Page images
PDF
EPUB

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 Twist. 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 stato 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 Tr its component PN is equal to rr sin @ or T. PE The corresponding component of the required stress is nr. PE, and involves equal forces in

Fig. 6.

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 distribution 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 result, 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 determination 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 conclusions of great value to the practical engineer.

We

67. "Hydrokinetic, Analogue to Torsion Problem: 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, 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,"—Camb. Phil. Trans., 1843.

"Conceive a liquid of density a 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 liquids 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 propositions 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 determine 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 :

[blocks in formation]

(1.) Elliptic Cylinder.—The plain and dotted curvilineal arcs are (fig. 7) "contour 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

r Fig. 7.

Conver

in the part of the prism above the plane of the diagram.

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

(2.) Contour lines for St Venant's "étoile à quatre points arrondis."-This diagram (fig. 8) shows the contour lines, in all respects as in case (1), for the case of a prism having for section the figure indicated. The portions of curve outside the continuous closed curve are merely indications of mathematical extensions irrelevant to the physical problem.

(3.) Contour lines of normal section of triangular prism, as warped by torsion, shown as in case (1) (fig. 9.

[merged small][ocr errors]
[merged small][merged small][ocr errors][merged small][merged small]
[blocks in formation]

Fig. 11.

analogy given in section 67), has been pointed out by M. de St Venant, with the important practical application, that strengthening ribs, or projections (see, for instance, the second of the annexed diagrams), such as are introduced in engineering to give stiffness to beams, have the reverse of a good effect when torsional rigidity or strength is an object, although they are truly of great value in increasing the flexural rigidity, and giving strength to bear ordinary strains, which are always more or less flexural. With remarkable ingenuity and mathematical skill he has drawn beautiful illustrations of this important practical principle from bis algebraic and transcendental solutions.

[blocks in formation]

(4.)

[blocks in formation]

Fig. 10.

(5.) Diagram of St Venant's curvilineal squares for which torsion problem is algebraically solvable. This diagram (fig. 11) shows the series of lines represented by the equation x3+ y2 — a(x1 – 6x2y2 + y1) = 1 − a, with the indicated values for a. It is remarkable that the values a = 0.5 and a = -(2-1) give similar but not equal curvilineal squares (hollow sides and acute angles), one of them turned through half a right angle relatively to the other.

[blocks in formation]

Fig. 12.-Diagrams showing torsional rigidities. Thus, for an equilateral triangle, and for the rectilineal and three curvilineal squares shown in the diagrams (fig. 12), he finds for the torsional rigidities the values stated. The number immediately below the diagram indicates in each case the fraction which the true torsional rigidity is of the old fallacious estimate (section 66),-the latter being the product of the rigidity of the substance into the moment of inertia of the cross section round an axis perpendicular to its plane through its centre of inertia. The second number indicates in each case the fraction which the torsional rigidity is of that of a solid circular cylinder of the same sectional area,

70. Torsional Rigidity less in proportion to cum of principal Flexural Rigidities than according to false extension (section 66) of Coulomb's Law.-Inasmuch as the moment of inertia of a plane area about an axis through its centre of inertia perpendicular to its plane is obviously equal to the sum of its moments of inertia round any two axes through the same point at right angles to one another in its plane, the fallacious extension of Coulomb's law, referred to in section 66, would make the torsional rigidity 71. Places of greatest Distortion in Twisted Prisms.—M. of a bar of any section equal to the product of the ratio de St Venant also calls attention to a conclusion from his of the modulus of rigidity to the Young's modulus into solutions which to many may be startling, that in his the sum of its flexural rigidities (section 61) in any two eimpler cases the places of greatest distortion are those planes at right angles to one another through its length.ints of the boundary which are nearest to the axis of the The true theory, as we have seen (section 67), always gives twisted prism in each case, and the places of least distortion a torsional rigidity less than this. How great the deficiency | those farthest from it. Thus in the elliptic cylinder the

substance is most strained at the ends of the smaller principal diameter, and least at the ends of the greater. In the equilateral triangular and square prisms there are longitudinal lines of maxinium strain through the middles of the sides. In the oblong rectangular prism there are two lines of greater maximum strain through the middles of the broader pair of sides, and two lines of less maximum strain through the middles of the narrow sides. The strain is, as we may judge from the hydrokinetic analogy, excessively small, but not evanescent, in the projecting ribs of a prism of the figure shown in (2) of section 69. It is quite evanescent infinitely near the angle, in the triangular and rectangular prisms, and in each other case, as (5) of section 69, in which there is a finite angle, whether acute or obtuse, projecting outwards. This reminds us of a general remark we have to make, although consideration of space may oblige us to leave it without formal proof.

