Inabsorbable Kinetic Energy
First, let us review the basic argument (Baˇzant 2001; Baˇzant and Zhou 2002). After a drop
through at least the height h of one story heated by fire (stage 3 in Fig. 2 top), the mass of the
upper part of each tower has lost enormous gravitational energy, equal to m0 gh. Because the
energy dissipation by buckling of the hot columns must have been negligible by comparison,
most of this energy must have been converted into kinetic energy
K = m0 v
2
/2 of the upper
part of tower, moving at velocity v. Calculation of energy Wc dissipated by the crushing of
all columns of the underlying (cold and intact) story showed that, approximately, the kinetic
energy of impact
K > 8.4 Wc (Eq. 3 of Baˇzant and Zhou 2002).
It is well known that, in inelastic buckling, the deformation must localize into inelastic hinges
(Baˇzant and Cedolin 2003, sec. 7.10). To obtain an upper bound on Wc , the local buckling
of flanges and webs, as well as possible steel fracture, was neglected (which means that the
ratio
K/Wc was at least 8.4). When the subsequent stories are getting crushed, the loss m0 gh
of gravitational energy per story exceeds Wc exceeds 8.4 by an ever increasing margin, and so
the velocity v of the upper part must increase from one story to the next. This is the basic
characteristic of progressive collapse, well known from many previous disasters with causes
other than fire (internal or external explosions, earthquake, lapses in quality control; see, e.g.,
Levy and Salvadori 1992; Baˇzant and Verdure 2007).
