I'm starting this to discuss with Oz (or anyone else) the assertion that Bazant's asymmetric crush down is false. I am quite sure that it is true (straight out of the math), and the following is my attempt to put it into simple to understand language, without the math.
There are only 2 factors that need to be true in order for Bazant’s “asymmetric crush down” approximation to be true. Both are, IMO, irrefutable.
I’ll address “asymmetric crush down” in this post, and save "Crush up" for a later one.
The two factors:
Here’s how it ties together.
__
First, a word on “conservation of momentum”.
This principle has been badly abused in several “collapse analyses” that I’ve seen.
Momentum is NOT conserved during the collapse. A moment’s thought will prove that to be true: The upper block starts out at a small velocity, and (relatively) small mass. Over time, the mass grows & the velocity grows. The momentum does NOT stay constant.
Momentum only stays constant as long as there are no other forces inputting energy into the system. In any vertical collapse, gravity is constantly inputting potential energy (mg ∆h) into the system (where ∆h = the drop from instant to instant).
Several analyses have asserted that momentum is conserved over the duration of the impact. If the collision between floors were instantaneous, then this would be true, because there would be zero time for any drop.
But the collisions are NOT instantaneous.
The collisions are spread over significant distance & significant times. During these intervals, gravity is adding energy (& momentum) to the falling mass.
A combination of factors made the collision virtually continuous, not discrete events.
The two primary factors are tilt of the upper block & the build-up of a large irregular mass of debris (roughly “bullet shaped”) below the upper block. The bullet shape results from the debris near the external walls of the building being preferentially ejected compared to the debris towards the center of the building.
This needs to be understood:
The collision at each floor would require a decrease in velocity (aka, a deceleration) ONLY IF the collision between floors were instantaneous.
For a non-instantaneous vertical collision, the energy added to the system by gravity can produce velocities during the collision that are less than, equal to, or greater than the velocity prior to the collision. Therefore, while the acceleration during the collision will always be less than it was before the collision, the magnitude of the acceleration can be positive, zero or negative.
It is NOT necessary that the velocity decrease (i.e., that there be a deceleration) during a non-instantaneous vertical collision.
Momentum is NOT conserved (locally) during this typed of collision.
[For the physics purists, and the merely curious, momentum is ALWAYS conserved. Always. In this case, if you include the entire system - the building & the planet earth - you can resolve this apparent contradiction.)
__
Back to Crush Down
First, I’ll use the “one floor collapse” idealization.
The figure below show an IDEALIZED version of an upper block after it has crushed down 5 stories & is about to crush down the 6th floor.
To the right, I’ve shown the impact velocity between each floor (n) & the floor beneath it (n+1).
Then the impact energy (in arbitrary units) of each collision.
Then the relative velocity between the Upper Block & the 1st crushed floor during each of the lower collisions.
Finally the energy of the collision between the Upper Block & the first crushed floor during each of the lower collisions.
For the calculations, I’ve assumed a starting mass of 12 floors, and each collision adds (1-EF)m mass to the descending mass (where EF = ejected fraction of material = 0.2). Lamda is Bazant’s “stretch”, considered a compaction factor. The crushed floors end up being about 0.15h thick, where h = 12’, the height of one floor. I've used the 0.65G measured acceleration for calculating velocities & energies.
Note that the "Impact Energy" column is the kinetic energy of the Upper Block plus Debris Layer at the moment of each collision (in arbitrary units: (= (m*v^2)/1000.) This is NOT the energy dissipated in each impact. This is the energy that the lower block would have to absorb if it were to be able to halt the collapse.
The energy consumed / dissipated in each impact is far lower than the kinetic energy of the Upper Block plus Debris Layers. At a minimum, it need only be the energy required to break 1/2 of the existent connections between the trusses & the truss support brackets. That’s all. A tiny amount.
In reality, a lot more energy goes into crushing & fracturing components, hurling things around, generating sound & wind, etc. The calculation of all these energy sinks is very complex, subject to all kinds of arguments. This is why it was very smart of Bazant to do his limiting case, to avoid these arguments.
But the energy absorbed per floor is limited to only that amount of crushing that can be done up to the time the truss bracket connections give way. After that, the floors are falling, and very little crushing takes place. It is necessary to constrain the debris both above & below in order to get any effective crushing to occur.
As stated above, the 2 columns at the right are the relative velocity & impact energies between the Upper Block & the first floor compacted debris, during all of the subsequent collisions.
