WTC: Rate of Fall (rate of crush)

[OneWhiteEye]

The columns would in reality go through three phases of distortion when buckling. As for Euler, that is only really applicable to the most slender of steel elements. Have you actually read the paper that Tony refers to? I did and I think it brings up many salient points.

Which paper is that?

 

[gerrycan]

Which paper is that?

I guess i will let Tony speak to that one. I'll stick to the root and branch stuff.

 

[OneWhiteEye]

I guess i will let Tony speak to that one. I'll stick to the root and branch stuff.

You said:

Have you actually read the paper that Tony refers to? I did and I think it brings up many salient points.

Surely you can tell me which paper you're referring to.

 

[gerrycan]

You said:



Surely you can tell me which paper you're referring to.

I don't have a link to it and I am not sure if it's public.

 

[Mick West]

I don't have a link to it and I am not sure if it's public.

Does it have a title?

 

[Tony Szamboti]

I believe gerrycan is referring to elastic and plastic axial deformation and then plastic hinging as the three energy sinks involved in the buckling process. This is mentioned in the Missing Jolt paper and in the January 2011 Le and Bazant paper published by the Journal of Engineering Mechanics.

 

[Jazzy]

I believe gerrycan is referring to elastic and plastic axial deformation and then plastic hinging as the three energy sinks involved in the buckling process. This is mentioned in the Missing Jolt paper and in the January 2011 Le and Bazant paper published by the Journal of Engineering Mechanics.

Well, that's a mistake for a start.

Elastic axial deformation (and bending for that matter) is NOT an immediate energy sink at all.

The energy could have been (and would have been) just as easily handed back and forth between the fixed and moving structures.

The only energy loss from this "sink" will be the production of sound. And not much energy is required to produce a large amount of noise, as you know only too well.

 

[Tony Szamboti]

Well, that's a mistake for a start.

Elastic axial deformation (and bending for that matter) is NOT an immediate energy sink at all.

The energy could have been (and would have been) just as easily handed back and forth between the fixed and moving structures.

The only energy loss from this "sink" will be the production of sound. And not much energy is required to produce a large amount of noise, as you know only too well.

If there was only elastic deformation you would have a point.

However, in this case the initial elastic deformation of the columns is not given back with regard to its effect on deceleration of the upper section, as the column deformation goes beyond it into the plastic range and then into buckling.

 

[Jazzy]

You are taking things out of context. The initial elastic deformation of the columns is not given back with regard to its effect on deceleration of the upper section as the column deformation goes beyond it into the plastic range and then into buckling.

No.

Elastic deformation is always given back, except for the noise it makes. You cannot count it twice.

How much work is done in a helical car spring?
.

 

[Tony Szamboti]

No.

Elastic deformation is always given back, except for the noise it makes. You cannot count it twice.

How much work is done in a helical car spring?
.

If a helical spring is compressed to the point where it yields and fails by an object compressing it, the energy drain on that object is comprised of the elastic and plastic strain energies of the helical spring. It can be calculated from the area under the stress strain curve.

The elastic energy is what was necessary to get to the plastic region and it is not given back once that point has been reached. The energy counted as plastic after the elastic limit is reached is the amount needed to continue the plastic deformation.

 

[Jazzy]

If a helical spring is compressed to the point where it yields and fails by an object compressing it, the energy drain on that object is comprised of the elastic and plastic strain energies of the helical spring. It can be calculated from the area under the stress strain curve.

But not all of the material was going to be made to yield that way. Nor would it ever be forced to yield in its entirety, and certainly it would not necessarily yield immediately.

The elastic energy is what was necessary to get to the plastic region and it is not given back once that point has been reached. The energy counted as plastic after the elastic limit is reached is the amount needed to continue the plastic deformation.

Yes, we know it isn't given back when the yield point is reached. That is beside the point.

Strain would have tended to prevent the onset of motion, certainly. But once the motion was underway, not all of the residual strain energy would have been available to arrest that motion by any means.

