Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/834

810 ELASTICITY Venant s conclusions of section 57, shows that, whatever the shape and the distribution of matter in the cross section of the bar, there are two planes at right angles to one another such that if the bar be bent in either of these planes the bending couple will coincide with the plane of flexure. These planes are called principal planes of flexure, and the rigidities of the bar for flexure in these planes are called its principal flexural rigidities. When the principal flexural rigidities are known the flexure of the bar in any plane oblique to the principal planes is readily found by supposing it to be bent in one of the principal planes and simultaneously in the other, and calculating separately the couples required to produce these two component flexures. The positions of the principal planes of flexure, the relative flexural rigidities, and the law of elongation and contraction in different parts of the cross section, are found according to the following simple rules : (1.) Imagine an infinitely thin plane disc of the same shape and size as the cross section loaded with matter in simple proportion to the Young s modulus in different parts of the cross section. Let the quantity of matter per unit area on any point of the disc be equal to the Young s modulus on the corresponding point of the rod when the material is heterogeneous : on the other hand, when the material is homogeneous it is more convenient to call the quantity of matter unity per unit area of the disc. Considering dill ereut axes in the plane of the disc through its centre of inertia, find the two principal axes of greatest and least moments of inertia, and find the moments of inertia round them. (2.) In whatever plane the bar be bent it will experience neither elongation nor contraction in the filament which passes through the centres of inertia of the crc-ss sections found according to rule (I), nor in the diameter of the cross section perpendicular to the plane of flexure. (3.) Thus all the particles which experience neither elongation nor contraction lie in a surface cutting the plane oi flexure perpendicularly through the centres of inertia of the cross sections. All the material on the outside of this cylindrical surface is elongated, and all on the interior is contracted, in simple propor tion to distance from it : the amount of the elongation or contrac tion being in fact equal to distance from this neutral surface divided by the radius of its curvature. (4. ) Hence it is obvious that the portions of the solid on the two sides of any cross section must experience mutual normal force, pulling them towards one another in the stretched part, and press ing them from one another in the condensed part, and that the amount of this negative or positive normal pressure per unit of area must be equal to the Young s modulus at the place, multiplied into the ratio of its distance from the neutral line of the cross section to the radius of curvature. The sum of these positive and negative forces over the whole area of the cross section is zero in virtue of condition (2). Their couple resultant has its axis perpendicular to the plane of curvature when this line is either of the principal axes (3) of the cross section ; and its moment is clearly equal to the moment of inertia of the material disc (1) divided by the radius of curvature. Hence the principal flexural rigidities are simply equal to the principal moments of inertia of this disc ; and the principal flexural planes are the planes through its principal axes and the length of the bar ; or taking the quantity of matter per unit area of the disc unity for the case of a homogeneous bar, we have the rule that the principal rigidities are equal to the product of the Young s modulus into the principal moments of inertia of the cross sectional areas, and the principal planes of flexure are the longitudinal planes through the principal axes of this area. 63. Law of Torsion. One of the most beautiful applications of the general equations of internal equilibrium of an elastic solid hitherto made is that of M. de St Venant to " the torsion of prisms." In this work the mathematical methods invented by Fourier for the solution of problems regarding conduction of heat have been most ingeniously and happily applied by St Venant to the problem of torsion. To reproduce St Venant s mathematical investigation here would make this article too long (it occupies 227 quarto pages of the Memoires des Savants fitranr/ens) ; but s. statement of some of the chief results is given (sections 65-72), not only on account of their strong scientific interest, but also because they are of great practical value in engineering ; and the reader is referred to Thomson and T ait s Natural Philosophy, sections 700710, for the proofs and for further details regarding results, but much that is valuable and interesting is only to be found in St Venant s original memoir. 64. Torsion Problem stated and Torsional Rigidity defined. To one end of a long, straight prismatic rod, wire, or solid or hollow cylinder of any form, a given couple is applied in a plane perpendicular to the length, while the other end is held fast : it is required to find the degree of twist produced, and the distribution of strain and stress throughout the prism. The amount of the twist per unit length divided by the moment of the couple is called the torsional rigidity of the rod or prism. This definition is founded simply on the extension of Hooke s law to torsion discovered experimentally by Coulomb, according to which a rod or wire when twisted within limits of torsional elasticity exerts a reactive couple in simple proportion to the angle through which one end is turned relatively to the other. The internal conditions to be satisfied in the torsion problem are that the resultant action between the substance on the two sides of any normal section is a couple, in the normal plane, equal to the given couple. This problem has not hitherto been attacked for seolotropic solids. Even such a case as that of the round wooden rod (section 61) with annual layers sensibly parallel to a plane through its length, will, when twisted, experience a distribution of strain compli cated much by its seolotropy. The following statements of results are confined to rods of isotropic material. 65. Torsion of Circular Cylinder. For a solid or hollow circular cylinder, the solution (given first, we believe, by Coulomb) obviously is that each circular normal section remains unchanged in its own dimensions, figure, and internal arrangement (so that every straight line of its particles remains a straight line of unchanged length), but is turned round the axis of the cylinder through such an angle as to give a uniform rate of tu ist equal to the applied couple divided by the product of the moment of inertia of the circular area (whether annular or complete to the centre) into the modulus of rigidity of the substance. For, if we suppose the distribution of strain thus specified to be actually produced, by whatever application of stress is necessary, we have, in every part of the substance, a simple shear parallel to the normal section, and perpen dicular to the radius through it. The elastic reaction against this requires, to balance it (section 43), a simple distorting stress consisting of forces in the normal section, directed as the shear, and others in planes through the axis, and directed parallel to the axis. The amount of the shear is, for parts of the substance at distance r from the axis, equal obviously to rr, if r be the rate of twist reckoned in radia?is per unit of length of the cylinder. Hence the amount of the tangential force in either set of planes is n-r per unit of area, if n be the rigidity of the substance. Hence there is no force between parts of the substance lying on the two sides of any element of any circular cylinder coaxal with the bounding cylinder or cylinders ; and consequently no force is required on the cylindrical boundary to maintain the supposed state of strain. And the mutual action between the parts of the substance on the two sides of any normal plane section consists of force in this plane, directed perpendicular to the radius through each point, and amounting to nrr per unit of area. The moment of this distribution of force round the axis of the cylinder

is (if do- denote an element of the area) nrffdar", or the