A Review of Energies and How They Are Dissipated of WTC 1 & 2, by Newton's Bit


 

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Appendix 1
Appendix 2

The purpose of this article is to provide an easily accessible resource in which to debunk the claim that World Trace Center buildings 1 and 2 would resist the collapse once started.

As of Fall, 2007, only one papers exists that claims that the collapses would be arrested once started. That paper is Gordon Ross’ entitled, “Momentum Transfer Analysis of the Collapse of the Upper Storeys of WTC 1”. In this paper Ross identifies sources of energy that push the collapse and energy sources which resist the collapse. He finds that there is more energy to resist the collapse than energy that would progress it. His conclusion is wrong because he doesn’t understand some rather simple and basic concepts of physics and engineering.

The following is the summary of energy sources both pushing the collapse and resisting the collapse produced by Ross (SIC):

Energy Summary:
The energy balance can be summarised as

Energy available;
Kinetic energy 2105MJ
Potential energy Additional downward movement 95MJ
Compression of impacting section 32MJ
Compression of impacted section 24MJ
Total Energy available 2256MJ

Energy required;
Momentum losses 1389MJ
Plastic strain energy in lower impacted storey 244MJ
Plastic strain energy in upper impacted storey 215MJ
Elastic strain energy in lower storeys 64MJ
Elastic strain energy in upper storeys 126MJ
Pulverisation of concrete on impacting floor 304MJ
Pulverisation of concrete on impacted floor 304MJ

Total Energy required 2646MJ
Minimum Energy Deficit -390MJ

This summary can be easily explained. In the energy available section, Ross adds up the sources pushing the collapse. There is 2105MJ of potential energy available for the collapse. This is calculated as 16 stories of the WTC falling 3.7m (one full story). The other minor additions are potential energy gains as the building compresses.

Ross identifies the energy that resists the collapse as energy required. This can be divided into three sections: Momentum Losses, Column Strain Energy, and Concrete Strain Energy. Momentum Losses is a calculation of kinetic energy losses; Column Strain Energy is the energy required to rupture the columns such that they do not provide any more support; Concrete Strain Energy is the energy required to pulverize 20% of the concrete slab into 60 micrometer particles.

In the end, Ross is very wrong. A more accurate summary, proven later in this paper, is as follows:


Kinetic Energy available: 2105MJ

Kinetic Energy gains:
Potential Energy of the Upper Block + 64MJ
Potential Energy of the Lower Stories + 41MJ

Kinetic Energy Losses (including strain rates):
Loss of Kinetic Energy due to Inelastic Collision - 123MJ
Loss of Kinetic Energy due to Mass-With-Spring - 414MJ

Elastic Strain Energy of the Lower Stories - 213MJ
Inelastic Strain Energy of the Lower Story - 171MJ
Elastic Strain Energy of the Upper Block - 71MJ
Inelastic Strain Energy of the Upper Story - 171MJ

Pulverized Concrete - 0MJ

Total Energy Available + 2210 MJ
Total Kinetic Energy Losses - 1163 MJ
Total Energy Balance + 1047 MJ


The result as seen is a total kinetic energy gain after the first impact is 1047MJ. This means that the the collapse of the tower will not be slowed down or stopped and the collapse front will accelerate.

Momentum Losses:
To understand momentum and losses associated with it, a basic knowledge of collisions is required. There is always conservation of momentum in collisions. This can be stated by: Mass*Velocity before a collision is equal to the Mass*Velocity after the collision. This is commonly referred to as the Law of Conservation of Energy.

There are two general types of collisions: inelastic and elastic. This is best explained with the help of animations.


An elastic collision, animation created by Simon Steinmann.


An inelastic Collision, animation created by Simon Steinmann.

Objects involved in an elastic collision have the same amount of momentum before the collision afterwards. They also have the same amount of kinetic energy (1/2*Mass*Velocity) before and after the collision.