One only has to consider what happened next when the top mass was finally arrested by ground zero: it bounced back aways. What made it do that?

But of course you cannot allow yourself the liberty of such thought experiments, and in your scenario an elastic steel structure must merely stop…

Your account is producing a list of absurdities.

 

[OneWhiteEye]

I guess i will let Tony speak to that one. I'll stick to the root and branch stuff.

I believe gerrycan is referring to elastic and plastic axial deformation and then plastic hinging as the three energy sinks involved in the buckling process. This is mentioned in the Missing Jolt paper and in the January 2011 Le and Bazant paper published by the Journal of Engineering Mechanics.

Satisfied with that answer, gerrycan?

 

[Tony Szamboti]

But not all of the material was going to be made to yield that way. Nor would it ever be forced to yield in its entirety, and certainly it would not necessarily yield immediately.


Yes, we know it isn't given back when the yield point is reached. That is beside the point.

Strain would have tended to prevent the onset of motion, certainly. But once the motion was underway, not all of the residual strain energy would have been available to arrest that motion by any means.

One only has to consider what happened next when the top mass was finally arrested by ground zero: it bounced back aways. What made it do that?

But of course you cannot allow yourself the liberty of such thought experiments, and in your scenario an elastic steel structure must merely stop…

Your account is producing a list of absurdities.

It is hard to know what to even make of this comment.

 

[Jazzy]

(Szamboti's) response has been a variety of:
1) - outright denial
2) - simply ignoring points and entire posts
3) - hiding behind anonymity of a source to automatically discredit their argument
4) - disappearing from the discussion permanently (only to appear somewhere else starting from the beginning)

It is hard to know what to even make of this comment.

That seems like a mixture of No.1 and No.2 to me.

[...]

 

[Mick West]

Chaps, let's leave the snide comments aside, and focus on claims, facts, science, maths, etc.

 

[OneWhiteEye]

Jazzy, I thought long and hard about this before finally relenting. It got especially hard when you quoted me on the four Szambotian pillars, but...

Tony is right on this matter. There, I said it. I've been extremely reluctant because you have been making some stellar arguments in this thread, and because you've been very supportive of my arguments, and because Tony needs to fight his own battles, and because I'll be damned if I'm going to assist him.

Elastic deformation is unrecoverable for column "snap-through" to full compaction, which is the basis of Tony's three-hinge calculations. That's not to say all stored strain energy is tossed to the aethers in the collapse, no, but elastic may as well be plastic for his model. And there's no problem there, basically SOP and theoretically correct. It's the application of the model itself which is the problem.

The tipping point was Tony's non-response to you. If you want an explanation/argument, I'll do that even if he doesn't.

 

[OneWhiteEye]

Jazzy, I do wish to be clear that your instincts are good and this does merit explanation. Even if a column is fully squashed, there will be elastic rebound when released, no doubt (see here). I'd ask you to consider several things which take this outside the domain of most everyday engineering.

With ductile materials, the energy consumed in severe plastic deformation can dwarf that of elastic deformation. A short displacement elastic response followed by large-scale plastic displacements is what happens with steel members. If the model calls for severe deformations, in many cases the elastic phase can be considered negligible and if so there's no harm in just throwing it away right off the top for energy purposes.

To illustrate this, I offer a digitization of the Szamboti-Johns load displacement curve...



... where I didn't bother with the initial elastic phase because it was too small to see. All of that curve is plastic deformation. Recovering the little >spoink< in the beginning isn't going to matter much.

That, of course, begs the question of whether the model is appropriate. But let's say it is for the moment. This model does indeed call for severe deformation as a prerequisite for continued collapse* so we are in the realm of plastic energy dissipation >> strain potential.

