WTC Towers: The Case For Controlled Demolition

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/20010911_WTC/WTC_LDEO_K IM.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

Re: WTC Towers: The Case For Controlled Demolition

schoenfeld.one@gmail.com 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".
>

No it doesn't.

Now bugger off and actually STUDY the engineering.

There was no 'steel mesh'.


Each floor was dislocated in its entirety and went into free fall.

The figures for the time delay between impact and collapse exactly match
the range of figures that the fire protection was designed to afford.

Its a classic case of progressive collapse

Re: WTC Towers: The Case For Controlled Demolition

schoenfeld....@gmail.com wrote:
> WTC Towers: The Case For Controlled Demolition
> By Herman Schoenfeld

1) What does this have to do with PHP development?
2) This "theory" has been demonstrated to be complete bollocks
already.

They really need to start teaching critical thinking skills in
schools.

Re: WTC Towers: The Case For Controlled Demolition

schoenfeld.one wrote:

> 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

If you're going to post conspiracy theories to comp.lang.php, at least
have the decency to port them to PHP.


define('G', 9.8);
$u = array();

function cascadeTime ($_H, $_N, $_J)
{
$j = $_N - $_J;
$n = $_N - $j;
$h = $_H / $_N;

$sum = 0;
for ($k=0; $k<=$n; $k++)
{
$z = u($_H, $_N, $_J, $k);
echo "\$z is $z\n";
$sum += (sqrt(($z^2) + (2 * G * $h)) - $z) / G;
}

return $sum;
}

function u_recursive ($_H, $_N, $_J, $k)
{
if ($k==0) return 0;

$j = $_N - $_J;
$n = $_N - $j;
$h = $_H / $_N;

$R = u_recursive($_H, $_N, $_J, $k-1);
return ( ($j+$k-1) / ($j+$k) ) * sqrt( ($R^2) + (2*G*$h) );
}

/* Original recusive definition of u() can hit up
* against PHP's function call stack. */
function u ($_H, $_N, $_J, $k)
{
global $u;
$u[$_H][$_N][$_J][0] = 0;

if (isset($u[$_H][$_N][$_J][$k]))
return $u[$_H][$_N][$_J][$k];

$j = $_N - $_J;
$n = $_N - $j;
$h = $_H / $_N;

for ($i=1; $i<=$k; $i++)
if (!isset($u[$_H][$_N][$_J][$i]))
$u[$_H][$_N][$_J][$i] =
( ($j+$i-1) / ($j+$i) ) *
sqrt( ($u[$_H][$_N][$_J][$i-1]^2) + (2*G*$h) );

return $u[$_H][$_N][$_J][$k];
}

echo "WTC1 " . cascadeTime(417, 110, 93) . "\n";
echo "WTC2 " . cascadeTime(417, 110, 77) . "\n";

?>

I think I probably made a mistake porting though, as my results don't
quite match yours. My Haskell isn't what it once was. I think the problem
is in u().

--
Toby A Inkster BSc (Hons) ARCS
[Geek of HTML/SQL/Perl/PHP/Python/Apache/Linux]
[OS: Linux 2.6.17.14-mm-desktop-9mdvsmp, up 13 days, 1:36.]

Sharing Music with Apple iTunes
http://tobyinkster.co.uk/blog/2007/11/28/itunes-sharing/

Re: WTC Towers: The Case For Controlled Demolition

schoenfeld.one@gmail.com wrote in
news:3b2c0e83-10d3-496e-8021-73d99f30ef36@d21g2000prf.google groups.com:

> WTC Towers: The Case For Controlled Demolition
> By Herman Schoenfeld

I'll file this with "The Case For A Moon Made Of Green Cheese"

Re: WTC Towers: The Case For Controlled Demolition

hehehehehe

Kye.

