Americans
don't know any more about physics and structural
architecture and engineering than the Soviet
population and Stalin knew about genetics. Today
"Lysenkoism"
is used as a metaphor to denote the corruption of
science in behalf of a social, political, or
ideological purpose.

The below Closure has been published in
Journal of Engineering Mechanics, July 2010
as a reply to the Anders Björkman
discussion paper
published at the same time. It can also be down
loaded aspdf.

The authors Zdenek P. Bažant, Jia-Liang Le,
Frank R. Greening and David B. Benson think that
skyscrapers collapse from top down as shown right,
i.e. a weak top C can crush an intact,
stronger bottom A by gravity alone.

They again demonstrate their complete ignorance
of structural damage analysis matters as
described by Björkman below the Closure
and, sometimes,
[within
brackets], in the Closure
itself:

The discusser's interest is appreciated.
[Thanks]
However, he presents no meaningful mechanics argument
against the gravity driven progressive collapse model of
our paper [Don't
I? - So what's wrong?]. His claim that
"the authors' theory is wrong" is groundless. Briefly,
the reasons are as follows:

Equations of Motion

The discusser claims that no differential equations
are required to model the collapse. This is incorrect.
The intuitive guesses emanating from his disconnected
quantitative estimates prove nothing. Although the
discusser uses some mechanics terms such as velocity and
acceleration, nothing can be deduced without actually
formulating and solving the equations of motion. If the
discusser rejects the differential equation form of the
equations of motion based on a smeared continuum
approximation, he could be credible only if he formulated
and solved discrete equations of motion.
[A one-dimensional
model - a line getting shorter/denser - described by a
differential equation - cannot describe a 3-D structural
damage model where the bottom part is 100X stronger than
the top part].

Energy Dissipation Sources

The discusser claims that the progressive collapse
model we developed in the paper does not consider the
energy required to compress the rubble. This claim is
absurd. [No, it
isn't! As described below, there is no energy available
to produce structural defects, ejection of parts,
etc.No structure can collapse from top down due
to rubble] He apparently overlooked that
this energy is included in parameter g
of Eq. (11). On p. 898 of the discussed paper it is
stated that, aside from the energy of comminution,
parameter g includes "the
energy of plastic fracturing deformations of floor
trusses with their connections and of horizontal steel
beams connecting the perimeter columns, the energy
dissipated by inelastic deformation and friction of
colliding fragments
[How to calculate
friction energy of colliding fragments??],
the energy of crushing the equipment, drywalls, perimeter
walls, furniture, piping, etc."

However, based on simple estimates of the surface
areas of all the fractures, fracture energies on these
surfaces, plastic strain magnitudes, magnitudes of
frictional forces in collisions, and frictional slip
distances, it transpires that the combined energy
dissipated by the aforementioned processes is much
smaller than the energies required for the comminution of
concrete into particles, for the ejection of air, dust,
and fragments at high speed (representing the work of F_a
and F_e). The reason for the dominance of the energy of
comminution of concrete is the extremely small size of
the particles, ranging from 10 to 100 mm
in size, which causes the combined surface energy of
these particle to be enormous. All these energies, in
turn, are small compared to the energy of plastic
buckling of the massive stocky columns (work of F_b), and
that energy is again smaller than the energy required to
accelerate downward the accreted stationary mass at the
crushing front [F_m in Eq. (5)].
[Hm, it would be
nice if these procesess can be shown in, e.g. a
laboratory!]

Therefore, it is not important to know parameter
g accurately. It was mentioned
in the paper that, within the range g
between 0.6 and 1, the calculation results for the motion
history, the time to hit the ground, and the amount and
size distribution of particles are virtually
indistinguishable. So it makes no sense to argue about
the precise energy dissipation by the aforementioned
secondary processes.
[It makes no
sense? It is not important! Come on, be
scientific!]

The overall energy balance is ensured by deriving the
differential equations of motion from an energy potential
[see Eqs. (25)-(30) of Bažant and Verdure 2007].
The necessity of gravitydriven progressive collapse is
demonstrated by the fact that the kinetic energy of
impact on each floor far exceeds the energy absorption
capability of the underlying columns [Eq. (3) in
Bažant and Zhou 2002].
[Well, it would be
nice to demonstrate a gravity driven collapse in a
laboratory!Do
it and win €1M].

