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In this one. Detailed analysis of collapse dynamics:

Posted by h_h (A) on April 1, 2006 at 08:13:32

In Reply to: The collapes violated the laws of physics? In what universe?! (nt) posted by Bersani on March 31, 2006 at 14:58:55:

FEMA Numbers Contradict Claim on Collapse Dynamics

In it's report on the construction of the WTC Towers and their collapse FEMA makes the following statement in their report from Section 2.2.1.5

“Construction of WTC 1 resulted in the storage of more than 4 x 1011 joules of potential energy over the 1,368-foot height of the structure. Of this, approximately 8 x 109 joules of potential energy were stored in the upper part of the structure, above the impact floors, relative to the lowest point of impact. Once collapse initiated, much of this potential energy was rapidly converted into kinetic energy. As the large mass of the collapsing floors above accelerated and impacted on the floors below, it caused an immediate progressive series of floor failures, punching each in turn onto the floor below, accelerating as the sequence progressed.”

First lets make an educated guess at both the mass and the velocity of the upper floors in WTC1. The FEMA report says the upper 20 floors contained 8x10**9 Joules of energy which if we take an average height for those 20 floors of (1368 + 1102)/2 = 1235ft or 375m. Gravitational acceleration is 9.8m/s. To calculate the Mass of the upper 20 floors we can use the relationship between PE and mass as follows:

PE = mgh and solving for m gives

m = PE/gh

Substituting the FEMA values and known constants we get

m = (8x10**9)/(375*9.8) = 2.176 x 10**6 kg

In order to compute the mass of the remaining 90 floors we do the same thing and arrive at an average height of 167m which gives us a mass of 2.39 x 10**8 kg or basically 100 times the mass of the top 20 floors. So actually we have 1% of the total mass of the structure collapsing on the remaining 99%. It seems a little hard to imagine that this could cause a total collapse but lets study the matter further using the numbers provided by FEMA.

When we look at the statement in the FEMA report concerning the potential energy being converted to kinetic energy it would seem that we are dealing with a very large quantity, i.e. 8 Billion Joules. However, the KE of a falling object is given by the expression

KE = ½ mv**2

When the mass of the upper floors first begins to move it is accelerated at a uniform rate of 9.8m/s. This means that the velocity at the end of one second would be 9.8m/s but it isn't the initial velocity, that is clearly 0. So we must calculate what the final velocity is after traversing one floors worth of distance or approximately 4m. We can use the relationship between position and acceleration to find the velocity after traversing 4m as:

v**2 = 2as where a is acceleration due to gravity and s is distance travelled

v**2 = 2*9.8*4 = 78.4 and taking the square root gives v = 8.85m/s. Now we are ready to compute the KE of the upper 20 floors after traversing the 4m to reach the first undamaged floor below. Plugging into the equation above yields 8.5 x 10**7 Joules or 85 million Joules of Kinetic Energy spread across the area of floorspace of the building which was 3972 square meters or nearly an acre. Now that we have that energy we would like to compare it to the energy that FEMA has allocated to the causes of the initial collapse.

The first energy source that affected the structure was the collision from the plane which we know was a 767 with an unknown velocity. However from the Boeing site we find that the maximum takeoff weight was 179,170 kg and the crusing speed was 530mph. We don't want to estimate a greater mass or speed in this case because we know the buildings withstood this impact and so therefore we want to minimize or estimate of the impact energy of the plane in order to be conservative concerning our estimates of minimum collapse energy. So estimating that the plane had only 10,000 gallons and the maximum capacity is 24,000 gallons we subtract 40,000 kg from the mass of the plane.
The passenger load was also only about 1/3 of capacity and assuming a passenger plus baggage weight of 200lbs each. We will generously subtract another 15,000 kg from the mass of the plane. So we have a plane mass of roughly 125,000 kg.

The next component of concern is the velocity of the plane. Now the govt has estimated a very high velocity for the impact on WTC-1 of 494mph. This is fine for estimating a large energy of impact but it one desires to minimize the impact energy then we would like a lower impact speed. A lower estimate from another source provided an impact speed of 275mph which is roughly 98m/s. So at the lower velocity we compute an impact energy of 5.25 x 10**8 Joules. If we use the government speed we get 3.4 x 10**9 Joules which is much greater.

Now comparing the energy at jet impact to the energy from the upper 20 floors we see that even our lower number of 525m Joules is far larger than 85m Joules of the upper 20 floors at 8.85m/s. Now one thing we have to consider about the energy from the airplane impact is that it was spread over an area that covered 4 floors so dividing the lower number by four gives us an impact energy of roughly 131m Joules on the low end and 975m Joules on the upper end. These energies are still larger so it is clear the building was able to absorb on a per-floor basis, far more energy than would be contained in the upper 20 stories collapsing.

