**Author:**

The author of this work, Gordon Ross, was born in Dundee, Scotland. He holds degrees
in both Mechanical and Manufacturing Engineering, graduating from Liverpool John Moores University, in 1984. He can be contacted
at gordonjross@yahoo.com.

**Summary: **

This paper examines the elastic loading and plastic
shortening phases of the columns of WTC 1 after impact of the upper 16 storeys of the building upon the lower storeys and
its effect on the momentum transfer after the collision. __An energy balance is derived showing that there is an energy deficit
before completion of the plastic shortening phase that would not allow the collapse to continue__ under the constraints
of this paper.

**Introduction: **

Previous analysis of the momentum transfer in the collapse
of the towers has viewed them as being floors suspended in space and have examined the momentum transfer as a series of elastic
or inelastic collisions, which are independent of each other. This type of analysis takes the momentum transfer out of the
context given by the other effects of the collisions. This is because this type of analysis assumes that the impacts have
an effect upon only the topmost storey of the impacted section. The reality of the situation is that the impacts would have
an effect upon several storeys in the lower section and for a valid analysis all of these momentum transfers must be included.

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 report by

Bazant & Zhou.[1]) In reality there would be some losses of energy due to residual strength within
the failing columns of the removed section, but these are ignored for the purposes of this analysis.

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.

**Analysis: **

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. We shall firstly examine this incremental
time period.

Bazant/Zhou [1] show in their analysis that elastic and plastic behaviour of a steel column under a dynamic
buckling load can be shown to consist of three distinct phases. These can be shown on a load against vertical deflection graph
and consist of an initial elastic phase, a shortening phase and a rapid plastic deformation phase.

1/ The elastic
phase shows a linear relationship between load and deflection up to the elastic limit. The load at this point is the failure
load and the deflection at the elastic limit for steel is generally 0.2% of the initial length.

2/ The shortening phase
allows for the same failure load to be applied until the vertical deformation reaches 3% at which point the column begins
to form buckle points.

3/ The third phase shows a rapid decrease in the load requirement to continue deformation, this
load necessarily being less than the failure load. This phase lasts until the total vertical deformation equals the original
length. In other words the column is bent in two.

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 in a time of 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,
or approximately 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.

Because these columns suffer a vertical deflection, the attached floors move downwards and they will
therefore have a velocity and momentum.

**Energy Losses: **

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 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.

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 sectionfiltered= 16 floors moving at (8.5 m/sec)

K. E. of falling sectionfiltered= 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.

**Energy
Balance:**

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. The kinetic energy would be augmented by potential energy released in the
further downward movement of the falling mass and if we assume that this falls through the full distance of the 3% shortening
phase of the impacted floor and the elastic deflection of the lower storeys, then the additional potential energy is

58*10*
g * (0.111 + .056) = + 95MJ.

The strain energy consumed by the impacted storey columns in the elastic phase and plastic
shortening phase can be calculated using the failure load. The failure load used throughout this analysis is derived using
the mass above the impact, 58 000 tonnes, and a safety factor of 4. Examination of the column geometry with reference to the
Euler equations show that this is an underestimation both of the failure load and the distance over which it would have to
act before failure, and this gives a gross assumption in favour of collapse continuation. A factor of 0.029 is included to
reflect the load profile over the 3% plastic shortening phase. The load profile exhibits a linear rise from zero to failure
load at 0.2% of the length, followed by a constant failure load over the next 2.8% of the length.

**Plastic
strain energy:**

58*10kg*4*g*3.7m*0.029= -244MJ.

A similar though slightly smaller figure would
be required for the first impacting storey in the upper falling section. Because this storey carried a lower load, 15 storeys,
than the impacted storey, 17 storeys, its designed capabilities would be proportionately smaller. Using this knowledge an
estimation can be made that the energy consumed by this storey would be,

(244 MJ * 15 / 17) = -215MJ .

The
elastic response of the lower storey columns within their elastic range would make further demands on the energy available
by absorption of energy in the form of strain energy. This can be estimated, using a safety factor of 4, a mass of 58000 tonnes,
a distance of 0.056metres, and a factor of 0.5 to reflect the load profile

58*10kg*4*g*0.056metres*0.5= -64MJ.