In addition to the issue of relative contributions between elastic and inelastic response, this model has a special feature - there's no "release" of the applied force while collapse continues. It continues stepwise though each story until complete or arrested. So there's no opportunity to recover any elastic potential energy until it ends. One could argue that the entire tower in this model acts like a single compressible column, so it could go to full compaction and then rebound (a bit), as you suggest above, but the important thing for the mechanics argument is that the energy is not energy of motion during the collapse. Gravitational potential which goes into strain potential is just as unavailable as that lost to plastic deformation, so long as it's locked up in that form.

One could also argue that the model would permit a step-wise local rebound at each story but the net effect on translational motion would be minimal. As the next set of columns going down enter the plastic phase, the compacted columns above could partially rebound mostly at the expense of the columns below. But this is essentially highly overdamped longitudinal oscillation which is doomed to end up as heat and so could be considered sinked immediately for all practical purposes.

Finally, on that point, it's important to remember that energy partitioned into any internal degree of freedom (vibration as well as random directional multibody translational KE) is, for the most part, not going to find its way back into uniform translational motion of the top. While the energy isn't lost to plastic deformation (directly), it's not part of the top's KE, either. Ever again.


* obviously one of its most glaring shortcomings! It is, to say the least, highly unrealistic. Again, how many axially crushed columns are seen in the debris pile? Angels, pins.

 

[Jazzy]

That was beautiful to receive. Thankyou.

 

[Mick West]

Again, how many axially crushed columns are seen in the debris pile? Angels, pins.

To that specific point, has anyone ever done any statistical analysis of the visible and/or recovered core columns? Like, how many were bent significantly? Is there any statistical trend in the visible failure modes? Or are the columns too infrequently visible?

 

[Mick West]

FYI
From a WTC drawing....."elevation of tree column"

Seems like the term had multiple uses.

From /release 9/WTC-FOIA/Banovic/Tree Mapping/

 

[OneWhiteEye]

That was beautiful to receive. Thankyou.

You're most welcome. Remember to return the favor when the time comes!

 

[OneWhiteEye]

To that specific point, has anyone ever done any statistical analysis of the visible and/or recovered core columns? Like, how many were bent significantly? Is there any statistical trend in the visible failure modes? Or are the columns too infrequently visible?

I'm not aware of such a sampling, at least in any formal sense. One person who certainly did survey a large number of columns as they came out of the pile was Dr. Abolhassan Astaneh-Asl. He probably saw a good percentage of core columns, at least briefly.

 

[OneWhiteEye]

Seems like the term had multiple uses.

Thank you. I've always heard the assemblies referred to as trees. Sometimes panels; mostly trees.

 

[Mick West]

I'm not aware of such a sampling, at least in any formal sense. One person who certainly did survey a large number of columns as they came out of the pile was Dr. Abolhassan Astaneh-Asl. He probably saw a good percentage of core columns, at least briefly.

Here's the HD overhead photo taken a few weeks after. I've cropped it and boosted the contrast.

And a crop of that:


Seems like there are a lot of core columns, and they are largely straight and unbroken 36' (?) lengths. Just popped apart at the splices.

 

[OneWhiteEye]

Seems like there are a lot of core columns, and they are largely straight and unbroken 36' (?) lengths. Just popped apart at the splices.

I'm not very good at 'Where's Waldo' type of exercises, but I only see a few instances of significantly deformed columns. It's remarkable how little distortion there is, all in all.

 

[Mick West]

Here's a close up on the scene:


What parts of this scene are consistent with the controlled demolition theory? What parts are inconsistent with progressive collapse?

 

[qed]

I only see a few instances of significantly deformed columns. It's remarkable how little distortion there is, all in all.

Because, as you regularly remind us,

What Bazant did was try to show that it collapses even in the best case scenario.

The idea of the upper part falling one story such that each of its columns meets its lower counter part perfectly (and without all the concrete breaking up etc.) is an ideal mathematical model.

The towers did not collapse like that.

• The upper columns missed or collided non-perfectly their lower counterparts.
• Concrete was falling through the weak floors below, breaking those floors.
• The falling massive blocks of concrete fell through the weak floors.

So in reality the floors and concrete were stripped from the columns leaving most the columns relatively undistorted. These unsupported columns fell shortly behind the concrete.

