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

## Abstract

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.

### Keywords

Nonlocal column Pontryagin’s principle Optimal shape

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### References

1. 1.
Lagrange, J.-L.: Sur la figure des colonnes. In: Serret, M.J.-A. (ed.) Ouveres de Lagrange, vol. 2, pp. 125–170. Gauthier-Villars, Paris (1868)Google Scholar
2. 2.
Clausen, T.: Über die Form architektonischer Säulen. Bull. Cl. Physico Math. Acad. St. Pétersbourg 9, 369–380 (1851)Google Scholar
3. 3.
Cox S.J.: The shape of the ideal column. Math. Intell. 14, 16–24 (1992)
4. 4.
Seyranian A.P., Privalova O.G.: The Lagrange problem on an optimal column: old and new results. Struct. Multidisc. Optim. 25, 393–410 (2003)
5. 5.
Seyranian A.P.: The Lagrange problem on optimal column. Adv. Mech. (Uspekhi Mekhaniki) 2, 45–96 (2003)Google Scholar
6. 6.
Keller J.: The shape of the strongest column. Arch. Rat. Mech. Anal. 5, 275–285 (1960)
7. 7.
Tadjbakhsh I., Keller J.B.: Strongest columns and isoperimetric inequalities for eigenvalues. J. Appl. Mech. ASME 29, 159–164 (1962)
8. 8.
Keller J.B., Niordson F.I.: The tallest column. J. Math. Mech. 16, 433–446 (1966)
9. 9.
Olhoff N., Rasmussen S.H.: On the single and bimodal optimum buckling loads of clamped columns. Int. J. Solids Struct. 13, 605–614 (1977)
10. 10.
Seyranian A.P.: On a problem of Lagrange. Mech. Solids (Mekhanika Tverdogo Tela) 19, 100–111 (1984)
11. 11.
Masur E.F.: Optimal structural design under multiple eigenvalue constraints. Int. J. Solids Struct. 20, 211–231 (1984)
12. 12.
Cox S.J., Overton M.L.: On the optimal design of columns against buckling. SIAM J. Math. Anal. 23, 287–325 (1992)
13. 13.
Atanackovic T.M., Seyranian A.P.: Application of Pontryagin’s principle to bimodal optimization problems. Struct. Multidisc. Optim. 37, 1–12 (2008)
14. 14.
Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
15. 15.
Sudak L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003)
16. 16.
Wang C.M., Zhang Y.Y., Ramesh S.S., Kitipornchai S.: Buckling analysis if micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. J. Phys. D Appl. Phys. 39, 3904–3909 (2006)
17. 17.
Lu P., Lee H.P., Lu C., Zhang P.Q.: Application of nonlocal beam models for carbon nanotubes. Int. J. Solids Struct. 44, 5289–5300 (2007)
18. 18.
Wang Q., Wang C.M.: The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18, 075702 (2007)
19. 19.
Challamel N., Wang C.M.: The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19, 345703 (2008)
20. 20.
Challamel N., Wang C.M.: On lateral-torsional buckling of non-local beams. Adv. J. Math. Mech. 2, 389–398 (2010)
21. 21.
Babaei H., Shahidi A.R.: Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method. Arch. Appl. Mech. 81, 1051–1062 (2011)
22. 22.
Atanackovic T.M.: Optimal shape of an Elastic rod in flexural-torsional buckling. Z. Angew. Math. Mech. ZAMM 87, 399–405 (2007)
23. 23.
Atanackovic T.M.: Stability Theory of Elastic Rods. World Scientific, River Edge (1997)
24. 24.
Eringen A.C.: Theory of nonlocal elasticity and some applications. Res. Mech. 21, 313–342 (1987)Google Scholar
25. 25.
Grinev V.B., Filippov A.P.: Optimization of Rods with Respect to Spectrum of Eigenvalues. Naukova Dumka, Kiev (1979)Google Scholar
26. 26.
Novakovic B.N., Atanackovic T.M.: On the optimal shape of a compressed column subjected to restrictions on the cross-sectional area. Struct. Multidisc. Optim. 43, 683–691 (2011)
27. 27.
Seyranian A.P.: Homogeneous functionals and structural optimization problems. Int. J. Solids Struct. 15, 749–759 (1979)
28. 28.
Vujanovic B.D., Atanackovic T.M.: An Introduction to Modern Variational Techniques in Mechanics and Engineering. Birkhäuser, Boston (2004)
29. 29.
Sage A.P., White C.C.: Optimum System Control. Prentice-Hall, New Jersey (1977)Google Scholar