The 9/11 Forum

As for the question how much of the fail energy will happen to be included in the "inelastic accretion energy", I suggest it depends somewhat on how much energy is dissipated by destroying just the necessary amount of supports for the structure to break loose and go down, relative to the 2/3 of potential energy dissipated in the inelastic collision as such.
Then I imagine that much will depend on the geometric and mechanical properties of the structure (including non-structural elements). In particular: To the extent that the structural connections whose breaking suffices to let collapse continue are probably mostly not hit directly by the advancing front, but forces propagate (as some kind of wave?) through intermediate materials (concrete decks, the lengths of trusses, beams and girders...) and arrive at the connections split-seconds later, how do we estimate how much inelastic deformation has already occured until finally the connections break free?
Intuition, that bitch, tells me most structural connections are not the last to absorb energy from any one floor impact, and that means, intuitively, that most of the fail energy is included in the 2/3 inelastic deformation energy.
I have no idea how to model this.


One hint, that may be obvious to you but not the casual reader: a converges towards g/3, thus becomes constant; but the force involved, and thus kinetic energy (and with it all other energies involved), momentum, pressures etc. increase proportional to mass, which increases by the mass of one floor per floor.


Oh got another question, which may show I am not well-read on the issue, or maybe it is a good one: I think I read in passing on this board that posters agree that, according to observation, both collapses converged on a constant velocity, not constant acceleration. What's wrong here?


A last nitpick: Your post of Sat May 05, 2012 9:35 pm (not sure what time zone that us - UT? CET?), which you (largely?) copied from the older thread, starts off with

Uniform density ρ given total height H and mass M:
ρ = H/M

Should be M/H, not?
And perhaps needs pointing out that this is density in a 1D-world.
This actually had me fumbling with the entire math of that post for about 10 minutes.

Oystein wrote:As for the question how much of the fail energy will happen to be included in the "inelastic accretion energy", I suggest it depends somewhat on how much energy is dissipated by destroying just the necessary amount of supports for the structure to break loose and go down, relative to the 2/3 of potential energy dissipated in the inelastic collision as such.
Then I imagine that much will depend on the geometric and mechanical properties of the structure (including non-structural elements). In particular: To the extent that the structural connections whose breaking suffices to let collapse continue are probably mostly not hit directly by the advancing front, but forces propagate (as some kind of wave?) through intermediate materials (concrete decks, the lengths of trusses, beams and girders...) and arrive at the connections split-seconds later, how do we estimate how much inelastic deformation has already occured until finally the connections break free?

More excellent commentary. Must return to this.

Intuition, that bitch, tells me most structural connections are not the last to absorb energy from any one floor impact, and that means, intuitively, that most of the fail energy is included in the 2/3 inelastic deformation energy.
I have no idea how to model this.

I think the thing which throws most people who've done basic physics is the notion that the coupling to ground means momentum is not conserved, so there's a tendency to throw away a powerful tool for arriving at an equation of motion - assuming momentum is conserved. While a reaction force is mediated from ground to the crush front, assuming strict momentum conservation will give an error, but that's working from the conservation principle only. If including the external force of ground coupling acting on the moving mass, I believe it can be linearly superposed on the inertial forces of collision. Obviously it can be so when the two are constrained to act in the manner of the slab model, where no displacement is allowed to occur during collision.

One hint, that may be obvious to you but not the casual reader: a converges towards g/3, thus becomes constant; but the force involved, and thus kinetic energy (and with it all other energies involved), momentum, pressures etc. increase proportional to mass, which increases by the mass of one floor per floor.

Yes, the two end up increasing in like manner.

Oh got another question, which may show I am not well-read on the issue, or maybe it is a good one: I think I read in passing on this board that posters agree that, according to observation, both collapses converged on a constant velocity, not constant acceleration. What's wrong here?

First, constant velocity is a special case of constant acceleration where the constant is simply zero instead of non-zero. But that's not the answer here, because the constant is in disagreement. The real answer points to the shortcoming of this model - there are no velocity-dependent forces included in the formulation which are independent of mass accumulation. We suspect the collapse became dominated by such forces when velocity became appreciable. The typical formulation will include one or more additive terms in powers of v, so these effects are superposed on the forces of the more primitive model. The actual roofline descent of WTC1 seems to match pretty well with the primitive model in the first 3 seconds, then velocity oriented sinks begin to take over.

