The 9/11 Forum

Let me give a concrete example. The model is (typical) discrete algebraic 1D slab with inelastic collisions and uniform (unit) floor mass. No fail energy is incorporated (momentum only) so it favors speed. A velocity-dependent mass reduction of the upper "block" /debris zone is performed at each step, otherwise mass accretes as normal. In this case, I'm examining an interior collapse, it may be floor-on-floor or chunk-vs-floor, no distinction is made - only driving mass is considered without regard for where it came from or how non-rigid it might actually be.

Parameters are set, step-wise crush calculations are performed, and the parameters which satisfy constraints are stored along with measures of fitness. The various inputs:

Constraint variables
- time to reach terminal velocity
- speed attained at that time
- time for collapse front to reach ground level
- (optional) elevation at which terminal velocity reached

Parameter variables
- (M0) initial driving mass (in story units)
- (s) stretch (compaction measure, dimensionless)
- (c) coefficient of mass reduction, used as follows:



where subscripts are step indices, M represents upper block mass, m the impacted story mass and v bar is the average velocity over the descent phase of the ith step.

The first set (constraints) are those things I need the best observational info on, it's what lends meaning to the effort. The actual implementation does not currently check to see that terminal velocity is achieved, it merely samples the velocity at a specified time which is to represent the onset of terminal velocity. To a greater or lesser degree it does correspond to onset of terminal velocity, depending on the interaction of constraints.

The second set: the coefficient of velocity-dependent upper block mass reduction is essentially unconstrained, it can wander as needed with no underlying physical motivation whatsoever. As are the other parameters, it's a constant throughout each trial but this one adds sufficient freedom to the calculation to make solutions possible, where they would not be otherwise. The initial driving mass M0 has a practical upper bound dependent on how much can really be above a given point, but that's also mostly free to roam. Stretch, it turns out, restricts solutions more as it increases, so has a natural upper bound set by the problem and a realistic lower bound of perhaps 0.05, given there will be shedding and/or lack of entrainment.

To start small, I've focused on a starting fail story of 96, but it is about to become a variable parameter, too.

The model does not necessarily capture the actual mechanics of collapse. Part of the purpose is to see whether a basic stepwise model can fit the observables with a minimum of goal-directed tweaking. Of course, the model needs a fair shake which means reasonable accuracy on the target constraints (e.g., if it really were only 22 m/s or 6 seconds to reach speed there would be more relaxed solutions found).

No adjustment in stretch is made for mass loss, but it's not clear where the thickness is shaved from the debris zone or whether density changes, etc. This is a weak point in the process because the solutions probably don't have much accretion over many of the later floors, thus stretch is no longer a measure of compaction and will be unnaturally small. Moreover, the choice of mass loss as opposed to force increase is somewhat arbitrary, and one can substitute for the other within reason. The driving mass over time given by the solutions does not need to be strictly interpreted as such but, if it is, the stretch will lose some physical meaning and might also be an impediment to a good fit.


Given constraints:

- 4.5 seconds to velocity sample time
- velocity greater than 25 m/s at sample time
- time to reach ground between 14 and 16 s

First pass is wide open (i.e., stretch down to zero, M0 up to 25) to get an idea of what's possible. The solutions, ordered by deviation from a target velocity of 28 m/s:


(M0) initial driving mass


(s) stretch


(c) coefficient of mass reduction



The first conclusion is there are no solutions at an initial driving mass M0 less than 12 stories. That means, for the other mutual constraints, most of the mass above floor 96 needs to be the (rigid) body impacting initially.

The second conclusion is that the stretch parameter is really hostile to solutions, with little possible above 0.13. That's quite a lot of compaction if taken literally, especially early on, but this is not possible because of the implied mass loss.

The coefficient is never especially interesting, but it's small and that's good.

Restricting the stretch to be above 0.05 and M0 less than 17 gives these results:


(M0) initial driving mass


(s) stretch


With the additional constraints on stretch, there are no solutions for M0 of 15 or less stories. Conversely, the constraint on maximum initial mass has diminished the upper end of stretch to less than 0.075, which as noted does not represent a compaction greater than 13x (except early in the collapse which is not realistic).

