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Old 09-06-2007, 02:10 PM   #1
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The pitfalls of statically determinant design

Statically Determinant

A statically determinant beam is a beam where the reaction forces can be found and equations of equilibrium can be solved. More at wikipedia.

So basically, if you remove a support from a determinant design, the structure fails. On the other side of the spectrum, statically indeterminate designs take a bit more to cause failure. That is, by design and definition, an indeterminate design is redundant.

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So the WTC towers. It's quite likely we've all seen video of it falling. There's a conspiracy theory floating about that it was demolished, because they fell straight down. First, search for demolition videos. Buildings with a high aspect ratio (that is, much taller than they are in any other dimension) typically DO NOT fall straight down. They twist and topple. As one section fails, other support members try to pick up the slack - but the local load is too great and uneven (causing non uniform collapse).

Okay, so that's out of the way. Lets look at the WTC design.

The WTC towers had a revolutionary design in that it was 100+ stories tall (read: high aspect ratio) with no support beams obstructing your view with the exception of the inner core and outer core structure.

So, how was it supported?

Think of two boxes. The inner core was a very strong structure at the center of each tower. It supported elevators, stairs, services, etc. Then, we have the outer core - what made the exoskeleton portion of the building. At the base of the inner and outer core was thicker material (steel) which tapered off with increasing altitude. This is completely fine as the base will have a greater axial load and bending moment to support.

Floors

Okay, here is where the design becomes revolutionary. Each floor was supported on steel beams. Pinned to the outer core and set on a roller on the inner core. Then, concrete was poured on top of this structure. Roller? There's a reason for this. Buildings with tall aspect ratios have to combat high wind loads. To accommodate this, the inner core was designed such that it would only see axial loading - it would never see a large bending moment transmitted from the outer core. This allows the floor and outer core to move with minimal load placed on the inner core.


On the WTC website, that has since been taken down - the designers showed that their statically determinant design allowed for unobstructed views on any floor of the tower. The design is statically determinant because each floor is supported at two locations - pinned at the outer core and placed on a roller on the inner core.

Here's a diagram from wikipedia


So basic recap

Inner Core - super strong, holds major services, axial load only Supports weight of building
Outer Core - Lightweight, resists wind loads

Both of these systems in themselves are statically indeterminate through use of trusses etc.

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So here's what happens....

Plane crashes into building - inner core is compromised, but is still capable of standing.
Fire - heat causes steel to de-temper NOT MELT as some conspiracy theorists would tell you. No, there's not enough heat to melt, but there's more than enough to de-temper thus changing it's structural properties.
Load - the load never changes, but the steel's ability to hold this load does (de-tempering)
Collapse

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The design itself was flawed. One would not need an airplane to cause such damage. There's a reason they don't make 'em like that anymore.

Just for comparison, here's the sears tower. This is a redundant design - and I would expect it will take much more damage to cause a collapse. Sure, you'd lose a few floors - but that's it.



Another example is the Oklahoma bombing. While the aspect ratio isn't quite as high... The entire structure didn't collapse - it sort of sheared away from the redundant support.

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Let me know if I was rambling or if anything needs clarification....
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Old 09-06-2007, 03:54 PM   #2
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Good explanation of the construction differences.

To clarify what I think you're stating: So in the Sears tower there are essentially nine vertical buildings where only some or part of each will fail and minimally affect the other sections?
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Old 09-06-2007, 04:06 PM   #3
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Quote:
Originally Posted by beatr911 View Post
To clarify what I think you're stating: So in the Sears tower there are essentially nine vertical buildings where only some or part of each will fail and minimally affect the other sections?
In a very basic sense, yes.

If you can imagine climbing a rope net with someone else underneath you. Every time s/he steps up, you will feel the net move (perhaps only slightly). But, if you were to cut one rung, the general structure is still there - but you don't tumble down and you feel the tension in the net shift.
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Old 09-06-2007, 04:10 PM   #4
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If I may try, the concentration of vertical support in the center of the building made it an easier target.
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Old 09-06-2007, 04:21 PM   #5
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Thanks for showing how this occurred and how buildings are otherwise built nowadays. A very good explanation if you taught me something on structural dynamics.

This method of failure in one sense was a blessing. Imagine if the towers fell sideways and took out many more city blocks causing other buildings to fail. Sheesh! I shudder at the thought!
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Old 09-06-2007, 04:40 PM   #6
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Very cool trebuchet03
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Old 09-06-2007, 07:07 PM   #7
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Ah, yeah, statically determinant. I remember that now. To set up the problem such that you could solve for the forces, you would end up with goofy looking structures that you would never see in real life, where one end of the beam was held by a pin, and one was on a roller... Oh, except that you would see it in the WTC buildings. Very interesting.
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Old 09-13-2007, 09:45 PM   #8
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I just ran across this page, I thought I'd post it here since it's related.
http://www.wpi.edu/News/Transformati...ing/steel.html
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Old 09-14-2007, 09:23 AM   #9
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Quote:
Originally Posted by DRW View Post
I just ran across this page, I thought I'd post it here since it's related.
http://www.wpi.edu/News/Transformati...ing/steel.html
So, I've heard of the sulfur eutectic theory too... And here's my two unofficial pesos...

1. Eutectic reactions are not novel - there's nothing new about them, they're used extensively in steel production. That location where all those lines intersect is called the eutectic point.


2.
Quote:
This liquid slag corroded through intergranular channels into the body of the metal, causing severe erosion and a loss of structural integrity.
This process takes many many hours at the relative low temperatures of burning jet fuel. Additionally, liquid doesn't just flow through in this process - elements (such as carbon, sulfur, Mg, Si, etc.) diffuse through interstitials (gaps in the crystal structure).

Now, perhaps it's just a prepared document... But the claim is either erosion or a change in crystal structure as a result of melting. These are two different things - but I think it's more of an editing issue than a research problem...

3.
Quote:
"The important questions," says Biederman, "are how much sulfur do you need, and where did it come from? The answer could be as simple--and this is scary- as acid rain."
I laughed at this, to be honest. Acid rain, is not a simple answer Unless acid rain was a problem inside the building A more simple answer, drywall. Burning drywall releases some noxious fumes - one of which is sulfur dioxide (one of the things that kill you in house fire smoke).

Given the few hours at relatively low temperatures, it seems highly unlikely that a eutectic phase change was the steel weakening culprit - especially when the temperature was high enough for more than a sufficient amount of time to de-temper the steel. More likely, if a great deal of these formations are found, it happened after the collapse when there was plenty of time...



Jet fuel alone burns just shy of 1800F. For low, or high carbon steel @ 1800F, we get austenite steel. Structural steel is martensite which you won't find on this table as martensitic steel is metastable, that is - it's unstable and slowly (1000's of years at room temperature) transforms into austenite (among other things).
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