Addressing Faulty Assumptions of Greening and Others in WTC Momentum Transfer Calculations

Brian Bray, of Studyof911.com, Sept. 23, 2006. Last updated Nov. 6, 2006.

There seems to be a common misconception that the falling upper floors of the Towers gained more and more mass and momentum as they crushed each floor, from the mass of each floor being added to the total amount of falling debris providing the theoretical “driving mass” of the collapses. The idea holds that if one floor fell onto the next, both floors would continue falling with double the total mass, and then a third floor would begin falling, tripling the initial falling mass, etc., etc., until a kind of “runaway” collapse is produced in which the falling mass consists of nearly a whole WTC Tower and cannot possibly be stopped or slowed by any remaining resistance.

An example of this can be found in Frank Greening's paper, Energy Transfer in the WTC Collapse[1], where he assumes each floor’s mass was simply added to the next floor’s mass whenever a floor collapsed.

We assume that both WTC building collapses began with an upper block of n floors collapsing onto a series of lower floors as in the “domino effect”. We shall refer to this process as the first stage of collapse. For this stage, (see equation 1), we have an initial mass nmf falling onto the floor below and becoming mass (n+1)mf. This new, enlarged, block of floors descends with velocity v2= {n/(n+1)}v1 through a distance hf at which point it strikes the floor below and becomes mass (n+2)mf moving at velocity {n/(n+2)}v2, and so on.

In this article I will show that this was not at all the case with either the collapse of WTC1 or WTC2, and that this theoretical model of collapse is unrealistic and impossible to accurately apply to either collapsing building.

I will show that calculations such as Greening’s should be adjusted so that the masses of both Towers are steadily and consistently removed during their collapses so that at least 80% of the total building masses will end up outside of their respective building footprints by the completion of their collapses (which assumes the final resting masses within the footprints constituted 20% of the total building masses, something which is seriously doubtful and therefore generous to Dr. Greening’s figures), rather than Greening’s assumption that 0% of the total masses was moved outside of the footprints, and that 100% stayed within them to contribute additional mass to the collapses of the trusses and their connections to the core and perimeter columns on each floor with an impossible amount of efficiency. This would obviously greatly impact figures such as any on floor momentum transfers during the collapses, even without additional considerations of Greening’s work. (Gordon Ross already has a paper outlining another error in Greening’s calculations, which allowed Greening to reuse energy already transferred down the building multiple times. This paper can be found here.)

Additional considerations for anyone that would like to attempt more realistic calculations on theoretical transfers of momentum and etc. will be mentioned in passing. These will require additional considerations for the deflection of broken-up floor slabs and dissociated columns and groups of columns, as well as the descension of the collapses onto floors which became steadily heavier and more solidly constructed (especially on the mechanical floors), thus requiring increasing amounts of energy to destroy each floor, by what should have been a force decreasing in energy after the destruction of each floor. I will not attempt to provide the actual means to accurately represent these additional variables in any calculations relating to the collapses, and I do not intend to offer any complete listing of all appropriate variables that should be taken into account. These are simply things that should be kept in mind as we generate simplified mathematical models of how a collapse could be allowed to propagate so rapidly and energetically.

Fig. 1. An image of debris being ejected from WTC2 in the form of perimeter columns, concrete dust, aluminum panels, and other material.

Researcher Jim Hoffman estimated in a January 2004 interview with Bonnie Faulkner that about 80 to 90 percent of the masses of the Towers ended up outside of their footprints.[2] Brent Blanchard of PROTEC authored a paper that appeared in August 2006 (arguing against demolition) suggesting that as much as 95% of the falling mass had landed outside of the Tower footprints.[3] How accurate are these figures?

We will consider a diagram from the “FEMA Report,” the WTC Building Performance Study, as well as an aerial photo of Ground Zero, and other photos, to get a rough idea as to how much mass was ejected outside of the Tower footprints during their destructions.

