at various depths. From these observations it is a simple matter to deduce the proportional strain and the stress. The results obtained are shown in Fig. 3336, and for purposes of comparison, Lamé's and Barlow's curves are inserted.

Built-up Cylinders. In order to equalize the stress over the section of a cylinder or a gun, various devices are adopted. In the early days of high pressures,

6

α

cast-iron guns were cast round chills, so that the metal at the interior was immediately cooled; then when the outside hot metal contracted, it brought the interior metal into compression. Thus the initial stress in a section of the gun was somewhat as shown by the line ab, ag being compression, and bh tension. Then, when subjected to pressure, the curve of stress would have been de as before, but when combined with ab the resulting stress on the section is represented by ef, thus showing a much more even distribution of stress than before.

FIG. 334

This equalizing process is effected in modern guns by either shrinking rings on one another in such a manner that the internal rings are initially in compression and the external rings in tension, or by winding wire round an internal tube to produce the same effect. The exact tension on the wire required to produce the desired effect is regulated by drawing the wire through friction dies mounted on a pivoted arm-in effect, a friction brake.

STRENGTH AND COEFFICIENTS OF ELASTICITY OF MATERIALS
IN TONS SQUARE INCH.

CHAPTER IX.

BEAMS.

THE beam illustrated in Fig. 334a is an indiarubber model used for lecture purposes. Before photographing it for this illustration, it was painted black, and some thin paper was stuck on evenly with seccotine. When it was thoroughly set the paper was slightly damped with a sponge, and a weight was placed on the free end, thus causing it to bend; the paper on the upper edge cracked, indicating tension, and that on the lower edge buckled, indicating compression, whereas between the two a strip remained unbroken, thus indicating no longi

tudinal strain or stress. We shall see shortly that such a result is exactly what we should expect from the theory of bending.

General Theory.-The T lever shown in Fig. 335 is hinged at the centre on a pivot or knife-edge, around which the lever can turn. The bracket supporting the pivot simply takes the shear. For the lever to be in equilibrium, the two couples acting on it must be equal and opposite, viz. W = px.

Replace the T lever by the model shown in Fig. 336. It

is attached to the abutment by two pieces of any convenient material, say indiarubber. The upper one is dovetailed, because it is in tension, and the lower is plain, because it is in

compression. Let the sectional area of each block be a; then, as before, we have WI = px. But p = fa, where ƒ = the stress in either block in either tension or compression;

Or

The moment of)

the external

forces

=

(the moment of the internal forces, or the internal moment of resistance of the beam

= stress on the area a × (moment of the two areas (a) about the pivot)

Hence the resistance of any section to bending-apart altogether from the strength of the material of the beam-varies directly as the area a and as the distance x, or as the moment ax. Hence the quantity in brackets is termed the "measure of the strength of the section," or the "modulus of the section," and is usually denoted by the letter Z. Hence we have

The connection between the T lever, the beam model of Fig. 336, and an actual beam may not be apparent to some

W

I

readers, so in Fig. 336a we show a rolled joist or I section, having top and bottom flanges, which may be regarded as the two indiarubber blocks of the model, the thin vertical web serves the purpose of a pivot and bracket for taking the shear; then the formula that we have just deduced for the model applies equally well to the joist. We shall have to slightly

FIG. 336a.

modify this statement later on, but the form in which we have stated the case is so near the truth that it is always taken in this way for practical purposes.

Now take a fresh model with four blocks instead of two. When loaded, the outer

end will droop down as shown by the dotted lines, pivoting about the point resting on the bracket. Then the outer blocks will be stretched and compressed, or strained, more than the inner blocks in the ratio

FIG. 337.

W

x y or ; i.e. the strain is directly proportional to the distance

show the extensions, and c, 4, show the compressions at the distances y, y1 from the pivot.

From the similar

have previously seen (p. 295) that when a piece
of material is strained (i.e. stretched or compressed),
the stress varies directly as the strain, provided the
elastic limit has not been passed. Hence, since the
strain varies directly as the distance from the pivot,
the stress must also vary in the same manner.
Let f = stress in outer blocks, and fi
in inner blocks; then-

= stress

FIG. 338.

Then, taking moments about the pivot as before, we have—