This not withstanding, Bazant shows that

• even if the columns did meet perfectly, the building would still have collapsed.

 

[qed]

@OneWhiteEye

It seems to me that central to Tony's argument is that a buckled column that is collapsing under a weight it cannot support, can become strong enough again to transfer this force downwards to the undamaged columns.

I understand that this can happen if the column fully compacts, but it seems that Tony is asserting some other way that the buckling column can become strong again.

Can you play devils advocate and explain to us how Tony sees the buckling column become strong again. I am not skilled enough yet to follow his argument.

 

[OneWhiteEye]

I understand that this can happen if the column fully compacts, but it seems that Tony is asserting some other way that the buckling column can become strong again.

No, it is via compaction. Putting aside a moment the issue I raised earlier about two of his citations dealing with axial failure modes other than hinge buckling, he's traditionally followed the derivation of Bazant which specifies compaction and rehardening as the final phase of buckling. Perhaps this statement introduced some doubt:

I believe gerrycan is referring to elastic and plastic axial deformation and then plastic hinging as the three energy sinks involved in the buckling process.

The compaction phase is omitted in this statement because it's not an energy sink, but it is the final phase in this process.


A bit of background on the notion of full compaction if anyone's interested...

This post on JREF by femr2 contains a highly idealized depiction of a column at the end of three-hinge buckling. You can see how (in theory) the top and bottom 'knees' make contact once it's completely folded. Tony's response affirms this is the point he expects deceleration to be observed, when the deformed stubs top and bottom contact.

Bazant's load-displacement graph shows this as vertical line at the end labeled "Rehardening":



From the analytical point of view, this final phase is necessary to make use of the so-called "Maxwell construction", which is just a fancy term for averaging the force over the interval. This is a purely theoretical construct to allow meaningful integration over the limits, but does mirror reality to a certain degree. Perhaps it's best explained by Keith Seffen in this paper which includes this text in the section Propagating Instabilities:

The precise variation does not matter, but it must exhibit a characteristic up-down-up profle, for this enables two areas, A1 and A2, to be respectively enclosed above and below the horizontal line of P*, as illustrated. The purpose in doing so, as detailed originally by Chater and Hutchinson (1984), leads to the Maxwell Construction, where the exact value of P* equates A1 and A2.

Going back to this FEA animation:



This terminates before full compaction, but use your imagination. Not nearly as clean as the ideal figure, but the principle is the same.

 

[OneWhiteEye]

I'd like to take a trip backwards a couple of pages in this thread...

In post #861, I said that "...in the paper you're reading [...] the load displacement response for the columns in the tower is not of the hinge buckling sort assumed by Bazant..." and posted an image which depicted the sort of failure mode currently embraced by Szamboti et al:



to which Tony responded:

The above is not what the energy calculations we have done were based on. The energy absorption calculations were based on three hinge buckling at the beginning of the collapse.
It is hard to understand why you would even put images on like you have, unless of course you don't understand or are trying to deceive.

Emphases mine.

Now, where would I get an idea like that? As I pointed out in my subsequent response (#866), the only two independent citations in Tony's paper for means of computing residual capacity refer to axial failure modes which are NOT three-hinge buckling. I included pictures and graphs from the references to prove my point beyond any shadow of doubt. It was apparent Tony had not read some of the citations given in his own paper. I said as much, and Tony denied it with this response:

I read the references...

Based on what I've already posted over a week ago, it's very unlikely you even skimmed them. What I'll show in the next post will remove all doubt; you didn't even skim the citations in your own paper. Perhaps being called on it here and being forced to lie to save face might be an explanation for why you suddenly disappeared, and also vindicate me for stating up front that you had a propensity for lying in your arguments.

... and you aren't only accusing me of somehow inflating the energy absorption of the columns in the towers, but also the 1st author of the paper, Gregory Szuladzinski...

Yes, indeed, let's have a look at the contributions of your co-author Szuladzinski, who cites himself in addition to the independent citations I covered above.

