The 9/11 Forum

David B. Benson wrote:Major_Tom offered two criticisms of the B&V crush-down equation. Neither hits the mark.

I think both are valid questions to raise. Further inspection may prove the objections to be untrue or irrelevant, but I think they need to be addressed.

...this crush-down equation is simply the continuous analogue of Frank Greening's floor-by-floor model;

In reality, the continuum limit with homogenous (or linearly varying) mass distribution is not much like the towers. Slabs of near the compaction size are closer. True or false?

Major_Tom objects to (2), implying, I think, two crushing fronts.

Yes, that's the idea.

Obviously Frank Greening does not think so...

Frank, what do you think? Is 'rigid body' a simplifying assumption, or is it dictated by the mechanics (exclude B&L for a moment, just your research)?

...and indeed Bazant & Le thoroughly remove that possiblity in the idealized, analytic case being considered just now.

They only prove their result for the case of homogeneous mass distribution and axial impact, which bears little resemblance to the actual situation. One can suppose it extends to the real situation, or that tilt somehow corrects it back to the valid domain of these assumptions, but that's all this is - supposition. How did the actual entanglement and subsequent punch-through dismember the upper block? What about any 1D analysis gives one confidence to proclaim the real eccentric impacts, possibly encompassing several floors, would act in any way similar to such analysis?

The whole point of both the exposition in B&V and Major_Tom's objections is rendered moot by the homogenization assumption;

The homogenization assumption is one of the things being scrutinized here. It is NOT representative of the tower's fine-grained structure. While I agree such detail should not be required for a global analysis, there is nothing I've seen that assures this is the case for early progression. In fact, what I have seen (barely above anectdotal at this point, but compelling all the same) indicates just the opposite. Initial conditions can have quite an influence, depending on how the system is modeled.

I don't expect you to buy into tinker-toy models, so let's talk about B&L. How much farther did the crush-up have to go before the plastic limit was exceeded (edit - removed text from here*)? We've got to be fair. Could the upper block drop a story on a (RIGID!) debris zone under one-third gravity and immediately arrest or even survive? You tell me. Looks like a pretty serious knife edge to me, regardless of the model. It seems that any significant asymmetry or uncertainty in parametric input has the potential to initiate a concurrent crush-up for at least a period. Tilt, non-uniform residual capacity, unknown damage... these things seem like major uncertainties.

How does the argument change when we're not talking about column-on-column impacts at all? Or an inhomogenous structure and mass distribution? These are valid questions.

I'm all for simplifying assumptions, genuinely; but I never lose sight that they are there for simplification and not improved physical description. If there is no difference, or the difference is acceptable for the purposes at hand, fine.

Edit - text removed:
" and by the same model's simplification, qualifies the resistance from the lowest columns in the upper block to be set to zero"

OneWhiteEye --- I've been thorugh all this before. Homogenization is fine when the tilt is taken into account; crushing proceeded on 3+ floors simultaneaously which is surely better represented by homogenization that by stepwise floor-by-floor model. However, both give essentially ythe same results; shagster actually went to the effort of running his own version of Greening's ideas using minifloors to demonstrate this; although, after some study, this is analytically obvious.

The issue of early crush-up never seems to die, does it? The problem is that it would have to proceed against the force of gravity, not with it. Instead what you seem to have noticed in frame 1007 is a lack of one dimensionality, with zone C west perimeter wall going outside the lower portion, yes? That actually does not trouble me, yet.

Now onto your video analysis: first of all, a small correction. Formal t0 is very close to frame 909 being about a tenth of a frame interval less.

Second, we don't know the tilt at 3.35 seconds (yet). Just to get started I'll assume it is enough so that 6 floors are simultaneouly being crushed, lowest to the south and highest to the north. With the south being, say, floor 86, the north is at floor 92. (I went and checked your post --- that's what you said, so far so good.) The problem is that my computer program says the center of the crushing front ought to be at floor 84, not the 86+3=89 approximation. That's a five stoy discrepancy. The program computes at that time having crushed 98-84=14 stories but from your visual analysis only 9 stories appear to have been crushed, 64% of the calculation.

