ELASTICITY 809 expand, but they had taken no account of the lateral shrinking or swelling which the filament must really ex perience in the bent bar. The subject first received satis factory mathematical investigation from St Venant. 1 He proved that the old supposition is substantially correct, with the important practical exception of the flat spring referred to in section 59 below. His theory shows that, in fact, if we imagine the whole rod divided parallel to its length into infinitesimal filaments, each of these shrinks or swells laterally with sensibly the same freedom as if it were separated from the rest of the substance and subjected to end pull or end compression, lengthening or shortening it in a straight line to the same extent as it is really lengthened or shortened in the circular arc which it becomes in the bent rod. He illustrates the distortion of the cross section by which these changes of lateral dimensions are necessarily accompanied in the annexed diagram (fig. 5), in which either the whole nor- j u mal section of a rect- i angular beam, or a rect angular area in the normal section of a beam of any figure, is represented in its strained and unstrained figures, with the central point O common to the two. The flexure is in planes perpendicular to YOY 1( and is concave upwards (or towards X), G, the centre of curvature, the direction the diaram. indicated, but too far to be included n The straight sides AC, BD, and all straight lines parallel to them, of the unstrained rectangular area become con centric arcs of circles concave in the opposite direction, their centre of curvature H being (articles 47, 48) for rods of india-rubber or gelatinous substance,or of glass or metal, from 2 to 4 times as far from O on one side as G is on the other. Thus the originally plane sides AC, BD of a rectangu lar bar become anticlastic 2 surfaces, of curvatures and P , in the two principal sections, if cr denote the ratio of Literal shrinking to longitudinal extension. A flat rectangular, or a square, rod of india-rubber [for which cr amounts (section 47) to very nearly , and which is susceptible of very great amounts of strain without utter loss of corresponding elastic action] exhibits this pheno menon remarkably well. 58. Limits to the bending of Rods or Beams of hard solid substance. For hard solids, such as metals, stones, glasses, woods, ivory, vulcanite, papier-mache", elongations and con tractions to be within the limits of elasticity must gene rally (section 23) be less than yi^. Hence the breadth or thickness of the bar in the plane of curvature must gene rally be less than y^- of the diameter of curvature in order that the bending may not break it, or give it a permanent bend, or strain it beyond its "limits of elas ticity." 59. Exceptional case of Thin flat Spring, too much bent to fulfil conditions of section 57. St Tenant s theory shows that a farther condition must be fulfilled if the ideal filaments are to have the freedom to shrink or expand as ex plained in section 57. For unless the breadth AC of the bar (or diameter perpendicular to the plane of flexure) be 1 M&moiresdes Savants Grangers, 1855, "De la Torsion des Prismes, avec des considerations sur leur Flexion," &c. 8 See Thomson and Tait s Natural Philosophy, vol. i. 128. very small in comparison with the mean proportional between the radius OH and the thickness AB the distances from YY X to the corners A , C , would fall short of the half thickness, OE, and the distances to B , D , would exceed it, by differences comparable with its own amount. This would give rise to sensibly less and greater shortenings and stretchings in the filaments towards the corners than those supposed in the ordinary calculation of flexural rigidity (article 61), and so vitiate the result. Unhappily, mathematicians have not hitherto succeeded in solving, possibly not even tried to solve, the beautiful problem thus presented by the flexure of a broad very thin band (such as a watch spring) into a circle of radius comparable with a third proportional to its thickness and its breadth. 60. But, provided the radius of curvature of the flexure is not only a large multiple of the greatest diameter, but also of a third proportional to the diameters in and per pendicular to the plane of flexure; then, however great may be the ratio of the greatest diameter to the least, the preced ing solution is applicable ; and it is remarkable that the necessary distortion of the normal section (illustrated in the diagram of article 57) does not sensibly impede the free lateral contractions and expansions in the filaments, even in the case of a broad thin lamina (whether of pre cisely rectangular section, or of unequal thicknesses in different parts). 61. Flexural Rigidities of a Rod or Beam. The couple required to give unit curvature in any plane to a rod or beam is called its flexural rigidity for curvature in that plane. When the beam is of circular cross section and of isotropic material, the flexural rigidity is clearly the same, whatever be the plane of flexure through the axis, and the plane of the bending couple coincides with the plane of flexure. It might be expected that in a round bar of seolotropic material, such as a wooden rod with the annual woody layers sensibly plane and parallel to a plane through its axis, would show different flexural rigidities in different planes, in the case of wood, for example, different according as the flexure is in a parallel or perpendicular to the annual layers. This is not so, however; on the contrary, it is easy to show, by an extension of St Venant s theory, that in the case of the wooden rod the flexural rigidity is equal in all planes through the axis, and that the plane of flexure always agrees with the plane of the bending couple, and to prove generally that the flexure of a bar of aeolotropic substance, and composed it may be of longitudinal fila ments of heterogeneous materials, is precisely the same as if it were isotropic, and that its flexural rigidities are calculated by the same rule from its Young s modulus, provided that the seolotropy is not such as (section 81) to give rise to alteration of the angle between the length and any diameter perpendicular to the length when weight is hung on the rod, or on any longitudinal filament cut from it. Excluding then all cases in which there is any such oblique oeolotropy, we have a very simple theory for the flexure of bars of any substance, whether isotropic or jeolotropic, and whether homogen-eous or not homogeneous through the cross section. 62. Principal Flexural Rigidities and Principal Planes of Flexure of a Beam. The flexural rigidity of a rod is generally not equal in different directions, and the plane of flexure does not generally coincide with the plane of the bending couple. Thus a flat ruler is much more easily bent in a plane perpendicular to its breadth than in the plane of its breadth ; and if we apply opposing couples to its two ends in any plane through its axis not either perpendicular or parallel to its breadth, it is obvious that the plane in which the flexure takes place will be more inclined to the plane of the breadth than to the plane of the bending couple. Very elementary statical theory, founded on St
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