Crushing of Columns

The discusser further claims that, for the
continuation of the crush-down phase, the columns in the
part C (upper part) must be assumed to be in contact with
the columns of part A (lower part). This claim is
erroneous. During the crush-down collapse, part C and
part B (compacted layer) are moving together as a whole
while part A is being crushed by the compacted layer B
[schematically, see Fig. 2(b) of the paper].
The energy condition for the crush-down phase to continue
is given by Eq. (6) of Bažant and Verdure (2007) (and the
gravity driven crush-down is actually guaranteed to occur
by Eq. (3) in Bažant and Zhou 2002).
[So Eq.(3) is
simply wrong].

Video and Direction of
Crushing

Observation of the upper margin of the cloud of dust
and smoke in the videos somehow makes the discusser
conclude that the tower top motion is caused by
"part C becoming shorter while
part A remains intact." This
is a delusion. Part A remaining intact would
violate the principles of conservation of momentum and of
energy. The writers' analysis of the initial two-way
collapse shows that the columns of part C get plastically
squashed by only 1% of their original length and
afterward the collapse proceeds in a one-way crush-down
mode (Bažant and Le 2008).
[Again, it would
be nice to show this in a laboratory or testing
institution; squash a column plastically 1% of its
original length and it produces a one-way crush down.
Evidently it is not
possble.
And then a
compacted layer of bits and pieces shall be formed! Let's
show it in a lab! Simple analysis
of any
video
of the 'collapse' shows top part C becoming
shorter and simply disappearing (!) before bottom part A
starts to be crushed ... and that no layer B is formed at
all. The video is actually a fake made by computer
generated images, CGI, + smoke added.]

The compacted layer cannot be expected to be seen in
the video record. Similar to construction demolitions, it
is not, and cannot be, located just under the upper
margin of the cloud because the rapidly ejected air and
dust spreads both downward and upward [Fig. 3(a) in
the paper].
[How can you
say that there is a compacted layer part B, when it
cannot be seen?]

Rubble Pile

Based on the profile of the rubble pile shown in Fig.
3(b) of the paper, the discusser estimates the rubble
density to have an unrealistic value (3.075 t/m³).
Since this figure is only schematic
[? it is very accurate],
his point is meaningless. Besides, he ignores the fact
that much of the rubble (characterized by mass shedding
coefficient k_out =
0.2 in the paper) has been ejected during the crush-down
[?? So first the
rubble is compacted and then it is ejected ... and your
1-D differential equation considers it?
LOL] and that the tall and narrow pile as
sketched exists only for a split second just before the
moment at which layer B hits the ground. At that moment,
the pile immediately begins to spread rapidly outside the
tower footprint. If one assumes the rubble pile density
to remain constant during spreading, a simple calculation
shows the rubble to spread about 60 m outside the tower
footprint. This gives for the rubble pile a slope of
about 20°, which agrees well with the typical slope
of rubble piles
[any
references?] seen in the demolitions of
buildings.

References

Bažant, Z. P., and Le, J.-L. (2008).
"Closure to 'Mechanics of progressive collapse:
Learning from World Trade Center and building
demolitions' by Zden?k P. Bažant and Mathieu
Verdure." J. Eng. Mech., 134(10), 917-923.

Bažant, Z. P., and Verdure, M. (2007).
"Mechanics of progressive collapse: Learning from
World Trade Center and building demolitions." J. Eng.
Mech., 133, 308-319.

Bažant, Z. P., and Zhou, Y. (2002).
"Why did the World Trade Center collapse? Simple
analysis." J. Eng. Mech., 128(1), 2-6.

Comments by Anders
Björkman(20 June
2010)

The above Closure must be regarded as the most
shameful Closure in structural damage analysis
history! It is suggested that "[Björkman]
presents no meaningful mechanics argument against the
gravity driven progressive collapse model of our
paper".

Actually no composite structure or building of any
kind can progressively collapse 'live
on (faked) TV'from top down to ground by gravity, i.e. a
moving top part C (about 20 concrete floors separated by
steel columns) dropping a short distance to start the
process crushes from above (!) a lower, intact part A1 of
identical structure into rubble/dust B that is 1/4 of A in
height according to Bazant & Co and then B and C crushes
A2, A3 and A4 into more B! This POUFF,
POUFF theory is a joke! Top part C, weaker than
bottom parts A1, A2, A3 and A4, cannot apply sufficient
energy so that elements of parts A1-4 only are destroyed.
Note further that A4 carries A3, A2, A1 and C prior
collapse, i.e. A4 is five times 'stronger' than C. Likewise
A3 is four times stronger than C, A2 three times stronger
than C and A1 twice as strong as C.

