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On the whole, therefore, we have
linear elongation-P(+), in the direction of the
applied stress, and

linear

contraction-P(-), in all directions per

pendicular to the applied stress.

(3)

and the same in amount for all the three ways in which a stress may be thus applied to any one cube, and for different cubes taken from any different positions in the solid. Hence the elastic quality of a perfectly elastic, homogeneons, isotropic solid is fully defined by two elements,-its resistance to distortion and its resistance to compression. The first has been already considered (section 43). The second is measured by the amount of uniform. pressure in all directions per unit area of its surface required to produce a stated very small compression. The numerical reckoning of the first is the compressing pressure divided by the diminution of the bulk of a portion of the sub-lineal contraction equal to 3k-2n of the longitudinal

stance which, when uncompressed, occupies the unit volume. It is sometimes called the "elasticity of bulk," or sometimes the "modulus of bulk-elasticity," sometimes the resistance to compression. Its reciprocal, or the amount of compression on unit of volume divided by the compressing pressure, or, as we may conveniently say, the compression per unit of volume per unit of compressing pressure, is commonly called the compressibility.

P

45. Strain produced by a single Longitudinal Stress (subject of Young's Modulus).-Any stress whatever may1 be made up of simple longitudinal stresses. Hence, to find the relation between any stress and the strain produced by it, we have only to find the strain produced by a single longitudinal stress, which, for an isotropic solid, we may do at once thus:-A simple longi- PIP tudinal stress P is equivalent to a uniform dilating tension P in all directions, compounded with two distorting stresses, each equal to P, and having a common axis in the line of the given longitudinal stress, and their other two axes

YIP

Fig. 4.

P

47. Hence "Young's Modulus",

and when the

9nk 3k+n' ends of a column, bar, or wire of isotropic material are acted on by equal and opposite forces, it experiences a lateral

2(3k+n)

dilatation, each reckoned as usual per unit of lineal measure. One specimen of the fallacious mathematics referred to in chap. xvi. of the mathematical theory below is a celebrated conclusion of Navier's and Poisson's that the ratio of lateral contraction to elongation by pull without transverse force is . This would require the rigidity to be of the resistance to compression, for all solids; which was first shown to be false by Stokes from many obvious observations, proving enormous discrepancies from it in many well-known bodies, and rendering it most improbable that there is any approach to a constancy of ratio between rigidity and resistance to compression in any class of solids. Thus clear elastic jellies and india-rubber present familiar specimens of isotropic homogeneous solids which, while differing very much from one another in rigidity Pip("stiffness"), are probably all of very nearly the same compressibility as water, which is about per atmosphere. Their resistance to compression, measured by the reciprocal of this, is obviously many hundred times the absolute amount of the rigidity of the stiffest of those substances. A column of any of them, therefore, when pressed together or pulled out, within its limits of elasticity, by balancing forces applied to its ends (or an india-rubber band when pulled out), experiences no sensible change of volume, though very sensible change of length. Hence the propor tionate extension or contraction of any transverse diameter must be sensibly equal to half the longitudinal contraction or extension; and such substances may be practically regarded as incompressible elastic solids in interpreting all the phenomena for which they are most remarkable. Stokes gave reasons for believing that metals also have in general greater resistance to compression, in proportion to their rigidities, than according to the fallacious theory, although for them the discrepancy is very much less than for the gelatinous bodies. This probable conclusion was Wertheim, who found the ratio of lateral to longitudinal soon experimentally demonstrated by change of lineal dimensions, in columns acted on solely by longitudinal force, to be about for glass and brass; and by Kirchhoff, who, by a well-devised experimental method, found 387 as the value of that ratio for brass, and 294 for iron. For copper it is shown to lie between 226 and 441, by experiments quoted below, measuring the torsional and longitudinal rigidities of copper wires.

any two lines at right angles to one another and tc it. The diagram (fig. 4), drawn in a plane through one of these latter lines and the former, sufficiently indicates the synthesis, the only forces not shown being those perpendicular to its plane:

