Title:

On Michell's theorem and its place in the study of framework optimisation

The first theorem pertaining to the minimum weight layout of pinjointed frameworks is due to J.C. Maxwell and shows that any framework which consists entirely of tension bars or entirely of compression bars is an optimum. However certain conditions must be satisfied by the external force system if such a framework is to be possible. Following Maxwell, A. G. M. Michell derived a theorem which gives sufficient conditions for a framework containing both tension and compression members to be an optimum: the bars must lie along lines of constant principal strain in a compatible virtual deformation of space, with the sign of the strain agreeing with the sign of the load in the bar. Displacement, strain and compatibility of strain are physical concepts which can be described by a mathematical model involving first and second order tensors. In Michell's theorem it is the mathematical rather than the physical model which is significant; this point is particularly important in connection with the optimum layout of frameworks on plane and curved surfaces. Lines of principal strain can, in general, be represented by three orthogonal families of curves in three dimensional space. The general properties of orthogonal curvilinear systems, and the particular properties of those which define layouts for minimum weight frameworks, are readily derived by means of the tensor calculus. The equilibrium equations for a continuum can be adapted to apply to the optimum frameworks and, again, tensor calculus is the most convenient analytical technique. The layout and equilibrium equations so derived apply to optimum frameworks both in three dimensional space and on plane and curved surfaces. The technique of linear programming provides another approach to the minimum weight design of pinjointed frameworks. The method is more direct in that it finds an optimum framework for a given force system but is restricted by the fact that the framework nodes are selected from a predetermined set: however, given the restricted nodal pattern, the optimum framework still satisfies the Michell criterion. The formulation of the framework design problem as a linear program enables some simple buckling constraints to be included and also allows fixed points of reaction to be specified, so that the external force system is in part dependent on the form of the framework (this represents a departure from the conditions imposed in the theorems of Maxwell and Michell, in which the bars are all assumed to be at either the limiting compressive or tensile stress, these being taken as material constants, and the external force system is completely specified). Linear programming problems can be efficiently solved on a digital computer, although, in the case of the framework problem, a considerable amount of extra computation is involved if the data is to be read in, and the results printed out, in a convenient form. Further developments in optimum layout of frameworks have been along the lines of mathematical programming, of which linear programming is a simple case, and away from Michell's theorem, which remains a unique and still fascinating contribution to structural design.
