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Old 10th March 2009, 02:53 PM   #1942
Join Date: Oct 2006
Posts: 3,454

I've noticed that you've declined to respond to my posts. I'd hate to think that it's because I'm a mechanical engineer, and you'd rather not talk to other MEs. I have to tell you that I found your last response ("read my paper") to be completely unsatisfactory. Since your paper & subsequent postings are riddled with errors.

I'm sure that others have given you their take. Here's mine.

Fundamental flaws in your analysis (i.e., "tripod" paper).

The "strain energy per floor" is significant to the total energy balance. It is irrelevant to the question of whether or not the collapse is halted. The only strain energies that play a part in halting the descent of the upper block are the local strain energies at the contact points between the upper & lower blocks.

A "rigid block" is not equivalent to a block with "infinite strain energy". It simply means that it moves as a unit. (An "indestructible rigid body" would fit your description.) Your objection to Bazant's "no crush up" simplification is correct. FOR THE MOMENT. When crush happens, it progresses both upwards and downwards. But very quickly after the crush begins, the open structure of the upper block becomes "impacted" with the debris of the collapse. The bottom edge of the upper block becomes almost a solid surface of impacted debris. This phenomenon does not occur for the lower block, because each floor's debris starts from a zero velocity. Since the upper block has been falling for several floors, it sweeps thru the crushing floor faster than the debris from that floor are descending, thereby incorporating its debris into the mass of the descending block or ejecting it sideways.

As a direct result of this, Bazant's assumption of a rigid upper block turns out to be very reasonable.

Nonetheless, he has incorporated improvements to this assumption, that allow immediate crush both upwards & downwards, in his 2008 paper.

In your tripod paper, you state: The amount of potential energy, PE, due to downward movement of the WTC 1 upper part mass was definitely too small to turn the lower structure into 100 000's of tiny pieces and dust.**

This is true. It is also irrelevant.

The KINETIC energy of the upper block did NOT break all of the lower block's pieces into dust & tiny debris. Neither did the kinetic energy of the upper block have to exceed the strain energy of even one single story in order to continue the crush. It did have to exceed the strain energy of a small portion of the structure that was far, far less (perhaps on the order of 10,000x less) than the total strain energy of one single story.

In order for the collapse to continue, the kinetic energy of the upper block had to exceed the sum of the strain energy of the thousands of comparatively small points of contact between the upper block & the lower block. For gross simplification, let's assume that these points of contact are principally 290 column stub ends contacting 290 thin concrete floor slabs. In all of these cases, the limited strain energy capacity of the concrete is going to result in a prompt failure of the concrete. Subsequent entrapment of the column end in the fractured concrete debris (e.g., rebar or cross trusses) will then torque the column one direction or another, fracturing its restraints (bolts and welds). And the descent of the upper story continues.

Again, the strain energy capacity of these small segments of the concrete floors and the bolts and welds is a miniscule portion of the total strain energy of all the components structures of that floor. And yet, once these components have been destroyed, the crush down continues.

In order for the collapse to be arrested, two requirements must be met.

1. The maximum local stress generated at each contact must be less than the ultimate strength of both contacting components.

2. The local strain energy in deformation at each contact must not exceed the strain energy capacity of the less "tough" structure.

If BOTH of these conditions are met at a sufficiently large number of points such that the sum of all the forces EXCEEDS the weight (m x g) of the upper block for a sufficiently long time that the integrated impulse, (the sum of forces - weight) x time, exceeds the momentum of the upper block, THEN the upper block will be brought to a halt.

In math terms, IF Integral[(Fi - mg) dt] > m v then the upper block will be brought to a halt, where Fi = local force, m = mass of upper block, v = downward velocity & g = grav. constant.

You must include the weight term in this equation because gravity is a force that is acting on the block throughout the collision. Viz., if v is epsilon greater than 0, then the force necessary to arrest the upper block is not zero. It is a value slightly greater than mg.

In energy terms, if the sum of all the individual strain energies of all the surviving contacts exceeds the kinetic energy PLUS the strain energy of static loading, then the block will be brought to a halt.

Again, if Integral[Ui] > KE + Us, then the block will be brought to a halt. Where Ui = all the local strain energies, KE = the kinetic energy of the upper block, and Us = the strain energy associated with the statically loaded case.

