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2019 | OriginalPaper | Buchkapitel

4. Constructing Michell Structures in Plane. Single Load Case

verfasst von : Tomasz Lewiński, Tomasz Sokół, Cezary Graczykowski

Erschienen in: Michell Structures

Verlag: Springer International Publishing

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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. 4.1–4.4. 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.

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Vorheriges Kapitel Theory of Michell Structures. Single Load Case

Nächstes Kapitel Selected Classes of Spatial Michell’s Structures

Carathéodory, C., & Schmidt, E. (1923). Über die Hencky-Prandtlschen Kurven. Zeitschrift für Angewandte Mathematik und Mechanik, 3, 468–475.View Article
Chakrabarty, J. (2006). Theory of plasticity (3rd ed.). Oxford, UK: Butterworth-Heinemann.MATH
Chan, A. S. L. (1960). The design of Michell optimum structures. Cranfield: The College of Aeronautics (Report 142).
Chan, H. S. Y. (1963). Optimum Michell frameworks for three parallel forces. Cranfield: The College of Aeronautics (Report AERO 167).
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).
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)).
Chan, H. S. Y. (1967). Half-plane slip-line fields and Michell structures. The Quarterly Journal of Mechanics and Applied Mathematics, 20, 453–469.View Article
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.View Article
Cox, H. L. (1965). The design of structures of least weight. Oxford: Pergamon Press.View Article
Darwich, W., Gilbert, M., & Tyas, A. (2010). Optimum structure to carry a uniform load between pinned supports. Structural and Multidisciplinary Optimization, 42, 33–42.View Article
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.View Article
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.MathSciNetView Article
Fung, Y. C. (1965). Foundations of solid mechanics. New Jersey: Prentice Hall.
Geiringer, H. (1937). Fondements mathématiques de la theorie des corps plastiques isotropes (Vol. 86). Mémorial des sciences mathématiques. Paris: Gauthier-Villard.
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.MathSciNetMATH
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.MathSciNetView Article
Graczykowski, C., & Lewiński, T. (2006b). Michell cantilevers constructed within trapezoidal domains - Part II: Virtual displacement fields. Structural and Multidisciplinary Optimization, 32(6), 463–471.MathSciNetView Article
Graczykowski, C., & Lewiński, T. (2007a). Michell cantilevers constructed within trapezoidal domains - Part III: Force fields. Structural and Multidisciplinary Optimization, 33(1), 27–46.MathSciNetMATH
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.MathSciNetView Article
Graczykowski, C., & Lewiński, T. (2010). Michell cantilevers constructed within a halfstrip. Tabulation of selected benchmark results. Structural and Multidisciplinary Optimization, 42(6), 869–877.View Article
Green, A. E., & Zerna, W. (1968). Theoretical elasticity. Oxford: Clarendon Press.MATH
Hemp, W. S. (1973). Optimum structures. Oxford: Clarendon Press.
Hemp, W. S. (1975). Michell framework for uniform load between fixed supports. Engineering Optimization, 1, 61–69.View Article
Hill, R. (1950). The mathematical theory of plasticity. Oxford: Clarendon Press.MATH
Hu, T.-C., & Shield, T. T. (1961). Minimum weight design of discs. Zeitschrift für angewandte Mathematik und Physik, 12, 414–433.View Article
Jacot, B. P., & Mueller, C. T. (2017). A strain tensor method for three-dimensional Michell structures. Structural and Multidisciplinary Optimization, 55, 1819–1829.MathSciNetView Article
Lewiński, T. (2004). Michell structures formed on surfaces of revolution. Structural and Multidisciplinary Optimization, 28(1), 20–30.MathSciNetView Article
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.MathSciNetView Article
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.MathSciNetView Article
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.MathSciNetView Article
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.View Article
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.View Article
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.MathSciNetView Article
Melchers, R. E. (2005). On extending the range of Michell-like optimal topology structures. Structural and Multidisciplinary Optimization, 29, 85–92.View Article
Mazurek, A. (2012). Geometrical aspects of optimum truss like structures for three-force problem. Structural and Multidisciplinary Optimization, 45, 21–32.MathSciNetView Article
Mazurek, A., Baker, W., & Tort, C. (2011). Geometrical aspects of optimum truss like structures. Structural and Multidisciplinary Optimization, 43, 231–242.View Article
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.
Mazurkiewicz, Z. E. (2004). Thin elastic shells. Linear theory. Warszawa (in Polish): Oficyna Wydawnicza PW.
McConnel, R. E. (1974). Least-weight frameworks for loads across span. Journal of the Engineering Mechanics Division, 100(5), 885–901.
Michell, A. G. M. (1904). The limits of economy of material in frame structures. Philosophical Magazine, 8(47), 589–597.MATH
Novozhilov, V. V. (1970). Thin shell theory. Groningen: Walters-Nordhoff.MATH
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.MathSciNetView Article
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.
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.MathSciNetView Article
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.MathSciNetView Article
Prager, W. (1978a). Optimal layout of trusses of finite number of joints. Journal of the Mechanics and Physics of Solids, 26, 241–250.View Article
Prager, W. (1978b). Nearly optimal design of trusses. Computers and Structures, 8, 451–454.View Article
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.View Article
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.MathSciNetView Article
Rozvany, G. I. N. (2010). On symmetry and non-uniqueness in exact topology optimization. Structural and Multidisciplinary Optimization, 43, 297–317.MathSciNetView Article
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.MathSciNetView Article
Sokół, T. (2011). A 99 line code for discretized Michell truss optimization written in Mathematica. Structural and Multidisciplinary Optimization, 43(2), 181–190.View Article
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.View Article
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.View Article
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.
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.MathSciNetView Article
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.MathSciNetView Article
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).
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.MathSciNetView Article
Sokół, T., & Lewiński, T. (2016a). Simply supported Michell trusses generated by a lateral point load. Structural and Multidisciplinary Optimization, 54(5), 1209–1224.MathSciNetView Article
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.
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.MathSciNetView Article
Watson, G. (1966). Theory of Bessel functions. Cambridge: Cambridge University Press.MATH

Titel: Constructing Michell Structures in Plane. Single Load Case
verfasst von: Tomasz Lewiński
Tomasz Sokół
Cezary Graczykowski
Verlag: Springer International Publishing
Buch: Michell Structures
Print ISBN: 978-3-319-95179-9

Electronic ISBN: 978-3-319-95180-5

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-319-95180-5_4

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