The FEMA Report on the WTC disaster gives us the following image[4] (fig. 2):

We will assume this FEMA representation is to scale (as it seems to be), and therefore real-life distances will be proportional to the image’s pixels. Examination of this graphic shows the basic inner (darker orange) circles surrounding WTC1 and WTC2 to be approximately 155 pixels in diameter each. The equation will give us the area of these circles, whereas r is the radius of each circle. The diameter of 155 will therefore be divided by 2 to get a radius of 77.5 pixels.

Therefore the total areas of the dark orange circles are 18,869.19 pixels each.

The space occupied by the building footprints themselves in the image is represented by squares of about 45 pixels, or 2025 pixels total area each.

Subtracting 2025 pixels from 18,869.19 pixels, we get 16,844.19 pixels for the total area of the dark orange circles around each building, excluding the areas of the buildings themselves, or the buildings’ footprint areas. Compared to these footprints, the rest of this dark orange circle, indicating only the heavy debris spread, has an area nearly nine times as large. Specifically, the building footprints compose only 10.73% of their total respective heavy debris areas represented by the dark orange circles. Notice that those circles denote only the area in which the majority of the support columns landed during the buildings’ destructions. Nearly 90% of this area (89.27% by the above figures) is outside of the footprints, according to FEMA.

This is excluding support columns ejected outside of this inner circle range (including the oval extensions from the circles in the directions of the faces of each building), and all other mass (such as aluminum panels), which was ejected even farther, some of which being denoted by lighter orange in the FEMA image. The concrete dust, representing what must have been an enormous amount of the concrete within the buildings, took to the air and became a massive dust cloud engulfing the local area after each collapse, and later coming to rest as fine powder all over lower Manhattan.

Fig. 3. An immense dust cloud over Manhattan from WTC1’s destruction. How much energy could this fluffy, pulverized material (formerly floor slabs) contribute to a floor-by-floor collapse, considering it was created during the collapse itself? On the other hand, how much energy would this take from the falling mass?

Fig. 4. A photo showing a stack of debris in the footprint area of WTC1.

The above image shows the total mass of WTC1 that actually remained within its footprint. Notice that the highest point in this pile is a group of box columns with their lateral bracing intact (cropped below). This appears to be a section of still-standing core structure, and thus could not have contributed energy to truss failures.

The following images show other views of the same pile, clearly showing the standing remains of WTC1’s core structure, again, representing mass irrelevant to our discussion, yet appearing to make up a large amount of that debris pile.

Fig. 5.

Fig. 6.

The pile of dissociated debris lying within WTC1’s footprint very likely consists largely of the core section of WTC1 that sank straight down after the destruction of the outer columns and trusses, known as the “spire”:

Fig. 7. WTC1’s remaining core structure, standing briefly before sinking vertically.

A video clip showing the vertical descent of this structure, in MPG format, can be found at http://st12.startlogic.com/~xenonpup/spire/spire_1.mpg. Since most of the spire appeared to sink straight down very rapidly, we can assume that this structure, which was clearly not involved in any “pancake” mechanisms of truss failure and perimeter column failure, landed within or immediately around the footprint area of WTC1, and thus can account for much of the mass seen within and around that pile. Because of the lack of influence this mass would have had on any collapsing truss systems (which would have already occurred by this point), this mass can also be excluded from whatever relevant mass may have actually landed within WTC1’s footprint (i.e., mass that would have actually contributed to the momentum of the theoretical floor-by-floor truss/floor collapses).

So we have noted a “stub” of still-standing core structure within the footprint of WTC1, and have noted the vertical sinking of the remains of WTC1’s core, which appears to have sank into WTC1’s footprint. This should account for a large amount of the debris seen within WTC1’s footprint, and this could not have contributed mass to failing the trusses that were allegedly propagating the building’s global collapse.

We are left to wonder exactly how much material did fall into WTC1’s footprint from the alleged collapse mechanism assumed by Greening of one floor falling onto the next, all the way down to the bottom.

Notice the lack of significant debris build-up in the footprint of WTC2 in the following image (WTC2 remains are in the foreground), even more so than for WTC1, when we would also expect the opposite for this building, if it were a pancake collapse.

Fig. 8. A portion of Ground Zero, with WTC2’s footprint in the foreground.

