Tuesday, June 30, 2009

Column resistance -- some thoughts on Chandler, Szamboti and Bazant

There has been some recent discussion on 911Blogger about one of David Chandler's videos on his YouTube channel.


I have some points to make here.

1. Chandler's argument, while correct as far as it goes, establishes only an upper bound on the column resistance. A better empirical estimate is roughly 0 - 0.1mg. This can be seen as follows.

The video measures the downward acceleration of the roofline of WTC1, during the first few seconds of collapse, and measures a downward acceleration of around 0.64g. Chandler argues, correctly, that this implies that the block is resisted by an upward force of (1-0.64)mg = 0.36mg, where m is the mass of the block. Yet this is puzzling, since the columns were (just a moment before) supporting the full weight (mg) of this block. (Moreover, assuming a safety factor of 3, they the columns were capable of supporting at least 3mg, almost 10 times their measured resistance.)

Chandler's video is, in my opinion, a useful device to attract attention to this important puzzle. However, while the reasoning is correct as far as it goes, the true situation is actually far more disturbing than even his argument suggests. (I assume that he is aware of this, but wants to keep the argument as simple as possible.) Let me explain why.

Chandler's argument is usefully seen as being based on an idealised model of the tower, where the floors below the airplane impacts have zero mass, and hence no inertia. Also, in his model there is no energy consumed in the pulverisation of the concrete to a fine powder. The only force resisting the descent of the upper floors (above the plane impacts) therefore is due to the steel columns. In reality, of course, the lower floors have inertia, and concrete requires force to crush it. Hence Chandler's quick argument greatly over-estimates the column resistance, and so should be regarded as an upper bound rather than an estimate. The true situation is even more worrying than his argument suggests. But what really was the column resistance?

The best way to measure this from the video evidence, I think, is to use a different idealised model, based on what Kenneth Kuttler has called a "floating floor" model. In this model, the floors have inertia, which resists the acceleration of the falling block. One also includes an energy drain for the crushing of concrete, as well as a variable drain for the deformation of the columns. (For convenience, it's best to calculate using energies rather than forces, although of course the work done by a given force is simply the integral of that force with respect to displacement.) In such a model one easily calculates the downward acceleration of the top block, for a given value of the column resistance. One then adjusts that resistance value to bring the model's behaviour into line with reality, as seen in the videos. This is then a reasonably accurate empirical estimate of the mean column resistance.

What value is obtained by such a procedure? I haven't yet finished my analysis, but it is certainly very low -- much less than Chandler's upper bound of 0.36mg. I believe it is in the range 0 to 0.1mg -- somewhere around that.

2. The papers by Zdenek Bazant on the WTC collapses provide an alleged theoretical estimate that roughly agrees with this empirical estimate. Bazant compares the potential energy released as the "top block" descends one story with the energy absorbed by the plastic deformation of the columns over the same distance. He concludes (without, unfortunately, showing much of his calculation) that the energy released is at least 8.4 times the energy absorbed, so that the collapse will inevitably progress. In other words, the columns absorb about 12%, at most, of the energy released. Now, this energy fraction is easily converted to a force, using the fact that the work done to the columns must equal the integral of the force applied to them, with respect to the distance fallen. In other words, the average resistance of the columns is at most 12% of the weight of the block. This is at the upper end of what is allowed by the empirical data.

3. We see therefore that Chandler's argument will not disturb Bazant in any way, who has already accepted a much lower value for the column resistance. On the other hand, if Chandler's empirical upper bound for the column resistance is worryingly low, then Bazant's theoretical estimate is far more so! One may even wonder whether Bazant's estimate is genuine, given that he doesn't explain how he obtained it.

These darker suspicions are supported by the fact that Bazant's figure lies at the upper end of what is empirically possible. If one were to invent a figure, say to reassure the engineering community, then this would be the perfect value to choose. One would choose the highest, i.e. most plausible, value that was consistent with the data.

4. How is a non-specialist like me to evaluate the reasonableness of Bazant's upper bound? One obvious way is to compare it with the estimates of other experts. I am aware of two of these, one due to Gregory Szuladzinski in Journal of Engineering Mechanics and the other from Tony Szamboti in "The Missing Jolt", Journal of 9/11 Studies.

The estimates are similar in approach. They calculate the energy absorbed from the elastic compression, then plastic shortening, then plastic buckling of the columns. The resistive force of a column drops sharply after it begins to buckle, but it seems to remain above about 25% of the yield stress until the thing breaks ("fractures").

The outcomes of the two calculations are also similar, giving average resistances above mg, certainly far in excess of Bazant's alleged upper bound of 0.12mg. Szuladzinski and Szamboti both predict collapse arrest.

5. In JEM, Bazant replies to Szuladzinski, finding some ten or so errors in his calculation. (Although there seems to be some padding here, as correcting some of these "errors" would increase, rather than decrease, the resistance.) It is hard for me, as a non-specialist, to judge the validity of these objections, although I suspect that they are minor quibbles. It is telling, I think, that Bazant does not correct the calculation, showing what it ought to be. This indicates to me that the outcome would not be too different.

To avoid such trifling objections, if that's what they are, it would be useful to calculate a robust lower bound for the column resistance. Can we have that, please? By "robust" I mean that all reasonable allowances for lowering the column resistance have already been made, so that it is the lowest possible value. It is quibble-proof, so to speak.

I repeat (to all relevant experts): Can we have that, please?