Thursday, July 5, 2007

Gordon Ross Shows Collapse Progression

The paper by Gordon Ross claiming that the WTC would self-arrest collapse
My initial response
Ross replies and implies "fake but accurate"

To Mr. Gordon Ross and the Journal of 9/11 Studies:

Regarding the reply to my criticism of the paper by Gordon Ross entitled, “Momentum Transfer Analysis of the Collapse of the Upper Storeys of WTC1”, I have demonstrated below the actual method in which to calculate the strain energy of the columns at the WTC.

For all compression members with eccentric loads or externally produced bending moments, the bending moment is magnified as the axial force approaches the ultimate compression capacity of the member. This is what is known as P-delta effects. As the column deflects due to the bending moment, the axial force becomes eccentric and creates another bending moment. This bending moment causes the column to deflect further which in turn means that the axial force creates yet another larger bending moment because the column has deflected outwards further. This continues until the column either reaches equilibrium or it fails. P-delta can be explained by the differential equation:

Mz = Magnified Moment (kip*in)
Mi = initial axial load (kip*in)
P = Axial force (kip)
y = initial deflection (in)
E = Modulus of Elasticity (29000ksi for steel)
I = Moment of Inertia (in4)

---Note, most of the calculations following are in the British Imperial Unit System. The numbers at the end will be converted to SI. For reference, a kip is 1000lb, and a ksi is 1000 psi.

A general solution of this equation will not be attempted here, as it is long and outside the scope of what this letter is intended to do. Following Timoshenko and Gere, the general solution is for a column with constant initial bending moments at the top at bottom is:


Pu = Axial Demand on the Column (kip)
Pn = Axial Capacity from Buckling (kip)

There is a limit to the interaction of bending and compression in a column. This is shown in the classic equation:


Mu = Bending Demand (kip*in)
Mn = Bending Capacity (kip*in)
e = eccentricity of vertical load (in)
Pu2 = Axial Load at an eccentricity e (kip).

In all real-world columns, there is always, at a minimum, a small but non-zero eccentricity between the axial force and the column. This is due to a variety of reasons, such as an unequal live load in two adjacent bays or a beam connection on the side of a column. For the purposes of this paper, let e equal a very small but non-zero number. As Pu/Pn approaches 1, the denominator of the term

approaches zero. The magnifier on the Mu/Mn term becomes infinity and the column stops being a compression member and becomes purely a bending member. In reality, the column would transition gradually at somewhere near 0.9 or 0.8 Pu/Pn depending on how eccentric the axial load was, however it is conservative in favor of collapse prevention, and easier math, to assume that this happens at Pu = Pn.

The strain energy of the column can then be calculated by separating the strain energy into two functions: the energy in compression and the energy in bending. These can be referenced from most engineering Strength of Materials books.

For the strain energy in bending, I am assuming 3 buckle points along the length of the column that rotate 30 degrees on each side. This is vastly conservative in favor of collapse prevention. In reality, columns will only go through an 8-12 degree rotation before failing. This of course varies depending on the column strength. The exterior columns were made of varying yield stress materials of up to 100 ksi. High strength steel such as this is more brittle than lower strength steel, such as the A36 wide-flanges, and will fail at a smaller rotation. Even then, a rotational fudge factor of 2.5 has been used to justify that the calculations are completely in favor of collapse prevention

To determine what the strain energy is, actual real column sizes are required. A guess on the cross-sectional area is not enough. To do this, columns based on the same data that Ross references are used to compute a cross-sectional area. The actual column size is then picked by using this cross-sectional area and the same nominal depth of the columns at WTC. The exterior columns were 14x14 box columns and the interior was made up of both box columns (of varying dimensions) and 14” nominal wide-flange shapes. Since the shapes of the interior box columns are not know, the interior columns are all assumed to be 14” nominal wide flanges.

Where (from Ross):
Weight Above = 59000tonnes
FudgeFactor = 4

There are 240 exterior columns. Assuming that the exterior is 50% of the total cross-section of steel, it follows that the cross-sectional area of steel of each is:

A similar shape is the HSS 14x14x1/2. The relevant properties to be used later in this paper are:
Ag = 24.6in2 r = 5.58in Z = 124in3 I = 743in3

There are 47 interior columns. The cross-sectional area of each is therefore:

A similar shape is a W14x370. The relevant properties to be used later in this paper are:
Ag = 109in2 Ix = 5440in3 Zx = 736in3 rx= 7.07in
Iy = 1990in3 Zy=370in3 ry=4.27in

The difference in strength between the two orthogonal axis of a wide-flange are shown with the x and y notation. In column failure, the column will always buckle in the weak-axis. Thus, the smaller numbers of the weak-axis are the relevant ones.

