On the optimal shape of compressed rotating rod with shear and extensibility

In this paper, the optimal shape of a compressed rotating rod which maintains stability against buckling is presented. In the rod modeling, extensibility along the rod axis and shear stress is taken into account. Using Pontryagin's maximum principle, the optimization problem is formulated with a fourth order boundary value problem. The optimally shaped compressed rotating (fixed-free) rod has a finite cross-sectional area on the free end. This shape is qualitatively different from that suggested by the Bernoulli–Euler theory with zero cross-sectional area on the free end. In addition, the Bernoulli–Euler theory overestimates the buckling load, and this effect is more significant in the optimally shaped rod than for the corresponding constant cross-sectional rod consisting of the same material volume and length. In order to show this effect, it is necessary to use a generalized constitutive model which takes real material properties, such as axial extensibility and shear stress into account. Particularly, the solution of this generalized problem, obtained for thin rods, approaches the classical solution predicted by the Bernoulli–Euler theory.