Hence if n denote the rigidity, and k the modulus of compression, or the modulus of bulk-elasticity (being the ame as tho reciprocal of the compressibility), the effect will e an equal dilatation in all directions, amounting, per unit of volume, to

[merged small][merged small][ocr errors]

. (1), compounded with two equal distortions, each amounting to (2), and having (section 43, footnote) their axes in the directions just stated for the axes of the distorting stresses.

n

46. The dilatation and two shears thus determined may be conveniently reduced to simple longitudinal strains by following the indications of section 43, thus:The two shears together constitute an elongation amountin the direction of the given force P, and equal ing to 12 contraction amounting to in all directions perpendicular to it. And the cubic dilatation implies a lineal

n

P

[blocks in formation]

48. All these results indicate rigidity less in proportion Poisson's theory. And it has been supposed by many to the compressibility than according to Navier's and theory as inapplicable to ordinary solids that it may be naturalists who have seen the necessity of abandoning that regarded as the proper theory for an ideal perfect solid, and as indicating an amount of rigidity not quite reached in any real substance, but approached to in some of the

"On the Friction of Fluids in Motion, and the Equilibrium and Motion of Elastic Solids," Trans. Camb. Phil..Soc, April 1845. See also Camb. and Dub. Math. Jour., March 1848.

"On the Elasticity and Viscosity of Metals" (W. Thomson), Pros, R. S.. May 1865,

9kn

3k+n

most rigid of natural solids (as, for instance, iron). But it | isotropic solid, or of sound waves in a fluid, is equal to is scarcely possible to hold a piece of cork in the hand the velocity acquired by a body in falling from a height without perceiving the fallaciousness of this last attempt equal to half the length of the proper modulus for the to maintain a theory which never had any good foundation. By careful measurements on columns of cork of various case; that is, the Young's Modulus for the first forms (among them, cylindrical pieces cut in the ordinary case, the modulus of rigidity (n) for the second, the way for bottles), before and after compressing them modulus of simple longitudinal strain (k+n) for the third, longitudinally in a Bramah's press, we have found that the modulus of compression for the fourth. Remark the change of lateral dimensions is insensible both with that for air the static "length-modulus of compression" at small longitudinal contractions and return dilatations, constant temperature is the same as what is often techniwithin the limits of elasticity, and with such enormous cally called the "height of the homogeneous atmosphere." longitudinal contractions as to or of the original length. 51. In reckoning moduluses there must be a definite It is thus proved decisively that cork is much more rigid, understanding as to the unit in terms of which the force is while metals, glass, and gelatinous bodies are all less rigid, measured, which may be either the kinetic unit or the in proportion to resistance to compression, than the gravitation unit for a specified locality, that is, the weight supposed "perfect solid"; and the practical invalidity of in that locality of the unit of mass. Experimenters the theory is experimentally demonstrated. By obvious hitherto have stated their results in terms in the gravitation mechanism of jointed bars a solid may be designed which unit, each for his own locality,-the accuracy hitherto shall swell laterally when pulled, and shrink laterally when attained being scarcely in any cases sufficient to requiro compressed, in one direction, and which shall be homo- corrections for the different intensities of gravity in the geneous in the same sense (article 40) as crystals and different places of observation. liquids are called homogeneous.

49. Modulus of Simple Longitudinal Strain.-In sections 45, 46, we examined the effect of a simple longitudinal stress in producing elongation in its own direction, and contraction in lines perpendicular to it. With stresses substituted for strains, and strains for stresses, we may apply the same process to investigate the longitudinal and lateral tractions required to produce a simple longitudinal strain (that is, an elongation in one direction, with no change of dimensions perpendicular to it) in a rod or solid of any shape.

