Reply #148: Scaling doesn't work that way

[William Seger]

I didn't check your arithmetic because you're completely off base with scaling, and worse, off track with what you would need to do to make the model accurate. If I have a cube that's 1 inch on each side, the cross-sectional area is 1 square inch and the volume is 1 cubic inch. If I double the length of each side to 2 (scaling 2x), the cross-sectional area would be 4 times larger and the volume would be 8 times larger.

At any rate, the problem is more severe than scaling linear dimensions versus square inches. Look at it this way: Assume that the towers (at least at the level of impact) were made of A36 steel. Under compression, that type of steel has a minimum yield strength of 36 kips (36,000 pounds of stress per square inch), but because of the factor of safety employed in the design, the normal load would have been less than that: According to the "reverse engineering" that NIST did, the perimeter columns had about 5x the minimum capacity necessary to resist their gravity loads (because they also needed to resist the wind loads), and the core columns had about 1.67x factor or safety (which is typical for a gravity frame). Without knowing anything about the number of columns or their dimensions or the loads they were carrying, that means that the perimeter columns were loaded to about 7200 psi (without any wind) and the core columns were loaded to about 21556 psi.

To make your model at least somewhat accurate, one important thing you need to do is get the stresses to be about the same stress per unit area, regardless of the number of columns or their dimensions or the scaling factor.

So, let's assume we can make your model with A36 steel hardware cloth so we don't have to worry about strength differences. From your numbers, I take it that the wires have a radius of 0.35 mm, which would be about .006 square inches per wire, or about 0.038 sq.in. for all 64 of them. So, just to get stresses similar to those in the tower, then, you would need a load of about 275 pounds on the perimeter columns.

Your model core columns are 1 mm radius, about 0.049 sq.in. each, or about 0.02 sq.in for the 4 of them. So, you need about 140 pounds of load on your core columns. (I've rounded all the numbers shown, but I used the full calculator precision for each calculation, so you may get slightly different numbers if you use the numbers I've shown.)

Right away, even if we used A36 steel, putting 415 pounds of load on your model is almost certainly going to cause a problem -- the same problem the towers would have if they were built like your model, without horizontal restraint at every floor: buckling. Buckling causes columns to fail well before their theoretical yield strength because they bend to the side. The model would surely just collapse immediately. But then, if you try to model the buckling behavior precisely, you couldn't do it with hardware cloth and coat-hangers, because column buckling behavior depends on the column shape and the "slenderness ratio" so it would be very hard to duplicate. Simply stated, your model's buckling behavior bears no resemblance whatsoever to that of the towers, which is just as much a problem as the lack of modeling the stresses appropriately.

Even if you did all that, the only thing you could hope to model would be the initial collapse -- and you would only be testing a failure mode (column buckling under heat alone) that is not very similar the one NIST found for the towers, because you're not modeling the sagging floors pulling the perimeter columns in, you're not accurately modeling the way loads would have been redistributed in the early stages of the collapse, and you're not modeling how welded and bolted pieces of steel would have become disconnected, destroying structural integrity. And then, unfortunately, you can't really model what happened after the collapse got started, because even if you scaled the masses involved, there isn't any way to scale the acceleration of gravity over the distances involved.

Does this approach to the analysis give you any insight into why your model doesn't tell us anything useful about the towers?