True in principle, but IMO unlikely to make a difference. We are, in effect, looking at the moment whan all the KE is absorbed by the two girders, which will be a local maximum of force and deflection. If that local maximum hasn't broken anything yet (with the connection being the obvious candidate to fail first), the girders will both recoil/unload elastically - and that goes to their other ends, too. Elastic waves will travel into and through the two columns and then mostly away. That's a real energy loss to our local system. If oszillations build up beyond the original amplitude, they better do so on the first cycle, or one of the first few. It seems very unlikely that this would actually happen from a singel fall-and-impact event.

Aren't we actually looking at how long that moment is, and finding a peak value? Also still interested to know if your 16X 2nd row value in your pdf table came from Nordenson or from a simply supported beam figure, or both maybe?

The answer is NO. Because the building failed internally first, and suffered a concealed failure of some sort, pseudoscientists like Tzamboti* feel free to theorise upon what couldn't have been seen, doing their best to imply, without proof of any sort, that the building was intentionally and malevolently brought down. (This is a fate i dearly wish I could bring upon him, of course.) His technique, the same that is so earnestly followed by creationists, is to theorise upon a specific and unobservable event, to prove it impossible to occur by natural means. He always tries to simplify to a simple event which he can mathematically prove to be impossible. But the building failed as a whole. There is no way he could ever deal with it, and his doubters are confronted with the same problem. Who knows the amount and direction of all the forces released upon the first failure within the building? It might have been an exploding fire extinguisher that momentarily overloaded a floor with a small overpressure. Or anything else. The forces could, and probably did, race throughout the whole building before creating the next event. No-one can gainsay his action (save to check his "calculations"). No-one ever will. It's rather like trying to be specific about the moment Life began upon the Earth. Just don't go there in the first place. You lend him a credibility he really doesn't deserve. * a worshipper of the "God of the Gaps". And a waster of his own existence.

It's Nordenson who is theorising. Tony is questioning it, and doing a fine job of that in my opinion. I am not engaging in the rest of your diatribe. Not useful and doesn't lend to the discussion, which if I may say, is one of the better and more illuminating exchanges that I have seen. Compliments to those on both sides for not allowing this to sink to the level displayed in the above quote.

Comments noted. I explained my position with explicit reasoning which you do not address. I'm well aware of the difference between series and parallel - AND how they relate to the specific scenario(s) under discussion . So I wont attempt to address your disagreement until and unless you post reasoning other than maths supporting it. Maths is not of itself a proof - maths will faithfully follow the model you apply the maths to. Whether right or wrong. I would be interested in your model and any reasoned argument you can present that shows it is better or more correct than mine. However I see that you agree with my conclusions: so I'll leave the point of disagreement on the back burner for now. Experience tells me that you and I will move towards agreement over time - if the issue is of any consequence:

Neither should have "theorised" anything. A collapse is chaotic, and by definition cannot be summarised accurately. It is certain that the analysis of any specific localised instance within the collapse will be fiction in some unknown respect. What we DO know is that the STEEL building was subjected to peripatetic fire for seven hours, two hours after the firemen knew it was about to collapse, as they had seen it progressively deform and sag. They were correct. Anything else is superfluous and more than that, fiction. Engineering is the science of the possible - and the REAL. NIST went exactly as far as it was possible to go in their analysis of WTC7's collapse. I like science fiction - but not fantasy.

Agreed .. but.... Agreed also - I served my apprenticeship moderating on a forum oriented towards evolutionary biology. The truthers <> creationists analogue is spot on in my experience. I've often recommended to truthers - including Tony - who call for a new investigation with para judicial powers that they read the transcript of Kitzmiller v Dover and what happened to Michael Behe. However on this occasion the para-"creationist" happens to be correct on one point. And that is a rare occurrence. And fun explaining. And I've even got a debunker or three disagreeing with me. Of course it is all irrelevant in the bigger picture. So what if Nordenson was wrong on one point of a paper? "Proving" that one - or a dozen - academics were wrong on a detail does not have the slightest effect on the true situation as you clearly recognise.

