` `
```We treat the heat transmitted from each square metre of floor space as a
point source midway between the floor and ceiling.  Splitting this up into a
number of sources at various heights to form a line or plane source, apart
from increasing complexity and lengthening computations, would slightly
reduce the total heat absorption for each column.  Moving the point source
away from the midheight position would increase the mean angle of incidence,
integrated over all points along the height of the receiving column surface,
and so reduce absorption.

An idealised isotropic emitter will lead to an inverse square law
attenuation when the absorbing object is very small in relation to the
as a proportion of the total transmission is equal to the area of the
receiving object presented normal to the flux divided by the curved surface
area of a sphere whose radius is equal to the transmitter - receiver
separation.  Since the total flux is divided into the total surface area of
the sphere, a doubling of separation or radius would quadruple the area of
the sphere and result in a fourfold decrease in the flux density.

In the situations we are concerned with, the absorbing objects are not very
small in relation to the separation, which renders the inverse square law
inapplicable.  The core columns, for example, are treated as rectangles for
radiation view purposes, presenting a length of ten feet from floor to
ceiling.  Part of the interfloor length was shielded from radiant flux.
For mass purposes, taking the length at ten feet would not allow for heat
conducted into the extra two feet.  On the other hand, assuming the full
mass of a twelve feet length per floor would be unfair to fire collapse
theories, since conduction would be insufficient to bring about the full
temperature increase throughout each interfloor end section.  The per floor
mass for each column was taken to be an eleven feet length of the relevant
WF shape, and the following confirms this as a fair measure,

Ten feet plus one foot either end is a mean distance of six inches for
conductive losses.  Thermal diffusivity (alpha) is the ratio of thermal
conductivity (k) to volumetric specific heat capacity (p * C) where p is
the density and C is the specific heat by mass.  For steel,
k = 40 W/m.K over [p = 7,860 kg/m^3 times C = 575 J/kg.K] gives a value of
8.85 * 10^-6 m^2/s for alpha.  If a bar of semi-infinite length with
insulated sides is "instantly" heated at one end, the distance x at which
half the temperature rise occurs is given by:
x = 2 * z * SQR(alpha * t)
where z is the value required to yield a Gauss error function of 0.5 and t
is the time in seconds.  Since erf(0.477) is close to 0.5, the distance is
2 * 0.477 * SQR(8.85 * 10^-6 * 6120) = 0.222 m = 8.74 ins, which is more
than the mean distance of all points along a 1 foot length from its
boundary.

Now admittedly, heat outflow from the ten foot section is reduced by the
fact that its temperature does not instantly increase to the maximum.  The
mean temperature throughout the 102 minutes in our heated, unprotected WTC
core column situation would range from barely more than the mean of ambient
and final temperature to around ambient plus three-quarters of the final
increase.  But the semi-infinite length model includes 0.1648 of the
temperature rise at 18 inches, 0.064 of the increase at 24 inches, 0.0206 of
the increase at 30 inches...  Beyond 12 inches, columns are assumed to be
heated by fires on adjacent floors, so the temperature rise over the 12
inches would be greater than predicted by semi-infinite length models.  And
if the conducted outflow from the floor under consideration did exceed the
12 inch limit, heat losses from the ten foot section would exceed that
calculated by taking the mean temperature of points within each 12 inch end
section.  Thus, in the unprotected steel calculations, eleven feet of column
was selected as a reasonable measure for the mass of steel to be heated.

The width of the rectangle presented was originally taken to be:
(W1 + W2) * 2 / pi
where W1 and W2 are the widths from looking head-on at the web or a flange,
respectively - denoted as "d" and "bf".  This calculated a mean effective
width presented, integrated over 360 degrees of possible viewing angle in
the horizontal plane.  With certain members such as spandrel plates, this
method was not really appropriate, and it was preferable to calculate the
amount of foreshortening in the vertical and horizontal plane.  It was
decided to calculate the rectangle width for each core column, for each
point source amongst the floor space, by calculating the view angle of each
visible side.  The amount of foreshortening was taken as the cosine of the
angle of incidence, which for both visible sides would sum to 90 degrees.
The total absorption predicted with this method was some 2% greater than
with the averaged foreshortening approach.

The total energy received does not tend to infinity as the separation tends
to zero.  Even in the case of a large flat absorbing object that is actually
touching a point source, it will only feel half of the total power output,
as 50% of the flux will be transmitted in the other direction.  The area of
the object divided by the area of the sphere would be approaching infinity,
when a flat object could not possibly receive more than 50% of total energy
transmitted.  But the distortion of a flat object against the curved surface
of the "sphere" becomes considerable, well before the transmitter - receiver
separation is in the order of the object dimensions.  A corrected formula
will feature the inverse square attenuation of object area divided by
"sphere" area for a distant source, and converge upon a 0.5 upper bound as
the source closes on the object.

Solving for a circular flat object is simpler than for a rectangle.  After
correcting for the flat object coming up against the curved surface of a
sphere, we find that the proportion of radiation received is the area of the
segment of a sphere divided by the area of a sphere.  This ratio is given by
0.5 * (1 - cos(theta))
where theta is the angle subtended at the centre of the sphere by lines to
the centre and edge of the flat object.  Cos(theta) is equal to
r / k = r / SQR(r^2 + a^2)
where r is the radius of the sphere and also the centre of sphere to centre
of flat circular object separation, a is the radius of the circular object,
and k is the diagonal from centre of sphere to circumference of circular
object.  Hence, the ratio of received / transmitted radiation is given by
rprop = 0.5 - 0.5 * (r / SQR(r^2 + a^2))
but for a rectangle, the variable a has a range of values.

Consider a rectangle with a line from its centre of gravity to perimeter,
initially in the twelve o'clock position, say.  If this is rotated through
90 degrees in a number of small steps, we can find the average length of the
line.  However, suppose we wanted to find the radius of a circle whose area
was equal to that of the rectangle.  Since the area of a circle function is
related to the square of the radius, the average length of the line in the
rectangle would not provide a circle of equal area.  It is the root mean
square value of this line which equals the radius of the equivalent circle.

So, the radiation view factor section of the program to determine
temperatures reached by each member included variables for length and width.
The former was taken as 11 feet for the core columns, the latter was
calculated allowing for foreshortening resulting from the view angle in the
x, y plane after determining likely sizes for individual columns.  For each
column, for each point within the floor area, the separation between point
source and column was calculated from the coordinates of source and column.

The formula
rprop = 0.5 - 0.5 * (r / SQR(r^2 + a^2))
as above was used in calculating the ratio of received to transmitted
energy.  But in order to allow for the 11 feet length and computed width
dimensions of the relevant member, a line was taken from the centre to the
perimeter of the rectangle and rotated through 90 degrees in a number of
steps.  For each step, the rprop function was calculated, and the value was
taken to be the mean of all rotation steps.  Applying the inverse function
to this mean would provide a weighted average for the variable a - the
equivalent of an "rms" value - although it was not necessary to calculate
this.  The operation was repeated for each combination of member and point
of source.  Hence, the flat rectangle to curved surface of sphere correction
was performed to a reasonable accuracy.

The energy density from the fire was taken to be the maximum possible, with
the burning rate just happening to coincide with the necessary rate for
consuming all of the floor's kerosene, aircraft and office combustibles, in
exactly 102 minutes.  The total heat released on this worst case floor (98)
came to 1.385 * 10^12 J in the model.  A previous calculation working with
each square foot of floor space had:

Non-core combustibles = 7 + 0.688 + 0.2 = 7.888 psf = 40.8 MJ/ft^2
Core combustibles = 2 + 0.688 + 0.2 = 2.888 psf = 14.93 MJ/ft^2

That core density was based on lowering the area from 11,919 to 7,151 ft^2
after allowing for 40% of area being taken up by shafts.  Rather than decide
which points were taken up by shafts, it was easier to multiply the energy
density by 0.6 and plot all points.  Our model pretends the shafts do not
exist to shield the columns, thereby exaggerating absorption.  The non-core
released 1.279 * 10^12 J, with a corresponding 1.068 * 10^11 J from the
core.  In metric, the total points plotted were x and y coordinates 0.5 to
62.5 which is 63^2 = 3969 m^2.  There were 1134 core points and 2835 office
area points.  Thus, the core energy density came to 94.2 MJ/m^2, and the
non-core was 451 MJ/m^2.

It was assumed that 40% of the energy flowed out of the building.  With a
mean adiabatic flame temperature of 1488 C after allowing for 68% combustion
efficiency with conditions ranging from stoichiometric to fuel-rich, actual
flame temperatures averaging 1000 C after transmitting heat to surroundings,
an exiting gases temperature of 669 C, and an average hot gases upper layer
temperature of 835 C, the total heat transmitted to the building was 6.84
MJ/kg of wood equivalent, with another 4.56 MJ/kg vented to the exterior.
Of the heat transmitted to the building, 4.16 MJ/kg was from the flames and
2.68 MJ/kg from hot gas or smoke upper layer.  So the program multiplied the
above energy density figures by 0.6, which took account of contributions
from the flames and smoke layer.

The original computations used a constant value for the specific heat of
steel.  This understated temperature increases at the low end and overstated
at the high end.  The final method, after dividing total heat absorption by
the mass of the column to obtain a J/kg measure, ran this through a banded
specific heat subroutine.  For steel, specific heat capacity is generally
regarded as 450 J/kg.K at 20 C, increasing to 725 J/kg.K at 750 C and up,
Around 730 C there is a brief but huge increase in specific heat associated
with the phase change from ferrite-pearlite to austenite.  The program
assumed 450 J/kg.K up to 125 C, 525 J/kg.K from 125 to 250 C, 575 J/kg.K
from 250 to 375 C, 630 J/kg.K from 375 to 500 C, 700 J/kg.K from 500 to
625 C, 720 J/kg.K from 625 to 730 C, up to 40,000 J/kg.K from 730 to 731 C
which allows for the phase change peak, back to 720 J/kg.K from 731 to
750 C, and 725 J/kg.K for 750 C and up.  We could have spread the peak out
to something like 4,000 J/kg.K from 725 to 735 C, but a five degree change
in predicted temperature would change predicted steel strength by less than
three percent.

For each core column, three alternative scenarios were considered, and
predicted temperatures inputted to a spreadsheet.  The first case assumed
fire resistant coating (FRC) to have been totally stripped off from all four
sides, leaving the member totally exposed.  The second case was of
one-quarter of FRC stripped off.  The remaining case assumed FRC totally
intact.

In the first scenario, the total heat absorbed by the steel member is taken
Most absorption is via radiant line of sight heat transfer from flames;
there is also radiation and convection from the smoke layer.  The sum of
these equates to the 60% of total energy deemed to be released before the
hot gases exited the building.

For some columns in this case, absorption was sufficient for the member to
approach the hot gas layer and flame temperatures of 835 and 1000 C, and an
850 C upper limit was imposed on the predicted temperature.  A more complex
calculation would take into account the limiting factor of Stefan's constant
as the difference between the fourth powers of radiator and absorber
temperatures becomes small.  The "850 C" predictions could easily be
