Advertisement

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

  • Tomasz Lewiński
  • Tomasz Sokół
  • Cezary Graczykowski
Chapter

Abstract

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.

References

  1. Achtziger, W. (1997). Topology optimization of discrete structures: An introduction in view of computational and nonsmooth aspects. In G. I. N. Rozvany (Ed.), Topology optimization in structural mechanics (Vol. 374, pp. 57–100), CISM courses and lectures. Wien New York: Springer.CrossRefGoogle Scholar
  2. Beghini, A., & Baker, W. F. (2015). On the layout of a least weight multiple span structure with uniform load. Structural and Multidisciplinary Optimization, 52, 447–457.CrossRefGoogle Scholar
  3. Bendsøe, M. P. (1995). Optimization of structural topology, shape and material. Berlin: Springer.CrossRefGoogle Scholar
  4. Bojczuk, D. (1999). Sensitivity analysis and optimization of bar structures. Monographs, Studies, Treatises, Kielce University of Technology, Kielce (in Polish).Google Scholar
  5. Bojczuk, D., & Mróz, Z. (1998). On optimal design of supports in beam and frame structures. Structural Optimization, 16, 47–57.CrossRefGoogle Scholar
  6. Bołbotowski, K. (2018). Theory of grillage optimization – a discrete setting (In preparation).Google Scholar
  7. Bołbotowski, K., He, L., & Gilbert, M. (2018). Design of optimum grillages using layout optimization. Structural and Multidisciplinary Optimization, 58, 851–868.CrossRefGoogle Scholar
  8. Cox, H. L. (1965). The design of structures of least weight. Oxford: Pergamon Press.Google Scholar
  9. Czarnecki, S., & Lewiński, T. (2013). On minimum compliance problems of thin elastic plates of varying thickness. Structural and Multidisciplinary Optimization, 48(1), 17–31.MathSciNetCrossRefGoogle Scholar
  10. Czarnecki, S., & Lewiński, T. (2017a). On material design by the optimal choice of Young’s modulus distribution. International Journal of Solids and Structures, 110–111, 315–331.CrossRefGoogle Scholar
  11. Czarnecki, S., & Lewiński, T. (2017b). Pareto optimal design of non-homogeneous isotropic material properties for the multiple loading conditions. Physica Status Solidi B: Basic Solids State Physics, 254, 1600821.CrossRefGoogle Scholar
  12. Dorn, W., Gomory, R., & Greenberg, M. (1964). Automatic design of optimal structures. Journal de Mécanique, 3, 25–52.Google Scholar
  13. Hemp, W. S. (1973). Optimum structures. Oxford: Clarendon Press.Google Scholar
  14. Heyman, J. (1959). On the absolute minimum weight design of framed structures. The Quarterly Journal of Mechanics and Applied Mathematics, 12, 314–324.MathSciNetCrossRefGoogle Scholar
  15. Gilbert, M., & Tyas, A. (2003). Layout optimization of large-scale pin-jointed frames. Engineering Computations, 20, 1044–1064.CrossRefGoogle Scholar
  16. Kozłowski, W., & Mróz, Z. (1969). Optimal design of solid plates. International Journal of Solids and Structures, 5, 781–794.CrossRefGoogle Scholar
  17. Lowe, P. G., & Melchers, R. E. (1972). On the theory of optimal constant thickness. Fibrereinforcement plates. International Journal of Mechanical Sciences, 14, 311–324.CrossRefGoogle Scholar
  18. Lowe, P. G., & Melchers, R. E. (1973). On the theory of optimal constant thickness, fibrereinforcement plates. II. International Journal of Mechanical Sciences, 15, 157–170.CrossRefGoogle Scholar
  19. Martinez, P., Marti, P., & Querin, Q. M. (2007). Growth method for size, topology, and geometry optimization of truss structures. Structural and Multidisciplinary Optimization, 33, 13–26.CrossRefGoogle Scholar
  20. Maxwell, J. C. (1872). On reciprocal figures, frames and diagrams of forces. Transactions of the Royal Society of Edinburgh, 26, 1–40.CrossRefGoogle Scholar
  21. Mazurek, A. (2012). Geometrical aspects of optimum truss like structures for three-force problem. Structural and Multidisciplinary Optimization, 45, 21–32.MathSciNetCrossRefGoogle Scholar
