Evidently it is practically impossible to make so thin a rim that it will collapse under the pull of a belt. As far as the theory of the rim is concerned, its proportion probably depends more upon the calculation for centrifugal force than upon anything else.
In order to separate this action from that of any other forces, let us suppose that the rim is entirely free from the arms and hub, and is rotating about its center. Every particle, by centrifugal force, tends to fly radially outward from the center. This condition is represented in Fig. 19. The tendency with which one-half of the rim tends to fly apart from the other is indicated by the force C1; and the relation between C1 and the small radial force c for each unit-length of rim can readily be found from the principles of mechanics. The case is exactly like that of a boiler or a thin pipe subjected to uniform internal pressure, which, if carried to rupture, would split the rim along a longitudinal seam.
The tensile stress thus induced per square inch can be found by simple mechanics to be:
P = 12 w v2 / g ; (17) or, since w = 0.20 pound, and g = 32.2 feet per second, p = 0.097 v2 ( say v2 /10) ; (l8) and, if p be taken equal to 1,000 pounds per square inch, which is as high as it is safe to work cast iron in this place, v = 100 feet per second. (19)
This shows the curious fact that the intensity of stress in the rim is directly proportional to the square of the linear velocity, and wholly independent of the area of cross-section. It is also to be noted that 100 feet per second is about the limit of speed for cast-iron pulleys to be safe against bursting.
If we wish to consider theoretically the rim together with the arms as actually connected to it, we get a much more complicated relation. This condition is shown in Fig. 20, where the rim, expanding more than the arms, bulges out between them. This makes the rim act something like a continuous beam uniformly loaded; but even then the resulting stress is not clearly defined on account of the variable stretch in the arms. Investigation on this basis is not needed further than to note that it is theoretically better, in the case of a split pulley, to make the joint close to the arms, rather .than in the middle of a span.