First let me say that the building did not collapse like the pancake theory suggests. In reality it was much more complicated with individual elements being overloaded before the rest of the structure was fully loaded, lateral loads induced from eccentric loading as well as multiple floors of columns failing at the same time. However Ross as well as Bazant and Zhou make these assumptions and I will not challenge them as they make the math easy.

Ross’s paper fails on his knowledge of buckling failure; buckling is the phenomenon which causes a member to bow in compression. However for the reader to understand what causes buckling, a rudimentary knowledge of the engineering stress-strain curve is required. Stress is a measurement of the force based upon the cross-sectional area of the member. Strain is the deformation (unit-less fraction) of that member.

Figure 1 - Stress vs. Strain

Figure 1 is a simplified graph of the stress-strain curve for steel with a yield point of 58ksi, which also occurs at a strain of 0.002 (or 0.2%). The yield stress, Fy, is the place on the graph where the curve becomes horizontal. The slope of the diagonal line from the origin to the yield point is known as the Modulus of Elasticity, E, which is 29000 ksi for structural steel.

The strain energy of a member is a function of the area under the Stress vs. Strain curve. Specifically, it is defined as:

This is nothing more than a fancy way of saying that the energy is the area under the curve of the axial force F(u) as a function of the displacement and. strain (u) curve. There is also a component of strain-energy for the bending (bowing due to buckling) portion of the column failure, however I will not get into it here. The above is just to show what Ross is doing in his calculations because he doesn’t really put it forward in understandable terms.

The actual derivation of buckling is non-trivial and I will not attempt to do it here. However, the AISC Manual of Steel Construction Specification has an entire section devoted to it and I will use its equations, only slightly modified to put the equations in terms of stress rather than axial load. First, there are two forms of buckling: inelastic and elastic (Euler). Elastic buckling occurs in very slender elements, Inelastic in short and stubby sections. The slenderness of a column is defined by ratio KL/r.

Where:

K = effective length factor, which will be assumed to be 1.0.

L = height of column between supports (in)

r = radius of gyration (in). (For those who have a knowledge of physics or engineering, r is the square root of the second moment of area, or moment of inertia over the cross sectional area).

When:

(Inelastic Buckling)

[AISC 13th E3-2]

Where:

[AISC 13th E3-4]

(Elastic Buckling)

[AISC 13th E3-3]

σcr is the critical stress in which the column yields and cannot resist any more vertical force. AISC provides a graph showing the relationship between the Design Stress, which is 0.9 times σcr and slenderness, shown in Figure 2. With a slenderness ratio of 0, the design stress is equal to the yield point times 0.9. The columns at or near the impact in the WTC had a slenderness ratio between 20 and 40. This means that critical stress is actually less than yield stress.

**Figure 2 – Slenderness vs. Design Stress.**

Buckling always occurs before the yield stress in a compression member. Ross doesn’t seem to think so. He describes a compression mode of failure, which buckles at 3% strain, “2/ The shortening phase allows for the same failure load to be applied until the vertical deformation reaches 3% at which point the column begins to form buckle points.” This is not correct. At 3% strain the column has all but disintegrated. It does not buckle at this point, it’s structurally non-existent.

Ross even acknowledges how important buckling is. A compression member cannot resist any additional vertical load after the critical stress (see figure 2) is reached. In fact, if the load is kept constant, the column will bow until it breaks from bending while resisting less and less axial force. This greatly reduces the strain-energy in that member. He even goes so far as to state this fact in his “Assumptions and Disregards” section. However, he then immediately says that buckling did not occur because the columns were not of sufficient length for Euler, or Elastic, buckling to take place, “Euler calculations show that columns of the dimensions used in the towers would not fail due to buckling over a length of one storey height, but would instead adopt a compressive failure mode.” He completely ignores a complete range of buckling with this statement, and with it, completely over-exaggerates the strain energy available in the columns.

Even with knowing how important buckling is, Ross doesn’t know what it really is. In the last part on buckling in his “Assumptions and Disregards” section, he states that, “…I have chosen a buckling failure mode as this mode has the lowest energy demand.” In reality, he has not picked a buckling failure mode. This is evident in his calculations (which I have other issues with, but will not get in to) where he attempts to take the energy from the full 0.2% to 3% at the yield stress. His calculations show that he has assumed the column to be in pure axial compression without buckling until 3% strain, where a normal member would have ruptured, and then says that he has chosen not to look at the energy from the stress-strain curve after the 3%, which doesn’t even exist structurally.

Gordon Ross does not have a clear understanding of structural engineering, and this is evident in his paper. His idea of the concept of buckling is incorrect. This in turn has led him to write a paper that completely over-exaggerates the structural capacity of the WTC towers and completely mislead a group of people who depended on him, as a professional engineer, to know what he was talking about. This is not his fault, or the fault of the education system which trained him. He is not a structural engineer; these concepts are not readily available to him. Even I have made mistakes on this concept before. We all make mistakes. This is one that could probably be corrected. This, however, is not about mistakes.