True enough as generic postulations BUT remember we are analysing a specific situation and considering one of two scenarios. The scenarios are "failure occurs" or "failure does not occur". If "failure occurs" it will occur on the first part of the first cycle - oscillation will never arise. The reasoning simple - as the impact occurs the force rises from zero towards a peak. If it reaches failure level failure occurs. Before the FIRST peak. (Or extremely improbably exactly at the peak - which is still failure.) So the stage of going past the peak and rebounding - the first cycle of oscillation - is never reached. By definition - if it doesn't fail in that first 1/4 of the first cycle of potential oscillation it will not fail on any second or later cycle. And that is why it is a parallel situation at that stage of the scenario - some energy down a dead end track which is exactly what Tony has postulated - some energy does NOT reach the lower beam - at that critical stage. Only if you define the scenario as per the reference Oystein gave me to the Wikipedia definition. i.e. you have "isolation". I defined the specific WTC 7 situation I was explaining - NOT force fitting a generic and inapplicable Wikipedia model. The objective - my objective - apply basic physics to the specific situation. This specific WTC 7 system as I have explained it never reaches oscillation in the scenario that leads to failure and as per my explanation the energy has two paths - one of them leads nowhere. So what happens to that "stored energy" IF there is no failure is a more complex issue. Which is why I avoided taking that path to prove my point. The more complicated scenario is IF there is no failure THEN the total two beams system would undergo oscillations - damped oscillations - as the system reverts to a static state. I didn't need to go there so I chose not to.

My major point was that seeing application of series springs in this arrangement doesn't work for me As for oscillations, yes highly damped, especially given the immediately following falling mass of floor and office contents.. However, OK let's grant the idea that the initial contact impulse is insufficient to fail the lower floor/girder. That lower girder is now subjected to the upper floor plus contents dumping onto it. Let's also assume the girder to be at 500 F. Does it fail now? This goes to Jazzy's point that this one simple mechanism of dropping girder is only part of the story. In the forensic analysis if such a simple major contributor to forces on a floor calculate out to many times what is sufficient to cause failure , there is no need to get more detailed. That, IMHO, is what Nordenson was seeing. If calcs show it to be close to sufficient then there is again little impetus to get more detailed. One might want to or one can simply assume other factors contributed enough to ensure failure( this is forensics, you know it did fail) If it is not very close then the most prudent thing may be to get more detailed in the mechanisms involved. What one does not do is run to an unseen, unevidenced, all powerful and secret special machination such as explosive demolitions( or Gawd)

A few things... The 13th floor girder did not impact the 12th girder.. it impacted the slab... which likely shattered and dissipated some of the energy. We don't know the condition or degradation of the 12th floor.... Maybe the beams dropped before the girder and damaged the floor below and caused the girder to tip? These calculations are a fool's errand. The prevailing wisdom of the collapse is that col 79 lost lateral support and buckled. The truth guys pitch all the interior columns being blown over 8 stories... including the facade which traveled unimpeded for an 8 story drop. Tony is trying to claim you can't strip the floors and girders from 79 enough to cause it to buckle. The collapsing NE quadrant floors and column 79 and 80 would also undermine TT1 and TT2 and promote a westward progression of failures below floor 8. That seems a plausible scenario looking at the vids. Should the fire have burned out before causing the thing to let go? Plaintiffs claim the defects say NO...

econ, the model I used has been layed out in this thread several times. Nordenson's Figure B5.1, plus my or your addedd scribblings to show where forces occur, and how the springs bend. I explained in detail how the energy is absorbed by both girders concurrently, wrote down the correct energy balance as an equation, noted correctly that the two springs interact with equal but opposite force (my second equation), and solved the two equations for force F. The result showed that the formula for series in springs is indeed the correct one for effective stiffness. If you didn't simply SEE that the springs are in series, then the result from the model build energy "flow" clinches it. There is a topological difference between springs in series and springs in parallel: "In mechanics, two or more springs are said to be in series when they are connected end-to-end, and in parallel when they are connected side-by-side; in both cases, so as to act as a single spring."[WP] The falling girder connects to the girder below only on one end. Their other ends connect to other parts of the system. That's the topological condition of springs in series. If they were parallel, their other ends would connect to the same element. There is a difference in the strains and the stresses of springs in series and parallel: "More generally, two or more springs are in series when any external stress applied to the ensemble gets applied to each spring without change of magnitude, and the amount strain (deformation) of the ensemble is the sum of the strains of the individual springs. Conversely, they are said to be in parallel if the strain of the ensemble is their common strain, and the stress of the ensemble is the sum of their stresses." In this case, the force (stress) on the two girders is the same, but they deflect (strain) differently -> springs in series. You disagreed with yourself. You were right that E1/E2 is proportional to K2/K1 (note the subscripts inversed). This is the opposite of your "hint": "if you put the springs in series the more flexibility - less stiffness - the upper one has the MORE energy passes through to the lower beam spring" -> lower stiffness in falling beam means lower energy in falling beam (<=> more energy in lower beam). E1/E2 proportional to K1/K2 is correct for springs in parallel, which we don't have here.

