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XVII.—An Analytical Theory of the Equilibrium of an Isotropic Elastic Rod of Circular Section

  • John Dougall


The main purpose of the following investigation is to improve the existing Approximate Theory of Beams by bringing it into closer relationship than has hitherto been done with exact solutions of the fundamental differential equations of equilibrium. In its aim, method, and results the paper closely resembles one already published in these Transactions on the corresponding Theory for Plates (vol. xli., 1904). It is less comprehensive than the latter paper, as it deals only with a beam of circular section; but the analysis is of such a nature that the chief results can be extended almost immediately to a beam of any form of section, on the assumption merely of certain Existence Theorems.



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page 895 note * Pochhammer, L.. J. f. Math., lxxxi. (1876), p. 33; Chree, C., Camb. Phil. Soc. Trans., xiv. (1889), p. 250; Tedone, O., Roma. Acc. Line. Rend. (5), xiii. (1904), p. 232; cf. Encyk. d. Math. Wiss., Band iv. S. 150.

page 898 note * Pochhammer, L., Untersuchungen über das Gleichgewicht des elastischen Stabes, Kiel, 1879.

page 900 note * Such a reference will be given briefly in the form, 1 (4).

page 900 note † Cf. Art. 11.

page 901 note * Art. 15, footnote.

page 902 note * Cf. Art. 11.

page 903 note * Another notation, also well known, is sometimes convenient, viz. we define ϵm as equal to 1 for m>0, but equal to ½ for m = 0. Thus = ½u0+u1+u2 + …. ad ∞.

page 904 note * Cf. Art. 50.

page 909 note * In this and next Article, (1) and (2) state the terms required in the expansions of the various functions in ascending powers of β.

page 912 note * The expansion of D0 begins with the term ½(v – 8)β8.

page 913 note * Love, Elasticity, Art. 130.

page 915 note * The definition of the function G is

page 921 note * When we come to calculate displacements due to a given force, the terms which vary according as p > or < p' disappear.

page 924 note * Love, Elasticity, Arts. 121, 169.

page 943 note * Pierpont, , The Theory of Functions of Real Variables, vol. i. p. 349.

page 956 note * Cf. a method due to Almansi, E., Encyklop. d. Math. Wiss., iv. 25, 13d.

page 956 note † Here b0 means the least modulus of a zero of Dm

page 959 note * Cf. Art. 47.

page 960 note * Cf. Art. 47.

page 964 note * Cf. Art. 43.

page 975 note * By a similar process to that given for Jm(mx), where x is real, by Graf, und Gubler, , Einleitung in die Theorie der Bessel'schen Functionen erster Heft, Bern, 1898, S. 96.

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Earth and Environmental Science Transactions of The Royal Society of Edinburgh
  • ISSN: 1755-6910
  • EISSN: 1755-6929
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