Heiwa,
Sorry for the delay in response. Real life intrudes.
Sorry for the length. Detail oriented.
In a crush down you apply a load from top and crush what is below.
Collapse? I assume you remove a support somewhere below and things above drops down.
NIST/Bazant use the words as equivalent and mix them to suit the circumstances.
I see. You appear to be defining a collapse as a total failure that starts at the bottom of the building & a crush down as a failure that starts at the top of the lower section.
Just out of curiosity, what would you call it if the building fails somewhere in the middle (i.e., say at the 5th story or so, like WTC7)?
Thus it is very important in structural damage analysis to identify the failures step by step and find the cause for each failure.
Where do I say the lower segment (?) will NEVER collapse? Segment? I have never used that word, ever.
Agreed that it is absolutely crucial to identify the component that fails first.
Segment: "portion of a larger entity". Interchangeable with your "part" or "structure" or "block".
Where do you say "will never collapse"? You've said it many times. Including in THIS reply. Below you say, "In the latter case part A will neither collapse nor be crushed down." These two statements are semantically equivalent.
__
My problems with your analysis are too numerous to count. A few of the more glaring points:
1. Ignoring initiation zone prior to collapse.
It is evident that, in your analysis, you need to define a fourth structure: the damaged part of the building PRIOR to crush down. What you refer to as the "initiation zone". By setting the bottom block at 97 stories, you have included the damaged 6 stories in Part A. You should change Part A to 93 stories, add a 6 story part D, the initiation zone prior to collapse. This segment is absolutely crucial when you are discussing your most critical point: "Why did the collapse not stop immediately?"
2. Incorrect unit of failure.
The building was constructed in units of "3 columns plus spandrel" welded assemblies, with each column being 3 stories high. The unit of the core column is a single 3 story high column. When these parts failed, they failed as units, as proven by the enormous percent of intact units in the debris pile.
The correct "unit of failure" is therefore the column, not the "floor".
3. Incorrect failure mode.
DURING THE CRUSH (i.e., ignoring for the moment the collapse initiation), the principle failure mode is not buckled columns. It is fractured joints. Due to the relative frailty of the bolts and fillet welds connecting the columns to the trusses compared to the enormous strength of the support columns, when a column failed, it detached from the lattice as one or more units, shearing all of its retaining bolts and welds.
If the failure mode had been buckled beams, a large amount of plastic deformation would have been evident in the parts in the rubble. This deformation was rarely seen. Without evidence of numerous bent & twisted columns, column buckling is untenable as a principle failure mode.
Therefore, the calculation of the strain energy capacity of the building's support columns is irrelevant, except as an absurdly overestimated "upper bound". The important calculation is the strain energy capacity of the parts that are going to fail: the connecting bolts & weld joints.
4. Ignoring column stagger
Because these units are 3 stories high, and because of the 1 story vertical stagger of adjacent units, a failure of any column on any given story will also compromise (i.e., destroy) the connections of that column's support 2 stories above and 2 stories below.
4a. The effect of column stagger on collapse initiation
In the case of the north tower, collapse appears to initiate at floor 95. But, as noted above, the columns that fail on floor 95 result in 2/3rds of all support lost for floors 96 & 94, plus 1/3rd of all support lost for floors 97 & 93. This means that, by the time the upper block descends to the 94th story (the one which you assert should have stopped the descent), fully 2/3rds of that story's supports have ALREADY failed. When those support connections fail, the entire lattice support structure on the 96th story has become massively unstable. Those floors are also in the zones that have had fires, and its columns are weak and have been subject to massive bending loads & creep.
Bazant et al. calculated that the strain energy of the columns on that floor would not be able to withstand the dynamic load of the upper block after it had descended 0.5 meters (IIRC). This was based on the strain energy of all the columns. There is no possible way that the enormously smaller strain energy of the BOLTS & WELDS of the surviving 1/3rd of the support lattice will absorb that same energy.
4b. Ignoring effect of columns stagger on collapse progression
This "unzipping" phenomenon applied to every story all the way to the ground. By the time the upper block reached any given floor, fully 2/3rds of its supports had already been destroyed.
