Structural and Multidisciplinary Optimization

, Volume 58, Issue 4, pp 1769–1782 | Cite as

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


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.


Curved beam Induced force Shape optimization Rib Instability 



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


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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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