Michell structures

Example

A Michell structure is a truss, or a limit of trusses with ever more members, of least volume of material. It must carry given loads to given supports. Here each member may carry stress of absolute value at most $\sigma > 0$. A member of length $L$ carrying an axial force $N$ then needs volume $\abs{N} L / \sigma$. So the task is to minimise $\sum_i \abs{N_i} L_i$ over all trusses in a design region that balance the loads.

Michell's criterion (1904) gives a sufficient condition for optimality. Suppose there is a continuous virtual displacement field $u$ on the design region, zero at the supports, and a number $\varepsilon_0 > 0$, such that:

1. the axial strain of $u$ along each tension member is $+\varepsilon_0$ and along each compression member is $-\varepsilon_0$;
2. the linear strain of $u$ in every direction at every point of the region has absolute value at most $\varepsilon_0$.

Then the truss has the least volume among all trusses in the region that carry the same loads. The check uses virtual work. For any admissible truss, the work $W$ of the loads on $u$ equals $\sum_i N_i \Delta_i$, where $\Delta_i$ is the elongation of member $i$ under $u$. Clause 2 gives $\abs{\Delta_i} \le \varepsilon_0 L_i$, so $\sum_i \abs{N_i} L_i \ge W / \varepsilon_0$. Clause 1 gives equality for the given truss.

Simplest instance: a force $P$ at a point $A$, pulling directly away from a fixed support point $O$. The single straight bar $OA$ is optimal. The uniform dilatation $u(x) = \varepsilon_0 (x - O)$ proves it, since its strain is $\varepsilon_0$ in every direction.

Michell also found the optimal cantilever for a point load carried to a circular support. Its members form two families of logarithmic spirals that cross at right angles.