Tomasz Sokół

On Basic Properties of Michell’s Structures

The paper is an introduction towards Michell’s optimum design problems of pin-jointed frameworks. Starting from the optimum design problem in its discrete setting we show the passage to the discrete-continuous setting in the kinematic form and then the primal stress-based setting. The classic properties of Hencky nets are consistently derived from the kinematic optimality conditions. The Riemann method of the net construction is briefly recalled. This is the basis for finding the analytical solutions to the optimum design problems. The paper refers mainly to the well known solutions (e.g. the cantilevers) yet discusses also the open problems concerning those classes of solutions in which the kinematic approach cannot precede the static analysis.

Validation of Numerical Methods by Analytical Benchmarks, and Verification of Exact Solutions by Numerical Methods

In this lecture, we discuss the validation of various numerical methods in structural topology optimization. This is done by computing numerically optimal topologies (e.g. for perforated plates in plane stress), using various numbers of ground elements and various volume fractions. Then the structural volume value is extrapolated for (theoretically) zero volume fraction and infinite number of ground elements, and this extrapolated value is compared with that of the analytical solution. As a related subject, volume-increasing effect of topology simplification is also discussed.

On the Numerical Approximation of Michell Trusses and the Improved Ground Structure Method

The paper deals with an improved method of solving large-scale linear programming problems related to Michell trusses. The method is an extension of the adaptive ground structure method developed recently by the author. As before, both bars and nodes can be switched between active and inactive states, but now, the adjoint displacements of inactive nodes, i.e. nodes in empty regions, are adjusted by solving an auxiliary optimization problem. This procedure is relatively cheap and allows a more efficient cutting off the design domain and a significant reduction of the problem size. Thus, the numerical results can be attained for denser ground structures than before, giving better approximations of Michell structures. It is especially important for 3D problems with multiple load conditions. The preliminary results of such problems are reported in the paper and clearly indicate high efficiency of the proposed method.

Constructing Michell Structures in Plane. Single Load Case

This chapter introduces the reader into the methods of construction of the planar frameworks being exact solutions to the Michell problems of optimum design. The layout of bars of these structures follows the trajectories of specific strain fields. The methods of their construction are given in Sects. 4.1–4.4. The simplest Michell structures are composed of straight and circular bars; they are described in Sect. 4.5. The next sections outline the construction of all available nowadays exact solutions of the Michell’s theory; some constructions are checked by the static method. The analytical results are compared with their numerical predictions found by the ground structure method.

Selected Classes of Spatial Michell’s Structures

The spatial Michell structures are solutions to the problem ( 3.85) in the kinematic setting and to the dual problem ( 3.95) in the stress-based setting. The optimality condition ( 3.84) referring to the spatial setting (\(n=3\)) differs from the plane setting (\(n=2\)), since in the plane problem, the number of the bounds is equal to the number of the adjoint principal strains and in some regimes the following equalities can be fulfilled \(\displaystyle \varepsilon _{II}=-\frac{\sigma _0}{\sigma _C}\), \(\displaystyle \varepsilon _{I}=\frac{\sigma _0}{\sigma _T}\), while in the spatial problem we may have \(\displaystyle \varepsilon _{III}=-\frac{\sigma _0}{\sigma _C}\), \(\displaystyle \varepsilon _{I}=\frac{\sigma _0}{\sigma _T}\), with \(\varepsilon _{II}\) attaining neither of the bounds. Thus, we expect that in many spatial problems the solutions will be either composed of fibrous membrane shells, or will be collections of planar structures not linked in the direction transverse to their planes. The well-known Michell’s sphere as well as other optimal shells of revolution subject to the pure torsion load belong to the first class mentioned, see Sects. 5.2 and 5.3. The rotationally symmetric Hemp’s structures (Sect. 5.1) belong to the second class.

Extensions of the Michell Theory

The present chapter concerns the extension of the Michell problems towards the multi-load cases and reveals the impact of the Michell theory on the other problems of topology optimization, especially those concerning the optimum material design.

Optimum Design of Structures of Finite Number of Bars. Single Load Variant

The central problem of the present chapter is the minimization of the compliance of bar structures. It turns out that the optimum design of trusses of minimal compliance leads to an auxiliary problem, which naturally appears in the problem of minimization of truss volume under the condition of stresses in bars being bounded from both sides. This auxiliary problem is a problem of linear programming, which makes it possible to construct optimal structures of huge number of bars. The numerical algorithm is based on the ground structure method. These optimum designs of huge number of bars are approximants of Michell’s structures to be discussed in the next chapter. The optimal designs of grillages (Sect. 2.5) are governed by more complicated equations. Only additional approximations imposed pave the way towards the linear programming problems thus delivering approximate solutions of Prager–Rozvany grillages, being flexural counterparts of Michell’s structures.

Theory of Michell Structures. Single Load Case

The Michell structures are solutions to the problems of optimum design put forward and discussed in Sects. 2.1 and 2.2 provided that the nodes of the trusses may be placed at arbitrary point of the design domain being a subdomain of the plane or of the Euclidean space. The present chapter introduces the reader to the main topic of the book.

Optimum Design of Funiculars and Archgrids

Michell structures are designed for a fixed load, applied at fixed places. However, in the engineering practice the load is usually linked with the structure and usually follows its current position, as, e.g. the self-weight, or the weight of snow. Thus, the majority of loads we face with are design dependent. A narrow class of such loads are transmissible loads acting along vertical lines, coinciding with the gravity field. This chapter is aimed at putting forward the theory of such optimal structures.

Selected Problems of Statics

In its original formulation, Michell’s (Philosophical Magazine, Series 6 8(47):589–597, 1904) problem reads as follows: find the framework of least volume in which state of stress \(\varvec{\sigma }\) satisfies the conditions \(-\sigma _C \le \min \lambda _i(\varvec{\sigma })\le \max \lambda _i (\varvec{\sigma }) \le \sigma _T\) and which a given load transmits to a given segment of the boundary of the design domain; \(\lambda _i (\varvec{A})\) represents ith eigenvalue of a symmetric matrix \(\varvec{A}\). The solution to this problem depends essentially on the ratio \(\kappa =\frac{\sigma _T}{\sigma _C}\) but does not depend on other material data. Therefore, this problem belongs to the class called the plastic optimum design.