The 9/11 Forum

Reference Closure to “Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions” by Zdene˘k P. Bažant and Mathieu Verdure, in particular pp 917-919.

The subject under consideration is the treatment given in those pages which attempts to formally justify the assumptions of a rigid upper block and exclusive crush-down followed by crush-up in the collapse of the towers. My opinion is that, while it is a very impressive treatment, it's hardly the final word on the subject. Over time, I intend to provide a number of objections and plausible loopholes which call into question the applicability of this analysis to the real situation in the towers and to subsequently test alternatives.

On the whole, this is an old subject, being discussed in innumerable threads here and elsewhere, frequently as an off-topic excursion. The latest such discussion on this forum ended here. It became a 'put up or shut up' kind of thing; while I may shut up at the conclusion of this exercise, it will only be after putting up.

The essence of my argument is that there are a very precise set of conditions implied by the treatment and that outside of these conditions there is no guarantee the B&L result holds. Some of the conditions are unknowns, estimable to within a reasonable value, others are clearly violated by the observed conditions. At what point, under what circumstances, does the result of crush-up proceeding only 1% into the first story of the upper block on initial collision fail to apply? What, then, are the consequences expected and do these match observables?

David B. Benson has kindly offered to evaluate the results of a competing model or change in parametric conditions in the existing model. It's now my duty to come up with a force function to permit analysis. This will come shortly, but I doubt that it will be just one! Hopefully, you're amenable to that, Dr. Benson. In any case, I'm not at that point yet, I want to discuss some of the background issues first, and expect this thread (or at least my end of it) to proceed ponderously for the most part. I've been working this angle for about two years, so I'm in no rush.

Of course, comments are welcome but be advised this thread is not a suggestion box. If there is a different model someone has in mind, it's their responsibility to formalize and present. Initially, my approach is to work within the framework Bazant provides to test the limits of his result, the premise being that there's no need to go any farther to achieve results I expect.

Claims:

1) The Bazant model can, without justification, assume rigidity of the upper block and also assume the consequence
2) B&L provides only a limited-case justification for employing this assumption
3) The actual conditions at initiation depart (radically in some cases) from these assumptions
4) Without a rigid block, d'Alembert's principle can not be used in formulating the equations of motion
5) While the result of such an assumption may correspond well to the observed overall dynamics, it may be insensitivity or even coincidence
6) Other models, even parameter adjustments to the existing one, may also give good correspondence to the dynamics
7) Other models may correspond better to the visual evidence

Can't know until you try.

From the first time I saw a video of the North tower collapsing, I wondered what happened to the upper section. It seems to disappear into the part below and only later is there obvious external signs of destruction below. It is a question seen over and over again in one form another, clearly a common impression to get when seeing videos which have been primarily taken from somewhere north, or else far away. Over time, and after seeing many more videos and still images, again and again, there's been nothing to shake that overall impression. If anything, the idea been bolstered quite a bit, though refined as well.

On the south side, progression outpaces the north. The upper section was known to tilt somewhat to the south. By contrast, a portion of the lower NW corner remained standing after the upper section had effectively passed, being knocked out from below. It's not difficult to support the notion that the tilting introduced a north-south asymmetry which resulted in the south lower section taking the preponderance of the beating, perhaps starting lower and earlier than the north. While it's not hard to imagine the upper north wall falling inside the lower, and having the south tuck inside the lower as well, the matter of having the upper block fit inside the lower laterally without presenting visual evidence of such is a tough pill to swallow.

Work done by Major_Tom has confirmed several large sections of perimeter above broke and fell away early in the collapse. Not only does this imply diminished capacity for the upper block, it does so at a critical time in which there is an opportunity for the damage to the upper block to proceed unabated in a more or less 'crush-up' fashion until such time as other factors come into play (hat truss).

Under the rigid block assumption, exclusive crush down must occur. This assumption was called into question and B&L is a response which purports to validate the assumption. I'm not calling into question whether this analysis was done correctly, I assume it is; neither do I question the correctness of the work of Bazant and partners in their other papers. If a rigid block is assumed, this is what you get. I question the applicability of the conditions assumed in the above-referenced B&L treatment, and therefore the applicability, not the correctness, of its outcome.

Much evidence, some very compelling, exists to show the integrity of the upper block, at least up to a hat-truss chunk in unknown orientation over time, was largely gone in only a fraction of time of the total collapse. When time permits, I want to dig up and link to some of these arguments, anyone else who wishes to do so, by all means. But this is just the support for requiring an explanation, not primary to the thread itself.

Does it matter? Personally, I don't believe that it does in this case, but beliefs are flaky things and even more so when not backed up by solid analysis. One thing that I do believe, and I'll stick to my guns on this because it is fundamental in physics: either a body is rigid and can (and must) be treated as rigid, or it isn't rigid therefore cannot AND must not be treated as such. Thus far, the only rigorous foray into answering this question, of which I'm aware, is B&L.

In the thread which led to starting this one, David B. Benson pointed out that Bazant does not disclose the form of the force function used in B&L. A couple things about that jump out immediately.

