# Ivan Němec et al (2018): Derivation of Simple Formulas for Collapse Upper Speed and Acceleration

#### Oystein

##### Senior Member
On Facebook, some lady named Jana Kárasková, from the Czech Republic I presume, presented me with an e-Mail from a certain prof. Němec to David Chandler, linking a paper Němec and colleagues had published in 2018:
A Contribution to Analysis of Collapse of High-Rise Building Inspired by the Collapses of WTC1 and WTC2: Derivation of Simple Formulas for Collapse Upper Speed and Acceleration

This is fresh, and I have not even browsed the paper, only the e-mail, which claims:
...we have derived the limit acceleration of progressive collapse of a high-rise building from its top. We have found that the theoretical upper limit of acceleration of falling mass, omitting any resistance of columns, and assuming that all the falling mass hit the lower floors, is one third of the gravitational acceleration. When assumed that half of the falling mass fell outside the building perimeter, the limit acceleration is one fifth of the gravitational acceleration. The acceleration limits were derived independently from both pertinent laws of mechanics, i.e. conservation of energy and conservation of momentum. The observed acceleration was faster than the theoretical limit. The needed energy was bigger than the potential energy of gravity of the buildings. The source of additional energy needed was not officially explained.

Earlier we have derived a general differential equation of progressive collapse of high-rise buildings and solved it for the case of the WTC 1 tower. We have shown that with regards to the basic laws of physics, the collapse, when started, for probable input parameters should arrest after 80-100 meters, or for a less realistic parameters the whole building could fall, but much more slowly than it was observed. We have described this in detail it in the book
Dynamics of Collapse of a High-Rise Building: Inspired by the Collapse of the Twin Towers of the WTC.
https://www.amazon.de/Dynamics-Coll...ert487p7i1tNjZV4pXqb7RZLP3nh_jmE78yGY3jaKJuIk

In the conclusion we declared that the official explanation of the WTC 1 and WTC 2 collapses is in contradiction with the fundamental laws of mechanics. ...
I have to admit, this work eluded me so far. A search for "Němec" or "Nemec" in this sub-forum came up empty.

The result of the paper, that acceleration derived from Conservation of Momentum, and ignoring resistance from columns, is 1/3 of g, is actually the same result that I got years ago when I modeled the WTC1/2 collapses as a series of collisions between falling accumulated upper floor slabs and static (magically suspended in air until hit) individual lower floor slabs. Except that my model only approached 1/3 g after a number of floors had already collasped and the collapse front had gained considerable speed. Before that, acceleration started at almost g and decreased from there.

So my model was, for a few suitably chosen floors early in the collapse, roughly in line with Chandler's measured "2/3 of g" during some time interval early in the collapse.

I wonder if the same is true for Němec's analytical model.

#### econ41

##### Senior Member
(1) He got the basic mechanism right and the dominant role that momentum gain plays in the "speed". (well he agrees with me on both points - so he is probably correct.. )
(2) I've never kept records but in the early pioneering days pre 2009 in mainstream (JREF) debate - both emerging "sides" - went down a false trail of macro motion measurement - implicitly assuming 1D approximations are valid. (Those pioneers who split from JREF in the 2008 schism went down a more esoteric version of the same false trail.)

1D approximations however framed or labelled are not applicable to Twin Towers because "tube in tube" is diametrically opposed to "1D approx". But in that early era there were a few momentum calcs by earlier pioneers. My memory is vague but F Greening was one. So they were chasing the right factor - momentum but NOT linking it to the true mechanisms whether we describe it as "debris missing the columns" or call it "ROOSD" and attract both technical and linguistic denials.

The name "Němec" is ringing a small bell at the back of my head but I could be thinking of our late colleague expert on metallurgy and thermite.

We have shown that with regards to the basic laws of physics, the collapse, when started, for probable input parameters should arrest after 80-100 meters, or for a less realistic parameters the whole building could fall, but much more slowly than it was observed.
Following abstractions and "parameters" whilst ignoring the real underlying mechanism derailed most of the Twin Towers debate up till around 2009-12.

PS a final thought - remember that Szuladzinski, Szamboti and Johns have claimed that Bazant & Zhou got their sums wrong and, with correct weight applied as per the B&Z "limit case" model Sz. Sz & J say the collapse would have arrested. So same conclusion but from two different approaches. And, AFAIK the Sz, Sz & J paper has never been falsified on that aspect. Moot point because the B&Z "initiation" is not possible > one of the 4 fatal errors in Bazant & Verdures "CD/CU"... but I digress too far.

