The 9/11 Forum

One more constriant on the form of the equations I described.

The acceleration must act in a physically realistic way at t(0) and t1.

So in the case of t1, f1''=f2'' ('' means second derivative over time in my notation, folks)


Also in the pre-collapse phase, you are talking about sinusoidal motion, no?

Mathematically this is probably best described by a fourier series?

Hard to fit sinusoidal motion as a polymonial.

There is probably a general measurable frequency the building is seen to move at?

This could be the base harmonic for the fourier series?

Or the equation pre-collapse could take the form of a dampened harmonic oscillator? As an oscillator subject to a sinusoidal repetitive force?

Sideways movement maybe but that up-down movement described is really weird.

Dr. G:

Thanks for the helpful comments!

Right back at ya.

Intuitively I would expect the switch-over to be very early on, say 1 - 2 seconds after t(0), but this then becomes coupled to the problem of choosing t(0).

Chicken and egg.

Thinking a bit more about my previous post makes me wonder if there really is something we could call t(0) anyway!

A different t(0) for every horizontal point on the roofline, at least, even if there were only an abrupt drop and not all this wandering. Didn't this thread start more or less on the subject of choice of t(0)?

The swaying I can accept as physically possible, but the up and down motion is very strange indeed.

I've gone back through frame by frame, over and over. While there may be some rebound in the big dark rectangle after the penthouse collapse, I think most of the vertical motion registered in the data is from swaying and deformation. The rectangle presents a large area which is subject to flexing as the penthouse goes through behind it.

My rough estimate is that the vertical motion spans about 2.5 meters or half a floor height. Can anyone confirm this number?

Depends on your t(0), of course! One measure I got was 9 pixels where a floor is 23 pixels at that horizontal location. No less than 6 pixels.

Perhaps the best we can do is take some arbitrary time, where the drop exceeds say 1 meter, and continues to increase monotonically thereafter, and make this our t(0). Another way to go would be to consider the Building 7 oscillations as the collapse initiation phase, and the monotonic descent as the collapse propagation phase!

That is reasonable and probably appropriate. There is a point where the face begins rapid distortion and the downward motion begins, frame 260-265 in the CBS video. I think that's the range for a global t(0) if one exists. The NW corner is simply the last to go around 2/3 of a second later. As if it's being dragged down...

Major_Tom wrote:Obviously in reality there is no point at which initiation ends and progression begins.

Nope, looks like it takes the better part of a second to get the whole thing moving.

Perhaps from what you are saying t(0) is also just another tool that we define. What transition does it actually represent physically?

Will that physical transition be expected to happen instantaneously? If not, then how can t(0) have a single definition?

Because the NW corner started down fairly abruptly, I think there was an unspoken assumption that it would provide a good discrete point in time for a one dimensional look at that corner. As it turns out, there's still hope for that provided we can get a grasp on the nature of the curve. In this sense, for whatever value it has, it would be defining t(0) as the time when the last of the structural integrity was lost - still approximate. Getting a measure of the bulk of the motion would give a different picture, a much earlier and fuzzier t(0). Useful, too, of course.

So maybe a decent math tool model would be a "3 structural phases" form where the phases are defined as

all t<t(o) called "pre-collapse" or....

t(0)<t<t1 called collapse initiation

and t>t1 called collapse progression

There is no physical reason why the functions representing the motion in each of these 3 regions need be related except that they meet smoothly at their boundaries.

This is also reasonable, makes perfect sense. There is an impression of continuity in the global event that one gets from looking at the video, a continuity that is yet to come through in measurements but will eventually, I have confidence. We're trying to measure a big glob of jello and say exactly when it began moving and how fast it was going. Getting the language together for that particular task is a bit daunting. This is why rigid bodies are such a popular assumption - but there shouldn't be too much surprise when the jello does not measure like a rigid body would.

I cannot emphasize enough that the building is non-integral by the time the NW corner drops; that is, it is a pile of debris approximately holding its last cohesive shape. There's nothing to crush up.

Major_Tom wrote:One more constriant on the form of the equations I described.

The acceleration must act in a physically realistic way at t(0) and t1.

So in the case of t1, f1''=f2'' ('' means second derivative over time in my notation, folks)

Not sure that's necessary.

