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

 Circular Section. By John Dougall, M.A., D.Sc. Communicated hy Prof. G. A. 

 Gibson. 



(MS. received April 1, 1913. Read November 3, 1913. Issued separately January 19, 1914.) 



The main purpose of the following investigation is to improve the existing Approxi- 

 mate 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 Trcmsactions 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. 



A complete solution is given of the problem of equilibrium of an infinite circular 

 cylinder subjected to the action of a force in any direction applied at a single point of 

 the body, whether at the surface or in the interior. 



Tlie problem of given surface tractions has been considered by Lame and Clapeyron, 

 PocHHAMMER, Chrbe, and Tedone.* PocHHAiMMER uses the method of particular solu- 

 tions to solve the problem for the case when the applied tractions are periodic functions 

 of 2, the axis of z being the axis of the cylinder. His solution is therefore not quite 

 general, not sufficiently general even to give a particular solution for the tractions 

 at the cylindrical surface in the case of a finite cylinder, since tractions independent 

 of z are not included. This defect, however, could be remedied very easily. 



In the present paper, the solutions for given surface traction follow up to a certain 

 point practically the same lines as Pochhammer's, but the use of Fourier's Integral 

 instead of Fourier's Theorem makes it possible to deal with surface conditions of quite 

 general form, and hence to deduce the solution for a single force at a point on the 

 surface. 



Previous writers have not dealt specially with the general case of internal force, 

 but this is considered in detail here, solutions for a force at any point of the body 

 being found in perfectly analogous forms to those for a force at a point on the 

 surface. The method is to begin with the well-known solutions for a single force 

 in an unlimited solid, to expand these in infinite series of integrals involving the 

 harmonic functions of the circular cylinder, and to balance the elementary terms of 

 these series. 



* L. PocHHAMMBR. J. f. Math,, Ixxxi. (1876), p. 33; C. Chree, Gamb. Phil. Soc. Trans., xiv. (1889), p. 250 ; 

 0. Tedone, Roma. Ace. Line. Rend. (5), xiii. (1904), p. 232 ; of. Encyk. d. Math. JViss., Band iv. S. 150. 



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