Note that in ship collisions, all the significant forces act horizontally. The weight & buoyancy forces cancel each other out. This is not true in a falling collision.

In summary, your errors are:

1. You can not determine global forces, stress, or strain energy a priori. You MUST determine them locally, and sum them to determine the global result (crush or no crush).

2. You can not AVERAGE forces, stresses or strain energies over bigger or smaller portions of the building. This ignores stress concentrations and local strain energy concentrations, which are absolutely crucial to progressive collapse.

3. You can not use the load carrying capability of the a column or beam that is properly constrained to determine the load carrying capacity of columns and beams that have had one or more connections removed. You implicitly do this when ever you bring up the issue of the Factor of Safety built into the structure. The factor of safety has meaning ONLY when the columns are undamaged and properly constrained. Without those constraints, the load carrying capacity of those beams drops by orders of magnitude. Especially for lateral loads. This is, BTW, the precise reason that the structures failed at the crush zone and not above or below it.

Other important issues you've not addressed in this paper:
4. The initial failure involves 3 floors, not one. The failure of a single column compromises the entire column, which is 3 stories high.

5. Your analysis does not allow for the significant effects of the stagger of adjacent columns. In other words, by the time that the crush down has reached any particular floor, fully 2/3rd of the columns and trusses supporting that floor have already been severely compromised. This damage occurred because those columns reach up 1 & 2 stories higher, and were damaged when those higher stories were destroyed.

These are the major errors in constructing your analysis. There are others.

Some examples:

1. "no evidence that the core structure displaced downward". Nonsense. The roof displaced downward. If the core did not, it'd be sticking out of the roof.

2 "no evidence of any simultaneously buckled visible, outside wall columns in the fire zone". Nonsense. Clear, unequivocal video images of massively buckled exterior columns.

3. All of your "rigid body" objections are straw-men arguments, since neither NIST nor Bazant intended them as any more than simplifying approximations. Rigid bodies are NOT required for either NIST's or Bazant's conclusions.

4. No "solid, intact columns below were overloaded by gravity only". Heated (ie., weakened), bent (i.e., unstable) columns buckled and had their connections snapped to initiate the collapse (or crush). After the crush down began, columns that had been massively compromised (by damage to already crushed upper floors) had their few remaining connections destroyed. The columns themselves were almost never overloaded, as proven by the absence of columns with massive plastic strain left in the debris.

5. The specific (& a bit deceptive, on your part) reason that your analysis says that the upper block collapses initially is that you've drawn the bottom "green line" that defines the upper block too low. You've drawn it to encompass the upper block, all of the impact floors, plus (it appears to me) a couple undamaged lower floors for good measure. You should redraw the blocks to define an undamaged upper block, an undamaged lower block and a damaged group of impact floors. If you do it like this, you'll find that the impact floors crush first. The upper block descends because the impact floors crush, the upper floors pack in, and the lower floors start to crush down.

Just about like Bazant's model says. How 'bout that?!

There's lots more that's flawed. That's enough for now.

** The source of energy to pulverize the contents and their contribution to slowing the descent of the upper block.

For any piece of the tower that was thrown clear of the footprint, the deformation that occurred when it hit the street (or other object) obviously did NOT contribute to slowing the descent of the upper block.

The energy to pulverize the contents into tiny pieces and dust was provided by the EXCESS potential energy of all the mass above any given piece, less the energy lost to disassembly, PLUS the potential energy of that piece above the ground.

A large portion of the concrete was held within the footprint of the towers by the intertwined rebar and was ground to dust within the churning mass of the crushing tower, thereby contributing to slowing the upper block. A smaller portion of the concrete was thrown clear of the towers and was reduced to dust as it collided with the street.

The bolts and welds were snapped as each floor disassembled, slowing the descent.

Most of the external columns were thrown out of the footprint, and their bending & damage did not contribute to the slowing of the upper block at all.

Almost all of the core beams were contained within the footprint. While the disassembly of the upper core columns may have contributed to the slowing the upper block, the lower 40 stories or so did not, as they were seen still standing in the North Tower.

The grinding into small pieces of the contents of the towers did contribute to slowing the descent of the towers.

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