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Harry K Harry K is offline
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Default WTC Towers: The Case For Controlled Demolition

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 -


- Show quoted text -


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