654. In one series of experiments cast-iron bars were taken one square inch in section and 10 feet long. They were stretched by various weights ranging from a little over 1000 pounds to 17000 pounds; the latter weight on an average broke the bars. A permanent set, indicating the limit of the elasticity, was obtained by about one-tenth of the breaking weight. When the stretching weight was a little over 1000 pounds the rod was lengthened 009 of an inch; when it was about 9500 pounds the rod was lengthened 1 of an inch; and when it was nearly 15000 pounds the rod was lengthened nearly 2 of an inch.

655. In another series of experiments similar bars were submitted to compression by various weights ranging from a little over 2000 pounds to a little over 37000 pounds; the latter weight on an average greatly injured the bars. A permanent set, indicating the limit of the elasticity, was obtained by about one-seventeenth of the injuring weight. When the compressing weight was a little over 2000 pounds the rod was shortened about '02 of an inch; when it was nearly 21000 pounds the rod was shortened rather more than 2 of an inch; and when it was about 37000 pounds the rod was shortened about ·4 of an inch.

656. It will be seen that Hooke's Law does not hold very exactly in the case of either of the series of experiments given in Articles 654 and 655. But we must observe that when the constraining weight is not extremely large the lengthening which it produces by stretching is numerically very nearly equal to the shortening which it produces by compression. Thus for example the constraining weight being about 4200 pounds both the lengthening and the shortening were about 039 of an inch. As the constraining weight increases the shortening becomes sensibly less than the lengthening; and this is in accordance with the statement of Art. 650 that the endurance of cast iron is much greater than the tenacity.

657. The material which is most extensively employed in the arts is wrought iron; it is obtained directly from cast iron by a process which removes the greater part of the impurities. The tenacity may be taken to be on an

average 23 tons per square inch; and the limit of elasticity as approximately half the tenacity.

658. Experiments give the following results with respect to the stretching of wrought iron bars one square inch in section and ten feet long. The stretching weight was at first 1262 pounds, and was successively increased by this amount until at last it was 30 times the original weight. Under the first weight the lengthening was rather more than 005 of an inch; under a weight 20 times as great it was about '11 of an inch: throughout this range the lengthenings followed Hooke's Law pretty closely. As the weight was increased beyond this point the deviations from Hooke's Law became very large: until under a weight 30 times as great as the first the lengthening was about 2.9 inches, that is about eighteen times as great as it would have been according to Hooke's Law.

659. The strength of wrought iron is not much affected by the increase of the temperature up to 350 degrees of the common thermometer. There has been a difference of opinion as to the influence of extreme frost; direct experiment does not seem to make out that the strength is less in cold weather; but there exists a popular notion that iron and steel are rendered more brittle by frost, and this receives some confirmation from the fact that the accidents on railways arising from the breaking of the rails and of the axles of the carriages are most frequent in winter.

660. An opinion prevails, and apparently on good grounds, that some change takes place in the constitution of wrought iron when it has been subjected to incessant jars for a long time; and that in consequence of this the strength is much diminished. The axles of railway carriages, and the chains of cranes, are cited as examples of this great deterioration. With respect to chains it is well established that in the course of time they change for the worse, and it is a rule in the War Department that all chains are to be passed periodically through the fire, and thereby annealed: thus the quality is restored, and the duration of the chain prolonged.

661. Steel is a combination of pure iron and carbon; its tenacity is far greater than that of wrought iron, ranging

from 30 tons to 50 tons per square inch. Moreover, when steel is tempered in oil the limit of elasticity is fully three times as great as that of wrought iron; and the steel will stretch much before rupture takes place. A good serviceable quality of steel is now manufactured, by what is called the Bessemer process, in an economical manner, and this is applied to many of the purposes for which iron was formerly used, especially where strength is to be combined with lightness.

