Why Did the World Trade
Center Collapse?
9/13/01
The towers of the
World Trade Center were designed to withstand as a whole the horizontal impact
of a large commercial aircraft. So why did a total collapse occur? The reason
is the dynamic consequence of the prolonged heating of the steel columns to
very high temperature. The heating caused creep buckling of the columns of the
framed tube along the perimeter of the tower, which transmits the vertical load
to the ground. The likely scenario of failure may be explained as follows.
In stage 1
(see the figure), the conflagration caused by the aircraft fuel spilled into
the structure causes the steel of the columns to heat to temperatures
apparently exceeding 800ºC (far higher than those in the ASTM fire standard).
The heating is probably accelerated by a loss of the protective thermal
insulation of steel during the initial blast. When structural steel is heated
to such temperatures, it exhibits significant creep, i.e., a slow increase of deformation
under load. Thus the effective stiff:ness of the columns is greatly reduced
and, as a result, many columns buckle (stage 2) and consequent1v lose their
load carrying capacity[1].
Once more than about a half of the columns in the critical floor that is heated
most suffer buckling (stage 3), the weight of the upper part of the structure
above this floor can no longer be supported, and so the upper part starts
falling down onto the lower part below the critical floor, gathering speed
until it hits the top of the columns of the underlying floor. At the moment the
upper part has moved down through the height of the floor it has an enormous
kinetic energy and a significant downward velocity. The vertical impact of the
upper part onto the lower part generates in the columns of the underlying floor
vertical loads that are much higher than the load capacity (stage 4), even if
these columns are not heated. So the columns of this floor buckle, too. This
progressive buckling under subsequent dynamic impacts is then repeated floor by
floor.
The details of
the failure process after the decisive initial trigger that sets the upper part
in motion are of course more complicated. For example, the upper part is
tilting as it falls; since the structure is a framed tube with floor beams of
large spans, the impacted floors may be collapsing ahead of the tube and thus
depriving the tube wall of its lateral support against global bucking. But
regardless of such and other details, the following two simple and crude
estimates of the overload ratio of the columns of the floor just below the
critical floor that; triggered the catastrophic chain of events can be made.
A short time
after the vertical impact of the upper part, but after the wave caused by
vertical impact has propagated downward, the lower part of the structure can be
approximately considered to act as a vertical spring (figure, right).
Neglecting the energy dissipation, particularly that due to the buckling of
columns, and equating the loss of gravitational potential energy of the upper
part due to its downwaxd displacement from the initial equilibrium position to
the point of maximum deflection of the lower part, considered to behave
elastically, one obtains the equation mg[h + (P/C)] = P2/2C. Its solution P = Pdyn, yields the following overload ratio due to
impact of the upper part:
Here h = height of critical floor columns (= height
of the initial fall of the upper part) » 3.7 m, m = mass of the upper part » 5.8 ´ 107 kg, C = spring constant of the lower part in
axial compression » 7.1 ´ 1010 N/m, g = gravity acceleration, and Po = mg = design load capacity. The input
numbers are estimates for the North Tower based on the typical properties of
this kind of buildings.
The second simple and crude
estimate of the initial overload ratio at the moment of impact is
where A = cross section area
of building, Eef = cross section stiffness of all columns divided by
A, r =
specific mass of building per unit volume. This estimate is calculated from the
elastic wave equation which yields the intensity of the step front of the
downward pressure wave caused by the impact if the velocity of the upper part
at the moment of impact on the critical floor is considered as the boundary condition.
The latter
estimate gives the initial overload ratio that exists only for a small fraction
of a second at the moment of impact. After the wave propagates to the ground,
the former estimate is appropriate.
In spite of
the approximate nature of this analysis, it is obvious that the elastically
calculated forces in columns caused be the vertical impact of the upper part
must have exceeded the load capacity of the lower part by at least an order of
magnitude.
Zdenĕk P. Bažant and Yong Zhou[2]
[Remitido por Fernando Martínez]