Advertisement

Constructing Michell Structures in Plane. Single Load Case

  • Tomasz LewińskiEmail author
  • Tomasz Sokół
  • Cezary Graczykowski
Chapter

Abstract

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. 4.14.4. 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.

References

  1. Carathéodory, C., & Schmidt, E. (1923). Über die Hencky-Prandtlschen Kurven. Zeitschrift für Angewandte Mathematik und Mechanik, 3, 468–475.CrossRefGoogle Scholar
  2. Chakrabarty, J. (2006). Theory of plasticity (3rd ed.). Oxford, UK: Butterworth-Heinemann.zbMATHGoogle Scholar
  3. Chan, A. S. L. (1960). The design of Michell optimum structures. Cranfield: The College of Aeronautics (Report 142).Google Scholar
  4. Chan, H. S. Y. (1963). Optimum Michell frameworks for three parallel forces. Cranfield: The College of Aeronautics (Report AERO 167).Google Scholar
  5. Chan, H. S. Y. (1964). Tabulation of some layouts and virtual displacement fields in the theory of Michell optimum structures. Cranfield: The College of Aeronautics (CoA Note Aero 161).Google Scholar
  6. Chan, H. S. Y. (1966). Minimum weight cantilever frames with specified reactions. University of Oxford, Department of Engineering Science Laboratory, Parks Road, Oxford, June 1966, No. 1, 010.66, p. 11 with 4 figures. (the paper cited for the first time in Sokół and Lewiński (2010)).Google Scholar
  7. Chan, H. S. Y. (1967). Half-plane slip-line fields and Michell structures. The Quarterly Journal of Mechanics and Applied Mathematics, 20, 453–469.CrossRefGoogle Scholar
  8. Chan, H. S. Y. (1975). Symmetric plane frameworks of least weight. In A. Sawczuk & Z. Mróz (Eds.), Optimization in structural design (pp. 313–326). Berlin: Springer.CrossRefGoogle Scholar
  9. Cox, H. L. (1965). The design of structures of least weight. Oxford: Pergamon Press.CrossRefGoogle Scholar
  10. Darwich, W., Gilbert, M., & Tyas, A. (2010). Optimum structure to carry a uniform load between pinned supports. Structural and Multidisciplinary Optimization, 42, 33–42.CrossRefGoogle Scholar
  11. Dewhurst, P., & Srithongchai, S. (2005). An investigaton of minimum-weight dual-material symmetrically loaded wheels and torsion arms. Journal of Applied Mechanics, Transactions ASME, 72, 196–202.CrossRefGoogle Scholar
  12. Dewhurst, P., Fang, N., & Srithongchai, S. (2009). A general boundary approach to the construction of Michell truss structures. Structural and Multidisciplinary Optimization, 39, 373–384.MathSciNetCrossRefGoogle Scholar
  13. Fung, Y. C. (1965). Foundations of solid mechanics. New Jersey: Prentice Hall.Google Scholar
  14. Geiringer, H. (1937). Fondements mathématiques de la theorie des corps plastiques isotropes (Vol. 86). Mémorial des sciences mathématiques. Paris: Gauthier-Villard.Google Scholar
  15. Graczykowski, C., & Lewiński, T. (2005). The lightest plane structures of a bounded stress level transmitting a point load to a circular support. Control and Cybernetics, 34(1), 227–253.MathSciNetzbMATHGoogle Scholar
  16. Graczykowski, C., & Lewiński, T. (2006a). Michell cantilevers constructed within trapezoidal domains - Part I: Geometry of Hencky nets. Structural and Multidisciplinary Optimization, 32(5), 347–368.MathSciNetCrossRefGoogle Scholar
  17. Graczykowski, C., & Lewiński, T. (2006b). Michell cantilevers constructed within trapezoidal domains - Part II: Virtual displacement fields. Structural and Multidisciplinary Optimization, 32(6), 463–471.MathSciNetCrossRefGoogle Scholar
  18. Graczykowski, C., & Lewiński, T. (2007a). Michell cantilevers constructed within trapezoidal domains - Part III: Force fields. Structural and Multidisciplinary Optimization, 33(1), 27–46.MathSciNetzbMATHGoogle Scholar
  19. Graczykowski, C., & Lewiński, T. (2007b). Michell cantilevers constructed within trapezoidal domains - Part IV: Complete exact solutions of selected optimal designs and their approximations by trusses of finite number of joints. Structural and Multidisciplinary Optimization, 33(2), 113–129.MathSciNetCrossRefGoogle Scholar
  20. Graczykowski, C., & Lewiński, T. (2010). Michell cantilevers constructed within a halfstrip. Tabulation of selected benchmark results. Structural and Multidisciplinary Optimization, 42(6), 869–877.CrossRefGoogle Scholar
