experiment gave 6.5 lbs.; hence 5.846.58986, showing an error of 1·0 · 89861014 or 10.14 per cent.

(871.) "Oval Tubes with Cross-joints, &c."-It should be observed that the multiplier 614 has been derived from experimental tubes without cross-joints, and with one longitudinal joint only. By analogy with cylindrical tubes (852) we may infer that for oval tubes on the large scale, having ordinary lap-joints, crossways and longitudinally, the multiplier would become 61.4 x 49.333.6 = 90, and hence we have the rule

(873.)" Tubes Slightly Oval.”—We have seen in (841) that theoretically the strain due to external pressure on a perfectly cylindrical tube is simply a crushing one, but that practically the strength is given by the rule in (845). With a flat oval tube the strain generated by external pressure is a transverse one, and the strength is given by the rule in (870). When the form differs very little from a perfect cylinder we may suppose that the strength will be given by the former rule rather than the latter. For such cases and for tubes without cross-joints we have the rule:

(874.)

P = 33.6 × (100 t)2·19 ÷ (L × d ̧),

d being the diameter of the osculating circle, or Au, as explained in (864), and the rest as in (845). For large tubes with ordinary lap-joints the rule becomes :

(875.)

P49.3 x (100 t)2:19 (L x d.).

With a certain ratio between the two diameters these rules will agree in their results with those for decidedly oval tubes in (870), as shown by Table 136; thus, with a tube 12 × 9 × 1, also with 24 × 22 × 1, and with 36 × 34.07 × 3, the two sets of rules agree. With ovals more nearly cylindrical than those sizes the rules in (845), (850) will govern the strength; but with flatter ovals the rules in (870), (872) will prevail. The best course in any doubtful case is to calculate by both rules and accept the lowest result as correct.

TABLE 136.-Of the CALCULATED STRENGTH of OVAL TUBES 10 feet long to RESIST EXTERNAL PRESSURE.

(876.) Thus with No. 35 in Table 132 we obtained 127.7 lbs. by one rule (870), and 274 lbs. by the other (845); and we may admit the lower result to be correct; a conclusion supported by experiment, which gave 127.5 lbs.

But in experiment No. 32 we had a tube very nearly cylindrical, there being a difference in the two diameters of inch only. We know beforehand that the rule in (870) will not apply to such a case. Nevertheless, for the sake of illustration,

we may try it, and we obtain P =

(100 x 125)219 x 61.4 (141) — 141) × 1411 × 5 = 1125 lbs. ! or 9 times the experimental pressure, which was 125 lbs. only. By the other rule (845) the osculating circle do 14112141 = 14.88 inches; then P = 33.6 × 252·5÷ (5 x 14.88) 114 lbs. ; hence 114125912, giving an error of 1.0·912088, or 8.8 per cent. only; here

=

=

evidently the rule in (848) is the more correct.

Again, in experiment No. 33 the two diameters differed only inch; here again the rule in (870) will give a pressure 252.5 × 61.4

greatly in excess; P =

(15-15) × 15 x 1.77 = 1281 lbs.! or nearly 6 times the experimental pressure, which was 220 lbs. per square inch. By the other rule we obtain d = 155 ÷ 15,3% = 16.07 inches, then P = 33·6 × 252·5 ÷ (1·77 × 16·07) =298 lbs. ; hence 298 220 = 1.35, or an error of 35 per cent. of course the lower result is obviously the more correct.

0

(877.) Table 136 gives the strength of oval tubes of boilerplate iron with different thicknesses, and 10 feet long, calculated by the two rules for the sake of comparing results. It is remarkable that not only does the ordinary rule give lower pressures for the tubes which are nearly cylindrical, a result that was to be expected, but does the same with the extremely flat tubes 12 x 1 and 12 × 2, shown by Fig. 195. In the absence of experiment we should instinctively suppose that both being such extremely flat tubes, and both having the same major diameter, there would be very little difference in the collapsing pressure, and that the new rule in (870) which gives only 12.92 11·75 = 1·10 or 10 per cent. difference in strength by col. 3, is more correct than the ordinary rule in (874), which gives double strength, or 11.8 to 5.9 by col. 2. Combining the two rules we obtain col. 5, in which the tubes from 12 x 12 to 12 × 93 have been calculated by the rule for cylindrical tubes, taking for the acting diameter the osculating circle do all the rest being taken from col. 3.

