Structural and Multidisciplinary Optimization

, Volume 52, Issue 4, pp 677–694 | Cite as

Rationalization of trusses generated via layout optimization



Numerical layout optimization provides a computationally efficient and generally applicable means of identifying the optimal arrangement of bars in a truss. When the plastic layout optimization formulation is used, a wide variety of problem types can be solved using linear programming. However, the solutions obtained are frequently quite complex, particularly when fine numerical discretizations are employed. To address this, the efficacy of two rationalization techniques are explored in this paper: (i) introduction of ‘joint lengths’, and (ii) application of geometry optimization. In the former case this involves the use of a modified layout optimization formulation, which remains linear, whilst in the latter case a non-linear optimization post-processing step, involving adjusting the locations of nodes in the layout optimized solution, is undertaken. The two rationalization techniques are applied to example problems involving both point and distributed loads, self-weight and multiple load cases. It is demonstrated that the introduction of joint lengths reduces structural complexity at negligible computational cost, though generally leads to increased volumes. Conversely, the use of geometry optimization carries a computational cost but is effective in reducing both structural complexity and the computed volume.


Truss Layout optimization Geometry optimization Multiple load cases 


  1. Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33:285–304. doi: 10.1007/s00158-006-0092-0 MATHMathSciNetCrossRefGoogle Scholar
  2. Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, BerlinGoogle Scholar
  3. Bendsøe MP, Ben-Tal A, Zowe J (1994) Optimization methods for truss geometry and topology design. Struct Optim 7:141–159. doi: 10.1007/BF01742459 CrossRefGoogle Scholar
  4. Chan ASL (1962) The design of Michell optimum structures. Tech. Rep. 3303, Aeronautical Research Council Reports and Memoranda, LondonGoogle Scholar
  5. Czarnecki S (2003) Compliance optimization of the truss structures. Comput Assist Mech Eng Sci 10:117–137MATHGoogle Scholar
  6. Darwich W, Gilbert M, Tyas A (2010) Optimum structure to carry a uniform load between pinned supports. Struct Multidiscip Optim 42:33–42. doi: 10.1007/s00158-009-0467-0 CrossRefGoogle Scholar
  7. Descamps B, Filomeno Coelho R (2013) A lower-bound formulation for the geometry and topology optimization of truss structures under multiple loading. Struct Multidiscip Optim 48(1):49–58. doi: 10.1007/s00158-012-0876-3 MATHMathSciNetCrossRefGoogle Scholar
  8. Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. Journal de Mècanique 3:25–52Google Scholar
  9. Gil L, Andreu A (2001) Shape and cross-section optimisation of a truss structure. Comput Struct 79:681–689CrossRefGoogle Scholar
  10. Gilbert M, Tyas A (2003) Layout optimization of large-scale pin-jointed frames. Eng Comput 20(8):1044–1064MATHCrossRefGoogle Scholar
  11. Graczykowski C, Lewiński T (2010) Michell cantilevers constructed within a half strip. Tabulation of selected benchmark results. Struct Multidiscip Optim 42(6):869–877. doi: 10.1007/s00158-010-0525-7 CrossRefGoogle Scholar
  12. Hemp W S (1973) Optimum structures. Clarendon, OxfordGoogle Scholar
  13. Hemp WS (1974) Michell frameworks for uniform load between fixed supports. Eng Optim 1:61–69CrossRefGoogle Scholar
  14. Lewiński T (2004) Michell structures formed on surfaces of revolution. Struct Multidiscip Optim 28(1):20–30. doi: 10.1007/s00158-004-0419-7 MathSciNetCrossRefGoogle Scholar
  15. Lewiński T (2005) Personal communicationGoogle Scholar
  16. Lewiński T, Zhou M, Rozvany GIN (1994) Extended exact solutions for least-weight truss layouts - Part I: cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375– 398MATHCrossRefGoogle Scholar
  17. Martínez P, Martí P, Querin OM (2007) Growth method for size, topology, and geometry optimization of truss structures. Struct Multidiscip Optim 33(1):13–26. doi: 10.1007/s00158-006-0043-9 CrossRefGoogle Scholar
  18. Mazurek A (2012) Geometrical aspects of optimum truss like structures for three-force problem. Struct Multidiscip Optim 45:21–32. doi: 10.1007/s00158-011-0679-y MATHMathSciNetCrossRefGoogle Scholar
  19. Mazurek A, Baker WF, Tort C (2011) Geometrical aspects of optimum truss like structures. Struct Multidiscip Optim 43:231–242. doi: 10.1007/s00158-010-0559-x CrossRefGoogle Scholar
  20. Michell AGM (1904) The limits of economy of material in frame-structures. Phil Mag 8:589–597MATHCrossRefGoogle Scholar
  21. MOSEK (2011) The MOSEK optimization tools manual.,
  22. Nagtegaal JC, Prager W (1973) Optimal layout of a truss for alternative loads. Int J Mech Sci 15:585–592CrossRefGoogle Scholar
  23. Parkes EW (1975) Joints in optimum frameworks. Int J Solids Struct 11(9):1017–1022MATHCrossRefGoogle Scholar
  24. Pichugin AV, Tyas A, Gilbert M (2012) On the optimality of Hemp’s arch with vertical hangers. Struct Multidiscip Optim 46:17–25. doi: 10.1007/s00158-012-0769-5 MATHMathSciNetCrossRefGoogle Scholar
  25. Prager W (1978) Nearly optimal design of trusses. Comput Struct 8:451–454MATHCrossRefGoogle Scholar
  26. Pritchard T, Gilbert M, Tyas A (2005) Plastic layout optimization of large-scale frameworks subject to multiple load cases, member self-weight and with joint length penalties. Proc of 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, BrazilGoogle Scholar
  27. Sokół T, Rozvany GIN (2012) New analytical benchmarks for topology optimization and their implications. Part I: bi-symmetric trusses with two point loads between supports. Struct Multidiscip Optim 46(4):477–486. doi: 10.1007/s00158-012-0786-4 MathSciNetCrossRefGoogle Scholar
  28. Spillers WR, Lev O (1971) Design for two loading conditions. Int J Solids Struct 7(9):1261–1267MATHCrossRefGoogle Scholar
  29. Vigerske S, Wachter A (2013) Introduction to IPOPT: a tutorial for downloading, installing, and using IPOPT.

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Civil and Structural EngineeringThe University of SheffieldSheffieldUK

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