Interesting contraption!
One horizontal element (a tray) - upper part C - shall one-way crush down 9 other trays connected to columns that in turn are connected some way, so that they also will break sequentially - the latter assembly is part A.
That is essentially correct.
Well, as part Co does not have any columns suggest an improvement.
Please be clear. Is this request for an "improvement" a suggestion based on your own personal preferences for what you'd like to see, or is it a requirement for winning your million dollar challenge?
It's your million dollars and of course you can impose any requirements you like. However, I will point out that no such requirement, that part A (or for that matter any part of the model structure) include columns and multiple floors, is stated in the terms listed in your Heiwa Challenge thread. Nor is any requirement stated that the crush-down must be one-way.
Similarly, the so-called Bjorkman Axiom states:
You cannot crush an isotropic or composite structure A by a part C of itself (C = 1/10 A) by dropping part C on A using gravity. Part C either bounces on A or gets damaged in contact with A and is stopped by A that is also damaged a little. It is quite basic and all due to forces. Materials and particulars of the elements of the structure A doesn't matter the least.
(Emphasis added.)
There is no mention that this axiom applies
only if part A (or indeed any part of the structure) includes columns. In fact quite the contrary. The bolded portion emphasizes that it applies to all structures however constructed, and the first sentence is specifically inclusive of isotropic structures. So, neither multiple floors in part C nor one-way crush-down is required to disprove, by counterexample, the stated claim.
I'll also point out that you yourself have offered models, in support of your claims, that do not include columns at all.
Part C to consist of two trays connected to columns as in part A. In order for part A to be 10 times part C, part C to consist of 20 trays.
So we have a tower of 22 trays (you can keep total height same as before).
I've explained the importance of scale in progressive collapse. You've just provided a good example of that importance.
Let's suppose I keep the height and overall construction methods the same but double the number trays.
If I kept the overall mass the same, each tray would be about half as heavy. The total gpe of the structure would remain approximately the same. The average load on each connector would remain approximately the same. But the number of connectors that must break for the structure to collapse would have doubled. That's a big change to the energy budget in the collapse dynamics.
If I doubled the number of trays while keeping the trays the same as in the original design, then the gpe would also be doubled, but the energy to break the connectors would be quadrupled, because there be twice as many connectors, and each one of them on average would have to be twice as strong as before to support the structure with the same safety margin. That's also a big change to the energy budget in the collapse dynamics.
If the World Trade Center were the same height but had had 650 floors (about 20" clearance between floors, just as you're suggesting for my demonstration), it might very well not have collapsed on 9/11. But it also wouldn't have been very useful.
So, unless you can state a reason why you think the number of floors should be doubled, or you're modifying your "axiom" or adding new previously-unstated conditions to your challenges, I will choose not to follow that suggestion.
However, if you think it's important, I will modify the design so that 1 tray is dropped on 10 trays instead of 1 onto 9. I've been interpreting the wording of your axiom and challenges as referring to 1/10th of the whole crushing the other 9/10ths of the whole. Perhaps because that's what you've said:
5. Before test 1/10th of the structure is disconnected at the top at h = 0.9 h without damaging the structure.
6. The lower structure, 0.9 h high is then called part A. The top part, 0.1 h high, is called part C.
7. Mass of part C should be <1/9th of mass of part A.
If you now insist that it only applies to 1/11th of the whole crushing the other 10/11ths of the whole, then I'll make that change in the design. (I am curious, though, why this distinction would be that important. Are you saying a wtc tower should not have collapsed if the upper block was 1/11th of the whole, but you
would expect it to collapse if the upper block was an entire 1/10th of the whole? Does that mean you accept that the "official theory" of progressive collapse accepted by structural engineers is sufficient to explain the collapse of the south tower, and only the north tower collapse is suspicious?)
Before drop C on A test, the model shall pass the lateral test, i.e. a force is applied sideways on C and the structure will deflect accordingly without falling apart.
Please review the parameters for the lateral test that are presented in the abstract, and tell me whether they are acceptable for your challenge. Or better yet, state the allowable range of parameters.
Is there a reason the sideways force should be on C, rather than on the interface between C and A as I've proposed?
What I've proposed is more than a small nudge; it's actually a rather large lateral impact. Perhaps you might be able to guess how I arrived at the values I proposed? (Hint: individually the weight and velocity I proposed have no particular significance. But if you multiply them together and divide them by another meaningful physical quantity...)