"Toby A Inkster" wrote in message
news:p0gq35-joj.ln1@ophelia.g5n.co.uk...
> schoenfeld.one wrote:
>
>> 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
>
> If you're going to post conspiracy theories to comp.lang.php, at least
> have the decency to port them to PHP.
>
> >
> define('G', 9.8);
> $u = array();
>
> function cascadeTime ($_H, $_N, $_J)
> {
> $j = $_N - $_J;
> $n = $_N - $j;
> $h = $_H / $_N;
>
> $sum = 0;
> for ($k=0; $k<=$n; $k++)
> {
> $z = u($_H, $_N, $_J, $k);
> echo "\$z is $z\n";
> $sum += (sqrt(($z^2) + (2 * G * $h)) - $z) / G;
> }
>
> return $sum;
> }
>
> function u_recursive ($_H, $_N, $_J, $k)
> {
> if ($k==0) return 0;
>
> $j = $_N - $_J;
> $n = $_N - $j;
> $h = $_H / $_N;
>
> $R = u_recursive($_H, $_N, $_J, $k-1);
> return ( ($j+$k-1) / ($j+$k) ) * sqrt( ($R^2) + (2*G*$h) );
> }
>
> /* Original recusive definition of u() can hit up
> * against PHP's function call stack. */
> function u ($_H, $_N, $_J, $k)
> {
> global $u;
> $u[$_H][$_N][$_J][0] = 0;
>
> if (isset($u[$_H][$_N][$_J][$k]))
> return $u[$_H][$_N][$_J][$k];
>
> $j = $_N - $_J;
> $n = $_N - $j;
> $h = $_H / $_N;
>
> for ($i=1; $i<=$k; $i++)
> if (!isset($u[$_H][$_N][$_J][$i]))
> $u[$_H][$_N][$_J][$i] =
> ( ($j+$i-1) / ($j+$i) ) *
> sqrt( ($u[$_H][$_N][$_J][$i-1]^2) + (2*G*$h) );
>
> return $u[$_H][$_N][$_J][$k];
> }
>
> echo "WTC1 " . cascadeTime(417, 110, 93) . "\n";
> echo "WTC2 " . cascadeTime(417, 110, 77) . "\n";
>
> ?>
>
> I think I probably made a mistake porting though, as my results don't
> quite match yours. My Haskell isn't what it once was. I think the problem
> is in u().
>
> --
> Toby A Inkster BSc (Hons) ARCS
> [Geek of HTML/SQL/Perl/PHP/Python/Apache/Linux]
> [OS: Linux 2.6.17.14-mm-desktop-9mdvsmp, up 13 days, 1:36.]
>
> Sharing Music with Apple iTunes
> http://tobyinkster.co.uk/blog/2007/11/28/itunes-sharing/

Re: WTC Towers: The Case For Controlled Demolition

"Toby A Inkster" wrote in message
news:p0gq35-joj.ln1@ophelia.g5n.co.uk...
> schoenfeld.one wrote:
>
>> 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
>
> If you're going to post conspiracy theories to comp.lang.php, at least
> have the decency to port them to PHP.

ROFL !!!

Re: WTC Towers: The Case For Controlled Demolition

On Dec 21, 1:57 am, Toby A Inkster
wrote:

> If you're going to post conspiracy theories to comp.lang.php, at least
> have the decency to port them to PHP.

Decency for the decent.

[...]

> $u[$_H][$_N][$_J][$i] =
> ( ($j+$i-1) / ($j+$i) ) *
> sqrt( ($u[$_H][$_N][$_J][$i-1]^2) + (2*G*$h) );


Try ($u[$_H][$_N][$_J][$i-1])^2.

>
> I think I probably made a mistake porting though, as my results don't
> quite match yours. My Haskell isn't what it once was. I think the problem
> is in u().



[...]

Re: WTC Towers: The Case For Controlled Demolition

On Thu, 20 Dec 2007 14:41:53 +0000, The Natural Philosopher
wrote:

>schoenfeld.one@gmail.com wrote:
>> WTC Towers: The Case For Controlled Demolition
>> By Herman Schoenfeld

Truther whack job sack of fucking dogshit.

Here's to hoping your wife, sister, mother or daughter goes flying out
the next 100 story window to splat on the sidewalk in the next attack
you scumbag rat fuck.


>No it doesn't.
>
>Now bugger off and actually STUDY the engineering.
>
>There was no 'steel mesh'.
>
>
>Each floor was dislocated in its entirety and went into free fall.
>
>The figures for the time delay between impact and collapse exactly match
>the range of figures that the fire protection was designed to afford.
>
>Its a classic case of progressive collapse

Re: WTC Towers: The Case For Controlled Demolition

schoenfeld.one wrote:

> Try ($u[$_H][$_N][$_J][$i-1])^2.

That seems to get me closer. Numbers still don't quite match up, but that
might be because PHP's floating point algorithms aren't as good as Hugs.

--
Toby A Inkster BSc (Hons) ARCS
[Geek of HTML/SQL/Perl/PHP/Python/Apache/Linux]
[OS: Linux 2.6.17.14-mm-desktop-9mdvsmp, up 3 days, 20:54.]

Sharing Music with Apple iTunes
http://tobyinkster.co.uk/blog/2007/11/28/itunes-sharing/