Fractal design for an efficient shell strut under gentle compressive loading

Because of Euler buckling, a simple strut of length $L$ and Young modulus $Y$ requires a volume of material proportional to $L^3 f^{1/2}$ in order to support a compressive force $F$, where $f=F/YL^2$ and $f\ll 1$. By taking into account both Euler and local buckling, we provide a hierarchical design for such a strut consisting of intersecting curved shells, which requires a volume of material proportional to the much smaller quantity $L^3 f\exp[2\sqrt{(\ln 3)(\ln f^{-1})}]$.