What If the Aircraft Impacted a Higher Floor?
Eq. (1) (Bazant and Zhou 2000) gives the dynamic overload ratio µ
0 =
Pdyn/
P0 of the lower part of tower caused by the vertical impact of the upper part after it falls down through the height of the critically heated floor. The value µ
0≈31 results when the impact occurs approximately 20 floors below the top, i.e., around the 90th floor. If the aircraft impacts a higher floor, say, the
nth floor, the axial stiffness
C of the lower part of tower gets reduced roughly in the ratio 90/
n, and the impacting mass m of the upper part of tower gets reduced roughly in the ratio (110-2
n)/20 where 110 = total number of floors in the tower. Considering that the critically heated floor (probably the same as the floor impacted by the aircraft) is, for instance, the fourth floor below the top, i.e., 110-
n=3, one gets from Eq. (1) a surprisingly large overload, µ
0=29, which would be fatal. But is it not strange that an aircraft impact so close to the top should destroy the whole tower? It is, and the explanation is two-fold:
• First, note that, in Eq. (1), P0 was defined as the design load capacity for the self-weight only, excluding the additional design axial load P1 caused in the columns by wind and dynamic loads (P0=mg). At 20 floors below the top, P0 may be roughly as large as P1 , i.e., P1 /P0≈1, which means that the total overload ratio, defined as µ=Pdyn /(P0+P1), is µ≈15. But on approach to the building top, the cross sections of columns are not reduced in proportion to its weight that they carry but are kept approximately constant, because of various stiffness, dynamic and architectural requirements, as well efficiency of fabrication. So, for n=3, P1 /P0≫1. Therefore, µ≪15, and thus a tolerable overload ratio, approximately µ≤2, may well apply in this case, depending on the precise structural dimensions and loads (not available at the moment of writing).
• Second, note that the analysis that led to Eq. (1) implies the hypothesis that the impacting upper part of the tower behaves essentially as a rigid body. This is undoubtedly reasonable if the upper part has the height of 20 stories, in which case the ratio of its horizontal and vertical dimensions is about 52.8/20×3.7≈0.7. But if the upper part had the height of only 3 stories, then this ratio would be about 5. In that case, the upper part would be slender enough to act essentially as a flexible horizontal plate in which different column groups of the upper part could move down separately at different times. Instead of one powerful jolt, this could lead to a series of many small vertical impacts, none of them fatal.
In theory, it further follows from the last point that, if people could have escaped from the upper part of the tower, the lower part of the tower might have been saved by exploding the upper part or weakening it by some ‘‘smart-structure’’ system so as to make it collapse gradually, as a mass of rubble, instead of impacting the lower part at one instant as an almost rigid body.