Curve optimization of tapered cantilever beams under tip loads

Minyi Zhu, Guobin Gong*, Jun Xia

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The study adopts the variational method for analyzing the cantilever tapered beams under a tip load as well as a definite end displacement, and further determining the optimized shapes and materials that can minimize the weights. Two types of beams are taken into account, i.e., the Euler-Bernoulli beam without considering shear deformation and the Timoshenko beam with shear deformation. By using the energy theorem and the reference of isoperimetric problem, the width variation curves and the corresponding minimum masses are derived for both beam types. The optimized curve of beam width for the Euler-Bernoulli beam is found to be a linear function, but nonlinear for the Timoshenko beam. It is also found that the optimized curve in the Timoshenko beam case starts from non-zero at the tip end, but its tendency gradually approaches the one of the Euler-Bernoulli beam. The results indicate that with the increase of the Poisson's ratio, the required minimum mass of the beam will increase no matter how the material changes, suggesting that the optimized mass for the case of Euler-Bernoulli beam is the lower boundary limit which the Timoshenko case cannot go beyond. Furthermore, the ratio r/E (density against Elastic Modulus) of the material should be as small as possible, while the ratio h2/L4 of the beam should be as large as possible in order to minimize the mass for the case of Euler-Bernoulli beam, of which the conclusion is extended to be applicable for the case of Timoshenko beam. In addition, the optimized curves for Euler-Bernoulli beam types are all found to be power functions of length for both tip point load cases and uniform load cases.

Original languageEnglish
Pages (from-to)1485-1500
Number of pages16
JournalAdvances in Applied Mathematics and Mechanics
Issue number6
Publication statusPublished - 2021


  • Curve optimization
  • Euler-Bernoulli beam
  • Isoperimetric problem
  • Tapered beam
  • Timoshenko beam
  • Variational principle


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