The 9/11 Forum

Major_Tom & OneWhiteEye --- The crushed zone B has a crushing front but eventually considerable, always growing, vertical thickness. Roughly, this is like a turbulent stream rushing down the side of a mountain. This has banks, corresponding to the perimeter walls. The WTC 1 it is clear that nothing (or very little) was vertical impacting the perimeter walls. Back to the analogy, the stream might civide around a large rock; I suppose this slightly impedes the progress of the stream, but certainly not noticably.

Droping the analogy, since zone B has vertical thickness, crushed materials are constantly rushing by a fixed elevation. So I am quite sure that standing portions of the core are indicative of the perimeter walls peeling away. (It would be nice to know which end of the 500 column line stands the highest; NE or NW?) (Even nicer would be to a very good estimate of the height, measured as length in West Street, of the large west wall sections which peeled off to fall over into West Street, essentially still intact.

My analysis has an early phase to set the intial drop of about 0.5 m and an intial speed of about 0.4 m/s before using the crush-down equation.

The other model I was going to propose is much harder to construct as the two walls are sized cardboard but otherwise similar in being a connection-breaking device.

David B. Benson wrote:The WTC 1 it is clear that nothing (or very little) was vertical impacting the perimeter walls.

True, except perhaps for the immediate period following initiation. Perimeter engagement, to whatever extent it existed, is distinct from later failure modes.

Back to the analogy, the stream might civide around a large rock; I suppose this slightly impedes the progress of the stream, but certainly not noticably.

Two things: 1) the remaining core gives me an impression of being eroded away at the top, the amount of wear decreasing going down, and 2) if the process is analogous to (turbulent) fluid flow around an obstacle, I'm not sure I agree with the conclusion - but am certainly willing to give it provisional acceptance.

David B. Benson wrote:OneWhiteEye --- I use the B&V crush-down equation with a resistive force proportional to

Bs^2

where B is the instantaneous mass of the crushed zone B and s is the material speed (not the crushing front speed, v, which is faster).

Did you also try a form including a term linear in s? If so, was the trial:

A) slightly better fit but not enough to justify the additional parameter
B) statistically insignificant difference when compared with linear term set to zero
C) worse fit
D) other

If (A), what is the criteria by which the factor of an additional parameter leads to disqualification when weighted against fit performance?

OneWhiteEye --- I tried a force linear in v some time ago and it did not work very well. Having never seen such a force discussed anywhere, I abandoned it.

There are various Information Criteria to help choose models based on goodness of fit and the number of parameters. Larger number of parametrs is penalized; quantitative Ockham's Razor.

Thanks for the prompt and informative reply. Let me know if the questions become burdensome; it seems there is an endless stream of them...

David B. Benson wrote:OneWhiteEye --- I tried a force linear in v some time ago and it did not work very well.

Was that c_2 * s^2 in combination with c_1 * v? No try with s, the material speed?

Having never seen such a force discussed anywhere, I abandoned it.

From http://tabitha.phas.ubc.ca/wiki/index.php/Dissipative_Forces:

Viscous Forces wrote:Here the frictional force increases as the first power of the relative speed between the surfaces and opposes the relative motion. Viscous friction is important for wet surfaces at small relative velocties

Would seem to apply to a laminar flow at low speed, at least.

There are various Information Criteria to help choose models based on goodness of fit and the number of parameters.

Things that you'd linked to before, Akaike and Bayesian information criterion?

Larger number of parametrs is penalized; quantitative Ockham's Razor.

Excellent answer! However, be sure Ockham doesn't get too bloodthirsty!

Edit: It's possible c_2 * s^2 in combination with c_1 * v may be better than c_2 * s^2 with c_1 * s. I could elaborate, but I won't just yet.

OneWhiteEye wrote:Let me know if the questions become burdensome; it seems there is an endless stream of them.

I enjoy it and learn as well.

No try with s, the material speed?

That is correct.

Would seem to apply to a laminar flow at low speed, at least.

Thank you, maybe I knew that once...

Akaike and Bayesian information criterion?

Yes.

Suggest this thread is re-named One-way Crush down Models!

It is very simple to model a One-way Crush down. Take an object A and put in on the ground and then another object C. You drop C on A and A is crushed.

If C can apply suffient energy PE at impact C with A and total strain energy SE that can be absorbed by A+C is less than PE and that C can absorb more strain energy than A and only deform elastically in the process, then A is crushed and C is not.

