On Mar 14, 1:24*pm, Harry K wrote:
On Mar 13, 9:51*pm, wrote:
On Mar 13, 9:41 pm, wrote:
On Mar 13, 9:53 pm, Harry K wrote:
On Mar 13, 1:17 am, wrote:
WTC Towers: The Case For Controlled Demolition
By Herman Schoenfeld
In this article we show that "top-down" controlled demolition
accurately accounts for the collapse times of the World Trade Center
towers. A top-down controlled demolition can be simply characterized
as a "pancake collapse" of a building missing its support columns.
This demolition profile requires that the support columns holding a
floor be destroyed just before that floor is collided with by the
upper falling masses. The net effect is a pancake-style collapse at
near free fall speed.
This model predicts a WTC 1 collapse time of 11.38 seconds, and a WTC
2 collapse time of 9.48 seconds. Those times accurately match the
seismographic data of those events.1 Refer to equations (1.9) *and
(1.10) *for details.
It should be noted that this model differs massively from a "natural
pancake collapse" in that the geometrical composition of the structure
is not considered (as it is physically destroyed). *A natural pancake
collapse features a diminishing velocity rapidly approaching rest due
to the resistance offered by the columns and surrounding "steel mesh".
DEMOLITION MODEL
A top-down controlled demolition of a building is considered as
follows
* * * * 1. An initial block of j floors commences to free fall.
* * * * 2. The floor below the collapsing block has its support structures
disabled just prior the collision with the block.
* * * * 3. The collapsing block merges with the momentarily levitating floor,
increases in mass, decreases in velocity (but preserves momentum), and
continues to free fall.
* * * * 4. If not at ground floor, goto step 2.
Let j be the number of floors in the initial set of collapsing floors.
Let N be the number of remaining floors to collapse.
Let h be the average floor height.
Let g be the gravitational field strength at ground-level.
Let T be the total collapse time.
Using the elementary motion equation
* * distance = (initial velocity) * time + 1/2 * acceleration * time^2
We solve for the time taken by the k'th floor to free fall the height
of one floor
* * * * [1.1] * t_k=(-u_k+(u_k^2+2gh))/g
where u_k is the initial velocity of the k'th collapsing floor.
The total collapse time is the sum of the N individual free fall times
* * * * [1.2] * T = sum(k=0)^N (-u_k+(u_k^2+2gh))/g
Now the mass of the k'th floor at the point of collapse is the mass of
itself (m) plus the mass of all the floors collapsed before it (k-1)m
plus the mass on the initial collapsing block jm.
* * * * [1.3] * m_k=m+(k-1)m+jm =(j+k)m
If we let u_k denote the initial velocity of the k'th collapsing
floor, the final velocity reached by that floor prior to collision
with its below floor is
* * * * [1.4] * v_k=SQRT(u_k^2+2gh)
which follows from the elementary equation of motion
(final velocity)^2 = (initial velocity)^2 + 2 * (acceleration) *
(distance)
Conservation of momentum demands that the initial momentum of the k'th
floor equal the final momemtum of the (k-1)'th floor.
* * * * [1.5] * m_k *u_k *= m_(k-1) *v_(k-1)
Substituting (1.3) and (1.4) into (1.5)
* * * * [1.6] * (j + k)m u_k= (j + k - 1)m SQRT(u_(k-1)^2+ 2gh)
Solving for the initial velocity u_k
* * * * [1.7] * u_k=(j + k - 1)/(j + k) SQRT(u_(k-1)^2+2gh)
Which is a recurrence equation with base value
* * * * [1.8] * u_0=0
The WTC towers were 417 meters tall and had 110 floors. Tower 1 began
collapsing on the 93rd floor. *Making substitutions N=93, j=17 , g=9.8
into (1.2) and (1.7) gives
* * * * [1.9] * WTC 1 Collapse Time = sum(k=0)^93 (-u_k+(u_k^2+74.28))/9.8 =
11.38 sec
* * * * * * * * where
* * * * * * * * * * * * u_k=(16+ k)/(17+ k ) SQRT(u_(k-1)^2+74.28) * * *;/ u_0=0
Tower 2 began collapsing on the 77th floor. Making substitutions N=77,
j=33 , g=9.8 into (1.2) and (1.7) gives
* * * * [1.10] *WTC 2 Collapse Time =sum(k=0)^77 (-u_k+(u_k^2+74.28))/9.8 =
9.48 sec
* * * * * * * * Where
* * * * * * * * * * * * u_k=(32+k)/(33+k) SQRT(u_(k-1)^2+74.28) * * *;/ u_0=0
REFERENCES
"Seismic Waves Generated By Aircraft Impacts and Building Collapses at
World Trade Center ",http://www.ldeo.columbia.edu/LCSN/Eq...C_LDEO_KIM.pdf
APPENDIX A: HASKELL SIMULATION PROGRAM
This function returns the gravitational field strength in SI units..
g :: Double
g = 9.8
This function calculates the total time for a top-down demolition.