+/- 50 C, but at 850 C some 91% of the original strength would have been
lost.  Whether 800 or 900 C, member failure would be expected.

One column in the spreadsheet recorded a fourth hypothetical temperature,
calculated purely on the basis of heat absorption, mass and heat capacity,
assuming that the flame and gas temperatures were sufficiently high that no
850 C limit was imposed.  This can be used for estimating temperatures of
columns or sections of columns with various other proportions of insulation
removed, e.g. one side of web is 50% FRC remaining before allowing for heat
flow to flanges.

So the spreadsheet included four columns for temperature: unprotected steel,
hypothetical case with no gas temperature limit; unprotected steel with
850 C upper limit; one side or one-quarter of FRC stripped off, combining
conduction through FRC and radiative / convective transfer; and all FRC
remaining, heat conducted through FRC.

The effect of the FRC is illustrated in curves such as FEMA 403, Appendix A,
Figs. A-9 and A-10.  The table below shows the salient details.  Two points
are taken from each curve, showing elapsed time in minutes and Celsius
temperature.  The final column shows the difference in temperature divided
by the time difference, i.e. the rate of temperature increase.

Unprotected:-
Member, test, time1, temp1, time2, temp2, degrees C increase/minute
Box column, E119, 6.5, 100, 19.5, 600, 38.5
Wide flange, E119, 5.5, 100, 18.75, 700, 45.3
Box column, UL 1709, 4.5, 200, 8.75, 700, 117.6
Wide flange, UL 1709, 4, 200, 7, 200, 166.7

Protected, 1" spray-applied fireproofing 0.116 W/m.K:-
Member, test, time1, temp1, time2, temp2, degrees C increase/minute
Box column, E119, 30, 90, 120, 240, 1.67
Wide flange, E119, 30. 125, 120, 330, 2.28
Box column, UL 1709, 30, 130, 120, 300,1.89
Wide flange, UL 1709, 30, 177, 120, 407, 2.56

Note that the 36" x 16" unprotected box column in the E119 test goes from
100 C at 6.5 minutes to 600 C at 19.5 minutes, a 500 degree increase in 13
minutes or 38.46 degrees/minute.  With 1" of spray-applied FRC, the same box
column in the E119 test goes from 90 C at 30 minutes to 240 C at 120
minutes, a 150 K rise in 90 minutes or 1.667 K/minute.  Hence, over the time
scale of interest, a 102-minute fire, the protection slows the temperature
increase by a factor of 38.46 / 1.667 which is some 23 times slower for the
protected case.  The protection has the greatest effect for the W14x193 in
the severe UL 1709 test.  Here, the unprotected increase is 166.7 degrees
Celsius / minute, the protected increase is 2.56 degrees / minute, giving
a 65-fold reduction in temperature rise.

The results of this analysis were calculated by FEMA assuming 0.116 W/m.K
for the thermal conductivity of the fireproofing.  From the above analysis,
for purposes of calculating the temperatures of WTC core columns after the
9/11 fires, it might initially seem reasonable to allow a factor of 20 times
as the reduction in temperature rise in the case where FRC is supposed
totally intact.  Then, in the intermediate case, if three sides or 75% of
FRC remains in place, absorption would have been reduced by one-twentieth
over three-quarters of the receiving surface.  This would put the absorption
rate at 0.25 + 0.05 * 0.75 = 0.2875 of the unexposed rate.  However, a more
detailed analysis determined the factor of 20 to be wholly unsustainable.
In our study, it was decided to compute conductive heat transfer for each
individual core column.

Core columns, where adjacent to office areas, public spaces, closets or
mechanical shafts, were protected with gypsum wallboard, of unknown type and
specification.  In these cases, some faces were protected with a spray
applied fire resistive material (SFRM).  Those columns located at elevator
shafts were protected with SFRM on all faces.  WTC 1 and 2 used Cafco
Blaze-shield.  The original installation (above WTC 1 floor 38) was with
Blaze-shield Type DC/F.

Upgrading work in the late 1990s certainly included the flooring system,
with the fireproofing thickness increased to at least 1 1/2 inches.  The
replacement fireproofing used for the upgrade was Blaze-shield Type II
throughout the WTC 1 impact zone.  It is possible that some Type II was
applied to the columns at this time, for patching.  There was little SFRM on
the metal decking; some parts were bare and others were coated to 1/4",
which may have been spillover from the trusses.  Types II and DC/F have
similar specifications with Type II having a better (lower) thermal
conductivity above 600 C and DC/F having a lower thermal conductivity below
200 C.  They are both portland cement based with the insulation primarily
provided by mineral fibers.

The manufacturer specifications for Cafco Blaze-shield Type II state a
thermal conductivity of 0.043 W/m.K at 24 C.  NIST published graphs of
thermal capacity and specific heat capacity versus temperature for
Blaze-shield (Condition of Thermal Insulation, Methodology, June 23, 2003,
Carino, Gross et al).  Thermal conductivity of fireproofing increases with
temperature.  According to the NIST curves, Blaze-shield II is up to
0.09 W/m.K at 100 to 200 C, 0.13 W/m.K at 400 C, 0.21 W/m.K at 600 C and
0.28 W/m.K at 800 C.

The data was obtained from tests carried out for NIST by Anter Laboratories,
Inc. of Pittsburgh (NISTNCSTAR1-6ADraft.pdf).  The NIST draft also provides
curves from Harmathy, 1983, for Blaze-shield DC/F.  Compared to the NIST /
Anter curves for DC/F, the Harmathy results for the same material place the
thermal conductivity lower, and specific heat capacity higher.  So if the
Harmathy data is accurate, then calculations based on the NIST / Anter
curves are discernibly too pessimistic on insulation performance.
Nevertheless, we shall use the NIST data for calculations.

It is of interest to note that WTC 7, which was not hit by any aircraft and
allegedly collapsed purely as a result of fire, used Monokote MK-5
fireproofing, which is a gypsum-based SFRM.  This outperforms Blaze-shield
Types DC/F and II in terms of thermal conductivity and specific heat
capacity, according to the NIST-sponsored Anter tests.

NIST's tests on gypsum panels showed these to outperform Cafco Blaze-shield,
particularly on specific heat capacity.  The specific heat curves feature
large peaks at around 150 C and 200 C.  NISTNCSTAR1-6ADraft.pdf, Appendix B,
includes the data sets used for these curves.  Selecting one of the panels
at random (their specific heats are very similar), a summing of J/kg
required over each temperature band to determine the area under the curve
obtained an integrated value of 2143 J/kg.K over the range 25 C to 600 C,
Thermal conductivity is also better than the Cafco Blaze-shields.  It is
higher at low temperatures, but then decreases to under 0.1 W/m.K around
250 C, and is still below 0.14 W/m.K at 600 C.  The average value over the
range 25 to 600 C was found to be about 0.116 W/m.K, which is the figure
used in the FEMA analysis.

The floor trusses originally had 3/4 inch of FRC; by September 2001 the
entire impact zone (floors 94 to 98) of WTC 1 had been upgraded to 1.5
inches.  Spandrels and girders had sufficient protection to achieve a
three-hour rating.  (FEMA 403, Chapter 2, p. 2-12).  The WTC Towers were
classified as Construction Class 1B, in the 1968 edition of the New York
City Building Code, which required a 3 hour fire endurance for the columns
and a 2 hour rating for the floor systems (NIST
WTC9-11_NistUpdateReportMay2003.pdf, p. 19).  A typical office floor had 43
tons of fireproofing in the non-core alone (2 to 4 psf, NIST Appendix D).

Port Authority of New York and New Jersey measurements in April 1999 were
reported (NIST, June 2004 Thermal Insulation Methodology) to have found an
average sprayed fire resistive material thickness of 0.8 inches for core
columns, and 1.0 inches for the core beams in WTC 1.  This was from floors 1
to 45.  The columns on these floors are necessarily much more massive with
a better [M/D] mass to heated perimeter ratio, compared to those on floors
94 and up.  (The inverse measure Hp / Wsteel is also used.)  Therefore, one
would expect a thicker coating on the upper floors, in order to provide an
equivalent fire resistance rating.

Port Authority correspondence of October 30, 1969, specified 2 3/16" for
core columns smaller than 14WF228, and 1 3/16" for the remainder.  NIST used
2.2" and 1.2" for their analysis.  For this study, it was decided to err on
the low side and simplify by assuming a constant 1 1/2" as the SFRM
thickness for core columns.  For all WTC 1 members around floor 98, thermal
conductivity and specific heat was deemed to be as illustrated in the NIST /
Anter curves for Blaze-shield II, which are more pessimistic than other
generally accepted specifications for Blaze-shield DC/F, II and
gypsum-based products.

The quantity of heat conducted per unit time is given by Fourier's Law of
heat conduction:

q = dQ / dt = -k * A * dT / dx

where Q is in J/s or W, k is the thermal conductivity in W/m.K, A is the
area of the conducting material in m^2, and dT / dx is the temperature
gradient in K/m.  For the latter, (T1 - T2) / x is frequently used, with T1
and T2 being the temperatures at the boundaries and x the thickness of the
material.

Our procedure computed the steel temperature in time steps of 1 second, with
heat flux proportional to the gas - steel temperature difference at each
step, allowing for the heat capacity of the steel and insulation material:

delta Q = ki * Hp * (Tg - Ts) / xi

where delta Q is the total heat conducted per increment (J), ki is the
thermal conductivity of the insulation (W/m.K), Hp is the heated perimeter
of the member (m), Tg and Ts the gas and steel temperatures (K), and xi the
thickness of insulation material (m).  For the steel, it was decided to
retain the banded specific heat subroutine rather than simply use a constant
value.  An amount was deducted from delta Q to allow for the specific heat
of the insulation material, with the remaining quantity of heat added to
total absorption by the steel.  The latter was divided by the mass per
lineal metre and put through the routine to calculate steel temperature at
each time step.

The gas temperature was set at 850 C.  For convected heat, if the upper
layer reached slightly above 850 C, the lower layer would be well below.
Radiant transfer from flames would vary greatly over the 102 minutes, with
particular areas burning for about 20 minutes.  At times the heat flux may
have been sufficient to raise the exposed side of the fireproofing in excess
of 850 C; at other times the local combustibles might not have ignited or
might have already burnt out with the fire too weak to sustain 850 C for the
fireproofing on some particular column.  Our method supposes that the 850 C
is maintained throughout the duration.  The initial steel temperature was
taken to be 25 C.

The insulation was deemed to have been raised by an average of 500 degrees
throughout the total 6,120 seconds.  With its specific heat averaging
1,200 J/kg.K (as per NIST chart), density of 256 kg/m^3 (manufacturer's
specifications sheet), and thickness of 0.0381 m = 1.5 ins, the product
500 * 1,200 * 256 * 0.0381 = 5,852,160 times the heated perimeter in metres
gave the total heat requirement over 6,120 time steps.  In the unsteady
state, with temperature varying in both space and time, heat flux through a
typical point within the insulation would exceed heat flux at the steel.
Fire engineering formulae tend to include a multiplication factor of 0.5
when allowing for the specific heat of the fireproofing.  The above heat
requirement was halved and divided by 6,120 seconds.  So the amount deducted
at each time step was 478 * Hp joules.  Compared to no allowance for SFRM
specific heat, this deduction had the effect of reducing the steel

There was the matter of what value to use for ki.  Initially, with the
fireproofing at room temperature, its thermal conductivity would be at a
minimum: 0.043 W/m.K according to specifications.  This would soon increase.
However, relatively early in the sequence, say at t = 200 seconds, a plot of
temperature versus distance within the fireproofing would be distinctly non-linear.```
```
A steep temperature gradient would exist at the exposed end with little
temperature increase at the midpoint.  (Such a curve applies, for example,
in the case of a 6 inch concrete slab after two hours of standard fire
exposure.  When the exposed surface is 880 C, the temperatures are about
475 C at a distance of 1 inch, 250 C at 3 inches, and 150 C at 6 inches
[Fire Protection Handbook, National Fire Protection Association, Quincy,
MA]).  At 1,500 seconds, the fireproofing would be approaching a
would be rising relatively slowly at each point, with the steel acting as a
heatsink for the shielded end.

