Example IV. Plate Butt-joint, with a pair of covering plates, doublerivetted. Fig. 125.

4 x Sectional area of rivet

Sectional area of plate between two holes in one row

Length of each covering-plate = 2 × overlap

from 3 to 31⁄2 c.

NOTE.-The length of a rivet, before being clenched, measuring from the head, is about 4 t for overlapped-joints, and 5t for butt-joints with covering-plates.

Example V Suspension bridge chain-joint. The chain of a suspension bridge consists of long and short links alternately. Each long link consists of one or more, say of n, parallel flat bars, of a shape resembling fig. 64, Article 138, placed side by side; each bar has a round eye at each end. Each short link consists of n + 1 parallel flat bars, with round eyes at their ends, which are placed between and outside of the ends of the parallel bars of the long links; so that the end of each long bar is between the ends of a pair of short bars. The eyes of the long and short bars at each joint form one continuous cylindrical hole or socket, into which a bolt or pin is fitted, to connect the links together. To break the chain at a joint, by the giving way of the bolt, that bolt must be sheared across at 2 n places at once. Hence, let S denote the total sectional area of the bars in a link, and d the diameter of the bolt; then S' 2n x 0.7854 d2 = 1·5708 n d'; and because S' should

=

S, we have

5

281. Fastenings of Timber Ties.-In timber framing, a tie may be connected with the adjoining pieces of the frame either by having their ends abutting against notches cut in the tie (as shown at A, A, fig. 81, Article 161), or by means of bolts or pins. In either case, the tie may yield to the stress in two ways,-by being torn asunder at the place where its transverse section is least (that is, where it is notched or pierced, as the case may be),-or by having the part beyond the notch, or beyond the bolt-hole, sheared off or sheared

out, as the case may be. In order that the material may be economically used, equation 1 of Article 280 should be fulfilled, viz. :—

This condition serves to determine the distance of the notch, or of the bolt-hole, or of the nearest bolt-hole where there are more than one, from the end of the tie, in the following manner :-

Leth be the effective depth of the tie, left after deducting the depth of the notch, or the diameters of bolt-holes, and d the distance of the notch, or of the nearest bolt-hole, from the end of the tie; then for a notch

In determining the number n, it is to be observed, that if two or more bolts pierce the same layer of fibres, the resistance to the shearing out of the part of that layer between the end of the tie and the most distant of the bolts is nearly the same as if that bolt existed alone; so that the most distant only of such a set of bolts is to be reckoned in using equation 3. In general, the piercing of the same layer of fibres by more than one bolt is unfavourable to economy.

SECTION 5.-On Resistance to Direct Compression and Crushing.

282. Resistance to Compression, when the limit of proof stress is not exceeded, is sensibly equal to the resistance to extension, and is expressed by the same "modulus of elasticity," already mentioned and explained in Articles 257, 265, 266, and 268. When that limit is exceeded, the irregular alterations undergone by the figure of the substance render the precise determination of the resistance to compression difficult, if not impossible.

283. Modes of Crushing.—Splitting, Shearing, Bulging, Buckling, Cross-breaking.-Crushing, or breaking by compression, is not a simple phenomenon like tearing asunder, but is more or less complex and varied, according to the texture of the substance. The modes in which it takes place may be classed as follows:

I. Crushing by splitting (fig. 126) into a number of prismatic fragments, separated by smooth surfaces whose general direction is nearly parallel to the direction of the crushing force, is characteristic

of hard homogeneous substances of a glassy texture, such as vitrified bricks.

Fig. 126.

B

Fig. 127.

Fig. 128.

Fig. 129.

II. Crushing by shearing or sliding of portions of the block along oblique surfaces of separation is characteristic of substances of a granular texture, like cast iron, and most kinds of stone and brick. Sometimes the sliding takes place at a single plane surface, like A B in fig. 127; sometimes two cones or pyramids are formed, like c, c, in fig. 128, which are forced towards each other, and split or drive outwards a number of wedges surrounding them, like w, w, in the same figure. Sometimes the block splits into four wedges, as in fig. 129.

