SECT. III.

ABSTRACTS OF PAPERS IN FOREIGN TRANSACTIONS AND PERIODICALS.

Equilibrium of Pulverulent Bodies. By PROF. J. BOUSSINESQ.1 (Mémoires couronnés et Mémoires des Savants étrangers, publiés par l'Académie royale des Sciences de Belgique, vol. xl., p. 1-180.)

Pulverulent bodies, such as sand or earth recently turned up, whose particles in sliding on one another experience no resistance except that arising from their mutual friction, are susceptible of many distinct modes of equilibrium.

The only one which has hitherto been studied is the limiting case when the mass is on the point of motion, and when the friction of the particles attains its maximum. Formulæ relating to this state were obtained as early as 1856 by Rankine, and subsequently by Levy and others.

But there is another kind of equilibrium equally important to consider, viz., that which is produced in the interior of a pulverulent mass confined by a wall sufficiently firm to prevent any disturbance of the particles. In this state the friction of the various strata on each other is generally less than in the other; just as, in a solid body in elastic equilibrium, the strains remain always less than those which would produce permanent alteration of its structure: the particles are then retained in position by their mutual actions less forcibly than if the wall were to give under their pressure, and they exercise upon it a thrust greater than the formulæ of Rankine and Levy give. This is the kind of equilibrium studied in the memoir. It is called "elastic equilibrium" because the pressures induced depend on certain small deformations, which the mass, supposed at first homogeneous and without weight, would undergo if it became, as it really is, heavy.

The bodies in question occupy a position midway between solids and fluids; for whilst solids and fluids, submitted to pressures varying from zero to considerable intensities, oppose to any given deformation a constant force-finite for the first, zero for the second -pulverulent bodies resist change of form with an energy so much the greater as the mean pressure to which they are subjected is

1 “Essai théorique sur l'équilibre des massifs pulverulents, comparé à celui de massifs solides et sur la poussée des terres sans cohésion," par M. J. Boussinesq, professeur à la faculté des sciences de Lille. Brussels, 1876.

greater; fluids almost when not compressed, they become rigid like solids under pressure. Their co-efficient of rigidity (co-efficient d'élasticité de glissement-Lamé's μ), instead of being constant, as in the case of solids, or zero, as in the case of fluids, appears to be proportional to the mean pressure p to which they are exposed. The Author deduces this from the expressions which represent in isotropic bodies the mean of the principal elastic forces (i.e. the mean pressure p with its sign changed), and the differences of these forces, in terms of the three principal deformations, 81, 82, 83. Retaining in all the results the terms of two dimensions in 81, 82, 83, and expressing that the matter under consideration ceases, for finite values of 81, 82, 83, to be subject to the action of tangential elastic forces when p is zero, he finds that the normal and tangential components, N1, N2, N3, T1, T2, T3 (according to Lamé's notation), of the pressure, per unit of area, on three elementary planes perpendicular to the axes, have the values (provided they do not exceed certain limits)

where m is a constant positive co-efficient of considerable magnitude, and where u, v, w designate the molecular displacements parallel to the axes of x, y, z. The same analysis proves that the cubic expansion (dilatation) may be neglected in comparison with the three linear expansions to whose algebraic sum it is sensibly

d u d v

dx

d w

+ + = 0 dy

d z

equal, or that the condition of incompressibility holds. The remaining three equations between the Ñ's and T's necessary for determining n, v, w, and p are

where p is the density of the mass, and X, Y, Z the components of gravity parallel to the axes.

The special conditions existing at the bounding surfaces are as follows:

1st. At free surfaces, the effective pressure on the outside layer is zero; for the atmospheric pressure around each grain does not influence the mutual action between it and contiguous grains.

2nd. At fixed boundaries (such as the posterior faces of retaining walls), which are sufficiently rough-as they generally are -to prevent motion in the particles adjacent to them, u, v, and w vanish ; and in the case of a surface infinitely smooth, the normal

component of the displacement and the tangential components of the pressures are zero.

The integrations of the equations are easy when the mass-of indefinite extent in all other directions-is limited above by a plane making an angle w with the horizon. The states of equilibrium of a pulverulent mass bounded in this way are infinite in number, and belong to one of two series, according to the values given to two of the arbitrary constants, c, c', introduced in the integration.

Any system of parallel material straight lines, situated in a vertical plane perpendicular to that of the upper slope, changes, through the small deformations it undergoes, into a family of concentric conic sections, similar and similarly situated, the axes of which bisect the four angles which are formed by a vertical straight line and the line of intersection of the upper surface of the mass with the plane aforesaid. These conic sections become circles of very large radius for straight lines parallel to the slope, and are reduced to parallel straight lines when one of the arbitrary constants c = 0.