It is not really 'one-way' as C is always affected - elastic deformation - but it is pretty near.

I would conclude that 'one-way' crush down is only possible, if C can absorb more strain energy only as elastic deformation than A can absorb totally (elastic & plastic deformation, failures, &c).

If C is then only 1/10th of A volume/mass wise and can only absorb 1/10th of A strain energy (A and C have same internal structure), then I would conclude C can never crush A in any model, size or scale.

David B. Benson wrote:

OneWhiteEye wrote:Would seem to apply to a laminar flow at low speed, at least.

Thank you, maybe I knew that once...

Dr. G is already on top of the viscous term:

Dr. G wrote:I have also added a viscous flow term that's proportional to the collapse velocity to allow for friction.

http://the911forum.freeforums.org/does-the-collapse-time-of-wtc1-indicate-assisted-collapse-t11-30.html#p809

Imagine a structure as per above sketch. The structure consists of elements m that are supported by columns with height h. Columns between m can carry all m above, i.e. the columns get stronger the lower they are.

A plane hits the structure in the side between 2 m. There are k off m elements above this damaged area. Let's call that top section C. After a while the columns between the 2 m fail and the structural assembly above (k off m elements), i.e. C, drops down height h and the bottom m of section C contacts another m (as shown in sketch). Let's call the assembly of many m below contact area section A. The impact is perfect = equal pressure between the two m everywhere (if unequal C may drop off on the side). Section A consists of at least 2k m!

What happens then?

Well, at contact kinetic energy applied is converted into forces that are applied to the columns ABOVE and below and they deform. The velocity of upper section C is reduced as kinetic energy is transformed into strain energy in sections C and A. There should be a visible jolt of section C!

One result, apart from the jolt, is that, e.g. the upper assembly/section C bounces, while section A is only compressed. This happens when the energy applied can only elastically deform columns.

Another result may be that a set of columns fail! Which one?

Answer: the weakest ones adjacent to the impact area, i.e. the columns ABOVE in section C.

The result is that another m in the C section drops down and maybe columns fail again. Which one?

Right! The columns ABOVE - still in section C.

This can take place k times and then we have m elements neatly stacked on top of section A like pancakes. Section C is compressed. No more m can drop!

There is no problem. Section A carried k off m elements before first impact and carries them again after k impacts. Only difference is that all columns in section C have failed.

NIST suggests that section A can only carry n off elements (n<k) after impact, but it is wrong. Section A can safely carry k off elements m! It did it for 30 years at WTC 1. Section A is simply stronger than section C.

One thing is absolutely certain. Section C can never destroy section A in a one-way crush down!

NIST suggests that section C one-way crushes down section A but ... it is wrong.

I have explained this also at http://heiwaco.tripod.com/nist3.htm .

Heiwa,

Another result may be that a set of columns fail! Which one?

Answer: the weakest ones adjacent to the impact area, i.e. the columns ABOVE in section C.

Not correct. The stress in the columns above the impact is lower due to less mass and less momentum in direct proportion to the designed strength. Also, the columns below are significantly damaged (in a more realistic scenario anyway) and consequently 7% weaker than designed.

Expecting a jolt is based on a misunderstanding of the Bazant and Zhou model. The B and Z model is a simplified scenario that is not meant to model the observed behavior but rather the most optimistic scenario. All other scenarios are more severe. In a realistic scenario, buckling is occuring over several floors and there is never any impact because the top is in continuous contact with the bottom.

Hambone wrote:Heiwa,

Another result may be that a set of columns fail! Which one?

Answer: the weakest ones adjacent to the impact area, i.e. the columns ABOVE in section C.

Not correct. The stress in the columns above the impact is lower due to less mass and less momentum in direct proportion to the designed strength. Also, the columns below are significantly damaged (in a more realistic scenario anyway) and consequently 7% weaker than designed.

Expecting a jolt is based on a misunderstanding of the Bazant and Zhou model. The B and Z model is a simplified scenario that is not meant to model the observed behavior but rather the most optimistic scenario. All other scenarios are more severe. In a realistic scenario, buckling is occuring over several floors and there is never any impact because the top is in continuous contact with the bottom.

Are you really sure? The stress in complete part C is zero before impact (due to free fall) and the stress in part A top columns below top m of section A prior impact is only due to one m, the other k m of section C are not applying any load on A. No static loads exists in interface A/C prior impact.

