strain is increased, or, is inversely proportional to the strain, this effect being due to defect of elasticity, for obviously with perfect elasticity the modulus would be the same for all strains, and the lines in the diagram would be horizontal and perfectly straight. (741.) 2nd. That this decrease in the Modulus is in inverse arithmetical ratio to the strain, as shown by the lines being approximately straight ones. Of course there are considerable irregularities due to errors of observation (not necessarily errors of the observer), but the general result is that the lines are straight.

(742.) 3rd. That the Modulus is inversely proportional to the size of bar, or rather to its least dimension, but not in arithmetical ratio; thus the line of the 2-inch bar is not exactly midway between those of the 1-inch, and 3-inch, but much nearer the latter, agreeing to some extent with the transverse strength, as shown by Table 142.

(743.) 4th. That the difference in the Modulus between castings of different sizes, but loaded to the same extent, varies directly as the strain, being a minimum with very small strains, and increasing progressively to a maximum with the breaking weight. This is shown in the diagram by the non-parallelism of the lines, which diverge from one another as the strain is increased: the line of the small bars being more nearly horizontal than those of the large ones, shows more perfect elasticity in the former, and that defect of elasticity is more influential on the Modulus Ep as the size of the bar is increased.

D

(744.) 5th. That in rectangular bars of unequal dimensions, the Modulus E, is governed potentially by the least dimension, rather than the greater, and does not differ materially from that of a square bar of that least dimension. Thus, the bar 1 x 2 has practically the same modulus as the bar 1 inch square, and the bars 6 x 1 and 3 x 1 occupy in the diagram the position of a bar 1 inch square or nearly so, not exactly, however, the 6 × 1 bar being somewhat affected by the larger dimension, which reduces its modulus below that of the 3 × 11⁄2 bar.

(745.) The straight lines F, G, H, J are intended to equalize the anomalies of the experiments and represent approximately

the mean Modulus E, for the different sizes of No. 2 Blaenavon cast iron, and from them we have obtained Table 114, which gives the combined effect of size of casting and defect of elasticity or ratio of the strain to the breaking weight, on the Modulus of that particular iron.

TABLE 114.-Of the MODULUS of ELASTICITY, by Deflection, E, of CAST-IRON BARS, showing the effect of Size of Casting, and Ratio of Strain to the Breaking Weight.

(746.) The effect of size of casting on the Modulus E, is also clearly shown by Table 118, where 1, 2, and 3-inch bars are all reduced to one standard for the purpose of direct comparison. Taking for illustration from col. 5, the same load on bars of all three sizes as nearly as possible, say 1210 lbs. for 1-inch bars, giving E1 = 11,767,600 lbs. by col. 4: for 1212 lbs. on reduced 2-inch bars ED 9,304,260 lbs.; and for 1260 lbs. on reduced 3-inch bars, Ep 7,995,000 lbs., &c., thus showing a great reduction as the size of the bar is increased.

ED

=

=

The effect of defect of Elasticity on the Modulus Ep is also clearly manifested with bars of all sizes, by the almost perfect regularity with which its value falls off as the strain is pro

gressively increased up to the breaking load, when the 1-inch bars give 9,738,700 lbs., the reduced 2-inch, 8,412,600 lbs.; and the reduced 3-inch, 7,614,440 lbs.; see (769).

(747.) The Diagram, Fig. 218, gives a general comparison of the three Moduli, namely, of Extension EE, Compression Ec, and Deflection Ep, all reduced to one uniform standard series of strains from 0 to the breaking loads increasing by th.

E

It will be observed that Eg and Ec are nearly parallel to each other up to 3rd of the breaking weight, although the actual compressive strain is then about 6 times the tensile. Beyond the strain of rd the modulus of Extension E continues to decrease with great regularity up to the breaking weight, but in an increasing ratio, as manifested by the line being curved. The line of Ec is remarkable, falling off at first in nearly arithmetical ratio so far as ths the breaking weight as is due with perfect elasticity, after which the Modulus decreases very rapidly up to about 547 of the Ultimate weight, and then returning nearly to its first rate of decrease. But it should be observed that beyond 547, we have no experimental evidence, and that the results are based on assumed compressions obtained by plotting all the experimental compressions in a Diagram, Fig. 215, and continuing the curve by judgment up to the Crushing strain. They are therefore more or less problematical and of uncertain accuracy, but it will be obvious from inspection that if the experimental curve were continued at its normal rate up to the breaking weight, the Modulus Ec would be reduced to nothing, which is manifestly incorrect, and indeed absurd (722).

(748.) The values of Eg and Ec were determined from bars 1 inch square, they therefore are strictly comparable with the value of E, from bars of the same size only, and we then find that instead of being intermediate between Eg and E as might have been expected (727), it is for all strains greater than either: hence it is expedient, whenever practicable, to use the particular Modulus adapted to the case, having been derived from similar cases. Thus, when the extensions are required, Eg should be used; for compressions Ec, and for deflections Ep, taking care, where accuracy is desired, to use the Modulus

specially applicable to the ratio of the strain to the breaking weight, kind of iron, and size of casting. It is advisable to adopt this course, not only with such materials as cast iron and timber whose elasticity is very imperfect, but also with the most perfectly elastic materials, such as wrought iron and steel, as may be seen by (726).

(749.) We may now give some illustrations of the application of the various Moduli Eg, Ec, E, to practice.

=

Say, we have a bar of cast iron 10 feet, or 120 inches long, loaded with a tensile strain of 4 tons per square inch, and we require the extension by that strain :-now by col. 10 of Table 88 the mean value of Eg is then 11,709,200 lbs., and the extension, with 4 tons, or 8960 lbs., will be 8960 11,709,200 000765 parts of the length, or 000765 × 1200908 inch. Now, say that when the bar is loaded with 4 tons, we require the effect of 4 cwt., or 448 lbs., more; then by col. 11, the Modulus when already loaded with 4 tons, is 9,472,000 lbs., hence the extension with 448 lbs. more, would be 448 ÷ 9,472,000 = 0000473 parts of the length, or ⚫0000473 × 120 = ·005676 inch.

The application of E to find the compressions by crushing strains is of course precisely similar to that of EE, the proper value of the Modulus being taken from col. 10 or col. 11 in Table 89.

To find the deflection of a rectangular bar, the value of ED must be selected, proper for the particular iron (739); ratio of the strain to the breaking weight (740); and size of casting (738); and we can then find the deflection by the rule:

In which the deflection in inches, w weight in pounds, 7 length between supports in inches, d = depth in inches, b = breadth in inches, ED

=

Modulus of deflection in pounds

per square inch. Thus with a wrought-iron bar in which d= 1 inch, b5 inches, 13 feet, or 162 inches, and w = 840 lbs. taking E, from col. 7 of Table 105 at 27,600,000,

=

840 × 162$ 27600000 × 4 × 11 × 51

= 1.743

the deflection becomes 8 = inches, &c. It is shown in (303) that the strength of Timber pillars may be found direct from the Modulus of Elasticity.

CHAPTER XIX.

ON PERMANENT SET.

(751.) "Defect of Elasticity."-The elasticity of all bodies is more or less imperfect, and manifests itself in two principal ways. 1st, by the extensions, compressions, deflections, &c., increasing more rapidly than the strains, whereas with perfect elasticity they would be simply proportional to those strains; the effect of this fact is shown in (604), (613), (688). The 2nd result of defective elasticity is that when once strained, the body never returns to its primitive form, but takes a "permanent set" varying in amount very greatly with the nature of the material and the extent of the strain; this will form the subject of the present chapter.

(752.) In earlier days it was assumed that within what was termed the limit of elasticity, or with strains less than about one-third of the breaking weight, the elasticity of ordinary materials, such as cast and wrought iron, timber, &c., was perfect, that is to say, the extensions, &c., being simply proportional to those strains, and giving no permanent set. But the refined experiments of Mr. Hodgkinson have shown that, although this may be practically true with some materials, such as wrought iron and steel, it is in all probability not strictly true with any; he found that the sets were generally proportional to the square of the strains, or nearly so: see (688) and Table 111, from which it would follow that if with a moderate strain the set were considerable, then, 1st, that with very slight strains the sets would be excessively small, and possibly unmeasurable, if not inappreciable; but, 2nd, that law would show that there must be sets with all strains, however small.