explain one another; for if K represent the transverse section of the column in square inches, and a the constant inclination of the plane of rupture to the base, then will K sec. a represent the area of the plane of rupture. So that if y represent the resistance opposed, by the coherence of the material, to the sliding of one square inch upon the surface of another *, then will yK sec. a represent the resistance which is overcome in the rupture of the column, so long as its height lies between the supposed limits; which resistance being constant, the pressure applied upon the summit of the column to overcome it must evidently be constant. Let this pressure be represented by P, and let CD be the plane of rupture. Now it is evident that the inclination of the direction of P to the perpendicular QR to the surface of the plane, or its equal, the inclination a of CD to the base of the column, must be greater than the limiting angle of resistance of the surfaces; if it were not, then would no pressure applied in the direction of P be sufficient to cause the one surface to slide upon the other, even if a separation of the surfaces were produced along that plane.

D

R

Let P be resolved into two other pressures, whose directions are perpendicular and parallel to the plane of rupture; the former will be represented by P cos. a, and the friction resulting from it by P cos. a tan. ; and the latter, represented by P sin. a, will, when rupture is about to take place, be precisely equal to the coherence Ky sec. a of the plane of rupture increased by its friction P cos. a tan. , or P sin. «= Ky sec. a+P cos. a tan. p, whence by reduction

It is evident from this expression that if the coherence of the material were the same in all directions, or if the unit of coherence y opposed to the sliding of one portion of the

*The force necessary to overcome a resistance, such as that here spoken of, has been appropriately called by Mr. Hodgkinson the force necessary to shear it across.

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

2a-4=1, or a

4

+2; whence it follows that a plane in

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 coherence y is not the same for every direction of the plane of rupture, or that the value of varies greatly in different qualities 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 also the angle of rupture.†

* Seventh Report of British Association, p. 349.

† 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 SECTION OF RUPTURE IN A BEAM.

407. When a beam is deflected under a transverse strain, the material 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, P2, P3. 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 sur

face 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 bordering 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; c1 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 extension or compression of the material (according as that distance is measured on the extended or compressed side), is therefore represented per square inch by

S

p, and the elastic force thus developed upon the element

S

AK of the section of rupture by PAK, so that the moment

S

of this elastic force about R is represented by AK, and the sum of the moments of all the elastic forces upon the sec

S

tion of rupture about the axis of rupture by p2AK*; 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

* 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.

rupture 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 P, about that point. Representing, therefore, by p, the perpendicular let fall from the point R upon the direction of P1, we have

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 accuSI C1

rately represented by the expression

This remark

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.