per square inch. That of a double-rivetted joint, per square inch of the iron left between the rivet holes (if drilled, and not punched), is the same; that of a single-rivetted joint somewhat less, as the tension is not uniformly distributed. It is convenient in practice to state the tenacity of rivetted joints in lbs. per square inch of the entire plate; and it is so stated in the annexed table, in which the results for rivetted joints are from the experiments of Mr. Fairbairn, and that for a welded joint from an experiment by Mr. Dunu. The joints of plate-iron boilers are single rivetted; but from the manner in which the plates break joint, the ultimate tenacity of such boilers is considered to approach more nearly to that of a double-rivetted joint than to that of a single-rivetted joint.

Wrought-iron plate joints, double-rivetted, the dia

meter of each hole being of the pitch, or dis-
tance from centre to centre of holes,....

35,000

Wrought-iron plate joints, singl, rivetted,

28,000

Wrought-iron boiler shells, with single-rivetted joints

properly crossed, .............

34,000

Wrought-iron retort, with a welded joint,

30,750

Cast-iron boilers, cylinders, and pipes (average),

16,500

Malleable cast-iron cylinders,

48,000

422. Thick Hollow Cylinders and Spheres.-The assumption that the tension in a hollow cylinder or sphere is uniformly distributed throughout the thickness of the shell is approximately true only when the thickness is small as compared with the radius.

Let R represent the external and r the internal radius of a thick hollow cylinder, such as a hydraulic press, the tenacity of whose material is f, and whose bursting pressure is p. Then we must

have

by means of which formula, when r, f, and p are given, R may be computed.

In the case of a hollow sphere the following formulæ give the ratios of the bursting pressure to the tenacity, and of the external to the internal radius:

SECTION III.-Of Resistance to Distortion and Shearing.

423. Distortion and Shearing Stress in General.-In framework and mechanism many cases occur in which the principal pieces, such as plates, links, bars, or beams, being themselves subjected to tension, pressure, twisting, or bending, are connected with each other at their joints by rivets, bolts, pins, keys, or screws, which are under the action of a shearing force, tending to make them give way by the sliding of one part over another.

Every shearing stress is equivalent to a pair of direct stresses of the same intensity, one tensile and the other compressive, exerted in directions making angles of 45° with the shearing stress. Hence

it follows that a body may give way to a shearing stress either by actual shearing, at a plane parallel to the direction of the shearing force, or by tearing, in a direction making an angle of 45° with that force. The manner of breaking depends on the structure of the material, hard and brittle materials giving way by tension, and soft and tough materials by shearing.

When a shearing force does not exceed the limit within which moduli of stiffness are sensibly constant, it produces distortion of the body on which it acts. Let q denote the intensity of shearing stress applied to the four lateral faces of an originally square prismatic particle, so as to distort it; and let be the distortion, expressed by the tangent of the difference between each of the distorted angles of the prism and a right angle; then

..(1.)

is the modulus of transverse elasticity, or resistance to distortion; of which examples are given in the tables, page 479.

One mode of expressing the distortion of an originally square prism is as follows:-Let a denote the proportionate elongation of one of the diagonals of its end, and a the proportionate shortening of the other; then the distortion is

› = 2 α.

The ratio

wrought iron and steel it is about

C of the modulus of transverse elasticity to the modulus E of direct elasticity defined in Article 420, page 493, has dif1 ferent values for different materials, ranging from 0 to 13

For

2

The ultimate shearing strength, or modulus of resistance to shearing, in other words, the intensity of the greatest shearing stress when the body is on the point of giving way,-is, in wrought iron and steel, and most other metals, equal, or nearly equal, to the tenacity in cast iron it is about once and a half greater than the tenacity; in timber it is nearly equal to the tenacity across the grain. (See the Tables, page 479.)

424. Strength of Fastenings and Joint-Pins.-The connecting pieces already referred to as being exposed to the action of a shearing force may be distinguished into fastenings, such as rivets, keys, wedges, gibs and cottars, and screws, by which two pieces are secured together so as to act as one piece; and joint-pins, by which two pieces are so connected as to be free to turn about the joint. It is obvious that the figure of a joint-pin, as well as that of the hole or socket in which it works, must be that of a surface of revolution, such as a circular cylinder; and that the fit, though accurate, must be easy, like that of an axle in its bearings. Most fastenings and joint-pins are exposed to a bending as well as to a shearing action, and in some cases the most severe stress is that arising from the bending action; but in other cases the most severe stress is that produced by the shearing load. These latter cases are as follows:-All rivets, keys, and other fastenings which are tightly jammed in their holes; all cylindrical joint-pins, fixed at one end, in which the length of the loaded part is less than onethird of the diameter; and all cylindrical joint-pins, fixed at both ends, in which the length of the loaded part is less than twothirds of the diameter.

