# Imperfections in a two-dimensional hierarchical structure

Fractal structures need very little mass to support a load; but for current designs, this makes them vulnerable to manufacturing errors.

*Physical Review E* 89, 23201 (2014)

D. Rayneau-Kirkhope, Y. Mao, R. Farr

Hierarchical and fractal designs have been shown to yield high mechanical efficiency under a variety of loading conditions. Here a fractal frame is optimized for compressive loading in a two-dimensional space. We obtain the dependence of volume required for stability against loading for which the structure is optimized and a set of scaling relationships is found. We evaluate the dependence of the Hausdorff dimension of the optimal structure on the applied loading and establish the limit to which it tends under gentle loading. We then investigate the effect of a single imperfection in the structure through both analytical and simulational techniques. We find that a single asymmetric perturbation of beam thickness, increasing or decreasing the failure load of the individual beam, causes the same decrease in overall stability of the structure. A scaling relationship between imperfection magnitude and decrease in failure loading is obtained. We calculate theoretically the limit to which the single perturbation can effect the overall stability of higher generation frames.

## More in Fractal structures

### Hierarchical space frames

A systematic way to vary the power-law scaling relations between loading parameters and volume of material aids the hierarchical design process.

### Towers of strength

The Eiffel tower is now a longstanding example of hierarchical design due to its non-trivial internal structure spanning many length scales.

### Hierarchical structures

The most efficient load-bearing fractals are designed as big structures under gentle loads ... a situation common in aerospace applications.

### Ultralight fractal structures

The transition from solid to hollow beams changes the scaling of stability versus loading analogously to increasing the hierarchical order by one.