Inelastic collisions result in the two bodies becoming one and having one velocity. Because of conservation of momentum there is a loss of kinetic energy. This can be easily explained from the above animation.

Before the collision, the object on the left has:
KE = ½*m*v^2

After the collision, we know that the velocity is halved and the mass is doubled, thus:
KE = ½*(2m)*(1/2*v)^2
KE = ¼*m*v^2

Or a total loss of 50% of the total kinetic energy, this loss kinetic energy goes into the internal energy of the colliding objects. Specifically strain energy (the objects going squish and becoming one), heat energy and sound energy. This loss of kinetic energy is not a loss of total energy and it is a mistake that Ross makes that will be explored later.

Ross attempts to calculate the kinetic energy lost by the collision. He states,
A simple conservation of momentum calculation, ignoring these movements, 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.

To estimate and illustrate the further momentum changes we can assume that the storey which is 25 storeys from the impact remains static and the velocity of the 24 affected storeys will vary linearly from the velocity of the falling section to zero.

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 +……+ 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 first thing that Ross states is that there is an inelastic collision between the 16 upper stories and the story that they are impacting at the top of the upper block. 16 stories with a velocity of 8.5m/s will become 17 stories with a velocity of 8.0m/s. This is true. And results in a loss of 123MJ of kinetic energy.

Ross then states that 24 stories below this story will also deflect. This is also true. However, from the discussion above, we know that Ross makes a critical mistake: there is not an inelastic collision between the intact stories below the collapse front and the upper block. It is an elastic collision. We know this because there is no permanent deformation of the lower columns, they stay in the elastic stress-strain region, and that the masses did not impact and combine into one larger mass. An accurate way to address this problem is to assume these additional 24 stories are part of a spring with mass. For this analysis, see Appendix 1.

The results of the spring-with-mass analysis:
Loss of Kinetic Energy due to Inelastic Collision - 123MJ
Loss of Kinetic Energy due to Mass-With-Spring - 414MJ

In addition the additional energy gained due to potential energy is:
Potential Energy of the Upper Block + 64MJ
Potential Energy of the Lower Stories + 41MJ

It is important to note that the Loss of Kinetic Energy due to Mass-With-Spring is a gain in kinetic energy in the bottom portion of the structure. This energy is not immediately dissipated and it will accumulate on each successive impact of the upper block on the lower block. This energy manifests itself as vibrations in the lower structure, which will ultimately reduce the total strength of the lower columns. However this is outside the scope of this paper as it relates to the continuing progressive collapse and not just the first impact.


Column Strain Energy:
The strain energy in columns is rather simple to calculate, however Ross fails to understand the basic concept of inelastic buckling. A detailed analysis of the energy required to completely rupture the columns can be seen in Appendix 2.

Elastic Strain Energy of the Lower Stories - 213MJ
Inelastic Strain Energy of the Lower Story - 171MJ
Elastic Strain Energy of the Upper Block - 71MJ
Inelastic Strain Energy of the Upper Story - 171MJ

Concrete Strain Energy:
The “pulvierisation” of concrete is perhaps the biggest flaw in Ross’ paper. He lists a total of 608MJ lost from this event. In reality, the loss of kinetic energy due to pulverized concrete is ZERO MJ. Why is this? It’s already been accounted for. Recall that the loss of kinetic energy from an inelastic collision. This means that some of the kinetic energy is converted into heat, some into sound and the vast majority into strain energy. The strain energy category includes this pulverized concrete. In reality, core columns punching through the floor slabs, or concrete debris doing the same, would cause an additional amount of interal energy to be added into the concrete slabs. However the Ross model only allows the columns to land directly on top of each other, as this is most conservative to collapse prevention.

A question might still remain for some readers as to why so much of the concrete in the tower was turned into a fine particulate dust. The answer to that question lies not in the impact of one floor against another floor, but in the eventual impact of the entire structure into the ground. At that point, almost all of the available kinetic energy will be converted into internal energies. Again, some into heat, some into sound, but the vast majority into strain energy.