Well, maybe the computed stretch of 0.295 is too high; 0.64x0.295=0.19 (which actually seems more believeable to me). That was the EY experimaental force; maybe should ignore it.
But EU is the same and EW has a stretch of 0.293, hardly different.
All three give supurb fits when ignoring the tilt and very good fits when an assumed tilt is added. So maybe quite a bit rests on actually determining the tilt.

The other possiblity is some crush-up. Assuming the stretch is ok, that's 5 stories of initial crush-up. That's more than I find credible.
Another possibility is that the resistance is being developed elsewhere.

What I would need, I think, is a by-eye, once each 10 frames, analysis of the SW corner to better understand the progression.

David B. Benson wrote:I've been thorugh all this before.

Yes, with me! By reflexivity, I have, too!

The issue of early crush-up never seems to die, does it?

Well, it sticks in my eye every time I do a frame-by-frame with one of these videos.

Homogenization is fine when the tilt is taken into account; crushing proceeded on 3+ floors simultaneaously which is surely better represented by homogenization that by stepwise floor-by-floor model.

This may be true, definitely can see how it could be so.

However, both give essentially ythe same results; shagster actually went to the effort of running his own version of Greening's ideas using minifloors to demonstrate this; although, after some study, this is analytically obvious.

I did the same thing with a simplified discrete crush-up like WTC7 using 47 and 470 stories, so I do know about this:

http://the911forum.freeforums.org/relation-collapse-time-and-collapse-energy-in-a-simple-model-t17.html#p1139

In the case of crush-down, I have little doubt it will give similar results when you define the top to be rigid! Did shagster do it in one DOF or two? What you may not realize is that the simple slab physics simulations show little difference in Zone B bottom coordinate while Zone C is crushing up on top. A little difference, not a lot. See this.

The problem is that it would have to proceed against the force of gravity, not with it.

Not at all. This is a fundamental misunderstanding of what I'm saying. The crush-up wave does not go up, it goes down, too! The most obvious external observable is the roofline dropping quicker; it's a crushup on top of a crushdown. As I said, it's crush-up in an accelerated frame of reference wrt to the ground. This non-inertial frame is accelerating downward at some value close to (2/3) in this case, or some appropriate value. This is the equivalent of the upper block crushing up in (1/3) g onto the 'ground level' of the top of Zone B. Look at the curves in the link I provided; nothing is going up. The net result is the roofline drops faster than it would otherwise, until crush-up terminates.

I'm not saying that because some slab simulations can crush up at (1/3) the force of gravity that the towers must necessarily have done so. I'm questioning the possibility and ramifications if it were so.

Instead what you seem to have noticed in frame 1007 is a lack of one dimensionality, with zone C west perimeter wall going outside the lower portion, yes?

Yes, fast side and slow side, to be approximate.

Just to get started I'll assume it is enough so that 6 floors are simultaneouly being crushed, lowest to the south and highest to the north. With the south being, say, floor 86, the north is at floor 92. (I went and checked your post --- that's what you said, so far so good.)

Yes, I can see some or all the difference being due to tilt, the south side undoubtedly leads. Because of this, I can argue that the advanced crush front is not too far off of what's expected in the Greening style model, and I have. It does seem to be a bit off, though, without some tweaking of a floor or two.

That's a five stoy discrepancy. The program computes at that time having crushed 98-84=14 stories but from your visual analysis only 9 stories appear to have been crushed, 64% of the calculation.

Thus the call for differential slices with different parametric input. It seems your solutions are closer to the leading front, certainly within a reasonable distance. Imagine for a moment two 1D models, the north half and the south half. This is very crude, but any less so than one 1D slice? The north half experiences some mixed crush while the upper block partially dissociates, and the south side is all crush down as the model predicts. The north side upper has a partially sheared perimeter and thus little integrity. The south side has the greater portion of the hat truss centered over it and possibly receives the bulk of it after punch through. Just a thought.