Taking two of the solutions - the top ranked (highest v at sample time) and a low rank at around 15 stories extra mass, these graphs compare upper block mass and velocity versus elevation in stories:

mass



velocity



Comments

4.5 seconds to get to 25 m/s is tight, given the mechanics. By making mass variable, beyond that of the fixed accretion scheme, it is possible to squeak in under dubious and very narrow parametric ranges of 15 <= M0 <= 17 and 0.05 <= s <= 0.075.

Since the current results indicate a single or two floor collapse is far from a solution, these are the possibilities:

- the observation is not really a crush front but some other phenomena
- the mechanics of the crush front don't correspond to this model
- a large, intact chunk of mostly rigid upper block is the driver, not slabs
- the constraints need to be relaxed
- stretch must be made non-constant, perhaps dependent on mass loss

The last two options are the first to explore, naturally, because they're less radical and within reach.

Stonking posts OWE. Will digest...(is very hot here recently, so have been relaxing )

Stinking? Why, you...

Ah, stonking. hahaha.

I'm still grappling with the application and interpretation of stretch when there's significant mass loss. I don't think it's as I characterized it above, but what's right I don't know. If the resistive force were being adjusted instead, the stretch would be directly applicable, therefore unrealistic in the ranges where solutions would be the most plentiful.

Still quite a mystery, initiation of what appear to be ROOSD conditions as seen out of the SW corner...



The scribbly line seems to keep up with the early collapse front along the NW corner well.

But notice how far ahead the SW corner propagation front is so early into the fall.

Remember, our own data gives the movement of the SW corner only about a 0.5 second lead over the NW corner movement.


If considered as a race between the SW ROOSD and the NW ROOSD, the SW corner got a huge jump over the first 6 floors and never looked back, winning by a wide margin.

The odd thing about this gif...how did it get such a large jump so early into the fall?

For what is basically a tiltless collapse initiation, the huge SW OSS jump makes no sense.

By the time the squiggly red line (fl 98) meets fl 92 (the large fire ejection) the SW OOS front already has a 4 or 5 floor lead.

If total upper portion release happens over an angle of less than 1 degree, how can the SW OOS front have a 4 to 5 floor lead at the end of the gif????

?????

MT could you add a fixed overlay showing the building floors before collapse perhaps as numbered ticks along one corner? I am curious to see if we can pick out what is colliding with what... This presentation seems to imply that the 98th (and above) remained intact and destroyed what it came in contact with.

Not my gif. No implications. The graphics are just to guide us.

We see a few interesting things:

1) We see that the upper perimeter was pushed out over the lower perimeter along the NW corner and we can actually see it sliding along the lower perimeter on the outside. It's speed matches the antenna descent, so this perimeter sheet is still firmly attached to the upper portion.

This sliding motion is what I also believe is happening along the west face, the north face and the NE corner. Large sheets snap along a single floor and slide just along the outside of the lower perimeter.

This is basically our more universal initial perimeter split mechanism for WTC1.

2) Interestingly, at the same time we can see the NE and NW corner rooflines falling inwards. The distance between the NE and NW corners is noticably increasing during the first 5 stories of falling. (Other angles show the north face leans inward, too. We used to call this north face leaning as the general "tilt" of the "upper block", but most of us are smarter these days. For the N face we were even able to measure this initial inward tilt as about 0.6 degrees at the time of the release of the NW corner.



Combine these two attributes and we see the process of how upper sheets can fall out and over the lower sheets while the upper roofline is being pulled inward at the same time. (And do so naturally, were some process to suddenly remove portions of the core.)


I really don't like typing in this little box...so...


These collapse initiation attributes along the WTC1 perimeter seem to be simultaneously true:

1) Outward splitting of the upper face from the lower face, mostly along a single floor.

2) Perimeter sheet "stripping" of the upper sheet sections as they slide just over the lower sheet, breakingh away from the building in 5-8 fl sections.

2) AN inward pulling of the roofline toward the center of the building in the moments leading up to visible initiation and as the upper portion fall over 6 stories.

And, related to this thread, since we know the tilt was minimal, how does the SW OOS collapse front shoot out to a 4 floor lead so early in the collapse?

I strongly suspect it did not begin at floor 98 like the NW OOS front.

Or earlier ?

A new approach: Typical OOS flooring reduced live loads by region...



Very interesting to compare the propagation rates in the OOS SW region and the OOS W region.

Can be viewed as a fluid flow with different regions experiencing different frictional forces. Notice how frictional force may be about 1/3rd greater between the 2 regions. Don't we observe something like this?

