I was trying leave this alone, honestly, but then someone has another go and calls me a wacko or a steaming poo-handler or something and I have to defend myself. Full steam ahead then.
I am talking primarily about the collapse
here (which hasn't been successfully modelled or proven by NIST), not the aeroplane impact and fire.
When looking at the resultant motion in any collision, you have to factor in the mass and hence the inertia of the objects in question, and the same applies in the case of the mass of the upper part of the towers falling onto the floors below, hitting each floor in turn and propagating a collapse that is only going to intensify as it proceeds due to the increasing amount (and therefore mass) of the falling material. Remember, you can't treat the body of the tower below the impact zone as a 'solid block', no matter how much you might wish to.
I said the tower body was "effectively a solid block" as a simplification, to get you to at least consider the resistance that it would provide and the time it would take to overcome this resistance.
If we assume that the upper section comprising 16 storeys falls under a full gravitational acceleration through a height of one (removed) storey, a distance of 3.7 metres we can calculate that its velocity upon impact will be 8.52 metres per second and have a kinetic energy due to its mass and velocity of 2.105 GJ. (Using the figure of 58000 tonnes as detailed in the pro-collapse report by Bazant & Zhou.
) In reality there would be some added losses of energy due to residual strength within the failing columns of the removed section, but we can ignore them here in favour
of the collapse theory.
Upon impact with the lower section the falling mass would deliver a force which would grow from zero, up to the failure load of the impacted storey columns, over a finite period of time and distance.
This force would also be felt by the columns below the storey which was first impacted.
The falling upper section with a velocity of no more than 8.5 metres per second at impact would meet resistance from the impacted columns and have as its first task the necessity to load these columns through their elastic range and thereafter through the plastic shortening phase.
Timewise, looking at the B&Z report, To shorten the columns of the first impacted storey by 3%, sufficient to complete the plastic shortening phase, a distance of about 0.111 metres, and allowing a constant speed of 8.5 metres per second, would take a minimum of 0.013 seconds.
The speed of the propagation wave through a medium is given by the general formula for wave propagation:
Velocity = Square root ( Bulk modulus / Density ),
and for structural steel is of the order of 4500 metres per second.
The propagation wave of the impact force would therefore travel a distance of 58.7 metres over 0.013 seconds. This means that during the time taken in the plastic shortening of the impacted columns, the same force would be felt at a minimum distance of 58.7 metres (~16 storeys,) from the impact. These storeys would thus suffer an elastic deflection in response to, and proportional to, the failure load applied at the impacted floor. These deflections would themselves take time and allow the propagation wave to move further downwards again affecting more storeys.
We can estimate the elastic deflection of these 16 storey columns as being in the range 0 to 7mm. The full elastic deflection of a 3.7m column, using the generally accepted figure of 0.2% of its original length is 7.4mm. The columns in the uppermost of these storeys would suffer almost their full elastic deflection since their failure load is similar though slightly greater than that of the first impacted storey. Those storey columns more distant from the impact would be of
a larger cross section, requiring higher loads to cause full elastic deflection. Using only half of the maximum elastic deflection, 56mm (16 * 7 / 2), is, again, an assumption in favour
of collapse continuation.
The elastic deflection of lower storeys would increase the distance through which the falling section would have to move in order to load the impacted column and complete its 3% plastic shortening. The time taken, again using a constant velocity of 8.5 m/sec would increase to about 0.02 seconds, and thus allow the propagation wave to move through and affect a further 8 storeys.
And yes, the structure - or rather, each floor as the weight of the collapsing material impacts on them in turn - will exert a force on the falling load equal to the force that load exerts on it, but that doesn't mean it will halt the motion of that load.
You will still get significant energy/momentum loss due to to the resistance of the structure.
