Review1 of Progressive Collapse of the World Trade Centre: a Simple Analysis by Keith Seffen
The paper is available: http://winterpatriot.pbworks.com/f/seffen_simple_analysis.pdf
Simple analysis indeed. This is complete buckle up-buckle down mechanics. He says it is idealized but takes the buckle-rebuckle collapse mode quite literally just as Dr Bazant does in his 2007-2008 collapse mechanics papers.
In the Seffen paper his equation of motion is eq #12. His whole derivation of his equation of motion is eqs 6 through 12. From that equation the claim of a g/2 acceleration asymptote follows.
He makes some bold statements about g/2 on page 17:
"The particular steady value of g=2 is a consistent property of any variable-mass system where the mass, initially at rest, is entrained by a non-impulsive action. An exposition of these general cases can be found in any comprehensive dynamics textbook, e.g. Meriam and Kraige (2003)."
If he is correct in saying that g/2 convergence is expected in a whole class of problems, that is worth double checking. He calls it a "variable-mass system where the mass, initially at rest, is entrained by a non-impulsive action."
Entrained by a non-impulsive action? A rubble-driven collapse is practically ALL impulsive. When he says non-impulsive, it's quite clear how it applies to the model he constructs, being a continuum mass distribution with an associated residual capacity. Whether or not one considers a conservative/non-conservative distinction, this aspect alone dictates the applicability of the model to an actual collapse, particularly the towers. I don't consider the stacked slab models to be very tower-like, but if I have to choose between the two as a better representation, I have to go with the slab. Even column rehardening effectively amounts to a(n inelastic) collision, which is impulsive.
In the simplest terms, Seffen is working a different class of mechanics problem without realizing the difference. We can call the two classes of problems the "g/2 class" and the "g/3 class". As the difference in these 2 classes of physical interaction is explored, the Seffen mistake will become clear.
Starting with an excerpt from Seffen's conclusion:
QUOTE
"The particular steady value of g=2 is a consistent property of any variable-mass system where the mass, initially at rest, is entrained by a non-impulsive action. An exposition of these general cases can be found in any comprehensive dynamics textbook, e.g. Meriam and Kraige (2003)."
There is no doubt in Seffen's mind he is doing the right thing. And, apparently, this is not some esoteric thing, it's common knowledge in the engineering mechanics community.
From Falling chains as variable mass systems...
QUOTE
"Recent experimental observations, that the free end of the U-chain falls faster than g [12,15,16] have contributed to an increasing interest on the behavior of systems such as ropes and chains. Such a behavior admits that the description of the system is based on the assumption of conservation of energy. The same hypothesis of conservation of energy also leads to the value of g/2 for the acceleration of the pile-chain,[10] instead of g/3 which is obtained in a non-conservative scenario.[1]"
Citation #1 is A. Cayley, “On a class of dynamical problems,” Proc. R. Soc. London 8, 506-511 (1857). Over 150 years old. It can be found here and, after an analysis of a discrete system of particles (a chain) quite a bit more involved than what we've done here, proclaims:
QUOTE
"...that the motion is the same as that of a body falling under the influence of a constant force -g/3."
This was identified in the previous article as the "non-conservative" approach. Without investigating further, it would seem the difference between g/2 and g/3 is the latter is not energy-conserving. These, then, represent the respective high-level choices of Seffen and Bazant and account for the difference in this particular metric.
The immediate question is: which approach is truly correct for the application at hand?
This has me thinking about the two simulation environments I've used, how they figure into this problem and why the results may be as they are.
TESTING WITH A GAME ENGINE SIMULATOR
The first is a game engine called PhysX. It is (or was a couple of years ago) a high end physics engine for games, not an engineering simulation program but quite adequate for simple systems. Naturally, I did a lot of testing of systems with known properties to verify validity, including some fun with 1D gases. The software is primarily oriented toward rigid body dynamics with support for a variety of 'joints' to connect bodies. It does have facility for soft body construction but that would be inappropriate for use here.
A rigid body is defined by these properties:
- mass
- static and dynamic surface friction coefficients
- collision restitution
- one (or a collection of) 3D shape primitive(s)
The bodies do not deform per se, though the surfaces allow (require) a small interpenetration depth, such that collision contacts are not singular. The only way to have true inelastic collisions in this framework is by decree. A restitution of 0 is fully inelastic while 1 is perfectly elastic. Therefore, the dynamics are calculated without concern for how energy is dissipated in collision, it's simply thrown away behind the scenes according to the specified restitution, which is in accordance with basic collision mechanics.
