In the BZ analysis of their scenario. The scenario is an axial impact. What I'm pointing out is that you haven't presented any other impact scenario which is more favourable to collapse arrest. You're disputing the analysis of the response of the structure to the BZ scenario, which is a reasonable line of enquiry. I'm simply asking that you stop representing it as something that it isn't.
So you are saying that the "scenario" is that of an axial impact, rather than that of an axial impact where the top is modeled as a rigid body (amongst many other assumptions, explicit and implicit). Is this correct?
If this is what you mean, are you using the word "scenario" in some precise manner that physicists would understand? Or are you essentially making much ado about nothing, by quibbling about a word whose English-language ambiguity allows one leeway to make interpretations according to one's desire?
When
I use the word "scenario", I am talking not just about a broad-brush stroke description, such as "top hits bottom, axially". I am talking about the specifics that BZ actually assumed in their analysis, such as "top was a rigid body". If I am using the word incorrectly, in a manner that physicists would agree is incorrect, please explain fully what the scientifically correct usage of the word "scenario" is.
I'll leave it to others with more engineering expertise to discuss (1) and (2), but would point out that failure of the structure at any weak point would prevent any transfer of elastic strain energy beyond the point of failure.
The word "failure" here is problematic. Even if the elastic limit is surpassed, complete structural failure need not follow. See, e.g., the work of Ari-Gur, et. al. Plastic flow begins after a certain lag time, and propagates at about 500 m/s in steel. Whether this leads to complete structural failure would depend on the details of the problem.
However, your reason (3) is invalid.
Kinetic energy of the lower block is not an energy sink if the collapse is arrested.
Well, it's an energy sink in terms of energy available to cause buckling in the the topmost story of the WTC base.
The movement downwards of the lower block has to stop for collapse to arrest, and this is achieved by converting the kinetic energy into strain energy. It's a two-stage process, in which initially the kinetic energy of the upper block is transferred into kinetic energy in the lower block, and subsequently the kinetic energy of both is transferred into strain energy of the spring.
No, this is too simple. The kinetic energy of the upper block get converted to both strain energy of the upper and lower blocks, as well as kinetic energy of the lower block. Even this is too simple a description, as the kinetic energy of "the" lower block is a function of time and space. Parts of it can be in motion, while other parts of it can be at rest. (This is not obvious from just looking at Goldsmith (c), but you can grok it by looking at Goldsmith (a)). Furthermore, as you can see in (c), at t = 5 L1 / 2C0, the velocity of the rod fixed at one end has reversed.
The first instance shown (t = L1/2c0) when velocity of the impacted rod is zero, has a stress well below its maximum stress (at t = 2L1/c0). At this time of maximum stress in the impacted rod, the velocity is also zero, but notice that this stress is less than the maximum stress in the impacting rod. Also notice that at this time, the impacting rod has reversed direction. It has kinetic energy, but not the sort that's going to contribute to peak stress in the other rod during this phase of it's motion
In such a collision, if the yield stress is achieved, it will occur in the impacting rod, not the impacted rod (we're only shown graphs for a few values of t, so maybe I'm drawing the wrong conclusion, but I doubt it.)
So, strictly speaking, you can say that for a sub time-interval of the collision, my complaint about BZ ignoring the energy sink represented by the kinetic energy of the base is invalid, but you cannot say that this is generally the case throughout the collision. The situation is dynamic and somewhat complicated, and ultimately you have to rely on mathematical descriptions to pin down peak stress. In fact, Kinetic energy of the top in a gravity driven collapse will be converted back and forth between kinetic energy and elastic strain energy of the top and bottom. In terms of comparing to BZ, the point of computing this is to determine the peak stress, at any point of the base, at any time.
Therefore, there is no need to consider kinetic energy of the lower block in determining whether the collapse is arrested; by definition, if energy is transferred into kinetic energy of the lower block and is not converted to some other form, then the collapse has not been arrested.
See above.
It's also worth noting that the boundary conditions for the upper and lower block are very different, in that the upper block is free to move at the top end whereas the lower is fixed at the bottom end. Force on the upper block can therefore potentially decelerate it, whereas any kinetic energy imparted to the lower block must then be entirely transferred into strain energy. Overall, it can't be valid to treat the two blocks as having identical responses.
Dave
Goldsmith's (c) scenario is for the problem of the impacted rod being fixed at one end. Perhaps you're looking at (a), instead?
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BTW, in (c), I just noticed that at t = 3L1/2c0, the non-zero velocity (= - 1/2 v1,0) is not graphed. Also, the velocity plot for t = 5L1/2c0 doesn't match the velocity drawings at the top, either.
And I also see what Heiwa was trying to explain to the children