A simple conservation of momentum calculation, ignoring these movements/deflections (in your favour again), would have, 16 falling storeys moving at 8.5 m/sec before impact, changing to 17 storeys moving at (8.5 * (16/17)) = 8 m/sec after impact. This does not reflect the fact that a minimum of 24 further storeys will be caused to move downwards at varying speeds. Assuming a storey 25 storeys from impact remains static,
Momentum before impact = 16 storeys moving at 8.5 m/sec
Momentum after impact = 17 storeys moving at V2 m/sec + 1 storey moving at 23/24*V2 m/sec
+ 1 storey moving at 22/24*V2m/sec +â€¦â€¦+ 1 storey moving at 2/24*V2 m/sec + 1 storey
moving at 1/24*V2m/sec 16*8.5 = V2 (17 + 11.5)
V2 = 16 * 8.5 / 28.5 = 4.8 metres per second.
The speed of the upper section would be reduced by the collision from 8.5 m/sec to a speed of less than 4.8 m/sec
rather than the 8 m/sec derived from a momentum calculation which does not include this factor. Note also that this reduction in speed would again give more time for the propagation wave to travel downwards through the tower columns and allow that more and more storeys are so affected.
Now think about the kinetic energy of a falling weight
The kinetic energy of the falling section would be similarly affected, but because of the velocity squared relationship, the reduction in kinetic energy would be more pronounced.
K. E. of falling section before impact = 16 floors moving at (8.5 m/sec)
K. E. of falling section after impact = 17 floors moving at (4.8 m/sec)
Percentage loss of K.E. = 1-(17 * 4.8/ (16 *8.5) * 100% = 66%
This is an underestimation of the energy loss, since the deceleration would allow more time for travel of the propagation wave and so allow more floors to be affected but even this shows an energy absorption of some 66%
of the total kinetic energy of the falling section.
Since there was only some 2.1 GJ available at the point of impact of the first collision, a loss of 66% would reduce this figure to 714 MJ.
You still have other energy losses due to:
- compression and using the same deflections as above and a value for mass proportionate to the number of storeys.
- elastic response of the lower storey columns within their elastic range
- plastic strain energy loss that would increase the further down the structure you go, as lower stories would necessarily have to support more load.
- compression and consequent movement of the storeys within the upper section
- (looking at Dr. Frank Greening
's report - someone who doesn't follow the "conspiracy theory" by the way)the energy required to pulverise approximately 50000 tonnes of concrete into the fine dust which was evident from the photographic and other evidence.
One kilogram of concrete at this particle size will have a surface area of 67 m^2.
Using Dr. Greening's figure for concrete fracture energy (100J/m^2) the
energy requirement for one floor itself would be ( 50*10^6kg / 110floors * 67m^2 * 100J/m^2 * 10% = ) 304 MJ!
It may be considered unlikely that a low velocity impact would expend large energies on pulverisation of materials, and this is more likely in later stages of the collapse. However, the large expulsions of dust were visually evident at early stages of the collapse.
Fig 1. "Poof."
Let me know if you'd like to see the calculations for all of the above. Then there's also:
- disconnection of the floor to column connections
- damage caused to spandrel plates or other structural elements
- crushing of floor contents
- any strain energy consumption during the initial fall through the height of one full storey
With only 2.1 GJ available at the point of impact of the first collision, the energy balance of the collapse moves into deficit during the plastic shortening phase of the first impacted columns
. There isn't enough energy to propagate the collapse, let alone all to account for any of the additional loss factors. A collapse driven only by gravity would not continue to progress beyond that point.
(I did get help from a structural engineer on the above, I'm sure you've figure that out for yourself though.)
My point is that it wouldn't hurt you to bring some of scepticism you bring to the official reports to all these youtube videos you're devouring.
Dude, I do. Some of them propound some great theories that are absolute bollocks. like:
- There were never any planes at all, they were missiles wrapped in holograms (this guy also thinks he is the messiah)
- The towers were brought down by an energy weapon from space.
This is not to say that all conspiracy theorists believe this, which is a generalisation a lot of people seem to leap to.
Stop calling me names and I will stop arguing!