RFC: Bazant and Zhou Simple Analysis refuted

[cmcaulif]

Lastly, even if we accepted all of the calculations here, it should be noted that B&Z computed a generous estimate by discounting the effects of fracture, etc. So a result showing the "corrected" energy fraction was 0.81 instead of > 1 would also not guarantee collapse arrest, though it would prove the B&Z model was oversimplified, and a more thorough analysis required to answer the question once and for all.

Thoughts?

I don't think using the overload factor would be appropriate for predicting arrest or collaspe because factors of safety would need to be taken into account. It would be much better to use an equation like Bazant and Zhou equation three, in which no safety factor is needed, and modify the equation to include the effects of other limit states such as fracture, or even possibly local stability modes of failure like web buckling, as well as elastic energy dissipation.

However, from what I have been able to gather from poking through a textbook by Bazant and another on stability of structures is that the approach taken in Bazant and Verdure may be the most complete that has been produced so far, in which the load-deflection diagrams for a story of the WTC are calculated, which allows for the snapthrough load to be determined, as well as the work done by the structure. This is similar to Seffen's approach and Maxwell construction as well.

 

[Major_Tom]

I'm not sure what you mean by this MT. Bazant's equation is valid because it satisfies conservation of energy (kinetic energy of the mass turned completely into strain energy of the spring at the max deflection).

True, but the implicit assumtion is that all of that energy is then transferred to the ends of the javelin-like columns of the lower section. Over and over.

Equation means "to equate". Equating the gained kinetic energy with the blow the lower columns receive is an assumption to the extreme.

It assumed direct column end-to-column end contact over and over again.

Assuming direct column end-to-column end contact after the first "buckling" or "hinging" is quite an assumtion. To assume that this happens over and over again seems ridiculous.


And besides, the entire spring argument loses all meaning when we consider that the large, large majority of core box columns seen in the rubble were very straight and clearly broke into smaller sections by cleanly breaking along their original column-to-column welds, manifesting no plastic, permanent destortion at all?

What meaning does the spring argument have if your "spring" breaks up cleanly into 38 foot sections before your spring equations even apply?

What meaning does the entire approach have if your "spring" has welds that give?

 

[Heiwa]

Excellent paper in many respects.

In my view the spring constant should be variable - or better - the tower below the initiation zone - should be represented by several springs with different spring constants - as a function of the total cross sectional area of the tower at the relevant level. You only need to make the tower below 10 springs.

Then we find that the 'softest' spring is located just below the initiation zone and that anything dropping on it - e.g. the tower above - will just bounce off!! All kinetic energy from above (it is easy to calculate) is evidently absorbed by the top spring as compression that then pushes some of the energy down to the spring below and the rest is used to push back the tower above.

Furthermore - the tower above is also a spring! So when it drops down and hits the top spring below, it also compresses itself and loses a lot of energy in that compression.

It will be a very soft collision when these two parts meet - and no global collapse of any kind, e.g. what you see on all videos of the collapse where 100X the kinetic energy available from the tower above makes the whole tower explode like a bomb was dropped on it.

Remember that when a spring breaks - it only breaks in one location. Not in 1000's of pieces.

 

[Dave Rogers]

Not an engineer, but a physicist, so I'm not claiming any particular expertise, but one thing strikes me as surprising here. The static loading of the top 13 storeys is 5.7MN, whereas you give the ultimate strength of the columns as 1.67GN. Is a safety factor of 290 really needed? This seems to suggest that the support columns were over-engineered by a factor of about 100, which doesn't look economically sound to me. If I got such an unlikely result I'd be tempted to re-examine my starting assumptions.

Dave

 

[GregoryUrich]

O.K, I should note that GU's value for L is incorrect, as is his value of A. Hence his value for k is wrong.

You need to consider where the spring begins, and it is not at floor 103 when the spring was broken at floor 95...............