A last nitpick: Your post of Sat May 05, 2012 9:35 pm (not sure what time zone that us - UT? CET?), which you (largely?) copied from the older thread, starts off with

Uniform density ρ given total height H and mass M:
ρ = H/M

Should be M/H, not?

Yes, *$&@!#. Thank you. You're the second person to catch it... but it shows you're reading! This time, I'm going to fix it in the pasted AND original version so I don't have to endure the embarrassment again.

And perhaps needs pointing out that this is density in a 1D-world.
This actually had me fumbling with the entire math of that post for about 10 minutes.

Yes, 1D only!

If you haven't seen this, it's recommended.

http://the911forum.freeforums.org/wtc-1-offered-no-resistance-t168.html

The greatest secret of the mechanics of the tower collapses. CDers would have a cow trying to argue against a formal paper showing NO structural resistance, yet no explosives. Hahahaha!!!

Now is probably as good a time as any to reveal that I (personally) have been the impediment to publication of Benson's paper, going on two (3?) years now. Posting this jibber-jabber instead of doing real work. He's a very patient man. Think it's time to deliver?

On the matter of mix of losses to inelastic impulse versus momentum transfer to ground, I'll work on a show and tell based on the mass-spring-damper graphic in this WP article on damping, SVG here.

The modification is to replace the Force arrow directed to the left by another mass block which will collide with the block attached to the spring (now F = -d(mv)/dt, change of momentum of block in collision). First considering the collision to be instantaneous, both the inelastic and elastic collision cases correspond to the basic collision interaction proposed for the slab model, with the difference that the spring is not fixed to an immutable constraint (except for the ground floor). That is, the problem can be separated into collision followed oscillation of the joined bodies or recoiling body, respectively.

(The parallel spring-damper as depicted in WP is NOT really a good approximation of the crushing, any type of crushing; the spring must break, and the damper must actually be nonlinear, but close enough)


Making the collision instantaneous simply assures there can be no coupling with the spring or damper as there's no displacement. The change in momentum of the two bodies is calculated first and used as initial conditions for the rest of the problem, t > t₀. If the displacement of the spring is very small during collision, this makes a suitable approximation, particularly if the impacted mass is at or near equilibrium position wrt the spring. Allowing the collision to take finite time over a finite displacement and be explicit about where the kinetic energy goes over time is simply a matter of acknowledging the brief coupling of the internal degrees of freedom of the two masses with the spring/damper components.

Thus the next stage of approximation is to replace the masses with mass-spring-damper subsytems, with different characteristic frequency and damping.

In the instantaneous case, all lost KE goes immediately to heat. In the coupled finite duration case, internal (vibratory) modes transfer some energy of motion to the spring/damper either conservatively or nonconservatively over time. Conservative potential will be an excited state of the system, but has nowhere to go except into the losses from damping either within the mass blocks or the damping from the "big" dashpot, acting along with the spring.

If the mass block subsystems have a very different characteristic frequency, the coupling between the two oscillatory degrees of freedom will be minimal, and the assumption of instantaneous collision should work very well. Moving to a continuous media is akin to reducing the mesh size for the mass-spring-damper subsytems to the limit of small size, though perhaps not exactly the same since a true continuous media may be the non-impulsive g/2 mechanics of Seffen.

One can easily cut to the chase for continuum cases in Bazant and Seffen.

OneWhiteEye wrote:

Intuition, that bitch, tells me most structural connections are not the last to absorb energy from any one floor impact, and that means, intuitively, that most of the fail energy is included in the 2/3 inelastic deformation energy.
I have no idea how to model this.

I think the thing which throws most people who've done basic physics is the notion that the coupling to ground means momentum is not conserved, so there's a tendency to throw away a powerful tool for arriving at an equation of motion - assuming momentum is conserved. While a reaction force is mediated from ground to the crush front, assuming strict momentum conservation will give an error, but that's working from the conservation principle only. If including the external force of ground coupling acting on the moving mass, I believe it can be linearly superposed on the inertial forces of collision. Obviously it can be so when the two are constrained to act in the manner of the slab model, where no displacement is allowed to occur during collision.