Thus a simple longitudinal straine is equivalent to a cubic dilatation e without change of figure (or linear dilatation be equal in all directions), and two distortions consisting each of dilatation fe in the given direction and contraction fe in each of two directions perpendicular to it and to one another. To produce the cubic dilatation e alone requires (section 44) a normal traction ke equal in all directions. And, to produce either of the distortions simply, since the measure (section 43) of each ise, requires a distorting stress equal to nxe, which consists of tangential tractions each equal to this amount, positive (or drawing outwards) in the line of the given elongation, and negative (or pressing inwards) in the perpendicular direction. Thus we have in all

normal traction-(k+n)e, in the direction of the given strain, and

(4).

normal traction-(-n)e, in every direction perpendicular to the given strain. Hence the modulus of simple longitudinal strain is k+n. 50. Weight-Modulus and Length of Modulus.-Instead of reckoning moduluses in units of force per unit of area, it is sometimes convenient to express them in terms of the weight of unit bulk of the solid. A modulus thus reckoned, or, as it is called by some writers, the length of the modulus, is of course found by dividing the weight-modulus by the weight of the unit bulk. It is useful in many applications of the theory of elasticity, as, for instance, in this result, which is proved in the elementary dynamics of waves in an elastic solid or fluid (chap. xvii. of the Mathematical Theory, below):-the velocity of transmission of longitudinal1 vibrations (as of sound) along a bar of cord, or of waves of simple distortion, or of simple longitudinal extension and contraction in a homogeneous

1 It is to be understood that the vibrations in question are so much spread out through the length of the body that inertia does not sensibly influence the transverse contractions and dilatations which (unless the substance have in this respect the peculiar character presented by cork, scotion 48) take place along with them.

The most useful and generally convenient specification of the modulus of elasticity of a substance is in grammesweight per square centimetre. This has only to bo divided by the specific gravity of the substance to give tho length of the modulus. British measures, however, being still unhappily sometimes used in practical and even in scientific statements, we too often meet with reckonings of the modulus in pounds per square inch or per square foot, in tons per square inch, or of length of the modulus in feet or in British statute miles.

The reckoning most commonly adopted in British treatises on mechanics and practical statements is pounds per square inch. The modulus thus stated must be divided by the weight of 12 cubic inches of the solid, or by the product of its specific gravity into 4335, to find the length of the modulus in feet.

To reduce from pounds per square inch to grammes per square centimetre, multiply by 70-31, or divide by 014223. French engineers generally state their results in kilogrammes per square millimetre, and so bring them to more convenient numbers, being 100000 of the incon

In sections 73-76 we shall see that changes of temperature produced by the varying stresses cause changes of temperature which, in ordinary solids, render the velocity of transmission of longitudinal vibrations sensibly greater than that calculated by the rule stated in the text, if we use the static modulus as understood from the definition there given; and it will be shown how to take into account the thermal effect by using a definite static modulus, or kinetic modulus, according to the circumstances of any case that may occur.

• This 'decimal being the weight in pounds of 12 cubic inches of water. The one great advantage of the French metrical system is that the mass of the unit volume (1 cubic centimetre) of water at its temperdegree of approximation for almost all practical purposes. Thus, acature of maximum density (3°-945 C.) is unity (1 gramme) to a sufficient cording to this system, the density of a body and its specific gravity mean one and the same thing; whereas on the British no-system the density is expressed by a number found by multiplying the specific gravity by one number or another, according to the choice of a cubic inch, pint, quart, wine gallon, imperial gallon, cabic foot, cubic yard, or cubic mile that is made for the unit of volume; and the grain, scruple, gunmaker's drachm, apothecary's drachm, ounce Troy, ounce avoirdupois, pound Troy, pound avoirdupois, stone (Imperial, Ayrshire, Lanarkshire, Dumbartonshire), stone for hay, stone for corn, quarter (of a hundredweight), quarter (of corn), hundredweight, or ton that is chosen for unit of mass. It is a remarkable phenomenon, belonging rather to moral and social than to physical science, that a people tending naturally to be regulated by common sense should voluntarily condemn themselves, as the British have so long done, to unnecessary hard labour in every action of common business or scientific work related to measurement, from which all the other nations of Europe have emancipated themselves. Professor W. H. Miller, of Cambridge, concludes, from a very trustworthy comparison of standards by Kupffer, of St Petersburg, that the weight of a cubic decimetre of water at temperature of maximum density is 1000-013 grammes.

veniently large numbers expressing moduluses in grammes | modulus be different in the two bars, their lengths must weight per square centimetre, but it is much better to reckon in millions of grammes per square centimetre.