From the basement Apollo. The initial collapse was between 97-98. So 97+6=103. I would agree the spring was damaged in the impact area but not broken.

 

[GregoryUrich]

Greg, what are you doing? Your paper was almost right before you changed it. You just added crap to give it a conclusion that you wanted.

The equation that you used is to determine the overload ratio of a mass impacting a spring. And then you go ahead and subtract out an event that happens after the peak impact load, why? The plastic energy E-plastic, dissipated by the columns happens AFTER the peak load is applied to them. You can't subtract the energy required to fail the columns before the mass even impacts the columns. This is the ******** that I'd expect Gordon Ross to perform, not you.

After your paper on mass and pe for the WTC I thought that you weren't a religious truther - dedicated to proving conclusion regardless of whether or not you have to fake the math to do it. Now I'm not so sure and it makes me sick.

Horsey dung!

The E plastic is during the initial collapse. That portion of the potential energy must be removed before the first impact. Bazant hand-waved this away as inconsequential. Oops!

I not sure what to make of the other bogus accusations.

 

[Dave Rogers]

True, but the implicit assumtion is that all of that energy is then transferred to the ends of the javelin-like columns of the lower section. Over and over.

Equation means "to equate". Equating the gained kinetic energy with the blow the lower columns receive is an assumption to the extreme.

It assumed direct column end-to-column end contact over and over again.

Assuming direct column end-to-column end contact after the first "buckling" or "hinging" is quite an assumtion. To assume that this happens over and over again seems ridiculous.

I suspect most people who've examined the situation really closely would agree with you. However, all that actually means is that Bazant's model goes to great lengths to assume that the impulse delivered to the lower structure is done so in such a way as to maximise the lower structure's ability to resist that impulse, i.e. axially on the columns. In a more realistic scenario, the columns impact on the floors, which are quite incapable of bearing even the static loadings, and will therefore collapse progressively. Without the cross-bracing of the floors, the columns are too slender to stand independently. Therefore, column-on-floor impact is energetically greatly in favour of collapse propagation compared to column-on-column impact.

And besides, the entire spring argument loses all meaning when we consider that the large, large majority of core box columns seen in the rubble were very straight and clearly broke into smaller sections by cleanly breaking along their original column-to-column welds, manifesting no plastic, permanent destortion at all?

What meaning does the spring argument have if your "spring" breaks up cleanly into 38 foot sections before your spring equations even apply?

What meaning does the entire approach have if your "spring" has welds that give?

It means that the "spring" collapses far more easily than Bazant predicts, because instead of the fracture energy for four plastic hinges per storey we only have to consider the fracture energy for a single weld every three storeys. Again, this is energetically greatly in favour of collapse propagation.

As Gregory very honestly admits, even if all his calculations are correct here (and I'm not entirely sure that's the case, as I've said), collapse progression is not disproved; it is simply shown that in Bazant's highly unrealistic simplification, which is strongly biased against collapse, then collapse has not been shown to be inevitable; and even in that scenario the force only falls short of that required by about 10%.

Dave

 

[GregoryUrich]

Oh, well done. You have a building with a 46Hz longitudinal mode. Home nobody did any joggine in there...or fast walking.

and a static deflection of almost .2 inches at the floor 97/98 interface.
That translates (simplifying as you did) to nearly 20 inches at ground level.
You think so? That, me lad, is a very high strain that the foundation has to take up...

Calculating the stiffness by reducing to area of the verticals is ignoring all the work the designers went to in order to make the actual structure stiffer than its component parts.

ETA: And your assumption 4 is incorrect.

I agree that the actual structure is stiffer in terms of resisting buckling and overturning moments but is the elastic energy really affected by the horizontal members and Vierendeel trusses?

I don't see how longitudinal mode has anything to do with axial compression. My simplification only needs to be correct in terms of absorbing energy.