In looking for a better analysis of the energy distribution on impact I found this article at

http://www.airdisaster.com/forums/archive/index.php/t-47343.html

This article speculates that 56% of the impact energy went into damaging the columns in the core. It unfortunately doesn't say how many floors of columns were assumed to be damaged but one would guess that at least two floors worth of columns were damaged and this article estimates that 24 of the 47 columns were damaged. So taking 56% of their estimate and dividing it by two leaves us with 1.07 x 10**9 Joules of energy absorbed by the core columns on two floors and yet that wasn't enough to cause a collpase. If we take our lower estimated speed and use the 56% number over two floors we get 1.47 x 10**8 or 147m Joules which is still 73% more energy than the upper 20 floors would contain at 8.85m/s.

We can end this analysis right here because we know that there wasn't enough energy in the impact to cause a collapse and we would expect that after the fires weakened the four impact floors that they had added even more energy. In fact, the FEMA report estimates that 1.15 x 10**9 W were produced by the fire in WTC 1. Allowing for 85 minutes of burn time and one half of this energy vented out of the building, we still get an additional 48 Billion Joules of energy. In order not to confuse this any further, I believe it's easy to see that the top 20 floors didn't contain enough energy to collapse even the next floor with generous assumptions.

Moreover, in a collision between two masses there is always kinetic energy lost in heat and deformation unless the collision is perfectly elastic. From the government estimates we have here, we can conclude with reasonable certainty that it would require the upper 20 floors collapsing to contain greater than 1.07 x 10**9 Joules of energy in order to even consider any subsequent collapse. This would require a velocity of 31m/s which would require that the collapse have progressed by 50m before impact which is 12 floors and anyone with eyes can see that there wasn't that much open space to fall through.

To take this collision another step, we can consider the columns in the core of the buildings. The amount of force required to cause a column to buckle is given by an equation called Eulers Formula and is given as

F = EI*pi**2/L**2

In this case E is the Youngs Modulus of the material, I is the moment of inertia for the beam type and L is the length of the beam. We are assuming what is called a simply pinned column in this case, which is what the colums on the interior floors of the building are.

There are several types of columns in the WTC core according to the NIST report on page 26 of chapter two. The largest on the floors above the impact zone are W14x730 which weigh 730 lbs per foot and are erected in 39 foot lengths so a single column weighs roughly 14 tons and spans three floors. According to the really poor NIST diagrams, there were at least four of these around the outside perimiter of the core and probably 4 in the center. The following table describes the force needed to initially deflect the column and the energy consumed in order to deflect it one meter.

W14x730 126Million Newtons bending force ==] 126 Million Joules per meter bent
W14x219 23.5 Million Newtons bending force ==] 23.5 Million Joules per meter bent
W14x61 6.4 Million Newtons bending force ==] 6.4 Million Joules per meter bent

There are 47 columns per three floor element of the building. The amount of energy to bend the 8 big columns by one meter is a Billion Joules. Compare that to the KE of the falling mass computed from the FEMA numbers, it's much much greater. Even allowing for a much larger falling mass, the total number of colums per element is 47. A conservative estimate of the buckling energy would be 2.1 Billion Joules. It's also critical to realize that once the KE has been expended in the deformation, it's converted to PE in the reshaping of the steel. That is what should produce a slowing of the falling mass because if KE = 1/2mv**2 and KE is reduced in the collision but m is constant, then v must decrease. Another way of looking at this is by applying the principle of the conservation of momentum. In any collision between two bodies where they are stuck together after the collision, the conservation of momentum equation is as follows:

m1*v1 + m2*v2 = (m1 +m2)*v3

In this case, m1 is the mass of the collapsing floors and v1 is the velocity of the collapsing floors. The mass that is impacted, m2, is for simplicity, considered as the next three floors for the purposes of a progressive collpase. So before the collision the velocity of m2 is 0 and we can rewrite this equation as

m1*v1 = (m1+m2)*v3

Now it's pretty easy to see that since m1+m2 is larger than m1, then v3 is less than v1 and as we would expect in a collapse where the energy needed to damage the floors below is coming from the KE of the floors above, the velocity of the falling floors must decrease at each impact. This is not what we see in the videos. The fact is that in order to have a progressive collapse, the falling mass must transfer sufficient KE to the undamaged structure to provoke the collapse before the combined masses can begin to move together.

The videos of the building collapse show that there was no decrease in velocity of the collapsing floors at any point. The velocity of the falling floors was increasing and thus the only conclusion is that the falling floors were never encountering any resistance. The reader can draw their own conclusion as to how the resistance of the lower floors was reduced to zero. The fact is, the government theory of a progressive collapse due to the KE of the falling mass is just not supported by the facts. There wasn't enough energy in the falling floors to cause the undamaged floors to collapse further and the actual physical mechanism proposed would have resulted in a collapse time on the order of 90 seconds.

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