At impact the kinetic energy is transformed into dynamic forces applied to C columns above and A columns below - they become suddenly loaded dynamically. On top of that the static loads are re-established: Top A columns carry (k+1) m as before, and bottom C columns carry (k-1) m as before.

Evidently the C columns are weaker than the A columns and as the dynamic force applied to A and C columns is same, the C columns fail before the A columns.

Re the Bazant/Zhou model: it has changed several times 2001-2008 but still they maintain that it is the top storey of section A that is compressed to lambda h at impact, while according my understanding it should be the bottom storey of section C that should suffer this fate - as outlined above. Regardless - the energy required to compress any storey, be it in sections C or A - is associated with a jolt or deceleration of part C = the energy required to compress one storey.

In real life you can never expect section C to impact section A perfectly and the expected result is then, evidently, that section C is displaced sideways ... and drops outside section A after some local failures occuring in section C lower part (and probably also in section A as forces are unsymmetrical, &c).

Whatever happens at impact, the Björkman Axiom appears to be valid: a structure A cannot be crushed by a part C of same structure (C=1/10A) due to gravity.
Anyway, this thread is just about Crush-down models (C crushing A, C=1/10A). As according my Axiom such one-way crush down is impossible in any size or scale, I just follow it out of curiosity. Evidently no model can be produced.

Nice illustration, Heiwa. I always appreciate clean graphics; like I said earlier, god don't love ugly, or something like that.

But one issue with sketches is that they can illustrate anything. Graphics which are driven by a logical or physical simulation are constrained to portray the generated results. I take it the core logic driving your diagram is this statement:

Evidently the C columns are weaker than the A columns and as the dynamic force applied to A and C columns is same, the C columns fail before the A columns.

Evidently the C columns will fail before the A columns - if the C columns are sufficiently weak as compared to A columns. And if there is only sufficient energy to destroy one set of columns, above or below. The sum of the static and dynamic forces you depict places (slightly) greater load on the columns below though, from any modest height, I think the dynamic load overwhelms the static load.

I ask you to consider what happens if sufficient KE exists to fail the columns both above and below, and/or the case if C columns are NOT weaker by the amount required to satisfy your axiom - and, by axiom, I mean your declaration that C columns are weaker than the A columns by the amount necessary to cause their failure first. This is an assertion, or given, not a principle derived from dynamics or even logic. You have defined a crush-up only system.

Beyond that, your model does not specifically address the situation where there is excess KE above what is required to break one set of columns. Excess KE would remain as velocity of the upper block. Upon the next collision, depending on values of m and h, the situation must be evaluated anew, and not rely on the repeated application of the previous result. At any point, if the energy is available to break two sets of columns then, by your axiom that load is distributed equally above and below, stories of both A and C are destroyed. The next step now has an upper block of mass reduced by the number of collisions dropping through 2h. And so on.

When I asked you what was wrong with my simulation, you indicated that forces were not properly distributed amongst the bodies. I disagree. The simulation is using a fairly accurate F=ma solver with a finite interpenetration skin width on rigid bodies and extremely stiff threshold-breakable springs for joints. It can accurately simulate


- a tipping stack or rigid rod
- multibody collisions using arbitrary shapes
- a newton's cradle
- a compound pendulum
- a fountain of particles
- n-body mutual gravitational interaction
- several thousand balls falling like sand grains in an hourglass
- a jenga tower collapse

in real time, or nearly so, provided it is set up with the correct parameters. Could you make a sketch of a compound pendulum of two rods connected with a hinge where length of the rod attached to bob is 1/10th the length of the other rod? The rods are initially collinear at 179 degrees deflection from equilibrium at bottom dead center (an inverted pendulum). I'd settle for a single snapshot at any time after t0 past the first period of a simple pendulum of the same dimensions.

----
Edit: changed 89 degrees to 179 to get an inverted pendulum, not that it matters to the argument. Could have just as easily said 1 degree; then small angle approximation is valid, not that it will help.

OneWhiteEye wrote:Nice illustration, Heiwa. I always appreciate clean graphics; like I said earlier, god don't love ugly, or something like that.


I ask you to consider what happens if sufficient KE exists to fail the columns both above and below, and/or the case if C columns are NOT weaker by the amount required to satisfy your axiom - and, by axiom, I mean your declaration that C columns are weaker than the A columns by the amount necessary to cause their failure first. This is an assertion, or given, not a principle derived from dynamics or even logic. You have defined a crush-up only system.