It seems like a very significant omission. After all, the entire outcome depends on this! If it didn't matter, anything could be chosen. But it is the core of the analysis, and I suspect it matters a great deal. Leaving this out essentially prevents independent confirmation; how was this response reviewed independently?

The absolute first step, for me, in attempting to explore the neighboring solution space is replicating the given result. I have to guess the original solution since it's not provided. This is a rather large hurdle - solutions in the neighborhood of what???

Fortunately, it may turn out to be pretty easy. I have my suspicions concerning the use of the Maxwell construction elsewhere... but this I need to check. Should my suspicions prove to be correct, that will spin off another round of criticism. If not correct, then it will spawn endless rounds of guessing and successive approximation to reverse engineer this most fundamental element of what was supposed to be a rigorous exposition.

In other threads, the presence or absence of jolts has been argued to death. The question of whether any are observable or not is being examined in detail at the present. It's clear that, should they exist, it's pretty difficult to prove that from the visual record, suggesting that there isn't nearly the magnitude expected from an axially-aligned impact involving the majority or all of the columns over short, discrete intervals.

Instead, resistance seems to be smeared out, something that might be expected from tilting and off-axis impacts. There is a good argument that tilting also tends to homogenize the mass distribution, making this assumption in the model closer to reality than a discrete distribution. On this I agree, tentatively, and will initially use a homogenous distribution.

Nevertheless, these factors are an automatic violation of the assumptions used to construct the analysis in B&L. On this basis alone, empirical evidence, it is out the window in terms of providing proof that crushing happened in the highly specific manner dictated by the results. No such disclaimer as to the inapplicability is given, neither is there an indication of the sensitivity of the outcome to mitigating adjustments or additions. It's a simple system which happens to bolster the confidence in a simple model, not a narrative for how it went.

Rigid bodies in axial impact make jolts at the roofline. Crushing bodies might not. Furthermore, the presence of a debris layer which can function as a rigid body acting on the portion below, but as a shock-absorbing cushion to the block above, is an unjustified leap of counter-intuitive hand-waving in need of some explanation.

F=ma (basic 1-D, cartesian metric) has 3 basic components: F, m and a. It is generally used to predict motion if F and initial conditions on mass m are known.

In the case of WTC1 we do not know F. We know "a" only through a limited set of data points during collapse initiation and a very rough guess of when large bulks of mass hit the ground only from seismic records.

So how does a person find evidence of either natural collapse or demolition through such limited data?

One attempt we notice in BV is that we assume an F (floor by floor column buckling in this case) and calculate an equation of motion based on that. If the motion matches the known data points, the author(s) then claim their choice of F was "on the money".

There are many problems to such reasoning and it is just circular logic if the problems are not addressed.

In the thread which led to starting this one, David B. Benson pointed out that Bazant does not disclose the form of the force function used in B&L. A couple things about that jump out immediately.

It seems like a very significant omission. After all, the entire outcome depends on this! If it didn't matter, anything could be chosen. But it is the core of the analysis, and I suspect it matters a great deal.

Yup. But how can anyone know F from just some building specs?

Like BV, you can guess the form F takes (equation form with some adjustable parameters), plug it in and see what parameter fit matches the data points.

But you are right, you can choose anything. You would use your own intuition to "make up" an F and see if it works. No need to stuff yourself in a shoebox at this point by believing one form is better than another until you test many guesses to see which force characteristics fit better than others.


Even after this, say you guess a great F. What does it tell you?


On the subject of jolts: For a long time I have expected to see some possible small jolts from floor collisions. I've never expected to see failed columns continue to hit columns below through end to end impacts, axial or no. In the core, once column displacement happens once, the columns act as spears colliding with floor slabs.

I do sympathise with Tony for trying to prove Bazant wrong because it was Bazant that set up equations of motion based on false claims (repeated column buckling), not Tony.

I am very interested to see if there are small jolts corresponding to 13ft floor separation. Oddly, it is only the demo ideas of yours truly, Major_Tom, that can explain such floor collisions along the north face. Natural perimeter faulure along multiple floors cannot explain such impacts from the first 13 ft drop. Symmetric and simultaneous perimeter kickout along the 97th floor upper windows can explain it. But maybe Trippy was only trippin' when he saw the jolts. Maybe not. Time will tell.

No such disclaimer as to the inapplicability is given, neither is there an indication of the sensitivity of the outcome to mitigating adjustments or additions. It's a simple system which happens to bolster the confidence in a simple model, not a narrative for how it went.

Exactly.

Major_Tom wrote:But how can anyone know F from just some building specs?

Doesn't seem possible. Therefore all the simplifications before even getting to the mechanics. Architects and engineers spend a great deal of time and effort to calculate loads and moments in an attempt to create a structure that will remain intact in the normal expected environment. If they spend any time at all in calculating the performance under suboptimal conditions, it's an attempt to box in the boundaries of a good design, not to see what the mechanics of failure are.

On an academic level and in the commercial arena, it is explored. So it's not like there's an absence of science on the subject. However, each circumstance is unique. It is doubtful that a single specific reserve capacity can be calculated for deviations from plan in anything but the most ideal situation. Once there is tilt and column-floor impacts, all bets are off in terms of whether the upper or lower or both fail at initiation. The argument of the upper block being a free body has some weight (forgive the pun) but the contrary notion of the lower portion being fixed also has some.

