The floor-to-column connections need to be strong enough to pull the perimeter inwards to the point of collapse initiation.
That depends on where in time, or with what event, you define the term "collapse initiation".
I am not sure you actually understand what
@econ41 explained: That the sagging floor trusses did NOT need to pull so strongly on the perimeter that the perimeter would already fail just because of the pull. That the sagging trusses did NOT need to pull in the perimeter the full 50 inches, or what was it, that were observed. That instead, after some initial tug-in, the already bowed columns, an account of having lost some of their capacity due to this bowing, would continue to bow because of the weight they already beared to begin with.
They were able to bear that weight as long as they were straight.
And still able to bear that weight as long as they were pulled in only a leeeeeeettle.
But at some point - when pulled in by 1/4 inch? 1 inch? 4 inches? - their capacity has dropped to the point where they no longer can carry the weight they had carried for 40 years.
And then, that weight drops, and pushes down, and THAT will continue to bow the columns even further. No more tugging from the trusses needed.
Do you understand that ANY
pulling inward with ANY force at all, no matter how small and trivial, results in a corresponding inward
bowing of the perimeter columns?
One Newton of force pulling inward (1 N is e.g. the weight (gravity force) of 0.1 liters of water - a force that most small children or frail elders can easily exert) deforms the perimeter columns by X meters of inward bowing. This is because every solid material has a stress-strain ratio where, starting at the origin 0|0, each positive stress causes a positive strain. The stress-strain curve is not linear, but steadily increasing at least as long as the strain is elastic. X is of course very small for small forces such as 1 N.
Now, the vertical load bearing capacity of a column is premised on it being a) vertical and b) straight. These two properties known jointly as "being in plumb". A simple, symmetric column, has its maximum load bearing capacity when it is in plumb. You pull it out of plumb by any amount, and load bearing capacity decreases accordingly. Do you underdstand that?
So, pull on a column laterally with even a small force, and you will bend it, and as a result decrease its load bearing capacity.
Of course, a small force resulting in a tiny bow should decrease capacity by just a tiny bit and be of no concern in a practical building.
Now, have multiple trusses sag and each pull inward with a force far greater than children or seniors could.
Do this even on two or more consecutive floors.
What are the possibilities? Let me enumerate them completely:
- Columns still do NOT visibly bow inwards, and DO retain sufficient capacity to hold the weight above
- Columns DO visibly bow inwards, but still DO retain sufficient capacity to hold the weight above
- Columns do NOT visibly bow inwards, but do NOT retain sufficient capacity (because inward boing, which is there but small enough to not be visible, is already so significant that column capacity drops below load)
- Columns DO visibly bow inwards, and do NOT retain sufficient capacity
(By "visibly" I mean measurable in the photos and videos that we have, with their limited resolution, contrast etc.)
We do not see the floors sag.
We only see columns. We assume that floor trusses at some point start to sag and tug. Here is a sequence of observations on video when that happens:
- At first (before truss sagging), the columns do not bow inwards
- then they start to bow inward but we can't see it yet (truss sag has begun)
- then, as they continue to bow inward, we start to see it
- then they continue to bow inward even more
- and finally we see them yield
You say "collapse initiation". In which of the 5 phases described above do YOU figure "collapse initiation" occurs, and what IS this "collapse initiation" event?
econ41 explained to you that at first, columns bow inward because of truss sag.
We can't see the inward bowing at first.
But the moment ANY, even invisible, inward boing occurs, the load bearing capacity of the column decreases
This causes a change in the stress:strain curve of the column - it gets steeper (the same stress causes a greater strain)
So, with the same vertical load on the column from above, because the strain in the column increases, that load drops in height. This drop in height is due to one or both of two factors: Column gets compressed vertically OR column bows more.
Initially, as the trusses start to tug inward ever so gently, I'd assume the column strain is wholly in axial compression.
But as the trusses tug more and more, and the columns bows more and more due to being tugged, and its strain due to vertical stress goes up and up, at some point the vertical stress will begin to cause some increased bowing in addition to the bowing caused by the sagging trusses.
This may happen before, just as, or after the inward bowing first becomes apparent on video.
And so it now continues that BOTH truss sag and vertical load cause the column to bow further.
And bow further...
And bow further...
Until "finally we see [it] yield" - my phase 5 above.
I would argue that the
latest possible moment that could pass as THE collapse initiation event is when the vertical load on the
already bowing columns starts straining the columns by bowing them inward further.