Here is another illustration.



assuming initial KE was just sufficent to damage the columns above or if contact/failures are unsymmetrical. Note that section A always carry the same amount of m before and after first and consequent impacts inside section C.

In case applied KE is sufficient to break columns both above and below impact interface, evidently these failures take place, but then we are no longer talking about a crush-down of A by C model. Both parts C and A will be damaged.

But again, as A columns get stronger the lower you crush (they carry more m statically) and are many more than available columns in part C, part C will be completely destroyed before part A and, in my opinion, destruction is arrested. It is realized that the dynamic impact forces will increase when impacts occur further down in part A and they will affect C as much as A.

Note that in this model no rubble is formed - to protect part C from the impact forces? - just horizontal elements m can displace due to failure of vertical support (without mass).

The worst thing than can happen with my model is that all elements m in C and A are piled up in a nice pile with all columns broken - no rubble. No crush-up of C is then possible. C was crushed much earlier in the process.

Part C can never crush down part A as suggested by BLGB, i.e. C remains intact during a one-way crush down of A in order later to be destroyed in a crush-up by a rubble part (B?) that previously protected it, i.e. the Bazant theory is not valid.

Maybe I should make an illustration of that?

Regardless - this model has nothing to with WTC 1 reality. The stronger elements (columns) in part A will destroy the weaker elements (floors) in part C already at initiation.

Heiwa wrote:In case applied KE is sufficient to break columns both above and below impact interface, evidently these failures take place, but then we are no longer talking about a crush-down of A by C model. Both parts C and A will be damaged.

Just to be clear, I've spent a good deal of time in the last couple of years arguing against exclusive crush-down, on message boards. When it comes to the towers, that is. So, in one sense, you're preaching to the converted. But when it comes to idealized models, the situation is different, it can go a lot of different directions depending on the setup. Most of my arguments were of the hand-waving variety prior to doing any simulations but are now finally starting to take shape in a more quantitative fashion. Regardless, I don't think anyone is claiming absolutely exclusive crush-down occurred (though I do have to scratch my head at B&L, which seems to claim just that).

If you require exclusive crush-down, that is not so easy to do. Nevertheless, I have done it in simulation. Ordinarily, I try not to repeat myself too much, but here goes:



There is even mass shedding, lots of it!

There are many reasons this is not an accurate simulation of any known structure, but it is a valid simulation. Exclusive crush down. But, as I've mentioned, there have been different trials where it was all crush-up. And bounces. And any mix of the above. The devil is in the details.

A small crush-up followed by all crush down is a common result:



Here there is a little crush-down, complete crush-up, then complete crush-down but, since there is only ONE connection in the upper block, you'll forgive me if I casually refer to this as crush-down. You'll note that it goes to completion.

Finally, to make liberal use of repetition as a debating tool, here again is a case of collision of identical blocks without a gravitational field imposed:



Equal damage, as expected. Gravity makes a big difference in introducing asymmetry. Fact.

But again, as A columns get stronger the lower you crush (they carry more m statically)...

At this point, I've not gotten to varying strength. Of course that is the case in a real building. But I noticed this was not part of your original criteria, which admitted equivalent construction above and below. Change the rules, change the results. I still think I can make a model crush down with a realistically varying DCR going up. I expect there to be some initial crush-up; I most certainly do. Go back and check B&L: for non-equal capacities, some crush-up IS expected.

Perhaps you've not visited all the other threads where I'm shouting from the rooftops -

"The upper block is destroyed early on!"

No, I guess not. But my reasons are not the same as your reasons. I think you need to get over a hump in which you do not recognize how and why crush-down is preferred and what the mix will be in any given situation. This is my opinion. To deny circumstances under which it can happen is to fail to grasp some of the more fundamental principles of dynamics in this 'vertical domino arrangement.'

Maybe I should make an illustration of that?

No need, see animation above.

Regardless - this model has nothing to with WTC 1 reality. The stronger elements (columns) in part A will destroy the weaker elements (floors) in part C already at initiation.

When, exactly, did varying strength enter the picture in your mind? After you proclaimed my earlier simulations invalid? Why is greater connection strength below now a neccessity to produce your expected results? Do you still claim the previous sims are invalid (yes/no will suffice)?

PS there are many drawbacks to my simulations, so far I'm the only one who has enumerated any of them.

OneWhiteEye --- B&L show little inital crush-up, not none at all. Since it is so small, the argument is that the crush-down only in B&V is a valid approximation.