If we regard the fireproofing as being comprised of a series of sections
proceeding from the exposed side along to the steel side, each section can
be regarded as an individual heat conductance.  Heat conductors in series
are like electrical resistors in parallel: the overall conductance is the
reciprocal of the sum of the reciprocals of each individual conductance.  An
individual conductance is Cn = kn / xn with Cn the conductance in W/m^2.K,
kn thermal conductivity in W/m.K, and xn the thickness of the conductance in
metres.  The overall conductance of a series of conductors is often denoted
as U, and so:

1/U = x1 / k1 + x2 / k2 + x3 / k3... = 1 / C1 + 1 / C2 + 1 / C3...

When the thermal conductivities kn of all conductances are equal, the
overall thermal conductivity for the series is also equal to kn.  However,
for unequal thermal conductivities, the overall k is less than the mean of
all the individual values for k.  For example, three equal thickness
conductances
1 / 3 + 1 / 5 + 1 / 7 = 1 / U
so to find the overall thermal conductivity of the three conductances block,
U is calculated and multiplied by the total thickness of 3 length units
giving an overall k of 4.437 which is less than 5.  The difference between
the overall k and the mean of the individual k values increases with the
variance of k values.  And of course, when the thermal gradient is sharpest
at the hot end of a material and thermal conductivity is positively
correlated with temperature, the mean temperature and k value is relatively
low.

Temperatures were calculated for various times and points within the
Blaze-shield material.  After dividing the total thickness into eight
sections and adding the individual conductances to find the overall thermal
conductivity for the series of conductances, it was decided to use a value
of ki = 0.09 W/m.K for the first 400 seconds, 0.14 up to 2,250 seconds, and
0.148 for the remaining time to t = 6,120 seconds.  FEMA's value of a
constant 0.116 W/m.K for insulation would probably be too low for the WTC
Blaze-shield material over the 6,120 second fire, unless we assume the
Harmathy, 1983, results.  But it would be just about right for gypsum
panels.

An SFRM thickness underestimate of one-third actually leads to a heat
transfer overestimate of more than one-half.  At any given time, the SFRM is
cooler and its overall thermal conductivity ki would have been
overestimated.  There is also the additional heat capacity of the neglected
insulation.  At 1.5 inches, we have probably underestimated the thickness of
most (particularly the lightest) of the core column SFRM.

The all FRC remaining calculation worked in 1 metre lengths of column.  The
no FRC radiative / convective transfer assumed the heat to be distributed
over 11 feet of column.  For the scenario with 75% of FRC remaining, the
total absorption from the no FRC case which had been computed and saved was
divided by four.  The total energy absorbed from the all FRC intact
conductive transfer version was multiplied by 0.75 * 11 feet / 3.2808 feet.
The sum of these subtotals was then treated as the total absorption by an
11 feet length of column, and put through the banded heat capacity routine.

The all FRC and no FRC cases represent worst case scenarios: maximum energy
density over the 6,120 seconds, zero shielding by adjacent members, burning
combustibles within a metre of all columns, etc.  But it must be admitted
that averaging the results from each to produce the third measure of
"3/4 FRC remaining" is an optimistic interpretation.  Thermal gradients
would exist within the steel, since its thermal conductivity and volumetric
heat capacity are both finite.  The exposed sections in the "3/4 FRC" case
would attain temperatures between the relevant figure shown for the 3/4 case
and for no FRC.  Conversely, the protected sections would reach temperatures
between those of the all FRC and 3/4 cases.  We shall be examining the
probable extent of the excess temperatures in due course, in the discussion
on the predicted results, and also in considering the likely damage to
fireproofing.

At floor 98, the box columns had all been replaced by WF columns.  NIST
(Appendix E, Fig. E-6) shows the specifications of typical rolled WF shapes
between floors 83 to 86.  The larger 14WF730 or W14x730 was used for
columns 1001, 1008, 501 and 508; the medium sized 14WF219 was used for
columns 607 and 906; and the small 14WF61 was used for column 705.  To
select the most likely member for each column on floor 98, the demand on
each core column was calculated by multiplying the total gravity load of
34,573 tons by the member's load area as a proportion of the total floor
area.  This was then divided by NIST's published pre-impact demand /
capacity ratio for 1WTC columns on floors 93 to 98 (which ranged from about
0.4 to just over 0.5 with a mean of 0.48), to find the required capacity.