I have been accused by Gordon Ross on a forum of not doing my research into the collapse, but to Gordon Ross: Sir, I accuse you of stepping outside the bounds of your expertise and publishing "research" in a field you have no knowledge of. You have acted unethically, specifically defined by the ASCE Code of Ethics, part 1 of Canon 2 “Engineers shall undertake to perform engineering assignments only when qualified by education or experience in the technical field of engineering involved.” And on part 1 of Canon 3, “Engineers should endeavor to extend the public knowledge of engineering and sustainable development, and shall not participate in the dissemination of untrue, unfair or exaggerated statements regarding engineering.” You have acted as a catalyst to the poison that has infiltrated the minds of many people of the public.

References:

Gordon Ross, Momentum Transfer Analysis of the Collapse of the Upper Storeys of WTC 1 http://www.journalof911studies.com/articles/Journal_5_PTransferRoss.pdf

Bazant and Zhou, Why Did the World Trade Center Collapse?—Simple Analysis1

http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/405.pdf

AISC Manual of Steel Construction, 13th Edition

ASCE Code of Ethics, https://www.asce.org/inside/codeofethics.cfm

**Update:**Fixed an error I had with the strain-energy equation.

Update2: Fixed an error with the critical stress equation. I mistranscribed slightly from the spec.

That's an excellent analysis and far more in depth than anything I considered when reviewing Ross's paper.

ReplyDeleteCan I just ask your opinion of my idle thoughts?

Just focusing on Ross's basic assumptions and results for one moment -

he assumes the downward velocity at the point of first collision with the uppermost intact floor to be 8.5 m/s.

He assumes a time period of 0.02 seconds

He calculates a final downward velocity of 4.8 m/s.

Now, by my rudimentary understanding of conservation of momentum, force = rate of change of momentum

If we say the mass of the falling building is M then change in momentum is 8.5M - 4.8M = 3.7M

over a time of 0.02 seconds gives us a force of

3.7M / 0.02 = 185 M or 185 times the mass of the falling part of the building.

Does this mean that he calculates that the lower part of the tower is withstanding a force 185 times the static load it would normally hold?

Or is it just me?

If those are his numbers, then yes. You are looking at F = m * a correctly.

ReplyDeleteAn important thing to remember is that a column cannot resist more load than it's capacity. Hence it won't transfer anymore load down to the rest of the structure than it's capacity. However with a total axial force 185 times that of it's capacity, p-delta effects, a.k.a. second-order effects control the axial capacity of said column. This is a reasonably complicated subject that involves differential equations, but it is something that would need to be addressed in a legitimate energy problem.

... 3.7M / 0.02 = 185 M or 185 times the mass of the falling part of the building."

ReplyDeleteIt should be 185 M *m/s^2 (185*M N) which is mean(over time) force acting during those 0.02s. You cannot compare force with the mass. So if you would like it to compare with "static" force acting you should compare this with 1M*g = 9.8 M*m/s^2.

So, this leads to conclusion that a force acting was 19 times higher than "static" load. This would be not a surprise since (withstanded!) dynamic loads can be an order of magnitude higher than static loads.

"Does this mean that he calculates that the lower part of the tower is withstanding a force 185(corr:19) times the static load it would normally hold?"

No, it is not withstanding it because the lower part and the upper part is destroyed in that process (no doubt about that) and stress is transferred to the upper and lower parts of the building (this is questioned in this post, however... see below). The force in question is simply acting.

To NewotonsBit: I would need to devote a lot of time to engage in meaningful discussion about this but I'm raising the following question: Am I getting it correctly that you suggest that little? (no?) energy is transfered to the upper/lower part of the building because columns are destroyed with stress and cannot transfer the energy?

This seems unreasonable to me, because: this is exactly the upper part that is PROVIDING energy to the collapse, so since every action has a reaction, since every heavy piece of the upper part is the cause of the crushing force, that means it has to be pushed back up. This force "back up" is transfered through the columns.

Do you suggest that all the columns in the upper part and in some signigicant lower part are disintegrating because of stress at the same moment?

Even if it happens so, what is most important here is how long "this moment" lasted because during that time of "disintegration" the energy IS transferred.

Not transferring the energy during a collision at all is possible only when "terminal velocity" is reached. (please refer to Wikipedia)

I cannot say if the collapse should arrest or not, but with my understanding, it should be significantly slowed down due to dynamic stress redistribution.

Correction: I meant not terminal velocity (of a falling object for example) but rather Hypervelocity

ReplyDeletehttp://en.wikipedia.org/wiki/Hypervelocity

m

ReplyDeleteYou are correct, the actual equivelent static load is 185/9.8. I shouldn't be attempting to do dynamics late at night.