Quick idea here, not sure if this is correct or goes anywhere:

The first collisions will bring the columns closer to maximum elastic capacity; in a limiting case to exactly maximum elastic capacity (thereby storing some of that initial kinetic energy).
Ignoring that this capacity is probably not really constant as collaps progresses and deconstructs the the assembly, it follows, as a first approximation, that coupling to the ground provides a constant average force against the collapsing front, resulting in an additional negative acceleration that decreases over time as the moving mass increases.

OneWhiteEye wrote:

Oh got another question, which may show I am not well-read on the issue, or maybe it is a good one: I think I read in passing on this board that posters agree that, according to observation, both collapses converged on a constant velocity, not constant acceleration. What's wrong here?

First, constant velocity is a special case of constant acceleration where the constant is simply zero instead of non-zero. But that's not the answer here, because the constant is in disagreement.

Right.

OneWhiteEye wrote:The real answer points to the shortcoming of this model - there are no velocity-dependent forces included in the formulation which are independent of mass accumulation. We suspect the collapse became dominated by such forces when velocity became appreciable. The typical formulation will include one or more additive terms in powers of v, so these effects are superposed on the forces of the more primitive model. The actual roofline descent of WTC1 seems to match pretty well with the primitive model in the first 3 seconds, then velocity oriented sinks begin to take over.

Just what ARE these velocity-dependent forces? You can't be talking about classical friction, or can you?

I think the velocity-dependent forces are already inherent in our model: As velocity increases, impacts with new building components at rest become "harder", AND these impacts become more numerous per time unit. Both these scale with v, and the combined effect with v² - but it is offset by the concurrent accretion of mass, which also scales with v, so in the end we are left with a term for average force that has v only to the first power.

OneWhiteEye wrote:Yes, *$&@!#. Thank you. You're the second person to catch it... but it shows you're reading! This time, I'm going to fix it in the pasted AND original version so I don't have to endure the embarrassment again.

You're welcome

You pretty much lost me with the post on springs and dampers - but that may in part be due to the time of night: past 2:30 am here

I started writing stream of consciousness prattle in reply to your prior post... then realized what crap it was.

What are the velocity dependent forces? Empirically determined from the data, more than anything else. Reasons? Ah, give me some time on that. It is postulated that the mechanics are that of a vertical avalanche. As over there, so over here, but the principles...

Increase in energy partitioned to fracture, rotational energy of those fragments, lossy expulsion of pressurized gas through orifices. Literally the sorts of things which are commonly associated with (unquantified) increase of entropy. The ensemble grows, and the states accessible to the system grows, and the energy tends towards equipartitioning. Seems to depend only on the square of velocity.

I rely on Dr. Benson's results, but that's provisional, not commonly accepted fact. Then layer some of my own interpretation on it. Consider it opinion only.

...may in part be due to the time of night: past 2:30 am here

Excellent. It might give me a chance to get my ducks in line, and catch any careless mistakes. I'll have better answers for you if you don't come up with them more quickly on your own...

I will caution you in advance, it all does fall apart at some point (pun intended). This I know. I don't let it trouble my mind too much. There's a difference between defending or comparing models, defending belief, or even having a belief. The models are cool, but not a one passes for a description so all do break down somewhere.

Example: where does the terminal velocity come from? Something other than the top block (mostly). You're going to ask this at some point. And I'm going to say, whatever crap went plowing through the interior, and it's not known with any sort of confidence. You take what you can get. How can you measure how long a non-existent object takes to get to ground? The block does start to show the rapid decrease in acceleration, but that's all. I'll refrain from editorial remarks. These are merely constructs, not truths.

Spherical chickens.

Correction: I stated capacity gradient doesn't matter, that's incorrect. Constant capacity doesn't matter, as you correctly divined.