The important thing is this: I only use a restitution of zero in "slab" style collapse modeling, therefore my systems are defined to be inelastic. Happily, the acceleration for uniform mass distribution converges on g/3, just as it is supposed to - for inelastic accretion of particles.
Results from the simulations are available here and reveal a clear g/3 asymptote within the acceleration.
It makes sense for a system of discrete rigid bodies to dissipate energy in this fashion; how else could it happen with 'rigid' bodies? But slabs aren't the same as a continuous or even contiguous distribution. In a continuous system, there are no collisions or, as Seffen puts it, impulsive forces.
What I think at this moment is - Seffen is right for his model, but his model is not right for the towers or any real building collapse.
Is plastic deformation conservative? Is comminution conservative? Fracture? No. Reducing effective capacity is not the same as dissipating energy into sinks. So there is the question of just how conservative his analysis is, or should be.
Seffen, or anyone for that matter, can be justified in taking the tower structure to be continuous since the vertical load bearing members satisfy that condition. However, when one examines the typical load-displacement curve of hinge buckling in axial compression, there is a profound (nearly step) change in resistive force at full compaction. This is tantamount to collision, and clearly intuition dictates it's not elastic.
A SECOND SIMULATION ENVIRONMENT
The second simulation environment is a port I did of an academic program which provides a nice variety of elements. On the most rudimentary level, it could be thought of as a mass-spring-damper system but it's really more powerful than that and, since I have source code, I can pretty much do anything I want within the framework, in terms of defining elements.
Unlike the first program, this one approximates a continuum (though the masses I use are point masses). Collisions are not native to the framework; to mimic collision a short range, high magnitude repulsive force field can be defined, and a plastic (non-conservative) phase can be included for inelasticity. I did not do that.
When I set up the 1D collapse scenario in this program, I used a parameterizable load displacement graph fashioned after Bazant. Unlike the rigid connections of the physics engine which had very short elastic then plastic travel before fracture, the connection between masses here was continuously available throughout compaction - a continuum.
As I've said, I've used agreement between the two programs in many trials to assert that the second one should also converge to g/3, but have not specifically checked. I may be wrong.
But I don't think so. I think the impulsive character of rehardening is what counts here, along with the plastic deformation prior. I don't see how N randomized impacts (including real ones, the building was not homogenous) changes the result as N goes large. Every possible mode of collapse - including perfect 1D axial compression - is impulsive in nature. All other 'real' failure modes are only more so.
Could Seffen have deceived himself with the Maxwell line?
It will be interesting to check the range of possibilities in the second simulator.
Seffen said "variable-mass system where the mass, initially at rest, is entrained by a non-impulsive action."
That would be: planets capturing moons, stars capturing planets, galaxies capturing stars, etc - conservative potential, no collisions. Also, apparently some configurations of chains! Though I'm beginning to doubt that, too. It's certainly hard to imagine a block of homogenous material conservatively compacting, without elastic rebound. Oxymoronic, I'd think.
Elastic strain potential is conservative and resistive force is a function of displacement, however shallow the slope (spring constant) may be. We could consider, in the limit, the spring constant to be zero, but that's the trivial solution. It avoids the paradox of being both elastic and plastic response simultaneously by being no response at all; undefined. I see no way to resolve the notion of a conservative force of non-zero constant magnitude with respect to displacement. This is typically called plastic response. A constant force is non-restoring. A non-constant force is not necessarily restoring but a constant force with increasing compression can't be.
Needless to say, in an irreversible collapse, there's no restoring force.
Whenever two previously isolated surfaces come into contact, it is impulsive by definition. But really, it's not about where the mass is - that just happens to be coincident with the electrostatic repulsion that is 'contact' in matter - it's where the force is, where potential gradient rises steeply. Therefore, two masses can be in indirect contact ever so tenously by a connecting member of insignificant relative mass which provides a strong restraining force keeping them at a fixed distance, as if they were in contact. Any external force applied to one mass would be felt by the other, so long as the connection is integral.
If this connecting member were overloaded by a compressive force or impulse, it could reduce in capacity as steel columns do, and then the bodies will approach each other. The member need not break or otherwise disappear, it need only become noodly and buckle, all the while still connecting the two bodies. At full compaction, there is a collision.
At a very, very large scale, with countless such bodies connected together, it may be tempting to call the system homogenous. At the microscale, however, we know it is not. And we know that the fundamental governing mechanics of compaction in this system is impulsive, and that fact is scale invariant. How would you know the intrinsic property of impulsiveness without examing the micro-detail?
I have a difficult time seeing where the conservative approach can be applied to any material in plastic compression or tension. Elastic, yes. Plastic and irreversible, no.
______________________________________________________________
1 Review originated by poster OneWhiteEye at this link