Abstract
An analysis is presented that calculates the temperature of the steel truss rods in the World Trade Center towers subject to a fire based on the building ventilation factor. The CIB correlation is used for the fire. Conduction analyses are made taking into account variable properties for the steel and the insulation. A structural failure model is described based on compression buckling of the truss rods due to a reduction in the Young's modulus. The computed times for the estimated failure or incipient collapse of the floors in both towers has been computed as 105±20 min for WTC 1 (north) and 51±9 min for WTC 2 (south), compared to the collapse times from the aircraft impact of 104 and 56 min, respectively. The insulation thickness and the difference of 19.1 mm () and 38.1 mm () between the two towers appear to have been the root cause of the collapses.


A
ceiling and wall surface area
Ao
ventilation opening area
D
diameter of the truss rod
E
Young's modulus
h
overall heat transfer coefficient
hconv
convective heat transfer coefficient
Ho
compartment height
kins
thermal conductivity of the insulation
L
length of the truss rod
P
critical buckling stress
r
radial coordinate, see Eqn. (1)
t
time
T
temperature
Tgas
gas temperature
α
thermal diffusivity
σ
Stefan–Boltzmann constant.


Article Outline
Nomenclature
1. Introduction
2. Analysis
2.1. Fire temperature
2.2. Heat transfer to the steel
3. The structure
4. Conclusions
Acknowledgements
References


1. Introduction
The fall of the World Trade towers has been initially reported and generally accepted as due to the collisions by aircraft and its fuel. Subsequent analyses and findings have dampened that cause and have pointed more at the fire involving the building contents [1 and 2]. It is estimated that about 29 Mg of jet fuel was in each aircraft and about 9.4 Mg of jet fuel was expended in the fireballs of the initial impacts, and the remaining jet fuel may have burned for several minutes more on a floor [2]. These calculations are based on standard formulas in the literature, and easily and clearly show that the jet fuel was only responsible for the ignition of the contents. The lack of clarity in the media is indicative of the lack of knowledge or reliance on the science of fire in reaching a quick answer. The recent FEMA report [1] could not determine the cause of the collapses. They cited a number of issues: (1) aircraft damage but acknowledge that the two buildings would have remained standing had they not received a fire load, (2) the jet fuel although it burned out quickly ignited the contents and caused rapid fully developed floor fires, (3) the type of steel truss floor system should be subject to more detailed evaluation. This is what we have done. They also cited that fireproofing needs to adhere under impact and fire, and connection performance needs to be better understood. In particular, two 15.9 mm () and two 25.4 mm (1″) diameter bolts to the exterior columns, and two 15.9 mm () diameter bolts to the interior columns connected the floor trusses. These connections have been questioned as weak links.

Instead, we have focused on the transverse and main trusses, and in particular on the 27.7 mm (1.09″) diameter truss rods. Our hypothesis is that the truss rods are the weak link because they have the lowest steel cross-sectional mass and the fire would increase its temperature the fastest. Therefore, we have based our analysis on a link between the temperature evolution and the buckling under restrained elongation.

Our hypothesis is illustrated in Fig. 1. We suggest that buckling of the transverse-truss compressed diagonal rods, which were adjacent to the main-truss, occurred first due to the combination of their restrained elongation and quick loss of buckling resistance induced by their fast temperature rise (marked as St. A in Fig. 1). With the buckling of these diagonals, the compressive forces in the transverse-truss middle diagonals were suddenly increased. As they had a similarly reduced buckling stiffness they buckled soon after (marked as St. , and the whole transverse-truss failed, sagging downwards (marked as St. C). Sagging of the floor deck followed, developing some membrane action in the steel deck, and possibly disconnecting the concrete within the floor span from the main-truss's upper bulk of concrete due to combined large tensile and shear stresses (marked as St. D). Due to the ductility and continuous nature of the deck itself and the continuous nature of the transverse-truss bottom chord, the floor probably did not fall down immediately, but rather stayed hanging from the main-truss bottom chord for some period. Subsequently, the buckled transverse-trusses and steel/concrete deck fell down on the floor below, thus aggravating loading conditions and enhancing its failure.


