coherence 7 opposed to the sliding of one portion of the mass upon another were accurately the same in every direction in which the plane CD may be imagined to intersect the mass, then would the plane of actual rupture be inclined to the base at an angle represented by the formula

since the value of P would in this case be (equation 634) a minimum when sin. (2a-4) is a maximum, or when

T

2a = or a= + ; whence it follows that a plane in

4

clined to the base at that angle is that plane along which the rupture will first take place, as P is gradually increased beyond the limits of resistance.

The actual inclination of the plane of rupture was found in the experiments of Mr. Hodgkinson to vary with the material of the column. In cast iron, for instance, it varied according to the quality of the iron from 48° to 58°*, and was different in different species. By this dependence of the angle of rupture upon the nature of the material, it is proved that the value of the modulus of sliding colierence 7 is not the same for every direction of the plane of rupture, or that the value of varies greatly in different quali

ties of cast iron.

Solving equation (634) in respect to y we obtain

from which expression the value of the modulus y may be determined in respect to any material whose limiting angle of resistance is known, the force P producing rupture, under the circumstances supposed, being observed, and alsc the angle of rupture.†

THE SECTION OF RUPTURE IN A BEAM.

407. When a beam is deflected under a transverse strain,

* Seventh Report of British Association, p. 849.

A detailed statement of the results obtained in the experiments of Mr. Hodgkinson on this subject is contained in the Appendix to the "Illustrations of Mechanics" by the author of this work.

the inaterial on that side of it on which it sustains the strain is compressed, and the material on the opposite side extended. That imaginary surface which separates the compressed from the extended portion of the material is called its neutral surface (Art. 354.), and its position has been determined under all the ordinary circumstances of flexure. That which constitutes the strength of a beam is the resistance of its material to compression on the one side of its neutral surface, and to extension on the other; so that if either of these yield the beam will be broken.

The section of rupture is that transverse section of the beam about which, in its state bordering upon rupture, it is the most extended, if it be about to yield by the extension of its material, or the most compressed if about to yield by the compression of its material.

In a prismatic beam, or a beam of uniform dimensions, it is evidently that section which passes through the point of greatest curvature of the neutral line, or the point in respect to which the radius of curvature of the neutral line is the least, or its reciprocal the greatest.

GENERAL CONDITIONS OF THE RUPTURE OF A BEAM.

---

408. Let PQ be the section of rupture in a beam sustaining any given pressures, whose resultants are represented, if they be more in number than three, by the three pressures P1, P., P. Let the beam be upon the point of breaking by the yielding of its material to extension at the point of greatest extension P; and let R represent, in the state of the beam bordering upon rupture, the intersection of the neutral surface with the section of rupture; which intersection being in the case of rectangular beams a straight line, and being in fact the neutral axis, in that particular position which is assumed by it when the beam is brought into its state bor. dering upon rupture, may be called the axis of rupture; AK the area in square inches of any element of the section of rupture, whose perpendicular distance from the axis of rupture R is represented by p; S the resistance in pounds

opposed to the rupture of each square inch of the section at P; c, and c, the distances PR and QR in inches.

The forces opposed per square inch to the extension and compression of the material at different points of the section of rupture are to one another as their several perpendicular distances from the axis of rupture, if the elasticity of the material be supposed to remain perfect throughout the section of rupture, up to the period of rupture.

Now at the distance c, the force thus opposed to the extension of the material is represented per square inch by S; at the distance p the elastic force opposed to the exten sion or compression of the material (according as that distance is measured on the extended or compressed side), is

S

therefore represented per square inch by p, and the elastic

force thus developed upon the element AK of the section of

S

rupture by PAK, so that the moment of this elastic force

about R is represented by pAK, and the sum of the mo

ments of all the elastic forces upon the section of rupture

about the axis of rupture by 'K;* or representing the

moment of inertia of the section of rupture about the axis of rupture by I, the sum of the moments of the elastic forces upon the section of rupture about its axis of rupture

SI

is represented, at the instant of rupture, by .+ Now the elastic forces developed upon PQ are in equilibrium with the pressures applied to either of the portions APQD or BPQC, into which the beam is divided by that section; the sum of their moments about the point P is therefore equal to the moment of R, about that point. Representing, therefore, by p, the perpendicular let fall from the point R upon the direction of P, we have

It will be observed, as in Art. 358., that the elastic forces of extension and those of compression tend to turn the surface of rupture in the same direction about the axis of rupture.

This expression is called by the French writers the moment of rupture: the beam is of greater or less strength under given circumstances according as it has a greater or less value.

409. If the deflexion be small in the state bordering upon rupture, and the directions of all the deflecting pressures be perpendicular to the surface of the beam, the axis of rupture passes through the centre of gravity of the section, and the value of c, is known. Where these conditions do not obtain, the value of c, might be determined by the principles laid down in Arts. 355. and 381. This determination would, however, leave the theory of the rupture of beams still incomplete in one important particular. The elasticity of the material has been supposed to remain perfect, at every point of the section of rupture, up to the instant when rupture is about to take place. Now it is to be observed, that by reason of its greater extension about the point P than at any other point of the section of rupture, the elastic limits are there passed before rupture takes place, and before they are attained at points nearer to the axis of rupture; the forces opposed to the extension of the material cannot therefore be assumed to vary, at all points of PR, accurately as their distances from the point R, in that state of the equilibrium of the beam which immediately precedes its rupture; and the sum of their moments cannot therefore be assumed to be acSI C1

curately represented by the expression

This remark af

fects, moreover, the determination of the values of h and R (Arts. 355. and 381.), and therefore the value of c,

To determine the influence upon the conditions of rupture by transverse strain of that unknown direction of the insistent pressures, and that variation from the law of perfect elasticity which belongs to the state bordering upon rupture, we must fall back upon experiment. From this it has resulted, in respect to rectangular beams, that the error produced by these different causes in equation (637) will be corrected if a value be assigned to c, bearing, for each given material, a constant ratio to the distance of the point P from the centre of gravity of the section of rupture; so that e representing the depth of a rectangular beam, the error will be corrected, in respect to a beam of any material, by assigning to c, the value mac, where m is a certain constant dependent upon the nature of the material. It is evident that this correction is equivalent to assuming cc, and assigning to S the value S instead of that which it has hitherto

been supposed to represent, viz. the tenacity per square inch of the material of the beam.

It is customary to make this assumption. The values of S corresponding to it have been determined, by experiment, in respect to the materials chiefly used in construction, and will be found in a table at the end of this work. It is to these tables that the values represented by S in all subsequent formulæ are to be referred.

410. From the remarks contained in the preceding article, it is not difficult to conceive the existence of some direct relation between the conditions of rupture by transverse and by longitudinal strain. Such a relation of the simplest kind appears recently to have been discovered by the experiments of Mr. E. Hodgkinson*, extending to the conditions of rupture by compression, and common to all the different varieties of material included under each of the following great divisions-timber, cast iron, stone, glass.

The following tables contain the summary given by Mr. Hodgkinson of his results:

The following table shows the uniformity of this ratio in respect to different varieties of the same material:

This discovery was communicated to the British Association of Science at their meeting in 1842; it opens to us a new field of theoretical research.