It is easy to see that the initial collision between the upper block & the first floor is a low-velocity, low energy collision, producing some damage to the Upper Block.
After that collision, the crushed mass of the first floor is moving downwards WITH the upper block.
This is a re-telling of the fact that the Upper Block does NOT impact the Lower Block. The lowest portion of the crushed debris layer impacts the uppermost floor of the Lower Block.
During subsequent collisions, the relative velocity between the upper block & the uppermost layer of the compacted debris is (approximately) zero. Approximately zero relative velocity means approximately zero damage inflicted between the two.
The impact velocity between the bottom of the debris layer and the about-to-be-crushed floor is high. It gets higher & higher as the upper block plus debris layer accelerates. This high relative velocity produces a massive amount of destruction between the bottom layer of debris and the top layer of the bottom block.
__
The second factor mentioned above is that, as soon as the supports for any subsystem (up to the entire floor) are ripped apart, those components begin to fall. Immediately. These components do NOT hang stationary in the air.
This is exactly the stupidity of Gage’s new fantasy of “the crushed & destroyed upper block” as shown here:
http://www.youtube.com/watch?v=FvuKUmK9eB0&t=2m07s
Gage draws some meaningless line at the crush floor, and then leaves the line stationary in the air. Even though Gage doesn't say it, that stationary line depicts floors getting crushed ... but still hanging motionless in the air.
He can only get away with this stupidity because he’s showing a video that hides the descent of the impact layer behind the cloud of debris. This allows him to claim, falsely, that the upper block is being destroyed by the collapse.
If he were to show a video from the NW corner, which gives a view thru the debris clouds, you’d see that the collapse front does NOT hang stationary in space, but begins to descend immediately, at some acceleration less than G.
So, Oz, this is my description of why Bazant’s “asymmetric crush down” is an accurate approximation to the actual events. It ain’t perfect. The upper block does not descend like an unblemished flower. These are violent collisions. But this model is 90 - 95% correct (for the first 10 stories or so).
Further, from the energy calculations, you can see that this only has to hold true for a couple of floors of descent in order to guarantee collapse to the ground.
This is my description of why the alternative theory (“equal crush up & crush down”) is completely wrong.
Please tell me what you disagree with.
There are only 2 factors that need to be true in order for Bazant’s “asymmetric crush down” approximation to be true. Both are, IMO, irrefutable.
I’ll address “asymmetric crush down” in this post, and save "Crush up" for a later one.
The two factors:
- The principle determinant of the amount of damage occuring in any two “similarly constructed assemblies” that collide is the relative speed of impact, with the damage proportional to the speed squared (the work done in causing the damage is proportional to the kinetic energy of impact).
- Once some structure has been ripped from its supports, it begins to fall immediately.
Here’s how it ties together.
__
First, a word on “conservation of momentum”.
This principle has been badly abused in several “collapse analyses” that I’ve seen.
Momentum is NOT conserved during the collapse. A moment’s thought will prove that to be true: The upper block starts out at a small velocity, and (relatively) small mass. Over time, the mass grows & the velocity grows. The momentum does NOT stay constant.
Momentum only stays constant as long as there are no other forces inputting energy into the system. In any vertical collapse, gravity is constantly inputting potential energy (mg ∆h) into the system (where ∆h = the drop from instant to instant).
Several analyses have asserted that momentum is conserved over the duration of the impact. If the collision between floors were instantaneous, then this would be true, because there would be zero time for any drop.
But the collisions are NOT instantaneous.
The collisions are spread over significant distance & significant times. During these intervals, gravity is adding energy (& momentum) to the falling mass.
A combination of factors made the collision virtually continuous, not discrete events.
The two primary factors are tilt of the upper block & the build-up of a large irregular mass of debris (roughly “bullet shaped”) below the upper block. The bullet shape results from the debris near the external walls of the building being preferentially ejected compared to the debris towards the center of the building.
This needs to be understood:
The collision at each floor would require a decrease in velocity (aka, a deceleration) ONLY IF the collision between floors were instantaneous.
For a non-instantaneous vertical collision, the energy added to the system by gravity can produce velocities during the collision that are less than, equal to, or greater than the velocity prior to the collision. Therefore, while the acceleration during the collision will always be less than it was before the collision, the magnitude of the acceleration can be positive, zero or negative.
It is NOT necessary that the velocity decrease (i.e., that there be a deceleration) during a non-instantaneous vertical collision.
Momentum is NOT conserved (locally) during this typed of collision.
