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Euler's Problem Revisited: 222 Years Later

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Abstract

This paper furnishes several closed-form solutions for the buckling eigenvalue problem of the columns with variable flexural rigidity along the axis. The column is either under uniformly distributed axial load or under concentrated compressive load.

Sommario. Questo lavoro fornisce diverse soluzioni in forma chiusa per il problema agli autovalori dell'instabilità di colonne con rigidezza flessionale variabile lungo l'asse. La colonna è soggetta ad un carico assiale unifor-memente distribuito oppure ad una forza di compressione concentrata.

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References

  1. Euler, L., ‘Sur la forces des colonnes’ Memoires de l'Academie Royale des Sciences et Belles Lettres, Berlin, 13 (1757) 252–282.

    Google Scholar 

  2. Engesser, F., ‘Über die Knickfestigkeit von Stäben veränderlichen Tr¨agheitsmomentes’ Zeitschrift der Üsterreichischer Ingenieur und Architekten Verein 34 (1909) 506–508 (in German).

    Google Scholar 

  3. Duncan, W.J., ‘Galerkin's method in mechanics and differential equations’ Aeronaut. Res. Committ. Reports Memor. 1798 (1937).

  4. Elishakoff, I. and Rollot, O., ‘New closed form solutions for buckling of variable stiffness column by Mathematica’ J. Sound Vibrat. 224 (1999) 172–182.

    Google Scholar 

  5. Euler, L., ‘Determinatio onerum quae columnae gestare valent’ Acta Academiae Scientiarum Petropolitanae 1 (1778a) 121–147 (see also Leonhardi Euleri Opera Omnia 17 Part 2 (1982) 232–251, Bern), (in Latin).

    Google Scholar 

  6. Euler, L., ‘Examen insignio paradoxi in theoria columnarum occurentio’ Acta Academiae Scientiarum Petropolitanae 1 (1778b) 146–162, (see also Leonhardi Euleri Opera Omnia 17 Part 2 (1982) 252–265, Bern), (in Latin).

    Google Scholar 

  7. Euler, L., ‘De altitudine columnarum sub proprio pondere corruentium’ Acta Academiae Scientiarum Petropolitanae 1 (1778c) 163–193 (see also Leonhardi Euleri Opera Omnia 17 Part 2 (1982) 266–293, Bern), (in Latin).

    Google Scholar 

  8. Panovko, Ya.G. and Gubanova, I., Stability and Oscillations of Elastic Systems: Paradoxes, Fallacies and New Concepts, Consultants Bureau, New York, 1965.

    Google Scholar 

  9. Greenhill, A.G., ‘Determination of the greatest height consistent with stability that a vertical pole mast can be made, and of the greatest height to which a tree of given proportions can grow’ Proc. Cambridge Phil. Soc. 4 (1881) 765–774.

    Google Scholar 

  10. Dinnik, A.N., ‘Buckling under own weight’ Proc. Don Polytech. Instit. 1 (Part 2) (1912) p. 19 (in Russian).

    Google Scholar 

  11. Willers, F.A., ‘Das Knicken Schwerer Gestaenge’ ZAMM 21 (1941) 43–51 (in German).

    Google Scholar 

  12. Engelhardt, H., ‘Die einheitliche Behandlung der Stabknickung mit Beruecksichtung des Stabeigengewichte in den Eulerfaellen 1 bis 4 als Eigenwertproblem’ Der Stahlbau 23(4) (1954) 80–84 (in German).

    Google Scholar 

  13. Frich-Fay, R., ‘On the stability of a strut under uniformly distributed axial forces’ Int. J. Solids Struct. 2 (1966) 361–369.

    Google Scholar 

  14. Eisenberger, M., ‘Buckling loads for variable cross-section member with variable axial forces’ Int. J. Solids Struct. 27 (1991) 135–143.

    Google Scholar 

  15. Shinozuka, M. and Yamazaki, F., ‘Stochastic finite element analysis: an introduction’ in: Ariaratnam, S.T., Schuëller, G.I. and Elishakoff, I. (eds), Stochastic Structural Dynamics, Elsevier Applied Science, London, (1988), pp. 241–292.

    Google Scholar 

  16. Elishakoff, I., ‘Inverse buckling problem for inhomogeneous columns’ Int. J. Solids Struct. 38 (2001) 457–464.

    Google Scholar 

  17. Elishakoff, I., ‘A closed form-solution for generalized Euler problem’ Proc. R. Soc. London 456 (2000) 2409–2417.

    Google Scholar 

  18. Timoshenko, S.P. and Gere, J.M., Theory of Elastic Stability, McGraw-Hill, New York, 1961.

    Google Scholar 

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Elishakoff, I. Euler's Problem Revisited: 222 Years Later. Meccanica 36, 265–272 (2001). https://doi.org/10.1023/A:1013974623741

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