Judy Wood's "Math"

[ellindsey]

I've updated my script with some numbers from the latest Greening paper. The current version factors in the energy required to destroy each floor (which includes the energy required for the observed pulverization of each floor).

With the numbers and assumptions I've plugged in, the script gives a collapse time of 19.2 seconds, which is still reasonably close to the observed collapse times. I have found that the collapse time is highly sensitive to how much mass we assume was lost from the collapse wave at each stack. The Greening paper states that at least 90% of the mass was retained in each collision, and that's the number I've plugged into the code at the moment. Feel free to copy this script and plug your own numbers in.

I don't claim that this is a particuarily accurate simulation of the actual collapses, merely a tool for visualizing how a progressive collapse occurs and estimating the time required.

from math import *

floors = 110.0 #110 stories
impactfloor = 95.0 #collapse initiation floor
floordist = 3.7 #in meters
gravity = 9.8 #meters per second squared
floormass = 3860000.0 #in kg, assume all floors have the same mass
massloss = 0.1 #amount of mass to fall off to sides on each impact
crushenergy = 603000000.0 #603MJ to destroy one floor

time = 0.0 #total accumulated collapse time
height = impactfloor * floordist #height of collapse wave
debrismass = (floors - impactfloor) * floormass #mass of debris falling
v = 0.0 #velocity of debris falling

while height > floordist and (v > 0.0 or time == 0.0):
    #Freefall one floor
    
    t = (-v + sqrt(v*v + 2.0 * gravity * floordist)) / gravity
    height = height - floordist
    time = time + t
    v = v + gravity * t
    
    #impact next floor

    #calculate impact energy, 1/2MV^2
    e = 0.5 * debrismass * v * v
    #subtract energy required to destroy floor
    e = e - crushenergy
    if e < 0: #collapse stalled, not enough energy to break floor!
        v = 0;
    else:
        #accumulate mass of next floor
        debrismass = debrismass + floormass
        #calculate resulting velocity
        v = sqrt((2 * e) / debrismass)
        #and spill some off the sides
        debrismass = debrismass * (1 - massloss)
        
if v > 0.0:
    print "Collapse reached ground at",time,"seconds"
else:
    print "Collapse halted at",height,"meters after",time,"seconds"

 

[R.Mackey]

I've updated my script with some numbers from the latest Greening paper. The current version factors in the energy required to destroy each floor (which includes the energy required for the observed pulverization of each floor).

With the numbers and assumptions I've plugged in, the script gives a collapse time of 19.2 seconds, which is still reasonably close to the observed collapse times. I have found that the collapse time is highly sensitive to how much mass we assume was lost from the collapse wave at each stack. The Greening paper states that at least 90% of the mass was retained in each collision, and that's the number I've plugged into the code at the moment. Feel free to copy this script and plug your own numbers in.

I don't claim that this is a particuarily accurate simulation of the actual collapses, merely a tool for visualizing how a progressive collapse occurs and estimating the time required.

Welcome, ellindsey. Nice model. It's good to see that even something so simple can get pretty close to the real thing.

If you want to add some detail to the model and perform a more detailed sensitivity analysis, here are a couple of other things you might try:

1. While overall collapse time is sensitive to the lost mass fraction, I suspect it's even more sensitive to lost mass at the initiation of collapse versus later in the sequence. Since the upper block is relatively undamaged at the start, mass retention can be assumed to be 100% for the first five or ten floors.

(Also, as I've argued repeatedly, "lost mass" isn't truly lost because in order for it to fall anything but vertically, it has to come in contact with the structure. The NIST "progressive collapse" postulates that the heavy steel elements remained tied by the sagging floors until secondary damage destroyed those connections completely; most loss over the sides is the facade and lighter office / interior materials squeezed out by the collapse.)

2. Your energy to destroy a floor is also constant. In reality higher floors will be easier to destroy due to lighter construction. Again, the total collapse time should be more sensitive to variation in the upper floors. You might try a floor energy function that is linear with respect to floor number and see what happens.

(Also, don't forget the first 10 floors or so were heavily damaged and weakened by the fires. This will also shave time off the real case that isn't represented here.)

 

[einsteen]

I'm working on the exact calculation (in very small steps), I got it for the end speed (about 51.8 m/s) but the time is a little bit tricky, it involves harmonic functions, the sum of squares can be calculated exactly.

I'm wondering what the mass distribution was of the building and the energy to break a floor. In the collapse the factor E_floor_break/m_floor is very important.

 

[T.A.M.]

[qimg]http://www.internationalskeptics.com/forums/imagehosting/thum_10761455202155aceb.jpg[/qimg]

The above is where my line of thinking is. If you look at it, there are 3 possible Free Fall times, depending on the height (d) you use in the equation...

d = 0.5*g*t*t

to give

t = square root of (d/5)

if you use top of building, free fall is = 9.13s
if you use the 98th floor, free fall is = 8.62 s
if you use the 104th floor (assumed center of gravity of top portion),
then free fall is = 8.88s

Seems like a small difference, but in fact it is aabout 1/2 a second difference, which out of 8-9 seconds is significant.

Once again, any help would be appreciated.

TAM


**The above is a duplicate of the one I posted on Loose Change thread, but I got no response. It refers to my speculation over what falling distance that NIST and others use to calculate the theoretical "Free Fall time" it would take the for the towers to fall, in free fall. Please, if anyone can help a guy with this, I am all ears.

TAM