52. "Resilience" is a very useful word, introduced about forty years ago (when the doctrine of energy was beginning to become practically appreciated) by Lewis Gordon, first professor of engineering in the university of Glasgow, to denote the quantity of work that a spring (or elastic body) gives back when strained to some stated limit and then allowed to return to the condition in which it rests when free from stress. The word "resilience" used without special qualification may be understood as meaning extreme resilience, or the work given back by the spring after being strained to the extreme limit within which it can be strained again and again without breaking or taking a permanent set. In all cases for which Hooke's law of simple proportionality between stress and strain holds, the resilience is obviously equal to the work done by a constant force of half the amount of the extreme force acting through a space equal to the extreme deflection.

53. When force is reckoned in "gravitation measure," resilience per unit of the spring's mass is simply the height that the spring itself, or an equal weight, could be lifted against gravity by an amount of work equal to that given back by the spring returning from the stressed condition. 54. Let the elastic body be a long homogeneous cylinder or prism with flat ends (a bar as we may call it for brevity), and let the stress for which its resilience is reckoned be positive normal pressures on its ends. The resilience per unit mass is equal to the greatest height from which the bar can fall with its length vertical, and impinge against a perfectly hard horizontal plane without suffering stress beyond its limits of elasticity. For in this case (as in the case of the direct impact of two equal and similar bars meeting with equal and opposite velocities, discussed in Thomson and Tait's Natural Philosophy, section 303), the kinetic energy of the translational motion preceding the impact is, during the first half of the collision, wholly converted into potential energy of elastic force, which during the second half of the collision is wholly reconverted into kinetic energy of translational motion in the reverse direction. During the whole time of the collision the stopped end of the bar experiences a constant pressure, and at the middle of the collision the whole substance of the bar is for an instant at rest in the same state of compression as it would have permanently if in equilibrium under the influence of that pressure and an equal and opposite pressure on the other end. From the beginning to the middle of the collision the compression advances at a uniform rate through the bar from the stopped end to the free end. Every particle of the bar which the compression has not reached continues moving uniformly with the velocity of the whole before the collision until the compression reaches it, when it instantaneously comes to rest. The part of the bar which at any instant is all that is compressed remains at rest till the corresponding instant in the second half of the collision.

55. From our preceding view of a bar impinging against an ideal perfectly rigid plane, we see at once all that takes place in the real case of any rigorously direct longitudinal collision between two equal and similar elastic bars with flat ends. In this case the whole of the kinetic energy which the bodies had before collision reappears as purely translational kinetic energy after collision. The same would be approximately true of any two bars, provided the times taken by a pulse of simple longitudinal stress to run through their lengths are equal. Thus if the two bars be of the same substance, or of different substances having the same value for Young's modulus, the lengths must be equal, but the diameters may be unequal. Or if the Young's

(Math. Theory, chap. xvii.) be inversely as the square roots of its values. To all such cases the laws of "collision between two perfectly elastic bodies," whether of equal or unequal masses, as given in elementary dynamical treatises, are applicable. But in every other case part of the translational energy which the bodies have before collision is left in the shape of vibrations after collision, and the translational energy after collision is accordingly less than before collision. The losses of energy observed in common elementary dynamical experiments on collision between solid globes of the same substance are partly due to this cause. If they were wholly due to it they would be independent of the substance, when two globes of the same substance are used. They would bear the same proportion to the whole energy in every case of collision between two equal globes, or again, in every case of collision between two globes of any stated proportion of diameters, provided in each case the two which collide are of the same substances; but the proportion of translational energy converted into vibrations would not be the same for two equal globes as for two unequal globes. Hence when differences of proportionate losses of energy are found in experiments on different substances, as in Newton's on globes of glass, iron, or compressed wool, this must be due to imperfect elasticity of the material. It is to be expected that careful experiments upon hard well-polished globes striking one another with such gentle forces as not to produce even at the point of contact any stress approaching to the limit of elasticity, will be found to give results in which the observed loss of translational energy can be almost wholly accounted for by vibrations remaining in the globes after collision.