The
downward movement of these floors in response to the impact will release additional potential energy due to their compression
and using the same deflections as above and a value for mass proportionate to the number of storeys, this will release

58*10kg
* 24/16 * g * 0.056metres / 2 = + 24 MJ.

Further energy losses are evident in an analysis of the compression of storeys
within the upper falling section. These storeys manufactured from columns with a smaller cross section than those at the impact,
would be unable to withstand the failure load present at the impact front and would suffer plastic deformation beyond their
elastic limit, but for simplicity, it is assumed that they suffer only their full elastic deflection. This is another large
assumption in favour of collapse continuation.

The total deflection would be 15 storeys multiplied by the elastic
deflection of 7.4mm, and strain energy consumed can be estimated as,

15*7.4*10*4*58*10*g/2= -126MJ.

Movement
of the storeys within the upper section will release additional potential energy due to their compression and consequent movement.
It is likely that this energy would manifest itself as failures within the upper section, but has nevertheless been added
as an energy available for collapse continuation. The uppermost storey will move downwards by 15 times the elastic deflection
whereas the lowest will remain static, both in relation to the impact point, giving additional potential energy as,

15*0.0074*58*10*g/2=
+32MJ

A considerable amount of energy would be required to pulverise the concrete into the fine dust which was
evident from the photographic and other evidence. To quantify this energy it is necessary to use the fracture energy value,
but this has a variable value dependent on, among other factors, the size of the concrete piece, and its constituents, most
notably, aggregate size. There is no typical value. In order to assess the energy consumed I will refer to the work of Dr.
Frank Greening [2]. It should be noted that Dr. Greening, like Dr. Bazant, does not, as yet, support the contention that the
tower collapse was caused by anything other than the damage caused by aircraft impact and subsequent and consequent fires.

The tower, using Dr. Greening’s figures, contained approximately 50000 tonnes of concrete, and the
assumption is made that only 10% of this was pulverised to a size of 60 micrometres. One kilogram of concrete at this particle
size will have a surface area of 67 m^2. We can now use Dr. Greening’s figure for concrete fracture energy of 100J/m^2
to show that the energy requirement for one floor 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.

__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

**Conclusion:**

The energy balance of the collapse moves into deficit during the plastic
shortening phase of the first impacted columns showing that there would be insufficient energy available from the released
potential energy of the upper section to satisfy all of the energy demands of the collision. The analysis shows that despite
the assumptions made in favour of collapse continuation, vertical movement of the falling section would be arrested prior
to completion of the 3% shortening phase of the impacted columns, and within 0.02 secondsafter impact.

A collapse
driven only by gravity would not continue to progress beyond that point.

The analysis shows that the energies expended during the time period of the plastic shortening of the first storey height
of the vertical columns is sufficient to exhaust the energy of the falling section and thereby arrest collapse. This however
is not the full extent of the plastic strain energy demand which exists. The next immediate task for the falling mass to continue
in its descent would be the plastic shortening within the remainder of the buckle length. As has already been stated a buckling
failure mode has a minimum length over which it can act and in the case of the towers would be several storey lengths. Each
additional storey length involved in the buckle would add a further demand of about 450MJ for a further downward movement
of 0.111metres. This also shows that collapse arrest is not dependent upon an expenditure of energy in concrete pulverisation,
since even if this expenditure were disregarded the input energy would be exhausted during plastic shortening of the second
storeys affected.

The analysis can be extrapolated to show that the energy expended within the plastic shortening
phase of a six storey buckle would ensure that a fall by the upper section through two storeys under full gravitational acceleration
would also be resisted at an early stage. A similar response would be elicited from an opposed three or more storey drop delivering
the same levels of energy at impact. It can be further envisaged that a collapse initiated by a fall through a greater number
of storeys, would be either arrested or significantly and noticably slowed when regard is taken for energy demands both in
the fall by the upper section, and by inclusion of demands identified but not quantified in this article. It should also be
noted that this analysis examines only the energy levels required up to a point in time during the plastic shortening phase.
Energy demands which involve further phases of the collapse mechanism, such as buckling of beams and disassociation of end
connections, spandrel plates and floor connections are further massive energy demands which must then be satisfied.

Assumptions
and disregards :