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 the "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
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.

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)



Simulates a top-down demolition of WTC 1 in SI units.

Simulates a top-down demolition of WTC 2 in SI units.

Why is this posted here? Am I missing something, or does this have
nothing whatsoever to do with Linux?
It looks like someone just posted a load of what is probably
conspiracy theory garbage to a handful of random groups (alt.fan.rush-
limbaugh, comp.os.linux.advocacy, alt.fifty-plus.friends,
alt.gossip.celebrities, microsoft.public.windowsxp.general)

"schoenfeld.one@gmail.com" <schoenfeld.one@gmail.com> stated in post
1194748691.850612.28350@e34g2000pro.googlegroups.c om on 11/10/07 7:38 PM:

No case needs to be made - unfortunately the towers have already been
demolished.


--
BU__SH__

On Nov 10, 8:33 pm, Snit <C...@gallopinginsanity.com> wrote:
....and "America's Mayor" hastily shipped out all the evidence for
recycling.

<bmillerjacobson@gmail.com> wrote in message
news:1194750100.619227.300180@o3g2000hsb.googlegro ups.com...


--
Chris, Sugar Hill, Barbados
A little learning is a dangerous thing, but it still beats total ignorance.

nrballard@gmail.com wrote:
He had to move it out of the way so new buildings could be erected !

No, wait...

[off-topic diatribe snipped]

This belongs in alt.conspiracy, not cola.

"bmillerjacobson@gmail.com" <bmillerjacobson@gmail.com> writes:

Hey look! There's a new Sheriff in town. Watch out High Plains Borefest
and Mark Kent - you're in danger of being usurped as the group's leading
self righteous gas bags.

--
You can tell how far we have to go, when FORTRAN is the language of
supercomputers.
-- Steven Feiner

On Nov 11, 8:34 pm, Hadron <hadronqu...@googlemail.com> wrote:
If that's how it came across, I would like to apologize. However, that
certainly wasn't my intent.

On Nov 12, 9:20 am, chrisv <chr...@nospam.invalid> wrote:
I'm kind of new, so I don't know who has the troll nature yet. I had
given him about a 75% chance of being a troll. One good thing about
that sort of apology is if someone is not a troll, it is acceptable,
and if they are, it denies them the opportunity to... well... troll
(can I use that as a verb?).

Didja use Excel for these figures?

--
Paul Hovnanian mailto:Paul@Hovnanian.com
------------------------------------------------------------------
mv Windows /dev/null

On Dec 15, 1:03 pm, "Paul Hovnanian P.E." <p...@hovnanian.com> wrote:
See Appendix A. The compiler used was WinHugs 98.