The other possiblity is some crush-up. Assuming the stretch is ok, that's 5 stories of initial crush-up. That's more than I find credible.

Even if 6-story sections are dislodged from the bottom of the upper block? That's a lot of partially supported flooring that certainly can't take much of a battering. Would such an upper block even be self-supporting on a flat, level surface? Contrast that to potential tuck-in and containment on the south side.

What I would need, I think, is a by-eye, once each 10 frames, analysis of the SW corner to better understand the progression.

Very likely.

So I show 2 fundamental problems with BV

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.


DBB says I miss the mark because

Major_Tom objects to (2), implying, I think, two crushing fronts. Obviously Frank Greening does not think so and indeed Bazant & Le thoroughly remove that possiblity in the idealized, analytic case being considered just now.

I do not think he'll ever understand this.

Every one of the demos in the following clip show that the upper block takes a beating during the crush-up, crush-down process.

Every example shows that crush-up happens at about the same rate as crush-down, about a 1:1 ratio.

http://www.youtube.com/watch?v=NwFHEoiUZ7o&NR=1

Yet here DBB claims this 1:1 destruction doesn't happen, can't happen.

We also see a similar thing happen to the WTC1 upper block, yet BV tells us this is not possible.

I'm sure I'll never be able to convince him, but most every other reader can see that he is wrong, so I'll just have to settle for that.



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.


To which he says I'm off the mark because

Also Major_Tom takes some sort of objection to (3) as B&V make it appear to depend upon column buckling.

Appear? The paper calculates F from column buckling. The BV eq of motion, eq 12, calculated F from column buckling.


Now, DBB will say the poor choice of F doesn't matter.


BV is nothing more than the derivation of crush down equations of motion using column buckling as an upwards resistive force.

We know that the upwards resistive force wasn't due to column buckling.


Now DBB says the equations of motion thus derived are still valid, though we know the assumption of F was totally wrong. Though they derived the equations using a particular F, and even though even David admits it is a poor choice, these equations are somehow still valid.

David, I know you will never be able to talk past this point. You can't because it's so fundamental to the paper. You will never be able to BS your way around the choice of F.

A simple version of complaint #2:

From physics 101:

BV sets out to solve the motion of WTC1 using the simplest of all equations F=ma in only one dimension (call it x).

It uses an F from column buckling and then uses F=ma to solve for x(t).



We actually know more than Bazant about the debris and the surviving core column group and the perimeter "peeling" (femr, give me a new word and I'll change it), so we know that F couldn't have been caused by buckling. The members of this forum pretty much know the column buckling floor by floor claim is bogus.


Now, DBB argues that our x(t) is correct, even though we admit we totally screwed up the derivation by being totally wrong about F.



Major Tom says "Whoa, guys, you totally screwed up the F. Your result is garbage."


But DBB says "x(t) is correct even though we totally screwed up F."

MT replies, "But how can you find x(t) using F=ma if you totally screw up F?"


DBB seems to treat the formula F=ma is magically self-correcting. Even though we make any poor, mistaken guess for F, the x(t) that pops out the other end works great.


I can't describe complaint #2 any simpler than this. I shouldn't have to.

David B. Benson wrote:B&V make four simplifying assumptions:
(1) one dimensional;
(2) energy dissipated only at the crushing front;
(3) known resisting force;
(4) homogeneous.

(1) well, WTC 1 was 3-D, so why not work in 3-D?
(2) when two structural assemblies collide, one of which is connected to ground, energy is dissipated in both complete assemblies and ground, e.g. as deformations and vibrations.
(3) if you don't know how the energy is dissipated, you cannot know the resisting forces (energy divided by displacement).
(4) none of structural assemblies was homogenous, rather the opposite; they were say 95% air, 4% concrete in horizontal layers and 1% steel and the lower structure easily carried the upper structure statically.

