Originally posted by Akareyon

Bazant/Zhou (2002), Bazant/Verdure (2007), Greening et al first calculate the force of the small hammer upon impact on the nail's head which would sink the nail a few millimetres into the wood. They say that the resulting force would be 31 times the weight of the small hammer, depending on the stiffness of the nail and the speed and weight of the hammer. Then they take a huge hammer which weighs 31 times as much as the small hammer and carefully put it on the nail's head to prove their point, because now, the nail slides into the wood. That's the whole trick, basically.

This is false.

While it is true that Bazant uses a homogenized equivalent force, it is NOT constant and it is derived via the well-accepted notion of a Maxwell construction from the non-constant load displacement response of steel columns in hinge buckling. Nor is the same peak initial force used throughout the collapse.

It seems I have to pull this graphic out a lot. Figure 3 from "Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions" (Bazant & Verdure):



In this load displacement graph, you can see the non-constant resistive force before your very eyes, so I trust there will be no further need to argue this point. From that curve with peak and tail-off, representing the well understood and empirically documented properties of columns undergoing three-hinge buckling, an energetically equivalent story-wise average force is obtained via integration and division by the displacement over the entire interval (that is to say - distance averaged).

This equivalent force is shown by the dotted line labeled "Maxwell Line" and is expressed in the diagram in units of mg, therefore scales with the load and capacity of each story. It is not constant, and it gets larger going down the structure.

The 31x overload is not propagated through the collapse. That figure refers to the insufficiency of the impacted supports to withstand the imposed dynamic loading. It falls straight out of fairly elementary calculations. It is not the force applied to the lower section since, by Newton's 3rd, there must be an equal and opposite force for each applied force, and the lower section simply was not capable of providing that resistance.

The magnitude of the Maxwell line in that diagram is 39% of the load, or 13% of the peak capacity assumed by Bazant. It means the average resistive force provided by columns which are overloaded to full compaction is about 40% of the static load above. Thus in this approximation, one would expect to see an average downward acceleration of about 0.6g and, funny, that's exactly what's observed in careful and detailed measurements of the early collapse displacement. However, similar measurements in the later progression unequivocally point to terminal velocity (i.e acceleration = zero) being reached fairly early on in the collapse. This would be due to velocity dependent sinks such as concrete comminution and perhaps other issues of mass retention within the leading crush zone footprint; nothing whatever to do with structural resistance.

As I noted a few weeks ago in another thread, all of this is moot since it's easily verified by direct viewing of any number of videos that the collapse mechanism was not an intact upper block buckling each and every column to full compaction all the way down.

I should clarify that my remarks above apply to the formulation of the resistive force. The applied force from above is dictated by the properties of the upper section and any accumulated debris. The force from static load of a given section is of course constant, but if there is accumulation of debris going down, then the mass - therefore the load - is increasing. Beyond that, the force which the upper section is capable of applying to the lower is dependent on momentum, which is composed of both mass and velocity.

Thank you for clarification, IrishWristwatch, and for perpetuating the idea that mass aggregation kept the collapse going (instead of "rubble" dampening any "crush" (up or down, who really cares)). You seem to have more than just a layman's insight into matters like Maxwell's demon, so, may I ask for your assistance in building a model that acts like the reverse bear trap from the SAW series? It goes like this: you activate it, than you give it a small momentum to start it, and even while you wonder what is going on, after just a few minutes, the thing goes off and your mouth stands open. Because that's exactly what the towers did, leaving us with our mouths open.

2.1 GJ triggered the potential energy of up to 1TJ.

Then along comes an expert and explains: yep, that's the way we build buildings, and he produces charts and diagrams and complex formulae that look all greek to us to show just how silly we've been to think that anything in the world could have stopped the collapse. Not even those professional desks and file cabinets that can usually be found in many offices, none of which were (allegedly) found in the rubble except for one.

While it is true that momentum is the product of mass and velocity, velocity goes squared in the equation for energy. When something hits something, the sum of all momentums in the system stay the same. Kinetic energy becomes internal energy. When you hit the nail on the head with a hammer, the nail will bend, the wood may get deformed, the hammer may get warmer, but the nail won't sink into the wood all by itself, and that's because of the friction force that it has to overcome. Velocity diminishes each time the mass has to accelerate or deform something else. Even if mass, in this case, becomes bigger by a tiny fraction because it takes another floor with it.

According to Newton's lex quarta, all the "upholding forces" in the towers need to be summed up and overcome by the fall of 58.000t from 3.7m height. It's not just one floor, it's the combined force of all 110 stories. The impact waves run through structural steel at 4500m/s, so all of this force would have been sent to the impact zone, while the impacting force would have gone the way of least resistance, maybe even cutting a bolt in the basement (actio_reactio). In nature, things take the least energy-consuming way through space-time, therefor, any impact would first shear the weakest bolts before bending a box column. Still, even every molekule in the way would slow down the falling block. But it didn't slow down. It accelerated downwards, just as shown in Fig 3+4a in "Mechanics of progressive collapse", because m*g was greater than F_c (the maxwell line or Force it needs to "crush" the plastic hinges) all the way down to Ground Zero.

