Optimal tensegrity structures in bending: The discrete Michell truss

This paper provides the closed form analytical solution to the problem of minimizing the material volume required to support a given set of bending loads with a given number of discrete structural members, subject to material yield constraints. The solution is expressed in terms of two variables, the aspect ratio, ρ^-1, and complexity of the structure, q (the total number of members of the structure is equal to q(q+1)). The minimal material volume (normalized) is also given in closed form by a simple function of ρ and q, namely, V=q(ρ^-1/q-ρ^1/q). The forces for this nonlinear problem are shown to satisfy a linear recursive equation, from node-to-node of the structure. All member lengths are specified by a linear recursive equation, dependent only on the initial conditions involving a user specified length of the structure. The final optimal design is a class 2 tensegrity structure. Our results generate the 1904 results of Michell in the special case when the selected complexity q approaches infinity. Providing the optimum in terms of a given complexity has the obvious advantage of relating complexity q to other criteria, such as costs, fabrication issues, and control. If the structure is manufactured with perfect joints (no glue, welding material, etc.), the minimal mass complexity is infinite. But in the presence of any joint mass, the optimal structural complexity is finite, and indeed quite small. Hence, only simple structures (low complexity q) are needed for practical design.