(39K)

Fig. 1. Diagram of suggested sequence of events triggering progressive collapse (truss drawings taken from FEMA report [1])



Simultaneously the fire challenged the main-trusses as well. Again, the thin diagonal rods heated much faster than the other members. However, only part of them was in compression and with various loading ratios. As the "truss" was not statically determinate (its upper and lower chords were highly continuous and did not include hinged joints at all), buckling of some compressed diagonals alone (marked as St. A1 in Fig. 1) would not necessarily cause immediate collapse; however, it would be a main trigger to further deterioration of capacity, leading eventually to total failure. Our hypothesis is that the whole collapse mechanism of this truss was as follows: Despite the slotted holes at the supports, the main-truss upper chord was too restrained and could not elongate freely. A horizontal force was induced at the supports, which were highly above the neutral line of the main truss's bending as a whole. This induced a high compressive force into the upper chord, and a superimposed bending moment on the truss as a whole, increasing its tendency to either strongly deflect downwards (with possible buckling of the upper chord in the vertical plane), or buckle in a transverse mode. However, since the upper chord had a much larger stiffness in the horizontal direction, transverse buckling did not occur. With most of the compressed diagonals already buckled, the "beam" or "truss" behavior did not exist any longer, and the upper chord acted as a "cable", from which all the other members are hanging downwards.

This increased the pressure on the inner edge of the end seats while increasing the tension in the bolts, actually bending and shearing the bolts due to the enlarged rotations. Eventually, the bolts failed as well, possibly rupturing the seat connections, and total collapse of the floor ensued, placing load on the floor below. With several floors simultaneously involved in fire, this would cause several floors to load a floor below. The connections at the columns could not tolerate the load and progressive collapse ensued even to floors that were not on fire. The progressive collapse mechanism is well reported and accepted, but the initiating event is not established. We offer the truss failure as described.

The following analyses were prompted by the FEMA report [1] that stated that the average thickness of the fireproofing on the trusses was 19.1 mm (), and were in the process of being upgraded to 38.1 mm (). On September 11, 2001, WTC 1 (north) had the higher level in the impact zone (floors 94–98 ), and WTC 2 (south) only had the higher level on floor 78 with impact over 78–84. The insulation was sprayed mineral fiber, reportedly Cafco DC/F similar to Isolatek Blaze-Sheild II having a value of thermal conductivity of 0.043 W/m K at 24°C. The DC/F is slightly higher at 0.046 W/m K.

2. Analysis
There are three parts to this analysis: (1) The fire, (2) The heat transfer to the steel, and (3) The structural failure. The fire depends principally on the ventilation, and secondly on the fuel type. The fuel loading (weight or energy available on the floor) will determine the duration of the fire. The heat transfer to the steel principally depends on the insulation, and the variation of thermal conductivity is crucial in such an analysis. The gas-phase heat transfer conductance is very high and the surface temperature of the insulation can usually be taken as the gas temperature; however, we did not make such approximations. Instead, we strove to calculate the heat transfer as accurately as possible, and the fire conditions were allowed to vary over a plausible range. The time of structural failure of the main-truss rods depends on the temperature-induced decreasing modulus of elasticity lowering the resistance to buckling. The time of structural failure of the transverse-truss rods depends on the combined effects of increasing compressive stresses stemming from temperature rise under elongation restraint, and the decreasing modulus of elasticity lowering the resistance to buckling.

2.1. Fire temperature
The fire temperature during fully developed conditions is computed from the correlations reported from the CIB test analyses [3]. These tests are of scales up to 1.5 m high and for wood cribs covering the floor. Fuels of higher heats of combustion and larger scale can produce higher temperatures, but this is only suggested theoretically [4].

The gas temperatures are based on Ao ranging from 124 and 267 m2 for WTC 1 [1] (p. 2–23), Ho equal to 3 m, and A based on a floor of 63.5 m by 63.5 m and a utility core of 26.5 m by 41.8 m that is about 4100 m2. With these values, the group A/AoHo1/2 can be estimated between 15 and 33 m–1/2. Fig. 2 provides a gas temperature range between 800°C and 1000°C.


(6K)

Fig. 2. Temperature range of exposure in WTC 1 & 2 [3].



The corresponding floor burning rates in wood are 14.5–18.7 kg/s or about 250 MW per floor. For typical floor loadings of 30 kg/m2, the fire duration per floor would be about 80–100 min. There is variation in this duration since the burning rate is based on a wood crib configuration; other configurations would vary. But the results suggest, based on the actual event that the duration was not much greater than the WTC 1 time to collapse. This would imply that had the steel structure survived longer, the fire duration time would have been less than the failure time, and no complete failure would have occurred. The loading and burning rate are key issues here and more information on the demography of the WTC furnishings, and research on different fuel types are needed.