It will be observed that small departures from the perfectly cylindrical form are not very influential on the strength: thus with a tube 12 x 11, we have 64.7770.792, or 92 per cent. of the strength of a perfectly cylindrical one 12 inches

diameter: another, 12 × 10, gives 58.92 70·7 = ·83, or 83 per cent.

The strength with inch thickness and a cylindrical 12-inch tube being 1.0, would be reduced to half with 12 × 8.3 inches; to one-third with 12 × 6 inches; to one-fourth with 12 x 43 inches; to one-fifth with 12 x 3 inches; and to one-sixth with 12 x 1 inch. These relative proportions, however, must not be taken as establishing general ratios for other sizes: for example, a tube 12 x 9 x collapses by col. 6 with 6082 of the pressure due to a 12-inch cylinder, and we might expect that the same ratio would prevail for other sizes where the proportions were the same; but the Table shows, col. 6, that with double sizes, or 24 x 18 x 1, the ratio is 3044, or half only; and with triple sizes, or 36 × 27 × 3, the ratio is 2030, or one-third only of that with a tube of one-third the size and thickness, &c.

(878.) Table 132 gives a general comparison of experimental with calculated strengths: Nos. 1 to 29 were calculated by the rule for jointless tubes (845); Nos. 30, 31 by the rule for lapjointed tubes (850); Nos. 32, 33 by the rule for slightly oval tubes (874); and Nos. 34, 35 by the rule for decidedly oval tubes (870). Omitting Nos. 25, 26, which were anomalous, and No. 28, which was not strained to the collapsing point, we have from 1 to 31 inclusive, five whose error (col. 6) = 0; 11 gave +errors, the sum of which = 131.4; 12 gave errors, the sum being 114.1. Hence 131.4 · 114.1 = +17.3, which with 28 comparable experiments gives an average error of 17.3 ÷ 28 +0.618, or less than g per cent.

(879.) "Factor of Safety."- For general purposes the Factor 6, as given by Mr. Fairbairn, may be usually admitted for Boilers with both internal and external pressures, but in practice a much lower Factor is very often permitted with both strains.

The two boilers Nos. 30 and 31 in Table 132 were intended for 40 lbs. per square inch, and were no doubt worked at that pressure, but, as shown by col. 4, the collapsing pressures were 127 and 97 lbs. respectively, giving as the value of the Factor, 12740 = 3.2 and 9740 = 2.42 only. It is shown in (78) that with internal pressures, the Factor commonly used in

the best practice varies from 3.45 to 2.76. Nevertheless, it is highly expedient wherever practicable to use Factor 6, and thus allow a wide margin for fluctuations in pressure, deterioration from rust, and other contingencies, which are unavoidable, and should thus be provided for.

CHAPTER XXII.

FACTOR OF SAFETY.

(880.) "Ratios of Breaking Weight, Proof Strain, and Working Load."-The Strength of Materials is usually determined by the ultimate or breaking weight of a specimen, and among the most important questions with which the Engineer has to deal is 1st, to determine the Ratio which the working load may safely bear to the ultimate strain,—or the “ Factor of Safety"; and 2nd, the "Proof Strain," or the extent to which work should be tested in order to prove the soundness of the materials, also the correctness of the design and perfection of workmanship in the case of complex structures.

In determining the proper value of the Factor of Safety so as to avoid unnecessary strength on the one hand, and risk of failure from inadequate strength on the other, it is necessary to consider, 1st, the varying conditions under which materials are strained; and 2nd, the variableness in the quality of the materials themselves. These will be very influential, so that the value of the Factor will not be constant for all cases, and the whole matter is thereby complicated considerably.

(881.) "Variable Conditions of Strain."-We may have, 1st, a perfectly dead load, or statical strain; 2nd, a rolling load in rapid motion, as in the case of a Railway Bridge, where the strain becomes more or less a dynamic one, but under certain limitations dependent on the horizontal velocity; 3rd, cases where the load is intermittent, being alternately laid on and taken off repeatedly, as in the case of cranes, single-acting pump-rods, &c.; 4th, alternating strains in opposite directions,