Scaling, Euler's Formula and My WTC Model

SEE IMPORTANT UPDATE BELOW

I described and took pictures of my 1:200 scale WTC tower model here on thios blog.

I found that my model was extremely resistant to any sort of collapse.

If (admittedly a large "if") my model was an accurate representation of a WTC tower, my findings would strongly imply that the towers were blown up by demolition.

There were already many reasons to think the WTC was blown, but I built the model with hopes of creating a convincing physical case for demolition.

Importantly, I built the model in good faith. I didn't know what to expect, and if it had collapsed in a heap, I would have reported that. One reason to do experiments, after all, is TO LEARN.

The tower was built out of steel wire and was a 1/200th scale version of a WTC tower-- in terms of basic proportions. I knew it was not a perfect model and I never claimed it was a perfect representation of a WTC tower. I built it because I was interested in the collapse mechanics. In terms of scaling down, I tried to make things roughly proportional, such as weight and column strength.

I was in fact worried about scaling issues-- that idea that although the structural elements were proportionally smaller, that there would be a different degree of strength. But I wasn't sure how to control for that, and some limited research I did on scale models failed to reveal what I should be worried about.

So I proceeded with the model and you can see the results here.

Predictably, revealing my model here on my blog induced cries of derision from the usual suspects. What an idiot I was, what a moron I was, how sad I was, how pathetic I was, how sad if I passed my idiocy to my children, etc etc.

Unfortunately, there was very little in the way of productive criticism as to what was wrong with my model-- and in fact that was what I was looking for.

Yesterday, I noticed a new comment saying how dumb I was, how I should have taken an engineering course, then mentioning the strength of an ant relative to a man.

Putting aside the gratuitous insults, the ant strength idea was something I could grab onto, and I did a little bit of reading on the strength of ants versus men. According to one site (that I can't locate right now), the apparent strength of ants has mostly do with the relationship between strength and the cross-sectional area of muscles.

So that wasn't very useful. But the same site also noted how a man scaled up to 100 feet would not be able to stand, because his bones could not stand the weight-- even if they were proportionally bigger.

Now that sounded like something relevant to my model.

So I did some reading on column load versus length, and found Euler's formula. Most relevant to my model:

Another bit of information that may be gleaned from this equation is the effect of length upon critical load. For a given size column, doubling the unsupported length quarters the allowable load.

What this would mean in simple terms is a column gets progressively weaker as its length increases, even if it is proportionally stronger.

Thus, let us say you have a two inch column of thickness 1 (Column A). It can support load X without buckling.

Now, let us say you have an eight inch column (Column B) of thickness 4.

What load can it take? According to Euler's formula, very approximately, F = K/L^2
Where F = load force, K is a constant and L is length. Basically load is proportional to the inverse of the length squared.

For the two columns, the constants should basically cancel out. For column A, F= K/1 and for column B, F= K/16-- simply in terms of length. A column 4 times longer than another column of the same thickness would hold only 1/16th the weight or load.

But column B is 4 times thicker, making it in essence 4 times stronger. So it could roughly hold 1/4th the weight or load. Thus, VERY SIMPLY, a column that increases in size proportionally in terms of thickness and length will hold less weight by the factor that it increases in length.

Getting to my model now. It was a 1/200 scale, meaning the columns were 200 times stronger to the analogous WTC columns if they were perfectly proportional in terms of size and strength.

And this would then explain the scaling effect quite well-- why my model was so resistant to collapse.

But were my columns proportional?

In fact, my columns were not perfectly proportional. They were 1.9 times smaller proportionally than a WTC outer column, in terms of cross-section. And I used less of them per outer wall (24) than the WTC (62). So that lowers their effective strength by 2.6 fold. Another important consideration was that I had only 1/5th the floors of the WTC (22 versus 110), thus making my columns proportionally 5 times longer than WTC columns (though this is somewhat counter-balanced by the about same degree of horizontal cross-bracing as the WTC outer columns. So we can lower this length factor somewhat-- say 2-fold. This means my shorter model columns were approximately 20-fold stronger than the WTC columns (200/(1.9 x 2.6 x 2)) taking these factors into consideration.

Other factors are that:
a) the WTC columns were box columns whereas my model columns were solid thin wires (and a larger cross-sectional area hollow column is significantly stronger than a solid column with a smaller cross-sectional area)
b) the outer wall columns in my model had over 100 times less steel per cross-section than an individual WTC outer column (0.9 mm round) at floor 80 (180 sq. mm proportionally versus roughly 18320 sq. mm)
c) the galvanized steel wire in my model was undoubtedly poorer quality than the steel used to make WTC columns.

I will factor in that 100-fold cross-section and calculate that my individual model columns were at least 5-fold weaker than the WTC columns!

SO-- in fact, my model WAS indeed significantly weaker than the WTC and STILL DIDN'T COLLAPSE.

AGAIN, I CHALLENGE ANYONE TO BUILD A SCALE WTC MODEL THAT COLLAPSES UNDER ITS OWN WEIGHT.

P.S. Made a mistake in calculating cross-section of steel in the model columns and significantly revised the conclusions to this post a couple of hours after posting.


UPDATE 12/31/06: this section "b) the outer wall columns in my model had over 100 times less steel per cross-section than an individual WTC outer column (0.9 mm round) at floor 80 (180 sq. mm proportionally versus roughly 18320 sq. mm)" was wrong and thus my overall conclusions were very wrong.

I made a major error in calculations-- actually my model columns were 32400 sq. mm in proportional cross-section-- making them 1.8-fold stronger than WTC columns. This obviously changes the overall equation and means my model was roughly 34 times stronger proportionally than the WTC. Thus my model is not a good system for studying collapse mechanicS AND NEEDS TO BE REDESIGNED.

