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In spite of these uncertainties, some estimates of the magnitude of E1, (the energyneeded to bring about the collapse of one floor), have been made. For example, Z. Bažant
et al. at Northwestern University, Illinois, have estimated that the maximum plastic
energy dissipated by the collapse of one floor, i.e. our quantity E1, is approximately equal
to 5.0 x 10^8 J. Unfortunately Bažant et al. do not give a detailed exposition on how this
value for E1 was derived, stating only that it is based on “approximate design
calculations” for one WTC tower /3/.
Another calculation that may be used to estimate E1 was published by G.C. Lee et
al. in a MCEER Special Report /4/. Lee et al. assume that 36 exterior columns on WTC 1
were destroyed by the Boeing aircraft impact and conclude (without giving
computational details) that the energy absorbing capacity of these damaged columns
“does not exceed 7230 kips-ft” or about 107 J. Based on this estimate, and remembering
that one complete floor has 236 exterior columns, it follows that the exterior columns
comprising one floor of a WTC tower have an impact energy absorbing capacity of about
7 x 10^7 J. From the relative cross-sectional area of a core column (0.1236 m2) compared
to an exterior column (0.0184 m2), we estimate that the effective strength of the core
columns is about 6.7 times higher than the effective strength of the exterior columns. A
consideration of the collapse of the 47 core columns therefore adds about 9 x 10^7 J of
energy absorbing capacity. Thus, based on Lee’s calculations, the total energy absorbing
capacity of the structural supports of one floor of the WTC is estimated to be about 1.6 x
10^8 J, which we equate to our quantity E1 while noting that this estimate is significantly
lower than Bažant’s value of 5.0 x 10^8 J. However, it appears that Lee’s results are based
on very rough estimates of the energies involved so that the level of agreement with
Bažant’s estimates is as good as might be expected in view of the approximations
involved.
A much better estimate for E1, and one that is based on a very detailed analysis of
the aircraft impact events, may be derived from a paper published by T. Wierzbicki et al.
at MIT /5/. These authors have calculated the energy dissipated by the wing of a Boeing
767 cutting through the exterior columns of a WTC tower and report a value equal
to 1.139 x 10^6 J
per column. On this basis, 2.69 x 10^8 J would be require to cut through all
236 exterior columns supporting one WTC floor. If we now assume, as previously
discussed, that the yield strength of the
core columns is about 6.7 times higher than the
yield strength of the
exterior columns, we estimate that an additional 3.60 x 10^8 J are
required to collapse the 47
core columns supporting each floor.
Thus, based on T.
Wierzbicki et al. calculation, we estimate a total of 6.29 x 10^8 J of impact energy was
required to collapse one WTC floor, a value that is remarkably close to Bažant’s estimate
of 5.0 x 10^8 J for the plastic energy dissipated by the collapse of one floor.
The fact that the values of E1 derived from Wierzbicki’s and Bažant’s studies are
quite similar is very significant because these author’s calculations were actually
undertaken for two different impact events: (i) The collision of a Boeing aircraft with one
floor of a WTC tower, and (ii) The collapse of a block of WTC floors onto the floor
below. Thus Wierzbicki considers floor support failure under
lateral impact loading
while Bažant’s considers the failure of the floor supports under
axial impact loading. The
fact that the energy calculated in each of these cases is about the same suggests that the
energy dissipated in a floor collapse is relatively insensitive to the mode of failure of the
support structures. This is a common observation in studies of collisions of large objects
involving complex structures such as aircraft, automobiles, trains, and ships.
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