And I had to do this post because it's #911 for the board.

crap, my post has disappeared and even not saved in drafts. Double derivate equal is not needed only the first one.

OneWhiteEye:

I cannot emphasize enough that the building is non-integral by the time the NW corner drops; that is, it is a pile of debris approximately holding its last cohesive shape. There's nothing to crush up.

Are you supporting NIST's claim that what we were looking at (at the moment of collapse of WTC 7) was an "empty shell"? I find that pretty hard to believe. Could the interior of the building totally cave in without there being some external signs of this degree of trauma visible from the outside?

Einsteen:

Sorry you lost your post! That was a problem for me once, a while ago ...... It appeared to happen when someone else was posting at the same time. I wonder if that means the second post is always lost. If this is true, is there some way it can be prevented because it is very annoying .......

Dr. G:

Today I'm really sick so I can't answer this with the support (or flourish) I prefer. Double whammy of near-bronchitis and severe back pain; everytime I cough, bad pain.

No, I don't support a claim of an empty shell. Yes, I see daylight through some of the windows after the penthouse passes but that's a relatively small region. It will take some visuals to support my position but the manner in which the top deforms just prior to the NW corner descent suggests there is no structural integrity left by that time - if you were to magically seize the upper 20 stories at this point and transfer them to solid flat ground I think they'd fall anyway. There are exterior 'pops' around the facade that indicate things are breaking violently. A line of these run up the SW corner, these are the famous WTC7 'squibs'.

The impression is that of a big block of jello having the foundation removed, nearly simultaneously, across the entire span.

einsteen:

I've been victim of that, too. I highly recommend what Major_Tom said elsewhere: select the whole post and copy before pressing submit. For long ones I just type it in a text editor and paste it in.

OneWhiteEye:

Hope you're feeling better!

A shot of your favorite alcoholic beverage always helps in such situations!

On the subject of the pre-collapse condition of WTC 7 you might find this interesting:

Yet another way to think about the collapse of WTC 7 is to compare it to the behavior of a high rise building in an earthquake. Based on some preliminary reading on this topic here are a few interesting observations:

· The San Fernando earthquake of February 1971 was ~ a magnitude 6 event. Nevertheless most high rise buildings within 20 miles of the epicenter survived with only minor damage. One such structure, the Kajima Building, recorded vertical motions of +/- 9 inches with peak accelerations of 0.2g.
· The Northridge earthquake of January 1994 was a 6.7 magnitude event. Vertical response data were recorded for 12 buildings within about 40 miles of the epicenter, including four steel frame buildings. The maximum measured roof acceleration was about 0.8g, and some of these buildings were badly damaged.
· J. F. Hall et al. have modeled the behavior of a 20 story steel frame building located near the epicenter of a Northridge-type earthquake, (See Earthquake Spectra 11(4), 569, (1995)). The simulation showed that welded beam-to-column and column-to-column splice fractures occurred in the lower part of the building during the peak motion eventually leading to global collapse of the modeled structure.
· What is most interesting about Hall et al’s. calculations is that the seismic event was modeled as a horizontal ground motion (pulse) lasting 4 to 5 seconds. However, the building took about 13 seconds to really begin its final fall. In the post-pulse period, from zero to 13 seconds, the roof underwent lateral (and vertical) oscillations with displacements up to 3 meters before global collapse set in!

Dr. G wrote:A shot of your favorite alcoholic beverage always helps in such situations!

That was some good advice (--hiccup--).

On the subject of the pre-collapse condition of WTC 7 you might find this interesting:

Yes I do. Very much so. I was into EQ visualizations before einsteen's smearogram sucked me into this full bore. Here's the Landers CA quake and aftershocks in 3D:



I've been thinking about the earthquake-related implications of all of this. I hope, if NIST is even half right, that structural engineers are, too. I don't know... no offense to those here who may be in that field, but they seem to be a lazy lot. (Edit: OK, maybe only a few that I've been exposed to)

I noted in particular the reaction of one individual (in this field) at another board to the NIST recommendations that came from the WTC7 study. Cried like a baby, he did. My feeling is, if you can't stand the heat, get out of the kitchen. In my field I don't have the luxury of designing only for the nominal case, I must account for even the hazards of deliberate misuse.

