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.

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

Sorry, you lost me at "t_k=(-u_k+(u_k^2+2gh))/g "

On Nov 5, 7:39 pm, wrote:
WTC Towers: The Case For Controlled Demolition
By Herman Schoenfeld
...

Uh, Herman, you're either a troll or a nutjob. Watch the films. See
the big shiny things hitting the buildings? Stop shaving your palms
with Occam's razor, it ruins the edge.

Mitch
(and surprise-surprise-surprise...gravity works in a predictable way
which is all your "case" shows.)

On Nov 6, 1:13 am, Mitch wrote:
On Nov 5, 7:39 pm, wrote: WTC Towers: The Case For Controlled Demolition
By Herman Schoenfeld

...

Uh, Herman, you're either a troll or a nutjob. Watch the films. See
the big shiny things hitting the buildings? Stop shaving your palms
with Occam's razor, it ruins the edge.

You can keep Occam's razor sharp if you store in one of those pyramid
thingys.

On Nov 5, 10:39 pm, 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 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.

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 -

In other news the Aliens put us here to observe us and to build them
Pyramids.

On Mon, 5 Nov 2007, wrote:

[snip]


Cilantro Mashed Potatoes

2-3 pounds potatoes
(I use yellow potatoes, unpeeled, your tastes may vary)
1 bunch fresh cilantro
2 Tbsp butter
1/2 C milk
salt to taste

Cut the potatoes into 1" chunks and cook until tender (your choice of
method, I sprinkle with water, sprinkle with salt and microwave them,
covered, for 10 to 15 minutes or until soft to the touch.)

Meanwhile: Stem the cilantro (leaf stems are OK) and process with the
butter until you have as smooth a paste as you can achieve.

Mash the potatoes well. Add milk as necessary to soften the mash. Fold
in the processed cilantro until throughly incorporated. The color change
and release of aroma will be dramatic, and the result is really very
yummy.

On Tue, 6 Nov 2007 14:08:30 -0500, "JA" wrote:


What's worse? A Conspiracy kook, or someone who TOP POSTS a single
line, then quotes a 170 line rambling diatribe

GFY! HAPPY?

You learn quicker than some...

On Nov 6, 1:00 pm, Gene Kearns
wrote:
On Tue, 06 Nov 2007 09:27:46 -0800, Bob Ward penned the following well
considered thoughts to the readers of rec.boats:

What's worse? A Conspiracy kook, or someone who TOP POSTS a single
line, then quotes a 170 line rambling diatribe

Uh, somebody that cross posts OT drivel to a gazillion newsgroups?

--

Grady-White Gulfstream, out of Oak Island, NC.

Homepagehttp://pamandgene.idleplay.net/

Rec.boats at Lee Yeaton's Bayguidehttp://www.thebayguide.com/rec.boats

Besides that, Gene, if you look at his calculations, it's quite a
piece! It shows that gravity does exist, and does so in a way that can
be resisted, but is very predictable!

On Nov 7, 10:23 am, Gene Kearns
wrote:
On Wed, 07 Nov 2007 13:50:52 -0000, penned the
following well considered thoughts to the readers of rec.boats:





On Nov 6, 1:00 pm, Gene Kearns
wrote:
On Tue, 06 Nov 2007 09:27:46 -0800, Bob Ward penned the following well
considered thoughts to the readers of rec.boats:

What's worse? A Conspiracy kook, or someone who TOP POSTS a single
line, then quotes a 170 line rambling diatribe

Uh, somebody that cross posts OT drivel to a gazillion newsgroups?

--

Grady-White Gulfstream, out of Oak Island, NC.

Homepagehttp://pamandgene.idleplay.net/

Rec.boats at Lee Yeaton's Bayguidehttp://www.thebayguide.com/rec.boats

Besides that, Gene, if you look at his calculations, it's quite a
piece! It shows that gravity does exist, and does so in a way that can
be resisted, but is very predictable!

It really wasn't worth my time to verify that figures don't lie, but
idiots can figure....

.... been there, done that, in the marathon "catenary" thread with the
master of Speedos....

--

Grady-White Gulfstream, out of Oak Island, NC.

Homepagehttp://pamandgene.idleplay.net/

Rec.boats at Lee Yeaton's Bayguidehttp://www.thebayguide.com/rec.boats - Hide quoted text -

- Show quoted text -

Oh, yeah! I surely didn't check all of his math, I do enough of that
for myself! But, if you look at the *essence* of his logic and calcs,
you'll find that, just like you say, the figures don't lie, they just
don't tell a whole lot!

On Wed, 07 Nov 2007 13:50:52 -0000, wrote:

On Nov 6, 1:00 pm, Gene Kearns
wrote:
On Tue, 06 Nov 2007 09:27:46 -0800, Bob Ward penned the following well
considered thoughts to the readers of rec.boats:

What's worse? A Conspiracy kook, or someone who TOP POSTS a single
line, then quotes a 170 line rambling diatribe

Uh, somebody that cross posts OT drivel to a gazillion newsgroups?

--

Grady-White Gulfstream, out of Oak Island, NC.

Homepagehttp://pamandgene.idleplay.net/

Rec.boats at Lee Yeaton's Bayguidehttp://www.thebayguide.com/rec.boats

Besides that, Gene, if you look at his calculations, it's quite a
piece! It shows that gravity does exist, and does so in a way that can
be resisted, but is very predictable!

If you can't dazzle 'em with brilliance, baffle 'em with bull****.

Gravity works - who 'da thunk it.