Spinning rods, elliptical disks and solid ellipsoidal bodies: Elastic and plastic stresses and limit spins

The analysis of the stresses in one-, two- and three-dimensional spinning bodies is discussed in a systematic and comprehensive way. First elastic solutions are derived for rods, for elliptical-shaped flat disks and for ellipsoidal solid bodies spinning about their sideways axes. Then the spins for first plastic yield are found in each case using each of the Tresca and the von Mises yield conditions. Then upper and lower bounds on the maximum allowable limit spins where the body would globally fail assuming perfectly plastic behavior are derived. The elastic solutions at first yield always give a lower bound to that limit spin, but global failure generally does not occur until the spin is increased. A way to calculate an improved lower bound is illustrated. Upper bounds are found in a simple and new way. The method uses the fact that the volume-averaged stresses can be calculated directly from the loadings without the need for any actual stress solutions, and then it is proved that the use of those average stresses in the yield functions always gives an upper bound to the limit loads. That use of the statically determinate average stresses to obtain meaningful plastic upper bounds to limit loads is though to be a new method, and can be applied to any shape. Finally, several finite element calculations are used to determine the quantitative relations between the lower and upper bounds and the actual limit spins for ellipsoidal bodies.The results are of interest in the spin of planetary bodies, where they explain the nature of an average-stress approximate method, and in the analysis of spinning bodies in general. In addition, the approach gives a very interesting example of the utility of the limit analysis approaches of plasticity theories.