Imagine, a person who designs hi-rises can't be bothered with making them strong enough to face a few hours of unfought fire on a limited number of floors! How ridiculous is that? I guess when LA has its inevitable 7+ magnitude quake, the entire skyline will be gone in less than 12 hours - even if all the buildings survive the shake itself - because water's not available, fires are everywhere, no one's fighting them, and the engineers couldn't be bothered to plan for the inevitable.

Dr. G wrote:Yet another way to think about the collapse of WTC 7 is to compare it to the behavior of a high rise building in an earthquake. Based on some preliminary reading on this topic here are a few interesting observations:

· The San Fernando earthquake of February 1971 was ~ a magnitude 6 event. Nevertheless most high rise buildings within 20 miles of the epicenter survived with only minor damage. One such structure, the Kajima Building, recorded vertical motions of +/- 9 inches with peak accelerations of 0.2g.
· The Northridge earthquake of January 1994 was a 6.7 magnitude event. Vertical response data were recorded for 12 buildings within about 40 miles of the epicenter, including four steel frame buildings. The maximum measured roof acceleration was about 0.8g, and some of these buildings were badly damaged.

These are incredible numbers. 0.8g is a lot! Thanks for sharing this data, it's something I've been curious about and didn't even know existed. I wonder what the displacements were for the 0.8g accelerations if 0.2g produced 18 inches peak-to-peak?

We know the lateral displacement of WTC7 was of similar magnitude in the wobble just prior to descent. It may not be outrageous to consider similar vertical fluctuations especially since the damage to WTC7 was catastrophic where the examples you cite were only severe. Still, the west side seems not to go up and down much to the eye, where I can see the sidesway. The lateral motion is sufficient to account for initial drift in the vertical data einsteen obtained because it would have the corner vertex wandering in and out of the pixel slice. The only real vertical motion suggested is from the big rect, and the geometry as deformation ensues is such that my measurement becomes unreliable as an indicator of fine motion. It may be OK as the geometric center of the rect, but that's not the same thing.

· J. F. Hall et al. have modeled the behavior of a 20 story steel frame building located near the epicenter of a Northridge-type earthquake, (See Earthquake Spectra 11(4), 569, (1995)). The simulation showed that welded beam-to-column and column-to-column splice fractures occurred in the lower part of the building during the peak motion eventually leading to global collapse of the modeled structure.
· What is most interesting about Hall et al’s. calculations is that the seismic event was modeled as a horizontal ground motion (pulse) lasting 4 to 5 seconds. However, the building took about 13 seconds to really begin its final fall. In the post-pulse period, from zero to 13 seconds, the roof underwent lateral (and vertical) oscillations with displacements up to 3 meters before global collapse set in!

That IS very interesting. Horizontal driving forces result in vertical displacements. Not too surprising when you think about it, the structure will fall into a resonance according to the preferred modes. Fascinating. Applied energy was stored temporarily as vibrational energy in the structure but the structure failed over time before all the energy could be dissipated.

I want to dissect einsteen's last set of data as it pertains to curve fitting. It should be noted that he posted the data in meters based on a height of 174m, and I've subsequently scaled it up by 186/174. This is no small thing when it comes to shaving hairs off of g and we must keep in mind a certain assumed error band around the story height. For the purposes of this examination, I've trimmed the first ~ 0.8 seconds off.

Here are the points over the entire range (links below are to full size images):


http://i38.tinypic.com/2vb7c7s.png

Zooming in on the first second (isn't that oxymoronic?):


http://i33.tinypic.com/2h4c3n9.png

You can plainly see one effect of integral pixel resolution, the stairstepped nature of the curve. What you don't see is in what I've trimmed and this is the bouncing between two values as the displacement transitions slowly between them. However, none of this is a problem. As einsteen pointed out, the oscillation reflects the true intermediate value by virtue of spending a proportion of time between the two integral bounding values. Also, several near slices can be averaged as well as a range of threshold masking values to give subpixel resolution. All in all, it's pretty smooth anyway and a decent enough resolution.

I've laid generated curves on the graph to compare to the point series:


http://i33.tinypic.com/dzfd3t.png

A constant acceleration of 9.8 is the lower bound in lime green, and a constant 11 is the upper bound in sky blue. For a freely falling body starting from rest, anything above the green line is the forbidden zone. The key here is starting from rest, the subject of our most recent discussions.