Hence, a fire temperature of 900°C was used in our calculations based on the mean in Fig. 2, and will commence and stay constant throughout. This is justified based on the rapidly developing fire from the jet fuel, and on the estimate that the fire duration on a floor is longer than the collapse time.

2.2. Heat transfer to the steel
The heat transfer analysis is a standard numerical calculation of the heat conduction equation for a composite cylinder of homogeneous insulation and steel. The governing equation is given as:

(1)


The gas-phase boundary condition uses a constant convective heat transfer coefficient of 20 W/m2 K and blackbody conditions for the radiant heat transfer. The latter is justified because of the large scale of the fire, and the convective component is a typical value for fire conditions and is negligible compared to radiation. The combined heat transfer coefficient is expressed as:
h=hconv+σ(Tgas2+T2)(Tgas+T). (2)


The heat transfer coefficient is in W/m2 K and the temperatures in the equation are in degrees Kelvin. These values used for the overall heat transfer coefficient have small effect on the final results since the insulation provides the significant resistance to heat flow. At the insulation-steel interface, we assume that there is no contact-resistance; we impose continuity of the temperature and conservation of energy across the boundary.
The properties of both the steel and insulation vary considerably with temperature and cannot be neglected. The manufacturer of the insulation states the conductivity is 0.046 W/m K at normal temperature, and the literature indicates an approximate linear increase [5 and 6]. We used:

kins=0.046+0.00024 (T−20) (3)


The thermal conductivity is expressed in W/m K and the temperature in this expression is in degree centigrade. The thermal diffusivity of the insulation is considered constant and equal to 3.1×107 m2/s [5]. The steel exhibits a strong variation both in thermal conductivity as well as in volumetric specific heat. We have carefully fitted the curves provided by Lie [7].
Note that for WTC 1 (north) we assign the insulation to be 38.1 mm () in thickness while for the WTC 2 (south), the insulation thickness is 19.1 mm () [1] (pp. 2–12). Eq. (1) is integrated with the Crank Nicholson implicit scheme starting from a uniform initial condition at 20°C throughout the steel and insulation. The temperature results are shown in Fig. 3.


(5K)

Fig. 3. Temperature of the steel truss rods for WTC 1 & 2.



It must be realized that there is an incipient delay time that must be added to the results in Fig. 3 since the ceiling membrane would retard the heating. It can be argued that the impact of the aircraft destroyed the ceiling partition, and this is certainly true in places. But where it has been destroyed, the fuel on the floor is also likely pushed away. The aircraft had a weight at impact of about 170 tons and each tower weighed about 750,000 tons; or the impact is like a sparrow hitting a 125 kg person [8]. Moreover the engines and landing gear are the dominant destructive missiles. The aircraft aluminum skin and the fuel would quickly dissipate its momentum. Hence, we believe the fuel from the furnishings and contents would be pushed to areas where there would be sustained fire, with no fire likely in the immediate impact area. These fuel piles would increase the fire duration in those areas, and first attack the ceiling membrane.

The acoustical ceiling tiles initially installed in the World Trade Center towers were of one type (Armstrong World Industries Inc.) and were not specified to be fire resistant. If not immediately destroyed, this membrane would likely retard the direct heating of the insulated truss by about 10 min, an estimation based on experience. Any reconstruction of the incidence needs to account for this membrane effect more completely.

It has been speculated that the impact knocked off the protective insulation. We do not believe that this occurred to any extent in the fire region since a calculation of the bare steel chord would suggest failure in 10–15 min. Again, in the impact area this may have happened, and testing needs to be done to assess the robustness of the insulation to impact.

3. The structure
The truss rods can be analyzed in tension and in compression at the mid-section. Compression can be shown to be the critical condition. The rods were A36 steel with a yield stress of 253 Mpa (36 ksi) and an ultimate stress of 422 MPa (60 ksi) [1]. The rods are nominally 0.889 m (35″) long. The critical buckling stress of the rod in compression is given as:

(4)


For E equal to 210,900 MPa (30,000 ksi), the critical buckling stress is 126 MPa (17.9 ksi). Since this is less than the yield stress, the member fails in buckling. With the design load stress as 126 MPa (17.9 ksi), if the overall safety factor is 2–3, then the actual load just before the fire may have induced a stress of about 42.2–63.3 MPa (6–9 ksi). With the rising of the rod temperature, it wishes to elongate, but the continuity of the upper and lower truss chords restrains this elongation. The reduction in E due to temperature to render this stress, 53±11 MPa (7.5±1.5 ksi), into a critical buckling stress must be about 70,300–105,500 MPa (10,000–15,000 ksi) [7]. The corresponding critical failure temperature is about 630–770°C, accordingly. With this temperature bound we estimate the time to failure for the towers to be given in Table 1 using a fixed fire temperature of 900°C.