Does this look like 110 floors of steel columns, braces and trusses, with 4”- and 5”-thick slabs of concrete, just fell straight down, floor onto floor, in a “pancaking” fashion?

Note that WTC2’s core also appeared to fall straight down, vertically, after the rest of the building, thus accounting for much of what was in its footprint as well.[5]

On August 24^th, 2006, Professor Jim Fetzer of the organization Scholars for 9/11 Truth interviewed licensed professional structural engineer of some 30+ years of experience, Charles Pegelow, who discussed pancake collapses, particularly in relation to steel structures.[6]

Here are some excerpts from that interview:

Charles Pegelow: As far as the pancake theory, that's not even applicable to steel structures. Actually, I think it seems like, some of the stuff I'm seeing is that NIST is sort of backing away from that theory now. [Note: Mr. Pegelow was correct, based on explicit statements NIST made on August 30^th of the same month.[7] Note that NIST is yet to define any other specific global collapse mechanisms for the Towers to date.]

First of all, it's a fully-welded structure. You may get deflections, and settlement, but you wouldn't get an immediate collapse on a floor unless you took out all the columns at the same time. …

There's been a lot of disinformation going around. One is that these weak bolts fractured and fell apart. Yeah, there was bolts holding both ends of these trusses in, but those are what's called erection bolts: you initially have to hook it together. And then afterwards, they come in, on the seat of the joist, which is most of the time just two angles back to back, you go in there and you run a seal weld around it. So these are welded up. …

So this pancake theory is something that doesn't hold up. It's a phenomenon that doesn't happen in steel structures. Pancake collapses happen mainly in certain types of concrete construction, and this is not all concrete, just certain types. And this is where you put your columns in, and you pour a slab, and around these columns you've got a pocket. And once its cured, by two days, you come in and pour another slab. And then you lift these slabs up, one by one, you see the pockets, the columns sliding in them, and then once you get them up there, then you tie them off to the columns.

And this is where sometimes there's problems. They're not tied off secure enough, or a lot of times these accidents happen at construction and by the time the last one gets there it falls down, but you understand they're not even connected to the columns except by some construction jacks at time. And then of course, the other times they get collapses, they had a lot of these in Mexico during the '70s on apartment buildings during earthquakes, so I don't know, I just have a feeling that their concrete wasn't mixed right, they cheated on as much rebar as they needed, or something like that. But anyway, that's a conventional pancake collapse. It doesn't happen in steel buildings.

Jim Fetzer: And Charles, when it's all said and done, don't you have a stack of floors there on top of one another, which is the reason it's called a "pancake"?

CP: Yeah. Yeah, they just, you're right. They just pancake down. Now, they're all busted up, but they don't turn to dust.

We notice in the interview that Mr. Pegelow describes pancake collapses as only occurring when there is a series of concrete slabs that are not properly connected to the support columns, such as when they are only being supported by construction jacks, or when the slabs are being raised and are tied off to the columns improperly in the process. The only constructions to pancake otherwise, are again concrete constructions, but after completion and during disasters such as earthquakes, as Mr. Pegelow states.

The floor slabs in the Twin Towers, however, were supported by steel trusses that were not only bolted, but also welded to the perimeter and core columns to which they connected. And further, the slabs would have put the heated trusses into compression during the fires, discouraging sagging, as Mr. Pegelow had also pointed out earlier in the same interview.

The following photograph provides an illustration of an authentic pancake collapse, taking place in a concrete structure, and leaving the support columns standing, and the concrete itself in large pieces.

Fig. 9. A concrete structure collapsed during a Mexican earthquake, leaving concrete slabs mostly intact and vertical support columns standing. Notice that nearly all of this building’s mass is within its footprint, with material lying outside of the footprint being the exception rather than the rule.

