Exact truss topology optimization: allowance for support costs and different permissible stresses in tension and compression—extensions of a classical solution by Michell

Abstract

The aim of this paper is twofold. It introduces a theory of Michell trusses taking support (reaction) costs into consideration, which is illustrated with a modified version of Michell’s best known example. Then eight other variations on this example are presented, with (i) equal or unequal permissible stresses in tension and compression, (ii) the structural domain consisting of a half-plane or a full plane, and (iii) the supports being two pins, or a pin and a roller. Previously noted shortcomings of Michell’s theory are highlighted and several fundamental properties of non-selfadjoint topology optimization problems discussed. The analytical solutions are verified to a high degree of accuracy by numerical results.

Appendix. A simple example, in which convex combinations of statically determinate optimal topologies are only valid in optimal plastic design

Consider a vertical point load P acting at point B in between two horizontal line supports (Fig. 8a). Its distance from the top and bottom supports is, respectively. 3L and L. The permissible stress in tension is assumed to be three times higher than the permissible stress in compression, $\sigma_T$ = 3$\sigma_{C}$. Using the optimality conditions in (9), for nonzero forces ($\left| F_{i} \right|>0)$ we get the adjoint strains of $\overline{\varepsilon}_y =1/\left( 3\sigma_{\rm C} \right)$ and $\overline{\varepsilon}_{\rm y} =-{1/}\sigma_{\rm C}$, and the ‘real’ strains of $\varepsilon_y =3\sigma_{\rm C} /E$ and $\varepsilon_y =-\sigma_C /E$, for tension and compression, respectively.

Fig. 8

An elementary example of topology optimization with different permissible stresses in tension and compression. a problem statement, b and c statically determinate optimal topologies which are valid for both elastic and plastic design, d convex combination of topologies under b and c, which is only valid for optimal plastic design

There exist two statically determinate optimal solutions, which are shown in Fig. 8b and c. These consist of an R-region and an O-region (region without member). However, in the O-regions (due to the compatibility requirement for the adjoint strains) the strains can take on only the limiting values in the inequality $-1/\sigma_C \le \overline{\varepsilon} \le 1/\sigma_T$ in (9). This means that these O-regions have the same adjoint strains as R-regions.

In a self-adjoint problem, any convex combination of statically determinate solutions is also an optimal solution in both plastic and elastic design. However it can be seen from Fig. 8d that in our current problem with unequal permissible stresses the ‘real’ (elastic) strains would be incompatible⁴ in such a convex combination (in Fig. 8d, the multipliers are 0.5 and 0.5). This implies that Fig. 8d represents an optimal plastic design, but not an optimal elastic design.

The material volume for all three solutions is the same, and takes on a value of $V=PL/\sigma_C$. This can be calculated by primal method (sum of products of bar lengths and bar cross sectional areas A $_i)$, or by the dual method (sum of scalar products of loads and adjoint displacements). In this example the latter reduces to $P\overline{v}_B$, where $\overline{v}_B$ is the (vertical) adjoint displacement at the point A (which takes on a value of $\overline{v}_B =L/\sigma_C$ in all three cases (Fig. 8b to d).

It was stated in a recent article (Rozvany 2011) that for trusses and grillages with equal permissible stresses in tension and compression the number of optimal solutions is either one or infinite. This was actually valid for both optimal elastic and optimal plastic design, because such problems are selfadjoint. It can be seen from the above example, that for non-selfadjoint problems, the above proposition is only valid for plastic design. For elastic design, non-selfadjoint problems may have a either one or a finite number of optimal solutions (but in most cases the optimal solution is unique).