All these modes of equilibrium are stable; for the wall may be supposed without weight, but held in position by an externally applied force, and by the pressure of the mass which it supports. If then the wall were to move towards or from the mass, the pressure of the latter upon it would increase or diminish, and this would exceed or be exceeded by the external force, so that the wall would tend to return to its original position.

Hence it is not necessary to make the thickness of a retaining wall greater than will enable it to support a pressure slightly exceeding that which corresponds to the least stable state of equilibrium.

Assuming the posterior face of the wall to be vertical (in which case the direction of the resultant pressure will be parallel to the upper slope), and to be 45°, the following table gives the greatest and least numerical values of K for different values of w.

TABLE I.

1= 0°, ± 10°, ± 20°, ± 30°, ± 40°, ± 45° Greatest value ∙1716, 1765, 1935, ·2320, ·3404, ·7071 Least of K 1, 9848, 9397, 8660, 7660, 7071

The equilibrium of such a mass will therefore be stable provided the wall can support a pressure, applied at one-third of its height, parallel to the upper slope, and slightly greater than Kpg L2, where K has the values 1716, &c.; but in order that the most stable state of equilibrium may be realised, the values of K must be taken from the lower line in the table.

For example, the retaining wall M ON M1, whose faces M O, M, N are vertical, would tend to turn about the front edge M1.

In order to determine the condition of equilibrium it is necessary to equate the moment of the weight of the wall about M1 to the moment of the pressure of the mass (P) about the same axis.

Calling p' the density of the wall, b its breadth, and h its height, the weight of an unit of length will be expressed by p' g h b, and its moment about M, by p' g b2 h. The point of application of the pressure P will be c, at one-third the height of the wall, and the direction of its line of action parallel to the upper slope, its moment about M1 is therefore

pg h2 K (3 h cos w b sin w).

By equating these expressions, and reducing, the value of the

1. Suppose the thickness of the wall is to be just sufficient to insure equilibrium, and assume for p' the value 3p, which will not in general be far from the truth, then, substituting for K its numerical value from the upper line of Table I., the following results are obtained

b (minimum) = .1953 •1866 •1802 1761 1786 • 2060 hstability)

2. If, on the contrary, b is to be such that the most stable mode of equilibrium may subsist, the value of K must be taken from the lower line in the table, and

A consideration of these figures proves that the practical rule of making the thickness of a retaining wall equal to one-third of its height insures in general sufficient stability.

The Author then proceeds to examine the condition of the mass when disruption commences, and deduces the equations relative thereto. He also obtains in polar co-ordinates the equations applicable to the limiting state of equilibrium of a mass, plastic or pulverulent, submitted to pressures very great in comparison with its weight.

M. L.

On the Limes and Cements of Casale, Piedmont, and Liguria. By G. SIGNORILE.

(Giornale del Genio civile, vol. xv., pp. 301-317.)

In the year 1846 the Author was instructed by the Italian Government to examine the limestones of Casale, Piedmont, and Liguria, with reference to their use on railways.

In 1850 the making of forty millions of bricks for the lining of the Giovi tunnel, led to further experiments on the calcining of the materials for the artificial mortar with pit coal, instead of with wood, which had been employed in the first instance. It resulted that lime which was excellent when calcined with wood, deliquesced if calcined with coal containing pyrites. The Author is now desirous to review the labours of thirty years, and to republish his earlier discoveries, bringing down the account so as to comprise those of the latest date.

The methods of investigation employed have been direct experiment, and chemical analysis. In order to avoid the numerous anomalies which occur in the action of hydraulic materials, certain precautions are indispensable. Perfect calcination has to be obtained in kilns of the ordinary structure, heated with wood, and fed from above. The tempering of the lime should be effected with the least quantity of water required to produce a stiff argillaceous consistency, similar to that required for making bricks when tempered. The lime should be kept under water, the temperature of which should vary between 51° and 70° Fahr.; a necessary precaution in the case of experiments, some of which last for six and others for twelve months or more.

The first sample of the immersed lime was tested by Vicat's needle; and the progressive hardening of the mortar was ascertained by means of the percussive apparatus designed by the same analyst. The durability of mortars composed of lime and sand was tested by forming them into small bricks, which were exposed to the atmosphere for periods, varying from six months to several years; the samples being always kept in a somewhat damp situation. The resistance offered to compression by these samples was measured by an ordinary machine for applying pressure by the attachment of weights to a lever, up to the point of crushing. Some of these experiments were made six months after the lime was burned; some after one, two, and even three years.