  21. Green, A. E., & Zerna, W. (1968). Theoretical elasticity. Oxford: Clarendon Press.zbMATHGoogle Scholar
  22. Hemp, W. S. (1973). Optimum structures. Oxford: Clarendon Press.Google Scholar
  23. Hemp, W. S. (1975). Michell framework for uniform load between fixed supports. Engineering Optimization, 1, 61–69.CrossRefGoogle Scholar
  24. Hill, R. (1950). The mathematical theory of plasticity. Oxford: Clarendon Press.zbMATHGoogle Scholar
  25. Hu, T.-C., & Shield, T. T. (1961). Minimum weight design of discs. Zeitschrift für angewandte Mathematik und Physik, 12, 414–433.CrossRefGoogle Scholar
  26. 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
  27. Lewiński, T. (2004). Michell structures formed on surfaces of revolution. Structural and Multidisciplinary Optimization, 28(1), 20–30.MathSciNetCrossRefGoogle Scholar
  28. Lewiński, T., & Rozvany, G. I. N. (2007). Exact analytical solutions for some popular benchmark problems in topology optimization II: Three-sided polygonal supports. Structural and Multidisciplinary Optimization, 33(4–5), 337–349.MathSciNetCrossRefGoogle Scholar
  29. Lewiński, T., & Rozvany, G. I. N. (2008a). Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains. Structural and Multidisciplinary Optimization, 35(2), 165–174.MathSciNetCrossRefGoogle Scholar
  30. Lewiński, T., & Rozvany, G. I. N. (2008b). Analytical benchmarks for topological optimization IV: Square-shaped line support. Structural and Multidisciplinary Optimization, 36(2), 143–158.MathSciNetCrossRefGoogle Scholar
  31. Lewiński, T., Zhou, M., & Rozvany, G. I. N. (1994a). Extended exact solutions for least-weight truss layouts - Part I: Cantilever with a horizontal axis of symmetry. International Journal of Mechanical Sciences, 36(5), 375–398.CrossRefGoogle Scholar
  32. Lewiński, T., Zhou, M., & Rozvany, G. I. N. (1994b). Extended exact solutions for least-weight truss layouts - Part II: Unsymmetric cantilevers. International Journal of Mechanical Sciences, 36(5), 399–419.CrossRefGoogle Scholar
  33. Lewiński, T., Rozvany, G. I. N., Sokół, T., & Bołbotowski, K. (2013). Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains revisited. Structural and Multidisciplinary Optimization, 47, 937–942.MathSciNetCrossRefGoogle Scholar
  34. Melchers, R. E. (2005). On extending the range of Michell-like optimal topology structures. Structural and Multidisciplinary Optimization, 29, 85–92.CrossRefGoogle Scholar
  35. Mazurek, A. (2012). Geometrical aspects of optimum truss like structures for three-force problem. Structural and Multidisciplinary Optimization, 45, 21–32.MathSciNetCrossRefGoogle Scholar
  36. Mazurek, A., Baker, W., & Tort, C. (2011). Geometrical aspects of optimum truss like structures. Structural and Multidisciplinary Optimization, 43, 231–242.CrossRefGoogle Scholar
  37. Mazurek, A., Carrion, J., Beghini, A., & Baker, W. F. (2015). Minimum weight single span bridge obtained using graphic statics. In Proceedings of International Association for Shell and Spatial Structures (IASS) Symposium, 17–20 August 2015 (11 pp.). Amsterdam, The Netherlands: Future Visions.Google Scholar
  38. Mazurkiewicz, Z. E. (2004). Thin elastic shells. Linear theory. Warszawa (in Polish): Oficyna Wydawnicza PW.Google Scholar
  39. McConnel, R. E. (1974). Least-weight frameworks for loads across span. Journal of the Engineering Mechanics Division, 100(5), 885–901.Google Scholar
  40. Michell, A. G. M. (1904). The limits of economy of material in frame structures. Philosophical Magazine, 8(47), 589–597.zbMATHGoogle Scholar
  41. Novozhilov, V. V. (1970). Thin shell theory. Groningen: Walters-Nordhoff.zbMATHGoogle Scholar
  42. Pichugin, A. V., & Gilbert, M. (2011). Optimum structure to carry a uniform load between pinned supports: Exact analytical solution. Proceedings of the Royal Society A, 467(2128), 1101–1120.MathSciNetCrossRefGoogle Scholar
  43. Pichugin, A. V., Tyas, A., & Gilbert, M. (2011). Michell structure for a uniform load over multiple spans. In 9th World Congress on Structural and Multidisciplinary Optimization, CD ROM.Google Scholar
  44. Pichugin, A. V., Tyas, A., & Gilbert, M. (2012). On the optimality of Hemp’s arch with vertical hangers. Structural and Multidisciplinary Optimization, 46(1), 17–25.MathSciNetCrossRefGoogle Scholar