The calculations for the HSS14x14x1/2 follow, please note that the method for calculating the axial compression capacity is inelastic buckling, as seen in my previous letter:

The calculation for the W14x370 are performed in the same manner, the results are:

The total strain energy is:

Or a 269% increase in the absolute maximum. The real column sizes are required to determine what the actual strain energy in the columns are. This calculation is grossly in favor of higher strain energy and even then it is 2.69 times less than the calculation in the Ross paper.

Ross said in his response, “The conclusion of my article, that the energy would be dissipated in many more areas outwith the uppermost storey has been broadly accepted and no serious challenge to that conclusion has been forthcoming.” The summary of the energy losses from his paper:

Energy available;
Kinetic energy 2105MJ
Potential energy Additional downward movement 95MJ
Compression of impacting section 32MJ
Compression of impacted section 24MJ
Total Energy available 2256MJ
Energy required;
Momentum losses 1389MJ
Plastic strain energy in lower impacted storey 244MJ
Plastic strain energy in upper impacted storey 215MJ
Elastic strain energy in lower storeys 64MJ
Elastic strain energy in upper storeys 126MJ
Pulverisation of concrete on impacting floor 304MJ
Pulverisation of concrete on impacted floor 304MJ
Total Energy required 2646MJ
Minimum Energy Deficit -390MJ

Momentum loss and pulverization of concrete: This is another large mistake in Ross’ paper. When two objects collide, their energy is conserved. The typical method to easily to determine this is to assume a plastic collision, where M1V1 = M2V2. This is what Ross has done. He shows that in a perfectly plastic collision, there is an energy loss of 1389MJ. This is also true. However, there is a very important assumption in this that Ross has completely ignored. This loss in kinetic energy is transferred into the plastic deformation of the two bodies. In other words, the difference in energy from impact IS the “pulverization” of the concrete. Ross has calculated the damage of each concrete plate twice. In contrast, in an elastic impact, where the objects are not damaged by the impact, have zero loss in kinetic energy. Energy cannot be created or destroyed, however Ross has allowed 1389MJ (12*107 kip*in) to vanish. Where do you assume this energy goes Mr. Ross? From this alone, there is no longer an energy deficit in your paper, but rather an energy gain from floor to floor.

Mr. Ross, your conclusions and sums and methods have been proven wrong. In my previous letter I offered you the chance to fix and update your calculations out of professional courtesy. Out of respect for your abilities, I said it would be easy for you to do. I had hoped that you would take a harder look at that issue, and take another look at the rest of your paper, but you have chosen not to do so. Your response was nothing more than, “fake but accurate”. This is a disgusting manner for any engineer to respond. Sir, retract your paper or fix your calculations.

Sunday, May 20, 2007

Failure of the “Truth” Movement’s Engineer

Gordon Ross is an engineer, in the mechanical field, that has put forth a paper showing how the towers of WTC 1 and 2 should have arrested the initial collapse of the building. This paper (available here) is used by many who have no knowledge of what actual engineering is as evidence to say that the building must have been brought down by controlled demolition. This is as part of some super secret government conspiracy to commit mass murder against its own population. His paper is in response to a paper by Bazant and Zhou ( available here) which shows how easily the upper sections would have smashed through the lower floors in a pancake collapse. Ross tries to duplicate the analysis conducted by Bazant and Zhou on the plastic strain capacity of the columns. He attempts to argue that the strain energy in the column should have been sufficient to completely arrest the collapse. I will only address this one critical point in his paper, buckling, which he does not appear to have even a basic understanding of.

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.

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).

(Inelastic Buckling)

[AISC 13th E3-2]

[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.

Gordon Ross, Momentum Transfer Analysis of the Collapse of the Upper Storeys of WTC 1

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

AISC Manual of Steel Construction, 13th Edition

ASCE Code of Ethics,

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.

Sunday, February 25, 2007

A talk with the author of "On Supporting the Iraqi Resistance"

I stumbled upon a Daily Kos Diary-Post by Heathlander, "On Supporting the Iraqi Resistance". I found just the title relatively offensive and the material contained within extremely biased in my viewpoint. I was actually downright furious over the claim that the insurgents were actively supported by more than a tiny fraction of the population. At first I wanted to write an extremely nasty email but then realized that it would be ignored. In the spirit of dialogue, I wrote a polite email, and it was answered.


In regards to your diary-post on Daily Kos, "On Supporting The Iraqi Resistance", I feel that you are mistaken about the “insurgency” in Iraq. Violence only begets violence. During the American Civil Rights movement, great leaders such as Martin Luther King Jr. stood up and demanded rights not given to him. In India, Gandhi fought tyranny and won.

I am sure you are familiar with these great men more than I am, and will not presume to lecture you about their great deeds. However, I will say that there is a great commonality with both of these leaders (and many like them), and that is the complete strategy of non-violence. They staged civil disobedience, sit-ins and protest rallies, but above all placed a sanctity on human life above all else. And because of that non-violence they gained legitimacy with a great many people who might not support them, but at least would not condemn them.