The surfaces of shearing make an angle with the direction of the crushing force, which Hodgkinson (who first fully investigated those phenomena) found to have values depending on the kind and quality of material. For different qualities of cast iron, for example, that angle ranges from 42° to 32°. The greatest intensity of shearing stress is on a plane making an angle of 45° with the direction of the crushing force; and the deviation of the plane of shearing from that angle shows that the resistance to shearing is not purely a cohesive force, independent of the normal pressure at the plane of shearing, but consists partly of a force analogous to friction, increasing with the intensity of the normal pressure.

Hodgkinson considers that in order to determine the true resistance of substances to direct crushing, experiments should be made on blocks in which the proportion of length to diameter is not less than that of 3 to 2, in order that the material may be free to divide itself by shearing. When a block which is shorter in proportion to its diameter is crushed, the friction of the flat surfaces between which it is crushed has a perceptible effect in holding its parts together, so as to resist their separation by shearing; and thus the apparent strength of the substance is increased beyond its real strength.

In all substances which are crushed by splitting and by shearing, the resistance to crushing considerably exceeds the tenacity, as an examination of the tables will show. The resistance of cast iron to crushing, for example, was found by Hodgkinson to be somewhat more than six times its tenacity.

III. Crushing by bulging, or lateral swelling and spreading of the block which is crushed, is characteristic of ductile and tough materials. Owing to the gradual manner in which materials of this nature give way to a crushing force, it is difficult to determine their resistance to that force exactly; that resistance is in general less, and sometimes considerably less, than the tenacity. In wrought iron, the resistance to the direct crushing of short blocks, as nearly as it can be ascertained, is from 2 4 to of the tenacity.

3

IV. Crushing by buckling or crippling is characteristic of fibrous substances, under the action of a thrust along the fibres. It consists in a lateral bending and wrinkling of the fibres, sometimes accompanied by a splitting of them asunder. It takes place in timber, in plates, and in bars longer than those which give way by bulging. The resistance of fibrous substances to crushing is in general considerably less than their tenacity, especially where the lateral adhesion of the fibres to each other is weak compared with their tenacity. The resistance of most kinds of timber to 1 2 crushing, when dry, is from to 2

3

of the tenacity. Moisture in the timber weakens the lateral adhesion of the fibres, and reduces the resistance to crushing to about one-half of its amount in the dry

state.

V. Crushing by cross-breaking is the mode of fracture of columns and struts in which the length greatly exceeds the diameter. Under the breaking load, they yield sideways, and are broken across like beams under a transverse load. This mode of crushing will be considered after the subject of resistance to bending.

284. A Table of the Resistance of Materials to Crushing by a Direct Thrust, in pounds avoirdupois per square inch, is given at the end of the volume. So far as that table relates to the strength of brick and stone, reference has already been made to it in Article 235. It is condensed from the experimental data given by various authorities, especially by Tredgold, Fairbairn, Hodgkinson, and Captain Fowke.

285. Unequal Distribution of the Pressure on a pillar arises from the line of action of the resultant of the load not coinciding with the axis of figure of the pillar, so that the centre of pressure of a cross section of the pillar does not coincide with its centre of figure, but deviates from it in a certain direction by a certain distance, which may be denoted by ro

In this case the strength of the pillar is diminished in the same ratio in which the mean intensity of the pressure is less than the

maximum intensity; that is to say, in a ratio which may be denoted by

That ratio may be found with a precision sufficient for practical purposes, by considering the pressure at any cross section of the pillar as an uniformly varying stress, as defined in Article 94. Consequently the following is the process to be pursued:

Find, by the methods of Article 95, the principal axes and moments of inertia of the cross section of the pillar; and thence determine the neutral axis conjugate to the direction of the deviation ro. Let be the angle made by that axis with the direction of the deviation ro; then the perpendicular distance of the centre of pressure from the neutral axis will be

Find the moment of inertia of the cross section relatively to the neutral axis, and denote it by I; then from equations 1, 2, and 4 of Article 94, it appears that if x be the greatest perpendicular distance of the edge of the cross section from the neutral axis in the same direction with x, the greatest intensity of pressure will be

P being the total pressure, and S the area of the section of the pillar. Consequently the ratio required is

Values of S, for certain symmetrical figures, and of I for the principal axes of these figures, have already been given in the table of Article 205, from which are computed the following values of the x, S

factor in the denominator of the preceding formula :—

I