The 9/11 Forum

Problem #3: Under the section "One-Dimensional Continuum Model for Crushing
Front Propagation" we are given 4 simplifying assumptions. Let's check out assumption #3:

3 The relation of resisting
normal force F
transmitted by all the columns of each floor to
the relative displacement u between two adjacent floors obeys a
known load-displacement diagram
Fig. 4, terminating with a
specified compaction ratio 
which must be adjusted to take into
account lateral shedding of a certain known fraction of rubble
outside the tower perimeter.

Sorry for the copy fonts.

This F is very important. Your results are as good as your choice of F.

He uses a floor by floor column buckling mechanism as his choice of F.

We know that most all core columns were found very straight in lengths of 36n where n=1, 2, 3 with ends broken cleanly along the welds.


We know the large, large majority of perimeter columns were found in very good shape, broken only along column and spandrel bolt connections. It's almost impossible to find a group of perimeter columns which were noticable buckled due to extreme downward force.


Despite all evidence to the contrary, the generized F in BV is derived by estimating floor by floor column buckling.


Why on earth would you expect accurate equations of motion to result from choosing the wrong F?

Problem #4: In the paragraph around equation (10) we get a well needed moment of comic relief.

[quote]First it needs to be decided whether crushed Zone B will
propagate down or up through the tower. The equation of motion
of Zone B requires that

F1 − F2 = s0g −
v·
10

where F1 and F2 are the normal forces
positive for compression
acting on the top and bottom of the compacted Zone B Fig. 2
c.
This expression is positive if Zone B is falling slower than a free
fall, which is reasonable to expect and is confirmed by the solution
to be given. Therefore F2F1 always. So, neither upward,
nor two-sided simultaneous, propagation of crushing front is
possible.


Here he theoretically proves that the upper block experienced no crush-up. It "isn't possible".

Yet we see many large perimeter chunks from the upper block of WTC1 ejected from the W, N and E sides just after collapse initiation. There is no evidence an upper block remained intact even 15 floors into the fall. Many of these pieces are 6 floors tall, one as tall as 11 floors. They are all noticably unbuckled and lead all heavy objects in the fall.

The evidence was so convincing that I believe every one of the regular posters with the exception of DBB admits the upper block was probably toast.

WTC1 was raining multi-floor unbuckled perimeter sections from the upper block yet BV claims that we must all be mistaken because crush up is impossible.

????

Instead of working this "stick" model with a minimum of 2 generalized coordinates as should be necessary, he writes his eq of motion with only one generalized coordinate by ignoring the destruction to the upper block we can all witness in video.


Q: How can these folks claim things that are so totally contrary to the physical evidence.....and get away with it?

Major_Tom wrote:BV can use only one coordinate because of the incorrect way they ignore upper block crush-up. (Totally intact, never crushes up, "fixed"). They "fix" the second coordinate. You can't do that.

The actual problem seems to require 2 coordinates just as OWE says.

It does require two. In all fairness, to skip ahead to another publication:

Closure to “Mechanics of Progressive
Collapse: Learning from World Trade
Center and Building Demolitions” by
Zdene˘k P. Bažant and Mathieu Verdure
March 2007, Vol. 133, No. 3, pp. 308–319.

by

Zdeněk P. Bažant and Jia-Liang Le

the matter is allegedly dealt with, specifically in answer to item #4 (Can Crush-Up Proceed Simultaneously with Crush Down?). The answer is "It can, but only briefly at the beginning of collapse, as mentioned in the paper." What follows then is a complex analysis that one must assume to be correct - for what it is - unless one is prepared to duplicate and check it. However, there are a number of things that jump out at me. I'm too dead on my feet to expound now, you're hard to keep up with, Major_Tom. It boils down to whether some of the assumptions accurately characterize the real situation, obviously.

I've said it before, it's worth saying again. I'm not at all confused where the model ends and reality begins, but I'm not so sure about the author. There is acknowledgement that the real affair was quite different in various ways, but then there's the insistence that significant early crush-up could not occur when it sure as hell looks like it did. Almost certainly did. I don't understand how that did not merit even a mention; perhaps he doesn't even know! It strongly suggests the author is wedded to theory such that the lines between theory and reality are blurred. Incredible intellect, without a doubt, wish I had a piece of that, but... It really becomes academic to the point of 'how many angels can dance on the head of a pin' - if there were angels, the argument might be worth double-checking!