By moment I mean a point in time, not an interval. The collision (strain energy increasing from 0 to its max) happens in an interval of only some fraction of a second. The moment we are looking at is the end of that interval, when KE=0 and the SEs are at their maximum. Sorry I missed your previous post. I chose 16x to get a result where F is close to Nordenson's shear capacity. I had an intuition that a factor of 2^3 or 2^4 might in fact appear somewhere, but I am far from sure on this. It's a coincidence that it's a factor of 2^4 that brings us close to shear capacity.

It looks very like Nordenson has used a simply supported beam to take his figures from then. Surely he should know better.

The contents are already included in the impact calculations (see beginning of Appendix B). If the connection doesn't fail under dynamic load, it won't fail under static load. Agreed: A result of F > capacity, as Nordenson got it, shows that cascading floor failure is probably, and given that cascading collapse did occur and must have initiated somewhere, this result supports the "girder failure" initiation hypothesis. A result of F ~= capacity means that, given the various margins of error, collapse propagation is still at least plausible or possible. No need to find a new hypothesis. On the subsequent floors, Nordenson did not consider the mass of the already collapsed floors that are now piling up. This is was VERY conservative. The critical impact is the very first. If the first connection (floor 12) fails, then all others down to the ground become increasingly likely to fail, too. Only a result of F << capacity is problematic for the "girder failure = initiation" hypothesis - in which case we might want to look for alternatives. There are several available before CD comes into play. As with NIST, proving Nordenson "wrong" doesn't make CD "right".

Explain? You are talking about the stiffness K = 7627 kips/in of the girder below? How would you advise to improve on this?

No, The bit where your pdf says "This table shows results for 5 cases â€“ the stiffness value Kf for the falling girder is varied" You take the Kf value on the second line as 59.2 Kip/in, which is 16 x 3.7 obviously. Is that not the same value as Nordenson used? Sorry if I am not following you correctly.

No, Nordenson didn't consider the stiffness of the falling girder at all. This is, as Tony put it (and the last line in PDF confirms) equivalent to infinite stiffness. The Nordenson-value that I approximately match is the shear capacity of the connection to column 79 of the girder below. Table B6.1 has the capacity in the second-to-last column: 632 kips on floor 12. My arbitrarily chosen K of 59.2 kip/in happens to result in an equivalent force F on the girder below - 639 kips - very close to capacity, and it happens to be 16x Tony's first K value for a cantilever beam. I included this mainly to show what order of magnitude would put us in "can't decide either way with great confidence" territory.

Thank you. I had presumed that Nordenson had used a simply supported beam and exaggerated the stiffness in doing so. I can see why you and Tony concurred now. Has Nordenson responded to anyone about this yet? Should be interesting, considering this was submitted to a court. I would imagine he may have to amend that in case it is ever cited.

Benthamitemetric had emailed Nordenson - we'll have to wait for her or him. In the end, the court didn't use the technical reports to support its ruling, so there doesn't seem a pressing need to submit a correction.

As the fire had been approaching from beneath one may assume it was worse. The building was bending over the gap torn out of it. Those long floors were sagging and attempting to apply bending loads to their internal columns through their fixings. Thirty-four floors up, the building would have been well off-plumb. [...] The steelwork was insulated, and would have taken longer to cool down than they took to heat up. Hours? A loaded complex steel structure on the point of failure is chaotic because steel freely transmits elastic strain energy throughout the structure it comprises. There is a "weather" of stress distribution over time and space which in any specific case can never accurately be modelled. Yet the collapse is inevitable, at some point. Mr. Tzamboti has you counting angels on a pinhead. I would like to see his load analysis of column 79, floor-by-floor. No, I wouldn't. LOL.

Tony: Where did you get I = 6,710 in^4? [ETA] Ok ok never mind, I found it, for example http://www.nucoryamato.com/staticdata/catalog.pdf page 17 [/ETA] It seems Nordenson has a different value: For the girder below, he used K = 3*E*I*L/(1 kip * a^2 * b^2) Where E = 29,000 ksi (page B32) L = 547 in (Table B5.1) a = 10 in (Table B5.1) b = 537 in (=L-a) and his result was K = 7627 kips/in Which we can rearrange the formula: I = K * (1 kip * a^2 * b^2) / (3*E*L) and insert I = 7627 kips/in * (1 kip * 537^2 * 10^2 in^4) / (3 * 29000 ksi * 547 in) = 4622 kip*in^4 Different number - and I have the unit "kip in^4" instead of "in^4" Help!? Nordenson's formula for K is wrong?! At least the one he wrote in B5.2 (page B33). Different number, different unit!?

Oystein, I = 6,710 in^4 is the correct area moment of inertia about the strong axis for the W33 x 130 shape. Nordenson has it listed for girder 44-79 at the 8th through 13th floors in his table on page 243 of Volume 2 of his report. Coincidentally, I was going to attach the Nucor-Yamato catalog for you to verify it and then saw you had a link to it. As for re-arranging Nordenson's formula of K = 3 * E * I * L / (1 kip * a^2 * b^2), to find the moment of inertia he actually used, I had questioned the use of this equation a few weeks ago in an e-mail discussion with DGM from the JREF forum. I didn't see where he defined "a" and "b". After he said they were the position of impact and the stiffness involved from each end of the beam, and that the offset from one end was in the table, I thought it seemed right. I would have to look at it again.

It is correct. I had to check the textbooks myself at the time I first gave serious attention to Nordenson - they are the offsets from either end as shown in Oystein's post.

I think the equation is correct also, but am not sure now why when Oystein tried to work backwards from Nordenson's stiffness value of 7,627 kips/inch, to get the moment of inertia (I), that it did not give him at least the value for the girder shown in the table as 6,710 in^4.

For some reason, Nordenson's stiffness formula gives his stiffness value of 7627 kip/in for a=bit more than 12 inches, where his table lists 10 inches. But it turns out this doesn't really matter much. When the falling girder is significantly less stiff, the resulting force is dominated by it, and you could vary the stiffness of the girder below through a very large range without altering significantly the result we're after.

I think the difference in length may have something to do with the angle. He says the impact is 10 inches to the east but on the girder that might be 12 inches along its longitudinal center. It doesn't matter much, relative to what we are trying to discern, for the reasons you mention.

Alright, folks, I think I got it: K(falling girder) = 81/2 E*I/L^3 = 48.15 kip/in I opened a thread in the Science&Technology section of ISF, hoping to attract the attention of some engineering type of person not usually lurking "9/11 CTs". Didn't really find any and helped myself, and here is my derivation: http://www.internationalskeptics.com/forums/showthread.php?postid=11088153#post11088153 (and following post). Can you look at this and check whether I make sense and did my multisubplitractions right? With that value, I get K(eff) = 47.85 kip/in and equivalent Force F = 576.5 kip. That's within 10% of Nordenson's connection capacity (632 kip)! In other words: Deep in "maybe" territory. Deflection is in the vicinity of 12 inches (formula for max. deflection of an unsymmetrically loaded beam is a bit complicated, haven't bothered yet), so average deflection would be anywhere between 4 and 12 inches, adding a bit of PE to the 46 kips * 83 in = 3818 kip*in gross PE Nordensen started out with. Also, being so close to the break point, we'd now might feel compelled to scrutinze all the values we have so far, consider that heat may ir may not have decreased shear capacity (how hot did the connection get in the various fire models?), that heat probably decreased girder stiffness (how hot did it get, on average, in the ARUP and NIST models?), and think hard about possible further energy sinks and whether or not they are significant. My hunch is that all these effects are small in magnitude (each affects the effective force by no more than 10% - order of magnutide - and some may cancel out). The only variable that I don't have a hunch about is the connection temperature (and whether or not the connection on floor 12 was already damaged somewhat). The girders got hot, the column did not - and the connection?

Well done sir. Now for the record on a recent side track: econ41 concedes a point - Oystein was right - you are right that the two springs are in series. - I was trapped by my insistence on "understand the model before you apply the maths" The credit/spoils go to the victor - The finesse of the distinction will remain lost here so no point me pressing it by trying to explain. -- unless somewhere in the future discussions someone needs to explain the issue rather than simply take it as "given" which was the basis of my concern. And it was only a side track - my ball park "engineer's guesses" supported by near zero maths now seem to be confirmed by your rigorous mathematics. Sorry - not my scene on checking DETAILED maths. At first glance I don't see any issues of concern with the "structure" of your analysis - within the order of accuracy/magnitude you are identifying. Which leads to the questions of accuracy you now raise: May I suggest that you revisit and focus the objective. It seems that Tony has been vindicated on identifying one factor where Nordenson's seems to have erred. The OP question is "does it demonstrate anything?" Anything about what? What scope? Do you want to show the effect within the scope of Nordenson's paper? OR in the broader context of the real event? Many questions being raised by other members here and elsewhere go outside the Nordenson scenario and its many simplifying and conservative assumptions. And if "mission creep" means going beyond Nordenson to the full situation of the real event - there will be a lot of Nordenson's assumptions which are implicit in what has been done so far which will need to be addressed. Significantly complicating the discussion. Sure - that has been the risk from the outset. Way back at the beginning of these discussions is was plausible that the flexibility of "falling spring" was a minor factor. We now know differently. But that was only one of the "other factors". Hence my comment - what is the objective? If "mission creep" says it becomes a full review of the whole relevant stage of the real event - it will be a big job. (That should qualify as "understatement of the week" ) If we stay with the OP then there are two areas of possible consequence for the Nordenson paper: A) "Technical" - so we have yet another paper with technical errors. Arguably there are more published papers with one or two errors than there are papers with no significant errors. ( And I dare not mention Bazant et various als.) The technical "bottom line" is IMO "So what?" Is the paper taken to be "published"? If so a response by counter assertions in another paper? Why? B) "Para legal" - the paper was produced to support a legal action which is concluded. Whether or not it was considered in evidence that door is closed. And no known appeals - and there is no way I can see that an error in technical evidence which was not relied on in the decision could get into an appeal. .

The equation you are using for stiffness (40.5EI/L^3) is very near that of a simply supported beam (where the stiffness is 48EI/L^3) and that would hardly seem likely with one end of girder 44-79 hanging out in space. I finished and ran my finite element analysis and it shows the stiffness of the assembly of the five beams attached at the east wall, three support beams attached at the north wall, with the girder just sitting unrestrained on its seat, has a natural frequency of 0.52 Hz. Fn = 1/2pi * SQRT(K/m) Stiffness (K) can then be found with the equation K = (Fn * 2pi)^2 * m The weight of the beams and girder is just over 20,000 lbs., so mass = 20,000/32.174 = 622 slugs so K = (0.52 * 6.28)^2 * 622 = 6,633 lbs./inch Using Nordenson's potential energy of 3,473,000 in-lbs. and the same standard equation he uses to find deflection P.E. = 1/2K *D^2 D = SQRT(2*P.E./K) = 32.4 inches Now using the standard equation Nordenson uses to find force F = K * D = 6,633 lbs./inch * 32.4 inches = 214,909 lbs. This is nearly three times less than the 632,000 lb. shear capacity of the seat and proves that the falling girder would not have sheared the seat and the northeast corner of floor 12 would not have collapsed if a girder at floor 13 came off its seat at column 79.

@ econ, I think I want to stick with the original topic of the thread - checking the validity of Nordenson's Appendix B, or even just a section thereof. It's interesting because the technical reports for the Aegis case have just been made available to us by benthamitemetric, and Tony did find a fault. Reminds us to remain skeptical and all. I want to reserve judgement until there is a bit more agreement on my results.

If the falling girder can't knock the one below loose... there is no progressive collapse from the falling 44-79 girder. So it seems to moot the whole discussion of whether the 44-79 moved off its seat from heat.

That indeed can be scrutinized to the same level of detail ETA: BUT remember how Mick West ticks and what the rules are here: No general discussions, stick closely to the topic and the single claim made in the OP.

Attached is a pdf with views of the FEA results. In case you are wondering the information on the upper left says 5.1693e-01 Hz, which I rounded up to 0.52 Hz. The vertical mode was the second mode. The first mode was side to side and it was 2.2661e-01, which would be 0.23 Hz, but isn't germane here.