in compression should be of a quality exhibiting a high modulus of elasticity. When a bar does not fail by deflection and cross breaking, its rupture may occur by shearing at some plane more or less inclined to the axis,

as at g h or fi in Fig. 54, representing part of a column of which a b is the axis. Taking

the action on the plane g h, make ce = the 9 load; this must be resolved in the directions of the line of fracture, and a line at right angles to it, the parallelogram being that shown at ckel; and in like manner for the plane fi, the parallelogram will be c n e m, ck and c n being the shearing forces respectively acting parallel to the planes

The maximum strain will be found to occur when the angle of the plane of rupture is 45 degrees to the axis of the bar, then the strain parallel to the plane of rupture

will be =

W 1.414'

and the area on which this strain acts

is the sectional area of the bar at right angles to its length, multiplied by 1414. It is evident, however, that this mode of rupture can only occur in columns of small length in ratio to the diameter, or least thickness, for if the resultant cm fall outside the base of the column, there must be bending strain. Taking the plane at the angle given above, and resolving it at the centre or axis, the base of the column must have a width equal to its height.

From experiment it is found such a column would crush with 36 tons for a square inch. The proportion of this

acting parallel to the plane of fracture would be

36

1.414

=

25.4 tons. The area of sheared section will be, for the bar 1 inch square, 1.414 square inches; hence the shearing

strain per square inch =

25.4 1.414

= 18 tons (nearly). This is

a very high standard, but the experiments upon which the formulæ were based were made upon metal of superior quality.

A considerable difference in strength will be found to exist between elements having flat properly bedded ends and such as have jointed ends, or ends upon which the column can turn, for the flat ends aid in resisting deflection; and it may be observed, as in the deflection of columns the extended side is that on which the compressive load is most directly resting, there will be a great tendency for the deflection, when it does occur, to happen suddenly, perhaps instantaneously, with rupture, and so pass unnoticed.

I will now insert the empirical formula, which have been derived from the experiments of Hodgkinson and others. The formula for timber is Love's, those for metal are Gordon's. The diameter or thickness is always to be measured the thinnest way of the column; thus, a column 6 inches by 4 inches would be said to have a thickness of 4 inches for the purposes of calculation.

Let W = breaking load in tons per sectional square inch of column; r = the length divided by the least diameter; C = ultimate resistance to compression in tons per square inch.

For Cast-iron cylinders, solid or hollow, flat ends,

For Cast-iron rectangular columns, flat ends, W=

For Wrought-iron solid rectangular columns, W =

For Angle, Tee, and Channel Iron, W =

19

дог

1+

900

For Mild Steel, solid round pillars, W =

For Strong Steel, solid round pillars, W=

CHAPTER X.

JOINTS AND CONNECTIONS.

THE strength of any structure is limited by that of its weakest part, and in order to obtain the most satisfactory results all the parts should be equally strong. In practice it is not possible to secure absolute equality of strength throughout our work, but this should be studied as closely as is practicable, and at all events care must be taken that the strength nowhere falls below a certain limit.

The student, having made himself proficient in the foregoing formulæ, can readily determine the area of the various elements of any structure he may have intrusted to his care; but when this is done there arises the question of arrangement of joints for the connection of the parts, and the transmission of strain from one to another. The sizes in which materials can be obtained have to be considered, and the joints arranged so as not to interfere with one another. The lengths in which bars and plates are rolled vary in different districts; thus plates 21 feet long are common in the Cleveland District, whereas about 16 feet rules in Staffordshire. A great deal depends upon the quality of the iron and its peculiar characteristics, and it is not advisable to insist on excessively long plates, for by extending the dimensions the fibre of the metal may be strained in manufacture, or in avoiding this the maker may be led to use a class of iron of a more yielding character,

and inferior in strength throughout. Other points also require regarding, such as convenience of handling and erection; for girder work generally 20 feet should be taken as the outside limit of length for plates; but angle irons of moderate section may be run up to 30 or 35 feet in length, but this is rather awkward to manage, and it is more convenient to keep to shorter lengths.

For the sizes of timber no general rules can be laid down of any practical utility. I will commence with the

a

f

Fig. 55.

d

joints of timber structures. The liability of timber to split along the line of the grain calls for great precautions in setting out the joints in this material. Joints in compression will be most satisfactorily made by butting the ends accurately together, as shown at a b, Fig. 55, and keeping them in juxtaposition by sur

rounding the joint with a box, shown in section at ef, and secured from slipping by bolts passing into or through the timber.

Another form of butt joint is shown at cd, in which the ends of the timber are stepped together and secured by bolts. This form is correct if the two pieces of timber are of exactly the same quality as regards elasticity; if not unequal straining may occur from the piece, say gh, being more compressible than the other tongue ji, when the one side yielding more than the other, the post will be more liable to deflect laterally. There is also more difficulty in insuring a fair bearing at i and h than there is in obtaining a uniform bearing in simple butt joints.

Joints subject to tensile stress will be generally more complex than those in compression, and in every case some sectional area will be lost. The most common method is by scarfing, as shown in Fig. 56. At ab is a plain scarf, bolted together by bolts passing through thin wrought