Response of shape optimization of thin-walled curved beam and rib formation from unstable structure growth in optimization

  • Yoshiki Fukada
  • Haruki Minagawa
  • Chikara Nakazato
  • Takaaki Nagatani
RESEARCH PAPER
  • 73 Downloads

Abstract

A thin-walled curved beam is a complex structure. Sectional deformation occurs due to induced out-of-plane force when the beam is bent. Bending stiffness is significantly lowered due to this deformation. Installation of ribs to support this induced force is often an effective countermeasure to ensure stiffness. This study examined shape optimization of an I-sectional curved beam. Ribbed structures were successfully created from non-ribbed structures by adding humps to the initial structure. It was discovered that instability of the shape optimization occurs under the influence of the induced force. Here, ‘instability’ refers to the amplification of initial perturbations similar to buckling phenomena. In the present case, the humps grew and formed ribbed structures. The bending stiffness of the ribs was significantly improved. In addition, simple thickening of flange parts also effectively improves the bending stiffness. As these two structural improvements progress simultaneously, branching of the optimization occur. This branching depends on the given volume constraint. A parameter study targeting volume observed branching to ribbed or thickened non-ribbed structures. This instability enables a leap from a non-ribbed to a ribbed structure in the optimization.

Keywords

Curved beam Induced force Shape optimization Rib Instability 

Notes

Acknowledgements

The present work did not receive any specific grants from funding agencies in the public, commercial, or not-for-profit sectors.