NIST Appendix E (Table E-3) lists the yield strength of core columns within
the aircraft penetration zone of floors 94 to 98.  Most were 36 or 42 ksi;
there were very few box columns or higher strength wide flanges.  Almost
two-thirds of core columns had a yield strength of 36 ksi; almost all the
remainder were 42 ksi.  With no box columns at floor 98, it was decided to
assume 16 WF shapes at 42 ksi and 31 at 36 ksi.  This allowed 42 ksi to be
used along the 24 columns at the perimeter of the core area, excluding the
two centremost columns on each of the four sides.  The nominal depth - the
first number shown in ddWFnnn or more typically Wddxnnn where dd is the
nominal depth in inches and nnn is the weight in pounds per linear foot -
was taken to be a constant 14 inches.  With the capacity having been
determined and the yield strength selected, the remaining specification nnn
was selected for the purposes of this study by taking the closest from a
list of preferred sizes.  For simplicity in selection, columns were treated
as short columns in which the axial capacity is equal to the yield strength
times the cross-section.

Under normal pre-impact conditions, the core columns achieved a sufficiently
low slenderness ratio that this short column approximation introduced errors
of only 1 to 5%, marginally favouring fire collapse theories by
underestimating member specification.  In reality, some columns -
particularly the lightest - would have been the next preferred size.  On
average, the demand / capacity ratio would have been as assumed.  But
underestimated mass to heated perimeter ratios would generate a small error,
leading to overstating of the predicted temperature rises.

The Euler buckling limit only applies to long columns.  In the intermediate
(and short) range, formulae such as Johnson's apply.  The axial load
capacity is given by the cross-section times the yield strength times a
factor dependent on the effective length, the radius of gyration, and the
Young's modulus and yield strength for the material.  This factor is very
close to unity for short columns.  The crossover point from intermediate to
long columns occurs at the critical slenderness ratio scr, and is dependent
only on Young's modulus E and the yield strength sigma_y:
scr = SQR(2 * pi^2 * E / sigma_y)

Taking Young's modulus for steel at 2.9 * 10^7 psi, 36 and 42 ksi steel have
critical slenderness ratios respectively of 126 and 117.  In the WTC the
interfloor length was 12 feet.  But with the floors securely fixing the
columns at both ends, an effective length constant of 0.5 applies, placing
the effective length at 6 feet.  The slenderness ratio s is the effective
length Le divided by the radius of gyration k.  The latter is given by
k = SQR(Iyy / A)
where Iyy is the minimum moment of inertia and A is the cross-section.  At
the intermediate-long or Johnson-Euler crossover point where the slenderness
ratio equals the critical slenderness ratio, the actual capacity is 0.5
times the yield strength times the cross-section.  Over the Johnson range of
intermediate (and short) columns, where the actual capacity Pcr divided by
the compressive capacity of yield strength times cross-section is 0.5 or
greater, this multiplying factor or ratio is given by:
Pcr / (sigma_y * A) = 1 - (Le / k)^2 * sigma_y / (4 * pi^2 * E)
which simplifies as
Pcr / (sigma_y * A) = 1 - 0.5 * s^2 / scr^2

As would be expected, it was found that members around floor 98 were less
substantial than those used for lower floors.  W14x730 would have provided
far too much redundancy on floor 98.  Here, W14x257 was selected as the most
likely size of the largest members (e.g. at the corners of the core area).
The smallest columns (704 and 705) were taken as W14x53.

For a 36 ksi W14x53, Iyy is only 57.7 ins^4, A is 15.55 ins^2, and the
radius of gyration is 1.926 inches.  Even if we assume 12 feet for the
effective length, the slenderness ratio is 74.77, which is well below the
critical 126.  This would make the multiplying factor 0.824.  If Le is taken
as 6 feet, the slenderness ratio is 37.38 (under 60 is regarded as within
the short column range) and the factor increases to 0.956.

For a 42 ksi W14x257, Iyy is 1290 ins^4, A is 75.41 ins^2, and the radius of
gyration k is 4.136 inches.  Taking the effective length at 6 feet, the
slenderness ratio is 17.41, well below the critical 116.7, and the
multiplying factor is 0.989.  Doubling Le would double s and reduce the
actual capacity to 0.955 of the compressive capacity.  Removing floors in
sufficient numbers would dramatically lower capacity.  At this point it is
worth calculating how many floors could be removed before producing global
instability.

WTC core column demand / capacity ratios averaged 0.48.  Let's evaluate the
required effective length to raise the slenderness ratio up to the critical
slenderness ratio, where the capacity becomes half that of the (almost)
infinitely short column.  For the 36 ksi W14x53, the radius of gyration was
1.926 inches.  In order for the slenderness ratio to equal the critical
slenderness ratio of 126, we require 126 = Le / k = Le / 1.926, so
Le = 126 * 1.926 = 242.7 inches.  The boundary conditions, unless the floors
are free to move as the column buckles, are fixed - fixed.  So the effective
length is half the actual length, which is 485.4 inches = 40.45 feet.
Hence, for this column, at pre-impact cold conditions, a length of three
floors or 36 feet would be suitably low.  Removal of three floors, raising
the length to 48 feet, should result in column buckling, with load
redistributed through the hat truss and remaining floor systems to
neighbouring columns.  If the W14x53 column had been heated close to a
critical temperature, say 500 - 600 C, or demand had already been
significantly increased due to redistribution from severed columns, then the
column might not withstand the loss of one or two floors.

Now for the 42 ksi W14x257.  The radius of gyration k was 4.136 inches, and
the critical slenderness ratio was 117.  We require 117 = Le / k =
Le / 4.136, so Le = 117 * 4.136 = 483.9 inches.  The length is 2 * Le =
967.8 inches = 80.65 feet.  Hence, for this column, at pre-impact cold
conditions, a length of six floors or 72 feet would be suitably low.
Removal of six floors, raising the length to 84 feet, should result in
column buckling.  The other columns are weaker, and by this point global
instability should result.

Let's take the case of a neighbouring column to the W14x53 (704 and 705),
which would probably be a 36 ksi W14x82 or similar.  For this member, Iyy is
148 ins^4, A is 24.06 ins^2, so from k = SQR(Iyy / A) we find the radius of
gyration is 2.48 inches.  Since the critical slenderness ratio is 126, the
half capacity of a perfectly short column occurs at 126 = Le / k = Le / 2.48
which places Le at 312.5 inches.  The actual length is twice this at 625
inches = 52.08 feet.  Hence, for this column, at pre-impact cold conditions,
a length of four floors or 48 feet would be suitably low.  Removal of four
floors, raising the length to 60 feet, should result in column buckling.  At
this stage the number of collapsed columns would probably be too low to
initiate global collapse, and loads would be redistributed to the more
massive columns.  However, if a number of perimeter and core columns had
already been taken out and the remainder had been heated so as to seriously
lower capacity, there would be serious risk of global instability and
collapse.

So under pre-impact cold conditions, removal of two floors should not cause
buckling of any columns.  With three floors taken out, columns 704 and 705
would collapse, redistributing load to neighbouring columns through the hat
truss and remaining floor systems.  After removing four floors, some more of
the smaller columns would fail, but the Tower would remain stable.  Removal
of five or more floors would risk instability and global collapse.  At
higher temperatures, the first column(s) would probably start to fail after
removal of two floors, with global instability and collapse a serious risk
after four floors had been taken out.  This is in line with the NIST
analysis (NIST Interim Report, Appendix D, D-12).

Judging by the NIST d/c ratios, the total cold capacity of all core columns
on floor 98 would have been 42,332 tons.  The total capacity of all members
selected (approximating as short columns) was 42,126 tons.  After rounding
errors, the total demand on all core columns totalled 20,127 tons,

In the ASCII version of this document, full results of the analysis for each
core column are presented below in comma-separated values (CSV).  Carriages
returns are only included at the end of each row.  This format may be copied
of 25 columns of variables, 2 rows to specify the header, 47 rows for the
core columns, and a final row for the total or mean value.

The latter totals or means are not weight-adjusted.  For example, the 42 ksi
total cross-section is 1026.4 ins^2 from only 16 columns, and the
corresponding 36 ksi total is 1142.26 ins^2 from 31 columns.  Multiplying
the weighted average yield strength of about 38.84 ksi by the total
cross-section gives the total cold capacity of 42,126 tons.

