The various theories based on these and other experiments are to be found in the following papers :-Dr. J. Weyrauch, C. E.,' 1880, vol. lxiii. p. 283; Gerber, Bay. A. I. V.,' 1874; Lippold, Organ,' 1879, vol. xvi. p. 22; Launhardt, Arch. I. V.,' 1873; Professor Mohr, 'Civil I.,' 1881; H. Tresca, T. Seyrig, E. E. Marchée, E. Trélat, A. Brull, H. Mattieu; Ing. Civ.,' Résumé, 1881, vol. ii. p. 39. Contraction of Test Pieces.-Continuing to follow the behaviour of a test piece, and this time until fracture takes place, it will be found that some materials, such as hard steel, manganese steel, bad iron, and gunmetal, show no contraction previous to rupture, while mild steel, iron, and brass and copper, do. The reasons for this contraction are not known. It has been suggested that contraction is due to local weakness, so that those metals which contract most are least uniform as regards tenacity; but if that were the case tensile tests with drilled samples would be very irregular, according as to whether the hole. was near a weak or a strong place. The following experiments on a mild steel plate whose tenacity was 28 tons show that this is not the case: The samples were shaped as shown in figs. 116, 117. Thickness of plates in.; diameter of hole 1 in., drilled. (See also E. Richards, I. and S. I.,' 1882, p. 43.) Another suggestion is, that the rise of temperature of a test piece weakens the part which first contracts more than the others which have accidentally not contracted. Dr. J. Kollmann's experiments (Ver. Gew.,' 1880, 2nd ser. vol. lix. p. 104) confirm this, for there it will be found that the contraction steadily increases from about 20% at an ordinary temperature to 90% at a red heat. Unfortunately for this view the tenacity does not show the same regularity. An explana M tion of this phenomenon is, therefore, still required. The subject will be referred to again when discussing compound stresses. Fractures. The next thing to be noticed in a fractured test piece of mild steel is, that when placed together only the edges touch, leaving a hollow, as shown in fig. 118. There can be but one explanation, viz. that the rupture started near the centre of the section, and that the outside fibres continued to stretch after this point was reached. This suggests the view that the stresses to be found in the centre of a test piece differ from those on the outside surfaces, and are also more injurious. An examination of almost any torn sample of mild steel will show that near the edge the surfaces of rupture are very much on the slant and have every appearance of having been produced by partial shear, suggesting the idea that this material gives way more readily under shearing stress than under tension (see pp. 165 and 168). FIG. 118 The same tendency is noticed when tearing a sample perforated as in fig. 119. The lines of fracture, instead of running as horizontal as possible, will be distinctly steeper than the angle at which the holes were drilled. grey to salmon-colour tints or yellow ones. Shearing Stresses. The investigation of shearing stresses is beset with various difficulties, one of the most important being the smallness of the strains. Direct experiments have therefore only been useful in determining the ultimate shearing strength of various materials. It has been found that the hardness of the metal into which the holes are drilled influences the results. It is therefore usual to groove the samples to be tested as shown in fig. 120. Most experiments on riveted joints include some carried out as above, and the general conclusion is that the shearing strength is less than the tensile strength. More interesting results are obtained if the material is exposed to torsion stresses. One end of a cylindrical bar is secured to the head of a lathe, while the other end is supported, if possible, on a knife-edge, and carries a lever, which is gradually loaded while the lathe head is being turned round. The twisting of the bar is measured with the help of a pair of mirrors m, m2, arranged as shown in fig. 121. A graduated scale is placed at a considerable distance from the test piece, and examined partly direct (x), and partly by doubly reflected light (y). Two scales instead of one are then visible through the telescope T, and it is their relative displacement which measures the angular deflection. This arrangement has been devised by the author and has proved to be very accurate. By plotting down the readings, curves are obtained, which may be called torsion diagrams, and represent the amount of twist which various torsion moments impart to a test bar. It is usual to reduce all these values so as to obtain the shearing stresses of the outside fibres and their angular displacement. The latter value is found by multiplying the angle of twist into half the diameter of the bar, and dividing by the distance of centres. The shearing stress of the outside fibre is found by the formula This Here M is the torsion moment, and d is the diameter of the bar. formula is only correct as long as the moments are strictly proportional to the twist; where this is not the case, and particularly when the limit of elasticity is passed, or the point of rupture reached, it gives wrong results. в с The actual shearing stress in the outer fibre is then found as follows: Let the heights of the line O B C (fig. 122) represent the distribution of the circumferential shearing stresses over the radius of a bar whose diameter is d = 2r, and which has been twisted through an angle in a length 1, while the torsion moment is M. d=27 FIG. 122 D DC represents the shearing stress in the outer fibre. Now if a thin film of metal, of the thickness dr, be machined off the circumference of the bar, and if it be twisted once more, till the stress in the (now reduced) outer fibre is again equal to D C, then, as the new curve O P C is similar to, but shorter than, the original one, the new twisting moment M, will have to be somewhat weaker than M, viz. M1 = M . =0. r r-dr (r-dr)3 while the angle 0 has increased from 0 to + do Now let a film of material, of the thickness dr, be placed round the bar, so that it is once more equal to its original diameter, 2r. Let this outer film be twisted till it has acquired a stress of The way to construct a curve which will show the actual shearing stress of the outside fibre, when the value o = 16. M has been previously determined, can be carried out as shown in fig. 123. The line o is copied from Platt and Hargraves's paper (C. E.,' 1887, vol. xc. plate 10, fig. 6). The faint tangential lines have been drawn to measure the has been constructed by proportional re value d. The line do duction, and to this has been added the respective values of e door 4 do with the help of which the curve has been constructed. The experimenters state that for this particular sample the elastic limit of shearing stress was 46,400 lbs. ; but an examination of the curve shows that it was already reached at 35,000 lbs., and that at 38,000 lbs. a very serious drop took place, and the material of the outside fibre had not recovered even when the deflection had increased to 0. Beyond this point it is always sufficiently accurate to adopt the value = 12. M π. That this is fairly correct will be seen from the following d3 experiments on tension, torsion, and direct shear (Platt and Hargraves, C. E.,' 1887, vol. xc. p. 408): Limits of Elasticity' Ultimate Strengths Estimated 16. M 12. M 5:4 Shear on 7:4 σε σ = Rivets These, D. Kirkaldy's, and some similar experiments by G. Berkley, 1868, V. Appleby ('C. E.,' 1883, vol. lxxiv. p. 268), the latter including compression tests, are apparently the most exhaustive ones that have yet been made to ascertain the relation which exists between compression, tension, and shearing stresses. The latter are certainly always smaller than either of the two former, amounting to from 50 to 90 %. Torsion tests carried out on bars from which the black scale had not been removed showed that it falls off when the limit of elasticity is reached, but even then only along a few axial lines, which implies that the elastic limit and breakdown point fall together. Bending Stress.-After the foregoing it will be unnecessary to analyse the behaviour of beams subjected to stresses beyond their elastic limit; suffice it to say that as the limits of elasticity of the top and bottom fibres are not necessarily reached at the same time, it is impossible to separate the one from the other; but, assuming that they did agree, then the stress in the outside fibre of a narrow, rectangular bar would be found by the following formula: S 6 h2b Here c stands for curvature. (&M + c.dM). From an examination of fig. 124, which represents a beam subjected to an excessive load, Q, it is evident that dM = Q, and, by carefully |