Bazant & Co however suggest that blocks A1, A2, A3
and A4, which each consists of abt 20 concrete floors
separated by steel columns, are destroyed/pulverized, when
the 20 floors of each block A1, A2, A3 and A4 are internally
impacted, floor by floor, by gravity and intact top block C
from above during the collapse and transformed into dust, B,
later found on the ground - block C destroying itself at the
end from bottom up ("Only gravity driven impact could
have produced the concrete dust as found on the
ground" - What Did and Did
not Cause Collapse of WTC Twin Towers in New York- Zdenek P. Bazant, Jia-Liang Le, Frank R. Greening and
David B. Benson, p 11 (2008)). It is terrible that
Bazant & Co support terrorism by producing 28 pages of
scientific, peer reviewed nonsense published in the ASCE
Journal of Engineering Mechanics ASCE , Vol. 134 (2008).

It is further suggested by Bazant & Co that "the
discusser claims that the progressive collapse model we
developed in the paper does not consider the energy required
to compress the rubble. This claim is absurd."

Energy required for
crushing

Absurd? So let's do an energy balance, when top part
C has crushed 41.84 meters of stationary part A
(density 0.25) and compressed it into 10.46 meters of
rubble B (density 1) at the beginning of the
'gravity collapse' as described in ref. [2]
of my original discussionpaper.

Or as Bazant & Co say: "If the discusser rejects
the differential equation form of the equations of motion
based on a smeared continuum approximation, he could be
credible only if he formulated and solved discrete equations
of motion" and"The energy condition
for the crush-down phase to continue is given by Eq. (6) of
Bažant and Verdure (2007) (and the gravity driven crush-down
is actually guaranteed to occur by Eq. (3) in Bažant and
Zhou 2002)".

OK, let's formulate and solve the discrete
equation of motion of top part C and the energy
associated with it. I have done it before!

The 53 meters tall top part C of WTC1 has cross
area 4 000 m² and volume 212 000 m^{3} and
weighs 53 000 000 kgs according Bazant & Co (i.e. 1 000
tons per meter in the Bazant 1-D line model). Top part
C actually consists of 1 000 000's of elements and
joints but Bazant simplifies them to one line!

Top part C drops down about 31.38 meters
in about 3.00 seconds as recorded from (fake -
animation!) videos. The top part C remains
intact (sic) according Bazant & Co: only
"the columns of part C get plastically squashed by
only 1% of their original length and afterward the
collapse proceeds in a one-way crush-down
mode". Bazant & Co ignore
1.000.000's
of weaker elements and joints that make up top
part C! In reality top part C should be
elastically deformed and seriously locally damaged
at contact.

Bazant & Co ignore the energy absorbed by
top part C as elastic and plastic
deformations. The drop corresponds to a loss of
53.000.000
x 31.38 x 9.82 kg m m/s² = 16.33 GJ of
potential energy. If the top part C
had fallen freely (no resistance) it would have had
a velocity of 24.82 m/s after 3.00 seconds, because
the kinetic energy of a block of 53 000 000
kgs at that speed is 16.33 GJ.

However, the average velocity of top part
C moving 31.38 meters in 3.00 seconds is 10.37
m/s, i.e. the actual velocity of C after 3.00
seconds is only about 20.74 m/s as observed from
(faked) videos (animations!), which corresponds to
11.40 GJ of kinetic energy (and 84% of
expected free fall velocity after 3.00
seconds).

Many observers point out that the acceleration,
read gravity collapse, is constant without any
initial and following jolts being observed, which
is quite strange. How can energy be transmitted in
multiple collisions = gravity collapse without
visible retardation of the moving top part
C?

However, only 69.8% of the available
potential energy has been converted into
kinetic energy - top part C is
accelerating - but not at 100% free fall
acceleration. Had top part C had velocity
24.82 m/s after 3.00 seconds, it would apply no
energy or force or pressure at all on lower part
A (like WTC 7
for 2.25 seconds during its destruction)!

According Bazant & Co remaining energy has
gone into deforming elements and ripping apart
joints in lower part A and compressing and
accelerating 41.84 meters (or 41 840 000 kilograms)
of stationary WTC1 lower part A at density
250 kgs/m^{3} into 10.46 meters of moving
rubble B at density 1 000 kgs/m^{3}
(the 31.38 meters drop of part C) to velocity 20.74
m/s. You do not need a differential equation of
a 1-D model to calculate that!

