There was a re
cent accident involving a test of the train that is to connect Kennedy International Airport with Jamaica, Queens. The accident involved three cars traveling on a curve on an elevated guideway. The train was loaded with sixteen concrete blocks of weight Wb =2000 lbs, simulating a loaded condition. Eight blocks were in each of the first two cars, and the third was empty. The cars had plywood floors, weighed Wc =55,000 lbs each when empty, and were 10-ft, 6-inches wide. Reports suggested that the blocks shifted, causing derailment and destruction of 150 ft of a concrete retaining wall. Items of interest are the shifting of the blocks, and what caused derailment. Some reports have indicated a speed of V=58 mph. The design operating speed is 60 mph. An operator riding the train was killed when he was pinned to the front of the front car by six sliding blocks.
Lacking many details of this accident, a definitive analysis is impossible. However, a preliminary analysis would indicate whether the accident happened as suggested, and indicate areas that need to be examined further. In Part 1, we will consider whether it is reasonable to conclude that the blocks could have shifted. In Part 2 we will consider whether shifting blocks could have caused the derailment.
This case is like others we have considered, except that it involves a load that may shift. Considering the forces acting on a block as shown in the figure, we see that if the block is not slipping, it has the same acceleration in the radial (outward) direction as the train. Thus, the centrifugal force acting on the block equals the mass times the acceleration (V²/r). If the block is unrestrained on a horizontal floor the friction force (f) is the only physical force acting in the radial direction. From Newton's 2nd law,
f = (Wb/g)(V²/r)
where "g" is the gravitational acceleration.
An equal and opposite force f is acting outward on the train, increasing the force on the rails and the tendency to tip.
As the speed increases, the f increases until at the critical speed, it reaches its maximum value fmax= µN, where N=Wb. At this critical speed, shifting of the block is impending and we obtain:
(Wb/g)(V²/r) = µ Wb
Note that the weight cancels out. This is why it is possible to design a curve in a road for a certain speed that will apply to all vehicles regardless of weight. At a higher speed, the block slips giving a radial acceleration larger than V²/r. Assume the static coefficient of friction µ=0.40 for concrete on plywood, and the radius of curvature of the track r=955 ft (a 6° curve). This gives the speed required for the blocks to slip, V=76 mph, greater than the normal operating speed, but certainly within possibility in a testing situation considering the assumptions that have been made.
Another factor that could contribute to shifting of the load is tilting (rolling) of the train to the outside of the curve. Such rolling could be due to "give" in the suspension, lifting of the wheels off the track, or both. This is a more complicated situation as shown by the figure. N and fmax would be smaller, allowing the load to shift at a lower speed. Assuming no vertical acceleration, and a tilt angle, T,
N = Wb cos T - (Wb/g)(V²/r) sin T
Then using the 2nd law again in the radial direction,
(Wb/g)(V²/r) = f cos T - N sin T
Using a tilt of 4°, and assuming the case where f= µN, gives a critical speed of V=68 mph, providing additional support to the theory that the blocks shifted prior to the derailment. Tilt makes a significant difference, and this possibility should be examined by analyzing the suspension and examining the dynamics of the train. We will consider the dynamics of the train in Part 2.