Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/835

ELASTICITY 811 Fig. 6. product of nr into the moment of inertia of the area round the perpendicular to its plane through its centre, which is therefore equal to the moment of the couple applied at either end. 66. Prism, of any shape constrained to a Simple Ttvist. Farther, it is easily proved that if a cylinder or prism of any shape be compelled to take exactly the state of strain above specified (section 65) with the line through the centres of inertia of the normal sections, taken instead of the axis of the cylinder, the mutual action between the parts of it on the two sides of any normal section will be a couple of which the moment will be expressed by the same formula, that is. the product of the rigidity, into the rate of twist, into the moment of inertia of the section round its centre of inertia. But for any other shape of prism than a solid or symmetrical hollow circular cylinder, the supposed state of strain requires, besides the terminal opposed couples, force parallel to the length of the prism, distributed over the prismatic boundary, in proportion to the distance PE along the tangent, from each point of the surface, to the point in which this line is cut by a perpendicular to it from O the centre of inertia of the normal section. To prove this let a normal section of the prism be represented in the annexed diagram (fig. 6). Let PK, representing the shear at any point P, close to the prismatic boundary, be resolved into PN and PT along the normal and tangent respectively. The whole, shear PK being equal to rr its component PN is equal to rr sin co or r.PE. The corresponding component of the required stress iswr.PE, and involves equal forces in the plane of the diagram, and in the plane through TP perpendicular to it, each amounting to nr. PE per unit of area. An application of force equal and opposite to the distri bution thus found over the prismatic boundary, would of course alone produce in the prism, otherwise free, a state of strain which, compounded with that supposed above, would give the state of strain actually produced by the sole application of balancing couples to the two ends. The re sult, it is easily seen, consists of an increased twist, together with a warping of naturally plane normal sections, by infinitesimal displacements perpendicular to themselves, into certain surfaces of anticlastic curvature, with equal opposite curvatures. In bringing forward this theory, St Venant not only pointed out the falsity of the supposition admitted by several previous writers, and used in practice fallaciously by engineers, that Coulomb s law holds for other forms of prism than the solid or hollow circular cylinder, but he discovered fully the nature of the requisite correction, reduced the de termination of it to a problem of pure mathematics, worked out the solution for a great variety of important and curious cases, compared the results with observation in a manner satisfactory and interesting to the naturalist, and gave con clusions of great value to the practical engineer. 67. " Hydrokinetic Analogue to Torsion Problem. 1 We take advantage of the identity of mathematical conditions in St Venant s torsion problem, and a hydrokinetic problem first solved a few years earlier by Stokes, 2 to give the following statement, which will be found very useful in estimating deficiencies in torsional rigidity below the amount calculated from the fallacious extension of Coulomb s law : 1 Extracted from Thomson and Tait, sections 704, 705. " On some cases of Fluid Motion," Cam*. PfM. Trans., 1843. " Conceive a liquid of density n completely filling a closed infinitely light prismatic box of the same shape within as the given elastic prism and of length unity, and let a couple be applied to the box in a plane perpendicular to its length. The effective moment of inertia of the liquid 3 will be equal to the correction by which the torsional rigidity of the elastic prism, calculated by the false extension of Coulomb s law, must be diminished to give the true torsional rigidity. " Farther, the actual shear of the solid, in any infinitely thin plate of it between two normal sections, will at each point be, when reckoned as a differential sliding (section 43) parallel to their planes, equal to and in the same direction as the velocity of the liquid relatively to the containing box." 68. Solution of Torsion Problem. To prove these pro positions and investigate the mathematical equations of the problem, the process followed in Thomson and Tait s Natural Philosophy, section 706, is first to show that the conditions of sections 63 are verified by a state of strain compounded of (1) a simple twist round the line through the centres of inertia, and (2) a distortion of each normal section by infinitesimal displacements perpendicular to its plane ; then find the interior and surface equations to de termine this warping ; and lastly, calculate the actual moment of the couple to which the mutual action between the matter on the two sides of any normal section is equivalent. 69. St Venant s treatise abounds in beautiful and instructive graphical illustrations of his results, from which the following are selected : (1.) Elliptic Cylinder. The plain and dotted curvilineal arcs are (fig. 7) "con tour lines " (coupes topographiques) of the section as warped by torsion ; that is to say, lines in which it is cut by a series of parallel planes, each perpendicular to the axis. The arrows indicate the direction of rotation Fi S- 7. in the part of the prism above the plane of the diagram. 3 "That is, the moment of inertia of a rigid solid which, as will be proved in vol. ii., may be fixed within the box, if the liquid be removed,

to make its motions the same as they are with the liciuid in it. "