arrow
Volume 13, Issue 6
Curve Optimization of Tapered Cantilever Beams Under Tip Loads

Minyi Zhu, Guobin Gong & Jun Xia

Adv. Appl. Math. Mech., 13 (2021), pp. 1485-1500.

Published online: 2021-08

Export citation
  • Abstract

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 $\rho/E$ (density against Elastic Modulus) of the material should be as small as possible, while the ratio $h^2/L^4$ 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.

  • AMS Subject Headings

49K35, 49S05, 74P05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{AAMM-13-1485, author = {Zhu , MinyiGong , Guobin and Xia , Jun}, title = {Curve Optimization of Tapered Cantilever Beams Under Tip Loads}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {6}, pages = {1485--1500}, abstract = {

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 $\rho/E$ (density against Elastic Modulus) of the material should be as small as possible, while the ratio $h^2/L^4$ 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0196}, url = {http://global-sci.org/intro/article_detail/aamm/19431.html} }
TY - JOUR T1 - Curve Optimization of Tapered Cantilever Beams Under Tip Loads AU - Zhu , Minyi AU - Gong , Guobin AU - Xia , Jun JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1485 EP - 1500 PY - 2021 DA - 2021/08 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0196 UR - https://global-sci.org/intro/article_detail/aamm/19431.html KW - Tapered beam, Euler-Bernoulli beam, Timoshenko beam, variational principle, isoperimetric problem, curve optimization. AB -

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 $\rho/E$ (density against Elastic Modulus) of the material should be as small as possible, while the ratio $h^2/L^4$ 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.

Minyi Zhu, Guobin Gong & Jun Xia. (1970). Curve Optimization of Tapered Cantilever Beams Under Tip Loads. Advances in Applied Mathematics and Mechanics. 13 (6). 1485-1500. doi:10.4208/aamm.OA-2020-0196
Copy to clipboard
The citation has been copied to your clipboard