Oystein wrote:Quick idea here, not sure if this is correct or goes anywhere:

The first collisions will bring the columns closer to maximum elastic capacity; in a limiting case to exactly maximum elastic capacity (thereby storing some of that initial kinetic energy).
Ignoring that this capacity is probably not really constant as collaps progresses and deconstructs the the assembly, it follows, as a first approximation, that coupling to the ground provides a constant average force against the collapsing front, resulting in an additional negative acceleration that decreases over time as the moving mass increases.

That's it. And that's why I had to correct my prior statement. This stuff confuses me left and right even after all this time, so I need to be more careful.

You have the correct idea, and I believe you've also stated a good reason why it makes sense to consider inelastic losses and capacity losses as separate things. Perhaps aspects which can act in conjunction or simultaneously, but distinct in cause and nature. More thoughts on the separation later.

Just what ARE these velocity-dependent forces? You can't be talking about classical friction, or can you?

Mostly not, but I wouldn't exclude it as a contributor a priori. I would, however, exclude it a posteriori since Dr. Benson's fitness tests indicated a considerably poorer result when including a straight velocity term, when weighed against the extra model complexity from the presence of the variable.

I think the velocity-dependent forces are already inherent in our model: ...

It does appear explicitly.

As velocity increases, impacts with new building components at rest become "harder"...

Yes, the energy dissipated to inelastic accretion per increment of mass increases.

... AND these impacts become more numerous per time unit.

Yes, very important.

Both these scale with v, and the combined effect with v² - but it is offset by the concurrent accretion of mass, which also scales with v...

I think this is the important distinction. The rate of accumulation scales with v, but the accumulated mass scales... differently with v. It's because they are related in the equation of motion. This is why I was careful to call for a velocity dependency which is independent of mass accrual.

m = ∫ρdy = ρ∫dy evaluated on [0,y] = ρy but y is a parametric function of t. It could as easily be written m = ∫ρdy = ρ∫(dy/dt)dt = ρ∫vdt evaluated on [0,t] and then power dependency on t is explicit. These (v,y) also are not independent but are related via the equation of motion, and v must remain inside the integral as it is a function of t. Under the assumption of a limit of constant acceleration, which is the region of interest, v = at and y = 0.5at², so the mass increases faster than the velocity can. With no more than the eq of motion, it has been shown the convergence is true, so the mechanical constraints ensure total mass accumulation limits velocity accrual. Well, the limit is mutual, but this expresses the relative magnitudes of growth.

Seems very bootstrappy, I know, but it's the only way I can get my mind around it.

..so in the end we are left with a term for average force that has v only to the first power.

Yeah, sort of. We have a mass which grows as the square (roughly, at limit) of time and a velocity that grows with time. If a force term is added which is independent of y, having only explicit dependence on v (implicit on t), then there is the possibility of the force increasing with velocity in such a way as to further limit acceleration, even to zero - terminal velocity.

Oh my. I just caught the Acceleration of the Falling Top Blocks at JREF. The you-Tony-DaveNMSR-ozeco conflagration.

Tony first looks at structural resistance, then considers momentum. Dave considers momentum only so far as I know. We are looking at both here, also, but so far homogenizing the result over stories at a time. Free drop/quasi-static initial descent, tilt/axial, reality/Bazant.

Apples, oranges, peaches, walnuts... all good food products from trees. They do not taste the same, though, no reason to expect them to. It's unreasonable to expect that a momentum-only analysis will match anything more than the gross overall features, which it does. No way am I going to get into this fray, that's old ground, but it's pertinent to this discussion since we're looking at how the difference matters and I think this is more fruitful than the same old same old which is like several people conversing with their backs to each other.


Here are some differences:

- momentum-only cannot lead to arrest
- accretion (both conservative and non) have dependency on velocity
- structural sinks, to first order and at these low velocities, are independent of velocity
- real structural sinks are dependent on position

Let's not forget this is all 1D! Were the towers 1D? No. Proceed with necessary caution.

Homogenization is successful but the degree dictates the applicability of the result. A momentum analysis like Dave's will not decide arrest, nor will it account for initial motion through the failed story, nor will it give an accurate trajectory of the roofline over time. But it does get into the ballpark of collapse times, and it might give a useful rough impression of the displacement over time taken as a whole.