Also, we expect discontinuity around the mechanical levels. A non-trivial amount? My first guess is maybe no for downward propagation, but for the lateral dust ejection pattern, yes.



AN excellent reason to watch for discontinuities or "funny business" (not so funny) around beam construction floors as femr already has in another thread.

That PATH live load diagram is odd. But it may reflect that the prefab assemblies were constructed of the same materials and would perform (fail) differently. The corner sections were supported on one side on transfer girders which from what we can see in the construction photos were simply the same short span truss found in the the short span zones. The design load appears to have been reduced by 20% over the long span load. The short span design load is increased because the span is reduced by 40% and they so they allowed 82PSF with consideration to the fact that the so called transfer girders were supporting the some of the corner loads.

In the real world its likely that these loading considerations were not carefully observed. Apparently the UPS system was located in the corner and reinforced accordingly.

I think that the take away is that with the same design throughout the tenant floors the corners were the areas which would fail first and the short span sections last under the same amount of over loads which is expected since the deal loads were more or less uniform. Note that the larger loads where on the long span side of the plan in the mechanical floors. There you find the heavy machinery and concrete pads to support and dampen the vibrations. So in a collapse the biggest loads can from above (mechanical floors) the areas which would fail first.

Major_Tom wrote:A new approach: Typical OOS flooring reduced live loads by region...

Awesome.

Can be viewed as a fluid flow with different regions experiencing different frictional forces. Notice how frictional force may be about 1/3rd greater between the 2 regions. Don't we observe something like this?

Yes. There are other factors to consider, but this is interesting.

A repost of something by Dr G:

Re: B&L Revisited - Rigidity and Crush Direction(s)
by Dr. G on Mon Oct 26, 2009 5:59 pm

Might I suggest that everyone take a look at the thesis by A.G. Vlassis entitled:

"Progressive Collapse Assessment of Tall Buildings"

Available on-line at:

http://eprints.imperial.ac.uk/handle/10044/1/1342

(Link doesn't work for me. Dr G? It asks me for a username and password.)

See especially Chapter 6.

Most of this material has also been published as a paper entitled "Progressive Collapse of Multi-Storey Buildings due to Failed Floor Impact" in Engineering Structures 31, 1522, (2009)

A quote from this paper is very pertinent to the present discussion:

"It can be concluded that in the event of a failure and subsequent impact of a single floor plate onto the floor plate below , the lower impacted system is highly unlikely to possess sufficient dynamic load carrying capacity to resist the imposed dynamic loads and prevent progressive collapse. .... This is particularly true when the falling floor completely disintegrates and falls as debris without retaining any residual strength or spanning capability

>>>>>>>>>>>>>>>>>

That title must mean he has been looking at this same problem for a while. Maybe his thoughts about progression rates are buried somewhere in the thesis.

A few posts later Dr G writes:

"According to Vlassis the "ductility supply" that would be available in the connectors (bolts and welds) is key to collapse propagation because it controls the "joint rotation demand", (theta), of typical beam - column connections as well as column - column splice/connections.

Seismically qualified connectors fail at rotations > 0.05 radians.

Vlassis estimates that for a single floor impact in a typical "tall building", the beam-to-beam impacts that would inevitably occur, involve connector deflections of 0.1 - 0.3 radians and would therefore most certainly result in connector failure."

and later, from the abstract:

"Furthermore, a methodology is developed, based on the proposed assessment framework for sudden column loss, to consider the impact of floor failure on a lower floor, in order to establish whether this would in fact trigger progressive collapse".

So why would we try to reinvent the wheel? He may be our ROOSD God.

If Dr. V's work is sound it is the underpinning for a ROOSD in the twin towers. Good catch MT and Dr. G. I'd go further and say the twins were a extremely susceptible to this compared to a typical 25' x 25' 3 D lattice type steel frame such as the Empire state building. In the typical frame you would have many more beam connections to sever/ destroy with less impacting mass and perhaps a smaller rotation on the joint because the beam length was shorter.

The column free tenant space which was so economical to build was the reason that the floors collapsed as easily and rapidly as they did. Hello Mr. Skilling and Robertson... would you care to comment?

From page 8 of this thread:



These are for the 41N, 43W ejections.

Question for achimspok or femr: Is the image in the gif of the west side ejection correct?

Are we sure that is not of the 77W ejection instead?