Dr. Greening,
Thank you for your answer. However, even understanding that it is a brief overview suitable for discussion in a forum, I have several concerns about your derivation.
NIST’s key statement concerning the WTC collapse-initiating event, is:
The change in potential energy due to the downward movement of
building mass above the buckled columns exceeded the strain energy
that could be absorbed by the structure. Global collapse ensued
This statement could be interpreted as referring to the quasi-static collapse initiation, i.e. the very first floor collapse, or as describing the early dynamic collapse, i.e. the second and following floor collapses. I only point this out because it is
vitally important not to confuse the two situations. Pursuant to your previous messages, we are only concerned with the former case, or what led to the first floor's collapse.
The NIST report suggests that the quantity Dd, the downward displacement in the upper section of each Tower, increased at a rate ~ 5 – 15 cm/hr after the aircraft impacts. This slow downward sagging of floors in the impact zone over a period of less than two hours, meant that a portion of the enormous potential energy stored in each Tower was slowly, but inexorably, converted into strain energy Es. This strain energy eventually exceeded the elastic limit of the structural steel and produced irreversible deformations of support columns immediately below the impact zones – columns that had a finite capacity to absorb strain energy.
There is a problem with this argument. Whether or not NIST makes this hypothesis, it is simply not correct to state that all of the increase in strain energy arises at the expense of gravitational potential energy.
What is missing is strain energy caused by the aircraft impact, and strain energy resulting from the fires. Let me consider only the latter, as the aircraft impact may have primarily destroyed everything it touched, the remainder never having reached the point of plasticity and rapidly returning to its new equilibrium. But the same cannot be said for the thermal energy.
As observed in both WTC towers, the fires led to buckling, one reason being thermal expansion. If we consider horizontal members exposed to heat, these members expand horizontally, exerting stress on columns and gradually assuming a bowed profile. This leads to significant strain throughout the structure, and this strain is
not related to gravitational potential energy. Similarly, once heat is removed, the members again contract, possibly reducing strain energy but also potentially
increasing strain energy still further, if the limits of elastic deformation were reached during heating -- a distinct possibility given the higher-than-normal loads experienced by the surviving structural elements while the fire raged.
If we restrict our view to vertical strain in the columns, this may well all be due to gravitational potential energy. However, this does not describe all of the damage or all of the displacement in the WTC towers. Heat is very much a factor here, and it can contribute much, much more energy.
Consideration of the conversion of potential energy into downward motion, first by reversible elastic yielding, then by irreversible column deformation, suggests the idea of a collapse-initiating energy equal to the maximum strain energy capacity Es(max) of the support columns on a single floor.
I have previously modeled the WTC collapse in terms of a quantity I called E1, the average energy needed to collapse one WTC floor. Clearly E1 is equivalent to the collapse-initiating energy Es(max) . In my report Energy Transfer in the WTC Collapse Events of September 11th 2001, Section 4.2, a value of about 0.6 gigajoules (0.6 x 10^9 joules) was estimated for E1. This is very close to the value estimated by Bazant.
I've read your paper and Bazant's. In your papers, you principally use energy E1 -- the collapse energy of a single floor -- as the threshold condition to collapse any
undamaged floor. This argument is most useful as a global collapse argument, hence why I called attention to the possible confusion above.
This is less useful in considering the
first floor, because it isn't undamaged. Nor are the floors truly homogeneous.
Energy E1 is not actually a constant. E1, dependent on the material properties of the steel as well as its size, will be a function of the floor as well as the temperature: E1 = E1(
h, T). E1 for the first floor will be significantly less than E1 for other floors due to its being heated, even discounting impact damage.
Taken in combination with the observation above, your derivation (1) undercounts the strain energy and (2) overestimates the energy needed to destroy the collapse floor.
Using NIST’s values of Dd, namely 10 cm for WTC 1 and 30 cm for WTC 2, in the formula Es = E1 = Mg(Dh) (with g taken as 9.81 m/s2), we find:
E1(WTC 1) = 0.637 x 10^8 Joules and E1(WTC 2) = 3.96 x 10^8 Joules
These values for E1 are problematic because E1 should be essentially the same for each Tower.
I would view this "problem" as a warning sign that your derivation is overly simplistic.
Regarding the "smouldering" fire:
It's too bad NIST had nothing to say on this, or the USGS, or FEMA. (And the EPA is another story)
I am not convinced though... smoldering is smoldering and that's what was needed to keep the fires alive.
We need a self-sustaining fire as much as a self-sustaining collapse
or why didn't the fire just GO OUT?
Whether or not the fire went out is a function of heat retention. I still don't see enough information to conclude that it must have gone out. As others have remarked, extreme low-level combustion in underground fires is hardly a new thing.
Like you say, though, pity NIST had nothing to say on this. Still, I'm hardly ready to throw out their ultimate conclusions.
