The 9/11 Forum

Comparing against an analytical result which arrests in the same time, the stepwise model had to have support capacities increased 3.7x. This has been added to the previous graph of stepwise results showing velocity versus displacement (the orange line):



With this capacity, the curve much more closely resembles the blue curve at zero g, much lower capacity.

A comparison of the velocity for the original arrest and the new stepwise arrest at 3.7x capacity. The arrest is gradual in the stepwise case, as is the norm.

A look at the KE of the physics sims only shows that the arrest is abrupt (already knew that) and that it occurs at a much higher energy than the standard collapse and while the KE is in a phase of rapid increase. What's going on here?

The previous graph compares KE in the time domain, but that's not an entirely fair comparison since the high speed collapse crushes much more structure per unit time. These two graphs compare the KE of the high speed and normal collapse against position. Still, not too revealing but this time we see the KE is actually comparable by the collapse position.

High speed:


Normal:


Note on the values/units - these are kg-m-s newtons, joules, etc, but I (typically) use unit masses because the dynamics are the same. However, this makes energy values very small, and of no use for comparison to other models which use skyscraper-like masses.

The amount of energy dissipated by the high speed cases dwarfs that of the normal collapse.



The rate at which the high speed cases throw energy away early on, without comminution or pulverization, is incredible! All momentum transfer, and it happens in the first few collisions. The configuration of rigid slabs is not realistic in this case because the inelasticity is by material property decree and not by virtue of something that could behave like that in real life.

Still it is amazing how quickly the dissipation slopes stabilize to something similar to a normal collapse. (although nearly constant for the high speed arrest)

Zooming in on the critical portion of time around arrest:



Suddenly there are big steps, where there were only smaller ones after the displacement stabilized. Where does this energy go?

Let's look at the momentum of the whole structure as well as the parts to see if there are any clues.

This is total momentum over the entire collapse period:


Zooming in on the arresting portion:



There is something interesting happening right after the downward steps, particularly just after 5.7s where something sort of slow and sustained is happening - momentum increases after an abrupt drop. There are signs of elastic rebound.

Turning to the momentum only for zone B (debris):



There's a bend downward at around 5.65 sec but a straight line starting just after 5.7. The debris zone is not responsible for the bend at that time.

Zone A, the 'fixed' lower portion:


The lower portion goes into motion at several points, greatest at the event (which we take to be a collision) just after 5.7s. There have been impacts all along, several occur on the graph besides the big spikes, but they produce little recoil. For some reason, the lower structure couples with the impacting bodies a few times late in the collapse. These recoils, as we shall see, dissipate a fair amount of energy as the response is highly damped - a realistic condition.

It is reasonable to expect the timing of impacts and the density of Zone B packing as being factors in the coupling of the lower portion.

The velocity of the top slab over this critical time shows very slight increase then begins to turn down right after 5 seconds. The collection of 'debris' slabs below has already begun to slow for reasons undetermined.

After re-running the same sim with higher resolution data (yes, it's repeatable to infinitesimal detail) and having spent some time examining the results, I've identified a timeline of conditions and events through the critical period. Inspired by achimspok's high information content graphics, I've tried to summarize the findings on a high resolution graph of the total dissipated energy over the critical period. It's a little much, but be thankful it isn't animated, too.


Large version: http://i46.tinypic.com/28v9bhv.png

The large version is much easier to see. I'll let it speak for itself for the time being, except to note that the causal chain has been traced back to a connection break at 5.108 that is associated with an unprecedented and sustained decrease in average debris slab density, which meant impacts/impulses on Zone A were spread out over an interval while this happened. The debris zone did not act as a rigid body, despite an increase in volume of less than a percent!

What specifically causes the onset of Zone B expansion is not known. There seems to be nothing special about the connection break that occurs at the time, it's just like the few before it. Some chaotic resonance effect...

Major_Tom wrote:In the extreme example (100x top mass) graphed in yellow above, you talk of "terminal acceleration".

If the lower block were taller (200 floors) rather than 100, do you think the yellow velocity graph will level off to some near constant "terminal acceleration" like the blue and red examples?

Major_Tom, it may amuse you to find out that Seffen's work might be in error as to the magnitude of terminal acceleration achieved, claimed to be g/2.

I smell fear. Food runs.

Life feeds on life feeds on life feeds on life feeds on life...

David B. Benson, elsewhere wrote: It appears that Seffan made a mistake in his derivation...

I'm trying to contain myself. This is a banner day.

Do you have a handy link to the Seffen paper ? Sure I had it, but...piling system

http://winterpatriot.pbworks.com/f/seffen_simple_analysis.pdf

Unofficial source, but I think this is the final. Haven't compared to my local copy.

Simple Analysis, indeed.

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OWE:

Nice work! Sorry for my prompt departure from this forum. Working around the clock on a problem at a nuclear plant. Hope to be back by October ......