References

  1. Akl W, El-Sabbagh A, Baz A (2008) Optimization of the static and dynamic characteristics of plates with isogrid stiffeners. Finite Elem Anal Des 44:513–523CrossRefGoogle Scholar
  2. Anderson CG (1950) Flexural stresses in curved beams of I-and box-section. Proc I Mech E 162(1):295–306CrossRefGoogle Scholar
  3. Azegami H (1994) Solution to domain optimization problems. Trans Jpn Soc Mech Eng, Ser A 60:1479–1486. (in Japanese)CrossRefGoogle Scholar
  4. Azegami H (2004) Solution to boundary shape optimization problems. In: Brebbia CA, de Wilde WP (eds) High performance structures and materials II. WIT Press, Southampton, pp 589–598Google Scholar
  5. Azegami H (2006) Solution to boundary shape identification problems in elliptic boundary value problems using shape derivatives. In: Tanaka M, Dulikravich GS (eds) Inverse problems in engineering mechanics II. Elsevier, Tokyo, pp 277–284Google Scholar
  6. Azegami H (2016) Shape optimization problems (in Japanese). Morikita Publishing Co., Ltd., TokyoMATHGoogle Scholar
  7. Azegami H, Kaizu S, Shimoda M, Katamine E (1997) Irregularity of shape optimization problems and an improvement technique. In: Hernandez S, Brebbia CA (eds) Computer aided optimization design of structures V, vol 28. Computational Mechanics Publications, Southampton, pp 309–326Google Scholar
  8. Azegami H, Takeuchi K (2006) A smoothing method for shape optimization: traction method using the robin condition. Int J Comput Methods 03(1):21–33MathSciNetCrossRefMATHGoogle Scholar
  9. Azegami H, Wu ZC (1996) Domain optimization analysis in linear elastic problems: approach using traction method. JSME Int J, Ser A 39:272–278Google Scholar
  10. Banichuk NV (1990) Introduction to optimization of structures. Springer, New York, pp 32–35CrossRefMATHGoogle Scholar
  11. Bletzinger K-U (2014) A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Struct Multidiscip Optim 49:873–895MathSciNetCrossRefGoogle Scholar
  12. Chancharoen W, Azegami H (2018) Topology optimization of density type for a linear elastic body by using the second derivative of a KS function with respect to von Mises stress. Struct Multidiscip Optim (issued on line:  https://doi.org/10.1007/s00158-018-1937-z)
  13. Fukada Y (2017) Stress redistribution as an effect of non-uniform in-plane laminate stresses in laminate composite plates. Compos Struct 159:505–516CrossRefGoogle Scholar
  14. Fukada Y, Minagawa H, Nakazato C, Nagatani T (2017) Structural deterioration of curved thin-walled structure and recovery by rib installation: verification with structural optimization algorithm. Thin-walled Struct 123:441–451CrossRefGoogle Scholar
  15. Imam MH (1982) Three-dimensional shape optimization. Int J Numer Method Eng 18:661–673CrossRefMATHGoogle Scholar
  16. Ji J, Ding X, Xiong M (2014) Optimal stiffener layout of plate/shell structures by bionic growth method. Comput Struct 135:88–99CrossRefGoogle Scholar
  17. Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: Proceedings of the 1995 IEEE conference on neural networks, IV. Piscataway, pp 1942–1948Google Scholar
  18. Kikuchi N, Chung KY, Torigaki T, Taylor JE (1986) Adaptive finite element methods for shape optimization of linear structures. Comput Methods Appl Mech Eng 57:67–89CrossRefMATHGoogle Scholar
  19. Kristensen ES, Madsen NF (1976) On the optimum shape of fillets in plates subjected to multiple in-plane loading cases. Int J Num Meth Eng 10:1007–1019CrossRefMATHGoogle Scholar
  20. Liu Y, Shimoda M (2014) Parameter-free optimum design method of stiffeners on thin-walled structures. Struct Multidiscip Optim 49:39–47MathSciNetCrossRefGoogle Scholar
  21. Liu Y, Shimoda M (2015) Non-parametric shape optimization method for natural vibration design of stiffened shells. Comput Struct 146:20–31CrossRefGoogle Scholar
  22. Luo J, Gea HC (1998) A systematic topology optimization approach for optimal stiffener design. Struct Multidiscip Optim 16:280–288CrossRefGoogle Scholar
  23. Rothwell A (2000) Explicit formulae for the flange efficiency of curved beams. Thin-Walled Struct 36:155–168CrossRefGoogle Scholar
  24. Schagerl M (2011) Analytical formulas for the effective width caused by bulging and flattening of curved aircraft panels. Proc IME C J Mech Eng Sci 225(10):2399–2412CrossRefGoogle Scholar
  25. Shimoda M, Azegami H, Sakurai T (1997) Numerical solution for min–max problems in shape optimization (minimum design of max. Stress and max. Displacement). Trans Jpn Soc Mech Eng, Ser A 63:158–165. (in Japanese)CrossRefGoogle Scholar
  26. Shimoda M, Tsuji J, Azegami H (2007) Non-parametric shape optimization method for thin-walled structures under strength criterion. WIT Transactions on the Built Environment (Computer Aided Optimum Design in Engineering X) 91:179–188Google Scholar
  27. Shintani K, Azegami H (2014) A design method of beads in shell structure using Non-Parametric shape optimization method. ASME Proceedings 34th Computers and Information in Engineering Conference 1A:17–20Google Scholar
  28. Sun SH, Yu TT, Nguyen TT, Atroshchenko E, Bui TQ (2018) Structural shape optimization by IGABEM and particle swarm optimization algorithm. Eng Ana Bound Elemt 88:26–40MathSciNetCrossRefGoogle Scholar
  29. Taylor JE, Bendosøe MP (1984) An interpretation for min–max structural design problems including a method for relaxing constraints. Int J Solids Struct 20:301–314MathSciNetCrossRefMATHGoogle Scholar
  30. Timoshenko S (1923) Bending stresses in curved tubes of rectangular cross-section. Trans ASME 45:135–140Google Scholar
  31. Timoshenko S (1955) Strength of materials part I elementary, section 84. Bending of curved tubes. Van Nostrand ReinholdGoogle Scholar
  32. Trompette P, Marcelin JL (1987) On the choice of objectives in shape optimization. Eng Opt 11:89–102CrossRefGoogle Scholar
  33. Westrup RW, Silver P (1958) Some effects of curvature on frames. J Aerospace Sci 25(9):567–572CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Toyota Motor CorporationSusono-shiJapan
  2. 2.Quint CorporationTokyoJapan

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