Archive of Applied Mechanics

, Volume 82, Issue 10–11, pp 1303–1311

Shape optimization against buckling of micro- and nano-rods

  • Teodor M. Atanackovic
  • Branislava N. Novakovic
  • Zora Vrcelj
Special Issue


In this paper, we analyze elastic buckling of micro- and nano-rods based on Eringen’s nonlocal elasticity theory. By using the Pontryagin’s maximum principle, we determine optimality condition for a rod simply supported at both ends and loaded with axial compressive force only. Thus, the problem that we treat represents a generalization of the classical Clausen problem formulated for Bernoulli–Euler rod theory. Several concrete examples are treated in details, and the increase in buckling load capacity is determined. In solving the problem numerically, we used a first integral of the resulting system of equations, which helped us to monitor error of the numerical procedure. Our results show that nonlocal effects decrease the buckling load of optimally shaped rod. However, nonlocal theory leads to the optimal rod with the cross-sectional area at the rod ends different from zero. This is important property since zero value of the cross-section at the ends, which optimally shaped rod according to Bernoulli–Euler rod theory has, is unacceptable in applications.


Nonlocal column Pontryagin’s principle Optimal shape 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Teodor M. Atanackovic
    • 1
  • Branislava N. Novakovic
    • 1
  • Zora Vrcelj
    • 2
  1. 1.Department of MechanicsUniversity of Novi SadNovi SadSerbia
  2. 2.Centre for Infrastructure Engineering and Safety, School of Civil and Environmental EngineeringUNSWSydneyAustralia

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