[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:
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
will reduce by the same value the load path of compression elements
In consequence Michell structures are minimum compliance trusses.
All bars of a truss are subject to a load of the same sign (tension or compression).
Required volume of material is the same for all possible cases for a given set of loads.
Michell defines minimum required volume of material to be:
Mixed tension and compression bars More general case are frames which consist of bars that both before and after the appropriate deformation, form curves of orthogonal systems.
A two-dimensional orthogonal system remains orthogonal after stretching one series of curves and compressing the other with equal strain if and only if the inclination between any two adjacent curves of the same series is constant throughout their length.
Note that straight line or a circle are special cases of a logarithmic spiral.
Michell provided several examples of optimum frames: In recent years a lot of studies have been done on discrete optimum trusses.
Significant contribution to the topic of discrete optimum trusses had William Prager who used the method of the circle of relative displacements to arrive with optimal topology of such trusses (typically cantilevers).
Later geometry of cantilevered Prager trusses has been formalized by Mazurek, Baker and Tort [5][6] who noticed certain geometrical relationships between members of optimal discrete trusses for 3 point or 3 force problems.