72. Strain at Projecting Angles, evanescent; at Re-entrant Angles, infinite, Liability to Crucks proceeding from Reentrant Angles, or any places of too sharp concuve curvature.-A solid of any elastic substance, isotropic or Holotropic, bounded by any surfaces presenting projecting edges or angles, or re-entrant angles or edges, however obtuse, cannot experience any finite stress or strain in the neighbourhood of a projecting angle (trihedral, polyhedral, or conical); in the neighbourhood of an edge, can only experience simple longitudinal stress parallel to the neighbouring part of the edge; and generally experiences infinite stress and strain in the neighbourhood of a re-entrant edge or angle; when influenced by any distribution of force, exclusive of surface tractions 'infinitely near the angles or edges in question. An important application of the last part of this statement is the practical rule, well known in mechanics, that every re-entering edge or angle ought to be rounded, to prevent risk of rupture, in solid pieces designed to bear stress. An illustration of these principles is afforded by the concluding example of torsion in Thomson and Tait's section 707; in which we have the complete mathematical solution of the torsion problem for prisms of fan-shaped sections, such as the annexed forms (fig. 13).

[blocks in formation]

The solution shows that when the solid is continuous from the circular cylindrical surface to its axis, as in (4), (5), (6), the strain is zero or infinite according as the angle between the bounding planes of the solid is less than or greater than two right angles as in cases (4) and (6) respectively.

73. Changes of Temperature produced by Compressions or Dilatations of a Fluid and Stresses of any kind in an Elastic Solid.-From thermodynamic theory1 it is concluded that cold is produced whenever a solid is strained by opposing, and heat when it is strained by yielding to, any elastic force of its own, the strength of which would diminish if the temperature were raised; but that, on the contrary, heat is produced when a solid is strained against, and cold when it is strained by yielding to, any elastic force of its own, the strength of which would increase if the temperature were raised. When the strain is a condensation or dilatation, uniform in all directions, a fluid may be

W. Thomson on "Thermo-elastic Properties of Matter," in

Quarterly Journal of Mathematics, April 1855 (republished in Phil, Mag. 1877, second half year.)

included in the statement. Hence the following propositions:

(1.) A cubical compression of any elastic fluid or solid in an ordinary condition causes an evolution of heat; but, on the contrary, a cubical compression produces cold in any substance, solid or fluid, in such an abnormal state that it would contract if heated while kept under constant pressure. Water below its temperature (3°-9 Cent.) of maximum density is a familiar instance. (See table of section 76.) (2.) If a wire already twisted be suddenly twisted further, always, however, within its limits of elasticity, cold will be produced; and if it be allowed suddenly to untwist, heat will be evolved from itself (besides heat generated externally by any work allowed to be wasted, which it does in untwisting). It is assumed that the torsional rigidity of the wire is diminished by an elevation of temperature, as the writer of this article had found it to be for copper. iron, platinum, and other metals (compare section 78).

(3.) A spiral spring suddenly drawn out will become lower in temperature, and will rise in temperature when suddenly allowed to draw in. [This result has been experimentally verified by Joule ("Thermodynamic Proper ties of Solids," Trans. Roy. Soc., 1858) and the amount of the effect found to agree with that calculated, according to the preceding thermodynamic theory, from the amount of the weakening of the spring which he found by experiment.] (4.) A bar or rod or wire of any substance with or with out a weight hung on it, or experiencing any degree of end thrust, to begin with, becomes cooled if suddenly elongated by end pull or by diminution of end thrust, and warmed if suddenly shortened by end thrust or by diminution of end pull; except abnormal cases in which with constant end pull or end thrust elevation of temperature produces shortening; in every such case pull or diminished thrust produces elevation of temperature, thrust or diminished pull lowering of temperature.

