Díaz Díaz, Jesús Ildefonso and Sauvageot, M.
(2010)
*Euler's tallest column revisited.*
Nonlinear Analysis: Real World Applications, 11
(4).
pp. 2731-2747.
ISSN 1468-1218

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Official URL: http://www.sciencedirect.com/science/article/pii/S1468121809002855

## Abstract

In 1757, Leonhard Euler started the study of the tallest column, i.e. the shape of a stable column with the symmetry of revolution, such that it attains the maximum height once the total mass is prescribed, buckling due to the effect of a load supported at its top. A more detailed analysis is due to Keller and Niordson in 1966 who formulated the problem in terms of an eigenvalue type problem under some coefficient constraints and also, by eliminating some of the unknowns, as a nonlocal boundary value problem for a p-Laplacian type operator with a negative exponent and with an infinite normal derivative in some of the boundaries. The main contribution of this work is the study of the existence and qualitative behavior of a weak solution completing the approach made by Keller and Niordson (developed merely by asymptotic analysis techniques). Under a suitable condition on the top load, we show that there exists a shape function a(x) for which the smallest eigenvalue is the largest one when a(x) is taken in a suitable class of shape functions (in contrast with the unload case according to a result due to Cox and McCarthy in 1998). We prove also that the nonlocal problem has a solution u such that u is an element of W(1,p)(0, 1) for any p is an element of [1, 3) but with u is not an element of W(1,3)(0, 1). We also give a sufficient condition for the uniqueness of the solution.

Item Type: | Article |
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Uncontrolled Keywords: | optimal shape; Euler's tallest column; Eigenvalue problem under coefficient constraints; Nonlocal problem; p-Laplacian type operator with a negative exponent; Infinite normal derivative |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15037 |

References: | [1] L. Euler, Sur la force des colonnes, Académie Royale des Sciences et Belles Lettres, Berlin (1757). Also in Leonhardi Euleri Opera Omnia, Scientiarum Naturalium Helveticae edenda curaverunt F. Rudio, A. Krazer, P. Stackel. Lipsiae et Berolini, Typis et in aedibus B. G. Teubneri, 1911. [2] J.L. Lagrange, Sur la figure des colonnes, in: M.J.-A. Serret éd. (Ed.), Oeuvres de Lagrange, Gauthier-Villars, Paris, 1968, pp. 125_170. [3] S.J. Cox, The shape of the ideal column, The Math. Intelligencer 14 (1992) 16_24. [4] Y.V. Egorov, On the Lagrange problem about strongest column, C.R. Acad. Sci. Paris, Sér. I 335 (2002) 997_1002. [5] J.B. Keller, I. Tadjbakhsh, Strongest columns and isopemetric inequalities for eigenvalues, J. Appl. Mech. (Trans. ASME) 29 (1962) 159_164. [6] S.J. Cox, M.L. Overton, On the optimal design of columns against buckling, SIAM J. Math. Anal. 23 (1992) 287_325. [7] J.B. Keller, F.I. Niordson, The tallest column, J. Math. Mech. 16 (5) (1966) 433_446. [8] S.J. Cox, C. Maeve McCarthy, The shape of the tallest column, SIAM J. Math. Anal. 29 (3) (1998) 547_554. [9] S.S. Antman, Nonlinear Problems of Elasticity, Springer-Verlag, New York, 1995. [10] J.I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985. [11] M. Comte, J.I. Díaz, On the Newton partially flat minimal resistance body type problems, J. Eur. Math. Soc. 7 (2005) 395_411. [12] A. Kufner, L.E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing, 2003. [13] T.M. Atanackovic, Optimal shape of column with own weight : Bi and single modal optimization, Meccanica 40 (2006) 173_196. [14] T.M. Atanackovic, Optimal Shape of a strongest inverted column, J. Comput. Appl. Math. 203 (1) (2007) 209_218. [15] J.I. Díaz, M. Sauvageot, On the Euler best column: A singular non local quasilinear equation with a boundary blowing up flux condition, communication to CEDYA05, Madrid, 19-23 sept. 2005. In CD-Rom Actas XIX CEDYA / IX CMA, Servicio de Publicaciones de la Univ. Carlos III, Madrid, 2005. [16] K.O. Friedrichs, Criteria for discrete spectra, Comm. Pure Appl. Math. 3 (1950) 439_449. [17] P.N. Jiky, Buckling analysis of pre-cracked beam_columns by Liapunov's second method, Eur. J. Mech. A Solids 26 (3) (2007) 503_518. [18] J.B. Keller, The shape of the strongest column, Arch. Ration. Mech. Anal. 5 (1960) 275_285. |

Deposited On: | 26 Apr 2012 07:58 |

Last Modified: | 06 Feb 2014 10:14 |

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