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On Mar 7, 1:48 am, schoenfeld....@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 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/20010911_WTC/WTC_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 -


Who cares , get over it.