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#### Oystein

##### Senior Member
(1) ...
(2) ...
(3) (I can't count)
"Our three - our FOUR chief weapons are..."
Thanks
The name "Němec" is ringing a small bell at the back of my head but I could be thinking of our late colleague expert on metallurgy and thermite.
Ivan Kminek. He was as a chemist more into polymers, iirc, and doing "thermite" only on the side.

Me too, but will take me some time. I whipped out this thread to preserve the links before I accidentally delete them. I am currently following a history lecture (Yale U history department) on the emergence of modern Ukraine, which atm takes up my daily hour of concentrated studying.

#### econ41

##### Senior Member
"Our three - our FOUR chief weapons are..."
Thanks

Ivan Kminek. He was as a chemist more into polymers, iirc, and doing "thermite" only on the side.
I tried searching ISF but for Kminic and Kminec. I should have realised "k" was more likely...
Me too, but will take me some time. I whipped out this thread to preserve the links before I accidentally delete them. I am currently following a history lecture (Yale U history department) on the emergence of modern Ukraine, which atm takes up my daily hour of concentrated studying.
I won't promise but the next day or so unless I get distracted.

#### Oystein

##### Senior Member
I did a first quick browsing before and during my commute to work today.

One of Němec' two approaches is something that I (and others) have done years ago, crudely, in the form of an excel sheet, numerically crunching the emergent acceleration...

Quick idea is that to consider only layers of mass that start out stationary (vertical velocity = 0) and then get hit by the falling mass from above, in a perfectly inelastic collision, such that the masses join and take on a common velocity governed by Conservation of Momentum.

Now while my model did this discretely - there would be a single layer of mass (a floor slab!) every 12 feet -, Němec throws nifty math at this and refines the discrete and finite in number floors slabs into a generalized, homogenous thing governed by smooth functions that you can analytically integrate and differentiate. Think of it this way: Instead of 110 floors, 12 feet apart, make it 220 floors of 1/2 te original floor mass, 6 feet apart, then 440 floors of 1/4th the mass, 3 feet apart .... and so forth, until you have infinitely many floors of mass approaching zero, spaced infinitesimally short distances approaching zero.

Němec shows - and I believe this is correct, as my own model approaches this value the further down you go in collapse progression - that such a smooth model collapse would occur at an acceleration of 1/3 of g.

This, he concludes, is significant as "other researchers" "measured" acclerations considerably greater than 1/3 g - which should be impossible, according to his model.

BUT: This value (1/3 g) is only (almost) true when the mass of the next "floor" is negligibly small compared to the falling mass that hits it.
This condition is true in Němec' smooth model, as the falling top "block" has a finite (and increasing) mass, while each imaginary "floor" has a mass indistinguishable from zero.
In my model, with discrete floors of finite mass, this is not true: For the North Tower, the falling mass would start out with the mass of 12 floor slabs, hitting 1 floor slab, and so that ratio is 1:12 - it's probably significant in the context of Němec' formulas, although I'd have to evaluate how much the acceleration would increase in such a case, according to Němec' math.

In reality, however, in the early stages at least, it's never the mass of the entire falling top block interacting with the next stationary floor: Initially, only one falling floor hits one stationary floor, truss seats may shear off on both, and the rest of the falling structure just continues falling near g. Then, two falling slabs hit one stationary - but as the two have already largely shorn off the columns, the effect of that 2-floors-on-1-floor collision again is hardly "felt" by the rest of the falling top structure.

This is why an observer of the first few seconds of collapse progression would measure the roofline (part of the steel exo-skeleton) to accelerate downwards faster than the accreting layer of compacted floor slabs does.

I kind of remember that Ansgar Schneider measured both the roof line and the progression of the collapse front, as made visible by rows of high-velocity dust ejections through the windows.

#### econ41

##### Senior Member
OK @Oystein - You posted just as I was starting to write a brief summary. So I will comment more fully.

Your response is broadly the same as mine but with a couple of distinctions that I will identify.
Put simply I think the approach of the paper is:
1) Correct to identify:
(a) Two stages of collapse. He identifies what I call 'initiation' and 'progression' and avoids the errors that others have made by not distinguishing stages. (As you are probably aware I now think it is necessary to distinguish four stages in order to support a complete qualitative explanation of the collapses. BUT those two are (nearly) sufficient for the purposes of their paper.)
(b) Momentum ac accumulation is the key factor dominating the "speed" of progression. AND
(c) his model is not correct - there must be some other factor in play. He accepts that the higher "speeds" around 2/3rds "G" are at least probably correct.