Two things to keep in mind:

The columns themselves can only resist a certain force, and cannot transfer anymore than that to the lower levels. The sum of the forces in the vertical direction is not equal to zero as this object is not at rest. If you wish to actually analyze what the force in the columns are, you can take the balance of kinetic, potential and strain energy over the initial axial displacement of the column (neglecting out of plane displacements due to the initial effects of buckling). You can determine the actual deacceleration of the block imposed by the columns in that case and what the actual force on the entire structure was.

I'm not providing a method to analyze energy transfer in a pancake collapse. It did not occur that way. There are many photos showing entire panels from the WTC intact and falling away from the tower. These were either sheared off, or broken at their splices.

NewtonsBit: I agree that full analysis would be very difficult. I for example imagine that the upper part shifts, or "walks" 0.5m to the side and columns no longer stand on each other, there would be a lot of tearing, shearing... still.

ReplyDeleteHowever, my general point is that a proper analysis of energy transfer has to include energy distribution through dynamic stress in the upper and lower part of the building. If one, in his computation, uses the whole weight of the upper part and assumes that, let's say, hat truss weight contributes to the crushing in the crush zone, then because of ACTION-REACTION he must admit that hat-truss "feels" the "kick" from below. The kick might be short lasting, but then it lasts as long, as hat-truss contributes to the crushing below. The energy IS transfered from hat truss to the crushing zone through the construction (or not and hattruss momentum does not influence the crushing at the moment). Now, WTC was a dynamic steel mesh of columns and trusses. In my opinion (it's only an opinion at this moment since no one did analysis of this type) a very significant amount of energy would be dissipated through stresses of joints, hat truss, columns all way up and all way down, their buckling or not. Bazant seems to have omitted that in his analysis - shame!

Gordon Ross in his paper criticizes Bazant, among other things, for not including this type of analysis in his paper. I agree with Ross that this was a very significant omission by Bazant. I feel that you were unfair with naming your blog post "Failure of..." even if your criticism of buckling analysis is in some way relevant (this I just don't know at the moment).

Ross's paper does not provide full account of the energy transfer either - this can be only done with performing rigid finite element analysis on a large cluster machine. I'd suggest supporting David L. Griscom and his call for full simulation the collapses, issued in his letter titled "Hand Waving the physics of 9/11"

My point about buckling is not that it's a complicated subject that needs to be addressed to cross t's or dot i's. It's a very basic structural concept. Ross even says he looks at "buckling", but he's not talking about the same thing. He doesn't know what the definition of buckling is. Thus, you need to not trust what he's saying purely on face value and do some of your own research.

ReplyDeleteI keep reading your post and checking references. Thanks.

ReplyDeleteVarious "experts" have written about the WTC collapse, the most credible of whom have been willing to put their real names on their work, show their credentials, publish their theories in peer-reviewed journals, etc. For example, we know that Dr. Bazant is a professor at Northwestern, Dr. Ross is Mechanical Engineer with a degree from Liverpool John Moores University, Dr Frank Greening is a Physical Chemist from Ontario, Dr. Judy Wood is a former professor at Clemson, etc. When people publish their work, they put their professional reputations on the line. As a fellow academic, I respect that. On the other hand, when people post anonymously on blogs like this, no matter how impressive-looking their mathematical "proofs" may be to the layman, it's hard to take them seriously. Mr. Bit, you even attack Dr. Ross's peer reviewers. But where are your peer-reviewed publications? Why don't you reveal yourself? Are you concerned that your credentials might not seem as impressive as your equations, or that your job title might lead some people to question your objectivity? Why don't you come on out of the closet, like the people whose theories you are attacking on these pages? I don't mean to be provocative or overly disparaging here; you may be, in fact, God's Gift to Physics and Structural Engineering. But if you really want to quash all the "conspiracy theories," then you're going about it in the wrong way. Hiding behind a pseudonym simply fans those flames, because it looks like you're trying to hide something.

ReplyDeleteWhat utter garbage posted by anonymous above. If I were to post a mathematical proof to a problem here, it would not matter who I was, who I work for or what my qualifications are. All that would matter is the quality of my work. The conspiracy nuts would have a problem that I'm anonymous? Too bad. Better that demonstrate my work to incorrect rather than whine about who I am and what my qualifications are.

ReplyDelete@ AnonymousJune 2, 2008 12:53 AM

Delete"The conspiracy nuts". I think you just showed your hand, thus negating any mathmatical workings you may have annonymously posted.

@ Anonymous June 2, 2008 12:53 AM

Delete"The conspiracy nuts". I think you just showed your highly biased hand, thus rendering all of 'your' mathmatics pretty much null and void. Good work!

Whose mathematics?

DeleteIs there any reason ;you haven't updated the flawed Bazant and Zhou paper? Considering that is paramount to your argument and the official story.

ReplyDeleteWhen will the Feds or sad debunkers correct Bazant and Zhou's paper? You know the entire house of cards that NIST relies upon.

This is a great site. Thank you for your information. I THANK YOU I SALUTE YOU IT,S A AMZING SITE.

ReplyDeleteThat's an excellent analysis and far more in depth than anything I considered when reviewing Ross's paper.

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ReplyDelete