Table 1. Time for critical events (minutes)




The time to failure suggested in the table complemented by the addition of 10 min to account for the suspended ceiling membrane gives times close to the actual failure events in both buildings. We find this result extremely telling and we believe forms the root cause initiating the collapses.

For structural steel elements in standard fire testing, the failure criterion ranges from 550–600°C. Therefore, we have also made computations for the truss rod to reach 600°C for fire temperatures of 800–1000°C. The times to reach these temperatures are roughly 75±12 min for WTC 1 and 32±6 for WTC 2. Hence, no matter how one estimates failure, the difference between the collapse times of the two buildings is of 44±7 min compared to the actual collapse time difference of 48 min. This strongly correlates with the insulation thickness.

Should one accept that indeed the generalized failure of the truss rods is the initiating event, then it follows that the consequent failure of the connections could be the result of the floor sagging and pulling on the connections downward thus exerting a shearing action that could fail them. From some circumstantial evidence shown in the FEMA report we see failure of the connections, as they were pulled apart rather than failure of weak elements of the connections such as bolts. Further, the images of the collapse of the WTC 1 (north) seem to show a burst of flame and smoke outward just before the collapse. This, in our view corroborates the idea of a generalized floor failure. The collapse of a floor would cause the smoke and fire to be pushed out of the building from the space below as video images seems to indicate. Upon loading one or two additional floors on the floor below, the whole structure would start failing as more and more floors fails and increased instabilities in the vertical unconstrained elements arise.

4. Conclusions
It appears that the insulation thickness on the truss rods was deficient and caused the heating of the steel that led to weakening and collapse. The variation of 19.1 mm () and 38.1 mm () thickness between the towers needs to be investigated further, and understood in light of fire safety design principles. The logical questions include: What was the basis of the original fire safety design and specification? Our estimate of the truss rod failure by temperature in the ASTM E 119 test for 38.1 mm () thick insulation is about 67–79 min corresponding to failure at 500°C and 600°C, respectively. Our estimate of fire duration was about 80–100 min. If these times are correct, it suggested that the fire safety design criteria need assessment. It has been implied that a two-hour criterion should have been used for this floor assembly. Certainly, some factor of safety needs to be incorporated in relating fire duration time with test ratings or computed failure times.

The truss rod failure is proposed as the initiating event. An important implication of this proposed mechanism points to the dominant effect of the insulation applied to the truss members. If one follows this theory, the reduced amount of insulation in the WTC 2 could have resulted in a premature collapse of that tower by almost one hour compared to WTC 1.

Acknowledgements
We appreciated checks on the early thermal calculations by Ali Rangwala. We acknowledge useful discussions with colleagues: C.C. Fu, J. Milke and F. Mowrer, and N. Shultz of VTEC Laboratories, Inc.


References
1. World trade center building performance study. Federal Emergency Management Agency, FEMA 2002.p. 403.

2. Torero JL, Quintiere JG, Steinhaus T. Fire safety in high-rise buildings, lessons learned from the WTC, Jahresfachtagung der Vereingung zur Forderrung des Deutschen Brandschutzez e. V., Dresden, Germany, 2002.

3. Thomas PH, Heselden AJM. Fully developed fires in single compartments—a co-operative research program of the Conseil International du Batiment. FR Note No. 923, Fire Research Station, UK, 1972.

4. Quintiere JG. Fire behavior in building compartments. Proceedings of the 29th International Symposium on Combustion. Sapporo, Japan: The Combustion Institute, accepted for presentation.

5. Kreith F, Bohn MS. Principles of Heat Transfer. 6th ed. US: Brooks/Cole, Thomson Learning, 2001. p. 44.

6. Reed RJ. North American Combustion Handbook. 3rd ed., vol. 1, Cleveland, OH: North America Manufacturing Co., 1986. p. 120.

7. Lie TT. Structural Fire Protection. New York: American Society of Civil Engineers, 1992.

8. Sometime Lofty Towers. Browntrout Publishers, 2001.

Does anyone know where the following was originally published? The absurdities are making my head hurt.