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Abstract:
This paper analyzes the relations between the theory of Michell structures, which is one of the most important theories in structural optimization, and some remarkable engineering structures, including selected high-rise buildings, large-scale roof coverings and long-span bridges. The first part of this study briefly presents the development of Michell's theory, its basic concepts, assumptions, and examples and fundamental features of Michell structures. Then, several untypical engineering structures that make use of said concepts are presented, including skyscrapers proposed by the Polish structural designer W. Zalewski and the international architectural office of Skidmore, Owings and Merill (SOM). Next, large-scale roof coverings of "Spodek" arena in Poland as well as selected bridges are thoroughly analyzed in the context of similarity to Michell structures. The conducted study reveals that considered structural forms of the analyzed structures follow some of the concepts known from Michell's theory and thus possess many features of the optimal structural designs.

Keywords:
topology optimization, Michell structures, high-rise buildings, large-scale roofs, long-span bridges

Abstract:
The paper delivers the benchmark results for the Michell cantilevers constructed within a half strip, for selected values of the σT /σC ratio, σT, σC being the admissible stresses in tension and compression, respectively.

Keywords:
Michell structures, Minimum weight design, Topology optimization, Trusses

Abstract:
The subject of the paper is an optimal choice of material parameters characterizing the core layer of sandwich plates within the framework of the conventional plate theory in which the core layer is treated as soft in the in-plane direction. The mathematical description is similar to the Hencky–Reissner model of plates with transverse shear deformation. Here, however, the bending stiffnesses and the transverse shear stiffnesses can be designed independently. The present paper deals only with optimal design of the core layer to make the plate compliance minimal. Two core materials are at our disposal, which leads to the ill-posed problem. To consider it one should relax this problem by admitting composite domains and characterize their overall properties by the homogenization formulae. The numerical approach is based on this relaxed formulation thus making it mesh-independent. The equilibrium problem is solved by the DSG3 finite element method. The optimization results are found with using the convergent updating schemes of the COC method.

Keywords:
Minimum compliance problem, Sandwich plates, Topology optimization

Abstract:
The paper concerns the Michell-like cantilevers transmitting a point load to a straight segment of a support. The feasible domain is of trapezoidal infinite shape, as in the previous parts of the paper. The ratio of allowable stresses in tension and compression is arbitrary, not necessarily equal to 1. The present, last part of the paper includes detailed geometric and static analyses of the optimal cantilevers for various admissible data, thus providing new benchmarks of topology optimization. All results are found by using analytical methods developed in the previous parts of the paper. Particular attention is put on the force field distribution within the fibrous domains. These force fields turn out to be defined in certain subdomains forming a static division. The volumes of the optimal cantilevers are computed in two manners: by direct integration of the density of fibres and summing it up with the volume of the reinforcing bars of finite cross sections, and by using the kinematic formula of Michell according to which the volume is proportional to the virtual work. The examples analysed prove that both approaches lead to identical results of the volumes, thus showing that the possible duality gaps vanish. The analytical solutions are verified by considering appropriately chosen sequences of trusses of finite number of joints converging to the exact Michell cantilevers.

Keywords:
Michell cantilevers, layout optimization, minimum weight design

Abstract:
This paper complements the analysis of geometric properties of the Hencky nets within the Michell cantilevers constructed in the trapezoidal domains by providing the analytical formulae for the force fields. The force field analysis introduces a new division of the cantilever domain and enables an alternative method for computing the optimal weights.

Keywords:
Michell cantilevers, layout optimization, minimum weight design

Abstract:
The present second part of the paper deals with the virtual displacement fields associated with the optimality conditions epsilon_1=1, epsilon_2 = -k, k=sigma_T/sigma_c, where sigma_T and sigma_C represent the allowable values of the tensile and compressive stress, respectively. The displacement fields vanish along a straight segment of a line support and are constructed within an infinite domain bounded by two half-lines. The displacement fields are provided by the integral formulae involving the Lamé fields found in part I of this paper. All the results are expressed in terms of Lommel-like functions. These results make it possible to determine the volumes of the optimal cantilevers designs within the feasible domain considered. Computation of the volumes along with analyses of concrete cantilevers will be the subject of part IV of the present paper.

Keywords:
Michell cantilevers, layout optimization, minimum weight design

Abstract:
The present paper is the first part of the four-part work on Michell cantilevers transmitting a given point load to a given segment of a straight-line support, the feasible domain being a part of the half-plane contained between two fixed half-lines. The axial stress sigma in the optimal cantilevers is assumed to be bounded by −sigma_C=< sigma<=sigma_T, where sigma_C and sigma_T represent the allowable compressive and tensile stresses, respectively. The work provides generalization of the results of the article of Lewinski et al. (Int J Mech Sci 36:375–398, 1994a) to the case of sigma_T unequal sigma_C. The present, first part of the work concerns the analytical formation of the Hencky nets or the lines of fibres filling up the interior of the optimal cantilevers corresponding to an arbitrary position of the point of application of the given concentrated force.

Keywords:
Michell cantilevers, layout optimization, minimum weight design

Abstract:
The paper refers to the problem of Michell (1904) of finding the lightest fully stressed structures, composed of possibly infinite number of members, trasmitting a given load to a support forming a circle. The point load can be located within or outside the circle. The known analysis by Hemp (1973) is enhanced here by disclosing the explicit formulae for the weights of the ribs and the interior (fibrous domain). The optimal weight can be found by two manners: by applying the primal integral formula involving the density of the reinforcement or by computing the work of the load on the adjoint displacement. One of the aims of the paper is to show that both these formulae are equivalent. This identity is essential since in the case of point loads the equivalence of the primal and dual formulations has not been proved till now. The analytically found layouts are confirmed by analysis of trusses (of finite number of joints) approximating the exact Michell-like solutions.

Keywords:
Michell cantilevers, layout optimization, minimum weight design