Reply #189: You're wrong about what Bazant calculated, and you are the one trying to use magic

[William Seger]

In order for the top columns in the impacted structure to bring the falling mass to a halt, they would need to do so before they buckle and can no longer provide any sensible resistance. That means they would need to decelerate the falling mass back to 0 within that time. The time cannot be magically spread out over an arbitrarily long time or distance, the way a car can be decelerated by the brakes. In fact, you can use that observation to make a rough estimate of the strength that would be required in those columns to arrest the falling mass. Since it's really a deceleration problem, you can use either the time or the distance involved to do that.

Structural steel can only compress "elastically" about 0.2% before it starts permanently deforming "plastically." The energy expended in plastic deformation, down to the point that the columns are bent completely in half, is actually the energy that Bazant calculates. But the resistive strength of a column drops off drastically at the point a column first begins to buckle and three "plastic hinges" form, as shown by the diagrams in Bazant's paper. At the point of first buckling (or actually, even before then), the columns would no longer be able to support even the dead load of the mass, even if its velocity had been reduced to 0 exactly at that point, so the weight alone would continue crushing the column. So, the buckling point is the limiting case: The columns must stop the fall before they buckle.

The buckling point of a steel column depends on its geometry, specifically its "slenderness ratio" which takes into consideration it's cross-sectional shape and total area, and its unsupported length. We can, therefore, estimate the necessary deceleration by estimating how much the length of the column could be compressed before buckling sets in. That would actually be somewhat different for each different column. However, we can work the deceleration problem backwards to see why there is no logical reason to expect the WTC columns to be able to stop the falling mass.

And the calculation is actually very simple if you use distances. If a mass falls with an acceleration of g for a distance of X feet, and then its velocity is smoothly brought back to 0 over a distance of Y feet, then the deceleration expressed in terms of a ratio to g is simply X/Y. For example if it falls 12 feet and the fall is smoothly arrested over a distance of 6 feet, that would be a deceleration of 2 "Gs" -- twice the acceleration of gravity. To stop it within 4 feet, you would need to apply a deceleration of 3 Gs. Being generous, let's say that the WTC columns could have supported (on average) 3 times the weight of the structure above, as a dead load. An impulse force is not exactly equivalent to a static weight, but as a first approximation, we can therefore estimate that the columns would need to stop the falling top of a WTC tower within a distance of roughly 4 feet. Since all we're looking for is a "yes or no" answer, let's be very conservative and estimate that the columns could provide a 4 G deceleration to the falling mass, so they have 3 feet to do it.

So, let's stop there for a minute and see where we are: Do you think the WTC columns could be shortened from 12 feet to 9 feet without buckling?