Selected Classes of Spatial Michell’s Structures

  • Tomasz Lewiński
  • Tomasz Sokół
  • Cezary Graczykowski


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.


  1. Bołbotowski, K. (2018). Theory of grillage optimization-a discrete setting (In preparation).Google Scholar
  2. Bołbotowski, K., He, L., & Gilbert, M. (2018). Design of optimum grillages using layout optimization. Structural and Multidisciplinary Optimization, 58, 851–868.CrossRefGoogle Scholar
  3. Bouchitté, G., & Fragalá, I. (2007). Optimality conditions for mass design problems and applications to thin plates. Archive for Rational Mechanics and Analysis, 184, 257–284.MathSciNetCrossRefGoogle Scholar
  4. Hemp, W. S. (1973). Optimum structures. Oxford: Clarendon Press.Google Scholar
  5. Jacot, B. P., & Mueller, C. T. (2017). A strain tensor method for three-dimensional Michell structures. Structural and Multidisciplinary Optimization, 55, 1819–1829.MathSciNetCrossRefGoogle Scholar
  6. Lewiński, T. (2004). Michell structures formed on surfaces of revolution. Structural and Multidisciplinary Optimization, 28(1), 20–30.MathSciNetCrossRefGoogle Scholar
  7. Lewiński, T., & Sokół, T. (2014). On basic properties of Michell’s structures. In G. I. N. Rozvany & T. Lewiński (Eds.), Topology optimization in structural and continuum mechanics (Vol. 549). CISM international centre for mechanical sciences. Courses and lectures. Wien: Springer.CrossRefGoogle Scholar
  8. Lowe, P. G., & Melchers, R. E. (1972a). On the theory of optimal constant thickness, fibrereinforced plates. I. International Journal of Mechanical Sciences, 14, 311–324.CrossRefGoogle Scholar
  9. Lowe, P. G., & Melchers, R. E. (1972b). On the theory of optimal constant thickness, fibrereinforced plates. II. International Journal of Mechanical Sciences, 15, 157–170.Google Scholar
  10. Michell, A. G. M. (1904). The limits of economy of material in frame structures. Philosophical Magazine, 8(47), 589–597.zbMATHGoogle Scholar
  11. Morley, C. T. (1966). The minimum reinforcement of concrete slabs. International Journal of Mechanical Sciences, 8, 305–319.CrossRefGoogle Scholar
  12. Prager, W. (1985). Optimal design of grillages. In M. Save & W. Prager (Eds.), Structural optimization. Volume 1. Optimality criteria (Vol. 1, pp. 153–200 and 308–317). New York and London: Plenum Press.CrossRefGoogle Scholar
  13. Prager, W., & Rozvany, G. I. N. (1977). Optimal layout of grillages. Journal of Engineering Mechanics, 5, 1–18.Google Scholar
  14. Rozvany, G. I. N. (1976). Optimal design of flexural systems. London: Pergamon Press.Google Scholar
  15. Sigmund, O., Zhou, M., & Rozvany, G. I. N. (1993). Layout optimization of large FE systems by new optimality criteria methods: Application to beam systems. Concurrent engineering: Tools and technologies for mechanical system design (pp. 803–819). Berlin: Springer.CrossRefGoogle Scholar
  16. Sokół, T. (2016). A new adaptive ground structure method for multi-load spatial Michell structures. In M. Kleiber, T. Burczyński, K. Wilde, J. Górski, K. Winkelmann, & Ł. Smakosz (Eds.), Advances in mechanics: Theoretical, computational and interdisciplinary issues (pp. 525–528). Boca Raton: CRC Press.CrossRefGoogle Scholar
  17. Sokół, T. (2017). On the numerical approximation of Michell trusses and the improved ground structure method. In Proceedings of the 12th World Congress on Structural and Multidisciplinary Optimisation, 5th–9th June 2017, Braunschweig, Germany.Google Scholar
  18. Zegard, T., & Paulino, G. H. (2015). GRAND3 - Ground structure based topology optimization for arbitrary 3D domains using MATLAB. Structural and Multidisciplinary Optimization, 52, 1161–1184.CrossRefGoogle Scholar
  19. Zhou, K. (2009). Optimization of least-weight grillages by finite element method. Structural and Multidisciplinary Optimization, 38, 525–532.CrossRefGoogle Scholar

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Authors and Affiliations

  • Tomasz Lewiński
    • 1
  • Tomasz Sokół
    • 2
  • Cezary Graczykowski
    • 3
  1. 1.Faculty of Civil EngineeringWarsaw University of TechnologyWarsawPoland
  2. 2.Faculty of Civil EngineeringWarsaw University of TechnologyWarsawPoland
  3. 3.Institute of Fundamental Technological ResearchPolish Academy of SciencesWarsawPoland

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