strain is increased. 4th. In the case of cast iron the Modulus is considerably affected by the size or least thickness of the casting, small castings having a higher Modulus than large ones from the same iron. We shall consider the effect of these facts separately. (726.) "Modulus of Extension, Compression, and Deflection."— Taking first the case of wrought iron, whose elasticity when not overstrained is nearly perfect, and of which we have the most perfect experimental knowledge, we find that the Modulus of Elasticity calculated Lbs. per Sq. In. From the Extensions up to 8 tons per square inch (Table 96) = 28,000,000 Clasticity" It will be observed that the Modulus of Deflection, which we supposed (724) to be a mean between those of Extension and Compression, namely 25,200,000 lbs., is in this case considerably greater. (727.) With cast iron, the comparison is obscured by defect of elasticity (688), but we can eliminate the effect of this source of complication by taking for all the strains the same fraction of the breaking weight, say 3rd, and we then have : Lbs. per Sq. In. From the Extensions with 24 tons per square inch (Table 88) = 12,646,500 Here the Modulus of Deflection, so far from being an arithmetical mean between those of Extension and Compression (724), is much greater than either. (728.) The principal value of the term "Modulus of Elasticity" is its supposed universal application to all the strains to which materials are subjected, so that the extension, compression, transverse flexure of a beam, and angular torsion of a shaft, &c., should all be calculated with the same constant by the use of appropriate formulæ, and we should be able to reason on any particular strain with data obtained from another kind of strain. (729.) We have seen, however, that this is only approximately true, and thus this distinctive advantage of the term "Modulus of Elasticity" is reduced considerably, in fact, it would be more correct to use the terms "Modulus of Extension," "Modulus of Compression," and "Modulus of Deflection," but the multiplication of terms would be objectionable; we shall therefore retain the old general term "Modulus of Elasticity," distinguishing the method by which it was determined, and to which alone it applies with absolute correctness, by characteristic affixes; thus Eg, Ec, ED will indicate the Modulus of Elasticity by Extension, Compression, and Deflection respectively. (730.) The Modulus found by experiment for any particular strain may be applied, as we have seen, with approximate correctness to other strains in the absence of more precise data, and this is very convenient in many cases. For instance, we have absolutely no experimental knowledge of the extension and compression of Timber, but experiments on deflection are very numerous, and E, derived from these may be used to determine the extension and compression approximately. (731.) "Differences in Value of Ey and Ec."-Table 91 gives a collective comparison of the relative elasticity of Cast and Wrought iron under equivalent tensile and compressive strains, and shows that at least with small strains those materials yield more to compression than to extension, and the effect of this on the Moduli E and Ec is shown by Tables 88, 89; 96, 98. Thus for Wrought iron with 1 ton per square inch Eɛ is 28,000,000 lbs., and Ec is 22,400,000 lbs.; similarly for Cast iron EE 13,449,400, and Ec 12,844,000 lbs. With Cast iron the case is affected by defect of elasticity, so that with strains greater than 2 tons per inch E, becomes the greater of the two. = = (732.) "Effect of Defect of Elasticity."-The elasticity of all bodies is more or less imperfect (688), and the effect of this fact on the Modulus of Elasticity may be illustrated clearly by the case of Cast iron, whose elasticity is very imperfect (688). Thus by col. 10 of Table 88, Eg is progressively reduced from 13,695,200 lbs., with ton tensile strain per square inch, to 9,223,500 lbs. with 7 tons, which is nearly the mean breaking weight. Similarly, by col. 10 of Table 89, Ec is progressively reduced from 12,844,000 lbs. with 1 ton, to 3,862,400 lbs. with 42 tons per square inch compressive strain. Table 108 gives the result of the experiments of Mr. Hodgkinson on 3 x 11 bars of Blaenavon Iron, which were made with special care, having friction rollers to support the ends, &c. The effect of defective elasticity is clearly shown by col. 11, the Modulus Ep being regularly and progressively reduced from 15,216,300 lbs. with th of the breaking load to 8,353,770 lbs. with breaking weight. (733.) This great variation in the Moduli with the ratio of the strain applied in proportion to the breaking weight, not only complicates the question, but also renders it necessary to distinguish the Modulus between two given strains, from that at a given strain. For instance, col. 6 of Table 88 gives the Extension by every successive half-ton throughout :—thus between 3 and 3 tons, the extension by that half-ton is 00010681 of the length, and the mean Modulus between those weights, or at the mean weight of 3 tons, will be 1120÷ .00010681 = 10,485,900 lbs. Then, between 3 and 4 tons the extension by that particular half-ton is 0001141, and the mean Modulus between those weights, or at a mean weight of 3 tons, is 11200001141 9,816,000 lbs. Having thus found