If the dynamics are different, what is the result? Faster, slower, or arrested?

There is acknowledgement that the real affair was quite different in various ways, but then there's the insistence that significant early crush-up could not occur when it sure as hell looks like it did. Almost certainly did. I don't understand how that did not merit even a mention; perhaps he doesn't even know! It strongly suggests the author is wedded to theory such that the lines between theory and reality are blurred.

There is no doubt of this.


OWE, imagine a person wants to find particle motion using F=ma but they use an F that is provably inaccurate.

Why on earth would they expect accurate equations of motion? It's silly, no?


In BV, why is the generalized force F derived by floor-by-floor column buckling? Why would anyone take the resulting equations of motion seriously if their choice of F is provably inaccurate?



(Yes, it is that bad. In F/m= a, if your F is wrong, your resultant a is garbage.)

Major_Tom wrote:Why on earth would they expect accurate equations of motion? It's silly, no?

I think Bazant would get a belly-laugh from my tack-welded vertical particle arrangement. I'm taking the time to read BV again to try to avoid saying anything (really) stupid, this is where the time is going right now. So far, no surprises.

Can anyone extract figures 2 and 3 from the BV paper linked and post them for me?

http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/466.pdf

They give the basic definition of the only generalized coordinate and the basic model of how they derive F.

Thanks.

The problems I lsted before can be reduced to 2 fundamental formal complaints which render the resultant equations in the paper invalid.


Formal complaint #1:

It is impossible to describe motion of both upper block and lower block using only one generalized coordinate.

The illustration below shows how BV attempts to explain the system motion using only one generalized coordinate.



How do they describe a system with 2 inherent degrees of freedom with only one coordinate?

They fix the distance Zo and do not allow it to vary during what they call "crush down phase".

The authors do this by claiming that crush-up of upper block C begins to occur only after the lower block A is completely crushed.

This claim is provably invalid for the following reasons:

a) WTC1 upper block was toast early on
b) The outside walls of the upper block actually fell out and over the lower walls on the E, N and W sides (probably on the S side too, but that is top secret). This means the upper block was milktoast (there was very little structure left early on).
c) Demos we have observed that use an upper and lower block see both their upper and lower blocks consumed in the collapse.


These observations invalidate the claim and means eq 12 in the paper cannot be used to describe crush-down.

There is in fact no single differential equation which can describe crush-down.

Even using only 1 physical dimension there must be at least 2 differential equations used with 2 variables to describe crush-up, crush-down motion.


Note: If I were to give this problem of crush-up crush-down to an undergraduate class of pretty advanced mechanics students and even give them the form that building resistance F(z) takes, very few of them would try to solve it using only one generalized coordinate (and the ones that do would be wrong). If I gave this problem for homework to a decent graduate class of physics students, I'd bet that nobody in the class would even attempt to solve it using only one degree of freedom.

Formal complaint #2:

Eq 12 in the paper uses a generalized force F to describe the intensity with which structural resistance tries to impede the collapse floor by floor.

The model used to calculate F is shown in the illustration below.



The magnitude of this force on any particular floor is calculated by assuming it is caused by collective buckling of the type seen on the right side of the illustration.


This assumption is wrong. Very wrong. This can be verified by noting:

1) Most all core columns were found very straight in lengths of 36n where n=1, 2, 3 with ends broken cleanly along the welds.

2) The large, large majority of perimeter columns were found in very good shape, broken only along column and spandrel bolt connections. It's almost impossible to find a group of perimeter columns which were noticable buckled due to extreme downward force.

3) The existence of large sections of the core that survived the initial collapse in both buildings.
For WTC1 the entire east-west length for a height of close to 70 stories and column pairs comprising at least 2/3rds the north-south length were witnessed to survive the initial collapse.

4) There are practically no core column sections that show bending remotely resembling the model shown on the right side of the illustration above.


This once again invalidates eq 12.

Since the purpose of the paper is to describe mechanics of progressive collapse as the title states, and since this mechanics is described in eq 12 of the paper, this is a serious boo-boo.

Major_Tom wrote:How do they describe a system with 2 inherent degrees of freedom with only one coordinate?

They fix the distance Zo and do not allow it to vary during what they call "crush down phase".

The authors do this by claiming that crush-up of upper block C begins to occur only after the lower block A is completely crushed.

This claim is provably invalid for the following reasons:

a) WTC1 upper block was toast early on
b) The outside walls of the upper block actually fell out and over the lower walls on the E, N and W sides (probably on the S side too, but that is top secret). This means the upper block was milktoast (there was very little structure left early on).
c) Demos we have observed that use an upper and lower block see both their upper and lower blocks consumed in the collapse.


These observations invalidate the claim and means eq 12 in the paper cannot be used to describe crush-down.

Actually upper part C with constant Zo (and B) should continue to crush down the ground (make a hole!) after part A has been destroyed accordinbg Bazant & Co.

Formal complaint #1: Mindful of the endless caveats, I agree with your assessment.

Formal complaint #2: Also in agreement. Inevitably, this leads to the notion that the real case required less energy, and I admit this is the angle I've been working from for a long time. Are there ways to dissipate more energy than this without expending the energy to buckle? There are some possibilities that have been bandied about over time, but by their nature (non-conservative, velocity dependent) I don't see how any of them could lead to arrest. I could be convinced by a good argument.

I want to be fair. A couple of Bazant's colleagues post here, and their paper is coming up next in the queue. You start swinging the machete and it can cut pretty close to home. These papers are good science and very useful stuff, I hold them in fairly high regard. Nonetheless, what you say above is well put and true, in my opinion. How to reconcile the two? It's really not a problem for me and never has been. I think the series of articles being examined have, on occasion, overextended somewhat. The peculiar thing, again, is the conclusion that crush-up did not happen on the basis of this analysis. Nowhere do I see acknowledgment of how sustained eccentric loading applied asymmetrically to the upper block could dramatically change the mix of crush-up/down. This seems obvious!

Moreover, on the reread, the reasons for the discrepancy between their model and simulation become clearer and I'm ever more convinced that the model falls short by a wider mark than the simulation. I'm not ready to back that up yet here, but the ducks are lined up in my mind and it's fairly straightforward, nothing earth-shattering. Where the sim still falls short (like the model) is the need to fully overcome the structural resistance below in order to have collapse continuation. This is the flipside of having a model where Zones B & C are effectively one mass, Zone A is also a unified block. No eccentricity, no cleaving, no propagation of failure without momentum transfer. These things are optimistic for survival.

Off in one direction + off in the other = about right?

Let's make a general, intuitive list of features an acceptable model of crush-up, crush down must include. I'd include:

1) Needs at least two generalized coordinates, at least two degrees of freedom.

2) Must deal with the very real phenomenon of crush-up in a realistic way

3) Cannot blindly group zones B and C masses as being able to impact and damage the lower structure equivalently.

4) Must deal with the interesting situation which results when the intact upper block (zone C) is totally "used up" midway into the collapse, possibly very early into collapse. What happens when, near floor 80 for WTC1, you have only zone B vs zone a? Only debris vs intact structure. To me this is the reality that Bazant and offshoots are careful to avoid. They invent an "eternal, omnipotent" zone C which we all know cannot exist (except DBB).


What would others like to add to the list? Any objections to my first 4 points?


I hope we can all admit that BV is dead. You cannot address the criticism offered and salvage the paper.




OWE, do you remember in the first femr model when he got the building to stop moving using energy sinks? I'll review, but do you know how he did that?

Note: I am expecially interested in the fourth point above. It basically is Balzac+30 or 60. Intuition tells me that there is no guarantee the collapse continues to the ground because zone C will not exist after just a few more floors. I suspect the moment when zone C is totally or effectively "used up" we have a major transition point in the dynamics of the problem. This is where conventional crushing ends and "erosion" begins. Hence zone B vs zone A which instinct tells me is totally different than what we had before.

I suspect any papers we can get by this "Yaramer" (sp) fellow will provide much needed reality to the discussion.


Also, We know zone A does provide structural resistance to being crushed. If it is not due to floor-by-floor column buckling, then what is it?

Major_Tom wrote:1) Needs at least two generalized coordinates, at least two degrees of freedom.

2) Must deal with the very real phenomenon of crush-up in a realistic way

3) Cannot blindly group zones B and C masses as being able to impact and damage the lower structure equivalently.

