(Note: this post follows my earlier and unsuccessful attempt to do the math testing whether the World Trade Centers collapsed at near free-fall rates just because of the force of the collapsing upper stories - as opposed to controlled demolition. John C. started emailing me his much better work on this. I'm basically just being a stenographer - someone could probably hold my hand and guide me through the math, but I prefer to just get this out on the Internet for anyone's use.)
John's first email analysis:
Some assumptions that are implicit to the analysis that I did and the effects of those assumptions are as follow:
- The analysis assumes that there is no material deformation of the structure. At first glance this would appear to be enough to cause my method to be invalid (after all there is a fair amount of material deformation), but I thought about it and if material deformation does take place then the analysis is invalid but the structure has already started to fail so I don't think it is that much of a restriction.
- The analysis assumes that the impact will be evenly distributed over the surface which it probably will not be. This would act to increase the dynamic stress.
- The analysis ignores the elastic compression in the tower. Since the height was so large this would probably be significant. By ignoring it the analysis that I have done here is conservative.
- I assumed that the steel structure was the same above the impact as it was at the impact. This is probably not the case (the supports were tapered) and again this would make our estimate conservative.
- I have also ignored the deformation and heat problems due to the jet striking the building. This would be very significant in the first floor or two but would be less significant as we go further down in the building.
- I should also note that after the first floor or two my analysis again goes out the window. First, you no longer have a solid (or even semi solid) object causing the impact but something like a wave of rubble. Also, as each floor falls you start to pick up kinetic energy. If the floor fails then it did not absorb all the potential energy. This leftover energy will be in the form of kinetic and at the next level you get potential and kinetic. I ignored the kinetic. What the net result of these two factors would be I haven't a clue, but I suspect that the kinetic energy would be such that it would eventually dominate the equation. Thus my analysis could show how it would start and then it is driven by the extra kinetic energy picked up as each floor collapsed.
As I said in my previous e-mail I guessed at a lot of values and came up with a crude estimate of 40 times the static load (actually it was 36). While this is a far cry from my initial guess of 2,000 (what is a factor of 50 between friends) It is still very significant.
However I wanted a better estimate and thus I went into it a little more this morning and it turns out that a lot of my assumptions (i.e. the area of steel, the height of each floor) don't matter. The important assumption is how much each floor weighs and I assumed that it was 1.5 times the weight of the steel. For the weight of the steel I used the density of steel x area of steel x height of the floor. If you don't like my 1.5 factor you can easily replace it with one of your own.
I find it hard to talk about math in an e-mail so I am attaching some scans of my work. If this is going to be put on the net you should probably go over it fairly well to check my assumptions, my equation manipulation and calculations (Editorial note: John is giving me way too much credit). My final answer is 32. Thus the dynamic load is 32 times the static load. I don't know about the WTC but for most structures a safety factor of 2 or 4 is used. I heard Dr. Thomas Eagar give a guess of a safety factor of 5. So my analysis is still 6 times this!!!
I will note that I am a mechanical engineer, not a civil one and someone with an advanced degree in structural mechanics could probably pick some holes in my analysis, but with the stipulations above I believe it is pretty sound.
PS I will probably be sending you e-mails with little comments on them as I think of things. For example I ignored the structural aspects of the building. For example there are areas on the floor that are designed to take only the load of the floor so even a static load of 10 times the expected load would cause failure - let alone the dynamic load!!!
John's second email:
I was going over my calculations trying to get better estimates and I realized a couple of things.
1) My h is too small (I used 3 meters but a better guess would probably be 3.7.
2) I underestimated my floor weight. I have a couple of references that say that each floor was about 4,500 t. I have no idea how they come up with this figure - it seems too high to me but the people who give it probably know more than me.
Anyway, I was looking for references and I came across this one. You have probably already read it but I found it interesting:
They used a similar approach to what I did (except they seem to have rolled up E, A and L into a single constant - probably a good idea which would save you integrating but I didn't have all the information to allow me to do it). However what I found interesting was their first estimate was a ration of dynamic load to static load of 31!!! So I think my rough value of 31.5 is actually pretty good (even though I have errors in my analysis it would appear that my errors cancelled out). In engineering sometimes it pays to be lucky instead of smart!!