Usually, it can easily explained how m*g could be greater than F_c. You just put a lot of mass on top. Or you hit something really fast, so that it is decelerated by a spring with 71GN/m, so that g=31*9.81m/s². But only for a fraction of a second, after a little {u}, it would diminish back to g, at least in any other isotropic structure, so that m*g is smaller than F_c as shown in Fig. 4b. Like the nail and the wood slow down the hammer.

m*g was greater than F_c on all 110 floors because F_c was smaller than m*g. In other words, F_c of the building had to be weakened to a point that it could not sustain itself. Because usually, architecture is about providing for a F_c at least twice as big as any expectable m*g so that Fig 4c applies (that's when you jump out of your bed on the floor).

This is as far as I can go as a layman to explain Dr. Bazants trick and I hoped for experts to join the discussion to explain in greater detail all the stuff that is going on mathematically in those papers instead of insisting that it can be called sound sience to pretend you'd just input a little energy while really secretly hiding "inside" energy 100-fold in the equations and silent premises. That's what the "experts" have done in economics for at least a hundred years to enslave the people and keep them dumb. But please don't try this all over again in physics, we've gone a too long way to to turn back to believing that earth is flat and lightnings are the wrath of malevolent gods.

Originally posted by Akareyon
According to Newton's lex quarta, all the "upholding forces" in the towers need to be summed up and overcome by the fall of 58.000t from 3.7m height.

Incorrect. Overload need only occur in the local region, then failure propagates downward. This is an iterative process, not an integrated one.

It's not just one floor, it's the combined force of all 110 stories.

Why stop there? Why not add the ground? If you do, there would never be a collapse of anything! The point is, localized failure occurs in materials undergoing plastic deformation and, due to the distribution of mass, this deformation is concentrated at the location of force application. When two vehicles collide head-on, does the rear bumper experience the same deformation as the front? Even if one of the cars is butted up against a wall? No.

The impact waves run through structural steel at 4500m/s, so all of this force would have been sent to the impact zone...

True, but not with the consequences you claim. First of all, the force transmission to which you refer is via elastic waves in the steel; plastic waves do not propagate through the medium. Therefore, by definition, the magnitude of force is limited to the elastic capacity of the medium. Second, in order for the force to displace mass at any point lower than the crush front, all the interstitial mass must also be displaced, so you're talking about the entire building displacing downward X amount to get the bottom to deform by X amount.

...while the impacting force would have gone the way of least resistance, maybe even cutting a bolt in the basement (actio_reactio).

Aside from the dodgy interpretation of the principle of least action made in language of force, I won't quibble with this except to say again that, in order to fail a bolt in shear, tension, or compression, it has to be displaced, and that means all the building above has displaced, too. Ya think that might break something up top first?

In nature, things take the least energy-consuming way through space-time, therefor, any impact would first shear the weakest bolts before bending a box column.

Now you're getting to the nitty gritty of why all this discussion about column buckling, peak capacity and FOS, and so on - is moot. The floor connections were NOT designed to support the entire tower load above, the columns were. If the debris manages to find itself on a floor pan (how could it not?), then we are concerned with the capacity of a floor, not the vertical members. Clearly the floors not only have significantly less capacity than the columns - that's why columns were used to hold the building up, not the floors - but the floor assemblies were of nearly uniform construction all the way down as opposed to getting stronger. And, one last thing: the floors were separated from each other so load applied to one was not applied to even the next one down.

Still, even every molekule in the way would slow down the falling block.

No. Every little molecule exerted a resistive force and that adds to the sum of forces. If the upwardly acting resistive force is less than the imposed force, it will accelerate downward. Period. The only way it will slow is if the resistive force exceeds the load.

But it didn't slow down.

There were a couple of noticeable jolts in there but, for the most part, you're right; it did not slow down.

It accelerated downwards, just as shown in Fig 3+4a in "Mechanics of progressive collapse", because m*g was greater than F_c (the maxwell line or Force it needs to "crush" the plastic hinges) all the way down to Ground Zero.

No, it did not accelerate all the way down. As I said above, terminal velocity has been measured for a good share of the leading progression. Bazant's model predicts acceleration all the way down. It did not happen.

I'm going to leave it at this. Something tells me I've not made a dent, and that's okay. Not necessary.

Just noticed your edit regarding impulse and momentum. I did not realize you were not a native English speaker (though I should have from your links). Your command of the language is quite excellent and I appreciate your efforts to discuss technical subjects in this manner. I'll be more mindful of this consideration in the future.