WTC Fires Data, HTML Table Format```
`WTC Fires Data, CSV Format (It's best to open these links in a new window or tab)`
```
Reviewing the results for column 501 in detail, the load area covered
x-coordinates 6 to 14 and y-coordinates 42.5 to 54 (in metres).  Hence, its
load area was 92 m^2 from a total floor area of 4,019 m^2.  At floor 98, the
total gravity load (we include floors 98 to 110 plus the roof) was 34,573
tons as previously calculated, so the axial demand on column 501 at floor 98
was 34573 * 92 / 4019 = 791.4 tons.  NIST shows the pre-impact demand /
capacity ratio at 0.5 for this column, so from 791.4 / 0.5 we want to make
its capacity close to 1582.8 tons.  A member selection program accepted
inputs of column number, x and y bounds of the load area, and NIST d/c ratio
as estimated from their diagram.  Variables calculated and displayed
included load area, share of load, axial demand, required capacity, and the
required cross-section and weight per linear foot for both cases of 36 ksi
and 42 ksi yield strength.

It had been decided to assume 501 was one of the 42 ksi columns.  At 42 ksi,
the required cross-section in ins^2 is the required capacity in tons divided
by 21.  So 1582.8 / 21 = 75.37 ins^2.  The cross-section in ft^2 times the
density of 490.7 lb/ft^3 gives the weight in pounds per linear foot.  So
75.37 * 490.7 / 144 = 256.8.  W14x257 is a recognised size and the obvious
choice.  To find the cross-section in ins^2 for each member, the previous
calculation was executed in reverse: divide the lb/ft rating by 490.7 and
multiply by 144, i.e. divide by 3.408.  So 257 / 3.408 = 75.41 ins^2
cross-section.  The short column approximation gives 42,000 * 75.41 =
3,167,220 lb = 1,583.6 short tons pre-impact cold capacity.  These
capacities were rounded to the nearest ton and inputted to the spreadsheet.

For a W14x257, its 'd' size - the width you view by looking normal to the
web - is 16.38 ins.  Its 'bf' size - as seen normal to a flange - is very
close to 16 ins.  Rather than totalling both widths and multiplying by
2 / pi to obtain an average width as integrated over all viewing angles,
foreshortening was calculated from the cosine of the angle of incidence for
both sides.  10 feet was assumed as the rectangle length.

The total heat absorbed after summing contributions from each m^2 of floor
space, given the condition of all fireproofing removed, was computed as
588 MJ.  After dividing this by the mass of an 11 feet section (1282 kg),
458,700 J/kg lies in the sharply peaking phase change region, with the
accumulated total from each temperature band of specific heat reaching
424,350 J/kg to raise the steel from 25 C to 730 C, and another 40,000
J/kg.K assumed required for the 730 to 731 peak before reverting to 720
J/kg.K from 731 to 750 C.  Hence, the predicted final temperature for the
totally exposed case was just under 731 C.

The heated perimeter for this column was calculated at 94.39 ins from
4 * bf + 2 * d - 2 * tw.  The data above doesn't include Hp or the web
thickness tw, but tw ranges from 0.37" for a W14x53 up to 1.175" for a
W14x257.  So the reader can readily calculate Hp quite accurately from the
bf and d specifications, possibly deducting an inch or two for 2 x tw.  This
program calculated the heat conducted through the insulation in time steps
of one second.  The gas temperature was a constant 850 C, with the steel
starting at 25 C and increasing with each second of simulated time.  Three
different values of k for the insulation were used to allow for increased
thermal conductivity with temperature at various stages of the fire.  After
deducting a quantity of heat with each time step to allow for the specific
heat of the insulation, the final total of heat absorbed was 32.8 MJ per
metre length which results in a final temperature of 203 C for the steel.
The 32.8 MJ becomes 110 MJ when normalised for an 11 feet section for direct
comparison with the all FRC removed case (588 MJ).  The case of three
quarters of FRC intact counts 3/4 of the 110 MJ plus 1/4 of the 588 MJ, for
a total absorption of 229.5 MJ per 11 feet section, resulting in a steel
temperature of 369 C.

For column 501 in the no FRC case, the final temperature of 731 C would
lower the strength of the column to 0.185 of its cold value.  The demand was
791 tons, cold capacity was 1,584 tons, cold d/c ratio was 0.499, hot
capacity was 293 tons, and hot d/c ratio was 2.699 (i.e. failure!).  In the
case of 3/4 of FRC intact, the final temperature was 369 C which lowers the
capacity to 0.87 times 1,584 tons which is 1,378 tons.  The final hot d/c
ratio was 0.574.  The case of all FRC intact leads to a final temperature of
203 C at 102 minutes, with the hot capacity being 1,584 times 0.95 which is
1,505 tons.  The final hot d/c ratio was 0.526.

In the 3/4 FRC remaining case, temperatures in the unprotected sections
would exceed the calculated figure depending on the thermal diffusivity for
steel and on the sizes of the unprotected areas.  As mentioned above, there
is a particular length associated with a given time duration and thermal
diffusivity.  The relevant time is 102 minutes or 6,120 seconds.  We take
the thermal diffusivity (alpha) of steel to be 8.85 * 10^-6 m^2/s assuming
mean values of 40 W/m.K and 575 J/kg.K for thermal conductivity and heat
capacity over the temperature range 25 to 850 C.  The distance x at which
half of the temperature rise occurs for a bar of semi-infinite length
'instantly' heated at one end, as explained above, is given by
x = 2 * z * SQR(alpha * t)
where z = 0.477, and we find the length is 0.222 metres = 8.74 inches.

Suppose e is the fully exposed temperature and f is the fully protected
temperature (C) in a particular member at t = 6,120 seconds.  The mean
temperature T of the member at this time for various proportions p of FRC
stripped away may be reasonably approximated from:
T = f + p * (e - f)

For the temperature f for a particular member, refer to the first predicted
temperature column in the above data, "final temp C, no FRC (or gas
temperature limit)".  The temperature e is shown in the fourth column,
"final temp, all FRC remaining".  If the averaged temperature T exceeds the
expected gas temperature, a limit can then be imposed.  This method, for
p = 0.25, gives slightly different results from those shown in the "final
temp, FRC lost 1 side" column, since the latter was based on heat absorption
and allowed for a variable specific heat with temperature.

The temperature T assumes an infinite thermal diffusivity for steel.  In
many cases, the average temperature will provide a good guide to the final
strength of the member.  The hot section yields and transfers load to the
colder section.  It is known that when supporting concrete slabs, beams
exposed to fire can develop temperature differences of some 200 C between
the upper, slab-protected and lower, exposed flange.  For a "fully" loaded
beam (fully loaded in the sense that designs do not tend to feature cold
demand / capacity ratios of worse than 0.5; the minimum redundancy or safety
factor would be two), the limiting temperature is generally taken as around
550 C for beams exposed on all four sides.  When supporting a concrete slab,
the limiting temperature for the lower flange is regarded as around 620 C.
These limiting temperatures also increase with decreasing load.

A 620 C lower flange and 200 C difference would place the average at 520 C,
suggesting 30 degrees as a reasonable correction to the mean temperature in
a member, for cases where the analysis predicts development of significant
temperature gradients.  We shall consider some of these scenarios in more
detail.

Let's suppose for the moment that the web and flanges of a WTC core column
are thermally isolated.  We'll take a worst case of "FRC lost, 1 side".
Suppose impacting debris is normal to a side of the web, and results in
total destruction of fireproofing on that side of the web (along most or all
of the height of a floor).  The protection is also stripped from the
flanges' inner sides adjacent to that side of the web, and from half of the
two debris facing ends of the flanges.  This conveniently divides the
cross-section into two: the web and half of each flange are left with very
close to half of their perimeter exposed, and the other flange halves are
fully protected.  The proportion of FRC removed is given by:
(d + bf - tw - tf) / Hp
and is actually more than 1/4; it is about 1/3 of the heated perimeter.

Most core columns were in fact aligned with the webs running north-south, so
the "Flight 11" debris approaching from the north of WTC 1 would have
impacted with one side of a flange.  The web and other flange would have
been essentially in the "shadow" of the exposed flange, hence protected.
However, columns 501 and 704 were each aligned the other way, with one side
of the web facing the incoming mass.  We'll start with 704, the worst case.

The web thickness (tw) of a W14x53 is only 0.37" which is very small in
relation to the characteristic length (at t = 6,120 seconds) of 8.74" for
steel, the distance at which half of a temperature rise at one end occurs
for a semi-infinite bar.  Thus, after 6,120 seconds, temperature difference
would be negligible for steel over a distance of 0.37 inches.  The same
applies for the flange thickness (tf) at 0.66 inches.  With fireproofing
removed from one side of the web, the final temperature T of the web would
be midway between the protected and exposed cases, at 0.5 * (410 + 1157)
which is 784 C.  With the thicknesses small in relation to the lengths (in
the W14x53 case d = 13.92" and bf = 8.06"), the proportion one small side
and one inner length to flange heated perimeter will always be close to a
quarter.  So the mean flange temperature at 6,120 seconds would be
f + p * (e - f) = 410 + 0.25 * (1157 - 410) = 597 C.

For this case, it is also helpful to initially consider each flange as split
into two isolated halves.  One half has half of the FRC removed and is at
410 + 0.5 * (1157 - 410) = 784 C; the other is at the 410 C "all FRC