5. Collapse
In your analogies (i.e., balls, sponges, etc), you frequently model the EXTREMELY nonhomogeneous lattice structure of the towers as a homogeneous solid. Widely spaced lattice structures never behave as homogeneous materials.
You similarly ignore the principle reason that the building collapses at the top floor and not someplace well below it: the extreme lateral forces that are impressed on all the components at the crush interface. These forces are translated into nearly vertical loads by the time they have been transmitted more than about 3 stories below the crush zone. These vertical loads are well tolerated compared to the lateral loads at the crush interface.
6. Strange conversion
You state: "Thus, when the roof line has dropped 35 m, 12.94 storeys ..." This conversion results in 2.7 meters/story (8.9' / story). This is far too small. On average, the 110 story (1368' / 417 meter) tall buildings had 12.4 feet/story, or 3.8 meters/story. By this measure, 35 meters would be 9.2 stories, not 12.94. My question is "Which number is correct? 35 meters? Or 12.94 stories?" (Note that Chandler's video tracks the roofline thru "approximately 32 meters, or 8 stories.")
7. Details of collapse forces
I will accept Chandler's calculation that the roofline falls at an approximate constant acceleration of ~0.64 g thru approximately 8 stories.
There are three principle forces that are slowing down the upper block's downward acceleration: air back pressure, the momentum transfer from the stationary floor to the upper block and the structural resistive forces of the lower segment.
As the upper block descends, the air pressure is going to increase as the speed of descent increases. The deceleration (and apparent force - actually impulse) due to momentum transfer is going to decrease, because the constant mass of each floor will be a decreasing percent of the momentum of the upper block's increasing weight and velocity. The structural resistive forces will stay approximately constant, because the number & size of the joints that must fail remains nearly constant.
The fact that the NET acceleration of the roofline stays approximately constant over the 8 stories of fall shows that the first & second effects approximately cancel each other out.
"The first crush"
Your paper claims that the upper block weighs 54,000 tons. I will accept that number.
When the 97th floor fails, its 3700 tons of rubble fall onto the 96th floor. You use the rated load of the 96th floor (918 Kg/m^2) to assert that the 96th floor ought to be able to handle this load.
Perhaps if it were placed as a static load. Perhaps if all the structural supports for the 96th floor were still intact. Perhaps if that debris had not arrived accompanied by approximately 98 massive "spears" (i.e., fractured, disconnected columns), driven by 54,000 tons of upper structure.
But as we've shown, none of these assumptions are true.
The rubble from the 97th floor has fallen approximately 3.6 meters producing a dynamic load. Approximately 2/3rds of the 96th floors supports have already been destroyed. And finally, descending with the debris are approximately 98 spears, approximately half of which will puncture the 96th story's concrete floor. Given these conditions, it is doubtful in the extreme that the floor will be able to withstand this first impact.
But let's be generous and assume that it does.
A few tenths of a second later, the upper block will arrive at the 96th floor with an another structural blow & an additional 3700 tons of debris from the 98th floor that has now fallen 7.6 meters at 0.64Gs.
Very quickly (by either the first or second story's impact), the connections of the 96th floor are going to give way.
8. Crush up is not symmetrical to crush down.
The dynamic force associated with 3700 tons having fallen 3.8 meters at .64G is applied to the 96th story's concrete floor, but NOT applied to the 98th.
Notice that ONLY the upper block packs in with debris, since it is descending faster than the stationary upper floor of the lower segment of the building and therefore gathers up the debris. The open spaces in the upper block are going to pack in with debris, and then present a nearly solid lower boundary to each successive story on the lower segment.
And, again, as the upper block hits each upper floor on the lower segment, fully 2/3rds of that floor's supports will already be destroyed.
Under these circumstances, a progressive failure to the street level IS inevitable.
Re: tk's comment that: "There is no mechanical theory that says that a smaller segment can never cause a collapse in a larger lower structure"...
Heiwa replies: topic is dropping a part C of a structure on another part A of similar structure. In the latter case part A will neither collapse nor be crushed down. I describe why in my articles.
Sorry, you have not described, proven or convincingly argued anything of the sort in your article. You have simply asserted it. And built your assertion on a foundation of mostly incorrect assumptions.
I'll await your response to this before getting into more details regarding those assumptions.
tk