The general results of the experiments, the details of which are presented in a tabular form, are as follows:

The limes of Casale, of Montferrat, and of Superga (near Turin) Occupy the first rank among true hydraulic limes, and are but little inferior to the best French limes investigated by Vicat.

Next come the limestones of Liguria, belonging to the brown Eocene calcareous beds. These are followed by those of the

LIMES AND CEMENTS OF CASALE, PIEDMONT, AND LIGURIA. 285

dolomitic rocks, which contain a certain proportion of clay. Finally come the imperfectly hydraulic limes of Tortona and Godiasco, which abound in the valley of the Borbera, under the mountains of Antola. All the above rocks, with the exception of the Superga, belong to the Eocene series, a formation which prevails in Emilia, Tuscany, and the Roman provinces. Throughout this system occur good materials for hydraulic mortars and cements. In the neighbourhood of Superga, limestone is found in the form of pebbles, or small boulders, in conglomerates of the middle tertiary or miocene epoch.

The difference in chemical constitution of the limestones in the same stratum is insisted on by the Author. Stone from the same quarry is generally regarded as of one quality, while the difference due to a few feet of vertical height in the layers from which the sample is taken may be very great. Thus at Casale there is a range of from 1 to 4 millimètres in the grade of hydraulic value of the lime; and the same is the case both at Superga and in Liguria. In the brown Eocene rocks an enormous difference occurs between the hydraulic value of successive strata, so that limes of excellent, of limited, and of extremely low value are intermixed, without evincing by any exterior characteristic the respective nature of each layer.

Another difficulty is the small increase of bulk of these limes when slaked. The slaking is troublesome, as the limes in question commence to soften before removal from the furnace, at certain temperatures, and in this state only continue to change very slowly, so as often to effloresce after use in building.

As to the chemical analysis of limestone, the Author states that ́ the method in present use is, to determine the insoluble residuum left, when the specimen is treated with boiling chloridic acid. This method, though easy of application, is not unlikely to prove misleading, owing to the facts that the residuum may contain extraneous matter, and that the particular molecular state of the silex, when in the well, which is a matter of much importance, cannot thus be ascertained.

The Author enters minutely into the theory of efflorescence in magnesian limestone cements. Proofs are cited as to the remarkable durability of the artificial cement before described, as used in railway works in the valley of the Scrivia in 1848. The Paper concludes with the declaration that magnesian lime, without the presence of either clay or silex, can never be truly hydraulic; and that while the best natural hydraulic cements never exceed a cohesive value indicated by 4.4 to 5 millimètres by the method of Vicat, artificial cements, with the mixture of a proper proportion of clay, obtain a value indicated by a resistance of 2.2 millimètres.

F. R. C.

Graphic Diagrams for the Strength of Teak Beams.
By GUILDFORD L. MOLESWORTH, M. Inst. C.E.

(Professional Papers on Indian Engineering, 2nd series, vol. vi., pp. 407–412, 5 diagrams.)

Mr. E. H. Stone, in his tables and diagrams "for facilitating the calculation of teak beams," employed separate diagrams for each span, and these diagrams involved curved lines. Mr. Molesworth has simplified the formula by expressing them in terms of the ratio of the depth to the span; and thus one diagram suffices for all spans. Further, by making the scale for spans proportional to the square of the span, and not as the span simply, the curves in the diagrams are replaced by straight lines. Scales are added for other timber besides teak.

The diagrams are constructed for transverse strength, for stiffness, and for weight; and for beams of two scantlings, in which the breadth is equal to half the depth, and to two-thirds of the depth. The formula employed to determine the working load, are

L = safe load in lbs. distributed;

W = weight, in lbs., of a beam of which the length is equal

to the span;

S = clear span, in feet;

T = modulus of rupture; being the breaking weight applied at the centre of a beam 12 inches long and 1 inch square, supported at both ends;

E = modulus of elasticity; or that load which, if applied to the centre of a beam 12 inches long and 1 inch square, supported at both ends, will produce a deflection of 1 inch;

12\3

when the

N

when the breadth

is half the depth; or, 0.02667
is two-thirds of the depth;

12\4

N

when the breadth

= co-efficient of stiffness = 0.02

N = number of times the depth is contained in the span ;

k = co-efficient of weight; varying with the depth of the

20 =

beam;

weight, in lbs., of a cubic foot of timber.