Bomber Crash into Empire State Building
On July 28, 1945, a ten-ton (10,000 kg) B-25 bomber crashed into the Empire State Building at an estimated 400 kph, into the north face of the 79th floor. Although several people were killed, the building remained standing (Levy and Salvadori, 1992).
To estimate the force applied to the building, we must estimate either the deceleration of the bomber as it crashed into the building (and use the force-acceleration method), estimate the distance it took the bomber to come to rest (and use work-energy), or estimate the time it took to bomber to come to rest (and use impulse-momentum). The bomber did not pass through the building, so a distance of 10 to 20 meters for the plane to come to rest could be used for calculations. If the bomber came to rest in 20 meters with a constant deceleration, the force would be about 4440 kilonewtons exerted over about 0.25 seconds.
All three of these methods are derived from F = ma. The acceleration is assumed to be constant and is integrated once for velocity and twice for distance traveled. With an initial velocity of 400 kph or 111.1 meters per second, a final velocity of zero, and an initial position of zero and a final position of 20 meters, the velocity equation becomes:
solving these two equations simultaneously gives a t of about 0.36 seconds and an acceleration of 308.6 meters per second per second (actually a deceleration, which will affect the direction of the force).
Force Acceleration: From F=ma,
Work Energy: Initial kinetic energy plus work done equals final kinetic energy:
where:
T1 = initial kinetic energy
U1-2 = work done between position 1 and position 2 (initial and final), and
T2 = final kinetic energy.
Initial kinetic energy is:
where m = mass (10,000 kg, given above) and v is initial velocity, of 111.1 m/s. Thus, the initial kinetic energy is 61,700,000 Newton-meters.
where F is the unknown force and d is the distance of 20 meters. The final kinetic energy is zero because the final velocity is zero, after the bomber comes to rest. Finally, to get the force:
Impulse-momentum: The initial momentum plus the impulse equals the final momentum, or:
For a constant force, this becomes:
Substituting previous values of 10,000 kg for m1 and m2, 111.1 m/s for v1, zero for v2, and 0.36 second for the time elapsed
we once more calculate:
This problem is useful for demonstrating the equivalence of the three methods. A bomber crashing into a building is an extremely unlikely occurrence. However, the consequences of the collapse of a large building are very grave. How should the profession guard against rare, but severe events? In a large and complicated project, is there an obligation to go beyond building code requirements? For example, for nuclear reactor containment vessels, what extreme events should engineers consider? What about buildings subject to terrorist attacks?
This case study was originally written in 1997 (Delatte, 1997), well before the terrorist attacks of September 11, 2001. The same methods may be applied to calculated the impact forces applied by any aircraft of known weight and speed, with an estimate of the average distance before the aircraft (or its parts) came to rest.
Reference: Levy and Salvadori, 1992.