The 9/11 Forum

Major_Tom wrote:How did that other guy get g/2?

In the big picture, he got g/2 by starting with a different equation of motion. I'm making a big deal of the asymptotic acceleration but that really falls out of the difference in the way the governing equations are formulated. I assume the g/2 figure is correct as determined by the analysis, the issue is how the analysis differs. Seffen goes into great detail on this, so it's not a big secret; he even has comparison graphs with the method indicating g/3 (though doesn't specifically mention that figure).

I've seized on these acceleration limits because they are easy signposts. I could even accept a host of different variations in initial behavior, but I'd expect simple momentum-only acceleration in the limit to agree no matter the method.

We have:

- Bazant says g/3
- the simple derivation of the OP (Cherapanov/shagster/me/etc?) says g/3
- a physics engine says g/3
- Greeningesque discrete algebraic model agrees with physics engine
- simple FEM program agrees (pretty well) with physics engine

The last two need to be checked specifically for g/3, but mutual agreement strongly suggests that's what will be shown. Only Seffen comes out differently, and he makes a point of the difference and why his way is correct.

If he's completely correct, then this would reveal a fundamental flaw in ANY solution framework which is conventionally Newtonian, as evidenced by results above in both the analytical and computational domains. That's hugely important, especially for simulations. 10% error?!?!

If he's correct about the formulation but wrong about the applicability, it's a big problem for his paper (only), which makes this 'improved' technique a centerpiece.

If he's wrong altogether, it's a huge blow for this relatively new spin in engineering mechanics. It would be an important refutation.

Right now, his derivation blisters my eyes. I've tried to follow it with various success many times. I understand what he's doing but there's an awful lot of manipulation and tidying between numbered equations. At the end, I just have to say "yeah" because I haven't comprehended every step. At some point, I'm going to set the problem up in Maxima and let it handle the symbolic math manipulation. I also have to go back to the cited work with Pesce. I did this once before and totally forgot, that's how unremarkable it was.

You already ran this situation with your toy variety model. In which post?

Question arises here
http://the911forum.freeforums.org/solid-mechanics-simulacra-of-the-toy-variety-t163-15.html#p3436

is examined here
http://the911forum.freeforums.org/solid-mechanics-simulacra-of-the-toy-variety-t163-15.html#p3437

is at least weakly confirmed here
http://the911forum.freeforums.org/solid-mechanics-simulacra-of-the-toy-variety-t163-15.html#p3456

More work has happened since to confirm the result for the physics engine. I need to do more in the other simulator. Things are done differently in each environment so, when there's agreement, there's confidence in correctness. But here we're seeing a swath of methods, analytical and computational, where the only real commonality is Newtonian foundation and they are all either 100% correct or substantially wrong.

In astrophysics and cosmology, the equivalent modification from Newton is Relativity! You know, the sort of thing that should be shouted from the rooftops in the engineering mechanics world. Hey, your car crash simulations are wrong! Your Kevlar vest simulations are wrong! This formulation should be the standard for this class of problems instead of classical Newtonian. Clearly it's not the case. Do engineers patch the results with empirical fudge factors, not knowing about this amazing new mechanics?

Or is the mechanics wrong?

For the equation of motion

a = g - v^2/z

We are saying the last term converges to 2g/3 for large t.

That is neat. v^2 in the numerator and z in the denominator increase at the exact same rate for very large t. The motion can be understood as a balancing act between v^2 and z.

.........................

Did anyone solve the diff eq for z(t)?

It isn't hard to see the shape of a, v and x from the simple relations expressed in the diff eq.

curvature = g - slope squared over position.

How many functions could look like that?

Major_Tom wrote:From the "toy" thread. OWE:

The simulator is not junk. It's a little difficult to reconcile a figure of 0.6g with 0.3g, however. Maybe WTC1 starts at 0.6g and goes to 0g, whereas this starts at g and goes to 0.3g, and it all comes out in the wash. ???

???

I was a little green on the subject at the time. Many more experiments and plenty of time to sleep on the results helped, too.

WTC1's collapse progression is definitely not described by this equation of motion, which closely approximates the action seen in simulations. I think the two factors mattering most (besides 1D-ness!) are interpenetration without accretion and velocity dependence in the resisting force. The former means conservation of momentum can't be directly used in the same way to start the derivation, and the qualitative result is less or no resistance in the first stage of collapse. The latter gives convergence on a=0, terminal velocity.

The simulations have not, to this point, explicitly incorporated a velocity dependence. With some limitation, it can be done, and I should like to. It can also be done easily in the analytical model, which is why it's attractive. It can't be done in the discrete algebraic method which uses story size displacements to calculate an energy cycle, except in the case of floor collapses which have intermittent free drops where it can be very grossly approximated.

Marrying the results of both types of analysis and simulation could be quite powerful. First, though, the analytical form has to catch up with the other two. There are already more advanced paths established by Bazant and Seffen, who don't agree. To understand which is better, I need to start from scratch. Besides, I may not wish to be yoked to all the same simplifications and assumptions either of them used.

Major_Tom wrote:For the equation of motion

a = g - v^2/z

We are saying the last term converges to 2g/3 for large t.

That is neat. v^2 in the numerator and z in the denominator increase at the exact same rate for very large t. The motion can be understood as a balancing act between v^2 and z.

Exactly.

Did anyone solve the diff eq for z(t)?

Not analytically, not me anyway, and I'm pretty sure that's what you're asking. No, I don't have an expression yet for z(t), only a graph. Needless to say, top of the list in this subject.

Major_Tom wrote:How many functions could look like that?

Maybe I've had my head in this too long, but it almost seems magical.

