Simple Model of 9/11/2001 World Trade Center Collapse

In summary, the collapse of the Twin Towers likely involved the outer walls of the top floors sliding in between the walls of the lower floors, generating extreme pressure, and the floors going down in free fall.
  • #1
Ackbach
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https://www.youtube.com/watch?v=xSVHWiZu8NM

One of my relatives sent me a link to a video (above) claiming that the WTC towers must have had internal demolition in order to fall as fast as they did. I think there are good reasons to oppose the theory, but I thought it as well to do some simple calculations. Here's a model, upon which I invite comment, based on the following assumptions:

1. The collapse started with only the top floor at rest.
2. Each fall occurs with no wind resistance.
3. When a collection of floors impacts the next floor, the collision is perfectly inelastic, and the Conservation of Linear Momentum applies (a simplifying assumption, since the system is actually not closed in the $y$ direction). Also, the target floor offers no inherent resistance to joining the floors above it due to structural weakness as recorded in the NIST investigation.
4. The floors have equal mass.
5. A WTC tower is $415\un{m}$ tall, and $110$ stories. We assume the stories are evenly spaced at $3.77\un{m}$.

With these assumptions, you get (I think) a relatively simple model: free-fall, collision, free-fall, collision, etc. The two relevant equations are the kinematics equation with no time: $v_{n+1}^2 = v_{n}'^2 + 2 a \Delta y$, and the Conservation of Linear Momentum for $n$ floors hitting the next floor: $(n+1) v_{n}' = n v_{n}$, where $v_{n}'$ is the velocity right after the collision, and $v_{n}$ right before. Solving the second equation for $v_{n}'$ yields
$$v_{n}'=\frac{n}{n+1} \, v_{n},$$
and plugging this into the kinematic equation yields
\begin{align*}
v_{n+1}^2&=\left(\frac{n}{n+1} \, v_{n}\right)^{\!\!2}+2 a \Delta y = \left(\frac{n}{n+1}\right)^{\!\!2} v_{n}^2 + 2 a \Delta y, \qquad \text{or} \\
v_{n+1}^2&=\left(\frac{n}{n+1}\right)^{\!\!2} v_{n}^2 + 2 a \Delta y.
\end{align*}
This is a nonlinear recurrence relation. We would like to get $v_{110}$ in terms of $v_{0}'=0$ - essentially knock out all the middle terms. Unfortunately, Wolfram Development Platform cannot solve this in closed form. However, we can simply run this loop and accumulate the times. The time is given as
$$\Delta t = \frac{v_{n+1}-v_{n}'}{a}=\frac{v_{n+1}-\frac{n}{n+1} \, v_n}{a}.$$
Running this $110$ times yields an accumulated time of $15.052\un{s}$. For reference, a pure free-fall would take
$$t=\sqrt{\frac{2 \Delta y}{a}}=\sqrt{\frac{2 (415\un{m})}{9.81\un{m/s}^2}}=9.2\un{s}.$$
So this answer seems reasonable.

As I explained to my relative, the theory in the video suffers (I think) from the following flaws:

1. The system is definitely not closed, since the force of gravity is acting in the direction of travel. It is an "outside force".
2. As the stories collect more and more mass as the collapse progresses, the incoming mass is much greater than the target mass. Think: instead of a ping-pong ball hitting another ping-pong ball when the first story hits the second, it's more like a bowling ball hitting a ping-pong ball as you get down even to the ten-from-top floor. It's actually not clear how many stories began each collapse.

Other considerations:

3. The NIST web page explains why the internal structure offered little to no resistance. The long and short of it is this: temperatures around 1000 deg C due to the jet fuel fires was enough to weaken the structural steel to $10\%$ of its original.
4. The building was designed to withstand significant static forces, which is quite different from withstanding dynamic forces such as one floor collapsing onto another.
5. There is no evidence of any explosions occurring during the collapse.
6. It is apparently NOT true that the top floor, in its entirety, hit the ground $11$ or $9$ seconds after the start of the collapse. Instead, as you get down to the 60th floor of WTC 1 and the 40th floor of WTC 2, there are significant pauses in the collapses, on the order of tens of seconds, before the rest of the buildings collapse. It is true that the outside portions of the top floors hit the ground then, but that's genuine free-fall!

Thoughts?
 
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  • #2
For starters, the comparison with towers of snow in the video is not a good comparison, since those towers are solid.
So yes, the top part would slide off, or otherwise we need to blow up the lower part, which is exactly what we see.

Anyway, here's my 2 cents with a slightly different model.

A floor consists mostly of outer walls and actual floor.
Looking at the video it seems to me as if the outer walls from the top floor slide in between the walls of the lower floors.
And when the base of such a wall hits the floor one level down, this would generate extreme pressure, that the lower level floor would not be able to noticeably withstand.
And once the edges of the floors have been crushed, those floors would indeed go down in free fall.
It also explains that the top of the tower comes straight down - it is guided by the surrounding walls.

Furthermore, I expect the outer walls of the Twin Towers to be designed to be extremely sturdy to give stiffness to the building.
And they would be carefully constructed on top of each other with a vertical structure of steel bars.
However, that steel would have melted locally where the collision with the plane occurred with the extreme amount of burning fuel that poured out.
(As I understand it, that was the one thing the towers were not designed for - such extreme amounts of burning fuel.)
So the walls above were suddenly in a 'good' position to 'slide' next to the walls on the levels immediately below.