Parameters:
* _H - the total height of building
* _N - the number of floors in building
* _J - the floor number which initiated the top-down cascade (the 0'th
floor being the ground floor)
cascadeTime :: Double - Double - Double - Double
cascadeTime _H _N _J *= *sum [ (- (u k) + sqrt( (u k)^2 + 2*g*h))/g | k-[0..n]]
* * * * * * * * * * * where
* * * * * * * * * * * * j = _N - _J
* * * * * * * * * * * * n = _N - j
* * * * * * * * * * * * h = _H/_N
* * * * * * * * * * * * u 0 = 0
* * * * * * * * * * * * u k = (j + k - 1)/(j + k) * sqrt( (u (k-1))^2 + 2*g*h )
Simulates a top-down demolition of WTC 1 in SI units.
wtc1 :: Double
wtc1 = cascadeTime 417 110 93
Simulates a top-down demolition of WTC 2 in SI units.
wtc2 :: Double
wtc2 = cascadeTime 417 110 77- Hide quoted text -
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For the people buying into the 'conspiracy' go over to
alt.conspiracy. *There are many threads there, at least 4 running now
in which all the BS theories are discussed. *Of course none of the
conspiracists will believe any of the debunking but it is good for
laughs.
Harry K- Hide quoted text -
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They really are a pathetic bunch, aren't they? * They like to focus on
every little nit separately, completely distort it and make outrageous
claims. * * None of these crackpots has put together a time-line and
story that runs from start to finish explaining what happened that
day. *Forget about controlled demolition. * What about all the planes
that left 3 airports that day and were flown into the buildings? * Who
flew them or is that just a myth too? * And if they were part of the
conspiracy, how would anyone know precisely where the planes would hit
the buildings and that the impact would not disrupt the allegedly pre-
planted explosives?
The official explanation does put together a whole picture that makes
sense. * *For this to have been some govt conspiracy, it would have
had to be the most elaborate one in history. *And the funniest part of
all, is these kooks try to blame it on the Bush administration. *If
they were gonna pull any conspiracy like this, why didn't they just
put some WMD's in Iraq? * That would be child's play compared to what
this alleged 911 conspiracy would have had to entail.
Sure.
If one can't explain every detail then the "official" conspiracy
theory is the truth?
What about the spire?http://youtube.com/watch?v=FWn8QNQWS...e=related-Hide quoted text -
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One does not have to explain _everything_ but mosst of it does have to
be. *If you are paying attention, the kooks explain absolutely
NOTHING. *They make assertions with nothing to back them up, ignore
reams of evidence, what they can't ignore they either distort or lie
about.
You are doing a good job of the latter.
Harry K- Hide quoted text -
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Exactly. If you take any sudden event, be it a bank robbery or auto
crash, subsequent interviews with eye witnesses almost always have
discrepancies. Someone will say guy A had a gun. Someone else may
say it's guy B. Somebody else will say there was no gun. If you
applied the standard of the conspiracy theorists to the bank robbery,
no one would ever get convicted. But if you look at ALL the evidence
fairly, then it's usually easy to figure out what happened.
Regarding these conspiracy kooks, my favorite was a show on TV couple
months ago. They re-examined the Oklahoma City bombing, because
there are conspiracy kooks claiming the exact same nonsense, which is
that the federal building couldn;t have been brought down by a simple
truck bomb. They claim there was another bomb well inside the
building. One key conspiracy nut is a retired airforce general,
who claims he was involved with weapons development with the airforce,
understands explosives, etc. He said by his calculations, the force
from the 1000lbs of amfuel explosive in the truck which was about 15ft
away, would have generated only a couple hundred pounds of force at
the key pillars holding up the front of the building. The official
investigation concluded that it was over a couple thousand pounds of
force.
So, the show got a Ryder truck, filled it with the same amount of the
same explosive, took it out to the desert and built concrete columns
placed the exact distance from the truck. They set it off with a
recording instrument to measure the force at the columns. Not only
were the columns destroyed, but the force recorded was dead nuts right
on to what the official report had concluded. I don't remember the
exact number but it was over 2000lbs and roughly 10X what the general
had calculated.
Presented with this result, the general said something to the effect,
"Well, it really doesn't matter, there had to have been a 2nd bomb
inside the building." That fits right in with one of the
characteristics of the conspiracy nuts listed above.
Among the other nit picking nonsense in the OKC case is that the
expolsion damage to the building wasn't symetrical. WTF? They expect
a truck bomb to make a perfect half circle imprint in damage to the
building. Construction experts explained that elevator shafts and
similar design details lead to the pattern being irregular.
But no matter what level of proof, the conspiracy theorists will never
be satisfied. I'm still waiting for their alternate explanation of
what they say happened on 9/11, start to finish. I don't want to
hear some lady in the basement heard a boom. I want to know the
plausible sequence of events, from start to finish, that explains what
happened.