It has been argued that the bottom section of WTC1 would have been able to withstand the impact of the top section dropped from a height of one floor such that the top section would just bounce on the lower part with both parts acting like a spring. For this to happen, of course the collision must be considered elastic.
Thus, the building is split into two sections. The top part is dropped onto the lower part such that the impact velocity is 7.1 m/s (a = 0.7g over a fall of 3.7m as observed in WTC1). Assume the bottom part is 384m high and the collision is elastic. Assume the entire lower part (and the upper part if you think that makes a difference) behaves as a linear spring.
In order to halt the top part before the elastic limit is reached, what would the minimum force required in the spring at maximum deflection?
Solve for force in terms of mg (the mass of the upper part times gravity).
Spring against spring.
Assume two springs 1 and 2!
Spring 1 has length L1 and spring constant C1 and spring 2 has length L2 and spring constant C2.
Spring 1 is fixed at one end and spring 2 is attached to the other end of spring 1.
A force F is applied to the free end of spring 2. What happens?
Evidently both springs compress and a reaction force -F develops at the fixed end of spring 1.
Compression d1 of spring 1 is d1 = F/C1
Compression d2 of spring 2 is d2 = F/C2
The energy E1 required to compress spring 1 is E1 = (d1)² C1/2
The energy E2 required to compress spring 2 is E2 = (d2)² C2/2
The total energy E to compress the two springs is E = E1 + E2
Now release spring 2 from spring 1.
Instead of applying a force to the two springs, spring 2 is now moving with a kinetic energy E and is colliding with the free end of spring 1. We assume a soft collision.
What happens then? Both springs compress as if a force F was applied to them and a reaction force -F develops at fixed end of spring 1.
What happens then? One possibility is that both springs decompress and that spring 2 bounces back! Another is that one of the springs break due to overload. A third possibility is that both springs break simultaneously but it is unlikely. If something breaks it is the weakest link.
What do you think?