  45. Pichugin, A. V., Tyas, A., Gilbert, M., & He, L. (2015). Optimum structure for a uniform load over multiple spans. Structural and Multidisciplinary Optimization, 52, 1041–1050.MathSciNetCrossRefGoogle Scholar
  46. Prager, W. (1978a). Optimal layout of trusses of finite number of joints. Journal of the Mechanics and Physics of Solids, 26, 241–250.CrossRefGoogle Scholar
  47. Prager, W. (1978b). Nearly optimal design of trusses. Computers and Structures, 8, 451–454.CrossRefGoogle Scholar
  48. Prager, W. (1985). Optimal design of trusses. In M. Save & W. Prager (Eds.), Structural optimization (Vol. 1, pp. 121–151). Optimality criteria. New York and London: Plenum Press.CrossRefGoogle Scholar
  49. Rojas, C. O., Bravo, J. C., & Espi, M. V. (2015). Near-optimal solutions for two point loads between two supports. Structural and Multidisciplinary Optimization, 52, 663–675.MathSciNetCrossRefGoogle Scholar
  50. Rozvany, G. I. N. (2010). On symmetry and non-uniqueness in exact topology optimization. Structural and Multidisciplinary Optimization, 43, 297–317.MathSciNetCrossRefGoogle Scholar
  51. Rozvany, G. I. N., & Sokół, T. (2012). Exact truss topology optimization: Allowance for support costs and different permissible stresses in tension and compression - Extensions of a classical solution by Michell. Structural and Multidisciplinary Optimization, 45(3), 367–376.MathSciNetCrossRefGoogle Scholar
  52. Sokół, T. (2011). A 99 line code for discretized Michell truss optimization written in Mathematica. Structural and Multidisciplinary Optimization, 43(2), 181–190.CrossRefGoogle Scholar
  53. Sokół, T. (2014). Multi-load truss topology optimization using the adaptive ground structure approach. In T. Łodygowski, J. Rakowski, & P. Litewka (Eds.), Recent advances in computational mechanics (pp. 9–16). London: CRC Press.CrossRefGoogle Scholar
  54. Sokół, T. (2016). A new adaptive ground structure method for multi-load spatial Michell structures. In M. Kleiber, T. Burczyski, K. Wilde, J. Gorski, K. Winkelmann, & T. Smakosz (Eds.), Advances in Mechanics: Theoretical, Computational and Interdisciplinary Issues (pp. 525–528). London: CRC Press.CrossRefGoogle Scholar
  55. Sokół, T. (2017). On the numerical approximation of Michell trusses and the improved ground structure method. In Proceedings of 12th World Congress on Structural and Multidisciplinary Optimisation, 05–09 June 2017, Braunschweig, Germany.Google Scholar
  56. Sokół, T., & Lewiński, T. (2010). On the solution of the three forces problem and its application to optimal designing of a class of symmetric plane frameworks of least weight. Structural and Multidisciplinary Optimization, 42(6), 835–853.MathSciNetCrossRefGoogle Scholar
  57. Sokół, T., & Lewiński, T. (2011a). Optimal design of a class of symmetric plane frameworks of least weight. Structural and Multidisciplinary Optimization, 44(5), 729–734.MathSciNetCrossRefGoogle Scholar
  58. Sokół, T., & Lewiński, T. (2011b). On the three forces problem in truss topology optimization. Analytical and numerical solutions. In 9th World Congress on Structural and Multidisciplinary Optimization, Book of Abstracts and CD-ROM (p. 76).Google Scholar
  59. Sokół, T., & Rozvany, G. I. N. (2012). New analytical benchmarks for topology optimization and their implications. Part I: Bi-symmetric trusses with two point loads between supports. Structural and Multidisciplinary Optimization, 46(4), 477–486.MathSciNetCrossRefGoogle Scholar
  60. Sokół, T., & Lewiński, T. (2016a). Simply supported Michell trusses generated by a lateral point load. Structural and Multidisciplinary Optimization, 54(5), 1209–1224.MathSciNetCrossRefGoogle Scholar
  61. Sokół, T., & Lewiński, T. (2016b). Solution of the three forces problem in a case of two forces being mutually orthogonal. Engineering Transactions, 64(4), 485–491.Google Scholar
  62. Tyas, A., Pichugin, A.V., & Gilbert, M. (2011). Optimum structure to carry a uniform load between pinned supports: exact analytical solution. Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 467(2128), 1101–1120.MathSciNetCrossRefGoogle Scholar
  63. Watson, G. (1966). Theory of Bessel functions. Cambridge: Cambridge University Press.zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations