circular pillar h=d, and for a rectangular pillar h = least

W

ки

W

(a)

(b)

(c)

Fig. 320.

breadth, since that is the direction the pillar will evidently bend in.

c is a constant depending on the manner of connection of the ends of the pillar, which we can see will affect the bending, for, looking at Fig. 320, in (a) we have a pillar with both ends free to turn, and it bends accordingly into one large arc, the bending moment at the centre

being Wx. Then in (b) one end of the pillar cannot turn, it bends into two arcs, and Wr the greatest B. M. is less, sincer is less. The decrease is still more in (c), where neither end can rotate, x being less still. The values of ƒ and c are as follows :—

We have here also given the results for cast-iron and timber, which are not ductile; but the formula applies to these as well as to the ductile ones; and it is the practical formula which in nearly every case is used, because pieces so short as not to bend at all are not often used compared with the longer pieces.

Working Strength of Pillars.—The value of W given by Gordon's formula is the crushing load, and

hence in calculating the size of a pillar for a given load a "factor of safety" must be allowed. This is a case in which accidental circumstances may produce large stresses, e.g. a pillar may be a little bent when set up, and then there is at once a bending moment produced at the bent part, hence the factor of safety should be large, and it is usually taken at 8 or more. To find then the size of a pillar to carry a load of 10 tons say, we should put W at least 80 tons.

Crushing of Short Pieces of Ductile Material. -When the length of a piece is not more than about 1 times its diameter, it will not give way by bending but in a manner we will explain.

In the first case, the elastic limit and modulus of elasticity are practically the same as for tension.

When the elastic limit is passed, then we have permanent set, and an increase of cross-sections, as opposed to the decrease in tension; and this increases with increasing loads, the stress per sq. inch on the section also increasing up to a certain point (see page 419).

After a certain stress is reached the load increasing produces an increase of section such that the stress remains fairly constant (page 419). And finally, when a certain compression is reached, the metal gives way by cracking round the circumference, as Fig. 321, which represents the compression of a block

of steel experimented on by Sir W. Fairbairn.

Fig. 321.

There is nothing corresponding to the local contraction in tension; the bulging of the block in the middle is not due to any local cause, but partly to the friction of the pieces by which the pressure is applied holding the ends together, and partly probably to another cause somewhat outside our present scope.

The curve of stress and strain is very similar to that for tension, but the results are not so definite. The

ultimate strength appears to be somewhat less than for tension; but since giving way is due to the formation of cracks, there is an accidental character which causes in some cases considerable discrepancies. This, however, is not of much consequence, because elastic strength is what we want in practice.

Rigid Materials-Tension.—Rigid materials are those possessing opposite characters to ductile materials ; so they can neither be drawn out nor hammered out, but break before any appreciable change of shape can be produced. A typical ex

but

5300

Fig. 322.

ample is cast-iron.

Fig. 322 shows the shape of the stress-strain curve, and it differs, we see, 16000 entirely from the case of ductile material.

In the first place there is no real elastic state at all, for permanent set is produced by all stresses except very small ones. Also, except for very small stresses, we have not

p=Ee,

p=Ee (1-ke) (k being a constant average value 209). Against this, however, we set the fact that this imperfectly elastic state continues right up to the breaking point, so there is no breaking-down point and no drawing out locally, the bar breaking across practically the full original area of section. There will be little error, however, in taking cast-iron to be perfectly elastic up to about one-third of the breaking stress, which is about 16,000 lbs. per sq. inch; so below 5300 lbs. per sq. inch the metal may be assumed to be elastic. Cast-iron is very variable in quality, so this can only be taken as an average value; but it is in all cases much weaker in tension than the ductile wrought-iron.

Compression of Rigid Materials.—Long pieces

we have already considered, so far as practical strength is concerned, but there is one point of difference which must be noticed, as compared with ductile material. There is an absence of perfect elasticity, and we have

p=Ee (1-ke),

but, moreover, whereas in ductile material E is the same in tension and compression, in rigid material E for compression is less than for tension (see table, page 435). Also the value of k is now about 40 instead of 209.

These results were obtained by Hodgkinson, long bars being compressed, and prevented from bending by enclosure in a frame.

For actual crushing we must have recourse to short blocks, as for ductile materials.

The load-strain curve is then found to be similar in shape to that for tension, but the breaking

load is very much higher, being five or

six times greater.

The blocks finally give, as shown in Fig. 323, by shearing or splitting on oblique planes.

The characteristics just mentioned are common to all rigid materials, and hence apply to the crushing of stone and of brick. For numerical results see the table at the end of the chapter.

Fig. 323.

Fibrous Materials - Wood and Ropes. We have given considerable space to the metals on account of their importance, hence we can only briefly glance at the present kind of materials.

Wood consists of fibres arranged longitudinally, but they are united together loosely; hence wood is strong against tension, the fibres each taking a full share, but can easily be split or sheared longitudinally, and under a compressive load gives way at a comparatively low stress by splitting.

The strength of wood is also affected by its condition; seasoned wood being elastic nearly up to the breaking point, while in green wood the elasticity is imperfect and the strength much less.

Ropes have the fibres arranged spirally, and when under tension, the fibres being pressed together, friction is developed sufficient to prevent their sliding over each other. One effect of this is, however, to weaken each individual fibre, so that a rope is not so strong as the fibres separately would be; nor is a cable so strong as the smaller ropes of which it is made up. The size of ropes is usually expressed by their girth, and hence the breaking load is also expressed in terms of this quantity. Thus

where

T=

C2
k'

T=breaking load in tons,
C=girth in inches,

k=constant.

k depends on the material, and also somewhat on the size, small ropes being, as we have seen, stronger comparatively.

Thus, for

Hemp, k=3.3 ordinary, and a little less for small ropes.
Iron wire, k=1, or rather more if C>6.

Steel wire, k=.5 or even less.

This gives the breaking load, and the “factor of safety" may be 5 for wire ropes and 6 for hemp.

Table of Strength and Weight.-The following table contains numerical values of the various physical constants referred to in Strength of Materials for some of the principal materials used. Working stress is omitted, its determination being explained on page 428, and its value not being constant but varying according to the factor of safety employed.