2) Falls short and gives wrong answers because, as you also identify, he is force-fitting his model to suit his mathematical approach.
So recall my preliminary caution about:
Following abstractions and "parameters" whilst ignoring the real underlying mechanism
... a problem that has confused much debate over the history of 9/11 discussion. (Bazant & Zhou followed by "Missing Jolt" seemed to set the trend on many derails pre ~2009-10. There were many more who "followed the leader".)

So let's see where we agree and where my perspective diverges from yours:

We agree the context setting of the paper and where its approach differs from your earlier similar efforts:
I did a first quick browsing before and during my commute to work today.

One of Němec' two approaches is something that I (and others) have done years ago, crudely, in the form of an excel sheet, numerically crunching the emergent acceleration...
By the way I recal several attempts at "momentum quamntifying" back in the day. I never gave them detailed examination because they failed my "test" of not fitting the real mechnaism.
Quick idea is that to consider only layers of mass that start out stationary (vertical velocity = 0) and then get hit by the falling mass from above, in a perfectly inelastic collision, such that the masses join and take on a common velocity governed by Conservation of Momentum.

Now while my model did this discretely - there would be a single layer of mass (a floor slab!) every 12 feet -, Němec throws nifty math at this and refines the discrete and finite in number floors slabs into a generalized, homogenous thing governed by smooth functions that you can analytically integrate and differentiate. Think of it this way: Instead of 110 floors, 12 feet apart, make it 220 floors of 1/2 te original floor mass, 6 feet apart, then 440 floors of 1/4th the mass, 3 feet apart .... and so forth, until you have infinitely many floors of mass approaching zero, spaced infinitesimally short distances approaching zero.

Němec shows - and I believe this is correct, as my own model approaches this value the further down you go in collapse progression - that such a smooth model collapse would occur at an acceleration of 1/3 of g.

This, he concludes, is significant as "other researchers" "measured" acclerations considerably greater than 1/3 g - which should be impossible, according to his model.

BUT: This value (1/3 g) is only (almost) true when the mass of the next "floor" is negligibly small compared to the falling mass that hits it.
This condition is true in Němec' smooth model, as the falling top "block" has a finite (and increasing) mass, while each imaginary "floor" has a mass indistinguishable from zero.
In my model, with discrete floors of finite mass, this is not true: For the North Tower, the falling mass would start out with the mass of 12 floor slabs, hitting 1 floor slab, and so that ratio is 1:12 - it's probably significant in the context of Němec' formulas, although I'd have to evaluate how much the acceleration would increase in such a case, according to Němec' math.
Fully agreed up till there.
In reality, however, in the early stages at least, it's never the mass of the entire falling top block interacting with the next stationary floor: Initially, only one falling floor hits one stationary floor, truss seats may shear off on both, and the rest of the falling structure just continues falling near g. Then, two falling slabs hit one stationary - but as the two have already largely shorn off the columns, the effect of that 2-floors-on-1-floor collision again is hardly "felt" by the rest of the falling top structure.
This is where we need three of the full four stages of the collapse mechanism. The issue is "How did ROOSD get started"? It is the "transition stage" and my hypothesis is NOT based on building up debris from a single floor impact. I have hypothesised that the "progression stage" was started by the impact of concentrated loads applied by the sheets of perimeter columns imposing a large proportion of the impulse of the Top Block onto the flooring and joists of the lower floors. (Actually "bi-directionally" - concurrent both "down" and "up") as can be seen from my second most commonly posted graphic:

... I won't repeat either the detailed explanation or the fatal consequences it has for Bazant & Verdure's "CD/CU"

BUT, if my hypothesis is correct ( ), it explains where the initial impetus came from. And it did not come from "single floor impacts accumulating".
This is why an observer of the first few seconds of collapse progression would measure the roofline (part of the steel exo-skeleton) to accelerate downwards faster than the accreting layer of compacted floor slabs does.
Yes. And it also highlights one of the big errors made (independently) by D Chandler and T Szamboti. They did not distinguish even the two main stages of collapse. And their measurements probably conflate all three - "initiation", "transition", and "progression". AND the differences between those mechanisms make any conflation AKA "homogenising vertical movements" totally invalid. It is the vertical equivalent error to the persistence with 1D approximations not applying to "Tube-in-Tube" >> leave that hobby horse for another day.
I kind of remember that Ansgar Schneider measured both the roof line and the progression of the collapse front, as made visible by rows of high-velocity dust ejections through the windows.
And IIRC he also conflated the stages. The same class of error.

Bottom line - methinks that between the three of us - the author(s) of the paper, your explanations of moment accumulation and my suggestions as to the starting trigger for "progression" >> we could be getting closer to the mark.

Thoughts???

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