Since the point series seems to be a bit steeper than g and largely lies above the green line, we must immediately consider three things: 1) the story height estimate may be too great 2) I lopped off too much of the beginning and 3) the initial velocity is not zero. #1 and #2 are reasonably likely, but I'll continue with the data as is.

The thin unbroken lines represent initial velocity increments in 10ths. The highest curve, which rivals that of 11m/s^s with no initial velocity, represents a mere 0.5 m/s. Thus we see that, for the first second, twiddling with the initial velocity can easily account for a non-physical fit result which assumes no initial velocity.

After adding these guide lines, let's pull back and see what's going on for the rest of the dataset:


http://i34.tinypic.com/vo4aqr.png

The point series runs away from all but the forbidden blue curve! Ouch!

Note: I did all this before falling ill, I'm just trying to flush, so to speak, before I get too far behind. The booze has given me renewed vigor.

A couple of things are evident from the last graph. There's a bump, and there's a shallowing of the curve in the latter half.

Let's have a closer look at the bump:


http://i38.tinypic.com/2i20cjd.png

It gets quite steep before it peaks and shallows out. If we split the curve there and do fits, what happens? As a nod to Dr. G's power fit, we have:

The entire set:

http://i37.tinypic.com/2q037nt.png

0 - 1.5 seconds:

http://i37.tinypic.com/2mq2tzt.png

1.5 seconds to end:

http://i35.tinypic.com/2d91mr7.png

The fit for the latter part is not all that different, in this method, from a fit on the entire set. But the first 1.5 seconds comes up a bit steeper, as would be expected. How a particular fitting method deals with this bump is important. For piecewise (say spline) fitting, something that could be useful to get an interpolation of the curve and thus specify acceleration at each point, there's going to be a pretty substantial swing in acceleration as it conforms to the details of this bump. If the bump is not real, this could lead to some unwarranted conclusions about the dynamics.

So the next big thing is to see how real this is. It may be beyond the scope of what I can do, but I'm going see if I can chase down einsteen's smear and see if there's any telltale signs of threshold drift due to brightness variance or smoke contamination.

OneWhiteEye,

I am having trouble with your (or Einsteen's!) calibration. I am getting a drop of 51 to 53 meters at NIST's 11 second mark. Now I believe that t(0) - remembering there may not be a "real" t(0) - should be set to ~ 7.2 seconds, hence "11 seconds" on NIST's timeline is 3.8 seconds into the collapse. So if we take the drop to be 52 meters at 3.8 seconds we have an average acceleration of 7.2 m/s^2.

The curves you have plotted show a drop of about 72 meters at 3.8 seconds in which case the average acceleration is very close to 10 m/s^2 or a tad greater than g. As you know I have an aversion to accelerations close to g, hence you can imagine how I feel about accelerations that are greater than g!

P.S. I was about to post when I saw you have added a comment on the "bump" in Einsteen's curve around the 1.6 second mark. Yes, I noticed that too.....

If the first motion involves the upper block (let's say this is the floors above the 8th floor) dropping on to the 8th floor after a fall of 4 meters, the next "break-through" would be the 10th and heigher floors dropping on the 9th floor so we would expect to see a speeding up after a drop of ~ 8 meters. But to figure this sort of detail out it is crucial to have the drop distance calibrated with the best possible precision.

Dr. G:

Hmmm, quite a discrepancy! I will re-check it front to back. Ugh, this why I hate leaving the pixel domain. What I can say is this: I used (very close to) the same scaling to get the horizontal data which matched NIST data:




(I like showing that, it makes me feel overly self-important, heh heh)

I did have to stretch the graphics so the scales matched on the overlay, but the important thing is the calculated peak to peak values are the same. Now, this is accuracy to inches so, while I'm not sure of my own calibrations, this would seem to verify the correctness. I was a little concerned about using a scale obtained from vertical measurement to do horizontal conversion, but the results seemed good. There could be a few percent error, I wasn't being too picky. But there's too much difference between your values.

All I did was scale up einsteen's data; do you have a smaller, but still large, discrepancy with his data?

I will double check everything.

PS Obviously einsteen didn't use my scaling, the reason I say they're comparable is because I'd earlier overlaid his with mine and they weren't very different in overall magnitude.