remaining" temperature.  So at 6,120 seconds, the thermally isolated web and
one half of each flange would be at 784 C with the remaining flange section
at 410 C.

The web is thinner than the flanges.  However, this is countered by the fact
that points along the web and inner flanges are slightly shielded from line
of sight radiation, whereas a point on the outer flange has 180 degrees of
view.  Thus, for a given level of fireproofing remaining, differences in web
and flange thickness would not lead to a substantial divergence of predicted
and actual temperatures.  Predicted temperatures would probably be slightly
low for the web and slightly high for the flanges.

The flanges, at 5.32 ins^2 each, total more than double the web
cross-section of 4.66 ins^2.  (These figures are about 2% low, neglecting
the extra material on the inner corners.)  The mean flange temperature is
597 C, with the web at 784 C, so connecting these could be expected to
result in a mean temperature below the average of 691 C.  There are two hot
flange halves totalling 5.32 ins^2 at 784 C, two colder flange halves
totalling 5.32 ins^2 at 410 C, and 4.66 ins^2 of hot web at 784 C.  The
expected mean temperature rise, with these all in thermal contact, would be:
[9.98 * (784 - 25) + 5.32 * (410 - 25)] / 15.3 = 629 degrees, making the
final mean temperature 629 + 25 = 654 C.

Given e = 1157 C and f = 410 C, the formula f + p * (e - f) requires that
the proportion p of FRC removed is about 32.7% to obtain the mean 654 C.
For the W14x53 the proportion (d + bf - tw - tf) / Hp is 0.353, which if
applied to f + p * (e - f) would predict 674 C.  The 654 C is a more
reasonable prediction here, since it makes an allowance for the fireproofing
remaining intact at the outer flange where absorption would be greatest, and
being stripped off at the web and inner flange where the flange affords some
shielding from infra red radiative transfer.  In the other case, where we
suppose FRC is totally stripped from the outer side of one flange, the
predicted temperature should be higher than f + p * (e - f) would predict.
Thus, the difference, between the worst case of debris impacting normal to a
web and the alternative of debris impacting normal to a flange, is slightly
reduced.

Since the web and flanges were in thermal contact to an extent governed by
time, length, and thermal diffusivity for steel, this is rather like a bar
of material with the temperature of one end raised by 784 - 410 = 374
degrees over the 102 minutes.  For steel, an instant heating of one end
would lead to half of the temperature rise at the associated length of 8.74
inches.  Now the maximum and minimum temperatures diverged over the
102-minute fire, so the temperature rise at the hot end averaged over the
102 minutes was only around half of the final increase relative to the
colder end.  However, this is countered by the "bar" being of limited length
rather than semi-infinite.  For our purposes, the proportion of increase for
an instantly heated semi-infinite bar provides a reasonable rough and ready
guide as to the temperature gradients that might be developed over a period
of 102 minutes.

So the 374 degree temperature difference would have been reduced by
approximately one-half, or 187 degrees, at an 8.74" separation point.  Let
us suppose the temperature at a web/flange connection is 654 C, the weighted
average final temperature.  If the length was very long in relation to 8.74
inches, the 374 degree difference would be reached at 102 minutes.  The hot
end(s) would be 654 + 187 C, and the cold end(s) would be 654 - 187 C.
Proceeding towards the web or the tip of the hot half of a flange, the
temperature gradient would be positive; proceeding towards the tip of the
cold half of a flange, the temperature gradient would be negative.

For a W14x53, the flange breadth bf is 8.06" so the web/flange connection to
flange tip is only 4.03".  At 102 minutes and 4.03 inches, a semi-infinite
bar would experience some 76% of the instantaneous temperature rise at the
exposed tip.  This suggests that after 102 minutes of temperatures diverging
according to the degree of FRC stripped from various points on the member,
the temperature gradients developed would be insufficient to enable even
half of the 187 degree difference resulting if the length involved were very
long.  With the web midpoint to web/flange connection being equal to
0.5 * d - tf = 0.5 * 13.92 - 0.66 = 6.3", this also suggests that the
thermally isolated temperature differences would be more than halved.  At
102 minutes and 6.3 inches, a point on a semi-infinite bar would experience
63% of an instantaneous exposed end temperature increase, i.e., only 37% of
the initial temperature difference across the 6.3 inches of material would
remain after the 102 minutes.

Let's suppose that the connected temperature differences are half of the
thermally isolated differences.  So the web/flange joint is at the 654 C
mean, the hot tips of the flanges and the web midpoint are at
654 + 0.5 * 187 = 748 C, and the colder flange tips are at
654 - 0.5 * 187 = 560 C.

A number of comparative strength of steel versus temperature plots were
consulted in order to determine the strength reduction factor for the
various temperatures.  FEMA Appendix A, Fig A-6 (Lie, 1992) shows a
relatively rapid decline compared to most published curves, which remain
flat until 250 to 400 C.  FEMA's curve has the 0.5 point at about 550 C,
whereas others such as Corus and the EC3 curve make it from 575 to 610 C.
At 750 C, for example, curves generally coincide with 0.17 as the strength
reduction factor.  Rather than take the mean of all curves, it was decided
to take the mean of the other curves, and then take the mean of this mean
and the FEMA version.  E.g., at 500 C FEMA has 0.57, and three other curves
have 0.8, 0.78, and 0.8.  The value is taken as the mean of 0.57 and 0.793
which is 0.68.  The table below, of temperature vs. relative yield strength,
shows the actual values assumed here.

Temp (C): 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
R.Strnth: 1, 1, .995, .995, .99, .99, .99, .985, .985, .98, .98
Temp (C): 110, 120, 130, 140, 150, 160, 170, 180, 190, 200
R.Strnth: .98, .975, .97, .97, .965, .965, .96, .955, .955, .95
Temp (C): 210, 220, 230, 240, 250, 260, 270, 280, 290, 300
R.Strnth: .95, .945, .94, .935, .93, .925, .925, .92, .915, .91
Temp (C): 310, 320, 330, 340, 350, 360, 370, 380, 390, 400
R.Strnth: .905, .9, .89, .885, .88, .875, .87, .86, .855, .845
Temp (C): 410, 420, 430, 440, 450, 460, 470, 480, 490, 500
R.Strnth: .835, .82, .805, .79, .775, .76, .74, .72, .7, .68
Temp (C): 510, 520, 530, 540, 550, 560, 570, 580, 590, 600
R.Strnth: .66, .635, .61, .585, .56, .535, .51, .48, .45, .425
Temp (C): 610, 620, 630, 640, 650, 660, 670, 680, 690, 700
R.Strnth: .4, .38, .36, .34, .32, .3, .275, .255, .235, .215
Temp (C): 710, 720, 730, 740, 750, 760, 770, 780, 790, 800
R.Strnth: .205, .195, .185, .18, .17, .16, .15, .14, .13, .125
Temp (C): 810, 820, 830, 840, 850
R.Strnth: .115, .11, .1, .095, .09

The web, the worst affected, ranges from 654 C to 748 C which corresponds to
strength reduction factors of 0.31 to 0.17, an average of 0.24.  The flanges
range from 654 C to 748 C along the partially exposed half, and 654 C to
560 C along the protected side.  The strength reduction factor for the
flange is 0.535 at the cold tip, 0.31 at the midpoint, and 0.17 at the hot
tip.  Although the flanges contribute most of the cross-section and
strength, let's suppose the overall strength reduction factor is the mean
of 0.31 and 0.24 which is 0.275.  The cold demand / capacity ratio for core
column 704 was 0.482, so failure would be expected at these temperatures.

A 0.275 strength reduction factor corresponds to 670 C, so here we are
allowing an extra 16 C for temperature gradients.  If we allow, say, an
extra 40 C on top of the 654 C, then 694 C would correspond to a 0.225 times
reduction in strength which is probably too much.  The proportion
(d + bf - tw - tf) / Hp = 0.353 for the W14x53, applied to f + p * (e - f)
predicts 674 C, with a strength reduction of 0.265.  So in this case the
calculation from proportion of FRC remaining provides a useful guide to the
final strength, if we accept that some allowance has been made for

The column was aligned so that the web was facing the oncoming debris.
Core columns 504, 505, 604 and 605, along with elevator shafts, would have
afforded some protection.  If the impact was sufficient to totally destroy
all fireproofing on the debris-facing side, then failure could have
resulted within 102 minutes of fire exposure.  However, the scenario
requires maximum energy density from the fire and significant aircraft
impact damage.  Floor(s) which suffered the greatest damage from aircraft
impact did not necessarily coincide with the floor(s) with the greatest
fires.  Here, we have concentrated on floor 98 for detailed calculations of
demand, column capacity, etc, but aircraft impact damage would seem to have
been greatest on floors 95 to 96.

On floor 96, demand / capacity ratio would have been as per floor 98 on
average, but members would be a little stronger.  As member capacity is
increased, the mass to heated perimeter ratio improves, so final steel
temperatures are lower for a given level of intact fireproofing.  Since any
column failure required both that aircraft impact damage and fire intensity
be sufficiently severe, then failure would have been more likely to occur on
floor 96.

The NIST paint analysis on recovered steel included core columns 605 (floors
98-99) and 603 (floors 92-93) [Project #3: Analysis of Structural Steel,
Update October 20, 2004, Frank Gayle].  These analyses indicated that both
of these columns remained under 250 C, and were consistent with an impact
model resulting in "fire proofing intact" for these columns at the named
floor.  Their fire model suggested column 605 on floor 98 experienced "some
surrounding fire", but peak temperatures were under 200 C.  Column 603,
floors 92 and 93, had a peak temperature of 100 C with "no significant fires
