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

John Dougall

doi:10.1017/S008045680001718X

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

Published online by Cambridge University Press: 06 July 2012

John Dougall

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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.

Type: Research Article
Information: Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 49 , Issue 4 , 1914 , pp. 895 - 978

DOI: https://doi.org/10.1017/S008045680001718X [Opens in a new window]
Copyright: Copyright © Royal Society of Edinburgh 1914

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

^{page 898 note *} Pochhammer, L., Untersuchungen über das Gleichgewicht des elastischen Stabes, Kiel, 1879Google Scholar.

^{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 ϵ_m 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. 349Google Scholar.

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

^{page 956 note †} Here b₀ means the least modulus of a zero of D_m

^{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 J_m(mx), where x is real, by Graf, und Gubler, , Einleitung in die Theorie der Bessel'schen Functionen erster Heft, Bern, 1898, S. 96Google Scholar.

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

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