 

[OneWhiteEye]

This is precious, I gotta tell you.

I excluded the Szuladzinski citations in my earlier critique because they weren't independent, a fairly arbitrary distinction. There are three which pertain to formulation of residual capacity:

---------------------
[3] Szuladzinski, G. Discussion of “Mechanics of Progressive Collapse: Learning from World Trade Center and
Building Demolitions” by Z.P. Bazant and M. Verdure. Journal of Engineering Mechanics, ASCE, Vol.134,
No.10, Oct.2008, pp.913–915.

[5] Szuladzinski, G. “Temporal Considerations in Collapse of WTC Towers”.Int. J. Structural Engineering, Vol.
3, No. 3, Feb 2012, pp.189–207.

[10] Analytical Service Pty Ltd. Large-deflection squashing of a steel column. Technical Note No.56, November
2008.
---------------------

It is interesting that there are only two independent citations for the residual capacity, but three self-citations.

#3 is a discussion submission to JEM and DOES employ a three-hinge buckling mechanism, as the paper states, but is acknowledged to be an early approach. I never said otherwise. The argument is shown to be flawed by Bazant, but the important thing is it's an old, failed argument.

#5 is behind a paywall. Can't say. If it is a three-hinge mechanism, it's undoubtedly the same one already dismantled by Bazant as being wildly overinflated.

#10 is a private study, not peer-reviewed, available here on the website of Szuladzinski's company. I'm going to claim fair use and reproduce Fig 4 here:



Checkmate. That's not three-hinge buckling, it's concertina/diamond folding of the type shown in the image I posted, the one Tony was so incredulous about.


So... at least three out of five references cited for residual capacity (including one from the co-author) employ the axial failure mode I indicated, not three-hinge buckling. The three-hinge argument made in at least one reference is discredited. Neither independent citation employs three-hinge buckling. The value for residual capacity used in the current paper is mentioned in conjunction with the two independent citations.

Where oh where did I ever get the idea that the current paper used concertina failure? Could it be from the three citations Tony never skimmed?

 

[OneWhiteEye]

Another reason I got the impression the SS&J paper did not use three-hinge buckling for the calculations is that it DOESN'T.

The energy absorption calculations were based on three hinge buckling at the beginning of the collapse.

This statement is false.

From the paper:

Prior to that, a square hollow section (SHS), was used in a simulation of a large-deflection squashing, Ref.[10]. The retention factor there was not less than η = 0.665, with this figure related to the use of the minimum, not the enhanced value of Fy.
...
A detailed calculation of the nominal column strength of the assembly and the expected resistance is provided in Appendix A.

Going to Appendix A:

Most columns under consideration had the SHS section for which the strength multiplier, as noted before was expected to be η = 0.5. However, most core columns had open sections, for which a smaller value would be expected. To be on the safe side, one can take η = 0.4 for all columns on the floor.

Notice that these are the only two instances where "SHS section" is mentioned. Somewhat confusing is that the multiplier value η = 0.5 was not "noted before" as claimed, not in the current paper nor in the technical note referenced (SHS squash test). The stated value in the paper was η = 0.665, and η values between that and 0.8 were mentioned in the reference. Nevertheless, the association with SHS sectioning makes clear that the origin of the value used was the SHS squash test, even if it is low-balled. The value is reduced to 0.4 because of the presence of open section columns which reference [3] (also NOT using three-hinge) are characterized as having values of 0.4 and below.

It doesn't matter that the η values are "low-balled", they are derived from calculations assuming a concertina/diamond folding pattern, as I first asserted.

 

[Tony Szamboti]

OneWhiteEye the energy absorption capacity values for the twin tower columns shown in the paper were calculated using the same three-hinge buckling mechanism as that of Bazant. The values are not for concertina compression as you are trying to assert. Below is from the paper.