If a 110-story concrete building were to pancake to its base, of course we could expect a lot of material to fall outside of the building’s footprint. But the problems with this are that, for one, concrete structures are not built 110-stories tall, and the WTC Towers were steel structures, welded, with concrete only over on floor pans over steel trusses. And for two, the incredible amounts of debris being ejected from the WTC Towers as they collapsed indicate that the great majority of their masses was not going towards collapsing further floors in the buildings (i.e. by falling onto them), but simply “falling” out of the buildings, through the air and landing elsewhere in the WTC Complex, or even hundreds of feet away in the roofs of other buildings. And we are talking about massive sections of columns that weighed thousands of pounds, and not simply aluminum panels or concrete dust, which of course traveled even farther.

Rather than anything like the previous photograph (figure 9), we observed the following at the WTC Complex after the Twin Tower collapses:

Fig. 10. Remains of WTC4 at Ground Zero, surrounded by a sea of core and perimeter columns from the Towers.

Fig. 11. Massive groups of perimeter columns ejected over at least 400 feet from WTC1 onto the Winter Garden building.

Fig. 12. To get a sense of the massiveness of the perimeter columns, compare the group of perimeter columns near the top of this image, still connected by spandrel plates, to the cranes and dump trucks beginning the clean-up effort.[8]

Here is a cross-sectional model representation of how a collapsed WTC Tower would have appeared if Greening’s momentum calculations were accurate, and all mass remained within the footprint of the building, causing each floor to crush and destroy the next with all of its contents, and then continue on with each successive floor without losing any mass:

Fig. 13. How a collapsed Tower would have appeared if all its mass went towards “pancaking”.

Greening assumes all steel, concrete, aluminum, etc., falls straight down with falling mass only increasing.

Other problems with Dr. Greening’s mathematical model include assuming no resistance to falling mass until hitting the next floor system (which suggests only the trusses were falling), but yet apparently including the mass of all vertical support columns within the same falling mass,

We consider the initiating event of a WTC tower collapse to be the failure of crucial steel support structures at the appropriate upper floor level of the building, followed by the free fall of the entire upper block of n floors through a distance hf = one floor height = 3.7 meters.

as well as assuming all WTC floors weighed the same (in reality, the top floors were much lighter). He also assumes that each collapsing mass “merges” into the mass it has just theoretically destroyed, thus seeming to re-combine the pulverized concrete and dissociated steel members for further, focused destruction of the remaining floors.

We will use this law for the non-elastic collision where the colliding masses essentially

merge into a single mass that continues to descend.

On the other hand, here is a cross-sectional representation of the WTC1 debris spread per FEMA:

Fig. 14. A more realistic representation of a collapsed Tower.

The gray clouds are representative not of smoke, but of the concrete dust that was produced, representing what must have been a significant amount of the total mass of that building, since each steel floor pan carried concrete slabs of 4” thickness, and each floor area within the core structure supported 5”-thick slabs.

Note that debris was spread much farther than actually implied in the above image; we are simply using FEMA’s diagram of major, non-concrete debris spread to be conservative. Also note that the heavy debris encircled the towers rather evenly, except in the four directions extending outwards from the faces of each building, where the debris was ejected farther. This suggests no significant leaning of the Towers during their collapses, but even ejection of material in roughly four primary directions, as if mass was being forced outwards and perpendicularly from the faces of the buildings, rather than falling this way or that as via gravity and simple deflection alone.

Destroyed floors did not re-combine to form a single, focused mass again, as suggested by Greening’s calculations, but broke up and were mostly ejected during the collapses, and thus contributed a small fraction of the total mass available to the continuing collapse, again, easily equating to 90% of the total mass being ejected throughout the collapses as shown by the heavy debris spread area alone in the FEMA diagram, and Ground Zero photos showing a clear lack of significant debris piles in the footprints of both buildings (remembering that most additional mass remaining within the footprints can rather safely be attributed to vertical core collapses after the perimeter and truss collapses, and not any type of pancake mechanism).

However, the mass that did continue downwards within either building, and could have contributed some theoretical “driving mass” to its collapse, would have been dissociated (disconnected steel members, destroyed concrete floor slabs; not a single, solid mass). This is important to remember, as dissociated material is easier to deflect than a single, focused force. As an example of this, would you rather have fifty pounds of gravel dumped onto your head from a height of 10 feet, or a single, fifty-pound rock dropped from the same height? A fifty-pound log, or fifty pounds of large woodchips? A fifty-pound steel beam, or fifty-pounds of nuts and bolts?