56. Examples of Resilience.-Example 1.-In respect to simple longitudinal pull, the extreme resilience of steel pianoforte wire of the gauge and quality referred to in section 22 above (calculated by multiplying the breaking weight into half the elongation produced by it according to the experimental data of section 22) is 6066 metre-grammes (gravitation measure) per ten. metres of the wire. Or, whatever the length of the wire, its resilience is equal to the work required to lift its weight through 179 metres.

Example 2.-The torsional resilience of the same wire, twisted in either direction as far as it can be without giving it any notable permanent set, was found to be equal to the work required to lift its weight through 13 metres.

Example 3.-The extreme resilience of a vulcanized india-rubber band weighing 12.3 grammes was found to be equal to the work required to lift its weight through 1200 metres. This was found by stretching it by gradations of weights up to the breaking weight, representing the results by aid of a curve, and measuring its area to find the integral work given back by the spring after being stretched by a weight just short of the breaking weight.

57. Flexure of a Beam or Rod.-In the problem o. simple flexure a bar or uniform rod or wire, straight when free from stress, is kept in a circular form by equal oppos ing couples properly applied to its ends. The parts of the bar on the convex side of the circle must obviously be stretched longitudinally, and those on the concave side contracted longitudinally, by the flexure. It is not obvious, however, what are the conditions affecting the lateral shrinkings and swellings of ideal filaments into which we may imagine the bar divided lengthwise. Earlier writers had assumed without proof that each filament, bent as it is in its actual position in the bar, is elongated or contracted by the same amount as it would be if it were detached, and subjected to the same end pull or end compression with its sides quite free to shrink or

expand, but they had taken no account of the lateral shrinking or swelling which the filament must really experience in the bent bar. The subject first received satisfactory 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 normal section of a rectangular beam, or a rectangular 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,, and is concave upwards (or towards

Fig. 5.

X),-G, the centre of curvature, being in the direction indicated, but too far to be included in the diagram. The straight sides AC, BD, and all straight lines parallel to them, of the unstrained rectangular area become concentric 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 rectangular bar become anticlastic surfaces, of curvatures

P

and

in the two principal sections, if σ denote the ratio of lateral shrinking to longitudinal extension. A flat rectangular, or a square, rod of india-rubber [for which σ 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 phenomenon remarkably well.

58. Limits to the bending of Rods or Beams of hard solid substance. For hard solids, such as metals, stones, grasses, woods, ivory, vulcanite, papier-maché, elongations and contractions to be within the limits of elasticity must generally (section 23) be less than. Hence the breadth or thickness of the bar in the plane of curvature must generally be less than 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 elasticity."

59. Exceptional case of Thin flat Spring, too much bent to fulfil conditions of section 57.-St Venant's theory shows that a farther condition must be fulfilled if the ideal filaments are to have the freedom to shrink or expand as explained in section 57. For unless the breadth AC of the bar (or diameter perpendicular to the plane of flexure) be

1 Mémoires des Savants Étrangers, 1855, "De la Torsion des Prismes,

avec des considérations sur leur Flexion," &c.

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, 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 preceding 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 precisely 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 solotropic 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 solotropic substance, and composed it may be of longitudinal filaments 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 solotropy 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 such oblique solotropy, we have a very simple theory for from it. Excluding then all cases in which there is any the flexure of bars of any substance, whether isotropic or solotropic, and whether homogeneous 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 VII.

102

Venant's conclusions of section 57, shows that, whatever | pages of the Mémoires des Savants Etrangers); but a

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 different 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 cross sections found according to rule (1), 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 of 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 proportion to distance from it: the amount of the elongation or contraction 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 norraal 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 ❘

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 Tait's Natural Philosophy, sections 700-710, 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 moment of the couple divided by the amount of the twist per unit length 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 æolotropic 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 complicated much by its moletropy. 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 twist 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.

T

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 perpendicular 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 radians per unit of length of the cylinder. Hence the amount of the tangential force in either set of planes is n 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) nτssdor3, or the

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