So my hypothesis is that we have 'swords' on the outside that pierce through the floors, leaving all floors effectively in free fall.
 
  • #3
Interesting analysis! I have watched some of these videos before and half of the claims revolve around physics and the other half are political/speculation. The former group is the one that interests me and I agree with you two. It is really hard to tell what is going on inside during the collapse as all we see is the outer beams breaking and a bunch of dust. I do not buy into the demolition theory but I think the building 7 collapse is more compelling that the twin towers. It looks way more like a demolition to me and I wish I had more engineering knowledge to understand the reports on the collapse.
 
  • #4
Just found on wiki:
They [NIST] also emphasized the role of the fires and found that sagging floors pulled inward on the perimeter columns: "This led to the inward bowing of the perimeter columns and failure of the south face of WTC 1 and the east face of WTC 2, initiating the collapse of each of the towers."

That seems to support my theory: the sagging floor broke the perimeter columns and drew the walls inside.
The weight of the plane may have helped for the floor to sag as well. I imagine that those floors were not designed to carry the weight of a plane.

And indeed, the video of building 7 looks different. That one does look more like controlled demolition.
Still, apparently NIST found:
Specifically, the window breakage pattern and blast sounds that would have resulted from the use of explosives were not observed.
which sounds like compelling evidence that it was not, leaving us to try and make sense of the physics of it.
 
  • #5
For the record, the 'tone' of that video is one of religious fanatics.
They do not want to listen to rational arguments, but they just want to propagate their own vision and their own hatred.
The comparison to a tower of snow is wrong, and can be seen as just propaganda.
The repeated references to 'high school physics' are wrong again, and are again just propaganda.
The reference to 'debunkers' that didn't defute even one argument sounds again as just propaganda.
They would not be listening to rational arguments in the first place.
Neither your model, nor my model would convince them of anything - they wouldn't be listening.

This is not objective science.
My own conclusion is that even scientists can definitely behave as religious fanatics.
 
  • #6
Yes, I fully agree. A few years ago I saw part of a documentary from a group of architects who felt that 9/11 was an "inside job" and unfortunately the way they bring up their case is through a complete overload of points that vary from engineering to political conspiracy. This kind of shifting arguments seems to be kind of common. If you challenge one point then the topic shifts to "well ok but there was this as well!". I think the most interesting thing is explaining how building 7 collapsed in such a way that looks like a free fall at first glance. All of the other stuff is not a clear proposition that is falsifiable.
 
  • #7
I like Serena said:
For the record, the 'tone' of that video is one of religious fanatics.
They do not want to listen to rational arguments, but they just want to propagate their own vision and their own hatred.
The comparison to a tower of snow is wrong, and can be seen as just propaganda.

I could be wrong, but I think there was stone underneath the snow, wasn't there? Maybe the video quality was just too poor to make it out.

I like Serena said:
The repeated references to 'high school physics' are wrong again, and are again just propaganda.
The reference to 'debunkers' that didn't defute even one argument sounds again as just propaganda.
They would not be listening to rational arguments in the first place.
Neither your model,

Not sure what my model shows, if anything. Injecting any realism into the model, such as wind resistance (compounded by having to push the air out of the space between the currently falling portion and the next floor), some structural resistance to being moved, etc., would only increase the total time.

I like Serena said:
nor my model would convince them of anything - they wouldn't be listening.

This is not objective science.
My own conclusion is that even scientists can definitely behave as religious fanatics.

In my experience, all people are religious in some fashion. Everyone worships something. The question is, what is worshipped? The greater the the object worshipped, generally, the more noble the person.
 
  • #8
That evil was a nameless evil, an evil whose name was Gnag the Nameless. - On the Edge of the Dark Sea of Darkness, by Andrew Peterson.

So- a "nameless evil" has a name?
 
  • #9
HallsofIvy said:
So- a "nameless evil" has a name?

This certainly wasn't the only howlingly funny thing in the introduction, but I must have laughed for ten minutes after reading that line.
 

FAQ: Simple Model of 9/11/2001 World Trade Center Collapse

What is the Simple Model of 9/11/2001 World Trade Center Collapse?

The Simple Model of 9/11/2001 World Trade Center Collapse is a simplified version of the events that occurred on September 11, 2001, when the Twin Towers of the World Trade Center in New York City collapsed after being struck by commercial airplanes.

Who created the Simple Model of 9/11/2001 World Trade Center Collapse?

The Simple Model of 9/11/2001 World Trade Center Collapse was created by Dr. Zdenek P. Bažant, a professor of civil engineering and materials science at Northwestern University, and his research team.

How does the Simple Model explain the collapse of the World Trade Center towers?

The Simple Model proposes that the collapse of the World Trade Center towers was a result of progressive collapse, where the weight of the upper floors caused the lower floors to fail and collapse. It also takes into account the effects of fire and the structural design of the buildings.

Is the Simple Model widely accepted by the scientific community?

The Simple Model has been extensively reviewed and validated by experts in the field of structural engineering and materials science. However, there are still some disagreements and ongoing research about the exact mechanisms of collapse on 9/11.

How does the Simple Model impact our understanding of the events on 9/11?

The Simple Model provides a scientific explanation for the collapse of the World Trade Center towers, which helps to dispel conspiracy theories and misinformation about the events of 9/11. It also highlights the importance of understanding and designing buildings to withstand extreme events, such as terrorist attacks or natural disasters.

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