A smaller impact would reduce expenses, so please let me know what the acceptable envelope is. You specified a "small" lateral impact which makes it sound like my proposed impact is much larger than is required, especially since my model is considerably slenderer than the wtc towers were.
Then the columns between part C and A to be cut, so that part C (two trays + columns) can be dropped on part A.
Suggest the cuts are at different angles and heights!
Part A top will thus consist of a tray with column stubs extending above the tray and part C will consist of a bottom tray with column stubs extending below the tray.
Thus, at impact interface area C/A the structures are equal.
What is the rationale for this change? The structures are already equal at impact interface in the current design, since there are no columns inside the footprint of any floor. Again, neither your challenge nor your axiom mentions column stubs as a
necessary condition for collapse arrest. And again I can point out that you have proposed and discussed several models of your own, all of them conspicuously lacking in column stubs.
As I said, it's your challenge and once you put up the million dollar stake you can specify any conditions you want when it comes to negotiating the formal challenge agreement. You can demand that the floors be coated with marmalade and supported by rubber chickens if that's your thing. However, adding major new conditions such as one-way crushing or angled column stubs as requirements after reviewing the plan wouldn't speak too well for your confidence that the broad claims you've made will stand up to fair testing. (If these are just suggestions, then that's not an issue. Hence, my request that you be clear on the distinction in the future.)
In any case, I should mention that cutting through the columns between C and A prior to the drop is not acceptable for safety reasons, as it would expose the personnel doing the cutting to undue risks to life and limb.
Yeah, that's the general idea.
The drop should be so that column stubs contact column stubs and as the stubs are angled, parts C and A will be given a little sideways motion - a tilt. This will then enable the column stubs to contact the top A tray and the bottom C tray, when C displaces further downwards.
Thus the part A columns will contact the part C bottom tray from below, etc.
Note that part C bottom tray will not contact part A top tray at contact. It is the part C column stubs that contact part A top tray.
And then you'll see what happens! No one-way crush down will take place.
The above appears to be a prediction of what will happen, when the whole point of the exercise is to observe what actually does happen.
Unless you're saying I must manipulate the drop (e.g. with careful placement of carefully angled column stubs) to make sure that what you predict should happen actually does happen? That would make little sense, since the challenge you've issued is to show otherwise.
Now I'm off on vaccation (or work).
Telling the difference is usually pretty easy. (And for the tough cases, the IRS has pretty good methods for figuring out which is work and which is vacation. But since you're not a U.S. citizen, I suppose that valuable service is not available to you.)
Now, here are some specific questions about the proposal that I'd like you to answer:
1. Are the guy ropes as described in the proposed design acceptable? Note that the proposed design, to keep costs reasonable, is considerably slenderer than the actual wtc towers were, so for safety the possibility of topping from near the base must be ruled out. Your rationale for collapse arrest does not involve toppling of part A, radical rotation of part A, or major horizontal displacement of part A, which the guy ropes would prevent from unexpectedly happening. (Part C is not attached to any guy ropes so it is completely free to tilt, rotate, bounce, or whatever the laws of physics determine.)
2. Are the nailed connections acceptable? Technically, nails (and most other construction fasteners other than welds or adhesives) attach things together by friction. Please clearly state whether or not the nailed connections in the proposal are acceptable under term 4 of the original challenge. (Note that in this design the nails are supporting shear loads as is typical of real-world nailed construction, and that nail failure is not expected to contribute to collapse in the design of the demonstration.)
3. How shall we proceed into the negotiation of a formal agreement between us regarding the challenge terms? I've made a reasonable start by describing the proposed demonstration in considerable detail. But it appears that you do have some doubts or objections regarding the details. I think the best way to proceed would be for you to begin a first draft of the contract, using what I've proposed as the guideline, and that as you do so you communicate with me to make clear any specific objections or additional requirements you are adding, so that we can negotiate precise final terms regarding what is and is not permissible in the demonstration.
4. Just to repeat from earlier, please clarify the acceptable parameter envelope for the side impact test, keeping in mind that the stated purpose of that test is to ensure that the structure is honestly self-supporting and that connections are fastened, not to prove that the structure is strong enough for any particular purpose.
I hope you have a pleasant (or profitable) vacation (or job). If continuing this discussion during that period is not practical, then I look forward to completing the necessary preliminary arrangements when you return.
Respectfully,
Myriad