A more realistic 1-D model to simulate WTC 1 (or any one-way crush down in a gravity field) is 110 point masses m connected by vertical springs. Top spring carries 1 m, bottom spring, connected to ground, carries 110 m. All springs deform elastically the same statically in the gravity field and break when overloaded 300%. Thus the bottom spring is 110 times 'stronger' than the top spring.
It is very simple to suddenly remove one spring anywhere in the model and allow the point masses above the removed spring to drop on the point masses/springs below ... and calculate what happens!
At impact top/bottom assemblies evidently all springs are affected (compressed!) and one spring may break (first failure) allowing a new drop of masses/springs to take place (and maybe a second failure will take place? Another spring breaks!).
The first failure is evidently of interest? Will it take place, and, if yes, where?
If no first failure takes place, upper part evidently just bounces on lower part.

It is quite easy to calculate the energy required for first failure. If the energy applied minus energy absorbed as compression of all springs and displacements of all point masses at/after impact exceed what one spring can absorb, that spring fails.

It would appear that the spring that always fails first, assuming suffient energy/drop height are available (no bounce) is the bottom spring of the upper, dropped part, i.e. the spring just above the removed spring.

A more realistic 1-D model to simulate WTC 1 (or any one-way crush down in a gravity field) is 110 point masses m connected by vertical springs.

I think the most realistic way to study how buildings crush is to watch examples of crushing.

The demos in the clip below tell us more about what actually happens than all these silly theories.

Why guess when we can just watch?

http://www.youtube.com/watch?v=NwFHEoiUZ7o&NR=1

Thanks to the French we can see if our theories are true.

OneWhiteEye wrote:Even if 6-story sections are dislodged from the bottom of the upper block? That's a lot of partially supported flooring that certainly can't take much of a battering. Would such an upper block even be self-supporting on a flat, level surface? Contrast that to potential tuck-in and containment on the south side.

This is a good point. It means that the lower few floors of zone C are weaker that further up and alos further down (except of the east side with the detachment from the lower portion). So the homogeneity assumption is violated. I'll ponder upon this as potentially important.

Major_Tom --- Kieth Seffen also uses one coordinate; his paper is also in ASCE J. Engg. Mechanics. I opine you need to rethink your position. The question is simply the extent to which the four simplifying assumptions are adequate to describe the evolution of position of the uppermost parts. Works d**n well for WTC 1 all the way to the bottom; not so for WTC 2 where it seems zone C fell off.

Statistician George Box once said "All models are wrong. Some are useful."

Regarding equation (12) in B&V, the righthand side F(z), representing the resistive force, is clearly an arbitrary real-valued function of a real variable. It represents an arbitrary force resisting motion and requires no further justification in my opinion. For some reason B&V felt it necessary to justify it in terms of column buckling, possibly because that is what first comes to mind for structural engineers. In BLGB other components to the resistive force are added. That is the way of scientific progress. I'm in the process of adding still another term, largely, but not entirely, supplanting the resistive terms used in BLGB. That is, I hope, further scientific progress.

Major_Tom wrote:I think the most realistic way to study how buildings crush is to watch examples of crushing.

I have seen many structures being locally crushed by other similar structures, also being damaged at the same time, dropping from above (or full speed from the side) but I have never seen a one-way crush down (small top part crushing big bottom part) followed by a one-way crush up of top part.
My experience is that both parts/objects in contact are affected by the energy applied and that the weaker part is getting more damaged than the stronger, etc, etc.

It is quite easy to do full scale experiements, e.g. put 10 40' containers* (each 8.5' tall) on top of each other and then drop another container on the stack and you will see that there is no one-way crushing.

(* steel framed box with strong floor but very thin sides used for transportation of goods).

The container stack is a useful example: Stack any number of fully loaded containers on top of each other until failure occurs! It is the bottom container that will collapse first, as all containers have same strength.

As I always say; anybody suggesting that you can one-way crush a structure from top to bottom by dropping a top part of same structure on it, doesn't know much about physics.

David B. Benson wrote:The question is simply the extent to which the four simplifying assumptions are adequate to describe the evolution of position of the uppermost parts. Works d**n well for WTC 1 all the way to the bottom; not so for WTC 2 where it seems zone C fell off.

