Knowing the radius, we can determine the ordinate of any point in the curve when the horizontal distance of the ordinate from the centre of the curve is known.

254. Camber in Plate Girders. In girders with continuous plate webs, the required camber may be practically produced by laying the web-plates on a temporary platform, stringing a line from one end to the other, and adjusting the plates so as to get their bottom edges at the different joints at the distances from the line as found from equation (18). The bottom edge of the plates will then approximately form the arc of a circle and the bottom angles, which have been previously punched or drilled, are bent to this curve, laid in their proper position on the web, and the position of the holes marked on it to correspond with those on the angles; the top angles are laid on in the same way,

* The truth of this formula may be demonstrated thus (fig. 206a) —— Let c1 d1 =y, d d1 =x,

cd=v,

after which the web is taken to pieces, and the holes in it

punched or drilled. When the

girder is afterwards put together it will be found to have the required camber.

255. Camber in Lattice Girders. -Fig. 207 represents a lattice girder with an exaggerated camber. If the lattice bars be equal to each other, the lines which connect the points of intersection of the lattices with the top and bottom flanges will, if produced, all meet at the same point, O, which is the common centre of curvature of the top and

Fig. 207.

-

bottom flanges. The panels into which the top flange is divided are longer than the corresponding panels of the bottom flange in the proportion of the radii of the two flanges.

In order to produce the required camber in a lattice girder, it is only necessary to determine ab and a, b1, and mark them off on the top and bottom flanges of the girder, the length of the lattices remaining the same throughout the girder. When this plan is adopted, it will be found that on putting the girder together it will have the camber which is desired.

Suppose, for example, we have a lattice girder 100 feet span and 10 feet deep, and it is required to give it a camber of 3 inches at the centre. Suppose the girder between the abutments to be divided into ten equal spaces, the lengths of the panels in the bottom flange being each equal to 10 feet.

From equation (17) we get

The panels into which the top flange is divided are, therefore, 0.02 foot or 0.24 inch longer than those in the bottom.

Another method of practically producing a camber in a lattice girder, is by keeping the top and bottom panels the same length, and making the lattice struts a little longer than they would be for a straight girder, and the lattice ties a little shorter.

=

Let f working stress per unit of section to which the lattice bars will be exposed,

E = the modulus of elasticity of the material.

Then, in order that the girder may become straight when the material is exposed to the stress f, the struts should be made longer than those for a straight girder in the proportion of to 1, and the ties should be shorter in the proportion of

f

(1

If, for example, a steel girder is designed so that its members are exposed to a working stress of 6 tons per square inch, and if the modulus of elasticity of the steel be 14,000 tons, and if the length of the lattice bars for a straight girder unloaded be 10 feet, 6 × 10 then the struts ought to be 10+ 14,000

=

10 feet 0.05 inch in

length, and the length of the ties should be 10.

11.95 inches, so that when the girder is fully loaded it may be quite straight.

CHAPTER XXI.

CONNECTIONS.

I. RIVETTED JOINTS.

256. Different Methods of Joining Plates by Rivets.-There are two principal methods of joining two plates or bars together by means of rivets or bolts. One is to make the plates overlap each other and rivet them in this position; the second method is to place the two ends flush together and connect them by

means of one or two strips overlapping each, and then rivet the whole together in this position.

Joints of the first description (see fig. 208) are termed lapjoints, and those of the second class (figs. 209 and 210) are termed butt-joints.

Fig. 208.

Fig. 209.

O oo o
Fig. 210.

257. Rivets in Single- and Double-Shear. In the joints shown in figs. 208 and 209, the rivets are in what is termed "singleshear," as each rivet can only be shorn at one section before the bars are pulled asunder. In the joint shown in fig. 210, the rivets are in "double-shear," as each rivet will have to be shorn across two sections before the bars can be pulled asunder. In addition to this shearing resistance the rivets confer upon the joint a further element of strength in the frictional resistance which they give to the plates. In the process of forming the rivet head by the machine or by hand, a certain amount of grip is given to the rivet on the plates; a further grip is obtained by the contraction of the rivet in cooling, this contraction pressing the plates powerfully together and causing a considerable tension on the rivet, so much so that in the case of long rivets the heads sometimes fly off. It is difficult to determine what value can be attached to the frictional resistance produced by this means, though cases have been known where the joint has been held together by this friction alone. In estimating the strength of a joint it is not usual to take this resistance into account, as in process of time, owing to the rusting of the plates and vibrations in the structure, the tension on the rivet may altogether disappear. When this frictional resistance is disregarded, the theoretic shearing stress on each rivet in a joint will be equal to the total stress on the bars divided by the number of sections of rivets that must be shorn in order to pull the bars asunder.

If P = total stress on the bars; then—

Practically, the shearing stress on each rivet may not be quite the same, one being subjected to a greater stress than another. However, if the holes are truly punched or drilled this difference of stress cannot be much.

258. Shearing Strength of Rivets. The resistance of wrought iron to a shearing stress is not so great as the ultimate strength of the material under a direct tensile stress; and, further, this resistance varies according to the direction in which the shearing action takes place. From Wöhler's experiments, it appears that the shearing strength of a bar or plate of wrought iron in a plane perpendicular to the fibre is equal to ths of its ultimate tensile strength in the direction of the fibres. The shearing strength, in a plane parallel to the direction of the fibres, is from 18 to 20 per cent. greater than the above, and is about equal to the tenacity of the iron. So far as the shearing strength of rivets is concerned, it will only be necessary to consider their strength in a direction at right angles to the fibre, so that if the tenacity of rivet iron be 23 tons to the inch, its shearing strength will only be 18.4 tons or thereabouts. It has also been shown by numerous experiments, that the shearing strength of a rivet in a punched hole is slightly greater than that in a drilled hole, the reason assigned being that the sharp edge of a drilled hole facilitates the shearing process.

If we adopt 4 as a factor of safety, about 4.5 tons per square inch will be the safe working stress for iron rivets in iron plates. As rivet iron is of a better quality and stronger than the plates, some engineers adopt the rule of making the total rivet area in a tensile joint equal to the net sectional area of the plate. It is, however, a much better practice to have the rivet area 10 per cent. greater than this.

Theoretically, a rivet in double-shear ought to be twice as strong as a similar one in single-shear; the balance of evidence, however, from numerous experiments, shows that rivets in single shear are rather more than one-half as strong as those in double shear.

Iron rivets in steel plates are not so strong as those in iron plates; their strength being about 16 tons to the inch. It is not safe, therefore, to allow a working stress of more than 4 tons to the inch on iron rivets used in steel structures.