WTC Towers: The Case For Controlled Demolition

Discussion in 'C++' started by schoenfeld.one@gmail.com, Mar 13, 2008.

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    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
    , Mar 13, 2008
    #1
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  2. wrote:
    > WTC Towers: The Case For Controlled Demolition


    Why has the 911 conspiracy theory become a religion to some? It's like
    all those muslims spamming all newsgroups with off-topic posts.

    > 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".


    Says who? You? At least not construction and demolition engineers.
    Why should we believe you instead of them? Are all the engineers in the
    world part of the conspiracy?
    Juha Nieminen, Mar 13, 2008
    #2
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  3. Juha Nieminen wrote:
    > wrote:
    >> WTC Towers: The Case For Controlled Demolition

    >
    > Why has the 911 conspiracy theory become a religion to some? [..]


    For the same reason you feel compelled to respond. Nothing better
    to do, would be my guess.
    Victor Bazarov, Mar 14, 2008
    #3
  4. Guest

    On Mar 13, 1:20 pm, Juha Nieminen <> wrote:
    > wrote:
    > > WTC Towers: The Case For Controlled Demolition

    >
    >   Why has the 911 conspiracy theory become a religion to some?


    As Victor intimated, don't let it bother you. Some people have
    checked out upstairs and left no forwarding address. Heck, the Flat
    Earth Society is still alive and well.
    , Mar 14, 2008
    #4
  5. OT : WTC Towers: The Case For Conspiracy Theorists

    On Thu, 13 Mar 2008 01:17:40 -0700 (PDT),
    wrote:

    >WTC Towers:
    >By Herman Schoenfeld



    A very fine case for... what ever it is you are making a case for, I'm
    sure, and as relevant to any of the other groups you selected as this
    one, no doubt about it..
    Brian Whatcott, Mar 15, 2008
    #5
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