In 2005, the Iraqi population provided their own self-government and independence with proudly raised purple fingers. This is the legitimate, democratically elected government in Iraq. Should a group of people in Iraq feel that the Coalition presence there is oppressive (as many do), then those people should petition their democratically elected government to ask the Coalition forces to withdraw. If the government there does ask the Coalition to withdraw and the Coalition refuses, it becomes obvious to the entire world that there is in fact an occupation in Iraq. Until that happens, however, one cannot say that the Coalition is occupying a country that has an independent and democratically elected government.

The "insurgency" in Iraq is not fighting an occupation, they are not fighting to gain any freedoms, they are not fighting to gain independence. They have already have these things. All they have to do to get what they want is to petition their government. Should they fail to get what they desire, there are many forms of redress available to them, first and foremost of these is voting for different people in the next election, followed by civil disobedience. Non-violent protest.

There is no legitimate "Iraqi Resistance". Every person who picks up a rifle or assembles an explosive belt with the intent to kill someone for disagreeing with them is not freedom-fighter, or a resistor, but a terrorist who is trying to change someone's mind by instilling fear.

Albuquerque, New Mexico

I called Heathlander "sir" even after seeing that Heathlander's name was Jamie. Being that Heathlander is British and the name Jamie is a gender-neutral there, I took the chauvinistic route and figured Heathlander to be a man. I wasn't corrected, so maybe my assumption was correct. I received a reply in the morning. I was also surprised at the politeness.


Thank you for your polite and detailed email.

I share your admiration for passive resistance. I personally am not sure that an armed resistance in Iraq is morally justified. But legally, it is legitimate, and when judging it we have to do so from the "reasonable man" perspective. I.e. could a REASONABLE Iraqi feel that they have to resort to violence to stop the destruction of Iraq. I think they could.

The elections in Iraq were not legitimate. No elections held under an unwanted foreign military occupation, in the midst of great violence and on the understanding that any future elected government would continue to operate under the foreign military occupation has ever been considered legitimate.

The Iraqi people overwhelmingly want the troops out. If they are ignored, they entitled to resist.


I was surprised at the argument over the legitimacy of the election. I've never felt that there was much of a challenge or a controversy over the election itself, only over the handling of the invasion, post-invasion security etc, etc. I decided to press the issue.

Thank you for your reply,

I would argue that the destruction in Iraq is a direct result of the violence. Many great minds who could be valuable leaders and ministers have been cowed by terrorism into not participating in government. However that is not a point I think either of us contest.

I would argue your point about the legitimacy of the current government. The UN itself had a hand in the elections, and it was independently (as best able) by outside sources. Though there were problems, such as the Sunni's boycotting it, in my mind it was a legitimate election, however I am sure that view it is not. May I ask what you feel the Coalition should have done to create a legitimate election in your mind?


And the follow up:


I don't think it's possible to have a legitimate election under a military occupation that is not wanted by the majority of the population, and that will continue to be around after the elections.

Of course the destruction in Iraq is the result of violence - but which violence? Firstly, there was the initial, devastating invasion. Then there were the many war crimes and massacres carried out by the occupying forces (Fallujah, for example). Then there is the sectarian conflict, the death squads and the criminal gangs. And then there is the insurgent violence. Certainly, the sum of all this is the mess that is Iraq today.

I think it seems clear what a legitimate resistance movement in Iraq would look like - it would be focused solely against the occupying forces and would not target civilians. Whether that exists in Iraq today, we cannot be sure.


This is the end of the correspondence, and I don't rightly feel that I can continue on with it and remain completely civil. My analysis, if Heathlander is an accurate representation of the current anti-war argument, is that the general sentiments of the anti-war crowd is

  1. There was no legitimate reason to invade in the first place.
  2. They feel that the population never wanted the invasion in the first place.
  3. The coalition is the source of the majority of the violence.
    1. Thus the argument that Bush is currently making, that the World is looking to see us responsibly not leave Iraq, is moot.
  4. An election supervised by a foreign army cannot be considered legitimate.
    1. Which follows that the only recourse for a people under a totalitarian government is self reform, be it a bloody revolution (impossible under Saddam) or a call for a general election (also not possible under Saddam).
At this point, I'm not sure how they feel a mad murdering dictator like Saddam should be removed. On the affairs of nations, I feel that violence is always a last resort, but it seems like the anti-war crowd would have rather left the populace of Iraq in bondage and fear under Saddam than in it's current state of freedom and violence.

This however is an extremely complex issue with many factions all with different goals and interests in causing violence in Iraq. It should be well noted that Heathlander is not equating al-Qaeda with the resistance, only the Sunni segment of the population (the same segment which boycotted the elections). Had Heathlander done otherwise, and said that al-Qaeda was a legitimate "resistor", I never would have started the conversation in the first place.