Newtons Bit did a more correct calculation of the spring constant and got 10 GN m which is very close to mine. I don't think it will affect the result significantly but of course I will check. Could you post that Newton?

 

[GregoryUrich]

At no point did Bazant and Zhou claim that the overload ratio would predict collapse or arrest. It is merely a basic estimate of the overload ratio based on a spring with mass collision equation derived by the authors and also found in mechanics of solids texts.

I really cant think of any way to use this equation to predict collapse or arrest either, since in order to use it, the assumption of evenly distributed load has been made, and the safety factors for the perimeter and core columns were different if I recall correctly.

Equation 3 is what you want for their prediction on collapse/arrest.

If you follow their argument, you can't get to equation 3 without failure at the first impact. The additional mgh never comes into play!

 

[The Doc]

I read some of this thread and now I feel stupid

Kudos to all of you. You're doing work that persons such as myself would never be able to do.

 

[GregoryUrich]

I am not really sure what point 1 (Paragraph 4) regarding internal strain energy within the upper striuucture itself has to do with external effects.

As I said--I couldn't get past the assumptions. Assuming the entire height of the building for AE/L totally ignores all the work the designers did--and if we use that value, then the buckling calcs will likely show that the building collapses at 1 g, static. (I claim no buckling expertise, here)

Additionally, E is the slope of the stress-strain curve from 0 strain to yield--not to ultimate. That value is 29.9e6 lbf/in^2.
For steel, the stress-strain curve sharply drops at yield, and then is very nearly flat (orders of magnitude lower than 29.9e6) in comparison to the 0-yield value)
If you wish to average, be consistent. The E for steel will make rubber look stout at that slope.
It is always best to keep things linear. Once the material reaches yield, everything becomes non-linear--material, geometry--all of it. Load paths are also no longer valid--they have to be revised.

In short, using a linear value for ultimate leads you to false conclusions.

Regarding point 1, the upper part is a spring also and absorbs energy during the collision. B & Z don't take this into account as I am trying to do. I am aware I am oversimplifying in that the upper spring will absorb less energy. Then again I am aware that all of the energy will not be available at the impact boundary.

This brings up another issue. Does anyone know how to establish the portion of the kinetic energy available at the impact boundary?

I understand your point regarding yield stress vs ultimate stress.

 

[GregoryUrich]

Not an engineer, but a physicist, so I'm not claiming any particular expertise, but one thing strikes me as surprising here. The static loading of the top 13 storeys is 5.7MN, whereas you give the ultimate strength of the columns as 1.67GN. Is a safety factor of 290 really needed? This seems to suggest that the support columns were over-engineered by a factor of about 100, which doesn't look economically sound to me. If I got such an unlikely result I'd be tempted to re-examine my starting assumptions.

Dave

F = mg; 32.8 x 10^6 x 9.81 = 0.321 GN

I think the safety factor is based on design load and ultimate strength. NIST refers to the "ultimate strength method". This is not necessarily based on the ultimate strength of the material. Can someone enlighten us here?

As Rwguinn points out, I should be using the yield strength. In that case the safety factor falls in the range 2-3.

 

[Heiwa]

Not an engineer, but a physicist, so I'm not claiming any particular expertise, but one thing strikes me as surprising here. The static loading of the top 13 storeys is 5.7MN, whereas you give the ultimate strength of the columns as 1.67GN. Is a safety factor of 290 really needed? This seems to suggest that the support columns were over-engineered by a factor of about 100, which doesn't look economically sound to me. If I got such an unlikely result I'd be tempted to re-examine my starting assumptions.

Dave

No - if the top part weighs 32 800 tonnes = 0.328 GN (with g = 10) and the cross sectional area of all columns at the initiation zone is 5.6 m² you get the static stress to be 58.6 MPa that is about 24% of the yield stress (248 MPa) or a FoS (up to yield) of 4. Standard design.
The ultimate strength of the 5.6 m² of columns is probably decided by buckling at yield stress, thus 1.39 GN (or 1.67 GN if yield is at 298 MPa).
If yield is at 298 MPa the static stresses are only 20% of yield and FoS is 5.
Quite strong structure, actually. No way that such strong (low stressed) structure will suddenly collapse and shatter in 1000 000's of parts when some sub-parts fail for any reason. It is as simple as that.