  22. Mazurek, A., Baker, W., & Tort, C. (2011). Geometrical aspects of optimum truss like structures. Structural and Multidisciplinary Optimization, 43, 231–242.CrossRefGoogle Scholar
  23. Morley, C. T. (1966). The minimum reinforcement of concrete slabs. International Journal of Mechanical Sciences, 8, 305–319.CrossRefGoogle Scholar
  24. Prager, W. (1977). Optimal lauout of contilever trusses. The Journal of Optimization Theory and Applications, 23, 111–117.CrossRefGoogle Scholar
  25. 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
  26. Prager, W. (1978b). Nearly optimal design of trusses. Computers and Structures, 8, 451–454.CrossRefGoogle Scholar
  27. Prager, W. (1985). Optimal design of grillages. In M. Save & W. Prager (Eds.), Structural optimization. Vol 1. Optimality criteria (pp. 153–200). New York and London: Plenum Press.zbMATHGoogle Scholar
  28. Prager, W., & Rozvany, G. I. N. (1977). Optimal layout of grillages. Journal of Structural Mechanics, 5, 1–18.CrossRefGoogle Scholar
  29. Rozvany, G. I. N. (1972). Grillages of maximum strength and maximum stiffness. International Journal of Mechanical Sciences, 14, 651–666.CrossRefGoogle Scholar
  30. Rozvany, G. I. N. (1976). Optimal design of flexural systems. London: Pergamon Press.Google Scholar
  31. Rozvany, G. I. N. (1989). Structural design via optimality criteria. The Netherlands: Kluwer Academic Publishers Dordrecht.CrossRefGoogle Scholar
  32. Rozvany, G. I. N., & Sokół, T. (2014). Validation of numerical methods by analytical benchmarks, and verification of exact solutions by numerical methods. In G.I.N. Rozvany & T. Lewiński (Eds.), Topology Optimization in Structural and Continuum Mechanics (Vol. 549), CISM International Centre for Mechanical Sciences. Vienna: Springer.CrossRefGoogle Scholar
  33. Sokół, T. (2011a). A 99 line code for discretized Michell truss optimization written in Mathematica. Structural and Multidisciplinary Optimization, 43, 181–190.CrossRefGoogle Scholar
  34. Sokół, T. (2011b). Topology optimization of large-scale trusses using ground structure approach with selective subsets of active bars. In A. Borkowski, T. Lewiński & G. Dzierżanowski (Eds.), 19th International Conference on Computer Methods in Mechanics CMM-2011 (pp. 457–458), Warsaw, 9–12 May 2011.Google Scholar
  35. Sokół, T. (2013). Numerical approximations of exact Michell solutions using the adaptive ground structure approach. In S. Jemioło & M. Lutomirska (Eds.), Mechanics and Materials, Sec. VI (pp. 87–98). Warsaw University of Technology.Google Scholar
  36. Steven, G., Querin, O., & Xie, M. (2000). Evolutionary structural optimization (ESO) for combined topology and size optimization of discrete structures. Computer Methods in Applied Mechanics and Engineering, 188, 743–754.CrossRefGoogle Scholar
  37. Stolpe, M. (2010). On some fundamental properties of structural topology optimization problems. Structural and Multidisciplinary Optimization, 41, 661–670.CrossRefGoogle Scholar
  38. Strang, G., & Kohn, R. V. (1983). Hencky-Prandtl nets and constrained Michell trusses. Computer Methods in Applied Mechanics and Engineering, 36, 207–222.MathSciNetCrossRefGoogle Scholar
  39. Zegard, T., & Paulino, G. H. (2013). Truss layout optimization within a continuum. Structural and Multidisciplinary Optimization, 48, 1–16.CrossRefGoogle Scholar
  40. Zegard, T., & Paulino, G. H. (2014). GRAND-Ground structure based topology optimization for arbitrary 2D domains using MatLab. Structural and Multidisciplinary Optimization, 50, 861–882.MathSciNetCrossRefGoogle Scholar
  41. 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

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

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

Personalised recommendations