3. COLUMN RESISTANCE IN LARGE-DEFLECTION COMPRESSION
The upper limit of compressive strength of steel columns is usually taken as Py = AFy, where
Fy is the yield strength and A is the section area. This value is reduced to the buckling load
Pcr, according to column slenderness [14]. However, if the columns are stout, as in the case
here, the difference between Py and Pcr is usually quite small. The peak resistance is
encountered at the outset, when a column is straight.

One of the methods used to assess the resistance offered by a slender column past the

elastic range is to treat it as a three-hinge mechanism as done by Bazˇant [1, 2]. The resisting

force decreases with deflection until the two arms of the mechanism become horizontal. At

that point there is a minimum resisting force, which is only a small fraction of its initial Pcr.
One can expect reasonable results from such a mechanism in a slender column, but only in

the early stages of deflection. When interactions between the walls of a column develop,

secondary resistance arises.

Because the columns in the WTC towers were stout by any criterion, one can expect that,

in the large deformation range, the columns would exhibit a minimum resistance being a

significant fraction of Pcr. Consequently, Szuladzn′ ski [3] employed the same approach, as
that of Bazant described above, except that he placed a higher estimate on the resisting

capability of columns. With this, he concluded that arrest of the downward motion would

take place quite promptly, just outside the zone affected by the aircraft impact.

 

[OneWhiteEye]

OneWhiteEye the energy absorption capacity values for the twin tower columns shown in the paper were calculated using the same three-hinge buckling mechanism as that of Bazant. The values are not for concertina compression as you are trying to assert. Below is from the paper.

I don't think you understand your own paper. Keep reading.

 

[OneWhiteEye]

Or, alternately, read my post above with comprehension.

 

[OneWhiteEye]

The one correction I'd like to make to what I said:


...they are derived from calculations assuming a concertina/diamond folding pattern...

should be:

...they are taken from FEA simulation modeling a concertina/diamond folding pattern...


which, in this case, may not be such a good thing.

Though the η value used is less than that obtained from the FEA, it is taken from the Szuladzinski results for SHS columns. The referenced Szudlanzinski technical note for SHS is a model of diamond folding. There's the picture reproduced right above your post.

 

[OneWhiteEye]

I really think you should explain why you were so incredulous earlier, at the very least. You said:

It is hard to understand why you would even put images on like you have, unless of course you don't understand or are trying to deceive.

Why is it hard to understand if at least three of five citations in your paper are for the failure mode depicted? And when the η value derived in Appendix A is mentioned in conjunction with the properties of SHS columns, for which your coauthor found the axial compression mode shown in the graphic above? I'll tell you this: if I am wrong, and I estimate the probability of that to be very very low, then the fault lies not with me but with the poorly written explanation which somehow leaves no doubt that my "erroneous" conclusion is correct.

You need to explain yourself since this is your paper. Instead of acknowledging the embarrassing array of evidence that your paper does not employ the results of three hinge buckling, you simply say I'm wrong. I think you're wrong and gave a detailed explanation of why. Your reply quoted the preamble of historical investigation which started with three-hinge buckling, something I already acknowledged. What matters is the source for the η value used in this paper. I make a very compelling case that it's based on the cited works by both your co-author and Korol et al and these do not concern three-hinge buckling.

Szuladzinski DOES have another technical note* describing (in pitifully weak detail) the results of a three hinge buckling FEA. Let me stress that: ONE. One non-reviewed proprietary FEA analysis, minimally documented. It is not an SHS column. It is not referenced by your paper. It may be the source of the energy dissipation figures and load displacement he used in the Bazant & Le discussion (also cited), but it's not used in your Appendix A.

Before you claim again that I just don't understand, I suggest you ask your co-author! Either the η value for SHS columns comes from Szuladzinski or it's entirely unsourced (i.e., made up). Could be both.



* Which, incidentally, is not very impressive. One thing at a time.

 

[qed]

@Tony Szamboti

I would like to understand your theory of arrest within one floor.

From attempting to read your paper, I understand that the initial conditions at t₀ (moment of beginning of collapse) are as follows.