(5.) An india-rubber band suddenly drawn out (within its limits of elasticity) becomes warmer; and when allowed to contract, it becomes colder. Any one may easily verify this curious property by placing an india-rubber band in slight contact with the edges of the lips, then suddenly extending it-it becomes very perceptibly warmer: hold it for some time stretched nearly to breaking, and then suddenly allow it to shriuk-it becomes quite startingly colder, the cooling effect being sensible not merely to the lips but to the fingers holding the band. The first published statement of this curious observation is due to Gough (Memoirs of the Literary and Philosophical Society of Manchester, 2d series, vol. i. p. 288), quoted by Joule in his paper on "Thermodynamic Properties of Solids" (Transac tions of Royal Society, 1858). The thermodynamic conclusion from it is that an india-rubber band, stretched by a constant weight of sufficient amount hung on it, must, when heated, pull up the weight, and, when cooled, allow the weight to descend: this Gough, independently of thermodynamic theory, had found to be actually the case. experiment any one can make with the greatest case by hanging a few pounds weight on a common indiarubber band, and taking a red-hot coal in a pair of tongs, or a red-hot poker, and moving it up and down close to the band. The way in which the weight rises when the redhot body is near, and falls when it is removed, is quite startling. Joule experimented on the amount of shrinking per degree of elevation of temperature, with different weights hung on a band of vulcanized india-rubber. and found that they closely agreed with the amounts calculated by Thomson's theory from the heating effects of pull, and cooling effects of ceasing to pull, which he hud observed in the same piece of india-rubber.

The

74. The thermodynamic theory gives one formula1 by | which the differential formula is applicable. For air and which the change of temperature in every such case may be alculated when the other physical properties are known:

[blocks in formation]

other permanent gases Boyle's law of compression and Charles's law of thermal expansion supply the requisite data with considerable accuracy up to twenty or thirty atmospheres. The result is expressed by the formula

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

and t the initial pressure and temperature of the gas; p the sudden addition to the pressure; and, as before, the elevation of temperature.

For the case of p a small fraction of P the formula gives

0-(k-1) t

.. (2.)

It is by an integration of this formula that (1) is obtained. For common air the value of k is very approximately 141. Thus if a quantity of air be given at 15° C. (t= 289°) and the ordinary atmospheric pressure, and if it be compressed gradually up to 32 atmospheres, or dilated to

of an atmosphere, and perfectly guarded against gain or loss of heat from or to without, its temperature at several different pressures, chosen for example, will be according to the following table of excesses of temperature above the primitive temperature, calculated by (1)."

TABLE SHOWING EFFECTS OF PRESSURE ON TEMPERATURE.
Air given at temperature 15° Cent. (289° absolute).

Value of
P+P.

24808

16

Elevation of temperature

produced by com-
pression.

95°

221

389

612
911

Value of Lowering of temperature
P+p.

produced by dilata

tion.

71°

125

166

196

219

and J, Joule's equivalent (taken as 42400 centimetres). In using the formula for a fluid, p must be normal pressure equal in all directions, or normal pressure on a set of parallel planes, or tangential traction on one or other of the two sets of mutually perpendicular parallel planes which (section 43) experience tangential traction when the body is subjected to a simple distorting stress; or, quite generally, p may be the proper numerical reckoning (Mathematical Theory, chap. x.) of any stress, simple or compound. When p is pressure uniform in all directions, e must be expansion of bulk, whether the body expands equally in all directions or not. When p is pressure perpendicular to a set of parallel planes, e must be expansion in the direction opposed to this pressure, irrespectively of any change of shape not altering the distance between the two planes of the solid perpendicular to the direction of p. When p is a simple tangential stress, reckoned as in section 43, e must be the change, reckoned in fraction of the radian, of the angle, infinitely nearly a right angle, between the two sets of parallel planes in either of which there is the tangential traction denoted by p. In each of these cases p is reckoned simply in units of force per unit of area. Quite generally p may be any stress, simple or compound, and e must be the component (Math. Th., chaps. viii. and ix.) relatively to the type of p, of the strain produced by an elevation of temperature of one degree when the body is kept under constant stress. The constant stress for which K and e are reckoned ought to be the mean of the stresses which the body experiences with and without p. Mathematically speaking, p is to be infinitesimal, but practically 75. When change of temperature, whether in a solid it may be of any magnitude moderate enough not to give or a fluid is produced by the application of a stress, the any sensible difference in the value of either K or e, corresponding modulus of elasticity will be greater in virtue whether the "constant stress" be with p or without p, or of the change of temperature than what may be called the with the mean of the two: thus for air p must be a small static modulus defined as above, on the understanding that fraction of the whole pressure, for instance a small fraction the temperature if changed by the stress is brought back to of one atmosphere for air at ordinary pressure; for water or its primitive degree before the measurement of the strain is watery solutions of salts or other solids, for mercury, for performed. The modulus calculated on the supposition oil, and for other known liquids p may, for all we know, that the body, neither losing nor gaining heat during the amount to twenty atmospheres or one hundred atmospheres application of the stress and the measurement of its effect, without transgressing the limits for which the preceding retains the whole change of temperature due to the stress, formula is applicable. When the law of variation of Kwill be called for want of a better name the kinetic modulus, and e with pressure is known, the differential formula is because it is this which must (as in Laplace's celebrated readily integrated to give the integral amount of the change correction of Newton's calculation of the velocity of sound) of temperature produced by greater stress than those for be used in reckoning the elastic forces concerned in waves and vibrations in almost all practical cases. To find the ratio of the kinetic to the static modulus remark that e, according to the notation of section 74, is the diminution of the strain due to the change of temperature 6. Hence if M denote the static modulus (section 41), the strain actually produced by it when the body is not allowed either to gain or lose heat is-ee, or, with ◊ replaced by its value according to the formula of section 74,

