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Implications of BLGB for Crushing Power
=======================================

Taking the height z to increase downwards,

z = 1/2 at^^2

taking a = 2/3g, we see that

z(t) = (1/2)(2/3)gt2

so z(1) – z(0) = (1/3)(9.8)(1)
= 3.3m
or not even 1 storey height

However,
z(10) – z(9) =
3.3(10^^2 – 9^^2)
= 62.7m
16 storey heights

So, before “crush-up”, the rate of floor crushings has increased by a factor of 16x.
The lowest column sections were about 16x stronger than the topmost ones. Call the difference, for the purpose of this calculation, 10x.

Therefore, before “crush-up”, crushing energy is being applied at a rate about 160x during 9s < t < 10s vs. 0s < t < 1s.

I expect such a wide disparity to show up in seismic records, somehow.



If we assume that ‘crushing energy in’ is proportional to ‘seismic energy out’, then if the seismic records should reveal that only 3% of KE dissipated as ‘storey crushing energy’ during “crush down”, and that this seismic energy is created at a constant rate, we can figure out what crushing energy per floor should be as an implicit function of height (and explicit function of time). I solve for the scenario of ignoring column strength differential, assuming that "crush down" takes 10 seconds.


Call the total energy expended (by dissipating KE) in crushing the bottom of a WTC tower, as a function of height z (z positive downward) "E_crush(z)"


z(t) = (1/3)g t^^2

therefore d(E_crush(z)) / dt = d(E_crush(z))/dz * dz/dt = E' * { (2/3)g t } ( just using shorthand E' = d(E_crush(z))/dz )

Interpreting the seismic record as noted above implies that d(E_crush(z)) / dt = k, where k is some constant to be determined

therefore E' * { (2/3)gt } = k
E' = 3k/(2gt)


Now, recall Greening's estimate of total KE or 10^^12J, as well as my measurements implying 3% dissipation into seismic energy during "crush-down" vs. 97%

dissipation into seismic energy during "crush-up" *.

(More carefully, looking at "Energy Transfer in the WTC Collapse" by Greening, he estimates total KE as 4.6 x 10^^11 J, and assumes 75% "available" to crush concrete. So, we can consider maximum KE available to create seismic waves to be 25% of 4.6 x 10^^11 J, or 1.5 x 10^^11 J. Not sure if 10^^12 was a revised
estimate. I'll use 10^^12 J for the purposes of this calculation.)

3% x (10^^12 J)} = 3 x 10 ^^10 J dissipated during "crush down"


Therefore, to find k, we multiply both sides of
d(E_crush(z)) / dt = E' * { (2/3)g t } = 3k/(2gt) { (2/3)g t } = k
by dt and integrate

note that integrating d(E_crush(z)) from t = 0 to t = 10 gives 3 x 10^^10 J (as per Greening estimate) and thus
E_crush(z(10 seconds)) - E_crush(z(0 seconds)) = k (10 - 0)

So 3 x 10^^10 J - 0 J = k(10)

therefore

k = 3 x 10^^9 J/s


=> E'(z(t)) = 3k/(2gt) = (1/t) (3/2) (3 x 10^^9)

therefore

d(E_crush(z)) / dt = k
=>

E_crush(z) = (3 x 10^^9 J/s) t + K2

At t = 0, E_crush(z) = 0, therefore K2 = 0


///////////////////////////////////////
// E_crush(z(t)) = (3 x 10^^9 J/s) t //
///////////////////////////////////////


By calculating t values corresponding to a given floor, we can determine the energy necessary to crush that floor. Using

z(t) = (.5)((2/3)g)t^^2 = 3.27 t^^2

t(z) = sqrt ( z / 3.27 )


For the first crushed storey
-----------------------------
after a fall of h,


t = sqrt ( 3.87 / 3.27 )
= 1.09 s


So E_crush(1st crushed storey) = (3 x 10^^9 J/s) ( 1.09 s) = 3.27 GJ


I'm frankly not sure whether to declare this in good agreement with Greening, et.al. estimate of .6GJ. I would think not, from the point of view that the earliest stage of the collapse yields the crispest data. Thus, a 5-fold difference here seems particularly bad.

OTOH, poor agreement here may be an artefact of the homogenization process, and, in fact, I expect BLGB assumptions to have more (but still questionable :-) ) validity for a collapse after it had progressed for a while.

BLGB is also in trouble with lower floors involved with "crush down" ....


For the last crushed story during "crush down"
-----------------------------------------------

after a fall of 83 h

t = sqrt ( 83 * 3.87 / 3.27 )
= 9.91 seconds


after a fall of 84 h


t = sqrt ( 84 * 3.87 / 3.27 )
= 9.97

So E_crush(84th crushed storey) = (3 x 10^^9 J/s) (9.97s - 9.91s) = .18 GJ


E_crush(1st crushed storey) / E_crush(84th crushed storey) = 18 x

This is very bad, considering the correct figure should be something like
E_crush(1st crushed storey) / E_crush(84th crushed storey) = .1 x


* recalls Benson's complaints about my drawing inferences from the graph in BLGB! I still have not looked into these complaints.


edited:

Changed the line "However, BLGB is in far more serious trouble with lower floors involved with "crush down" .... "

to

BLGB is also in trouble with lower floors involved with "crush down" ....

since being off by a factor of about 5 over seems worse than being off by a factor of about 3 under. It is the ratio of values of upper and lower storeys that may indicate "serious trouble", more than anything else.