In order that the shearing stress on a connecting piece may be uniformly distributed over the cross-section, it is necessary that the fastening should be held so tight in its hole or socket that the friction at its surface may be at least of equal intensity to the shearing stress; and then the intensity of that stress is represented simply by PA; P being the shearing load, and A the area which resists it.

But when the connecting piece fits easily, as must always be the case with joint-pins, the greatest intensity of the stress, to which the strength of the connecting piece must be adapted, exceeds the mean intensity P÷ A, in a ratio which depends on the figure of the cross-section; and whose values, for the ordinary figures, are

for rectangular cross-sections, 3;

for circular and elliptic cross-sections, i

and the sectional area must accordingly be made greater in that

ratio than the area which would have been sufficient had the connecting piece fitted tightly.

The chief kinds of connecting pieces, to which these principles have to be applied, will now be considered separately.

425. Rivets are made of the most tough and ductile metal. (See, for example, "Rivet Iron," in pages 460 and 482.)

The ordinary dimensions of rivets in practice are as follows:Diameter of a rivet for plates less than half an inch thick, about double the thickness of the plate.

For plates of half an inch thick and upwards, about once and ahalf the thickness of the plate.

Length of a rivet before clenching, measuring from the head = sum of the thicknesses of the plates to be connected + 24 × diameter of the rivet.

The longitudinal compression to which a rivet is subjected during the operation of clenching, whether by hand or by machinery, tends to make it fit its hole tightly, and thus to produce uniform distribution of the stress; but as such uniformity cannot be expected to be always realized, it is usual to assume, in practice, that there is a deviation from uniformity of shearing stress sufficient to neutralize the greater toughness of the metal in the rivets than in the plates which they connect; and, therefore, the distance apart of the rivets used to connect two pieces of metal plate together is regulated by the rule, that the joint sectional area of the rivets shall be equal to the sectional area of plate left after punching the rivet holes. This rule leads to the following algebraical formula :

Let t denote the thickness of the plates;

d, the diameter of a rivet;

n, the number of ranks of rivets ;

it being understood that the rivets which form a rank stand in a line perpendicular to the direction of the tension which tends to pull the plates asunder.

c, the pitch, or distance from centre to centre of the adjoining rivets in one rank; then

Each plate is weakened by the rivet holes in the ratio

.(1.)

(2.)

In "single-rivetted" joints, n = 1; in "double-rivetted" joints, n = 2, and the two ranks of rivets form a zig-zag; in "chain

rivetted" joints, n may have any value greater than 1. A singlerivetted joint is weakened by unequal distribution of the tension on the plate in the ratio of 4: 5.

Suppose that in a chain-rivetted joint the pitch c from centre to centre of the rivets is fixed, so as not to weaken the plates below a given limit; then in order to find how many ranks of rivets there should be,-in other words, how many rivets there should be in each file,—the following formula may be used :—

426. Pins, Keys, Wedges, Gibs, and Cottars.-These fastenings are, like rivets, themselves exposed to a shearing ioad, while they serve to transmit a pull or thrust from one piece in framework or mechanism to another; and the rule for determining their proper sectional area is the same, with this modification only, that it is safest in most, if not in all cases, to allow for the possibility of an easy fit, according to the rule stated at the end of Article 424, page 497.

In order that a wedge, key, or cottar may be safe against slipping out of its seat, its angle of obliquity ought not to exceed the angle of repose of metal upon metal, which, to provide for the contingency of the surfaces being greasy, may be taken at about 4°. (Article 309, page 349.)

427. Bolts and Screws.-If a bolt has to withstand a shearing stress, its diameter is to be determined like that of a cylindrical pin. If it has to withstand tension, its diameter is to be determined by having regard to its tenacity. In either case the effective diameter of the bolt is its least diameter; that is, if it has a screw on it, the diameter of the spindle inside the thread. It is to be observed, however, that in order to provide for possible irregularities in the distribution of the stress, it is customary to use for screws a very large factor of safety, ranging from 12 to 15; the mean intensity of the working stress on wrought-iron screws being only about 4,000 lbs. on the square inch, or 2.8 kilogrammes on the square millimètre.

The ordinary form of section of the thread of a fastening screw is an isosceles triangle with the angles rounded; and according to the proportions recommended by Mr. Whitworth, the angle at the summit is 55°, making the height of the triangle = 0.96 of its base. One-sixth of that height is taken away by the rounding of the edge of the thread, and another sixth by the rounding of the bottom of the groove, leaving two-thirds, or 0.64 of the base; and as the base of the triangle is the pitch of the screw, the projection of the thread is 0.64 of the pitch.