Shape optimization against buckling of micro-and nano-rods

Archive of Applied Mechanics

Volumn 82, Issue 10-11, 2012, Pages 1303-1311

  Atanackovic, Teodor M ^a   Novakovic, Branislava N ^a   Vrcelj, Zora ^b  

^a UNIVERSITY OF NOVI SAD   (Serbia)

^b UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Nonlocal column; Optimal shape; Pontryagin's principle

Indexed keywords

AXIAL COMPRESSIVE FORCES; BUCKLING LOADS; CROSS SECTIONAL AREA; ELASTIC BUCKLING; FIRST INTEGRAL; NON-LOCAL EFFECT; NONLOCAL; NONLOCAL ELASTICITY; NONLOCAL THEORY; NUMERICAL PROCEDURES; OPTIMAL SHAPE; OPTIMALITY CONDITIONS; PONTRYAGIN'S MAXIMUM PRINCIPLE; PONTRYAGIN'S PRINCIPLE; ROD THEORY; SIMPLY SUPPORTED; SYSTEM OF EQUATIONS; ZERO VALUES;

BUCKLING; ELASTICITY; NANORODS; OPTIMIZATION;

PROBLEM SOLVING;

EID: 84870251083     PISSN: 09391533     EISSN: 14320681     Source Type: Journal    
DOI: 10.1007/s00419-012-0661-1     Document Type: Conference Paper

References (29)

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)
2. Clausen, T.: Über die Form architektonischer Säulen. Bull. Cl. Physico Math. Acad. St. Pétersbourg 9, 369-380 (1851)
3. Cox, S.J.: The shape of the ideal column. Math. Intell. 14, 16-24 (1992)
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. Seyranian, A.P.: The Lagrange problem on optimal column. Adv. Mech. (Uspekhi Mekhaniki) 2, 45-96 (2003)
6. Keller, J.: The shape of the strongest column. Arch. Rat. Mech. Anal. 5, 275-285 (1960)
7. Tadjbakhsh, I., Keller, J.B.: Strongest columns and isoperimetric inequalities for eigenvalues. J. Appl. Mech. ASME 29, 159-164 (1962)
8. Keller, J.B., Niordson, F.I.: The tallest column. J. Math. Mech. 16, 433-446 (1966)
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. Seyranian, A.P.: On a problem of Lagrange. Mech. Solids (Mekhanika Tverdogo Tela) 19, 100-111 (1984)
11. Masur, E.F.: Optimal structural design under multiple eigenvalue constraints. Int. J. Solids Struct. 20, 211-231 (1984)
12. Cox, S.J., Overton, M.L.: On the optimal design of columns against buckling. SIAM J. Math. Anal. 23, 287-325 (1992)
13. Atanackovic, T.M., Seyranian, A.P.: Application of Pontryagin's principle to bimodal optimization problems. Struct. Multidisc. Optim. 37, 1-12 (2008)
14. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
15. Sudak, L.J.: Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281-7287 (2003)
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. 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. 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. Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19, 345703 (2008)
20. Challamel, N., Wang, C.M.: On lateral-torsional buckling of non-local beams. Adv. J. Math. Mech. 2, 389-398 (2010)
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. Atanackovic, T.M.: Optimal shape of an Elastic rod in flexural-torsional buckling. Z. Angew. Math. Mech. ZAMM 87, 399-405 (2007)
23. Atanackovic, T.M.: Stability Theory of Elastic Rods. World Scientific, River Edge (1997)
24. Eringen, A.C.: Theory of nonlocal elasticity and some applications. Res. Mech. 21, 313-342 (1987)
25. Grinev, V.B., Filippov, A.P.: Optimization of Rods with Respect to Spectrum of Eigenvalues. Naukova Dumka, Kiev (1979)
26. Novakovic, B.N., Atanackovic, T.M.: On the optimal shape of a compressed column subjected to restrictions on the crosssectional area. Struct. Multidisc. Optim. 43, 683-691 (2011)
27. Seyranian, A.P.: Homogeneous functionals and structural optimization problems. Int. J. Solids Struct. 15, 749-759 (1979)
28. Vujanovic, B.D., Atanackovic, T.M.: An Introduction to Modern Variational Techniques in Mechanics and Engineering. Birkhäuser, Boston (2004)
29. Sage, A.P., White, C.C.: Optimum System Control. Prentice-Hall, New Jersey (1977)