The shape of the strongest column is arbitrarily close to the shape of the weakest column
Author:
David C. Barnes
Journal:
Quart. Appl. Math. 46 (1988), 605-609
MSC:
Primary 73H05; Secondary 49F99, 73K05, 73K40
DOI:
https://doi.org/10.1090/qam/973378
MathSciNet review:
973378
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We reconsider the problem of determining the shape of the strongest column having a given length $l$ and volume $V$. Previous results [13,7] have given optimal shapes for which the cross section vanishes at certain points. Although these results are mathematically correct, Theorem 1 below explains what is wrong with these anomalous shapes.
- David C. Barnes, Buckling of columns and rearrangements of functions, Quart. Appl. Math. 41 (1983/84), no. 2, 169–180. MR 719503, DOI https://doi.org/10.1090/S0033-569X-1983-0719503-4
- David C. Barnes, Extremal problems for eigenvalues with applications to buckling, vibration and sloshing, SIAM J. Math. Anal. 16 (1985), no. 2, 341–357. MR 777472, DOI https://doi.org/10.1137/0516025
- David C. Barnes, Extremal problems for eigenvalue functionals, SIAM J. Math. Anal. 16 (1985), no. 6, 1284–1294. MR 807909, DOI https://doi.org/10.1137/0516092
- David C. Barnes, Extremal problems for eigenvalue functionals, SIAM J. Math. Anal. 16 (1985), no. 6, 1284–1294. MR 807909, DOI https://doi.org/10.1137/0516092
- Earl R. Barnes, The shape of the strongest column and some related extremal eigenvalue problems, Quart. Appl. Math. 34 (1976/77), no. 4, 393–409. MR 493674, DOI https://doi.org/10.1090/S0033-569X-1977-0493674-3
T. Clausen, Über die Forme architektonisher Saulen, Bulletin Physico-Mathématiques et Astronomiques, Tome 1, pp. 279–294 (1849–1853)
- Joseph B. Keller, The shape of the strongest column, Arch. Rational Mech. Anal. 5 (1960), 275–285 (1960). MR 128160, DOI https://doi.org/10.1007/BF00252909
- Joseph B. Keller and Frithiof I. Niordson, The tallest column, J. Math. Mech. 16 (1966), 433–446. MR 0213089, DOI https://doi.org/10.1512/iumj.1967.16.16029
- M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, Amer. Math. Soc. Transl. (2) 1 (1955), 163–187. MR 0073776, DOI https://doi.org/10.1090/trans2/001/08
J. L. de Lagrange, Sur la figure des colonnes, Miscellanea Tourinesia (Royal Society of Turin), Tomus V (1770–1773), p. 123. Also Oeuvres, 2 (1770–1773) pp. 125–170
- M. K. Myers and W. R. Spillers, A note on the strongest fixed-fixed column, Quart. Appl. Math. 44 (1986), no. 3, 583–588. MR 860907, DOI https://doi.org/10.1090/S0033-569X-1986-0860907-8
N. Olhoff and S. H. Rasmussen, On single and bimodal optimum buckling of clamped columns, Internat. J. Structures 13, 605–614 (1977)
- I. Tadjbakhsh and J. B. Keller, Strongest columns and isoperimetric inequalities for eigenvalues, Trans. ASME Ser. E. J. Appl. Mech. 29 (1962), 159–164. MR 137381
- Stephen P. Timoshenko, Theory of elastic stability, Engineering Societies Monographs, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1961. 2nd ed; In collaboration with James M. Gere. MR 0134026
D. C. Barnes,a Buckling of columns and rearrangements of functions, Quart. Appl. Math. 41, 169–180 (1983)
D. C. Barnes, Extremal problems for eigenvalues with applications to buckling, vibration and sloshing, SIAM J. Math. Anal. 16, 341–357 (1985)
D. C. Barnes, Extremal problems for eigenvalue functionals, SIAM J. Math. Anal. 16, 1284–1294 (1985)
D. C. Barnes, Extremal problems for eigenvalue functionals, Part II, SIAM J. Math. Anal., to appear
E. R. Barnes, The shape of the strongest column and some related extremal eigenvalue problems, Quart. Appl. Math. 34, 393–409 (1977)
T. Clausen, Über die Forme architektonisher Saulen, Bulletin Physico-Mathématiques et Astronomiques, Tome 1, pp. 279–294 (1849–1853)
J. B. Keller, The shape of the strongest column, Arch. Rat. Mech. Anal. 5, 275–285 (1960)
J. B. Keller and F. I. Niordson, The tallest column, J. Math. Mech. 16, 433–446 (1966)
M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on Lyapunov zones of stability, Trans. Amer. Math. Soc. 2, 163–187 (1955)
J. L. de Lagrange, Sur la figure des colonnes, Miscellanea Tourinesia (Royal Society of Turin), Tomus V (1770–1773), p. 123. Also Oeuvres, 2 (1770–1773) pp. 125–170
M. K. Myers and W. R. Spillers, A note on the strongest fixed-fixed column, Quart. Appl. Math. 44, 583–588 (1986)
N. Olhoff and S. H. Rasmussen, On single and bimodal optimum buckling of clamped columns, Internat. J. Structures 13, 605–614 (1977)
I. Tadjbakhsh and J. B. Keller, Strongest columns and isoperimetric inequalities for eigenvalues, J. Appl. Mech. 29, 159–164 (1962)
S. P. Timoshenko and J. N. Gere, Theory of Elastic Stability, McGraw-Hill, New York, 1961
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73H05,
49F99,
73K05,
73K40
Retrieve articles in all journals
with MSC:
73H05,
49F99,
73K05,
73K40
Additional Information
Article copyright:
© Copyright 1988
American Mathematical Society