it still contributes to structural damage and weakening of the columns.

Originally Posted by Gordon Ross, pg. 4
The strain energy consumed by the impacted storey columns in the elastic phase and plastic shortening phase can be calculated using the failure load. The failure load used throughout this analysis is derived using the mass above the impact, 58 000 tonnes, and a safety factor of 4.
I question this assumption, uncited. You can bury nearly anything in a fudge factor of 4, which seems high. There are more detailed descriptions of the WTC design available, so there's no need to guess.

Originally Posted by Gordon Ross, pg. 5
Energy Summary:
The energy balance can be summarised as
Energy available;
Kinetic energy 2105MJ
Potential energy Additional downward movement 95MJ
Compression of impacting section 32MJ
Compression of impacted section 24MJ
Total Energy available 2256MJ

Energy required;
Momentum losses 1389MJ
Plastic strain energy in lower impacted storey 244MJ
Plastic strain energy in upper impacted storey 215MJ
Elastic strain energy in lower storeys 64MJ
Elastic strain energy in upper storeys 126MJ
Pulverisation of concrete on impacting floor 304MJ
Pulverisation of concrete on impacted floor 304MJ
Total Energy required 2646MJ

Minimum Energy Deficit -390MJ
(emphasis added)

I'd call this a smoking gun. Ross in his energy balance equation has double-counted the floor collapse energy. He's assuming not only the floor getting hit has to collapse, but also the lowest floor of the falling block collapses as well (bolded items). Well, that could be, but if both of them collapse, you're not dropping that weight one floor -- you're dropping it two floors. Energy available DOUBLES.

I also note that the calculated loss due to "momentum losses," e.g. the allegedly harmless acceleration of lower floors due to plastic deformation, is greater than half of the energy budget. That is a heck of a big correction. As noted above, I disagree that you can simply throw this energy away, because it is still contained in the structure.

As noted before, all of the other real-world conditions that are hard to simulate -- asymmetric damage before collapse, asymmetric impact, anisotropic deformation caused by fires -- are not included in this paper. Far from being a "conservative" estimate as is claimed, this model, even if taken at face value, would not be entirely conclusive.

A final point that Ross has not addressed is that the floor that the upper stories fell upon was not in blueprint condition! It was immediately below the raging fire that collapsed the impact floor, suffered deformation from proximity to the impact floor, and was heated enough to weaken its yield strength. Again, even if we take Ross's numbers as correct, but add another floor's worth of gravitational energy, we still get collapse initiation.

To conclude, this is way better than the usual CT fodder but would still fail peer review. He's shown his math and assumptions, and I credit him for that. But I reject several of his assumptions, I don't understand why he complicates the energy balance equation, and even if true his "energy deficit" is too small to be a definitive disproof of collapse. And that's for the tower that was hit more gently.

ETA: Missed that Ross also double-counted the concrete-crunching energy. In fact, neither floor would have to be pulverized before collapse could initiate -- the floors are not holding up the structure. They can be crushed later.

Bottom line, Ross is way off, even with his own numbers.

Last edited by R.Mackey; 5th August 2006 at 04:55 PM.
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