TABLE 68.-Of the COMPARATIVE STRENGTH of COMMON and STIRLING'S TOUGHENED CAST IRON, in Girders 16 feet long, &c. Fig. 79.

results: thus, taking for the values of M92 ton for the breaking weight, and 3rd of 92, or 3067 for the safe weight, then calculating the former by the special rule and the latter by the ordinary rule, the safe load comes out considerably more than 3rd of the breaking weight; in fact more than in some cases. For instance: for hollow square beams in (346) the breaking weight by special rule 2 tons, but by ordinary rule with M1 = ·92 we obtained 3·119 tons: obviously with MT = ⚫3067, or 3rd of 92, we should have 3·119 ÷ 3 = 1·0397 ton, which is more than half the breaking weight as calculated by the special rule, namely, 2 tons.

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Again: with the equal-flanged girder in (350), the calculated breaking weight by the special rule 6197 lbs., and the safe load by ordinary rule would be 10630 ÷ 3 = 3543 lbs., which again is considerably more than half the breaking weight by special rule and experiment, instead of 3rd, as it would have. been if both had been calculated by the same rule.

(355.) This reasoning is certainly correct for cast iron with very light strains up to 4th of the breaking weight, as shown in (617); as the load is increased beyond that point, the ordinary rules give results more and more in excess of the true strength until the breaking weight is attained, when, as shown in (353), it becomes more than 50 per cent. On the other hand, the special rules may be taken as perfectly correct for the breaking weight; but as the load is reduced they give results in defect of the true value, the error increasing until with 4th of the breaking weight it becomes 33 per cent.

It will be evident from this that with of the breaking weight, which is usually adopted for the safe load, the strength would be intermediate between those given by the two rules. It is therefore not quite correct to assume that the ordinary rules will give the working load with "Factor" 3.

(356.) In Fig. 81 we have a Diagram in which the combination of the two rules is represented graphically, the results by the special rule being shown by the line A, and by the ordinary rule by the line B, the latter being throughout 50 per cent. in Now, we have seen that the ordinary rule is correct with 4th of the breaking weight, and the special

excess of the former.

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4,

rule with the full breaking weight: then the line C, which connects those two points, should give the correct load throughout. Thus, at 4th the breaking weight, or at D, the line C is exactly intermediate between A and B, and the true strength is an arithmetical mean between the two rules. For example: the hollow square beams in (346) came out 2 tons breaking weight by the special rule, therefore at 4th, or with Factor we have 2 ÷ 4 = 0·5 ton working load. By ordinary rule the breaking weight was 3.119 tons, and for the working load we have 3.1194 = 78 ton. Then, as by Diagram, the true load forth the breaking weight is an arithmetical mean between those results, and becomes (0·5+0·78) ÷ 2 = ·64 ton. Or we might have done it another way, by dividing the difference between the two rules into two equal parts; adding one of those parts to the calculated strength by the special rule, or, deducting it from that by the ordinary rule, we should obtain the same result. Thus, in our case (78 ·5 + ·14 = ·64; or 78 · 14 = the breaking weight is 264 and yet the beam is strained to 4th the effect being to add ·64 ÷ ·5 = 1·28, or 28 per cent. to the working load.

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•5)÷2 · 14, and 64 as before. In this case,

3.13 times the safe load, only of the breaking weight:

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It seems paradoxical to say that a beam loaded with 4th of the breaking load is not strained to 4th of the breaking weight, but this is due to the peculiar character of the material. Thus, say we have a beam whose breaking weight 100, then when loaded with 25 it seems to be self-evident that it will be strained toth of the breaking weight, but we have shown that to produce that strain we shall require a load of 1·28 ÷ 4 = 32 instead of 25. See (133) and (504).

(357.) For 3rd the breaking weight the diagram shows that the difference between the two rules must be divided into three equal parts, and the true strength will then be found by adding one of those parts to the special rule, or deducting two of the same from the ordinary rule. Thus, taking the same example as before, we have 2 ÷ 3 =·6667 ton by special rule, and 3.119 ÷ 3 = 1·0397 by ordinary rule: the difference = 1.0937 -·6667 = 0·373. Then 373 ÷ 3 = · 1243 is to be added to

6667, and we obtain

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6667 +1243. = 791 ton:-similarly

two of those parts, or 1243 x 2 = 2486, deducted from 1·0397 gives 1.0397-2486 791, as before; the ratio of which to the breaking weight, as found by special rule and experiment, is 27912.528 to 1.0 instead of 3 to 1, as by the ordinary method of procedure. The effect of this is to add 791 ÷·6667 =1.1865, or 18.65 per cent. to the working load:-taking a beam whose breaking weight 100 tons, then, when strained to 3rd, the load will not be 100 ÷ 3 = 33 33 tons, but 118.65 339.55 tons.

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(358.) The application of these principles to practice is very simple; for instance, the breaking weight of the girder, Fig. 79, was found in (351) to be 40·5 tons, as calculated by the special rule; then if we adopt 4 as the factor of safety, the working load would be 40.54 10 12 tons by the ordinary course, but by (356) we obtain 10∙12 × 1·28 = 13 tons nearly. If we adopt 3 as the Factor, we have 40 5÷3 = 13.5 tons in the usual way, which by (357) becomes 13.5 × 1.1865 = 16 tons, &c.

(359.) "Ratio of Square to Round Bars."-It is shown in (519) that the theoretical ratio of the strength of square to round bars of the same dimensions is 1·7 to 1·0; but this is strictly true for those cases only where the tensile and compressive strengths and the corresponding extensions and compressions are equal to one another, and this, as we have seen, is not realised perfectly with any materials, except with very light strains (617). Under these circumstances, experiment alone can determine the real ratio for the Breaking weight, &c. (360.) For cast iron we have the experiments of Mr. W. H. Barlow, which were made with direct reference to this question: the results are given in Table 69. It will be observed that the round bars were not made of the same linear dimensions as the square ones, but rather of the same sectional area. The object of this was possibly to avoid the complications due to the size of the casting, which, as shown by (932), is very influential on the transverse strength of cast iron. Thus, a bar 2 inches square would have the same area as another 2 inches diameter, and presumably there would be equality of strength so far as that is affected by the size of casting.

T

In the first set of experiments the bars were about one square inch in area the mean value of Mr from five square bars = 2530 lbs., and from five round ones 1697 lbs., the ratio being 25301697 1.491 to 1.0. In the second set the bars were about 4 square inches in area: four square bars gave Mr 2173 lbs., and nine round ones = 1399 lbs., the ratio being 21731399 = 1.553 to 1.0.

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TABLE 69.-Of EXPERIMENTS on the RELATIVE TRANSVERSE STRENGTH of SQUARE and ROUND BARS of CAST IRON, all 5 feet long.

2530 ÷ 1697 = 1.491 to 1; Ratio of square to round.

2173 ÷ 1399 = 1·553 to 1; Ratio of square to round.

(361.) The mean ratio from the whole of these experiments, 23 in number, is 1.522 to 1.0, or nearly 1.5 to 1.0, instead This may be taken, therefore,

of 1.7 to 1.0, as due by theory. as the real ratio for cast iron.