 

[Dave Rogers]

F = mg; 32.8 x 10^6 x 9.81 = 0.321 GN

Must have picked up the wrong mass from somewhere, sorry. Stundie nomination, anyone?

As Rwguinn points out, I should be using the yield strength. In that case the safety factor falls in the range 2-3.

In which case, doesn't that mean that you get a ratio of 1.35-2, hence the collapse propagates?

Dave

 

[GregoryUrich]

For those of you beating up on Gregory, please be civil... I find problems with his paper too, but so what? It's properly posed, so we have to think about the replies, and we'll learn something. This alone sets it apart from the vast majority of Truther idiocy, about which one can only say, "huh??"

Anyway, as others have noted, the spring constant computed here is not supportable. I don't see any way for the whole 110 stories (or even the whole 90-odd stories of the "lower block") to participate elastically, let alone in compression.

So what is the correct value of the spring constant? I'd be tempted to hand-wave and say that we should only treat a column down to the next splice, i.e. three stories worth, but this is probably wrong too.

B&Z's estimate of C is also hand-wavey, of course. However, they choose what they call an optimistic value, i.e. all of the columns participate equally in opposing the collapse. I'm extremely reluctant to accept an order-of-magnitude adjustment to this value without more careful thought.

Also, regarding Ultimate Strength, what's computed in Gregory's writeup is the ultimate compressive strength, not the ultimate buckling strength, so it's overestimated. (Yes, B&Z's estimate is coarse and not quite correct either. Remember their paper came out in three days and with few actual WTC design details.)

Lastly, even if we accepted all of the calculations here, it should be noted that B&Z computed a generous estimate by discounting the effects of fracture, etc. So a result showing the "corrected" energy fraction was 0.81 instead of > 1 would also not guarantee collapse arrest, though it would prove the B&Z model was oversimplified, and a more thorough analysis required to answer the question once and for all.

Thoughts?

I would guess B & Z used around 10 floors in calculating "C" as I checked it against the actual cross-sectional area.

Your last point is essentially what I say in the conclusion.

 

[GregoryUrich]

Must have picked up the wrong mass from somewhere, sorry. Stundie nomination, anyone?



In which case, doesn't that mean that you get a ratio of 1.35-2, hence the collapse propagates?

Dave

There are still other issues, but ignoring them, yes.

ETA: Ratio = 1.29 actually

 

[einsteen]

This strain stuff etc is indeed specialists work, I even didn't know (until Apollo20 told) that the crushing force of a plane's fuselage goes with the square root of the radius. But from pure energy equations etc also something can be said about the "overdesign". It is interesting for example that if you take David B. Benson's estimation of the average resistive force in the beginning (or the energy value E1 divided by h in the discrete model) that this is roughly 1/3 of the static force, i.e. the force to keep the top section in the air. In reality the force is not constant but will grow linearly in the beginning and finally give the complex stress-strain curve. The integration of that force over the distance should equal that same value E1. If you assume (for simplicity) that is a simple linear function that drops to zero after a maximum value (a saw-tooth function) on a small interval [0,a], 0<a<h then you could theoretically determine the maximum value.

 

[metamars]

Horsey dung!

The E plastic is during the initial collapse.

That portion of the potential energy must be removed before the first impact. Bazant hand-waved this away as inconsequential. Oops!

I don't know what you mean. In BZ, plastic hinge rotations occur as a result of impact. During the assumed free fall through h, impact has not occurred, yet.

