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Funny m and its Tower Introduction
Since the WTC towers at NY were
destroyed on 9/11 2001 'live
on (faked) TV',
there is an ongoing discussion whether steel
structures just can one-way crush down or
progressively collapse from top to bottom due to
some local failures up top with the result that the
complete structure becomes rubble - as shown
right. It is also discussed at 1.18.50 in
DVD2
- a film about the 911 incident. According some people, incl.
religious fundamentalists and terrorists,
it is a natural phenomenon that, however, cannot be
modelled or explained by structural damage
analysis, like, e.g. ship
collisions. In order
to clarify matters I have designed the Funny
m structural assembly that you can use to build
a Tower that you then can try to crush ... just for
fun. Purpose is to establish what
spring, if any, breaks first when a Funny m
assembly, unrealistically, is dropped , free falls
and then contacts another Funny m assembly
in a Tower like structure or ground
itself: Funny m Funny m is a simple
structural 3-D assembly that consists of a
horizontal element with a mass m (e.g. a
floor) supported via solid connections by four
vertical elements s (e.g. columns) that can
compress like springs before breaking. Each
s carries m/4. The height of this structural
assembly is h. A picture of the Funny
m assembly is seen right. Due to mass m the springs
s compress elastically d =
0.03h. The structural Funny m
assembly is really funny or at least the spring
elements. They can compress 0.09h elastically and
0.1h plastically before they break (in this
example). It means you must put on 3 m for the
springs to start deforming plastically! If you drop a number of Funny m assemblies, let's call it part C, on another number of Funny m assemblies, let's call it part A, of a Funny m tower structure, it doesn't slip off at contact, as you would expect, but it will 'fuse itself' to it, and both parts C and A will absorb the energy applied at the contact. If a spring breaks after compression/failure and m drops, parts C and A remain 'fused' after next contact. However, if no spring break at contact, there is no fusing, but part C bounces. Very funny! The Problem! Historically, when you drop a small weak structural part C (apply energy!) on a stronger, bigger structural part A, C breaks up, while there may be some local damages to A, unless C simply bounces on A. NIST suggests however in its report of WTC 1 destruction on 9/11 that, if you apply energy on the top (!) of a structure - e.g., a part C of the structure is dropped free fall on the remainder of the similar but stronger structure, part A - global collapse ensues, when the potential energy applied at contact plus potential energy released due to failures exceed the strain energy that can be absorbed by the structure. A professor Z P Bazant maintains the same things in numerous peer reviewed papers published in the USA. Bazant suggests that the smaller, weaker part C remains intact (!) and one-way crushes down the bigger, stronger part A. This article will explore, if this funny suggestion applies to a tower of n Funny m assemblies! Right we see a Funny m
structure/tower on ground with n = 22
Funny m assemblies. The total mass of this structure
is n m (or 22 m) The potential energy of each m
is its distance above ground times g, where g is
gravity acceleration. The total Potential Energy,
PE, stored in the structure, relative ground,
is the sum of the PE of each m or n * m * n
* h * g /2 or PE = (n²mhg)/2
.......................................................................(1) The spring elements adjust
themselves to the number of m carried as explained
above. Thus the spring elements below the top m can
just carry one m. The bottom spring elements can
carry n m, i.e. they are n times "stronger" than
the top springs. The bottom springs can
absorb n times more strain energy than the top
ones! This means that all springs
compress equal distance d in the funny tower under
static load. Compressive stress is same throughout.
It also means that a spring above will
always break before a spring below, if you
add extra m on top without adjusting the
springs. The springs are really funny!
They can compress 0.09h elastically and 0.1h
plastically before they break (in this example) as
already explained. In the intact tower right all
springs compression is d = 0.03h, i.e. hardly
visible. Thus n m will compress the tower n d or
0.03nh. The spring constants C for all
springs are thus known. Each spring can in fact
carry 3 times more m than it is certified for,
before it starts to deform
plastically. Thus, the total elastic and
plastic Strain Energy, SE, stored in the
intact tower structure (the springs) is known. It
is actually the total mass of the tower, n m, times
g times 0.03h! It is also known that the total
structure can absorb 3 times the elastic strain
energy before any spring is overloaded, but then -
of course - the extra loads or masses must be
applied uniformly inside the whole structure! The
maximum SE that the Funny m Tower
then can absorb uniformly is SEmax =
0.09nmgh...............................................................
(2) or with n = 22, SEmax =
1.98mgh. It is very little compared to PE
- but we are just considering the
springs. Now we just suddenly apply
energy at specific locations! Top, bottom and in
the middle! Experiment 1 - Top
loading! We first remove, suddenly, the
springs below element n-1, #21 in this case, the
tower structure decompresses accordingly, and
then we
drop
the top (n/11) m (part C = 1/10 of part A) or 2 m
on the tower below (part A) or 20 m distance
h! It is assumed that part C really
contacts part A below and compresses part A …
and that part A compresses part C and the
ground. The energy applied by part C on
part A at the collision is (n/11)mhg or 2mgh
or 200/(11n) % of the total Potential Energy,
PE, of the tower. It is not much! It seems
to be about SEmax (2) !!! (The parameters have been
adjusted to get this result) These questions are best answered by removing other springs! Experiment 2 - Bottom
loading! Let's remove the springs below
the bottom #1 m and
drop
the upper part on ground h below! Then part
C consists of n=22m, it becomes completely
decompressed under free fall and the energy applied
by part C on ground (part A!) is 22mgh. It is 11
times more than in Experiment 1! Evidently the
ground applies energy and force on part C! Which
springs in part C break first, if any? Experiment 3 - Middle part
loading! We can also suddenly remove, say
springs between levels 6 and 12, so that part C -
11 m -
drops
5 h 'free fall'. Then part C, again completely
decompressed, applies 55mgh energy on part A - 6 m.
