Advertisement

A general boundary approach to the construction of Michell truss structures

  • P. Dewhurst
  • N. Fang
  • S. Srithongchai
RESEARCH PAPER

Abstract

The goal of the present study is to provide a building block approach which will enable the synthesis of new Michell truss structure solutions. Curved support boundaries for Michell truss structures are categorized into four types. Each type is graphically illustrated as a simple example structure. A general matrix operator method is developed to solve the layout of each type. Numerical solutions that use the matrix method are compared with analytical solutions in a case study that comprises complimentary logarithmic spirals on a circular arc boundary. To illustrate the applications of this study, numerical layout solutions on a circular support boundary are explored that produce a family of globally-optimal Michell cantilever solutions for the support of a distributed load along a straight cantilever flange.

Keywords

Michell truss structures Curved boundary Matrix operators 

References

  1. Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224CrossRefMathSciNetGoogle Scholar
  2. Bendsoe MP, Sigmund O (2004) Topology optimization: theory, methods and applications. Springer, BerlinzbMATHGoogle Scholar
  3. Chakrabarty J (2006) Theory of plasticity. Elsevier Butterworth-Heinemann, AmsterdamGoogle Scholar
  4. Chan ASL (1960) The design of Michell optimum structures. The College of Aeronautics, Cranfield. Report No. 142Google Scholar
  5. Cox HL (1958) The theory of design. Aeronautical Research Council 19791Google Scholar
  6. Dewhurst P (1985) A general matrix operators for linear boundary value problems in slip-line filed theory. Int J Numer Methods Eng 21:169–182zbMATHCrossRefGoogle Scholar
  7. Dewhurst P (2001) Analytical solutions and numerical procedures for minimum-weight Michell structures. J Mech Phys Solids 49:445–467zbMATHCrossRefGoogle Scholar
  8. Dewhurst P (2005) A general optimality criterion for strength and stiffness of dual-material-property structures. Int J Mech Sci 47:293–302CrossRefGoogle Scholar
  9. Dewhurst P, Collins IF (1973) A matrix technique for constructing slip-line field solutions to a class of plane strain plasticity problems. Int J Numer Methods Eng 7:357–378zbMATHCrossRefGoogle Scholar
  10. Dewhurst P, Srithongchai S (2005) An investigation of minimum-weight dual material symmetrically loaded wheels and torsion arms. J Appl Mech 72:196–202zbMATHCrossRefGoogle Scholar
  11. Ewing DJF (1967) A series-method for constructing plastic slipline fields. J Mech Phys Solids 15:105–114zbMATHCrossRefGoogle Scholar
  12. Fang N (2003) Slip-line modeling of machining with a rounded-edge tool, part I: new model and theory. J Mech Phys Solids 51:715–742zbMATHCrossRefGoogle Scholar
  13. Fang N, Dewhurst P (2005) Slip-line modeling of built-up edge deformation in machining. Int J Mech Sci 47:1079–1098CrossRefGoogle Scholar
  14. Graczykowski C, Lewinski T (2006) Michell cantilevers constructed within trapezoidal domains. Part I. Geometry of Hencky nets. Struct Multidiscipl Optim 32:347–368CrossRefMathSciNetGoogle Scholar
  15. Graczykowski C, Lewinski T (2007a) Michell cantilevers constructed within trapezoidal domains. Part III: force fields. Struct Multidiscipl Optim 33:1–19CrossRefGoogle Scholar
  16. Graczykowski C, Lewinski 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. Struct Multidiscipl Optim 33:113–129CrossRefMathSciNetGoogle Scholar
  17. Hemp WS (1958) Theory of the structural design. The College of Aeronautics, Cranfield. Report No.115Google Scholar
  18. Hemp WS (1973) Optimum structures. Clarendon, OxfordGoogle Scholar
  19. Hill R (1950) The mathematical theory of plasticity. Clarendon, OxfordzbMATHGoogle Scholar
  20. Johnson W, Sowerby R, Venter RD (1982) Plane strain slip line fields for metal deformation processes. Pergamon, OxfordzbMATHGoogle Scholar
  21. Lewinski T, Rozvany GIN (2007) Exact analytical solutions for some popular benchmark problems in topology optimization II: three-sided polygonal supports. Struct Multidiscipl Optim 33:337–349CrossRefMathSciNetGoogle Scholar
  22. Lewinski T, Rozvany GIN (2008a) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains. Struct Multidiscipl Optim 35:165–174CrossRefMathSciNetGoogle Scholar
  23. Lewinski T, Rozvany GIN (2008b) Exact analytical solutions for some popular benchmark problems in topology optimization IV: square-shaped line support. Struct Multidiscipl Optim 36:143–158CrossRefMathSciNetGoogle Scholar
  24. Lewinski T, Zhou M, Rozvany GIN (1994) Extended exact solutions for least-weight truss layouts. Part I. Cantilever with a horizontal axis of symmetry. Part II. Unsymmetric cantilevers. Int J Mech Sci 36:375–419zbMATHCrossRefGoogle Scholar
  25. Martinez P, Marti P, Querin OM (2007) Growth method for size, topology, and geometry optimization of truss structures. Struct Multidiscipl Optim 33:13–26CrossRefGoogle Scholar
  26. Michell AGM (1904) The limits of economy of material in frame structures. Phil Mag 8:589–597Google Scholar
  27. Prager W (1958) A problem of optimal design. In: Proceedings of the union of theoretical and applied mechanics, WarsawGoogle Scholar
  28. Prager W, Rozvany GIN (1977) Optimization of the structural geometry. In: Bednarek AR, Cesari L (eds) Dynamical systems (proc. int. conf. Gainesville Florida). Academic, New York, pp 265–293Google Scholar
  29. Rozvany GIN (1996) Some shortcomings in Michell’s truss theory. Struct Optim 12:244–250CrossRefGoogle Scholar
  30. Rozvany GIN, Gollub W (1990) Michell layouts for various combinations of line supports. Part I. Int J Mech Sci 32:1021–1043zbMATHCrossRefGoogle Scholar
  31. Rozvany GIN, Bendsoe MP, Kirsch U (1995) Layout optimization of structures. Appl Mech Rev 48:41–118CrossRefGoogle Scholar
  32. Rozvany GIN, Gollub W, Zhou M (1997) Exact Michell layouts for various combinations of line supports. Part II. Struct Optim 14:138–149CrossRefGoogle Scholar
  33. Rozvany GIN, Querin OM, Logo J, Pomezanski V (2006) Exact analytical theory of topology optimization with some pre-existing members or elements. Struct Multidiscipl Optim 31:373–377CrossRefGoogle Scholar
  34. Srithongchai S, Dewhurst P (2003) Comparisons of optimality criteria for minimum-weight dual material structures. Int J Mech Sci 45:1781–1797zbMATHCrossRefGoogle Scholar
  35. Taggart DG, Dewhurst P (2008) A novel topological optimization method. In: 8th world congress on computational mechanics (WCCM8), Venice, ItalyGoogle Scholar
  36. Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896CrossRefGoogle Scholar
  37. Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, LondonzbMATHGoogle Scholar
  38. Zhou M, Rozvany GIN (2001) On the validity of ESO type methods in topology optimization. Struct Multidiscipl Optim 21:80–83CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringUniversity of Rhode IslandKingstonUSA
  2. 2.College of EngineeringUtah State UniversityLoganUSA

Personalised recommendations