4.1 Michell trusses

Introduction

The Michell Truss PFUG studies a variational problem posed by mechanical engineer Anthony George Maldon Michell in the early part of the last century;what configuration of bars and cables is needed to withstand a system of balanced point forces is most economical?Suppose that some collection of forces with their points of application (point forces) is given and you are asked to build a structure which can withstand these forces.Your choice of materials are bars and cables. Bars can withstand compression (forces parallel and pointing inward) but will breakunder extension (forces parallel and pointing outward). Cables can withstand extension but will buckle under compression.You may also choose the strength of the bar or cable; the strength is the largestcompression or extension the bar or cable respectively can withstand. A structure built of such cables and bars will be called a truss.

The first question the PFUG has addressed is whether one can build such a truss at all. Notice that a bar or cable by itself canwithstand only one set of point forces; that of two opposite forces applied to the endpoints of the bar or cable. These point forces have the property that if the points of application were attached to a stationary body, the center ofmass would not accelerate and the body would not rotate. This property is called balanced. Since point forces can be linearly superimposed, a necessary condition that a set ofpoint forces be withstood by a truss is that it be balanced. Below we show that being balanced is also a sufficient condition; if the set of point forces is balanced then there is a truss withstanding it.

Next, the PFUG formulated necessary conditions for a truss to be economical. The cost of a bar or cable is its strength times its length. The cost of a truss is definedto be the sum of the cost of bars and cables constituted by the truss. In general, there are many trusses which withstand a given a set of point forces.The question is, which is most economical, i.e. which truss costs the least? In general this is a very difficult question to answer because a minimizing sequence,which we know to exist from the first part, may converge in a larger class of measures than described here and it is difficult to formulate any class of perturbations of a truss.Below we describe one class of perturbations on trusses with corners which yields a surprising necessary condition for economy. A set of perturbations to be studiedin the future are also described.