Again, this may happen before we even notice any inward bowing, as we first notice inward bowing, or at any moment during the time interval that we see inward bowing up until the moment before the columns yield.
From mere observation of videos, we can't pinpoint that exact moment of collapse initiation. We can't see how much of the bowing is caused by lateral tugging, and how much by vertical stress.
So where, exactly, do YOU place that moment of collapse initiation?
The point I am trying to make is that, if you put that moment as late as the moment where columns yield, or at which inward bowing is so pronounced we can measure its progress,
then NO, the "floor-to-column connections" do NOT "need to be strong enough to pull the perimeter inwards to the point of collapse initiation". They only need to be strong enough to pull the perimeter inwards just enough for the perimeter's capacity to drop just enough so that the already existing vertical load would continue to strain them out of plumb.
As econ41 said: The trusses do not need to pull the perimeter 50 inches inward. Perhaps a pull-in of as little as half an inch will do.
That is, there's an interesting ratio between the (lateral) strength of the perimeter structure and the strength of the floor assembly and its connection to the perimeter.
That's partly where they intuition that the buildings were "held together" by the total structure (including the floors) comes from. This strength was of course completely destroyed by the falling upper section impacting the floor assemblies (as
@Oystein explains). It's just a matter of building something (at a manageable scale) that actually behaves this way.
I do not disagree with this, but want to caution that the real structure of course reacts with more complexity to the changing stresses.
You say "the buildings were "held together" by the total structure (including the floors)". Totally correct of course.
This "total structure (including the floors)" is indeed what pushed the "strength of the perimeter structure" to what it was: The floor structure supported the perimeter structure laterally every 12 feet of height.
The perimeter structure had its own in-plane lateral support structure - the spandrels on each floor between any two columns. But it had no lateral support in the perpendicular direction - inward and outward. That was the structural purpose of the floor trusses. The cold steel trusses would prevent the perimeter both from bowing inward and from bowing outward.
If one floor truss was taken out, or even started to sag and pull inward (that would happen already if you merely cut the bottom chord of a floor truss - if you want to develop a realistic hypothesis of intentional demolition, that is the exact point where I would start!), then the trusses to its left and right, and above and below, would still be there to resist that local inward bowing, and I am certain that this is then enough redundancy to not cause any trouble.
Quite possibly, the structure would also survive if all trusses on one floor sagged.
But the reality of 9/11 was that there were fires raging on half a dozen and more floors at once, weakening several consecutive floor systems, making perhaps more than one sag and tug, and even those that did not sag would provide less support in the inwards-direction on account of being heated. And so that simultaneous failure and/or weaking of several adjacent floors left the perimeter laterally unsupported over not one, not two but multiple floors.
And I am sure you know about Euler buckling: Increasing the unsupported length of a column by a factor N decreases its capacity by a factor of N².
Have one floor sag instead of acting as lateral support increases unsupported column length by a factor of 2, and decreases column capacity by a factor of 4.
Have one more adjacent floor fail its lateral support role, even while not yet itself actively sagging, and unsupported length increases by another factor of 1.5, and capacity drops by another factor of 2.25.
This effect comes on top of the increasing strain:stress ratio due to pulling things out of plumb - and further complicates the task of pinpointing the "collapse initiation" event.
Perhaps a single floor sagging would have to pull in the perimeter by 2 inches to cause collapse, but 2 floors sagging would need to pull in by only half an inch, and three floors by mere millimeters. After that, increased strain caused by equal weight testing doubly decreased capacity becomes dominant.
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The simple, real world
experiment you can do is still the empty beverage can that you can stand on if you are careful and not too heavy, and that also your child, weighing not even half of what you weigh, coulds stand on, but if you tap the side of the can ever so lightly, it will collapse and crumple up completely instantly.
(Caveat: I have crushed plenty of soda cans that way in my teenage years in the 1980s. It was a fun thing to do, required a bit of balancing skill and was thus something to show off on the schoolyard. However, I am pretty sure that today's soda cans, in a quest to save costs and material and amount of garbage, have much thinner, lighter walls than they used to in the 1980s. I have not tried this experiment in decades. As a matter of fact, I have not even drunk from a beverage can in years, because for environmental reasons, such cans are hardly in use anymore here in the EU. I personally haven't bought any since the 1990s I am sure. The only ones I ever get to see nowadays seem to be for energy drinks - yikes! )