the Modulus at 31 and 3 tons, that at the mean weight of 3 tons may be found, and becomes (10,485,900 9,816,000) 2 = 10,150,950 lbs., as given in + ÷ col. 11, but between 0 and 31 tons, the Modulus by col. 10 is 12,040,900 lbs. The meaning of this is, that between 0 and 3 tons per square inch a bar of cast iron extends in a gradually increasing ratio to the strain, but at such a mean rate that if thenceforth it were continued uniformly proportional to the strain, 12,040,900 lbs. per square inch would stretch the bar to double its original length, or, in other words, each pound would stretch the bar 1200500th of the length. But when already loaded with 3 tons per square inch, a small further strain would stretch the bar 11th of the length per pound. Cols. 10 and 11 of Table 88, and cols. 10, 11 of Table 89, have been calculated in this way. (734.) The elasticity of wrought iron is practically perfect nearly up to the "limit of Elasticity" (692), and as a result, the Modulus of Elasticity is constant within that limit. This is shown by Table 96, where Eg by direct experiment in col. 5 is nearly constant up to 9 or 10 tons per square inch tensile strain, the small differences being due to errors of observation which are unavoidable. The mean value is given by col. 7 at 28,000,000 lbs., and as constant up to 8 tons. Similarly, Table 98 gives E, nearly constant by direct experiment in col. 5, its mean value being given at 22,400,000 lbs., and as constant up to 11 tons per square inch by col. 7. The same results are given by Table 106, where E, is nearly constant, up to the limit of Elasticity, the mean of the whole of the experiments up to that point being 27,645,000 lbs., agreeing nearly with the mean Modulus 27,603,000 lbs. derived from general observations (726). Beyond the strains named, the Modulus falls off rapidly and irregularly, and is in fact a question of time, as may be inferred from the Tables, which show that with greater strains, the extensions, &c., go on increasing even with constant weights. D (735.) The elasticity of Steel is nearly perfect up to the "limit of Elasticity":-we have no experiments giving the value of Eg or Ec, but that of E, is given by Table 107, and is nearly constant with strains less than the "limit of Elasticity" as calculated by the rule d2 x b x 5600 L = w. The mean value of E, for cast steel calculated from the mean deflection in col. 4 of Table 105 by Rule (724) is 30,146,600 lbs. per square inch, col. 7:-that for shear steel being 31,169,000 lbs. ; and it should be observed that these results were obtained from the deflections of bars of unwrought and untempered steel. With tempered spring steel we may admit an elasticity practically perfect, and that the Modulus is constant. (736.) The elasticity of Timber is very imperfect, and the Modulus of Elasticity very variable, as shown by Table 112, which gives the value of E, from two experiments by Mr. E. Clark on American Red Pine: the Modulus of Elasticity E, is in col. 4 reduced progressively from 1,712,350 lbs. with th to 583,350 lbs. with the breaking weight. The value of the Modulus for the same kind of Timber is given by col. 7 of Table 105 at 1,623,450 lbs., which is correct for strains up to the Safe load, or say th of the Breaking weight (888). (737.) The effect of defective elasticity on the Moduli Eg Ec, Ep is shown graphically by Diagrams, Figs. 214, 218, in which all the experimental strains have been reduced to fractions of the Ultimate, or breaking weight, so as to render the results directly comparable with one another. If the elasticities were perfect, all the lines indicating the value of the Modulus would have been horizontal. (738.) "Effect of Size of Casting."-In searching for the Modulus Ep for cast iron, from the experimental deflection of square and rectangular bars of various sizes, another, and perhaps an unexpected complication is discovered: the modulus is found to vary considerably with the size of the bar, or more correctly with the least dimension, in the case of rectangular bars. We found in (932) that the transverse strength of rectangular bars of cast iron is inversely proportional to the size of the casting, bars 1, 2, and 3 inches square having specific strengths in the ratio 1, 7519, and 6364 respectively, and experiments have shown that the Modulus ED is similarly affected. (739.) In order to show distinctly the effect of size alone, it is necessary to clear the subject from complications arising from the varying elasticities of different kinds of iron by selecting and comparing the experiments on one and the same kind of iron, varying only in size. Then again, to eliminate the effect of defective elasticity (688), it is necessary to bring the strains in all cases to equality, by reducing them to fractions of the ultimate strain or breaking weight. This is done in Diagram, Fig. 214, where the Moduli given by Tables 114, 118, from the experiments of Mr. Hodgkinson, on bars of different sizes, but all of the same kind of iron, namely, Blaenavon No. 2, are plotted to the same scale, and the actual strains being reduced to fractions of the breaking weight, the effects of the size of casting, defect of elasticity on the different sizes, &c., are made manifest. This diagram shows: (740.) 1st. That the Modulus of Elasticity E, decreases as the |
