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(This is an initial suggestion, originally written on Sept 11 2001 (with some minor subsequent changes) on one possible reason for failure, and should not be regarded as official advice.)
World Trade Center collapse
The structural integrity of the World Trade Center depends on the closely spaced columns around the perimeter. Lightweight steel trusses span between the central elevator core and the perimeter columns on each floor. These trusses support the concrete slab of each floor and tie the perimeter columns to the core, preventing the columns from buckling outwards.
After the initial plane impacts, it appeared to most observers that the structures had been severely damaged, but not necessarily fatally.
It appears likely that the impact of the plane crash destroyed a significant number of perimeter columns on several floors of the building, severely weakening the entire system. Initially this was not enough to cause collapse.
However, as fire raged in the upper floors, the heat would have been gradually affecting the behaviour of the remaining material. As the planes had only recently taken off, the fire would have been initially fuelled by large volumes of jet fuel, which then ignited any combustible material in the building. While the fire would not have been hot enough to melt any of the steel, the strength of the steel drops markedly with prolonged exposure to fire, while the elastic modulus of the steel reduces (stiffness drops), increasing deflections.
World Trade Center collapse
Modern structures are designed to resist fire for a specific length of time. Safety features such as fire retarding materials and sprinkler systems help to contain fires, help extinguish flames, or prevent steel from being exposed to excessively high temperatures. This gives occupants time to escape and allow fire fighters to extinguish blazes, before the building is catastrophically damaged.
It is possible that the blaze, started by jet fuel and then engulfing the contents of the offices, in a highly confined area, generated fire conditions significantly more severe than those anticipated in a typical office fire. These conditions may have overcome the building's fire defences considerably faster than expected. It is likely that the water pipes that supplied the fire sprinklers were severed by the plane impact, and much of the fire protective material, designed to stop the steel from being heated and losing strength, was blown off by the blast at impact.
Eventually, the loss of strength and stiffness of the materials resulting from the fire, combined with the initial impact damage, would have caused a failure of the truss system supporting a floor, or the remaining perimeter columns, or even the internal core, or some combination. Failure of the flooring system would have subsequently allowed the perimeter columns to buckle outwards. Regardless of which of these possibilities actually occurred, it would have resulted in the complete collapse of at least one complete storey at the level of impact.
Once one story collapsed all floors above would have begun to fall. The huge mass of falling structure would gain momentum, crushing the structurally intact floors below, resulting in catastrophic failure of the entire structure. While the columns at say level 50 were designed to carry the static load of 50 floors above, once one floor collapsed and the floors above started to fall, the dynamic load of 50 storeys above is very much greater, and the columns at each level were almost instantly destroyed as the huge upper mass fell to the ground.
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