Right now, the treatment of the OP does not include:

- energy sinks of any kind other than momentum
- non-uniform density
- finite compaction ratio
- mass loss

and that's a lot missing. When it gets all of that, I suspect it will look like Bazant. Any variation would hopefully be a more general framework for examining accretion problems than the WTC-centric stuff so far.

I agree that the earlier confusion about whether momentum is conserved can be resolved by including earth in the system.

If the focus is only on the building and earth is forgotten, a confusion will arise concerning momentum conservation.

Going from discrete to continuous is an interesting problem.

.................................

This can be easily demonstrated by including earth in the system as the lowest slab in the building which goes to infinite mass.

With each collision the earth is playing a part. If the mass of the earth was rather finite relative to the rest of the system, the contribution from the earth would be more obvious.

As earth goes to infinite mass, people will tend to forget it is playing any part at all because nobody sees the earth "react" with each collision.

The momentum question can be completely resolved by considering the earth contribution to the problem to be twofold:

Gravitational contribution (as gravitational mass)
Inertial contribution (as a giant mass unit in the system)


The first provides the gravitational field. The inertial aspect can be modeled as a single slab of mass m at the base of the building.

m can be varied to the point that it represents the mass of the earth. Then everyone can see how momentum is ultimately conserved.

I am sure I am not telling anyone anything new, but maybe it could be mentioned that any momentum the building mass picks up during the collapse is stolen from the earth. As the building mass accelerates towards earth, the earth accelerates towards the building. As the mass comes to rest on the ground, momentum is returned.

Not that this changes any calculations appreciably...

Yes, momentum is conserved, just not in the local system under consideration, the towers. The earth becomes a mechanical fiction - immutable ground. In our calculations, we don't let the earth rise up to meet the falling tower top, ever so slightly.

Interestingly enough, the issue about Seffen's treatment goes deeper than that. Actually, it's an entirely different thing. Seffen conserves momentum, just like everyone else, but he also conserves energy in accretion, which is different. He knows the tower is coupled to ground and deals with it accordingly. He stops his analysis with the final velocity and decrees impact with ground! The issue is, is the accretion conservative or non-conservative?

When we were having those discussions with Hambone about whether additional sinks should be considered above and beyond inelastic collision, he took the position that the sink of inelastic collision by definition included those other sinks. I felt otherwise, and have since (in the toy simulacra thread) shown otherwise. By contrast, Seffen only uses the other sinks and assumes no losses to non-conservative impulse.

An example of conservative, non-impulsive accretion would be a planet capturing a moon. The original mass of the planet is replaced by planet+moon. but no collision is involved. By contrast, a moon colliding and fusing with a planet is definitely an impulse, and inelastic.

Seffen claims the tower collapses (and, by extension, any building's progressive collapse) are conservative processes, and he handles the dynamics by assigning a reduced (residual) capacity. The reasoning IS good for a true continuum, but we must consider that, while the column lengths are mostly contiguous, the primary mode of accretion in the real tower collapses was collisional. Not only floors and loose debris, I claim axial compression of steel columns with rehardening (which is part of Seffen's model, apart from the continuum aspect) IS an impulsive action. I intend to demonstrate this as soon as is reasonably possible.

This call - impulsive or non-impulsive - is either right or wrong, no in between, no room for fudging. Something like the acceleration limit is a distinct signpost. Because of the interesting and somewhat non-inuitive properties of the respective solutions, they will converge on different limits no matter what explicitly position-dependent sinks are incorporated.* That is, while it may be impossible to distinguish between governing mechanics in the early phases of progression, given arbitrary parameters, it is always possible to distinguish over a sufficiently long interval, regardless of parameters.

With that in mind, if there were a super-mega-skyscraper which could sustain a very long collapse, we'd be able to determine empirically whether Seffen or Bazant is correct. Neat, huh?

My money would be on Bazant, if it weren't already on Benson! His vertical avalanche force does what neither Bazant or Seffen's model does (natively): it predicts acceleration convergence to zero, and this appears to be more in line with observed properties.


* I don't remember if I've covered that yet here... the limit is independent of overall mass, initial starting mass fraction AND residual capacity.

I realize that I've been scattering my ideas in three threads in two forums. The last post looks like a quantum leap from my prior post in this thread - because it is. I've absorbed a lot on this subject since starting the thread, not documented all of that, and mostly documented at physforum and the toy simulacra thread.

I wouldn't have been able to construct that post a month ago. All of this is a learning experience, that's the purpose. Now that I'm clear on some of these things, earlier confusion seems quite distant. I know what Seffen did, I know what Bazant did, I know how other work (including Benson's and mine) fit in to the picture.

I wonder if Galileo's Square-Cube law would not place a maximum on skyscraper height.

There is a limiting factor in the cross section area of the columns to support the loads not to mention the area that the vertical conveniences consume in the floor plan. The the former the columns will at some point consume too much real estate not to mention fabrication and material handling for the massive lower columns.

The question is what is the justification for super tall buildings? One reason could be very high real estate values and high density land occupancies... with the only way to increase density is building up. But this does begin to strain all the systems which support these high density occupancies.

I think that current analysis would not justify these monsters.

SnowCrash wrote:I wonder if Galileo's Square-Cube law would not place a maximum on skyscraper height.

Yes, I'm quite sure. There's a limit between strength of materials and self weight. Consider the core of the earth is reputed to be iron-nickel. Sure, it's melted, but it would still be roughly spherical because mutual gravitational attraction of the mass elements exceeds the stiffness of the material. It goes into a spherical minimum-energy shape, like a raindrop, but not because of surface tension, rather due to internal stress.