W. Thomson, "Dynamical Theory of Heat" (§ 49), Trans, R.S.E., March 1851, and "Thermoelastic Properties of Matter," Quarterly Journal of Mathematics, April 1855 (republished Phil. Mag. 1877, second half year).

Ibid., Part vi. §§ 97, 100, Trans. R.S.E., May 1854. According to the scale there defined on ther no dynamic principles, independently of the properties of any particular substance, t is found, by Joule and Thomson's experiments, to agree very approximately with temperature centigrade, with 274° added.

But we have no knowledge of the effect of pressures of several thousand atmospheres in altering the expansibility or specific heat in liquids, or in fluids which at less heavy or at ordinary pressures are "gases."

M

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

where K denotes the thermal capacity of a stated quantity of the substance under constant stress, and N its thermal capacity under constant strain (or thermal capacity when the body is prevented from change of shape or change of volume). For permanent gases, and generally for 'fluids approximately fulfilling Boyle's and Charles's laws as said above, k is proved by thermodynamic theory to be approximately constant. Its value for all gases for which it has been measured differs largely from unity, and probably also for liquids generally (except water near its temperature of maximum density).

M' K

M' K

On the other hand, for solids whether the stress considered be uniform compression in all directions or of any other type, the value of differs but very or little M N from unity; and both for solids and liquids it is far from constant at different temperatures (in the case of water it is zero at 3°.9 Cent., and varies as the square of the difference of the temperature from 39 at all events for moderate differences from this critical temperature, whether above or below it). The following tables show the value of and the value of 0 by the formula of sec. 74, for different fluid and solid substances at the temperature 15° Cent. (289° absolute scale). The first table is for compression uniform in all directions; the second, necessarily confined to solids, is for the stress dealt with in "Young's Modu. lus," that is, normal pressure (positive or negative) on one set of parallel planes, with perfect freedom to expand or contract in all directions in these planes. A wire or rod pulled longitudinally is a practical application of the

latter.

THERMODYNAMIC TABLE I.

[ocr errors]

Pressure equal in all directions-Ratio of Kinetic to Static BulkModulus. Temperature 15° C. (289° absolute) J-42400 centimetres.

Bubstance.

Density -p.

Thermal Capacity Expanper unit albility

Lowering of Temperature produced by a pull

Static Young's Modulus

Deduced value of

of one

in grammes

M'

N

[ocr errors]

per

mass

[ocr errors]

gramme

[blocks in formation]
[blocks in formation]

-JKP

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

77. Experimental Results.-The following tables show determinations of moduluses of compression, of Young's modulus, and of moduluses of rigidity by various experimenters and various methods. It will be seen that the Young's moduluses obtained by Wertheim by vibrations, longitudinal or transverse, are generally in excess of those which he found by static extension; but the differences are enormously greater than those due to the heating and cooling effects of elongation and contraction (section 76), and are to be certainly reckoned as errors of observation. It is probable that his moduluses determined by statio elongation are minutely accurate; the discrepancies of those found by vibrations are probably due to imperfections of the arrangements for carrying out the vibrational method:

[merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]
[blocks in formation]

te2M\-1

metre

te

=M.

[blocks in formation]
[blocks in formation]

Glass, different specimens. Brass, different specimens.

Mean 150 x 106 Mean 350 x 106

114 x 10

1.0040 1-22

[blocks in formation]

Glass, fint

9-942

•1770 000026 000000340

[blocks in formation]

Brass,

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small]
[blocks in formation]

Amaury and Descamps, Comptes Ren dus, tome xvii. p. 1564 (1869).

Everett's Illustrations of the CentimetreGramme

Second System of Units. Wertheim, Ann. de Chim., 1848.

Authority.

Wertheim, Annales

de Chimie, 1848.

Everett's Ill.

of the Centimetre-GrammeSecond System

240 x 106

Brass, drawn

[ocr errors]

373 x 106

834 x 106

Iron, wrought Copper

785 x 106

542 x 106

of Units.

456 x 108

« EelmineJätka »