near".

This is a very reasonable model, and is by far the most likely scenario.
Our results for core column 605 on floor 98 predicted 354 C at 102 minutes
if all fireproofing was intact.  But this assumed a worst case fire, with
all of the exposed side of the FRC remaining at 850 C throughout the 102
minutes.  In addition, the fireproofing thickness for this column was
probably 2 3/16" not 1 1/2" as used to obtain the 354 C result.

The 50x row of columns were the most exposed core columns (after the
putative "Flight 11" had smashed its way through the exterior columns, floor
slabs, etc), followed by the 60x row and the 70x row...  If debris had not
even dislodged fireproofing from column 605, it would have been unlikely to
strip the insulation from columns 704 or 705.  On floor 98 it is quite
possible that all the fireproofing remained intact.  Floors 95 and 96
probably saw some damage, particularly to the 50x row, and one or two of
these core columns might have been severed if directly hit by an engine.

Let's turn to core column 705.  This was also likely to have been a W14x53
at Floor 98, and if upgraded to a W14x68, say, for floors 95 or 96, our
predicted temperatures would not be gross overestimates provided the
requirements of fire intensity and impact damage were satisfied.

In contrast to column 704, 705 was aligned as were most of the core columns,
with the web running north-south and the flanges normal to the impacting
debris.  Hence, the most likely damage to fireproofing would have been to
the outer side of one of the flanges.  We shall consider a scenario with
fireproofing totally stripped off the outer side of a flange.

Here, the proportion of the heated perimeter with FRC removed is bf / Hp
which is only 0.136 for a W14x53.  Using f + p * (e - f) we have p = 0.136,
f = 410 C, and e = 1127 C which would predict a final temperature of 508 C.
(The computed e is slightly down from the 1157 C of column 704, due to
rounding errors arising from co-ordinate selection.)  508 C would correspond
to a strength reduction factor of 0.66, i.e. no failure.  However, the
alternate case with the web FRC stripped is slightly better when one allows
for the partial shielding of the web and inner flanges.  Insulation on the
outer flange has a greater effect, since this is more exposed to radiative
transfer than the web or inner flange.

If we imagine the partially exposed debris-facing flange as thermally
isolated from the web and undamaged flange, the former has half of its FRC
removed whilst the remaining section is fully protected.  The protected web
and flange is at 410 C, and amounts to 4.66 ins^2 plus 5.32 ins^2 which is
9.98 ins^2.  The exposed flange is 5.32 ins^2 at 410 + 0.5 * (1127 - 410) =
769 C.  If we connect these together and calculate a weight-adjusted average
temperature, we have [9.98 * (410 - 25) + 5.32 * (769 - 25)] / 15.3 = 509 C
temperature rise to give a final temperature of 534 C.  At a strength
reduction factor of 0.6, the column should survive.

If we were to add 40 degrees as an allowance for thermal gradients within
the member, the strength reduction factor becomes 0.495.  Column 705 had a
cold demand / capacity ratio of 0.482, so 574 C as the equivalent mean
temperature would be a borderline case.  Under normal conditions it would
probably still hold up.  On average, the steel would exceed its nominal
strength, and in most cases a member has enough reserve capacity to sustain
a demand / capacity ratio greater than one.  In a minority of cases, the
actual strength is less than the nominal strength.  The impact damage
clearly included perimeter columns, with some load transferred to the 50x
row of core columns and relatively little extra to other core columns.  If
some core columns had been severed by the impact, then the increased load on
705 would increase its chances of collapse.

After 704 and 705, the next hottest predicted temperatures were for core
columns 601 and 608.  Let's take 601 which is marginally above 608 after
co-ordinate rounding errors.  These columns are both aligned as 705 and
virtually all of the other columns, so less of the fireproofing would have
been damaged.  We'll suppose that the whole outer surface of one flange
becomes exposed, as in the case of 705.

601 was assumed to be a W14x120 in this analysis.  As explained, it may have
been slightly heavier on floor 95 or 96, and the W14x120 choice treats the
member as a perfect short column with no safety factor to allow for the
slenderness ratio.  The proportion p of the heated perimeter with FRC
removed is bf / Hp which is 14.67 / 86.46 = 0.17 for a W14x120.  From
f + p * (e - f) where p = 0.17, f = 317 C, and e = 1132 C we obtain 456 C as
the predicted final temperature.  With 0.765 of cold capacity retained,
no collapse would be observed.  But the more detailed calculation should
predict a higher temperature.

The protected web and flange is at 317 C, and amounts to 7.43 ins^2 plus
13.79 ins^2 which is 21.22 ins^2.  The exposed flange has half its FRC
remaining, and amounts to 13.79 ins^2 at 317 + 0.5 * (1132 - 317) = 725 C.
If we connect these together and calculate a weight-adjusted average
temperature, we have [21.22 * (317 - 25) + 13.79 * (725 - 25)] / 35.01 =
453 C temperature rise to give a final temperature of 478 C.  At a strength
reduction factor of 0.725, the column should survive.  Adding an extra 40
degrees as an allowance for thermal gradients within the member, the
strength reduction factor becomes 0.64.  Column 601 had a cold demand /
capacity ratio of 0.512, so 518 C as the equivalent mean temperature would
be unlikely to lead to collapse.

Generally, adding 60 C to the calculated temperature from proportion of
heated perimeter should predict a strength reduction factor which is at
least as severe as should be expected.  Column 804 is another to consider.
For a W14x74, the ratio p = bf / Hp = 10.07 / 67.72 = 0.149.  From
f + p * (e - f) where f = 367 C and e = 906 C we obtain 447 C as
the predicted final temperature.  Adding 60 C suggests taking the strength
reduction factor for 507 C which is 0.665.  Collapse would be unlikely as a
result of fire.

Although the four corner core columns were the heaviest of these members at
W14x257, they were all aligned such that debris would be likely to damage
a greater proportion of fireproofing.  The proportion p for these columns
would be p = (d + bf - tw - tf) / Hp = (16.38 + 16 - 1.175 - 1.89) / 94.41
= 0.31.  So from f + p * (e - f) where f = 203 C and e = 731 C we obtain
367 C as the predicted final temperature.  The FRC remains on the outer web,
where it has a relatively high protective effect, and is stripped from the
web and inner flange which are partially shielded in any case.  These
columns would not collapse due to fire.

Of the 47 core columns, there are 22 where the predicted final temperature
for "FRC lost on one side" was over 450 C, including 7 where it was over
500 C.  These 22 might just have possibly suffered fire-induced collapse if
all the following conditions had been met, but the chances of this are so
low that almost all can be ruled out:

(i) Local fires would need to be sufficiently severe, with 1.385 * 10^12 J
released on the relevant floor over 102 minutes.
(ii) Debris-induced damage to fireproofing on the relevant floor would need
to be total along the full breadth of the debris-facing side and throughout
most of the height of the floor for the relevant column.
(iii) Aircraft impact damage to columns on the relevant floor and the
resulting redistribution of load would need to be sufficiently severe in
order to bring about significant impairment of cold demand / capacity ratio
for the relevant column.  At the same time, it was clearly not sufficient
for aircraft impact alone to bring about immediate global collapse.
(iv) Fireproofing, where intact, would need to be 1 1/2" not 2 3/16" as per
the design specifications.

To expand a little on the first condition, in order for each member to
absorb the computed energy quantities, it would be necessary that the fires
were sufficiently distributed across the particular floor that minimum
separation of heat source and steel was within a metre or so.  The program
analysed the proportion of total heat received over various separations.
For column 501, for example, points within a 2.5 metre radius contributed
31.7% of the total; points within a 5 metre radius contributed 53.6%; points
within 10 metres provided 74% of the total; and points within 20 metres
provided 89.7% of absorption.  Another analysis examined the effects of a
more uneven distribution, where it was supposed that fires remained some
specified minimum separation from the receiving column.  Here, it was
proposed that the total heat transmitted across the floor was unchanged, but
points with a separation below the specified minimum were taken to have a
separation equal to that minimum.

For column 501, with the sub-5 metre separation points moved out to 5
metres, total absorption decreased by 43.4% from 588 to 333 MJ.  The three
calculated final temperatures decreased to 498 C (exposed), 282 C (3/4 FRC
intact), and 203 C (protected.  This assumed sufficient flux always existed
to maintain the exposed edge of FRC at 850 C, so this temperature -
determined by the Hp / Ws ratio - was unchanged.  If we want to allow for a
lower mean FRC exposed edge temperature, maybe over 850 C at times and below
for longer periods when fires were concentrated at other areas on the floor,
then the heat conducted through the insulation would have to be revised
downwards.

The worst case was probably the relatively light inner core column 704, a
W14x53, with 705 a close second.  Being in the inner core, these columns
absorbed less heat - and we have neglected the shielding of elevator shafts,
stairwells, other core columns, partitions, etc, so the predicted member
temperature rises are likely to have been overestimated.

For column 704, the lower combustibles density within the core resulted in
the sub-2.5 metre points contributing only 16.7% of the total.  The 20 to 30
