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9-11 free-fall

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No one came up with the facts for the World Trade Center North Tower.
I googled and made the following assumptions to entertain the conspiracy theorist's claims.

I found that the claimed 9.4 seconds of collapse, can not prove if the building was in a free fall. The sensitivity of varying assumptions did not substantially change the time to collapse. Free fall is 80% of the fall time. Doubling or tripling the fall time is very unbelievable. The first 5% of the distance to fall takes up the first 20% of time. This makes it difficult to spot when the collapse begins and the timer can be off one or two seconds.

Assuming the column failures were at the fire and all gave out at the same time. (unlikely). The initial floor level failure might have stopped at the failed floor, if the columns were to symmetrically crumple like an aluminum soda can, for about 4 feet of a 12 foot column length. The initial floor level failure might have also stopped at the failed floor, if the columns were able to resist buckling at 6 times the DL+LL and buckling would be limited to a 1 foot drop. If a design is based on the column safety factor of 12/23 then the additional strength needed to prevent a complete collapse is about 3. More than the seismic omega factor!

Why allow an increased risk of loss per failure of one column when buildings are taller? I am surprised. Columns for a low building puts a limited floor area at risk. Live loads are reduced at a column when the tributary area above is large. It seems that for a high rise, loads should be increased when a larger area is at risk.

Assumptions

*The 1362 foot North Tower was 110 stories. Floors 94 through 98 were initially damaged.
*average story height of 12.4 feet
*all floors equal equal weight to allow not knowing the weight.
*columns minimum buckling strength might be at a force (23/12)*weight above.
*ignore variations in live loads.
*Conspiracy theorists claimed fall time is around 9.4 seconds
*Only static floor masses and buckling columns resist the fall.
*Only the columns directly below a floor, being impacted, are buckling.

Method and Findings

Some simple physics and a 15 min. spread sheet of 94 floor impacts and column sets buckling added up to the following.

Initial floor level of failure is assumed a free fall without column resistance. Velocity is reduced as the accumulating falling mass impacts the next static floor mass. Acceleration is reduced or reversed as buckling resistance, a constant upward force, reduces affects of gravity. Further free falls occur between elevations where buckling stops resisting and the impacting with the next floor below.

Free fall is about 8.5 seconds. (a falling steel ball)

Conservation of momentum reduces the velocity at each floor impact of the falling floor weights above.
Time increased 1.5 seconds subtotal is 10 seconds.

Buckling columns assumed to hold a constant resistance post buckling for a short distance. Time increased from 10 seconds to times varying with distance of constant resistance.
1 foot: time = 0.3 seconds (most reasonable)
2 foot: time = 0.8 seconds (2 feet is not reasonable considering the 36" box column might have a plate buckling failure tearing welds,rather than the bowing of its length.)
5 foot: time = no collapse of floors below.

The first 5% of the 94 floors takes about 20% of the total fall time. The most inaccurate part of this problem is to determine the beginning time. Because the beginning time is plus or minus one second, the difference between the time of a free fall and an energy absorbing is not measurable.

Response to the email.

Recall the quote "One conspiracy theorist argued that the fall was faster than a free fall." What if it was meant that free fall should include conservation of momentum, only excluding energy absorption. 9.4 sec is faster than 10 seconds.

Recall another argument "A fall time noticeably longer than free fall is not likely in this case because how can you stop the inertia of 10 and 20 stories once they are in motion?". The beginning of the answer is that the static floors and buckling columns below will slow the fall. I want to know how much?

The earthquake slow-falling-story could also be applied to the towers in that hot columns should not all simultaneously disintegrate, abruptly. The fire was not symmetrical and had most likely had been past the maximum temperatures and already cooling.

I thought that an abrupt failure might be due to the floor supporting clips at the outside columns, but they would not all fail together unless it was the core first failing and equally pulling at the exterior clips.

If the collapse did "funnel the debris in", it makes it believable that the core columns failed 1st. The symmetrical response suggests that the failure occurred far below. This allows unequal column failures to average out to all core columns above, causing the symmetrical failure above to begin. There must have been something going on below.

The fall time does not prove that the tower was in a free fall and my 15 minute spread sheet failed to rule it out.

--
David B. Merrick, Structural Engineer


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