[For the physics purists, and the merely curious, momentum is ALWAYS conserved. Always. In this case, if you include the entire system - the building & the planet earth - you can resolve this apparent contradiction.)
__
Back to Crush Down
First, I’ll use the “one floor collapse” idealization.
The figure below show an IDEALIZED version of an upper block after it has crushed down 5 stories & is about to crush down the 6th floor.
To the right, I’ve shown the impact velocity between each floor (n) & the floor beneath it (n+1).
Then the impact energy (in arbitrary units) of each collision.
Then the relative velocity between the Upper Block & the 1st crushed floor during each of the lower collisions.
Finally the energy of the collision between the Upper Block & the first crushed floor during each of the lower collisions.
For the calculations, I’ve assumed a starting mass of 12 floors, and each collision adds (1-EF)m mass to the descending mass (where EF = ejected fraction of material = 0.2). Lamda is Bazant’s “stretch”, considered a compaction factor. The crushed floors end up being about 0.15h thick, where h = 12’, the height of one floor. I've used the 0.65G measured acceleration for calculating velocities & energies.
Note that the "Impact Energy" column is the kinetic energy of the Upper Block plus Debris Layer at the moment of each collision (in arbitrary units: (= (m*v^2)/1000.) This is NOT the energy dissipated in each impact. This is the energy that the lower block would have to absorb if it were to be able to halt the collapse.
The energy consumed / dissipated in each impact is far lower than the kinetic energy of the Upper Block plus Debris Layers. At a minimum, it need only be the energy required to break 1/2 of the existent connections between the trusses & the truss support brackets. That’s all. A tiny amount.
In reality, a lot more energy goes into crushing & fracturing components, hurling things around, generating sound & wind, etc. The calculation of all these energy sinks is very complex, subject to all kinds of arguments. This is why it was very smart of Bazant to do his limiting case, to avoid these arguments.
But the energy absorbed per floor is limited to only that amount of crushing that can be done up to the time the truss bracket connections give way. After that, the floors are falling, and very little crushing takes place. It is necessary to constrain the debris both above & below in order to get any effective crushing to occur.
As stated above, the 2 columns at the right are the relative velocity & impact energies between the Upper Block & the first floor compacted debris, during all of the subsequent collisions.
It is easy to see that the initial collision between the upper block & the first floor is a low-velocity, low energy collision, producing some damage to the Upper Block.
After that collision, the crushed mass of the first floor is moving downwards WITH the upper block.
This is a re-telling of the fact that the Upper Block does NOT impact the Lower Block. The lowest portion of the crushed debris layer impacts the uppermost floor of the Lower Block.
During subsequent collisions, the relative velocity between the upper block & the uppermost layer of the compacted debris is (approximately) zero. Approximately zero relative velocity means approximately zero damage inflicted between the two.
The impact velocity between the bottom of the debris layer and the about-to-be-crushed floor is high. It gets higher & higher as the upper block plus debris layer accelerates. This high relative velocity produces a massive amount of destruction between the bottom layer of debris and the top layer of the bottom block.
__
The second factor mentioned above is that, as soon as the supports for any subsystem (up to the entire floor) are ripped apart, those components begin to fall. Immediately. These components do NOT hang stationary in the air.
This is exactly the stupidity of Gage’s new fantasy of “the crushed & destroyed upper block” as shown here:
http://www.youtube.com/watch?v=FvuKUmK9eB0&t=2m07s
Gage draws some meaningless line at the crush floor, and then leaves the line stationary in the air. Even though Gage doesn't say it, that stationary line depicts floors getting crushed ... but still hanging motionless in the air.
He can only get away with this stupidity because he’s showing a video that hides the descent of the impact layer behind the cloud of debris. This allows him to claim, falsely, that the upper block is being destroyed by the collapse.
If he were to show a video from the NW corner, which gives a view thru the debris clouds, you’d see that the collapse front does NOT hang stationary in space, but begins to descend immediately, at some acceleration less than G.
So, Oz, this is my description of why Bazant’s “asymmetric crush down” is an accurate approximation to the actual events. It ain’t perfect. The upper block does not descend like an unblemished flower. These are violent collisions. But this model is 90 - 95% correct (for the first 10 stories or so).
Further, from the energy calculations, you can see that this only has to hold true for a couple of floors of descent in order to guarantee collapse to the ground.
This is my description of why the alternative theory (“equal crush up & crush down”) is completely wrong.
Please tell me what you disagree with.
via International Skeptics Forum http://ift.tt/1SlcoWb
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