UPDATE 2-- 1/1/07-- OOPS again. Realized I factored in the scale of the columns twice, and the first time going the wrong way-- so I need to take away a factor of 1.9. Also found another mistake, meaning I need to redo the overall calculations.

So, my model columns were 200X stronger than the WTC columns due to scale.

They were 1.8x stronger proportionally, in terms of cross-section of steel.
I had 2.6x fewer columns per wall and 5x few floors meaning the effective strength of the model columns was 5x weaker. Except my model columns had similar cross-bracing to the WTC, meaning we need to lower the 5x figure. Let's be generous and cut this in half to 2.5x.

Thus we have 200 times 1.8 divided by 2.6 divided by 2.5 = 55.

Meaning that VERY ROUGHLY the model columns were 55 times stronger proportionally than the WTC columns. In reality the model is probably less than this-- due to weaker steel and inferior column shape. But the bottom line is the model is far too strong proportionally, and needs to be redesigned.

The simplest way would be to redo the model design with even smaller wires than what I used.

One more attempt at inducing collapse

I picked up up the top section shown lying on the floor to the right here:

I raised this section of the tower two feet over the top of the tower stump, then let it drop straight down onto the tower.

There was no collapse. The only thing that happened was the bottom of the top section got a little bent in.

Construction Details

The basic construction unit was a steel wire grid composed of 1/2 inch squares. The steel wire was approximately 0.65 mm thick. At 1:200 scale, this would translate to a 5 inch thick solid steel column-- smaller than a WTC outer column but more steel.

My 12 inch model wall had 24 columns. This translates to about 1/3 of the columns on a WTC wall (62). However my columns WERE about 2.4 times thicker in terms of steel than a WTC wall at the 80 floor (2.1 inches of steel-- 13/16 inch x2 plus 1/4 inch x2), and since I had only 1/3 the outer columns, my outer walls were overall WEAKER than a WTC wall-- at least at the 80th floor.

The outer wall was somewhat WEAKER but the inner core in my model was made up of about 96 x 1 mm columns. It is hard to know how this compares to the real WTC core, as I don't have the dimensions of the WTC core columns. I did have more columns overall than in the WTC core, and I could make the assumption that my core was somewhat stronger than the real WTC core.

The core was roughly the same dimensions as the real WTC core-- in my model it was 4 inches on one side versus 8 inches on the long side. This translates to 67 feet by 133 feet, which is very close to the WTC core dimensions.

The floors were one sheet of the steel wire grid, and my floors did not extend into the core area. The floors only spanned the distance between the outer walls and the core. The floors did not have concrete or any stiffener or truss system. Overall, the floors were probably comparable in strength to a WTC floor. Each floor was attached to each wall in two places by 0.5 mm wire, and to the core in two places by 0.5 mm wire. Thus, the floors were not anchored tightly to the structure, and this favored a collapse mechanism.

There were only 22 floors, which makes the tower weaker compared to the WTC with 110 floors. Each floor adds significant strength and rigidity to the structure.

My WTC model was divided into three sections, at 2.2 foot intervals, similar to the real WTC which had mechanical and elevator floors dividing the tower into thirds. The lowest section had an outer wall of three sheets of steel grid, the middle section had an outer wall of two sheets of steel grid and the top third had an outer wall of one sheet grid. Between the sections, I had a triple-ply steel grid floor which was to mimic the solid steel beams of the mechanical and elevator floors. My core was similar in the middle and top sections, but had an extra set of "columns" in the lowest base section.

In terms of overall load, the weight of my tower was approximately 105 pounds. Since my tower was a 1:200 scale model, we can multiply that weight by 200 in 3 dimensions-- 115 X 200 x 200 x 200 = 8.4 x 10^8 pounds = 420,000 tons. This is actually very similar to the estimated weight of one WTC tower (between 250K and 500K tons).

My tower was 19% structural steel, which is very similar to the figure I've heard for a WTC tower (20% structural steel).

SUMMARY: My model was similar in overall structure to the WTC, but had many fewer floors. Fewer floors with more space between should have favored a rapid collapse. The floors were not tightly and strongly anchored to the vertical elements, and this should have favored collapse.

The outer walls of my model were somewhat weaker in proportion to the WTC. Possibly the inner core section of my model was relatively stronger than the real WTC, though my model core did not have a well-ordered inner structure. My core was therefore not as structurally sound as the real WTC should have been.

12/10/06

(click on pictures twice to enlarge)

Steel frame before loading. This has three outer walls intact revealing the inner core and a few floors. The fourth wall has not been attached yet:




Complete loaded tower. This is a 1:200 scale model-- 1 foot wide by 6.5 feet tall. It only has 22 floors though:

The frame is about 30 pounds of steel wire, it was loaded with 50 pounds of sand in bags and 35 pounds of ceramic tile-- the weight was fairly evenly distributed.


"Plane" damage-- severe case-- about 1/3 of the core is severed:

side view:



To mimic fire damage, all the columns on one side were cut along with all the columns of half of the core. The tower was quite stable with this damage:


I could get the tower top to lean over if I pulled on it VERY HARD:




The tower was NOT going to fall down on the "impact side"-- but I could tilt the tower on the other side of the "plane damage" more easily. The tower here was stable-- in no danger of collapsing:


I cut some more of the core columns, and the top leaned at a 90 degree angle:




I cut all of the core columns and the top fell some more:



A very unexpected result!!!!! This thing was stable as could be!



Finally I cut all the columns on one floor and the top fell to the ground. Surprisingly (joke), the top stayed intact after falling!