4) Must deal with the interesting situation which results when the intact upper block (zone C) is totally "used up" midway into the collapse, possibly very early into collapse.

Good list. I think satisfying #2 could be a consequence of satisfying #1 and #3, and conditions expressed #3 and #4 have some overlap in that the influence of a non-rigid Zone B may increasingly dominate according to the thickness. The discontinuity of #4 may, in practice, be fairly smooth.

What happens when, near floor 80 for WTC1, you have only zone B vs zone a? Only debris vs intact structure. To me this is the reality that Bazant and offshoots are careful to avoid. They invent an "eternal, omnipotent" zone C which we all know cannot exist (except DBB).

The lower perimeter did effectively contain a large portion of the rubble, for a time. Bypassing most of the perimeter and some of the core, in terms of buckling, while still involving the majority of the building mass. In the quick and dirty model, having minimal shedding in the first few seconds most closely matches observation. I see the top essentially merging into the zone below, requiring the destruction of one or both in some proportion, and most of that mass is still there inside the perimeter. Intuitively, I'd be very suspicious of an analysis that concluded this rubble could be brought to rest and contained by the lower section in stable static equilibrium. So, yes, a transition (perhaps gradual until consumption) and one that might involve fairly dramatic changes in acceleration over a smallish period of time and perhaps little difference in the overall collapse time versus the Bazant model.

OWE, do you remember in the first femr model when he got the building to stop moving using energy sinks? I'll review, but do you know how he did that?

By dissipating a lot of energy in a step wise fashion combined with presumably more accurate mass distribution than homogeneity. I can't say whether the order of subtraction represents reality but, if structural is up against KE, then the structure bounced. In an axial 1D model, there is a very short window of displacement where the lower structure has the opportunity to arrest the impinging mass. If it does not, the structure is failed, no matter how much energy can be dissipated by descending through the story. Many of these dissipative effects are velocity dependent - they won't happen in great measure if there isn't sufficient velocity. Simply assigning energy dissipated per story is way off the mark; once I remarked how I don't trust my sims on the dynamics unless there's a good head of steam. What does have to happen at some story is the impinging load must be stopped, if there is to be arrest. This has to come from the elastic response of the lower section and not from (e.g.) concrete crushing. The 1D model draws upon the total capacity of what's below, and that doesn't reflect reality, either.

Also, We know zone A does provide structural resistance to being crushed. If it is not due to floor-by-floor column buckling, then what is it?

If it's moving fast enough to bust things up instead of just entraining them and executing most of the 'crush' towards the bottom, this will slow the descent. Gas expulsion becomes significant at appreciable velocities. Rubble loses kinetic energy to unrecoverable internal degrees of freedom. Everything but the vertical column crushing energy has a velocity dependency, I think, and even that would at higher impact velocities but we're excluding a lot of it through observation.

Work does not allow me the time to devote to this right now, but it is a fascinating discussion.

Edit: imagine the top 20 floors rubble-ized, distributed uniformly on the 80th floor, the middle 10 shed but enough of the perimeter left to contain the rubble like a dumpster in the sky. Would it stand?

Keep it simple, stupid: KISS! Take any composite structure with primary load bearing steel columns and secondary, mostly concrete, floors hanging on the steel columns and drop a small top part of this structural assembly on the lower part and see what happens (in 3D)! In my opinion the primary structural elements will damage the secondary structural elements where they are in contact and the latter will fail but still hang on to the former ones ... and after less than 1 second (in any scale) failed secondary and intact primary elements will get entangled into one another and further failures will be arrested due to friction between elements in contact.

It is the Björkman Axiom re structures!

I do believe surface friction from entanglement can be a significant factor in reducing early descent rate. If the lower structure were capable of supporting this randomly imposed load which has no integrity, then it might be possible for the progressive resistance provided by entanglement to arrest the upper portion without exceeding the overall peak capacity of the lower portion.

But I don't believe that. Not for a minute, not in these towers. Great effort is taken in skyscrapers to ensure load bearing surfaces are in full contact and good plumb. That would not be necessary if a general structure could support its design load arranged in 'any which way' and no concern for welds or fasteners! Instead, I believe, such an arrangement would quickly find the angle of repose for a pile of debris, slowed only by the breaking of connections on the way down.