You're jumping ahead, but you should note that I do
not make this assumption at all. I make no claims about the upper block's rigidity whatsoever.
Step 2 of my collapse model considers conservation of momentum, not energy. That law, and my application of it, holds no matter how flexible the upper block is. No matter how much it squashes in on itself, its momentum remains the same.
The only place in the model that compaction can affect the equations is in Step 3. The compaction force is no greater than the resistance of the lower columns, and lasts for no longer than it takes those columns to buckle. The amount of compaction that occurs from this contribution is nonzero, but negligible. This was discussed by Dr. Bazant in his blanket response to naive criticisms of his earlier papers.
I also point out that in your model, your 0.2 and 0.3 yield forces are wrong. You have been challenged to back these up in
another thread, and you have lost the challenge.
I am just assisting you getting your model right and not jumping at all (like my grandchildren in my bed!). In Step 2 the upper part C should not be shown as a block M = k
m but as an assembly of floors/columns where the lowest floor (or rather the ceiling + floor trusses) of part C with mass
m collides with the top floor of part A, also with mass
m. You should also consider indicating the failed columns between parts C and A (and the columns of part C). The assumed perfect impact floor/floor is not realistic.
Momentums are nice to work with in solid mechanics but in structural analysis we generally work with deformations/strains/energies, particularly so in damage analysis, where the original structure changes after failures.
In Step 3 of your model the uppermost floor of part A is added to the the upper part C - still a block, now with M = (k + 1)
m "M has gotten bigger". That is not possible! This is the
'pan cake' theory, where part A upper columns only are broken like spaghetti and the part A floors are just pushed down (and accelerated!) one by one by a
rigid upper part C, which NIST has abandoned long time ago.
Collision contact between two similar structures, parts C and A, does not result in that! You must study what happens to part C and all its elements and adjust your model accordingly.
Re 'yield forces' (sic) in my papers, you probably mean the load of the upper part C (M = k
m) and resulting
static compressive forces/stresses in the columns at interface parts C/A prior destruction. They are described at
http://heiwaco.tripod.com/nist0.htm#2 .
The compressive stresses are about 0.2 and 0.3 of yield stress in perimeter/core before plane impact and about 0.3 in east/west walls just prior destruction.
Evidently the local stresses in the columns will increase, if part C columns actually impact on part A columns, as you suggest in your model, and it is also described further down in my paper. Then the dynamic forces/stresses are a function of energy input and deformations (and the static stresses are temporaily 0 = columns cut off)
My latest calculations indicate that the total stresses in the columns will not increase >3 of the previous static ones, which means that only elastic deformations will take place = the bounce! Reason being that the structure is pretty flexible and that you can argue about the drop height/energy involved. If you increase the energy input so that the columns fail locally, the upper part C (M = k
m) will slide off part A and we are back to my final model, where the columns slices the floors ... in both parts C and A. Collapse arrest should follow pretty soon.
I enjoyed your show for Hardfire. Hardfire, not knowing anything about physics or mechanics or structural analysis and sluring about conspiracy theories, looked a bit confused when you started to talk about the derivate of velocity (acceleration), etc. Keep it simpler!
It is only the velocity of the moving mass (the energy) at collision that is of any importance and produces failures. If that energy is not totally consumed at the gravity fed collision, evidently the moving mass will continue to accelerate and produce more collisions and the rubble it produces need also be accelerated, etc, etc, but I can assure you that a small part C of similar structure as bigger part A cannot simply destroy part A in a crush down.
Hopefully you will agree to that, when you have got your model right?