Actually the BLGB theory doesn't work for WTC1 either, which I describe in my paper to be published at JEM. The uppermost part, C, is evidently destroyed first (as seen on all videos) and cannot therefore produce the alleged one-way crush down of 10X bigger bottom part, A, the latter apparently being destroyed from top to bottom by energy/forces applied locally inside A blowing out sections of wall panels in all directions and producing a cascading fountain of debris.

Heiwa --- But nobody can see it! Zone C simply disappears into the obscuring dusts. Not sufficient reason to assume it is being crushed first. If sufficiently close to homogeneous, then from Bazant & Le it is not being crushed at all.

OneWhiteEye --- To approximately account for the imhomogeneity in the lower part of zone C, I started the collapse at floor 98+4=102. Then the crushing front is at floor 88.543 at 3.3217 sweconds and at floor 87.867 at 3.4106 seconds and also the runout to 15 seconds is reasonable, with the crushing front in the fourth down basement level.. So it seems the the B&V crush-down equation is quite flexible.

The fit isn't quite as good, but this is because I only used the fast simplex method rather than going on the the slow program based on Powell's method.

Regarding equation (12) in B&V, the righthand side F(z), representing the resistive force, is clearly an arbitrary real-valued function of a real variable. It represents an arbitrary force resisting motion and requires no further justification in my opinion. For some reason B&V felt it necessary to justify it in terms of column buckling, possibly because that is what first comes to mind for structural engineers.

I don't think you followed the argument in the paper.

Of your 4 simplifying assumptions, check out #3..

From the paper:

(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)

DBB, have you looked at figure 4? That's column buckling.


I assume you are answering my complaint #2 by saying normal force F in BV is a generic, arbitrary expression and is not related to column buckling? Then what is simplfying assumption #3?


And as for the first complaint you believe in the indestructable upper block based on a theory even though demolition planners have been smashing upper blocks against lower blocks for a few years now and even have a special name for this style of demoltion.

We can see the upper block take quite a beating in every one of these cases http://www.youtube.com/watch?v=NwFHEoiUZ7o&NR=1 and crush itself with the lower block in what appears to be a 1:1 ratio. Every case.


I do not have the patience that OWE does so I'll stop here. It is unfair that so many demos are available for viewing which are perfectly designed to answer the crush up, crush down question yet you refuse to view them or be honest about them.

David B. Benson wrote:Heiwa --- But nobody can see it! Zone C simply disappears into the obscuring dusts. Not sufficient reason to assume it is being crushed first. If sufficiently close to homogeneous, then from Bazant & Le it is not being crushed at all.

In my opinion upper part C disappears horizontally as ripped away perimeter wall panels and debris are thrown 100 meters in all directions and neither it nor part B - the compacted rubble below part C - exists, e.g. 10 seconds into 'crush down:

I really wonder if you have studied any videos or pictures of the WTC 1 destruction/controlled demolition?
The energy required to produce the cascade of debris is probably 1000X the kinetic energy that part C could have applied if intact!

Major_Tom --- I follow the paper very well, thank you. Do note that in BLGB the resising force becomes more complex. That's scientific progress for you.

I'm video challenged, so give up on that, if you will please.

Assuming homogeneity, Bazaant & Le show thaqt zone C is almost industrucible. That's mechincs for you. The sturcture obviiously was not homogeneous and you have, in other threads, shown some distruction along the west and north walls. In of itself that mass loss is not important, but it does mean the floor trusses in those areas have been weakened. So an average of about 4--6 stories above floor 98 do not come close to satisfying the homogeneity condition. Fine. consider then that zone C is from floor, say, 102 up. To keep the equation simple, assume crush-down begins from there. As I mentiioned in this thread yesterday, this works well enough to match the additional observations by OneWhiteEye.

Heiwa --- About 80% of the mass of the tower was in the basement levels below the tower. You are letting the dust obscure your vision.

I've certainly seen plenty of still pictures of the collapse, thank you.
My program calculates the excess energy available beyond that in the KE and consumed by the resistive force(s). There is, after the first second or so, ample available to throw all the materials the distance thrown and do lotsa heating as well.