Intuitively, one would likely opt for the dissociated material always, and for a good reason: broken-up material is not going to exert as much energy upon a single point upon an impact, because the smaller pieces are less focused and more easily deflected; their energy is not focused solely downward upon a single point, but upon multiple points in multiple directions. If a falling solid object impacts something, though the great majority of the object may not actually make contact, the rest of its mass will nonetheless transfer energy efficiently. This is the same principle that allows a hammer to deliver much more impact force than a thin cylinder of the same diameter, though only the flat face of the either may actually make contact. With dissociated material, even such as separated steel beams and pulverized concrete, this is cannot be the case, and the material would simply be deflected rather than delivering the amount of energy that would be available if the material made up only a single solid mass.

Again, Greening’s calculations assume only single masses falling, and never dissociated materials, let alone dissociated materials that are mostly falling over the sides of the collapsing buildings and are thus providing absolutely no mass to the alleged pancake collapses. These assumptions are not only unrealistic, but they are also extremely generous to pancake collapse theory, whereas a rigorous proof of even the possibility of such a theory should allow for the very realistic probabilities of less-than-ideal circumstances. Instead, Greening makes assumptions that prove his model to be impossibly generous to the theory that he is attempting to defend.

Brief mention will also be made of the design of the Twin Towers. As with all skyscrapers, the structures are stronger at the bases than at the tops. This is very simply because the lower structure has to support both its own weight, and the weight of everything above it, while the upper structure only has to support its own weight. Structurally, this translates into thicker, heavier columns nearer the bases, and thinner, lighter columns nearer the uppermost floors. Though Greening assumes that each floor weighed the same in his calculations (he simply divides the building weights by the number of floors), the first floors to drop would have had equal or less structural mass than the floors onto which they dropped. Mechanical floors were an exception, since the mechanical floors (the windowless bands on the Towers in roughly intervals of thirds: 41^st and 42^nd floors, and 75^th and 76^th floors) had reinforced trusses, with I-beams spanning between the perimeter and core columns, and apparently encased in concrete. Floors 1 through 9 seem to also have been reinforced.[9]

Therefore, a progressive collapse would meet increasing opposition from one floor to the next, as the core columns thickened, and upon reaching the mechanical floors and lowest floors the Towers. This would indicate that the collapses would require more and more energy to continue without losing significant velocity, but instead, we observed little to no collapse velocity loss from initiation to completion with either Tower, despite thickening columns, dissociated falling material, and most of the total mass being ejected outwards from initiation to completion, and not falling straight down into the footprints.

This collapse behavior would seem more than a little counter-intuitive without considering additional sources of energy within the Towers, and yet it is an issue that has never been addressed in any form by any investigative releases relating to the Twin Tower collapses.

The following image was cropped from a high-resolution photograph taken on September 23^rd, 2006, and hosted on the Internet by the NOAA.[10]

Fig. 15.

The images that follow are included for further study of heavy debris ejections from the WTC Tower collapses, representing mass that became unavailable for “pancaking” during the collapses.

Fig. 16. WTC1 destruction.

Fig. 17. South Tower “collapses”.

Fig. 18. More South Tower destruction.

Fig. 19.

Fig. 20.

Fig. 21.

Fig. 22.

Fig. 23.

Why wasn’t there more mass at the bases of the Towers, if the floors simply fell down, one floor to the next, to the next? How did nearly all of the mass of each Tower end up outside of its respective footprint?

Regardless, the fact that so much mass moved to outside of the Towers’ footprints must be considered if one is to realistically calculate any momentum transfers from the force of one falling floor’s mass onto the next floor, and then in turn onto the next, and so forth. There simply could not have been a linear progression in terms of mass, momentum, etc., from each floor to the next, as individuals like Dr. Greening have assumed. At least 80%, (and probably closer to 90%) of the masses of either building were moved out of the way of the collapses themselves, during the collapses, as demonstrated by the relatively small piles of rubble in each of the Towers’ footprints after their collapses were complete.