Abstract
Optimal shape of an elastic rod loaded by extensional force is determined. It is assumed that the rod is described by a classical Bernoulli–Euler rod theory. The optimality conditions are obtained by using Pontriyagin's maximum principle. It is shown that the optimal shape (cross-sectional area as a function of an arc length) is determined from the solution of a nonlinear second-order differential equation. The solution of this equation is given in the closed form. It is shown that for the same buckling force, the savings of the material are of the order of 30%. An interesting feature of the problem is that for certain values of parameters, there is no optimal solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alekseev VM, Tihomirov VM, Fomin SV (1979) Optimal control. Nauka, Moscow (in Russian)
Antman SS (1995) Nonlinear problems of elasticity. Springer, Berlin Heidelberg New York
Atanackovic TM (1997) Stability theory of elastic rods. World Scientific, River Edge, NJ
Atanackovic TM, Djukic DS (1989) Buckling by extension: stability boundary and post-critical behavior. Dyn Stab Syst 4:81–94
Atanackovic TM, Djukic DS, Strauss AM (1992) Instability by extension of an extensible elastic column. ZAMM 72:209–218
Bizeno CB, Grammel R (1953) Technische dynamik. Springer, Berlin Heidelberg New York
Chow S-N, Hale JK (1982) Methods of bifurcation theory. Springer, Berlin Heidelberg New York
Greenev VB, Filippov AP (1979) Optimization of rods. Naukova Dumka, Kiev
Sage AP, White CC (1977) Optimum control systems. Prentice-Hall, New Jersey
Vujanovic BD, Atanackovic TM (2004) An introduction to modern variational methods in mechanics and engineering. Birkhaüser, Boston, MA
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jelicic, Z.D., Atanackovic, T.M. On an optimization problem for elastic rods. Struct Multidisc Optim 32, 59–64 (2006). https://doi.org/10.1007/s00158-005-0583-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-005-0583-4