A fully dynamic analysis will utilize calculus to make dynamical evolution mathematically precise. If you study, e.g., the Ari-Gur/Singer paper, the question of subtracting a net plastic energy before or after a net elastic energy doesn't arise. Not in a simple way such as being discussed, anyway.

From their experimental results, seen in Fig. 2a, http://metamars.i8.com/ , at the particular spot in the rod that measurements were made, we can see that an elastic deformation and plastic deformation only overlap for about 1.5 msec. The elastic pulse preceeds the plastic pulse by about .5 msec. So, at least for the Ari-Gur/Singer scenario, it looks like you can say that, in some sense, you can subtract a portion of elastic energy before plastic energy.

After the elastic compressive pulse has passed, the plastic flow continues, along with it's associated energy sink.

I interpret this all as meaning that during collision, locally speaking, you can have plastic deformation simultaneously with an elastic overload condition, or without it, but immediately following.

Even if you can integrate respective energy sinks over all time and over all length of the struck rod/column, I don't think that will tell us what we want to know.


The real questions for axial strike WTC models, such as BZ, are
1) should plastic deformation have arisen, at all?
2) if so, was the plastic deformation sufficient to fail the columns?


Your calculations, both the new and previous one, are intriguing because the overload ratios you calculate are close to 1. However, in either case, we have to take this as suggestive, and look to a deeper theory.

It'd be nice to have Ari-Gur and Singer participate in these discussions....

 

[einsteen]

Although I didn't read his paper I already think I know what he means

A free fall of one story leads to

E_kin=(1/2)Mv^2=Mgh [v=sqrt(2*g*h)]

But I think he means that E1 should be subtracted in the beginning, or maybe not E1 but E1' because that story was much weaker than the 'intact' stories below, that leads to a velocity not sqrt(2*g*h), but

sqrt(2*[g*h-E1'/M])

Or simply not a drop in vacuum but a drop in a demolished story. That is correct of course. But it doesn't matter, 8.5 m/s or 7.3 m/s

 

[GregoryUrich]

I don't know what you mean. In BZ, plastic hinge rotations occur as a result of impact. During the assumed free fall through h, impact has not occurred, yet.

A fully dynamic analysis will utilize calculus to make dynamical evolution mathematically precise. If you study, e.g., the Ari-Gur/Singer paper, the question of subtracting a net plastic energy before or after a net elastic energy doesn't arise. Not in a simple way such as being discussed, anyway.

From their experimental results, seen in Fig. 2a, http://metamars.i8.com/ , at the particular spot in the rod that measurements were made, we can see that an elastic deformation and plastic deformation only overlap for about 1.5 msec. The elastic pulse preceeds the plastic pulse by about .5 msec. So, at least for the Ari-Gur/Singer scenario, it looks like you can say that, in some sense, you can subtract a portion of elastic energy before plastic energy.

After the elastic compressive pulse has passed, the plastic flow continues, along with it's associated energy sink.

I interpret this all as meaning that during collision, locally speaking, you can have plastic deformation simultaneously with an elastic overload condition, or without it, but immediately following.

Even if you can integrate respective energy sinks over all time and over all length of the struck rod/column, I don't think that will tell us what we want to know.


The real questions for axial strike WTC models, such as BZ, are
1) should plastic deformation have arisen, at all?
2) if so, was the plastic deformation sufficient to fail the columns?


Your calculations, both the new and previous one, are intriguing because the overload ratios you calculate are close to 1. However, in either case, we have to take this as suggestive, and look to a deeper theory.

It'd be nice to have Ari-Gur and Singer participate in these discussions....

The freefall assumption is false. I.e. the plastic hinges will arise in the first columns to fail. That is what I am trying to take account of. B and Z state that this energy is insignificant due to damage and heating, and therefore a freefall is assumed. I show that there is very little damage at the initial collapse floors (97-98) and that heating cannot be considered a factor based on the evidence.

I agree that a full dynamic analysis is required. This is what NIST has done but we have no way of examining their method.