Michell structures

Michell structures are structures that are optimal based on the criteria defined by A.G.M. Michell in his frequently referenced 1904 paper.^[1]

Michell states that “a frame (today called truss) (is optimal) attains the limit of economy of material possible in any frame-structure under the same applied forces, if the space occupied by it can be subjected to an appropriate small deformation, such that the strains in all the bars of the frame are increased by equal fractions of their lengths, not less than the fractional change of length of any element of the space.”

The above conclusion is based on the Maxwell load-path theorem:

${\displaystyle \sum _{}l_{p}f_{p}-\sum _{}l_{q}f_{q}=C}$

Where ${\displaystyle f_{p}}$ is the tension value in any tension element of length ${\displaystyle l_{p}}$, ${\displaystyle f_{q}}$ is the compression value in any compression element of length ${\displaystyle l_{q}}$ and ${\displaystyle C}$ is a constant value which is based on external loads applied to the structure.

Based on the Maxwell load-path theorem, reducing load path of tension members ${\displaystyle \textstyle \sum _{}l_{p}f_{p}}$ will reduce by the same value the load path of compression elements${\displaystyle \textstyle \sum _{}l_{q}f_{q}}$ for a given set of external loads. Structure with minimum load path is one having minimum compliance (having minimum weighted deflection in the points of applied loads weighted by the values of these loads). In consequence Michell structures are minimum compliance trusses.