Descent curve and the phases of collapse of WTC 7
We examine four WTC 7 descent curves, labeled "C," "E," "N," and "O," either anonymously published, or confidentially communicated to us. Descent curve describes apparent height of a collapsing building as a function of time. The set "C" suggests that there are three active phases of collapse. Phase I is a free fall for the first $H_1\simeq28$m or $T_1\simeq2.3$s, during which the acceleration $a$ is that of the gravity, $a=g=9.8$m/s$^2$. In Phase II, which continues until drop $H_2\simeq68$m, or $T_2\simeq3.8$s, the acceleration is $a\simeq5$m/s$^2$, while in Phase III which continues for the remaining of the data set, $a \simeq -1$m/s$^2$. We propose that the collapse of WTC 7 is initiated by a total and sudden annihilation of the base (section of the building from the ground level to $H_1$), which then allows the top section (building above $H_1$) to free fall during Phase I, and then collide with the ground in Phase II and III. The total duration of the collapse, assuming that Phase III continues to the end, is in the range $7.8-8.6$s. We derive a physical model for collision of the building with the ground, in which we correct the "crush-up" model of Bažant and Verdure, J. Engr. Mech. ASCE, {\bf 133} (2006) 308, and estimate the magnitude of the resistive force in the top section. We compare our findings to those of NIST investigators and find an agreement with respect to the distribution of damage in the primary zone. We conclude that the building was destroyed in a highly controlled fashion.