However "the compacted layer cannot be
expected to be seen in the video record"
according to Bazant & Co (sic)! So we are
simply asked to believe that top part C,
remaing intact (!), in 3.00 seconds
compresses/accelerates 41.84 meters of upper
section of lower part A into a 10.46 meters
rubble layer B that we cannot be expected to
see! Are Bazant & Co serious? A weaker top
part C destroying a stronger structure lower
part A below ... and we cannot see it?

Summary of energies
involved

So the following is a summary of what happens
during the 3.00 first seconds of lower part
A gravity collapse/crushing from above of WTC 1
according to Bazant & Co:

1. Top part C (mass 53 000 000 kg) moves
down 31.38 meters crushing lower part A
below and accelerates to and gets final velocity
20.74 m/s. Top part C is virtually undamaged
and not deformed.

"The
compacted layer (part B) cannot be expected to be
seen in the video record" according to Bazant
& Co.

Note - photo
above is a fake from a stupid animation! Evidently
no intact upper part C or compacted rubble part B
can destroy lower, intact part A. C and B are just
smoke/dust added by the
animators!

2. Uppermost 41.84 meters of lower part A (mass
41 840 000 kg) becomes 10.46 meters of compressed
rubble B (not seen!) that moves down 15.69 meters and
also accelerates to and gets velocity 20.74 m/s.

3. After 3.00 seconds of lower part A collapsing
due to gravity, top part C and rubble B -
rubble B not visible (sic) - with combined mass
94.840.000
kg have velocity 20.74 m/s that corresponds to 20.4 GJ of
kinetic energy as per Bazant & Co: "During the
crush-down collapse, part C and part B (compacted layer) are
moving together as a whole, while part A is being crushed by
the compacted layer B [schematically, see Fig. 2(b) of
the paper]".

4. The potential energy applied to (lost in) the
process is top part C moving down 31.38 meters and
41.84 meters of lower part A crushed down 15.69
meters (53 000 000 x 31.38 + 41 840 000 x 15.69) x
9.82 = 22.8 GJ, which means that only 22.8 - 20.4 = 2.4 GJ
of energy is used to compress
41.840.000
kg of lower part A into rubble B or 2 400 000
000/41 840 000 = 57.4 J/kg or 57 kJ/ton or 0.016 kWh/ton WTC
1 structure.

In fact - applying 0.016 kWh of energy on one ton
of any material/structure steel/concrete during say 3
seconds would not produce much effect at all. Just some
elastic deformations and local damages at contact points.
Easy to verify in any laboratory or university or material
testing company.

But acording to Bazant & Co: "...aside from the
energy of comminution, parameter g
includes "the energy of plastic fracturing deformations of
floor trusses with their connections and of horizontal steel
beams connecting the perimeter columns, the energy
dissipated by inelastic deformation and friction of
colliding fragments, the energy of crushing the equipment,
drywalls, perimeter walls, furniture, piping, etc.""

Another energy balance can be done after 12
seconds (dynamic state) - when WTC 1 lower part A
is completely crushed down into rubble B that
impacts solid ground (that is not affected!) - just
before crush-up.

5. Top part
C (mass 53 000 000 kg) has then moved
down 268 metres crushing part A below to abt 90
meters of rubble
B in 12 seconds. As average velocity is
22.3 m/s, the final velocity is say 44.6 m/s or
61.5% of free fall velocity.

6. Complete 358 meters of lower part A
(mass
358.000.000
kg) has become 90 meters of
compressed rubble
B (not seen due to smoke) that has moved
down 134 meters. It has also velocity 44.6 m/s.

7. The potential energy applied to (lost in) the
process is top C moving down 268 meters and 358
meters of part A crushed down 134 meters (53 000
000 x 268 +
358.000.000
x 134) x 9.82 = 610.6 GJ. The moving mass
C +
B with
height 143 meters is
411.000.000
kg with velocity 44.6 m/s after 12 seconds
corresponding to 407.8 GJ of kinetic energy
(that is applied to the ground). It means that
202.8 GJ of energy was used to crush down
350.000.000
kg of lower part A
into rubble B
or 579 J/kg or 10 more than crushing the first
41.84 meters of part A. It is still much too little
to crush anything.

Thus a complete energy balance shows without
doubt that WTC 1 was destroyed very quickly - 12
seconds according to videos of the collapse -
with extra energy than potential energy applied
from top added during destruction.

Actually, any energy balance shows that no
building of any kind can be crushed down from above
(sic) by a little top piece of itself + gravity.
The videos of the WTC1 collapse is simply a faked
animation shown 'live on TV' to fool the
viewers. In reality WTC1 was destroyed or blow
apart from bottom up using very strong conventional
means.

