Advertisement

Optimal design of cantilevered elastica for minimum tip deflection under self-weight

  • Raymond H. Plaut
  • Lawrence N. VirginEmail author
Research Paper

Abstract

The optimal distribution of material to minimize the vertical deflection of the free end of a horizontal cantilever is determined. The beam is only subjected to its own weight. Large deflections are considered, and the structure is modeled as an inextensible elastica. A minimum-area constraint is included, and is active in a region near the tip. After the problem is formulated, numerical results are obtained with the use of a shooting method. The moment of inertia is assumed to be proportional to the area or its square or cube. The results depend on this relationship, the minimum-area constraint, and a nondimensional parameter depending on the beam’s density, length, and modulus of elasticity. In the numerical results presented, if the minimum area is 1/20 of the area of the uniform beam, the tip deflection for the optimal design is 78–89% smaller than that for the uniform beam. An experiment is conducted and the data are in close agreement with the numerical results.

Keywords

Optimal beam Cantilever Self-weight Minimum deflection Large deflections Elastica 

Notes

Acknowledgements

The authors are grateful to the reviewers for their helpful comments.

References

  1. Ansola R, Canales J, Tárrago JA (2006) An efficient sensitivity computation strategy for the evolutionary structural optimization (eso) of continuum structures subjected to self-weight loads. Finite Elem Anal Des 2:1220–1230CrossRefGoogle Scholar
  2. Atanackovic TM (2006) Optimal shape of column with own weight: bi and single modal optimization. Meccanica 41:173–196MathSciNetzbMATHCrossRefGoogle Scholar
  3. Atanackovic TM (2007) Optimal shape of a strongest inverted column. J Comput Appl Math 203:209–218MathSciNetzbMATHCrossRefGoogle Scholar
  4. Atanackovic TM, Glavardanov VB (2004) Optimal shape of a heavy compressed column. Struct Multidisc Optim 28:388–396MathSciNetCrossRefGoogle Scholar
  5. Bahder TB (1995) Mathematica for scientists and engineers. Addison-Wesley, ReadingGoogle Scholar
  6. Bellido JC, Donoso A (2009) Two-material optimal design for nonlinear elastica. Appl Math Lett 22:459–463MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bruyneel M, Duysinx P (2005) Note on topology optimization of continuum structures including self-weight. Struct Multidisc Optim 29:245–256CrossRefGoogle Scholar
  8. Chern J-M (1971) Optimal structural design for given deflection in presence of body forces. Int J Solids Struct 7:373–382zbMATHCrossRefGoogle Scholar
  9. Cox SJ, McCarthy CM (1998) The shape of the tallest column. SIAM J Math Anal 29:547–554MathSciNetzbMATHCrossRefGoogle Scholar
  10. Dixon LW (1967) Pontryagin’s maximum principle applied to the profile of a beam. J R Aeronaut Soc 71:513–515Google Scholar
  11. Farjoun Y, Neu J (2005) The tallest column—a dynamical system approach using a symmetry solution. Stud Appl Math 115:319–337MathSciNetzbMATHCrossRefGoogle Scholar
  12. Hornbuckle JC (1974) On the automated optimal design of constrained structures. PhD thesis, University of FloridaGoogle Scholar
  13. Karihaloo BL (1987) On minimax optimum design of flexural members in presence of self-weight. Mech Struct Mach 15:17–28CrossRefGoogle Scholar
  14. Karihaloo BL, Hemp WS (1983) Maximum strength/stiffness design of structural members in presence of self-weight. Proc R Soc Lond A 389:119–132CrossRefGoogle Scholar
  15. Keller JB, Niordson FI (1966) The tallest column. J Math Mech 16:433–446MathSciNetzbMATHGoogle Scholar
  16. McCarthy CM (1999) The tallest column—optimality revisited. J Comput Appl Math 101:27–37MathSciNetzbMATHCrossRefGoogle Scholar
  17. Novakovic BN, Atanackovic TM (2009) Optimal shape of a heavy elastic rod loaded with a tip-concentrated force against lateral buckling. Int J Struct Stab Dyn 9:383–390MathSciNetCrossRefGoogle Scholar
  18. Sadiku S (2008) Buckling load optimization for heavy elastic columns: a perturbation approach. Struct Multidisc Optim 35:447–452CrossRefGoogle Scholar
  19. Stadler W (1978) Natural structural shapes (the static case). Q J Mech Appl Math 31:169–217MathSciNetzbMATHCrossRefGoogle Scholar
  20. Tadjbakhsh I (1968) An optimum design problem for the nonlinear elastica. SIAM J Appl Math 16:964–972zbMATHCrossRefGoogle Scholar
  21. Virgin LN (2007) Vibration of axially loaded structures. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Mechanical Engineering and Materials ScienceDuke UniversityDurhamUSA

Personalised recommendations