┃
┠───────      c+1     ─
⎞                     ⥌  3.7m
⎬┄┄┄┄┄┄┄┄ →  c        ─
⎠
┠───────      c-1
┃

Critical floor c was sagging and induced the straddling column to buckle. At t₀ the buckled column was no longer strong enough to support the weight of the floors above. At this point the system becomes dynamic and the floors c+1 and above begin to subside.

(1) Am I correct in these initial conditions?
(2) What (and how) do you calculate the slenderness ratio of the column to be?

From reading your paper, I understand that the system must stabilize at time t₁ before floor c+1 reaches floor c (unless pre-weakened).

(3) Can you sketch us a picture of what the column and floor would look like at t₁ (or textually if that is too much).

 

[Tony Szamboti]

I really think you should explain why you were so incredulous earlier, at the very least. You said:



Why is it hard to understand if at least three of five citations in your paper are for the failure mode depicted? And when the η value derived in Appendix A is mentioned in conjunction with the properties of SHS columns, for which your coauthor found the axial compression mode shown in the graphic above? I'll tell you this: if I am wrong, and I estimate the probability of that to be very very low, then the fault lies not with me but with the poorly written explanation which somehow leaves no doubt that my "erroneous" conclusion is correct.

You need to explain yourself since this is your paper. Instead of acknowledging the embarrassing array of evidence that your paper does not employ the results of three hinge buckling, you simply say I'm wrong. I think you're wrong and gave a detailed explanation of why. Your reply quoted the preamble of historical investigation which started with three-hinge buckling, something I already acknowledged. What matters is the source for the η value used in this paper. I make a very compelling case that it's based on the cited works by both your co-author and Korol et al and these do not concern three-hinge buckling.

Szuladzinski DOES have another technical note* describing (in pitifully weak detail) the results of a three hinge buckling FEA. Let me stress that: ONE. One non-reviewed proprietary FEA analysis, minimally documented. It is not an SHS column. It is not referenced by your paper. It may be the source of the energy dissipation figures and load displacement he used in the Bazant & Le discussion (also cited), but it's not used in your Appendix A.

Before you claim again that I just don't understand, I suggest you ask your co-author! Either the η value for SHS columns comes from Szuladzinski or it's entirely unsourced (i.e., made up). Could be both.



* Which, incidentally, is not very impressive. One thing at a time.

The column energy absorption capacity values found for twin tower columns and described in the paper, as a function of the critical buckling load, are for three hinged buckling, as explained in the attached.

If you just can't wrap your head around the fact that stout columns like those in the twin towers would have a high average resistance as a function of their critical buckling load, during a three hinged buckle, and absorb large amounts of energy, you can talk to the first author of the paper himself about it. He is on Skype.

 

[OneWhiteEye]

The column energy absorption capacity values found for twin tower columns and described in the paper, as a function of the critical buckling load, are for three hinged buckling, as explained in the attached.

Do you even read my posts?

I SAID:

Szuladzinski DOES have another technical note* describing (in pitifully weak detail) the results of a three hinge buckling FEA. Let me stress that: ONE. One non-reviewed proprietary FEA analysis, minimally documented. It is not an SHS column. It is not referenced by your paper. It may be the source of the energy dissipation figures and load displacement he used in the Bazant & Le discussion (also cited), but it's not used in your Appendix A.

The technical note you attached is exactly the one I'm talking about. I not only read that one, but I've read about half the technical notes on his site.

Technical note 64 is NOT cited in your paper. Technical note 56 IS cited; it's for SHS columns, it is NOT three hinge buckling, and SHS columns are mentioned in your Appendix A. I've said this, what, three times now?

If you still don't get it...

Oh, I get it, alright. I just don't believe you do. And it's your paper.

...and just can't wrap your head around the fact that stout columns like those in the twin towers would absorb enormous amounts of energy during a three hinged buckle...

I get that, too. Problem for you is no significant amount of ANY sort of buckling occurred.