There are, in particular, two systems of equidistant and parallel straight lines, originally at right angles and inclined, the one to the vertical, and the other to the horizontal, at an angle e, which after deformation of the mass remain straight, parallel, and unchanged in length, but are turned with respect to one another sin w through a small angle The squares which they m cos (w 2 €) originally formed by their intersections become diamonds, and the ultimate form of the mass may be arrived at by supposing it originally divided into indefinitely thin layers, inclined at an angle e to the vertical, which slide in their respective planes in such a manner, that if one be considered as fixed, any other situated at a distance D in front of it will be displaced downwards by a quantity D sin w The case in which c = 0 comprises an infinite m cos (w number of modes of equilibrium, since there still remains the arbitary constant c', is the most interesting, as it is the only one in which, by properly determining e', the conditions existing at a boundary, whether smooth or rough, are found to hold throughout the extent of any plane cutting the upper slope in a horizontal straight line, in other words, inclined to the vertical at the angle e; and it is the only one in which the particles in this plane remain immovable during the deformation of the mass. Hence equilibrium will still subsist, if the material on one side of the plane be replaced by a retaining wall having this plane for its posterior face. This is naturally the mode of equilibrium produced where such a wall really exists, and it will be the same for two directions of the wall at right angles to each other.

2 €)

When, therefore, the posterior face of a retaining wall is rough, and inclined to the vertical at an angle e, the settlement of the mass takes place by displacement parallel to the face, and is, as

already stated, equal, for any particle, to the product of its distance sin w from the face by the constant factor

m cos (w - 2€)

Admitting the existence of the special condition set forth above at the surface of the wall, the pressure R on an unit of area at a depth L (measuring along the face), and the angle 4, which its line of action makes with the prolongation of the normal thereto, toward the interior of the wall, are given by sin w

R = Kpg L

tan $1 =

cos (w — 2 €)'

cos 2 (we) sin 1

If, on the contrary, the wall were mathematically smooth, the

values of tan 4, and K would be 0 and

sin €

tan (e

w)

The total pressure P on an unit of width of the wall is Kpg L2 (where K has one of the values given above), and the point of application of its resultant, whose direction is parallel to R's, is at a depth L (L being in this case the total height of the wall measured along the face,

=

h

COS €

In obtaining these results the limit of elasticity of the matter has not been taken into consideration. Now, just as in solving the problem of the equilibrium of a perfectly elastic solid, under the action of given forces, it is necessary to express that the greatest deformation at any point must be less than that which will cause a permanent set, so it is necessary, in this case, to express that the greatest linear extension at different points of the mass must not exceed that limit which it cannot surpass without danger of disruption (éboulement).

Pulverulent bodies, being without cohesion, are incapable of transmitting tensions, whence it follows, by the analysis, that the

greatest linear extension 8, must be less than the ratio

1

1

2 m

The

limit of elasticity, being thus less than can always be ex

2 m'

O and generally), whose value for each particular kind of matter 2

must be determined by experience. This angle may be called the angle of internal friction of the mass.

It therefore appears that the only states of equilibrium possible are those in which the condition p > 0 and ò̟ >

the imperfect elasticity of the matter, are satisfied. A first consequence of the imposition of these new conditions is to make the constant c vanish, i.e. to reduce all the possible

modes of equilibrium of an indefinite mass to those which can subsist in a mass bounded by a plane wall, and further such modes of equilibrium, which then depend on a single parameter ‹, say, sin2 w are shown to be possible only when cos2 (w - 2 c) sin2 d'

Their number, unlimited so long as the inclination of the upper slope to the horizon is zero, becomes more and more restricted as that inclination becomes greater, and when w = they are reduced to one only; when exceeds & equilibrium becomes impossible. Thus the theory explains the impossibility of a pulverulent mass existing with a slope whose inclination exceeds the angle of friction of the matter of which it is composed.

The formulæ already obtained depend on special conditions, and relate to the case of a mass, originally without weight and free from pressure, which, on becoming heavy, takes a new state of equilibrium without the layer next the retaining wall having moved, if the wall be rough, or having moved out of its own plane, if it be smooth. Practically, however, in forming such a mass against a rough immovable wall already built, the particles contiguous to the wall only remain stationary so long as they are but slightly compressed. But the addition of successive quantities of earth or sand subjects them to increasing pressure, and causes them to move through finite distances, the result being an entirely different state of equilibrium. This state is that in which the internal stability of the mass is greatest, or in which the maximum extension & has at each point its least value compatible with the degree of resistance of the wall. The analysis shows that 8, attains its sin w minimum value + 2 m which corresponds to the most stable state of internal equilibrium. If, however, the wall is not sufficiently firm to allow of the mass attaining its maximum stability, the state of equilibrium produced is that in which the whole resistance of the wall is utilised. In these cases the value of the parameter 8, which defines the mode of equilibrium, is determined by the relation

€1

when € =

W

2

tan 2 € =

cos 2 cos 2 (w e)

This, then, is the value of €

Whether the stability of the mass attain its maximum or not, the direction and intensity of the resultant pressure on a wall inclined at an angle to the vertical are given by PK pg L2,