The 5 m between parts C and A also
drop
and add 15mgh energy on part A. Total energy
applied at 6 contacts - which take some time - is
70mgh. It is 35 times more than in Experiment
1! Which springs break then first? The top ones
in part A or the bottom ones in part C? Answers In Experiment 2 it is
evidently a spring in part C that breaks first,
unless all springs in big part C just compress,
apply a big force on ground, the ground applies a
big force on part C … that bounces. Please
note that bottom #1springs are 22 times stronger
than top #22 springs. One question remains
unanswered! How much energy is absorbed by ground
and how much of it is absorbed by part C? In Japan
they put shock absorbers between towers and ground
to avoid earthquakes to shake towers into pieces
from below. Evidently the ground can absorb plenty
of energy. In Experiment 1 it is
evidently the top #22 springs in part C that break
first, if any (compare Experiment 2), when
both parts C and A first compress, apply forces on
one another, as the C springs are the weakest. The
energy available, 2mgh, may be split 50/50
to parts C and A and big part A can handle
1mgh. Top #20 springs in part A are three
times (!) stronger than the #22 C springs, etc. Top
#22 m of part C may then drop on part A, as #21 m
did a little time before and the same thing will
happen again: part A, decompressed due to top #22
springs having failed, part A compresses again (!)
when #22 m contacts, applies a force on the #22 top
m of part C that bounces, as there are no more
springs above to break. Part C cannot
crush part A! You could say that part A
crushed part C! This is what normally
happens! In Experiment 3 it is
again the weaker springs in part C that breaks
first but as the energy applied is big also springs
in part A may break at the same time or a little
later. This effect is what we call Controlled
Demolition, CD. So a Funny m tower may
be demolished by CD! Conclusions Suggestions by NIST and
professor Z P Bazant that you can one-way
crush down and destroy a steel framed structure
part A by dropping a little top part C on it due to
lack of strain energy that can be absorbed is utter
nonsense. Towers are built so they are stronger
at bottom than at top and energy applied will also
be transmitted to ground and to horizontal
elements! Also, when something drops on a
structure, it is not the total mass of the dropped
part C that matters! It is the energy applied and
the associated forces and what damage they do that
count. It seems NIST has watched too much
TV! The myth about the Twin Towers
had its genesis in the immediate
aftermath. This does not prevent other clowns and true believers of fundamentalism at Internet forums to continue to maintain that steel tower structures globally collapse due to gravity, when a little part drops free fall (!) due to local structural failures up top. They even try to build structures of different sizes from 1 to 10 meters height to prove it - with various built in weaknesses at the bottom plus a heavy top that can be dropped free fall. This is very good! As they will always fail - the structure will collapse by itself during assembly - they might finally come to the conclusion that a global collapse of a normal steel tower structure after a drop of a small part C on the remainder big part A due to gravity alone is impossible. In reality real steel structures do not behave as outlined above - only springs compressing vertically! In reality top parts do not suddenly drop free fall, and, if it happens, the springs s will punch holes in the m elements or slip off and the m elements will get entangeled into one another, friction develops and arrest follows, etc. I have described it here! In reality a steel structure also contains much more strain energy, strength, than the one in the columns. There are horizontal elements and numerous connections, which all deforms in tension and compression and absorb energy. Explanation: To destroy a steel structure you really have to manually cut all supporting elements, the columns/springs of a big section using energy other than the one supplied by gravity and stored inside the building as per Experiment 3. That WTC 1 and WTC 7 were destroyed using energy other than provided by gravity on 9/11 should be clear to anybody that does not believe in fairy tales. But we are living in a funny world full of religious fundamentalists of all types and that's not funny. They need their myths to go about their evil ... or holy, in thier eyes ... work. Recommended
homework Many persons agreeing to above
still maintain that a one-way crush down is
still possible (!!) because it is not the springs
s that fail but the connections between the
springs (columns) and m (floor)! They are the
same throughout the structure! And at WTC 1 on
9/11 it was not 2 m but 14 m that dropped on the
top #97 m (and 96 other m below) and overloaded it,
etc, etc. This is the so called pancake theory. For
that to be valid all connections c between
14 masses (floors) and springs (columns) in upper
part C must suddenly fail … and that is not
possible. There were 1000's of connections c
in WTC 1 upper part C. Even the fundamentalists do not
believe that! Recommended homework is however to
make a Funny m tower, where eight
connections c fail, i.e. the springs (above
and below) get detached from m due to
gravity loads at contact part C with part A - see
picture right - while the springs s remain
intact. The homework includes explaining
connections c, incl. details, safety
factors, intact load transfers (m to
s), failure mode(s) and why c would
fail in the first place and why various m should
drop and the time table when the two ms
contact lower structure. Anders Björkman Heiwa Co and it will be published here! For the Advanced Student The Funny m assembly is very simple. Only one m and four identical springs s. You can also join 9 Funny ms to a Super Funny m assembly shown below:
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