OneWhiteEye --- I'm willing to consider several force functions.

Major_Tom --- One chooses a force function F0 and determines how well that fits the data. The one chooses F1 to see how well that fits the data. If F1 fits much better than F0 then the weight of the evidence suypports F1 highly more than F0; there is a large literature on how to do this and the meaningfulness of the results. It not than anything new is being done here.

All --- While not considered (much) when the towers were designed, it is now required for large structures in earthquake prone areas to carry out a thorough major earthquake study, up to the point of collapse. So structural engineers now routinely use com[puter simulations to shake the design hard enough for inelastic behavior to occur.

David B. Benson wrote:OneWhiteEye --- I'm willing to consider several force functions.

Excellent.

All --- While not considered (much) when the towers were designed, it is now required for large sturctures in earthquake prone areas to carry out a thorough major earthquake study, up to the point of collapse. So structural engineers now routinely use com[puter simulations to shake the design hard enough for inelastic behavior to occur.

Good thing and good to know. Up to the point of failure. Once it gets moving, you need a physicist on staff.

OneWhiteEye wrote:Once it gets moving, you need a physicist on staff.

What happens is that one buys a copy of a modern FEA which is capable of handling progressive collapse. For certain structures even that has to be part of the design, although that is ratther rare AFAIK.

but the problem is that it is not possible to know the real force functions. We measure y(t) which is full of errors due to limited data. If we find the fn that fits best we cannot determine f as function of distance, in reality that will be a high peak that decays quickly but we can only guess energy drops per distance.

einsteen wrote:but the problem is that it is not possible to know the real force functions.

That is true, in general, for all of empirical science. However, there are only a few dissipative force function which approximate reality. Certainly possible to try all of those.

We measure y(t) which is full of errors due to limited data.

The Sauret video and OneWhiteEye's treatment of it is of excellent quality. I could wish for a longer interval of observation, but if wishes were horses beggars would ride.

If we find the fn that fits best we cannot determine f as function of distance, ...

Turns out it depends upon mass and speed, not distance.

...but we can only guess energy drops per distance.

I compute, on the side, energy expended in resistance, kinetic energy and potential energy made available by the drop. The sum of the first two is always less than the energy available.

Sorry, I don't actually understand the point you were trying to make.

Einsteen:

What we do know, from roofline drop measurements, is Mgh. This loss in PE is countered by a buildup of elastic and inelastic strain energy for each floor. For the elastic portion, at least, the stored strain energy is 0.5Fh, where F is an equivalent spring force.

Energy conservation requires that the kinetic energy of the falling mass is:

K.E. = h(Mg - F/2)

The falling mass comes to rest when the KE is zero. Also it follows that the peak resisting force needed to bring the upper mass to rest is twice the weight of the falling mass.

So the collapse is only propagated if the KE is > 0.

Elastic and inelastic strain buildup in steel frame structures such as WTC 1 & 2 can absorb a lot of energy, but only up to a fracture limit.

When a steel frame structure fractures and starts to break up, kinetic energy is created; in disproportionate collapse this process, once started, is very rapid and corresponds to an essentially instantaneous, "explosive' release of strain energy.

Dr. G wrote:Elastic and inelastic strain buildup in steel frame structures such as WTC 1 & 2 can absorb a lot of energy, but only up to a fracture limit.

And then a local failure develops in the weakest element of the structure. Question remains what kind of force is applied on the structure in question. If the force, F, is applied by a small part, C, of the structure being dropped on the remainder of the structure, A, I am happy to inform that A also applies force -F on C with known results: C is decelerated, C is deformed, C absorbs energy, C may bounce or suffer local failures.

In no case part C can one-way crush part A as per the famous axiom re structural damage analysis stating: A smaller part of an isotropic or composite 3-D structure, when dropped on and impacting a greater part of same structure by gravity, cannot one-way crush down the greater part of the structure.

Reason is simply that C, having same structure as A but being smaller than A, cannot absorb more strain energy than A; C is simply arrested by A in the form of bounce, deformation and local, structural failures.

Heiwa:

The verinage technique of demolition PROVES without doubt that a smaller part of an isotropic or composite 3-D structure, when dropped on and impacting a greater part of same structure by gravity, can indeed one-way crush down the greater part of the structure.

Dr. G wrote:Heiwa:

The verinage technique of demolition PROVES without doubt that a smaller part of an isotropic or composite 3-D structure, when dropped on and impacting a greater part of same structure by gravity, can indeed one-way crush down the greater part of the structure.

One-way crush? Actually the verinage technique is to quickly but manually destroy/remove a large number of primary support at midheight in the structure allowing the rather big top part to drop on a similar size bottom part, the latter probably weakened a little by further removing some elements there, whereby both parts are destroyed at impact. It may work for rather flimsy, concrete buildings that cannot absorb much strain energy but not for, e.g. steel structures of any kind and isotropic structures, which can absorb much more strain energy.
To suggest that the verinage technique, e.g. supports your ideas in the BLGB paper is not correct.