metre range contributed 27.8% compared to 4.1% in the case of 501.  Not
surprisingly, moving the sub-5 metre points for column 704 out to a 5 metre
separation had less effect, with total absorption reduced by only 22.7%
(down to 158.15 MJ).

Column 704 was the worst case being the lightest member and aligned for the
maximum impact damage to fireproofing.  Conditions (i) and (ii) above, fire
intensity and fireproofing damage, would have been necessary.  The analysis
predicted that the strength reduction factor at 102 minutes would be
sufficient to induce collapse without (iii) being a requirement.  But (iv),
insulation thickness, was still relevant.  FRC thickness could have been
raised slightly.  But at 2 3/16", if the situation was significantly
different to condition (iv) being met, then (iii) would have been required
to degrade cold demand / capacity ratio.

Core column 705 was the next worst case, with the alignment less conducive
to insulation damage.  At 534 C predicted final temperature, with an extra
40 C possibly allowed to compensate for the effects of thermal gradients, a
satisfying of conditions (i), (ii) and (iv) would render it a borderline
case.  Relatively little additional stress as a result of (iii) would have
been necessary to bring about in collapse.

Of the remaining core columns, the 80x, 90x and 100x rows can be ruled out.
Their siting in the southern half of WTC 1 would make it very unlikely that
conditions (ii) and (iii) could have been met.  Apart from 704 and 705,
there were another four (601, 604, 605, and 608) with predicted final
temperatures in excess of 500 C for 25% of fireproofing removed.  All four
were aligned with the flange normal to impacting debris, meaning that total
destruction of fireproofing on the exposed side would only amount to 17% of
the heated perimeter.  All conditions (i) to (iv) above would have to be
met, with increasingly severe core column damage models (iii) required with
decreasing predicted final temperatures.

Before determining the likely degree of damage to fireproofing, we shall
consider predicted absorption and final temperatures for the remaining
steel.

For the perimeter columns, the cross-section is taken as 10.4 ins^2 per
column, placing the mass at (10.4 ins^2 * 144 ins / 1728) * 490.7 pcf =
425 lb = 193 kg per column per floor.  (The spandrel plates, between
columns, will be treated separately.  The above neglects extra thickness of
spandrels where they are an integral part of the inner web, which would add
some 3.4% to the mass.)  The two column widths presented are 15.75 ins and
13.5 ins, making the mean width presented from a range of viewing angles
equal to (W1 + W2) * 2 / pi = 18.62 ins = 0.473 metres.  Length was taken as
3.048 metres for the radiation view factor, as for the core columns.  At
550 J/kg.K a net absorption of 550 * 193 = 106.15 kJ is required per degree
C rise in temperature.

With many of the north facade columns taken out, perimeter column 111 would
be most vulnerable.  Its pre-impact demand/capacity ratio was 0.2.
Post-impact, this could have ranged from 0.7 to marginally greater than 1.0,
depending on the damage to core columns and on how well the spandrel plates
and floor slabs redistributed the load across to perimeter columns 101 to
110.  Columns 112 to 148 sustained varying degrees of damage across five
floors.  Let us suppose the worst - that these columns were all completely
taken out.  Yet there was no partial collapse of the section of wall hanging
in tension above the severed columns.  Load was successfully transferred via
Vierendeel action of the wall frame.  In the bizarre event that the load of
the severed columns was transferred to 111 and 149 with no further
redistribution towards the corners, column 111 would have had a load area
from (11.9, 54) to (31.7, 63.4) which is 19.8 * 9.4 / 4,019 m^2 = 4.63% of
the total 34,573 tons which is 1,601 tons.  With an average perimeter column
yield strength of 64.2 ksi, the capacity was 64,200 * 10.4 ins^2 = 334 tons,
indicating immediate collapse.  With no immediate collapse after impact,
such a bizarre selective redistribution did not occur, and load was
transferred to adjacent columns via the spandrel plates and floor slabs.

If columns 101 to 111 shared a load area of 31.7 by 9.4 which is
298 m^2 / 4,019 m^2 = 7.41% of the total, the load is 34,573 * 0.0741 =
2,562 tons over 11 columns = 233 tons per column to give a 0.7 d/c ratio.

This is not allowing for severed core columns.  If core column 504 had been
severed, for example, and assuming one-quarter of its 495 tons demand was
transferred to columns 101 to 111, the load is then 2,562 + 124 = 2,686 tons
over 11 columns = 244 tons per column, a d/c ratio of 0.73.

So, selecting perimeter column 111 at co-ordinates (12.4, 63.4), the total
energy absorbed (under the absolute worst case fire scenario of
1.385 * 10^12 J released over 102 minutes across a single floor) was
computed and found to be 300.2 MJ, had fireproofing been totally stripped
off.  In this case - and assuming no loss to the exterior - the heat
quantity would be ample to take the column very close to the flame or gas
temperature.  With a quarter of FRC stripped off, the final temperature
would be 838 C, and with all FRC intact the temperature would only reach
166 C.  For the core columns, it was assumed that fires on adjacent floors
prevented conductive losses, and net absorption was 60% of the total heat
liberated with 40% of the heat flowing out to drive the smoke plume.  In the
case of the perimeter columns and spandrel plates, much of the absorbed heat
would be re-radiated to the cooler exterior.  The inner web would conduct
and transmit to the flanges and outer web, which in turn would transmit to
the external atmosphere.

To be fair, the fire collapse theorist might argue that if we take 60% as
the proportion of liberated heat that remained in the building and if the
perimeter retained less than 60%, then the core columns, trusses, etc,
should have retained more than 60%.  We assume 1488 C as the mean adiabatic
flame temperature after averaging a range of equivalence ratios from
stoichiometric to fuel-rich, and the exiting gases temperature is taken as
669 C.  (669 - 25) / (1488 - 25) comes to about 44%, reflecting the fact
that retaining 40% of the heat maintains over 40% of the temperature rise
due to increase of specific heat with temperature.  The 40% of energy which
drives the smoke plume is essentially convected out rather than radiated.
60% of the total energy released per floor - 8.31 * 10^11 J out of
1.385 * 10^12 J - is lost from the flames and hot gases / smoke layer to the
building, before the combustion products are vented out.  We shall calculate
the small proportion of energy radiated from the perimeter, sum the total
building absorption, and check that errors understating predicted
temperatures are smaller than the errors which overstate them.

Let us repeat the computations for some other perimeter columns.  Column
number 136, at co-ordinates (37.8, 63.4), is of particular interest, since
NIST recovered this panel (floor 98 section) and found no mud cracking of
paint.  NIST's paint study concluded that "Most perimeter panels (157 of 160
locations mapped) saw no temperature T lower than 250 C" and "Exposure to
fire does not necessarily lead to high temperatures in insulated steels".
For demand / capacity calculations above, we assume that column 136 is
removed.  However, it was severed at floors 94 - 95 and present at floor 98,
helping to redistribute load through Vierendeel action with the spandrels.

As a result of the aircraft entry hole, the remaining north facade perimeter
columns were the most vulnerable in terms of demand / capacity ratio.  But
these members were the least vulnerable to impact damage to fireproofing,
since the fireproofing was on the blind side of the incoming projectiles.
The greatest risk of impact damage to fireproofing and possible invisible
collapse would have been associated with core columns 503 to 506, and the
trusses in the immediate areas of impact.  All remaining 1WTC perimeter
columns and spandrels would have sustained little or zero damage to
fireproofing, and this is consistent with the NIST paint study.  Perimeter
column temperatures of 250 C or less would indicate at worst, negligible
damage to fireproofing.

With the computation for perimeter column 136, the total absorption assuming
no losses to the exterior and FRC totally removed, was found to be slightly
higher than for column 111, at 314.2 MJ.  This would take the column close
to the gas temperature, and the same would occur with one-quarter of FRC
removed.  For all FRC intact, the final temperature would be 173 C.  At
maximum, perimeter column 159 would absorb 228.8 MJ, column 130 would absorb
315.7 MJ, column 330 would absorb 363.1 MJ, and column 359 would absorb
269.3 MJ.  The mean absorption per column per floor was close to 300 MJ, and
therefore the total absorption for all perimeter columns excluding
transmission to the exterior was 236 times 300 MJ equalling 7.08 * 10^10 J.

For analysis the spandrel plates were divided into 52" x 24.25" sections,
between each pair of 15.75" perimeter inner webs, per floor.  The mass of
each section was (52" * 24.25" * 3/16" / 1728) * 490.7 pcf = 67.1 lb =
30.5 kg.```
```[To be continued...]
```

Ongoing research at present suggests that the only real vulnerability would have been with the trusses on two floors that were subject to the onslaught of impacting debris. In any case, the central area of one floor, up to the core, was probably taken out by the initial impact. Previous analysis (and confirmed by the NIST study) found that it would be necessary to totally remove four or five floors - depending on core column temperatures - before there would be any possibility of global instability. A partial failure of trusses on two floors and partial local floor collapse would not bring down the building. Hence, the probability of 1 in 12,175 against each (of three) high-rise collapses will be found to be valid.