662. The engineer being furnished with information as to the strength of the materials which he has to use, must be guided by experience as to the greatest demand which he will make on that strength in his constructions. Of course for safety he will keep far below the extreme limit which is theoretically allowable. Thus the average tenacity of wrought iron may be taken as 23 tons to the square inch; but in practice it is not considered prudent to calculate on more than 5 tons: and in the case of chains which are liable to a sudden impulse, as the chains of cranes, it would be unwise to rely on as much as 5 tons. In practice there are three terms used for different degrees of strength of materials; namely ultimate strength, proof strength, and working strength. The ultimate strength may be taken to be measured by the constraint which will destroy or damage the material in a specified way; the proof strength as measured by the greatest constraint which is consistent with safety; and the working strength as measured by the greatest constraint allowed by prudence and experience. The ultimate strength may be 2 or 3 times as great as the proof strength, and 10 times as great as the working strength. Constraint equal to the proof strength might not produce any harm in a single short trial, but it might by long continuance or by frequent operation.

663. Numerical results slightly different from those adopted in the present Chapter may be found in works of good authority; and from the nature of the subject it cannot be expected that all experiments will be in exact agreement. The older writers naturally could not attain the same accuracy as their successors; but it is difficult to account for the wide discrepancy sometimes to be found

between their statements and those which are at present received. For instance Dr Young says "The weight of the modulus of the elasticity of a square inch of steel...is about 3 million pounds..." The modern value is at the lowest nearly ten times this: see Art. 641. Again, he says "Oak will suspend much more than fir; but fir will support twice as much as oak..." According to modern authority oak will support nearly twice as much as fir: see Art. 648.

664. A few interesting remarks may be quoted from Dr Young. "The strongest wood of each tree is neither at the centre nor at the circumference, but in the middle between both; and in Europe it is generally thicker and firmer on the south east side of the tree. Although iron is much stronger than wood, yet it is more liable to accidental imperfections; and when it fails, it gives no warning of its approaching fracture...... Wood, when it is crippled, complains, or emits a sound, and after this, although it is much weakened, it may still retain strength enough to be of service. Stone sometimes throws off small splinters when it is beginning to give way: it is said to be capable of supporting by much the greatest weight when it is placed in that position, with respect to the horizon, in which it has been found in the quarry."

LVI. STRENGTH OF BEAMS.

665. We have spoken of the strength of materials in the preceding Chapter; in the next place it would be proper to enquire into the strength of structures formed of these materials: we shall however confine ourselves almost entirely to one simple case, that of beams placed in a horizontal position, either fixed at one end or supported at both ends. The engineers now use the term girder for a beam supported at both ends, and cantilever for a beam fixed at one end; the beam in both cases is supposed to be subjected to transverse strain, as for instance, to that produced by a weight.

666. Suppose a horizontal rectangular beam to have one end firmly fixed, and at the other end let a weight W be hung; we neglect the weight of the beam itself. By

the action of the weight the beam will be drawn out of the horizontal position; the diagram is intended to shew the new form of the beam: it is much exaggerated for the sake of distinctness. The beam is supposed to be built into a wall, on the right hand side of the vertical line AC, or to be

otherwise fixed, and we are concerned with only the portion on the left-hand side of AC. The boundaries AB and CD both become curved, AB being longer, and CD shorter, than when they were both horizontal. We must understand distinctly what is meant by the length, the depth, and the breadth of the beam. The length is the distance from end to end, namely the distance from A to B in the original position. The depth is the distance from the upper surface to the lower; it is the straight line AC in the diagram. The breadth is the distance from the front to the back; it is not shewn in the diagram, being perpendicular to the plane of the paper.

667. Somewhere between AB and CD a line exists which was originally horizontal, and of the same length as it is in the bent form; we will denote it by EF. This line is called the neutral line in the surface ABDC. The assemblage of the neutral lines in all the sections parallel to ABDC is called the neutral surface.

668. Now take the portion of the beam which stands out from the wall, denoted by ABDC, and consider the forces which keep it in equilibrium: these are the weight W, and the actions along the imaginary section represented by AC. The actions will be partly in the vertical plane, denoted by AC, and partly at right angles to it. The former must on the whole be equal to the weight W, and we shall not require to take any more notice of them; the latter are very important for our purpose. The part of the beam above the neutral surface is lengthened; its