Elastic Scaling of Small Structures

As aerospace designers strive to build smaller systems, it is important that they understand scaling laws to take full advantage of the inherent strength of small structures. Simple geometric scaling yields masses that scale in proportion to ℓ3. However, in the process, the stress levels decrease, and the materials are not used to full advantage. Also, the resistance to buckling increases as the length decreases. With "elastic scaling," as the dimension parallel to the predominant load shortens, the dimension normal to the main load is thinned down even faster. This preserves a constant factor of safety with respect to the critical buckling load. The structural mass decreases even faster than ℓ3 and the material is used more effectively than for simple geometric scaling. Examples abound in nature, from tree trunks to bones. Several such examples will be shown to illustrate this type of scaling. Even with elastic scaling, the stress levels continue to decrease as the size is reduced. An extension of elastic scaling with more than one dimension normal to the main load-bearing direction is considered. The possibility of scaling the different lateral dimensions differently in an attempt to preserve constant stress in the material as the object shrinks is investigated. It is shown that no systematic scaling can achieve this goal, although some useful insight is developed. A related issue is the minimum gage problem. When one attempts to use elastic scaling (or even geometric scaling), one discovers that as the size decreases, the materials required become too thin to handle. Techniques for addressing this difficulty will be discussed.