Figures by
Bažant, Z. P. explaining gravity driven crush-down
collapse

To shred a 1 000 kg car into pieces in a modern recycling
plant may require 32 kWh energy, i.e. 200 to 2 000 times
more energy! Bazant & Co suggest that extremely
little energy is required to pulverize and compact
steel/concrete lower part A into
rubble B. They have no
idea how much energy is required to break elements of steel
or concrete. To suggest that 0.016 kWh/ton can initiate
the destruction of the WTC 1 structure of steel/concrete is
criminal.

11 storeys of WTC 1 destroyed in 3
seconds!

41.84 meters of lower part A, destroyed in 3
seconds, is about 11 storeys of WTC 1 each weighing about 3
700 tons consisting of 4 000 m² of a concrete
floor held by >350 trusses connected via >700 bolted
joints to 47 core and >240 perimeter columns. So only 3
700 x 0.016 = 59.2 kWh of energy is required to destroy
one floor of WTC 1. Or it is suggested that the
columns have spliced connections every 3 floors so to
rip apart >2 100 bolted truss joints and >287 spliced
columns joints are done by about 180 kWh applied in less
than a second. It does not add up. To break bolted/spliced
joints requires much more energy. Easy to verify in any
laboratory!

Idiots believe
otherwise

Many
idiots belive that, if you drop a mass of steel
shaped into long, delicate strands (such as a bird cage or
the top part C of WTC 1 full of air) on a similar but
slightly stronger structure below (lower part A of
WTC 1 - another bird cage?) and apply 57 J per kilogram,
then one can be certain that only the delicate structure
below (i.e. lower part A) will be
fractured & crushed, while the even weaker structure
above (i.e. top part C) remains intact. In fact
top part C will just bounce on bottom lower part
A as A and C can easily absorb >5 700 J/kg as elastic
deformations. Bazant & Co are only supported by such
idiots that are also found at NIST. Plenty of crazy people
around!

Top part C should have rested on lower
part A

Actually you need about 2 000 times more energy per
kilogram structure to crush the structure in the WTC 1
collision or any structure and it cannot be applied by
gravity from above! So in reality only 1/1 000th of
both parts C and A would have been damaged at the
collision on 911 and then top part C would have rested on
top of lower part A! It happens in any collision between
two multi components/members/joints structural parts C and
A!

Both parts C and A get their components in contact
locally damaged and no rubble B is formed. It is easy
to verify in any laboratory. Drop any C on A (same structure
and C< A) and C always comes to rest on top of A. No
gravity driven collapse of a structure from top down is
possible under any circumstances. The 911 terrorists
overlooked that simple fact! But they seem to be supported
by US authorities, agencies (NIST) and professors!

It is sad to conclude that the Bazant & Co Closure is
as useless as the original
paper. It actually confirms that Bazant & Co
have a hidden agenda with its false information. What can it
be? To support terrorism!

However, I like the last sentence of the Bazant & Co
Closure: "This gives for the rubble pile a slope of about
20°, which agrees well with the typical slope of rubble
piles seen in the demolitions of
buildings".

Yes - the rubble pile really looks as if the
building was demolished by some planned
means.

Comments are always welcome to
anders.bjorkman@wanadoo.fr

If you disagree with above and can describe a structure
or building that collapses from top down, top C crushes
bottom part A into a rubble pile B using say 57 or 579 J/kg,
you can win Euro 1 000 000:- by participating in the
Heiwa Challenge!

A winner? Another
loser!

A structure A top C of which crushes A in a gravity
collapse is described here
- it is suggested that"progressive
collapse, when wrought, is quite ordinary and regular and
not due to extraordinary, possibly conspiratorial,
influences". It seems very simple ... because the
description is 100% nonsense based on Bazant's early
falsifications ... and does not work in reality. The false
suggestion by K.A. Seffen is that - the instability
(to create a gravity collapse of a
structure) is formed by the dynamical overloading
of the storey immediately below the weakened columns,
i.e. the uppermost elements of part A is dynamically
overloaded by contact of top part C. However, top
part C (a small part of A) cannot apply sufficent energy
(and pressure) on part A below under any conditions to
initiate and maintain further destruction of A. It really is
a simple as that. I do not wonder why K. A. Seffen
(kas14@cam.ac.uk) hasn't
collected Euro 1 000 000:- in the Heiwa
Challenge ! He is another loser!