Dan Minette wrote:
Thanks for this Dan, some comments below:
That's not really accurate. To believe this theory one needs to accept that
the physics of rigid objects can be counterintuitive at times. I'm not sure
why this would be hard to accept. I gave one model of this, which didn't
get much response, but I'll try another. I'm going to construct a toy model
(a very simplified steel girder building) and show how the physics works
with this building. The actual building is more complicated, of course, but
that's what the engineering & architecture finite element analysis programs
(I believe those are the kinda programs that are used) are for.
Anyways, my building is build of these items:
Rigid Steel Beams,
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| O O |
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Bolts
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Rigid steel floors
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_/ /_
/ /
/ /
/ /
/ /
/_ _/
/_______________/
The beams are bolted together at the corners, and the floors are put on top
of the horizontal beams. Only the bolts have any give to them, and that
give is rather small.
What keeps the building from collapsing? It's the bolts. Now, in reality,
there is more than 1 bolt per corner, and the corners are often welded
together. But, the basic physics of the question is not affected by the
difference.
The building is held up by the sheer strength of the bolt. Lets assume that
the building about the floor we are considering has a mass of 10,000 metric
tons. That means there is a force of about 2.5*10^7 Newtons on each of the
bolts (10,000 metric tons is 10^7 kg, acceleration due to gravity is 9.8
m/s^2, 4 bolts).
Apparently the building was around 500,000 tonnes total, so what happens
if you halve the mass of each floor?
Let's have the bolts rated to 2.5*10^8 Newtons. That means that the
shearing force applied to the bolts needs to be 2.5*10^8 Newtons before the
bolts break.
I assume this is a number you made up to demonstrate the idea, rather
than an actual number. Thats ok, I guess there would be a lot more than
4 bolts too, I am just wondering how realistic it is. I will try to find
some numbers when I get time. I might have to do a spreadsheet !
So, let's have the floor immediately above the floor we are considering
collapse. Let's also assume that the distance between floors is 4 meters.
The top of the building, with a mass of 10,000 metric tons free falls those
4 meters.
Given the height of the towers (417m) and 110 stories, no argument there.
It takes about 0.9 sec to free fall those 4 meters. The velocity after the
fall is about 9 m/sec. So, we have a rigid object falling at 9 m/sec
hitting another rigid object. The give, we are assuming, is in the bolts.
But hang on, 110 stories, 0.9 seconds per story? isn't that at least 99
seconds for the whole thing? And yet the total time is more like 10
seconds isnt it? So is that time right, and if it isn't, then presumably
the velocity you are quoting is too high? If it takes say 10 seconds to
fall 400m, each floor is maybe more like 0.1 of a second. So the
velocity will be more like 1m/sec not 9 m/sec wont it?
The critical question is "how far will the bolts bend before breaking?" The
reason for this is that, for the floor we are considering to hold the
falling mass, it must decelerate it to a stop before the mass falls enough
to bend the bolts beyond the breaking point.
Steel, as others observe, doesn't bend much over short intervals. Let's
assume a stress strain relationship for our toy model where 2.5*10^8 Newtons
of force strains the bolts 5 cm. Any greater force, even 1 more Newton,
breaks the bolts.
Again I will see if I can find some numbers for the stress/strain thing.
But how are we looking if the mass is half and the velocity maybe 10%?
Given this, the falling floors are decelerated by the floor we are
considering until the deceleration of the mass is at 9 g's (we have to
consider the constant gravity force of 1 g). At this point, the bolts
break. Now, if we assume a linear stress-strain relationship (simplifies
the problem, but isn't essential), we can see the change in velocity of the
falling mass due to the resistance of the bolts.
Doing some handy-dandy math, with 1 msec steps, we find that the bolts shear
in less than 60 msec. During that time, the velocity of the falling mass
was reduced by about 0.25 m/sec from free fall.
By the time the mass falls another floor, it's speed increase, so it takes
less time to shear the bolts: about 40 msec....during which time the
velocity was reduced by another 0.17 m/sec from what free fall would be.
For each floor, the reduction in speed is less, as the bolts are broken in
less time.
I hope this was a straightforward explanation of the basic physics involved.
If anyone sees any problems with it, I'd be happy to review them. It is
true that I've simplified the problem, but going through the physics of the
simplification in my head, it doesn't change the nature of the problem.
I understand it is simplified, and I will try to do my own back-of the
envelope thing, but any comments on the above, maybe we can work
together to work out a good model.And then we can agree (or not) as the
case my be. There is also the issue of the core, which is not so much a
stack of pancakes as a vertical beam. I don't see how we can apply this
model to that. And yet it looks like the core collapsed first, and all
the way to the ground, not leaving a spire in the middle like you might
expect. How do we explain that?
Now, one may argue that this sort of analysis should be done without the
simplifying approximations of rigid beams and floors. It has been...that's
what the professional analysis is. I agree with them, which isn't
surprising....it just means that the first order approximations I made are
fairly reasonable.
So, I think we've gotten to the point where, in order to still say "I cannot
accept" the conventional explanation, then you will have to reject my basic
physics argument. I'd be very interested to see what flaws you might see in
this argument.
One other thing that puzzles me. Why do people bother paying a fortune
for controlled demolition experts? Just loosen a few bolts on the top
floor, or hell, pour a bit of kero out, light and run away fast. And the
whole thing folds up on itself. Given the money I might go into the
business myself.
Anyway, thanks, It is a good talking point you have started.
And look, I thinks it about 95% likely it happened as its told. Even
Bush is not that insane. But there are still things that trouble me
about it.
Andrew
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