The stepwise algebraic model, with or without support sinks, is only an approximation to resolution of one story's granularity, so is inadequate for jolts as well - aside from the obvious 1D ramifications of axial alignment. You'd really have to get the tweezers out to predict a jolt for the 1D axial case, following the aggregate load displacement response. Then you'd have some ideal 1D result.

Where I find value is in understanding the basic principles of the presumed inelastic accretion as a foundation for subsequent refinement. There is a velocity sensitivity, it is invariant with respect to total mass, it does preserve these characteristics going to the continuum limit. That's pretty powerful information. Not nearly all that's necessary to move to a realistic collapse model, but good foundation.

Even in a storywise computation, it pays to consider sub-story dynamics to get a better result when adding fail energy. That is, subtract the sinks in the order in which they're encountered, since the collision losses are not commutative. That may seem obvious but it's easy to overlook, even when you need to come up with a velocity value to plug in.

Likewise, in an inelastic slab collision, a freefall drop of one story results in a greater velocity delta than a restrained descent; the results of the momentum-only model cannot be superposed with another having only work done against supports, the sinks must be computed together.

Velocity dependent sinks apart from momentum cannot be accurately analyzed with this method. A mean value approximation can be done, but is almost worthless at story granularity.

But saying they need to be computed together is not the same as saying fail energy is subsumed under inelastic accretion losses. Again, it's useful to consider the approaches Bazant and Seffen took, which are not the same and which produce similar but not identical results. This is jumping to the continuum case but, except for computational limitations associated with granularity (time and/or space subdivision), discrete and continuous are congruent.

Bazant assumes non-conservative (inelastic) accretion, where Seffen assumes conservative accretion. Both consider losses to residual capacity to be an additional force acting on the upper body. Seffen's approach uses a modified Lagrangian (Pesce) and is said to be non-impulsive. I argue that Seffen is correct for his homogenized model but not for anything else, certainly not the towers. Do the collapses look smooth and creamy or do they seem impulsive? Non-impulsive accretion is like a planet capturing a moon, simply the work done to accelerate the mass(es).

About the only thing I'd intuitively classify as non-impulsive (in a 1D model) in the kinematic sense is the period of three-hinge buckling, but even this is highly plastic and nearly fully lossy in the real world. If three hinge buckling were approximated by mass rods with lossless hinges, then I'd say YES the acceleration is non-impulsive. BUT, upon rehardening (aka full compaction in the 1D model), there is column end banging column end, definitely impulsive. Erases the gains of the prior conservative accretion mostly if not totally. Moreover, the column lengths are a lesser proportion of mass.

What's left? Any discete surface impacting another is impulsive, even particulate collisions in the air. I say Bazant is right and Seffen is wrong. At least it's closer to Bazant than Seffen.

With regard to whether inelastic accretion is all encompassing, no it's not. Both of them agree, despite the differences in treatment, that impulsive inelastic losses are distinct from residual capacity. Seffen just thinks that that the inelastic part of accretive losses can be ignored. The elastic part remains, and capacity is added as a force term to that. The opposite of what you were first wondering (can support crushing be ignored in a fully inelastic model).

I think all that's necessary to satisfy yourself that this is true, yet your intuition remains solid, is that the momentum component dominates the losses for all but very strong structures in the skyscraper class, yet it cannot cause arrest in itself. If you believe arrest is ever possible (yes, drop one soda can on another), then you already know the energy to fail supports is a separate concern from momentum losses. The presence of the former, however, will affect the magnitude of the latter.

Your intuitions about energy dissipation at the collapse front, no coupling of magnitude past the weakest link in a chain, etc, are all good.

During a collapse in which a hypothetical upper block is accelerating downward, the lower section (on average) unloads. Think about it.

Only elastic waves get propagated and, at every material discontinuity (column ends, weld planes), a certain amount of energy is reflected back. Oscillation, vibration, yes; piledriver at the bottom, no. I suspect it was very loud inside. Louder closer to the crush front, and probably some vibration where constructive interference might snap fasteners and fracture welds, here and there, before impact. Maybe. Low energy overall, and most dissipated before ground.